Research Article A New Model for Predicting Dynamic Surge...
17
Research Article A New Model for Predicting Dynamic Surge Pressure in Gas and Drilling Mud Two-Phase Flow during Tripping Operations Xiangwei Kong, 1 Yuanhua Lin, 2 Yijie Qiu, 1 Hongjun Zhu, 2 Long Dong, 3 and Yanchao Chen 4 1 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500, China 2 CNPC Key Lab for Tubular Goods Engineering, Southwest Petroleum University, Chengdu, Sichuan 610500, China 3 Daqing Normal University and Petro China Daqing Oilfield Company, Daqing, Heilongjiang 163000, China 4 No. 1 Oil Production Plant, Petro China Daqing Oilfield Company, Daqing, Heilongjiang 163000, China Correspondence should be addressed to Xiangwei Kong; [email protected] and Yuanhua Lin; [email protected] Received 31 August 2013; Accepted 28 November 2013; Published 8 January 2014 Academic Editor: Lu Zhen Copyright © 2014 Xiangwei Kong et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Investigation of surge pressure is of great significance to the circulation loss problem caused by unsteady operations in management pressure drilling (MPD) operations. With full consideration of the important factors such as wave velocity, gas influx rate, pressure, temperature, and well depth, a new surge pressure model has been proposed based on the mass conservation equations and the momentum conservation equations during MPD operations. e finite-difference method, the Newton-Raphson iterative method, and the fourth-order explicit Runge-Kutta method (R-K4) are adopted to solve the model. Calculation results indicate that the surge pressure has different values with respect to different drill pipe tripping speeds and well parameters. In general, the surge pressure tends to increase with the increases of drill pipe operating speed and with the decrease of gas influx rate and wellbore diameter. When the gas influx occurs, the surge pressure is weakened obviously. e surge pressure can cause a significant lag time if the gas influx occurs at bottomhole, and it is mainly affected by pressure wave velocity. e maximum surge pressure may occur before drill pipe reaches bottomhole, and the surge pressure is mainly affected by drill pipe operating speed and gas influx rate. 1. Introduction Prospects of petroleum industry are the exploration and development of high pressure, formations pressures uncer- tainties, and abnormal permeability reservoirs [1, 2]. Man- aged pressure drilling (MPD) is oſten used in formations that have a narrow window between formation pore pressure and fracture pressure. Applying variations of MPD maintains the bottomhole pressure within the narrow window during the entire operation process, including drilling, and connections, tripping. To prevent formation influx or lost-circulation problems, wellbore pressures should be kept as constant as possible during the entire drilling process [3–5]. Tripping is a constant operation in the MPD, and moving drill pipe is accompanied by a displacement of the drilling mud in wellbore [6], so the movement of drill pipe in a wellbore filled with drilling mud can generate an additional pressure. e pressure produced by the downward movement of drill pipe is called surge pressure, while the pressure that decreased in the upward movement of drill pipe is called swab pressure. For the unstable moving velocity of the drill pipe, the generated surge pressure will affect the equilibrium relation of the pressure system in the wellbore. e bottomhole pressure fluctuates with the change of the surge pressure during the tripping operation. As the bottomhole pressure should be constant during the trip progress in MPD, it is necessary to accuratly calculate the surge pressure. e circulation loss or the other complex accidents can also be caused by the surge pressure for the high velocity of the drill pipe. e high drilling mud velocity and surge pressure may damage reservoir [7]. Many research data show that 25% of accidents Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 916798, 16 pages http://dx.doi.org/10.1155/2014/916798
Transcript of Research Article A New Model for Predicting Dynamic Surge...
Research ArticleA New Model for Predicting Dynamic Surge Pressure in Gas andDrilling Mud Two-Phase Flow during Tripping Operations
Xiangwei Kong1 Yuanhua Lin2 Yijie Qiu1 Hongjun Zhu2
Long Dong3 and Yanchao Chen4
1 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation Southwest Petroleum UniversityChengdu Sichuan 610500 China
2 CNPC Key Lab for Tubular Goods Engineering Southwest Petroleum University Chengdu Sichuan 610500 China3Daqing Normal University and Petro China Daqing Oilfield Company Daqing Heilongjiang 163000 China4No 1 Oil Production Plant Petro China Daqing Oilfield Company Daqing Heilongjiang 163000 China
Correspondence should be addressed to Xiangwei Kong m13880214723163com and Yuanhua Lin yhlin28163com
Received 31 August 2013 Accepted 28 November 2013 Published 8 January 2014
Academic Editor Lu Zhen
Copyright copy 2014 Xiangwei Kong et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Investigation of surge pressure is of great significance to the circulation loss problem caused by unsteady operations inmanagementpressure drilling (MPD) operationsWith full consideration of the important factors such as wave velocity gas influx rate pressuretemperature and well depth a new surge pressure model has been proposed based on the mass conservation equations and themomentum conservation equations duringMPD operationsThe finite-difference method the Newton-Raphson iterative methodand the fourth-order explicit Runge-Kuttamethod (R-K4) are adopted to solve themodel Calculation results indicate that the surgepressure has different values with respect to different drill pipe tripping speeds and well parameters In general the surge pressuretends to increase with the increases of drill pipe operating speed and with the decrease of gas influx rate and wellbore diameterWhen the gas influx occurs the surge pressure is weakened obviously The surge pressure can cause a significant lag time if the gasinflux occurs at bottomhole and it is mainly affected by pressure wave velocity The maximum surge pressure may occur beforedrill pipe reaches bottomhole and the surge pressure is mainly affected by drill pipe operating speed and gas influx rate
1 Introduction
Prospects of petroleum industry are the exploration anddevelopment of high pressure formations pressures uncer-tainties and abnormal permeability reservoirs [1 2] Man-aged pressure drilling (MPD) is often used in formations thathave a narrow window between formation pore pressure andfracture pressure Applying variations of MPD maintains thebottomhole pressure within the narrow window during theentire operation process including drilling and connectionstripping To prevent formation influx or lost-circulationproblems wellbore pressures should be kept as constant aspossible during the entire drilling process [3ndash5] Trippingis a constant operation in the MPD and moving drill pipeis accompanied by a displacement of the drilling mud inwellbore [6] so themovement of drill pipe in a wellbore filled
with drilling mud can generate an additional pressure Thepressure produced by the downwardmovement of drill pipe iscalled surge pressure while the pressure that decreased in theupward movement of drill pipe is called swab pressure Forthe unstable moving velocity of the drill pipe the generatedsurge pressure will affect the equilibrium relation of thepressure system in the wellbore The bottomhole pressurefluctuates with the change of the surge pressure during thetripping operation As the bottomhole pressure should beconstant during the trip progress in MPD it is necessary toaccuratly calculate the surge pressure The circulation lossor the other complex accidents can also be caused by thesurge pressure for the high velocity of the drill pipe Thehigh drilling mud velocity and surge pressure may damagereservoir [7] Many research data show that 25 of accidents
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 916798 16 pageshttpdxdoiorg1011552014916798
2 Mathematical Problems in Engineering
in the drilling operation are caused by surge pressure It issuggested that excessive surge pressures have initiated lostcirculation formation fracturing and well kick problemsduring the drilling operation This may result in expensivedrillingmud treating costs and cause otherwellbore problems[8] Obviously the accurate calculation of surge pressure isrelated to the safety of the drilling process [9] Also thereis a particular relationship between reasonable drilling muddensity well structural design and surge pressureThereforethe surge pressure is very important basic parameters indrilling design [10] So far the calculation of surge pressurein drillingmud and gas is still worth continuing and in-depthresearch
A number of field studies investigate the effects of well-bore geometry on surge pressures anddrillingmudproperties[11] However the first valid reports from surge pressure werenot published until 1960 and the method to calculate thesurge pressure began to apply to drilling industry [12] Atheoretical analysis of couette flow of power-law fluids forsurge pressure was conducted by Chukwu in 1989 [13] Modelfor both slot flow and concentric wellbore flowwas developedfor a stationary outer pipe and a steady axial motion of aconcentric inner pipe The inner pipe was assumed to beeither plugged and of uniform geometry or open at thebottom After the year Fan [14] presented a method foraccurate calculation of the surge pressure caused by drillingmud viscosity during tripping in a vertical wellbore in 1990Themodels of laminar flow velocity profile and surge pressureof Newtonian and power law liquid in the wellbore aredeveloped and the drill pipe operating speed and drillingmud properties controlled for the safety of operation Inthe same year the method for calculating surge pressureunder stable conditions was proposed by Zhou et al [15] andthe problem existing in the calculation curves for adhesivefactor value was cited In this research the drilling mudis mainly composed of the power-law fluid and Binghamfluid The calculation method for surge pressure under stableconditions is applicable by taking the compressibility ofdrilling mud and expansion of wellbore into considerationIn the article of Chukwu [16] a laminar couette flow modelfor power-law fluids through a slot was considered Usingthe mathematical relationships developed knowledge of thedrilling mud rheological properties drill pipe operatingspeed and wellbore geometry is needed to determine thesurge pressure With consideration the compressibility ofdrilling fluids and wellbore surge pressure was calculatedby Jiang [17] By using theoretical analysis and experimentalresults it demonstrated that surge pressure is a functionof well depth the combination of drilling tools wellborediameter properties of drilling mud drill pipe operatingspeed and acceleration of drill pipe movement and so forthNext a calculation method of surge pressure caused bydrilling mud viscosity while the drill pipe moves througha Caisson drilling mud in wellbore was given by Wang etal [18 19] The coefficients of surge pressure under differentconditions are plotted These results are useful in the controlof drill pipe operating speed and the design of drillingmud properties In the computational model flow rate ofdrilling mud is considered as one-dimensional flow and
the method of characteristics and finite-difference techniqueare used to solve the basic equation of unsteady state flowMeanwhile a theoretical model is presented by Fan et al [20]for predicting dynamic surge pressure based on the theoryof unsteady state flow in 1995 The basic partial differentialequations for describing dynamic surge pressure are givenFan methods are proposed for determining boundary andinitial conditions when considering well parameters drill bitjet nozzles of drill bit variability of tripping velocity andacceleration An example for the application of the modelwas also given [21 22] Bjoslashrkevoll et al [23] developed aneasy to use transient surge swabmodel by taking into accountdampening of pressure peaks due to fluid compression andelasticity of open wellbore In this model the contributionsfrom acceleration are also included When there is littlegel strength in the fluid the transient model reproducedrelevant measured data with adequate accuracy RecentlyNygaard et al [24] present a new method for coordinatedcontrol of pump rates and choke valve for compensatingsurge pressure during tripping operations Fedevjcyk et al[25 26] searched that wellbore diameter variations and theuse of drill pipe accessories might cause changes to theannular cross-section space between the drill pipe and theborehole This study proposes a mathematicalnumericalmodel to simulate the surge pressure problem in wellborewith variable cross-section areasThe fluid flow yielded by themovement of drill pipe is considered to be one-dimensionalcompressible and transient In order to predict the surgepressure caused in the horizontal well drilling process Sunet al [27] created models for predicting surge pressure basedon the general theory of hydrostatic drilling mud mechanicsand specifically described the flowing physicalmodel towardssurge pressure in horizontal wellbore by taking the effect ofdrill pipe eccentricity on the flowing law of drilling fluid intoconsideration Lebele-Alawa and Oparadike [28] analyse theeffects of surge pressure on pipe flow The surge pressureinvestigated is that propagated by the emergency relief-coupling valve (ERV) connected to a loading system carryingcrude oil from four flow stations Kong et al established thecalculation model for fluctuation pressure in gas-liquid two-phase flow during tripping operations but the model is justapplicable to steady state conditions [29] Crespo and Ahmed[30] reported recently the results of an experimental studyaimed at investigating the effects of drill pipe speed drillingmud properties and wellbore geometry on surge pressuresunder laboratory conditions
Although a large number of experiments under con-trolled laboratory conditions and modeling studies wereconducted in the past to investigate surge pressures theexisting surge pressure models rarely take the impact of thegas into consideration and thus will inevitably affect theprecision calculation for the surge pressure during MPDoperations If the gas influx occurs at the bottom of wellthe surge pressure may be reduced greatly The calculationmethod for predication of surge pressure in two-phase is notas accurate as the prediction of single-phase drillingmud dueto its simplicity In conclusion we must take into accountthis problem rationally and placemore emphasis on the surgepressure in two phases during MPD operations
Mathematical Problems in Engineering 3
Drilling mud
Gas
Drill pipe
Cross-section 2
Cross-section 1
Casing
p + dp + d
z
120579
p
Figure 1 The schematic of gas and drilling mud two-phase flowduring tripping operations
The object of the present work is to present a newcalculation model for predicting surge pressure during trip-ping operations which is applicable for gas drilling mudtwo-phase fluid In this paper in addition to the pressuretemperature and the void fraction in the wellbore the com-pressibility of the gas phase the wave velocity in real-timethe changes of wellbore parameter and three scenarios arealso taken into consideration By introducing themomentumconservation equations and the mass conservation equationsin MPD operations pressure wave velocity in real-time gas-drilling mud equations of state (EOS) and a new model forpredicting dynamic surge pressure in gas drilling mud two-phase during tripping operations is developed Numericalsolutions difference method and characteristics method areobtained to solve the model The model can be used topredict the change of the surge pressure at any well positionat different influx rate drill pipe operating speed and drillingparameter
2 The Mathematical Model
21 The Basic Equation Drilling mud contains clay cuttingsbarite other solids and so forth The solid particles aresmall and uniformly distributed therefore drilling mudis considered to be a pseudohomogeneous liquid and thenatural gas influx is considered to be the gas phase Asshown in Figure 1 the wellbore is filled with gas and drillingmud two-phase fluid The downward movement of drill pipealong wellbore filled with drilling mud can create a surge ofpressure The two-phase drilling fluid and the passagewayare all compressible and expansibleTherefore the two-phasedrilling fluid flow induced by the trip is unstable especiallyin the deep well
To establish the model about surge pressure during tripoperations the following assumptions are made
(i) the two-phase flow is treated as one-dimensional
(ii) no mass transfers between the gas and drilling mud
(iii) the flow pattern in wellbore is either bubble or slugflow
(iv) the annulus is concentric
When drilling strings are moving in the wellbore fullof drilling mud the pressure gradient equations the massconservation equations and the momentum conservationequations along the flow direction in the wellbore is com-bined to study the surge pressure in tripping operations
The mass conservation equation and momentum conser-vation equation are shown as follows
119901119860 + 120588119898119892(119860 +
120597119860
120597119911sdot119889119911
2) 119889119911 sdot sin 120579 + (119901 +
120597119901
120597119911sdot119889119911
2)120597119860
120597119911
sdot 119889119911 minus (119901119860 +120597 (119901119860)
120597119911119889119911)
minus 1205910119883119889119911 = 120588
119898(119860 +
120597119860
120597119911sdot119889119911
2) 119889119911 sdot
119889V119889119905
(1)
1
120588119898119892
120597119901
120597119911+1
119892(120597V120597119905
+ V120597V120597119911) +
119891V |V|8119898119892
= 0 (2)
The total pressure drop gradient is the sum of pressuredrop gradients due to potential energy change kinetic energyand frictional loss From (2) the equation used to calculatepressure gradient of gas drilling mud two-phase flow withinthe wellbore can be written as
119889119901
119889119911= 120588119898119892 sin 120579 minus
120591119908120587119863
119860minus 120588119898V119898
119889V119898
119889119911 (3)
Assuming the compressibility of the gas is only relatedwith the pressure in the wellbore the kinetic energy andacceleration term in the equation above can be simplified to
120588119898V119898
119889V119898
119889119911= minus
120588119898V119898V119904119892
119901
119889119901
119889119911= minus
119882119898119902119866
1198602119901
119889119901
119889119911 (4)
Substituting (3) into (4) the total pressure drop gradientalong the flow direction within the wellbore can be expressedas
119889119901
119889119911=
120588119898119892 + 120591119891
1 minus119882119898119902119866 (1198602119901)
(5)
The continuity equations can be obtained directly basedon (1)
1198882
(119875 119879119863 120601 119908 119871119892)120597V120597119911
+1
120588119898
119889119901
119889119905+ V
120597119901
120597119911= 0 (6)
where wave velocity 119888(119875 119879119863 120601 119908 119871119892) changes in real-timewhen the gas influx occurs at the bottom of drilling well
The equations of motion can be obtained directly basedon (2)
1
120588119898
120597119901
120597119911+120597V120597119905
+ 119892119889V119889119911
+119891V |V|2119863
= 0 (7)
22 Pressure Wave Velocity Only consider the deformationof wellbore caused by surge pressure the elastic constant ofdrill pipe is
120573 =2
1198631Δ119901
(1 minus 119906119904
119864119904
Δ1199011198632
1
11986322minus 11986321
1198631+1 + 119906119904
119864119904
1198632
2minus 1198632
1Δ119901
11986322minus 11986321
1
119863)
(8)
4 Mathematical Problems in Engineering
The elastic constant of well wall is
120573 =2
119864119891
(1 + 120583119891) (9)
The elastic constant between well wall and drill pipe is
120573 =2
11987722minus 1
[1198772
2
119864119891
(1 + 119906119891) +
1
119864119904
(1198772
1+ 1
11987721minus 1
minus 119906119904)] (10)
The elastic constant between casing and drill pipe is
120573 =2
11987722minus 1
[1198772
2
1198643
(1198772
3+ 1
11987721minus 1
+ 1199063) +
1
1198642
(1198772
1+ 1
11987721minus 1
minus 1199062)]
(11)
where 1198771= 11986321198631 1198772= 11986331198634 and 119877
3= 11986341198633
The density of gas and drilling mud mixture is describedas the following formula
120588119898= 120601119866120588119866+ 120601119871120588119871 (12)
The small perturbation theory is also applied to thesolution of wave velocity model According to the solvablecondition of the homogenous linear equations that thedeterminant of the equations is zero the equation of pressurewave can be expressed in the following form [31]
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(120588119892+ 119888119901120601119866120588119871
V2119904
1198882119892
)119908120601119892
1198882119892
[1 minus 119888119901120601119871]V2119904
1198882119871
119908 minus[120601119892120588119892119896 + 2119888
119901120601119892120601119871120588119871
V119904
1198882119871
119908] 22119888119901120601119892120601119871120588119871
V119904
1198882119871
119908
minus120588119871119908
1 minus 120601119866
1198882119871
119908 0 minus119896 (1 minus 120601119866) 120588119871
120588119871V2119903119896 (minus120601
119866119888119901+ 119888119903minus 119888119894+ 1198881198982) minus120601119866119896 [1 minus 120601
119871
119888119901V2119904
1198882119871
+ 119888119894
V2119904
1198882119871
]
120601119892(120588119892+ 119888V119898120588119871)119908
minus119894 (3
4
119888119863
119903120588119871120601119892V119904+4
119863119891119892119908120588119892V119892)
minus119888V119898120601119892120588119871119908 + 119894 (3
4
119888119863
119903120588119871120601119892V119904)
120588119871V2119904119896 (120601119871119888119901minus 2119888119903minus 1198881198982) minus119896(120601
119871+ 119888119903120601119892
V2119904
1198882119871
) minus119888V119898120601119892120588119871119908 + 119894 (3
4
119888119863
119903120588119871120601119892V119904)
120588119871[120601119871+ 120601119892119888V119898]119908
minus119894 (3
4
119888119863
119903120588119871120601119892V119904+4
119863119891119871120588119871V119871)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 0
(13)
where 119888119894= 03 119888
119901= 025 119888
1198982= 01 and 119888
119903= 02
The real value of wave number is determined by thepressure wave velocity 119888(119875 119879119863 120601 119908 119871119892) and pressure wavevelocity in the gas drilling mud two phase flow is
119888 (119875 119879119863 120601 119908 119871119892) =
1003816100381610038161003816119908119877+
(119896) minus 119908119877minus
(119896)1003816100381610038161003816
2 (14)
23 Physical Equations
231 Equations of State for Gas The equation of state (EOS)for gas can be expressed as follows
120588119866=
119875
(119885119866sdot 119877 sdot 119879)
(15)
where 119885119866is the compression factor of gas
The formula presented by Dranchuk has been used tosolve the compression factor under the condition of low andmedium pressure (119875 lt 35MPa) [32]
119911119866= 1 + (03051 minus
10467
119879119903
minus05783
1198793119903
)120588119903
+ (05353 minus026123
119879119903
minus06816
1198793119903
)1205882
119903
(16)
where 119879119903= 119879119879
119888 119901119903= 119901119901
119888 120588119903= 027119901
119903119885119866119879119903
The formula presented by Dranchuk and Abou-Kassemhas been adopted to solve the compression factor under thecondition of high pressure (119875 ge 35MPa) [32]
119911119866=006125119875
119903119879minus1
119903exp (minus12 (1 minus 119879minus1
119903) 2)
119884 (17)
where 119884 is given by
minus 006125119875119903119879minus1
119903exp (minus12(1 minus 119879minus1
119903)2
) +119884 + 119884
2
+ 1198843
+ 1198844
(1 minus 119884)3
= (1476119879minus1
119903minus 976119879
minus2
119903+ 458119879
minus3
119903) 1198842
minus (907119879minus1
119903minus 2422119879
minus2
119903+ 424119879
minus3
119903) 119884(218+282119879
minus1
119903
)
(18)
232 Equations of State for Drilling Mud Under differenttemperature and pressure the density of drilling mud can beobtained by the empirical formulas If 119879 lt 130
∘C the densityof drilling mud can be obtained by the following equation
120588119871= 1205880(1 + 4 times 10
minus10
119875119871minus 4 times 10
minus5
119879 minus 3 times 10minus6
1198792
) (19)
Mathematical Problems in Engineering 5
If 119879 ge 130∘C the density of drilling mud is
120588119871= 1205880(1 + 4 times 10
minus10
119875 minus 4 times 10minus5
119879 minus 3 times 10minus6
1198792
+ 04(119879 minus 130
119879)
2
)
(20)
233 Correlation of Temperature Distribution The tempera-ture of the drilling mud at different depth of the wellbore canbe determined by the relationship presented by Hasan andKabir [33]
119879 = 119879119890119894+ 119865 [1 minus 119890
(119911119887ℎ
minus119911)119860
]
times (minus119892 sin 120579119892119888119869119888119901119898
+ 119873 + 119892119879sin 120579)
+ 119890(119911119887ℎ
minus119911)119860
(119879119891119887ℎ
minus 119879119890119887ℎ)
(21)
24 Flow Pattern Analysis Based on the analysis of flowcharacteristics in the closed drilling system it can be safelyassumed that the flow pattern in wellbore is either bubbleor slug flow The pattern transition criteria for bubbly flowand slug flow given by Orkiszewski are used to judge the flowpattern in the gas-drilling mud two-phase flow [34]
For bubbly flow
119902119866
119902119898
lt 119871119861 (22)
For slug flow
119902119866
119902119898
gt 119871119861 119873
119866119881lt 119871119904 (23)
where 119902119898is the volumetric flow rate of two-phase flow and
119902119898= 119902119866+ 119902119871
The dimensionless numbers 119871119904and 119871
119861are defined as
119871119904= 50 +
36119873119866119881119902119871
119902119866
119871119861= 1071 minus
07277V2119898
119863
(24)
where
119873119866119881
= V119904(120588119871
119892120590119904
)
025
(25)
Flow parameters such as void fractionmixed density andvirtual mass force coefficient are discussed for specific flowpattern
The correlation between void fraction and drilling mudholdup is expressed as
120601119866+ 120601119871= 1 (26)
241 Bubble Flow The average density of the mixture forbubble flow is
120588119898= 120601119871120588119871+ 120601119866120588119866 (27)
The void fraction for bubble flow is determined by thefollowing equation
120601119866=1
2[
[
1 +119902119898
V119904119860minus radic(1 +
119902119898
V119904119860)
2
minus4119902119866
V119904119860]
]
(28)
The friction pressure gradient can be obtained from thefollowing formula
120591119891= 119891
120588119871V2119871
2119863 (29)
242 Slug Flow The distribution coefficient of gas in thedrilling mud phase is determined by
1198620=000252lg (103120583L)
119860138minus 0782 + 0232lgVm minus 0428lgA
(30)
The average density of the mixture for slug flow is
120588119898=119882119898+ 120588119871V119904119860
119902119898+ V119904119860
+ 1198620120588119871 (31)
The void fraction for slug flow is determined by thefollowing equation
120601119866=
119902119866
119902119866+ 119902119871
(32)
The friction pressure gradient can be obtained from thefollowing formula
120591119891= 119891
120588119871V2119898
2119863(119902119871+ V119904119860
119902119898+ V119904119860+ 1198620) (33)
25 Wellbore Characteristic Analysis Effective diameter pro-posed by Sanchez [35] is
119863 =120587 (1198632
2minus 1198632
1) 4
120587 (1198632+ 1198631) 4
= 1198632minus 1198631 (34)
The effective roughness of the wellbore can be expressedas
119896119890= 1198960
1198632
1198632+ 1198631
+ 119896119894
1198631
1198632+ 1198631
(35)
With 119863119894and 119863
119900being the diameters of inner pipe and
outer pipe respectively the 1198960and 119896
119894are the roughness of
outer pipe and the inner pipe respectively
6 Mathematical Problems in Engineering
Gas
Drilling pipe
Drilling string
Drilling mudI
IIIII II
Q1
Q2 Q3 Q2
(a) OPOP
I
IIIII II
Q1
Q2 Q3 Q2
(b) CPOP
0
I
IIIII II
Q2 Q3 Q2
L2
L1Q1
(c) OPWP
Figure 2 Different pipe end conditions in trip operations
3 Boundary Conditions and InitialConditions of the Model
When the drill pipe runs into the wellbore filled with drillingmud and gas at high velocities there are three scenarios forthe movement of the drill pipe They are closed pipe withoutpumping (CPOP) open pipe without pumping (OPOP)and open pipe with pumping (OPWP) respectively Thethree scenarios are illustrated in Figure 2 As drill pipe ismoved downward into a wellbore the drilling mud willmove upward To analyze the transient flow in the wellborecaused by the downward movement of drill pipe accordingto (6) and (7) and calculate the surge pressure at that timeit is indispensable to obtain the definite conditions of themodel for the three scenarios that is initial conditions andboundary conditions
We established coordinate systems of the hydraulic sys-tem in the wellboreThe coordinate systems have their originat the end of the drill pipe The flow rate of gas and drillingmud two-phase in open borehole annulus (comprised bydrill pipe and the casing) and drill pipe were respectivelydefined as 119876
1 1198762 and 119876
3 Respectively the corresponding
passageways filled with gas and drilling mud two-phase fluidwere marked as I III and II andthe pressure is 119875
1 1198752 and
1198753 The positive direction of passageway I was towards the
bottomhole while the positive directions of the passagewayIII and II were towards the wellhead As seen in Figure 2 119871
1
was the length from the origin to the bottomhole and 1198712was
the length from the origin to the wellhead By calculating thevelocity of two phases flow the new model for surge pressurein gas-drilling mud two-phase during the movement of drillpipe can be deduced [36]
In actual hydraulic system of the well many pipelineseries are included By assuming the length between thenodes and origin as 119871
3 the boundary consistency conditions
are provided by
119876119869+1
119894(1198713) minus 119876119869
119894(1198713) = V119901sdot Δ119860
119901119869+1
119894(1198713) = 119901119869
119894(1198713)
(119894 = 1 2 3) (36)
According to the relationship between the pressure andvelocity at the wellbore the boundary conditions can bedetermined as follows
119901119894(1198712) = 0
119876119894(1198711) = 0
(119894 = 2 3) (37)
The effective cross-area of jet nozzles is
119860119890=1
4120587
119899
sum
119894=1
1198892
119894 (38)
31 OPOP Scenario The following equation presents theinitial conditions of the flow rate and pressure for OPOPscenario
119901119894(119911) = 0
119876119894(119911) = 0
(119894 = 1 2 3) (39)
The boundary conditions of the flow rate and pressure are
1198761(0 119905) + 119876
2(0 119905) + 119876
3(0 119905) = V
119901(119905) (119860
0minus 1198603)
1199011(0 119905) = 119901
2(0 119905)
1199011(0 119905) minus 119901
3(0 119905)
=120588119898
2120583119898
(1198763(0 119905)
119860119890
+ V119901(119905))
10038161003816100381610038161003816100381610038161003816
1198763(0 119905)
119860119890
+ V119901(119905)
10038161003816100381610038161003816100381610038161003816
(40)
32 CPOP Scenario The following equation presents theinitial conditions of the flow rate and pressure for CPOPscenario
The initial conditions of the flow rate and pressure are asfollows
1199011(119911 0) = 0 119876
1(119911 0) = 0
1199012(119911 0) =
11989121199111205881198981198762
119908
811989821198602
1198762(119911 0) = 119876
119908
1199013(119911 0) =
1198913(1198712minus 119911) 120588
1198981198762
119908
811989831198603
1198763(119911 0) = 119876
119908
(41)
Theboundary conditions of the flow rate and pressure canbe written in the following form
1198761(0 119905) + 119876
2(0 119905) = V
119901(119905) 1198600
1198763(0 119905) = minusV
119901(119905) 1198603
1199011(0 119905) = 119901
2(0 119905)
(42)
33 OPWP Scenario As shown in Figure 2(c) the OPWPscenario is discussed below
Excluding atmospheric pressure the initial conditions atthe wellhead and bottom hole are
1199011(119911 0) = 0 119876
1(119911 0) = 0
1199012(119911 0) =
11989121199111205881198981198762
119908
811989821198602
1198762(119911 0) = 119876
119908
1199013(119911 0) =
1198913(1198712minus 119911) 120588
1198981198762
119908
811989831198603
1198763(119911 0) = 119876
119908
(43)
Mathematical Problems in Engineering 7
The boundary conditions of the flow rate and pressure areas follows
1198761(0 119905) + 119876
2(0 119905) = V
119901(119905) (119860
0minus 1198603) + 119876119908(0 119905)
1198763(0 119905) = minusV
119901(119905) 1198603+ 119876119908(0 119905)
1199011(0 119905) = 119901
2(0 119905)
(44)
4 Solution of the United Model
Referring to Figure 4 the variable 119909 is defined as a short pipeand the constraint is 119871
119909lt 119888119909sdot Δ119905 The variables 119910 and 119911 are
short pipe which are made up of wellbore and drill pipe andthe constraint is 119871
119910lt 119888119910sdot Δ119905 119871
119911lt 119888119911sdot Δ119905 The variables
119909 + 1 119910 + 1 and 119911 + 1 are defined as a long pipe and theconstraint is 119871
119909+ 1 lt 119888
119909+ 1 sdot Δ119905 119871
119910+ 1 lt 119888
119910+ 1 sdot Δ119905 119871
119911+ 1 lt
119888119911+ 1 sdot Δ119905 The implicit finite-difference method is applied to
calculate the short pipe and characteristicsmethod is appliedto calculate the long pipe The pressure and flow rate can beobtained at the end of drill pipe as follows
[1199012119905]2minus 120588119898[119888119910+1
119860119910+1
119876119910+1
2119905]
2
minus [1199013119905minusΔ119905
]2minus 120588119898[119888119910+1
119860119910+1
119876119905minusΔ119905
]
2
+120588119898Δ119905
8[119888119910+1
119898119910+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
2
= 0
[1199012119905]3minus 120588119898[119888119911+1
119860119911+1
119876119911+1
2119905]
3
minus [1199013119905minusΔ119905
]3minus 120588119898[119888119911+1
119860119911+1
1198763119905minusΔ119905
]
3
+120588119898Δ119905
8[119888119911+1
119898119911+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
3
= 0
[1199012119905]1minus 120588119898[119888119909+1
119860119909+1
119876119909+1
2119905]
1
minus [1199013119905minusΔ119905
]1minus 120588119898[119888119909+1
119860119909+1
1198763119905minusΔ119905
]
1
+120588119898Δ119905
8[119888119909+1
119898119909+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
1
= 0
[1199012119905+ 1199011119905]2+ 11986311[119876119910
2119905+ 119876119910
1119905]2
+ 11986312= 0
[1199012119905minus 1199011119905]2+ 11986411[119876119910
2119905minus 119876119910
1119905]2
+ 11986412= 0
[1199012119905+ 1199011119905]3+ 11986321[1198762119905+ 119876119911
1119905]2
+ 11986322= 0
[1199012119905minus 1199011119905]3+ 11986421[1198762119905minus 119876119911
1119905]2
+ 11986422= 0
[1199012119905+ 1199011119905]1+ 11986331[1198762119905minus 119876119909
1119905]1
+ 11986332= 0
[1199012119905minus 1199011119905]1+ 11986431[1198762119905+ 119876119909
1119905]1
+ 11986432= 0
1
2
3
i minus 1
i
i + 1
N minus 2
N minus 1
N
Bottomhole
Wellhead
3
3
2
1
2
3
c+y+1cz+1
Δsx+1
Δsx+1
Ly = Lz
Lx
cminusx+1
z
Figure 3 Schematic diagram of the solve
[1198761119905]1+ [1198761119905]2+ [1198761119905]2minus V119901(119905)
sdot (1198600minus 119860119894) = 0
[1199011119905]1minus [1199011119905]3minus
120588119898
2(119906119860119890)2
times [1198761119905]3+ 119860119890V119901(119905)
10038161003816100381610038161003816[1198761119905]3+ 119860119890V119901(119905)
10038161003816100381610038161003816= 0
119876119910+1
2119905minus 119876119910
2119905minus V119901(119905)
sdot Δ119860 = 0
[1199011119905]1= [1199011119905]2
11986311=1205881198981198882
119860sdotΔ119905
119871119869
11986312=120588119898
119860sdot119871119869
119871119869
(45)
The119863119896119895 119864119896119895can be expressed as follows
119863119894119905= minus (119901
119894+1119905minusΔ119905+ 119901119894119905) + 11986311(119876119894+1119905minusΔ119905
minus 119876119894119905minusΔ119905
)
119864119894119905= minus (119901
119894+1119905minusΔ119905+ 119901119894119905) + 11986411(119876119894+1119905minusΔ119905
minus 119876119894119905minusΔ119905
)
+120588119898119891119871119869
16119898(V119894+1119905minusΔ119905
+ V119894119905minusΔ119905
)1003816100381610038161003816(V119894+1119905minusΔ119905 + V
119894119905minusΔ119905)1003816100381610038161003816
(46)
where 119896 = 1 2 3 and 119895 = 1 2Obtaining the analytical solution of the mathematical
models concerned with flow pattern void fraction param-eters and pressure drop gradient are generally impossible formost practical in two-phase flow In this paper the Runge-Kutta method (R-K4) is used to discrete the new model forsurge pressure At different wellbore depths we can obtainpressure temperature gas velocity drilling mud velocity and
8 Mathematical Problems in Engineering
void fraction in MPD operations by application of the R-K4The solution of pressure drop gradient equation (3) can beseen as an initial-value problem of the ordinary differentialequation
119889119901
119889119911= 119865 (119911 119901)
119901 (1199110) = 1198750
(47)
With the initial value (1199110 1199010) and the function 119865(119911 119901)
(3)ndash(5) can be obtained
1198961= 119865 (119911
0 1199010)
1198962= 119865(119911
0+ℎ
2 1199010+ℎ
21198961)
1198963= 119865(119911
0+ℎ
2 1199010+ℎ
21198962)
1198964= 119865 (119911
0+ ℎ 119901
0+ ℎ1198963)
(48)
where ℎ is the step of depthThe pressure on the nod 119894 = 119894 + 1can be obtained by
1199011= 1199010+ Δ119901 = 119901
0+ℎ
6(1198961+ 21198962+ 21198963+ 1198964) (49)
In the present work the surge pressure calculation modelsolved by personally compiled code on VBNET (Version2010) The schematic diagram of solve is shown in Figure 3and the solution procedure for the surge pressure in thewellbore is shown in Figure 4 At initial time the wellheadtemperature wellbore structure well depths gas and drillingmud two-phase properties and so forth are known On thenode 119894 the pressure gradient wave velocity varieties in real-time temperature and the void fraction can be obtained byadopting R-K4 The density of gas can be obtained by gasEOS (15) and the density of drilling mud can be obtained byequations (19) and equations (20) Then the surge pressureat different depths of wellbore in tripping operations can besolved by (45)Theprocess is repeated until the surge pressurein the whole wellbore has been obtained [37 38]
The developed model takes full consideration of the gasinflux and wave velocity in real-time Owing to the complexconditions of wellbore in tripping operations measurementof surge pressure in the actual drilling process is very difficultIn order to verify the new model for surge pressure thepredicted surge pressure is compared with the results ofprevious experimental investigations presented by Junfang intripping operations in Figure 5 [39] The comparisons revealthat the developed new model fits well with the experimentsdata Thus the new model can be used to accurately predictsurge pressure on different scenario and gas influx rate intripping operations The value of surge pressure calculatedby adopted Burkhardtrsquos steady method is larger than theexperiments data
Table 1 Parameters of calculation well
Type Property Value
Mud Dynamic viscosity (Pasdots) 0056Density (kgm3) 1460
Gas Relative density 065Viscosity (Pasdots) 114 times 10minus5
StringElastic modulus of string (Pa) 207 times 1011
Poisson ratio of string 03Roughness (m) 154 times 10minus7
Surface condition Surface temperature (K) 298Atmosphere pressure (MPa) 0101
5 Analysis and Discussion
The drilling system described is a closed system Theschematic diagram of gas influx process in tripping opera-tions is illustrated in Figure 6 The drilling mud is pumpedfrom surface storage down the drill pipe returns from theannulus and travels back through surface processing wheredrilling solids are removed to surface storage
The well used for calculation is a gas well in XinjiangUygur Autonomous Region China The wellbore structurewell design parameters (depths and diameters) gas anddrilling mud two-phase properties (density and viscosity)and operational conditions of calculation well are displayedin Table 1 [40]
The drilling mud mixed with gas is considered as a two-phase flow medium The surge pressure in the gas-drillingmud are calculated and discussed by using the establishedmodel and well parameters As shown in Figure 7 accordingto the drill conditions such as the drill pipe velocity gasdrilling mud two-phase flow velocity the total length of drillpipe and drill pipe is 1000mwe can account for the changingrules of surge pressureThe following examples illustrate thatthe surge pressure is affected by different well parametersdrilling mud density gas influx rate and well depth in thetripping operations
51 The Effects of Various Drilling Parameters on SurgePressure Figure 7 presents the influence of acceleration ondrilling mud velocity drilling mud velocity and drill pipeoperating speed with respect to the drilling parameters Themovement of drill pipe starts at the depth of 1000m Withthe movement of the drill pipe the drilling mud also movesin the opposite direction along wellbore at the same timeThe drill pipe operating speed reaches the maximum valueat 23 seconds and stops at 30 seconds In order to make acomparison of the surge pressure in different well parametersthe analysis and compared Figures 8ndash15 based on the data ofFigure 7
Figures 8 and 9 graphically show the distributions ofvoid fraction and variations of wave velocity along the flowdirection in the wellbore When gas influx occurs at thebottom of well some gas invades into the wellbore andmigrates form the bottomhole to the wellhead along the flowdirection At low gas influx rate it is extremely obvious that
Mathematical Problems in Engineering 9
Start
Initial conditions
Grid division
Solve equation (21) for temperature
Calculate k1 minus k4Pi+1 = Pi + ΔP
zi+1 = zi + h
Solve equation (12) for 120588m
Solve equation (14) for c
Determine the time and the step
Enter the pipe velocity
Spline smoothing
Satisfy the boundary conditions and initial conditions
Δt rarr 0
Analysis conditions at the end of pipe
Solve pressure and flow at boundary points
Solve pressure and flow in pipe points
T = T + Δt
NoT lt Tmax
YesEnd
Figure 4 Solution procedure for surge pressure in MPD operations
80 16 24 32 40
00
05
10
15
20
Valu
e
t (s)
Pipe velocity (ms)This model (MPa) Experimental data (MPa)
Burkhardt data (MPa)
Figure 5 Experimental verification by comparison with previousexperiment data
the void fraction and wave velocity first slightly change in acomparatively smooth value then change sharply It is becausethe volume of gas expanded rapidly with the decreasingof pressure near the wellhead the void fraction increasessharply and at the same time the wave velocity decreasesobviously At a high gas influx rate the wave velocity tends toincrease because the void fraction in thewellhead is increasedto a high extent In conclusion the wave velocity is sensitiveto the void fraction and the void fraction is dominated by
MudMotor
Choke
z
DAPC
Casing
Drill pipeAnnulus
Drill collar
Formation
RCD0
800m
1200m
4000m
1778mm 78 mm ) times G105
2159mm times G105120601
120601 (120601
1270mm 78 mm ) times G105120601 (120601
Figure 6 Schematic diagram of gas influx process in tripping
influx rate and pressure in the wellbore especially the influxrate
As shown in Figure 9 within the range of low gas influxrate the wave velocity decreases significantly It is becausethe compressibility of the gas increases remarkably andthe medium appears high elasticity though the density ofgas-drilling mud two-phase flow changes slightly With theincrease of gas influx rate and corresponding increase of thevoid fraction in the wellbore the compressibility of the two-phase unceasingly increases which promotes themomentumand energy exchange in the interface Therefore the wavevelocity continuously decreases
10 Mathematical Problems in EngineeringVa
lue
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
minus05
minus10
Pipe acceleration (ms2)Pipe velocity (ms)Fluid velocity (ms)
Figure 7 Drill pipe velocity variation at different accelerations
0 800 1600 2400 3200 4000H (m)
100
80
60
40
20
0
minus20
G(
)
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
120601
Figure 8 Void fraction distribution at different gas influx rates
0 800 1600 2400 3200 4000H (m)
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
1500
1200
900
600
300
0
c (m
s)
Figure 9 Wave velocity variation at different gas influxes
0 10 20 30 40 50 60
25
20
15
10
05
00
minus05
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
P (M
Pa)
t (s)
Figure 10 Surge pressure variation at different gas influxes
Figure 10 illustrates the effect of gas influx on the surgepressure with the change of gas influx rate As shown inFigure 10 under different gas influx rates (0m3h 2427m3hand 8312m3h resp) the maximum surge pressure mayoccur before drill pipe reaches bottomhole With theincreases of the gas influx the corresponding surge pressuredecreases Considering the gas influx in the calculation of thesurge pressure in the wellbore the surge pressure weakensobviously As the gas influx rate decreases the surge pressureat bottom of the wellbore is reduced gradually while surgepressure can cause a significant time lag if the gas influxoccurs at the bottom of well The lag time is mainly affectedby wave velocity
Figure 11 presents a change of surge pressure in thewellbore at different scenario (OPOP CPOP and OPWPresp) According to Figure 11 surge pressure is increasedin CPOP scenario by comparing with the surge pressure inOPOP scenarios It is because the speed of the drilling mudincreases remarkably in CPOP scenario the surge pressurebecomes larger obviously Especially when the maximumsurge pressure occurs a strong increase of surge pressure isobserved at OPWP scenario If the movement of drill pipeis stop immediately the surge pressure will not disappearimmediately and the surge pressure will exist and changeslightly for short while
Figures 12 and 13 show that the surge pressure varies withwellbore diameter caused by the drill pipe operating speedaccording to Figure 7 The surge pressure decreases with theincrease of wellbore diameter It is because the drilling mudflow rate decreases with the increases of wellbore diameterBased on (1) and (2) surge pressure decreases at each positiondue to the decrease of drilling mud flow rate Surge pressurecaused by converting of potential energy into kinetic energyand then kinetic energy back into potential energy over andover again During the tripping operations especially in slimwellbore drilling process the speed of drill pipe movementshould be appropriately reduced The surge pressure in
Mathematical Problems in Engineering 11
OPWPCPOPOPOP
t (s)0 10 20 30 40 50 60
16
12
08
04
00
P (M
Pa)
Qg = 2427m3h
Figure 11 Surge pressure variation at different scenarios
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 2427m3h
Figure 12 Surge pressure variation at different wellbore diameters(gas influx 119876
119892= 2427m3h)
Figure 13 is prominently larger than that in Figure 12 Thisis because both the decrease of the wave velocity and theincrease of the compression in gas drilling mud two-phasedue to the gas influx result in the comparatively large surgepressure
Figure 14 illustrates that the surge pressure varied withthe length of drill pipe The surge pressure caused by shorterdrill pipe (whose the total length is 1000m) is smaller thanthe longer drill pipe (whose total length is 3000m) beforethe maximum surge pressure occurs This can explain thatsurge pressure is affected by the friction force of drill pipe andwellbore Compared with the longer drill pipe the frictionforce caused by shorter drill pipe is largerTherefore the surgepressure gradually decreases As a result the surge pressure isgreater at 1000m
Figure 15 illustrates that the surge pressure varied withthe drilling fluid density At the bottom of well the surge
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 0m3h
Figure 13 Surge pressure at different borehole diameters (gas influx119876119892= 0m3h)
t (s)0 10 20 30 40 50 60
P (M
Pa)
16
12
08
04
00
Qg = 2427m3h
L = 1000mL = 3000m
Figure 14 Surge pressure variation at different well depths
pressure increases with the increase of drilling fluid densityThe changes are significant in the maximum surge pressureThis is because the surge pressure is mainly affected by wavevelocity density and velocity of gas drilling mud two-phaseAccording to (6) and (7) the surge pressure is increased as thedrilling fluid density increasesThemaximum surge pressureoccurs before drill pipe reaches bottomhole
52 The Effects of Gas Influx Rate on Surge Pressure Figures16 17 and 18 illustrate that the surge pressure varied withthe increase of the gas influx rate Figure 17 shows the surgepressure with the gas influx rate of 2427m3h at the bottomof well Figure 18 shows the surge pressure with the gas influxrate of 8312m3h at the bottom of well Mainly affected bywave velocity density and velocity in gas-drilling mud twophase the surge pressure is greatest at the gas influx rate of0m3h Figure 16 illustrates an example of a minimum wavevelocity at the gas influx rate of 8312m3h at the bottom
12 Mathematical Problems in Engineering
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
Qg = 2427m3h
= 1500kgm3=
= 2000 kgm3=
120588m120588m
Figure 15 Surge pressure variation at different drilling fluid densi-ties
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 0m3h
Figure 16 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 0m3h)
of well As the gas influx rate increases the surge pressuredecreases When gas influx occurs the lag time is lengthenedremarkably compared with the condition of no gas influxThe longer the distance is the greater the lag time will beThe lag time is mainly determined by the propagation time ofpressure wave in the wellbore
Figures 19 and 20 illustrate the surge pressure caused bydifferent operating speeds of the drill pipe at various gasinflux rates at the bottom of well As drilling mudmixed withinflux gas not only reduces the pressure wave velocity butalso reduces the density of themixture density surge pressuremainly affected by wave velocity andmixture density reducesdrastically with the increase of gas influx rate at the bottomof well Figures 19 and 20 are compared to Figures 17 and 18
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 2427m3h
Figure 17 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 8312m3h
Figure 18 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 8312m3h)
it can be concluded that the operating speed is also a majorfactor of surge pressure
53The Effects of Drill Pipe Operating Speed on Surge PressureFigure 21 illustrates the effect of drill pipe speed on the surgepressure The change of surge pressure is smaller in gasdrilling two-phase than in single-phase drilling mud Thesurge pressure becomes smaller and closes to zero after themovement is stopped due to greater friction force existing inwell wall during drill pipe operating
6 Conclusions
With full consideration of the important factors such as wavevelocity in real-time the compressibility of the gas phasegas influx rate pressure temperature and well parametersa new surge pressure model has been proposed based on the
Mathematical Problems in Engineering 13Va
lue
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 2427m3h
Figure 19 Surge pressure and fluid velocity variation at differentpipe velocities (gas influx 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 8312m3h
Figure 20 Surge pressure and fluid velocity variation at differentgas influxes (gas influx 119876
119892= 8312m3h)
30
24
18
12
06
00
Valu
e
t (s)0 10 20 30 40 50 60
Pipe velocity (ms)Pipe velocity (ms)Surge pressure (MPa)
Surge pressure (MPa)
Qg = 2427m3h
Figure 21 Surge pressure variations at different pipe velocities (gasinflux 119876
119892= 2427m3h)
mass conservation equations the momentum conservationequations characteristic line method and the gas-drillingmud equations of state (EOS) duringMPDoperations Solvedby the difference method and fourth-order explicit Runge-Kutta method the model is used to predict surge pressurefor different drill pipe operating speed and gas influx rates inMPD operations The main conclusions can be summarizedas follows
(1) An accurate mathematical model to predict surgepressure after gas influx is of great importancebecause of the effects of surge pressure on operationalsafety and success The new model provides betterprediction of surge pressures in comparison with theexisting experiment
(2) The surge pressure tends to increase with the increaseof drill pipe operating speed and decrease of gas influxrate andwell diameter Considering that the gas influxoccurs the surge pressure weakens obviously Withthe increases in well depth the maximum allowabletripping speed gradually become smaller the surgepressure increases at the bottom of well and drillingsafety window is accordingly narrower The surgepressure can cause a significant time lag if the gasinflux occurs at the bottom of well and the timelag is mainly determined by the propagation time ofpressure wave in the wellbore
(3) The surge pressure obviously varies at different sce-narios (OPOP CPOP and OPWP resp) duringthe tripping operations Surge pressure increased inCPOP scenario than in OPOP scenario Especiallywhen the maximum surge pressure occurs a strongincrease of surge pressure is observed at OPWPscenario By running the new surge pressure modeldrilling engineers can identify potential surge prob-lems and optimize tripping operations for differentscenarios as to avoid complicated accidents to happenin wellbore during drilling operations
(4) The calculation of surge pressure in gas drillingmud two-phase is more complex than conventionalcalculation in single drillingmudDrill pipe operatingspeed in drilling operations should be strictly con-trolled and the surge pressure should be consideredin the design The greatest advantages of new modelare the ability to reduce nonproductive time (NPT) indrilling
Nomenclature
A Wellbore effective cross-area (m2)1198600 The outer drill pipe cross-area (m2)
1198602 Wellbore cross-area (m2)
1198603 The inner Drill pipe cross-area (m2)
119860119890 Effective cross-area of jet nozzle (m2)
119888 Wave velocity (ms)119889119904 One discrete length of wellbore (m)
14 Mathematical Problems in Engineering
119863 Wellbore effective diameter (m)1198631 The inner diameter of casing (m)
1198632 The outer diameter of casing (m)
1198633 The inner diameter of drill pipe (m)
1198634 The outer diameter of drill pipe (m)
119863119896119895 Differential equation coefficients
119864119896119895 Differential equation coefficients
119864119904 Modulus of elasticity of the drill pipe
(MPa)119864119891 Modulus of elasticity of formation (MPa)
1198642 Modulus of elasticity of casing (MPa)
119891119866 Shear stresses coefficient of gas interface
119891 Friction coefficient1198912 Wellbore friction coefficient
1198913 Drill pipe friction coefficient
119891119871 Shear stresses coefficient of drilling mud
119865119911 Pressure applied on the two-phase flow
(N)119892 Acceleration due to gravity (m2s)119892119888 Conversion factor
119892119879 Geothermal temperature gradient (Km)
119896119890 Wellbore effective roughness (m)
1198960 Roughness of outer pipe (m)
119896119894 Roughness of inner pipe (m)
119867 The depth of well (m)1198711 The length from pipe bottom to hole
bottom (m)1198712 The length from pipe bottom to well
head (m)119871119892 Flow pattern119898 Hydraulic radius (m)1198982 Wellbore hydraulic radius (m)
1198983 Drill pipe hydraulic radius (m)
119873 Parameter combiningThompson energyeffects
119875 Pressure (MPa)1198750 Initial pressure of wellbore (MPa)
1198752 Pressure within the empty wellbore
(MPa)1198753 Pressure within the drill pipe (MPa)
119875119888 Critical pressure (kPa)
119875119871 Pressure of drilling mud (MPa)
119875119903 Reduced pressure
119875119895119905 Pressure at node (119895 119905) (MPa)
119902119898 Volumetric flow rate of two phase (m3s)
119902119866 Velocity of the gas (m3s)
119902119871 Velocity of the drilling mud (m3s)
119876119871 Mud flow rate within the wellbore(m3h)
1198761 Drilling mud flow rate within hole(m3h)
1198762 Mud flow rate within the wellbore(m3h)
1198763 Mud flow rate within the drill pipe(m3h)
119876119908 Pump flow rate (m3s)
119876119895119905 Flow rate at time 119905 and nod 119895 (m3s)
119877 Constant of EOS (JKgK)
119904 The length of wellbore along wellbore(m)
119905 Time (s)119879 Temperature (K)119879119888 Critical temperature (K)
119879119903 Reduced temperature
119879119890119894 Undistributed temperature at a depth
(K)119879119890119887ℎ
Undistributed temperature of wellhead(K)
119879119891119887ℎ
Undistributed temperature at bottom-hole (K)
V Velocity of drilling mud and gas (ms)V119901 Velocity of drill pipe (ms)
V119898 Gas and drilling mud flow velocity (ms)
V119866 Gas flow velocity (ms)
V119904 Slip velocity (ms)
V119904119892 Superficial gas velocity (ms)
V119871 Drilling mud flow velocity (ms)
119882119898 Mass flow velocity (kgm3)
119911119866 Gas deviation factor
1199110 Initial calculation height of wellbore (m)
119911119887ℎ Total well depth from surface (m)
119911 Variable of distance from surface (m)
Greek Letters
120588119866 Gas density (kgm3)
120591119891 Frictional pressure gradient (Pam)
120591119908 Frictional pressure coefficient
120588119898 Gas and drilling density (kgm3)
1205880 Density under atmospheric pressure (kgm3)
120588119871 Drilling mud density (kgm3)
120590119904 Surface tension (Nm2)
120601119866 Gas void fraction
120601119871 Drilling mud holdup
120588119903 Reduced density
120583119871 Viscosity of drilling mud (Pasdots)
120579 The ground angle (rad)120583119891 Poissonrsquos ratio of formation
120583119904 Poissonrsquos ratio of drill pipe
1205832 Poissonrsquos ratio of casing
Δ119860 The change of the across areas (m2)
Subscripts
CPOP Closed pipe without pumpingDAPC Dynamic annular pressure controlEOS Equations of stateERV Relief-coupling valveMPD Managed pressure drillingOPOP Open pipe without pumpingOPWP Open pipe with pumpingR-K4 The fourth-order explicit Runge-KuttaRCD The rotating control device
Mathematical Problems in Engineering 15
Subscripts of Graph
119888 Wave velocity in two-phase flow (ms)119863 Wellbore effective diameter (m)119867 Depth of wellbore (m)119871 The length from drill pipe and drill pipe (m)119876119892 Gas influx rate at the bottomhole (m3h)
120601119866 Void fraction ()
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 51274170 andQC2012C128) Without their support this work would nothave been possible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of AirInjection in Annuli to manage pressure during CementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling Conferenceand Exhibition San Diego Calif USA March 2012
[4] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the Norh Africa Technical Conference andExhibition Cairo Egypt February 2012
[5] J Saponja A Adeleye and B Hucik ldquoManaged-pressuredrilling (MPD) field trials demonstrate technology valuerdquo inProceedings of the IADCSPE Drilling Conference Miami FlaUSA February 2006
[6] G Liu ldquoPredicting surge pressures that result from runninglinersrdquoWorld Oil vol 222 no 4 p 92 2001
[7] K Zhou J Yang B Zhong and L Tang ldquoSurge pressure inwellborerdquo Oil Drilling and Production Technology vol 12 no 2pp 1ndash10 99 1990
[8] R S Rudi Rubiandini ldquoNew formula of surge pressure fordetermining safe trip velocitiesrdquo in Proceedings of the SPE AsiaPacific Oil and Gas Conference and Exhibition pp 16ndash18 ausOctober 2000
[9] J Li Z Huang K Li and W Wu ldquoThe influence of collaron surge pressure caused by the drilling fluid viscous forceunder pumping conditionrdquoResearch Journal of Applied SciencesEngineering and Technology vol 5 no 5 pp 1781ndash1785 2040ndash7459 2013
[10] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvancedMaterials Research vol 468ndash471 pp 1736ndash1742 2012
[11] F Crespo and R Ahmed ldquoSurge-and-swab pressure predictionsfor yield-power-law drilling fluidsrdquo SPE Drilling amp Completionvol 27 no 4 pp 574ndash585 2012
[12] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[13] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquo PHDThesis vol 50 no 9 p 143 1989
[14] H Fan ldquoDevelopment of application software for dynamicsurge pressure while trippingrdquo Petroleum Drilling and Technol-ogy vol 23 no 4 pp 11ndash14 1995
[15] K Zhou J Yang B Zhong and L Tang ldquoSurge pressurein wellborerdquo Journal of Petroleum Exploration and ProductionTechnology vol 12 no 2 pp 1ndash10 1990
[16] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquoDissertation Abstracts International Bvol 50 no 9 1989
[17] Z Jiang ldquoStudies of wellbore surge pressure by a dynamicmethodrdquo Journal of Petroleum Exploration and ProductionTechnology vol 13 no 3 pp 39ndash44 97 1991
[18] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[19] D Johnston M Bracken and J Thornton ldquoAssessing pipecondition useful remaining life and maximum operating andsurge pressures when to change the pipe or control thepressurerdquoMagazine of the International Water Association vol14 no 5 pp 43ndash44 2012
[20] H Fan Y Chu and X Liu ldquoPrediction for wellbore dynamicsurge pressure while tripping a drill piperdquo Journal of ChinaUniversity of Petroleum vol 19 no 5 pp 36ndash41 1995
[21] C K Burgess J E Miller and K D Vondervort ldquoDownholesurge pressure reduction system and method of userdquo World9848 143 1998
[22] J P Allamon J E Miller and A M Macfarlane ldquoDownholeSurge Reduction Method and Apparatusrdquo 2004
[23] K S Bjoslashrkevoll R Rommetveit B Aas H Gjeraldstveit andA Merlo ldquoTransient gel breaking model for critical wellsapplications with field data verificationrdquo in Proceedings of the2003 Offshore Mediterranean Conference February 2003
[24] G H Nygaard E Johannessen J E Gravdal and F IversenldquoAutomatic coordinated control of pump rates and choke valvefor compensating pressure fluctuations during surge and swaboperationsrdquo in Proceedings of the IADCSPE Managed PressureDrilling and Underbalanced Operations Conference pp 120ndash125March 2007
[25] J V Fedevjcyk S L M Junqueira and C O R Negrao ldquoSurgeamp swab pressures in wells with cross-section changesrdquo BoletimTecnico da Producao de Petroleo vol 4 no 2 pp 221ndash234 2009
[26] S Tian G H Medley and C R Stone ldquoUnderstandingyield point effect onpressure surges critical to managing deepdifficultMPDwellsrdquoDrilling Contractor vol 65 no 4 p 46 4850 53 2009
[27] Y Sun Q Li and J Zhao ldquoNew method of predicting surgepressure apply to horizontal well based on caisson flowrdquoNaturalScience vol 2 no 12 pp 1394ndash1399 2150ndash4091 2010
[28] B T Lebele-Alawa and F E Oparadike ldquoAnalysis of the effectsof valve propagated pressure surge on pipe flowrdquo Engineeringvol 3 no 11 pp 1098ndash1101 2011
[29] X Kong Q Yuan and Y Qiu ldquoThe research of two-phasefluctuation pressure in wellbore tripping operationrdquo InnerMongolia Petrochemical Industry TE242 2011
[30] F Crespo and R Ahmed ldquoA simplified surge and swab pressuremodel for yield power law fluidsrdquo Journal of Petroleum Scienceand Engineering vol 101 pp 12ndash20 2013
16 Mathematical Problems in Engineering
[31] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
[32] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[33] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86-87 pp 127ndash136 2012
[34] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[35] M J SanchezComparison of Correlations for Predicting PressureLosses in Vertical Multiphase Annular Flow University of TulsaTulsa Okla USA 1972
[36] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[37] Y Lin X Kong Y Qiu and Q Yuan ldquoCalculation analysis ofpressure wave velocity in gas and drilling mud two-phase fluidin annulus during drilling operationsrdquo Mathematical Problemsin Engineering vol 2013 Article ID 318912 17 pages 2013
[38] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[39] H Junfang ldquoBalanced drilling and well controlrdquo PetroleumIndustry Press vol 1 article 33 1992
[40] W A Bacon Consideration of compressibility effects for applied-back- pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Va USA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
in the drilling operation are caused by surge pressure It issuggested that excessive surge pressures have initiated lostcirculation formation fracturing and well kick problemsduring the drilling operation This may result in expensivedrillingmud treating costs and cause otherwellbore problems[8] Obviously the accurate calculation of surge pressure isrelated to the safety of the drilling process [9] Also thereis a particular relationship between reasonable drilling muddensity well structural design and surge pressureThereforethe surge pressure is very important basic parameters indrilling design [10] So far the calculation of surge pressurein drillingmud and gas is still worth continuing and in-depthresearch
A number of field studies investigate the effects of well-bore geometry on surge pressures anddrillingmudproperties[11] However the first valid reports from surge pressure werenot published until 1960 and the method to calculate thesurge pressure began to apply to drilling industry [12] Atheoretical analysis of couette flow of power-law fluids forsurge pressure was conducted by Chukwu in 1989 [13] Modelfor both slot flow and concentric wellbore flowwas developedfor a stationary outer pipe and a steady axial motion of aconcentric inner pipe The inner pipe was assumed to beeither plugged and of uniform geometry or open at thebottom After the year Fan [14] presented a method foraccurate calculation of the surge pressure caused by drillingmud viscosity during tripping in a vertical wellbore in 1990Themodels of laminar flow velocity profile and surge pressureof Newtonian and power law liquid in the wellbore aredeveloped and the drill pipe operating speed and drillingmud properties controlled for the safety of operation Inthe same year the method for calculating surge pressureunder stable conditions was proposed by Zhou et al [15] andthe problem existing in the calculation curves for adhesivefactor value was cited In this research the drilling mudis mainly composed of the power-law fluid and Binghamfluid The calculation method for surge pressure under stableconditions is applicable by taking the compressibility ofdrilling mud and expansion of wellbore into considerationIn the article of Chukwu [16] a laminar couette flow modelfor power-law fluids through a slot was considered Usingthe mathematical relationships developed knowledge of thedrilling mud rheological properties drill pipe operatingspeed and wellbore geometry is needed to determine thesurge pressure With consideration the compressibility ofdrilling fluids and wellbore surge pressure was calculatedby Jiang [17] By using theoretical analysis and experimentalresults it demonstrated that surge pressure is a functionof well depth the combination of drilling tools wellborediameter properties of drilling mud drill pipe operatingspeed and acceleration of drill pipe movement and so forthNext a calculation method of surge pressure caused bydrilling mud viscosity while the drill pipe moves througha Caisson drilling mud in wellbore was given by Wang etal [18 19] The coefficients of surge pressure under differentconditions are plotted These results are useful in the controlof drill pipe operating speed and the design of drillingmud properties In the computational model flow rate ofdrilling mud is considered as one-dimensional flow and
the method of characteristics and finite-difference techniqueare used to solve the basic equation of unsteady state flowMeanwhile a theoretical model is presented by Fan et al [20]for predicting dynamic surge pressure based on the theoryof unsteady state flow in 1995 The basic partial differentialequations for describing dynamic surge pressure are givenFan methods are proposed for determining boundary andinitial conditions when considering well parameters drill bitjet nozzles of drill bit variability of tripping velocity andacceleration An example for the application of the modelwas also given [21 22] Bjoslashrkevoll et al [23] developed aneasy to use transient surge swabmodel by taking into accountdampening of pressure peaks due to fluid compression andelasticity of open wellbore In this model the contributionsfrom acceleration are also included When there is littlegel strength in the fluid the transient model reproducedrelevant measured data with adequate accuracy RecentlyNygaard et al [24] present a new method for coordinatedcontrol of pump rates and choke valve for compensatingsurge pressure during tripping operations Fedevjcyk et al[25 26] searched that wellbore diameter variations and theuse of drill pipe accessories might cause changes to theannular cross-section space between the drill pipe and theborehole This study proposes a mathematicalnumericalmodel to simulate the surge pressure problem in wellborewith variable cross-section areasThe fluid flow yielded by themovement of drill pipe is considered to be one-dimensionalcompressible and transient In order to predict the surgepressure caused in the horizontal well drilling process Sunet al [27] created models for predicting surge pressure basedon the general theory of hydrostatic drilling mud mechanicsand specifically described the flowing physicalmodel towardssurge pressure in horizontal wellbore by taking the effect ofdrill pipe eccentricity on the flowing law of drilling fluid intoconsideration Lebele-Alawa and Oparadike [28] analyse theeffects of surge pressure on pipe flow The surge pressureinvestigated is that propagated by the emergency relief-coupling valve (ERV) connected to a loading system carryingcrude oil from four flow stations Kong et al established thecalculation model for fluctuation pressure in gas-liquid two-phase flow during tripping operations but the model is justapplicable to steady state conditions [29] Crespo and Ahmed[30] reported recently the results of an experimental studyaimed at investigating the effects of drill pipe speed drillingmud properties and wellbore geometry on surge pressuresunder laboratory conditions
Although a large number of experiments under con-trolled laboratory conditions and modeling studies wereconducted in the past to investigate surge pressures theexisting surge pressure models rarely take the impact of thegas into consideration and thus will inevitably affect theprecision calculation for the surge pressure during MPDoperations If the gas influx occurs at the bottom of wellthe surge pressure may be reduced greatly The calculationmethod for predication of surge pressure in two-phase is notas accurate as the prediction of single-phase drillingmud dueto its simplicity In conclusion we must take into accountthis problem rationally and placemore emphasis on the surgepressure in two phases during MPD operations
Mathematical Problems in Engineering 3
Drilling mud
Gas
Drill pipe
Cross-section 2
Cross-section 1
Casing
p + dp + d
z
120579
p
Figure 1 The schematic of gas and drilling mud two-phase flowduring tripping operations
The object of the present work is to present a newcalculation model for predicting surge pressure during trip-ping operations which is applicable for gas drilling mudtwo-phase fluid In this paper in addition to the pressuretemperature and the void fraction in the wellbore the com-pressibility of the gas phase the wave velocity in real-timethe changes of wellbore parameter and three scenarios arealso taken into consideration By introducing themomentumconservation equations and the mass conservation equationsin MPD operations pressure wave velocity in real-time gas-drilling mud equations of state (EOS) and a new model forpredicting dynamic surge pressure in gas drilling mud two-phase during tripping operations is developed Numericalsolutions difference method and characteristics method areobtained to solve the model The model can be used topredict the change of the surge pressure at any well positionat different influx rate drill pipe operating speed and drillingparameter
2 The Mathematical Model
21 The Basic Equation Drilling mud contains clay cuttingsbarite other solids and so forth The solid particles aresmall and uniformly distributed therefore drilling mudis considered to be a pseudohomogeneous liquid and thenatural gas influx is considered to be the gas phase Asshown in Figure 1 the wellbore is filled with gas and drillingmud two-phase fluid The downward movement of drill pipealong wellbore filled with drilling mud can create a surge ofpressure The two-phase drilling fluid and the passagewayare all compressible and expansibleTherefore the two-phasedrilling fluid flow induced by the trip is unstable especiallyin the deep well
To establish the model about surge pressure during tripoperations the following assumptions are made
(i) the two-phase flow is treated as one-dimensional
(ii) no mass transfers between the gas and drilling mud
(iii) the flow pattern in wellbore is either bubble or slugflow
(iv) the annulus is concentric
When drilling strings are moving in the wellbore fullof drilling mud the pressure gradient equations the massconservation equations and the momentum conservationequations along the flow direction in the wellbore is com-bined to study the surge pressure in tripping operations
The mass conservation equation and momentum conser-vation equation are shown as follows
119901119860 + 120588119898119892(119860 +
120597119860
120597119911sdot119889119911
2) 119889119911 sdot sin 120579 + (119901 +
120597119901
120597119911sdot119889119911
2)120597119860
120597119911
sdot 119889119911 minus (119901119860 +120597 (119901119860)
120597119911119889119911)
minus 1205910119883119889119911 = 120588
119898(119860 +
120597119860
120597119911sdot119889119911
2) 119889119911 sdot
119889V119889119905
(1)
1
120588119898119892
120597119901
120597119911+1
119892(120597V120597119905
+ V120597V120597119911) +
119891V |V|8119898119892
= 0 (2)
The total pressure drop gradient is the sum of pressuredrop gradients due to potential energy change kinetic energyand frictional loss From (2) the equation used to calculatepressure gradient of gas drilling mud two-phase flow withinthe wellbore can be written as
119889119901
119889119911= 120588119898119892 sin 120579 minus
120591119908120587119863
119860minus 120588119898V119898
119889V119898
119889119911 (3)
Assuming the compressibility of the gas is only relatedwith the pressure in the wellbore the kinetic energy andacceleration term in the equation above can be simplified to
120588119898V119898
119889V119898
119889119911= minus
120588119898V119898V119904119892
119901
119889119901
119889119911= minus
119882119898119902119866
1198602119901
119889119901
119889119911 (4)
Substituting (3) into (4) the total pressure drop gradientalong the flow direction within the wellbore can be expressedas
119889119901
119889119911=
120588119898119892 + 120591119891
1 minus119882119898119902119866 (1198602119901)
(5)
The continuity equations can be obtained directly basedon (1)
1198882
(119875 119879119863 120601 119908 119871119892)120597V120597119911
+1
120588119898
119889119901
119889119905+ V
120597119901
120597119911= 0 (6)
where wave velocity 119888(119875 119879119863 120601 119908 119871119892) changes in real-timewhen the gas influx occurs at the bottom of drilling well
The equations of motion can be obtained directly basedon (2)
1
120588119898
120597119901
120597119911+120597V120597119905
+ 119892119889V119889119911
+119891V |V|2119863
= 0 (7)
22 Pressure Wave Velocity Only consider the deformationof wellbore caused by surge pressure the elastic constant ofdrill pipe is
120573 =2
1198631Δ119901
(1 minus 119906119904
119864119904
Δ1199011198632
1
11986322minus 11986321
1198631+1 + 119906119904
119864119904
1198632
2minus 1198632
1Δ119901
11986322minus 11986321
1
119863)
(8)
4 Mathematical Problems in Engineering
The elastic constant of well wall is
120573 =2
119864119891
(1 + 120583119891) (9)
The elastic constant between well wall and drill pipe is
120573 =2
11987722minus 1
[1198772
2
119864119891
(1 + 119906119891) +
1
119864119904
(1198772
1+ 1
11987721minus 1
minus 119906119904)] (10)
The elastic constant between casing and drill pipe is
120573 =2
11987722minus 1
[1198772
2
1198643
(1198772
3+ 1
11987721minus 1
+ 1199063) +
1
1198642
(1198772
1+ 1
11987721minus 1
minus 1199062)]
(11)
where 1198771= 11986321198631 1198772= 11986331198634 and 119877
3= 11986341198633
The density of gas and drilling mud mixture is describedas the following formula
120588119898= 120601119866120588119866+ 120601119871120588119871 (12)
The small perturbation theory is also applied to thesolution of wave velocity model According to the solvablecondition of the homogenous linear equations that thedeterminant of the equations is zero the equation of pressurewave can be expressed in the following form [31]
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(120588119892+ 119888119901120601119866120588119871
V2119904
1198882119892
)119908120601119892
1198882119892
[1 minus 119888119901120601119871]V2119904
1198882119871
119908 minus[120601119892120588119892119896 + 2119888
119901120601119892120601119871120588119871
V119904
1198882119871
119908] 22119888119901120601119892120601119871120588119871
V119904
1198882119871
119908
minus120588119871119908
1 minus 120601119866
1198882119871
119908 0 minus119896 (1 minus 120601119866) 120588119871
120588119871V2119903119896 (minus120601
119866119888119901+ 119888119903minus 119888119894+ 1198881198982) minus120601119866119896 [1 minus 120601
119871
119888119901V2119904
1198882119871
+ 119888119894
V2119904
1198882119871
]
120601119892(120588119892+ 119888V119898120588119871)119908
minus119894 (3
4
119888119863
119903120588119871120601119892V119904+4
119863119891119892119908120588119892V119892)
minus119888V119898120601119892120588119871119908 + 119894 (3
4
119888119863
119903120588119871120601119892V119904)
120588119871V2119904119896 (120601119871119888119901minus 2119888119903minus 1198881198982) minus119896(120601
119871+ 119888119903120601119892
V2119904
1198882119871
) minus119888V119898120601119892120588119871119908 + 119894 (3
4
119888119863
119903120588119871120601119892V119904)
120588119871[120601119871+ 120601119892119888V119898]119908
minus119894 (3
4
119888119863
119903120588119871120601119892V119904+4
119863119891119871120588119871V119871)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 0
(13)
where 119888119894= 03 119888
119901= 025 119888
1198982= 01 and 119888
119903= 02
The real value of wave number is determined by thepressure wave velocity 119888(119875 119879119863 120601 119908 119871119892) and pressure wavevelocity in the gas drilling mud two phase flow is
119888 (119875 119879119863 120601 119908 119871119892) =
1003816100381610038161003816119908119877+
(119896) minus 119908119877minus
(119896)1003816100381610038161003816
2 (14)
23 Physical Equations
231 Equations of State for Gas The equation of state (EOS)for gas can be expressed as follows
120588119866=
119875
(119885119866sdot 119877 sdot 119879)
(15)
where 119885119866is the compression factor of gas
The formula presented by Dranchuk has been used tosolve the compression factor under the condition of low andmedium pressure (119875 lt 35MPa) [32]
119911119866= 1 + (03051 minus
10467
119879119903
minus05783
1198793119903
)120588119903
+ (05353 minus026123
119879119903
minus06816
1198793119903
)1205882
119903
(16)
where 119879119903= 119879119879
119888 119901119903= 119901119901
119888 120588119903= 027119901
119903119885119866119879119903
The formula presented by Dranchuk and Abou-Kassemhas been adopted to solve the compression factor under thecondition of high pressure (119875 ge 35MPa) [32]
119911119866=006125119875
119903119879minus1
119903exp (minus12 (1 minus 119879minus1
119903) 2)
119884 (17)
where 119884 is given by
minus 006125119875119903119879minus1
119903exp (minus12(1 minus 119879minus1
119903)2
) +119884 + 119884
2
+ 1198843
+ 1198844
(1 minus 119884)3
= (1476119879minus1
119903minus 976119879
minus2
119903+ 458119879
minus3
119903) 1198842
minus (907119879minus1
119903minus 2422119879
minus2
119903+ 424119879
minus3
119903) 119884(218+282119879
minus1
119903
)
(18)
232 Equations of State for Drilling Mud Under differenttemperature and pressure the density of drilling mud can beobtained by the empirical formulas If 119879 lt 130
∘C the densityof drilling mud can be obtained by the following equation
120588119871= 1205880(1 + 4 times 10
minus10
119875119871minus 4 times 10
minus5
119879 minus 3 times 10minus6
1198792
) (19)
Mathematical Problems in Engineering 5
If 119879 ge 130∘C the density of drilling mud is
120588119871= 1205880(1 + 4 times 10
minus10
119875 minus 4 times 10minus5
119879 minus 3 times 10minus6
1198792
+ 04(119879 minus 130
119879)
2
)
(20)
233 Correlation of Temperature Distribution The tempera-ture of the drilling mud at different depth of the wellbore canbe determined by the relationship presented by Hasan andKabir [33]
119879 = 119879119890119894+ 119865 [1 minus 119890
(119911119887ℎ
minus119911)119860
]
times (minus119892 sin 120579119892119888119869119888119901119898
+ 119873 + 119892119879sin 120579)
+ 119890(119911119887ℎ
minus119911)119860
(119879119891119887ℎ
minus 119879119890119887ℎ)
(21)
24 Flow Pattern Analysis Based on the analysis of flowcharacteristics in the closed drilling system it can be safelyassumed that the flow pattern in wellbore is either bubbleor slug flow The pattern transition criteria for bubbly flowand slug flow given by Orkiszewski are used to judge the flowpattern in the gas-drilling mud two-phase flow [34]
For bubbly flow
119902119866
119902119898
lt 119871119861 (22)
For slug flow
119902119866
119902119898
gt 119871119861 119873
119866119881lt 119871119904 (23)
where 119902119898is the volumetric flow rate of two-phase flow and
119902119898= 119902119866+ 119902119871
The dimensionless numbers 119871119904and 119871
119861are defined as
119871119904= 50 +
36119873119866119881119902119871
119902119866
119871119861= 1071 minus
07277V2119898
119863
(24)
where
119873119866119881
= V119904(120588119871
119892120590119904
)
025
(25)
Flow parameters such as void fractionmixed density andvirtual mass force coefficient are discussed for specific flowpattern
The correlation between void fraction and drilling mudholdup is expressed as
120601119866+ 120601119871= 1 (26)
241 Bubble Flow The average density of the mixture forbubble flow is
120588119898= 120601119871120588119871+ 120601119866120588119866 (27)
The void fraction for bubble flow is determined by thefollowing equation
120601119866=1
2[
[
1 +119902119898
V119904119860minus radic(1 +
119902119898
V119904119860)
2
minus4119902119866
V119904119860]
]
(28)
The friction pressure gradient can be obtained from thefollowing formula
120591119891= 119891
120588119871V2119871
2119863 (29)
242 Slug Flow The distribution coefficient of gas in thedrilling mud phase is determined by
1198620=000252lg (103120583L)
119860138minus 0782 + 0232lgVm minus 0428lgA
(30)
The average density of the mixture for slug flow is
120588119898=119882119898+ 120588119871V119904119860
119902119898+ V119904119860
+ 1198620120588119871 (31)
The void fraction for slug flow is determined by thefollowing equation
120601119866=
119902119866
119902119866+ 119902119871
(32)
The friction pressure gradient can be obtained from thefollowing formula
120591119891= 119891
120588119871V2119898
2119863(119902119871+ V119904119860
119902119898+ V119904119860+ 1198620) (33)
25 Wellbore Characteristic Analysis Effective diameter pro-posed by Sanchez [35] is
119863 =120587 (1198632
2minus 1198632
1) 4
120587 (1198632+ 1198631) 4
= 1198632minus 1198631 (34)
The effective roughness of the wellbore can be expressedas
119896119890= 1198960
1198632
1198632+ 1198631
+ 119896119894
1198631
1198632+ 1198631
(35)
With 119863119894and 119863
119900being the diameters of inner pipe and
outer pipe respectively the 1198960and 119896
119894are the roughness of
outer pipe and the inner pipe respectively
6 Mathematical Problems in Engineering
Gas
Drilling pipe
Drilling string
Drilling mudI
IIIII II
Q1
Q2 Q3 Q2
(a) OPOP
I
IIIII II
Q1
Q2 Q3 Q2
(b) CPOP
0
I
IIIII II
Q2 Q3 Q2
L2
L1Q1
(c) OPWP
Figure 2 Different pipe end conditions in trip operations
3 Boundary Conditions and InitialConditions of the Model
When the drill pipe runs into the wellbore filled with drillingmud and gas at high velocities there are three scenarios forthe movement of the drill pipe They are closed pipe withoutpumping (CPOP) open pipe without pumping (OPOP)and open pipe with pumping (OPWP) respectively Thethree scenarios are illustrated in Figure 2 As drill pipe ismoved downward into a wellbore the drilling mud willmove upward To analyze the transient flow in the wellborecaused by the downward movement of drill pipe accordingto (6) and (7) and calculate the surge pressure at that timeit is indispensable to obtain the definite conditions of themodel for the three scenarios that is initial conditions andboundary conditions
We established coordinate systems of the hydraulic sys-tem in the wellboreThe coordinate systems have their originat the end of the drill pipe The flow rate of gas and drillingmud two-phase in open borehole annulus (comprised bydrill pipe and the casing) and drill pipe were respectivelydefined as 119876
1 1198762 and 119876
3 Respectively the corresponding
passageways filled with gas and drilling mud two-phase fluidwere marked as I III and II andthe pressure is 119875
1 1198752 and
1198753 The positive direction of passageway I was towards the
bottomhole while the positive directions of the passagewayIII and II were towards the wellhead As seen in Figure 2 119871
1
was the length from the origin to the bottomhole and 1198712was
the length from the origin to the wellhead By calculating thevelocity of two phases flow the new model for surge pressurein gas-drilling mud two-phase during the movement of drillpipe can be deduced [36]
In actual hydraulic system of the well many pipelineseries are included By assuming the length between thenodes and origin as 119871
3 the boundary consistency conditions
are provided by
119876119869+1
119894(1198713) minus 119876119869
119894(1198713) = V119901sdot Δ119860
119901119869+1
119894(1198713) = 119901119869
119894(1198713)
(119894 = 1 2 3) (36)
According to the relationship between the pressure andvelocity at the wellbore the boundary conditions can bedetermined as follows
119901119894(1198712) = 0
119876119894(1198711) = 0
(119894 = 2 3) (37)
The effective cross-area of jet nozzles is
119860119890=1
4120587
119899
sum
119894=1
1198892
119894 (38)
31 OPOP Scenario The following equation presents theinitial conditions of the flow rate and pressure for OPOPscenario
119901119894(119911) = 0
119876119894(119911) = 0
(119894 = 1 2 3) (39)
The boundary conditions of the flow rate and pressure are
1198761(0 119905) + 119876
2(0 119905) + 119876
3(0 119905) = V
119901(119905) (119860
0minus 1198603)
1199011(0 119905) = 119901
2(0 119905)
1199011(0 119905) minus 119901
3(0 119905)
=120588119898
2120583119898
(1198763(0 119905)
119860119890
+ V119901(119905))
10038161003816100381610038161003816100381610038161003816
1198763(0 119905)
119860119890
+ V119901(119905)
10038161003816100381610038161003816100381610038161003816
(40)
32 CPOP Scenario The following equation presents theinitial conditions of the flow rate and pressure for CPOPscenario
The initial conditions of the flow rate and pressure are asfollows
1199011(119911 0) = 0 119876
1(119911 0) = 0
1199012(119911 0) =
11989121199111205881198981198762
119908
811989821198602
1198762(119911 0) = 119876
119908
1199013(119911 0) =
1198913(1198712minus 119911) 120588
1198981198762
119908
811989831198603
1198763(119911 0) = 119876
119908
(41)
Theboundary conditions of the flow rate and pressure canbe written in the following form
1198761(0 119905) + 119876
2(0 119905) = V
119901(119905) 1198600
1198763(0 119905) = minusV
119901(119905) 1198603
1199011(0 119905) = 119901
2(0 119905)
(42)
33 OPWP Scenario As shown in Figure 2(c) the OPWPscenario is discussed below
Excluding atmospheric pressure the initial conditions atthe wellhead and bottom hole are
1199011(119911 0) = 0 119876
1(119911 0) = 0
1199012(119911 0) =
11989121199111205881198981198762
119908
811989821198602
1198762(119911 0) = 119876
119908
1199013(119911 0) =
1198913(1198712minus 119911) 120588
1198981198762
119908
811989831198603
1198763(119911 0) = 119876
119908
(43)
Mathematical Problems in Engineering 7
The boundary conditions of the flow rate and pressure areas follows
1198761(0 119905) + 119876
2(0 119905) = V
119901(119905) (119860
0minus 1198603) + 119876119908(0 119905)
1198763(0 119905) = minusV
119901(119905) 1198603+ 119876119908(0 119905)
1199011(0 119905) = 119901
2(0 119905)
(44)
4 Solution of the United Model
Referring to Figure 4 the variable 119909 is defined as a short pipeand the constraint is 119871
119909lt 119888119909sdot Δ119905 The variables 119910 and 119911 are
short pipe which are made up of wellbore and drill pipe andthe constraint is 119871
119910lt 119888119910sdot Δ119905 119871
119911lt 119888119911sdot Δ119905 The variables
119909 + 1 119910 + 1 and 119911 + 1 are defined as a long pipe and theconstraint is 119871
119909+ 1 lt 119888
119909+ 1 sdot Δ119905 119871
119910+ 1 lt 119888
119910+ 1 sdot Δ119905 119871
119911+ 1 lt
119888119911+ 1 sdot Δ119905 The implicit finite-difference method is applied to
calculate the short pipe and characteristicsmethod is appliedto calculate the long pipe The pressure and flow rate can beobtained at the end of drill pipe as follows
[1199012119905]2minus 120588119898[119888119910+1
119860119910+1
119876119910+1
2119905]
2
minus [1199013119905minusΔ119905
]2minus 120588119898[119888119910+1
119860119910+1
119876119905minusΔ119905
]
2
+120588119898Δ119905
8[119888119910+1
119898119910+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
2
= 0
[1199012119905]3minus 120588119898[119888119911+1
119860119911+1
119876119911+1
2119905]
3
minus [1199013119905minusΔ119905
]3minus 120588119898[119888119911+1
119860119911+1
1198763119905minusΔ119905
]
3
+120588119898Δ119905
8[119888119911+1
119898119911+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
3
= 0
[1199012119905]1minus 120588119898[119888119909+1
119860119909+1
119876119909+1
2119905]
1
minus [1199013119905minusΔ119905
]1minus 120588119898[119888119909+1
119860119909+1
1198763119905minusΔ119905
]
1
+120588119898Δ119905
8[119888119909+1
119898119909+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
1
= 0
[1199012119905+ 1199011119905]2+ 11986311[119876119910
2119905+ 119876119910
1119905]2
+ 11986312= 0
[1199012119905minus 1199011119905]2+ 11986411[119876119910
2119905minus 119876119910
1119905]2
+ 11986412= 0
[1199012119905+ 1199011119905]3+ 11986321[1198762119905+ 119876119911
1119905]2
+ 11986322= 0
[1199012119905minus 1199011119905]3+ 11986421[1198762119905minus 119876119911
1119905]2
+ 11986422= 0
[1199012119905+ 1199011119905]1+ 11986331[1198762119905minus 119876119909
1119905]1
+ 11986332= 0
[1199012119905minus 1199011119905]1+ 11986431[1198762119905+ 119876119909
1119905]1
+ 11986432= 0
1
2
3
i minus 1
i
i + 1
N minus 2
N minus 1
N
Bottomhole
Wellhead
3
3
2
1
2
3
c+y+1cz+1
Δsx+1
Δsx+1
Ly = Lz
Lx
cminusx+1
z
Figure 3 Schematic diagram of the solve
[1198761119905]1+ [1198761119905]2+ [1198761119905]2minus V119901(119905)
sdot (1198600minus 119860119894) = 0
[1199011119905]1minus [1199011119905]3minus
120588119898
2(119906119860119890)2
times [1198761119905]3+ 119860119890V119901(119905)
10038161003816100381610038161003816[1198761119905]3+ 119860119890V119901(119905)
10038161003816100381610038161003816= 0
119876119910+1
2119905minus 119876119910
2119905minus V119901(119905)
sdot Δ119860 = 0
[1199011119905]1= [1199011119905]2
11986311=1205881198981198882
119860sdotΔ119905
119871119869
11986312=120588119898
119860sdot119871119869
119871119869
(45)
The119863119896119895 119864119896119895can be expressed as follows
119863119894119905= minus (119901
119894+1119905minusΔ119905+ 119901119894119905) + 11986311(119876119894+1119905minusΔ119905
minus 119876119894119905minusΔ119905
)
119864119894119905= minus (119901
119894+1119905minusΔ119905+ 119901119894119905) + 11986411(119876119894+1119905minusΔ119905
minus 119876119894119905minusΔ119905
)
+120588119898119891119871119869
16119898(V119894+1119905minusΔ119905
+ V119894119905minusΔ119905
)1003816100381610038161003816(V119894+1119905minusΔ119905 + V
119894119905minusΔ119905)1003816100381610038161003816
(46)
where 119896 = 1 2 3 and 119895 = 1 2Obtaining the analytical solution of the mathematical
models concerned with flow pattern void fraction param-eters and pressure drop gradient are generally impossible formost practical in two-phase flow In this paper the Runge-Kutta method (R-K4) is used to discrete the new model forsurge pressure At different wellbore depths we can obtainpressure temperature gas velocity drilling mud velocity and
8 Mathematical Problems in Engineering
void fraction in MPD operations by application of the R-K4The solution of pressure drop gradient equation (3) can beseen as an initial-value problem of the ordinary differentialequation
119889119901
119889119911= 119865 (119911 119901)
119901 (1199110) = 1198750
(47)
With the initial value (1199110 1199010) and the function 119865(119911 119901)
(3)ndash(5) can be obtained
1198961= 119865 (119911
0 1199010)
1198962= 119865(119911
0+ℎ
2 1199010+ℎ
21198961)
1198963= 119865(119911
0+ℎ
2 1199010+ℎ
21198962)
1198964= 119865 (119911
0+ ℎ 119901
0+ ℎ1198963)
(48)
where ℎ is the step of depthThe pressure on the nod 119894 = 119894 + 1can be obtained by
1199011= 1199010+ Δ119901 = 119901
0+ℎ
6(1198961+ 21198962+ 21198963+ 1198964) (49)
In the present work the surge pressure calculation modelsolved by personally compiled code on VBNET (Version2010) The schematic diagram of solve is shown in Figure 3and the solution procedure for the surge pressure in thewellbore is shown in Figure 4 At initial time the wellheadtemperature wellbore structure well depths gas and drillingmud two-phase properties and so forth are known On thenode 119894 the pressure gradient wave velocity varieties in real-time temperature and the void fraction can be obtained byadopting R-K4 The density of gas can be obtained by gasEOS (15) and the density of drilling mud can be obtained byequations (19) and equations (20) Then the surge pressureat different depths of wellbore in tripping operations can besolved by (45)Theprocess is repeated until the surge pressurein the whole wellbore has been obtained [37 38]
The developed model takes full consideration of the gasinflux and wave velocity in real-time Owing to the complexconditions of wellbore in tripping operations measurementof surge pressure in the actual drilling process is very difficultIn order to verify the new model for surge pressure thepredicted surge pressure is compared with the results ofprevious experimental investigations presented by Junfang intripping operations in Figure 5 [39] The comparisons revealthat the developed new model fits well with the experimentsdata Thus the new model can be used to accurately predictsurge pressure on different scenario and gas influx rate intripping operations The value of surge pressure calculatedby adopted Burkhardtrsquos steady method is larger than theexperiments data
Table 1 Parameters of calculation well
Type Property Value
Mud Dynamic viscosity (Pasdots) 0056Density (kgm3) 1460
Gas Relative density 065Viscosity (Pasdots) 114 times 10minus5
StringElastic modulus of string (Pa) 207 times 1011
Poisson ratio of string 03Roughness (m) 154 times 10minus7
Surface condition Surface temperature (K) 298Atmosphere pressure (MPa) 0101
5 Analysis and Discussion
The drilling system described is a closed system Theschematic diagram of gas influx process in tripping opera-tions is illustrated in Figure 6 The drilling mud is pumpedfrom surface storage down the drill pipe returns from theannulus and travels back through surface processing wheredrilling solids are removed to surface storage
The well used for calculation is a gas well in XinjiangUygur Autonomous Region China The wellbore structurewell design parameters (depths and diameters) gas anddrilling mud two-phase properties (density and viscosity)and operational conditions of calculation well are displayedin Table 1 [40]
The drilling mud mixed with gas is considered as a two-phase flow medium The surge pressure in the gas-drillingmud are calculated and discussed by using the establishedmodel and well parameters As shown in Figure 7 accordingto the drill conditions such as the drill pipe velocity gasdrilling mud two-phase flow velocity the total length of drillpipe and drill pipe is 1000mwe can account for the changingrules of surge pressureThe following examples illustrate thatthe surge pressure is affected by different well parametersdrilling mud density gas influx rate and well depth in thetripping operations
51 The Effects of Various Drilling Parameters on SurgePressure Figure 7 presents the influence of acceleration ondrilling mud velocity drilling mud velocity and drill pipeoperating speed with respect to the drilling parameters Themovement of drill pipe starts at the depth of 1000m Withthe movement of the drill pipe the drilling mud also movesin the opposite direction along wellbore at the same timeThe drill pipe operating speed reaches the maximum valueat 23 seconds and stops at 30 seconds In order to make acomparison of the surge pressure in different well parametersthe analysis and compared Figures 8ndash15 based on the data ofFigure 7
Figures 8 and 9 graphically show the distributions ofvoid fraction and variations of wave velocity along the flowdirection in the wellbore When gas influx occurs at thebottom of well some gas invades into the wellbore andmigrates form the bottomhole to the wellhead along the flowdirection At low gas influx rate it is extremely obvious that
Mathematical Problems in Engineering 9
Start
Initial conditions
Grid division
Solve equation (21) for temperature
Calculate k1 minus k4Pi+1 = Pi + ΔP
zi+1 = zi + h
Solve equation (12) for 120588m
Solve equation (14) for c
Determine the time and the step
Enter the pipe velocity
Spline smoothing
Satisfy the boundary conditions and initial conditions
Δt rarr 0
Analysis conditions at the end of pipe
Solve pressure and flow at boundary points
Solve pressure and flow in pipe points
T = T + Δt
NoT lt Tmax
YesEnd
Figure 4 Solution procedure for surge pressure in MPD operations
80 16 24 32 40
00
05
10
15
20
Valu
e
t (s)
Pipe velocity (ms)This model (MPa) Experimental data (MPa)
Burkhardt data (MPa)
Figure 5 Experimental verification by comparison with previousexperiment data
the void fraction and wave velocity first slightly change in acomparatively smooth value then change sharply It is becausethe volume of gas expanded rapidly with the decreasingof pressure near the wellhead the void fraction increasessharply and at the same time the wave velocity decreasesobviously At a high gas influx rate the wave velocity tends toincrease because the void fraction in thewellhead is increasedto a high extent In conclusion the wave velocity is sensitiveto the void fraction and the void fraction is dominated by
MudMotor
Choke
z
DAPC
Casing
Drill pipeAnnulus
Drill collar
Formation
RCD0
800m
1200m
4000m
1778mm 78 mm ) times G105
2159mm times G105120601
120601 (120601
1270mm 78 mm ) times G105120601 (120601
Figure 6 Schematic diagram of gas influx process in tripping
influx rate and pressure in the wellbore especially the influxrate
As shown in Figure 9 within the range of low gas influxrate the wave velocity decreases significantly It is becausethe compressibility of the gas increases remarkably andthe medium appears high elasticity though the density ofgas-drilling mud two-phase flow changes slightly With theincrease of gas influx rate and corresponding increase of thevoid fraction in the wellbore the compressibility of the two-phase unceasingly increases which promotes themomentumand energy exchange in the interface Therefore the wavevelocity continuously decreases
10 Mathematical Problems in EngineeringVa
lue
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
minus05
minus10
Pipe acceleration (ms2)Pipe velocity (ms)Fluid velocity (ms)
Figure 7 Drill pipe velocity variation at different accelerations
0 800 1600 2400 3200 4000H (m)
100
80
60
40
20
0
minus20
G(
)
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
120601
Figure 8 Void fraction distribution at different gas influx rates
0 800 1600 2400 3200 4000H (m)
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
1500
1200
900
600
300
0
c (m
s)
Figure 9 Wave velocity variation at different gas influxes
0 10 20 30 40 50 60
25
20
15
10
05
00
minus05
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
P (M
Pa)
t (s)
Figure 10 Surge pressure variation at different gas influxes
Figure 10 illustrates the effect of gas influx on the surgepressure with the change of gas influx rate As shown inFigure 10 under different gas influx rates (0m3h 2427m3hand 8312m3h resp) the maximum surge pressure mayoccur before drill pipe reaches bottomhole With theincreases of the gas influx the corresponding surge pressuredecreases Considering the gas influx in the calculation of thesurge pressure in the wellbore the surge pressure weakensobviously As the gas influx rate decreases the surge pressureat bottom of the wellbore is reduced gradually while surgepressure can cause a significant time lag if the gas influxoccurs at the bottom of well The lag time is mainly affectedby wave velocity
Figure 11 presents a change of surge pressure in thewellbore at different scenario (OPOP CPOP and OPWPresp) According to Figure 11 surge pressure is increasedin CPOP scenario by comparing with the surge pressure inOPOP scenarios It is because the speed of the drilling mudincreases remarkably in CPOP scenario the surge pressurebecomes larger obviously Especially when the maximumsurge pressure occurs a strong increase of surge pressure isobserved at OPWP scenario If the movement of drill pipeis stop immediately the surge pressure will not disappearimmediately and the surge pressure will exist and changeslightly for short while
Figures 12 and 13 show that the surge pressure varies withwellbore diameter caused by the drill pipe operating speedaccording to Figure 7 The surge pressure decreases with theincrease of wellbore diameter It is because the drilling mudflow rate decreases with the increases of wellbore diameterBased on (1) and (2) surge pressure decreases at each positiondue to the decrease of drilling mud flow rate Surge pressurecaused by converting of potential energy into kinetic energyand then kinetic energy back into potential energy over andover again During the tripping operations especially in slimwellbore drilling process the speed of drill pipe movementshould be appropriately reduced The surge pressure in
Mathematical Problems in Engineering 11
OPWPCPOPOPOP
t (s)0 10 20 30 40 50 60
16
12
08
04
00
P (M
Pa)
Qg = 2427m3h
Figure 11 Surge pressure variation at different scenarios
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 2427m3h
Figure 12 Surge pressure variation at different wellbore diameters(gas influx 119876
119892= 2427m3h)
Figure 13 is prominently larger than that in Figure 12 Thisis because both the decrease of the wave velocity and theincrease of the compression in gas drilling mud two-phasedue to the gas influx result in the comparatively large surgepressure
Figure 14 illustrates that the surge pressure varied withthe length of drill pipe The surge pressure caused by shorterdrill pipe (whose the total length is 1000m) is smaller thanthe longer drill pipe (whose total length is 3000m) beforethe maximum surge pressure occurs This can explain thatsurge pressure is affected by the friction force of drill pipe andwellbore Compared with the longer drill pipe the frictionforce caused by shorter drill pipe is largerTherefore the surgepressure gradually decreases As a result the surge pressure isgreater at 1000m
Figure 15 illustrates that the surge pressure varied withthe drilling fluid density At the bottom of well the surge
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 0m3h
Figure 13 Surge pressure at different borehole diameters (gas influx119876119892= 0m3h)
t (s)0 10 20 30 40 50 60
P (M
Pa)
16
12
08
04
00
Qg = 2427m3h
L = 1000mL = 3000m
Figure 14 Surge pressure variation at different well depths
pressure increases with the increase of drilling fluid densityThe changes are significant in the maximum surge pressureThis is because the surge pressure is mainly affected by wavevelocity density and velocity of gas drilling mud two-phaseAccording to (6) and (7) the surge pressure is increased as thedrilling fluid density increasesThemaximum surge pressureoccurs before drill pipe reaches bottomhole
52 The Effects of Gas Influx Rate on Surge Pressure Figures16 17 and 18 illustrate that the surge pressure varied withthe increase of the gas influx rate Figure 17 shows the surgepressure with the gas influx rate of 2427m3h at the bottomof well Figure 18 shows the surge pressure with the gas influxrate of 8312m3h at the bottom of well Mainly affected bywave velocity density and velocity in gas-drilling mud twophase the surge pressure is greatest at the gas influx rate of0m3h Figure 16 illustrates an example of a minimum wavevelocity at the gas influx rate of 8312m3h at the bottom
12 Mathematical Problems in Engineering
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
Qg = 2427m3h
= 1500kgm3=
= 2000 kgm3=
120588m120588m
Figure 15 Surge pressure variation at different drilling fluid densi-ties
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 0m3h
Figure 16 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 0m3h)
of well As the gas influx rate increases the surge pressuredecreases When gas influx occurs the lag time is lengthenedremarkably compared with the condition of no gas influxThe longer the distance is the greater the lag time will beThe lag time is mainly determined by the propagation time ofpressure wave in the wellbore
Figures 19 and 20 illustrate the surge pressure caused bydifferent operating speeds of the drill pipe at various gasinflux rates at the bottom of well As drilling mudmixed withinflux gas not only reduces the pressure wave velocity butalso reduces the density of themixture density surge pressuremainly affected by wave velocity andmixture density reducesdrastically with the increase of gas influx rate at the bottomof well Figures 19 and 20 are compared to Figures 17 and 18
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 2427m3h
Figure 17 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 8312m3h
Figure 18 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 8312m3h)
it can be concluded that the operating speed is also a majorfactor of surge pressure
53The Effects of Drill Pipe Operating Speed on Surge PressureFigure 21 illustrates the effect of drill pipe speed on the surgepressure The change of surge pressure is smaller in gasdrilling two-phase than in single-phase drilling mud Thesurge pressure becomes smaller and closes to zero after themovement is stopped due to greater friction force existing inwell wall during drill pipe operating
6 Conclusions
With full consideration of the important factors such as wavevelocity in real-time the compressibility of the gas phasegas influx rate pressure temperature and well parametersa new surge pressure model has been proposed based on the
Mathematical Problems in Engineering 13Va
lue
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 2427m3h
Figure 19 Surge pressure and fluid velocity variation at differentpipe velocities (gas influx 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 8312m3h
Figure 20 Surge pressure and fluid velocity variation at differentgas influxes (gas influx 119876
119892= 8312m3h)
30
24
18
12
06
00
Valu
e
t (s)0 10 20 30 40 50 60
Pipe velocity (ms)Pipe velocity (ms)Surge pressure (MPa)
Surge pressure (MPa)
Qg = 2427m3h
Figure 21 Surge pressure variations at different pipe velocities (gasinflux 119876
119892= 2427m3h)
mass conservation equations the momentum conservationequations characteristic line method and the gas-drillingmud equations of state (EOS) duringMPDoperations Solvedby the difference method and fourth-order explicit Runge-Kutta method the model is used to predict surge pressurefor different drill pipe operating speed and gas influx rates inMPD operations The main conclusions can be summarizedas follows
(1) An accurate mathematical model to predict surgepressure after gas influx is of great importancebecause of the effects of surge pressure on operationalsafety and success The new model provides betterprediction of surge pressures in comparison with theexisting experiment
(2) The surge pressure tends to increase with the increaseof drill pipe operating speed and decrease of gas influxrate andwell diameter Considering that the gas influxoccurs the surge pressure weakens obviously Withthe increases in well depth the maximum allowabletripping speed gradually become smaller the surgepressure increases at the bottom of well and drillingsafety window is accordingly narrower The surgepressure can cause a significant time lag if the gasinflux occurs at the bottom of well and the timelag is mainly determined by the propagation time ofpressure wave in the wellbore
(3) The surge pressure obviously varies at different sce-narios (OPOP CPOP and OPWP resp) duringthe tripping operations Surge pressure increased inCPOP scenario than in OPOP scenario Especiallywhen the maximum surge pressure occurs a strongincrease of surge pressure is observed at OPWPscenario By running the new surge pressure modeldrilling engineers can identify potential surge prob-lems and optimize tripping operations for differentscenarios as to avoid complicated accidents to happenin wellbore during drilling operations
(4) The calculation of surge pressure in gas drillingmud two-phase is more complex than conventionalcalculation in single drillingmudDrill pipe operatingspeed in drilling operations should be strictly con-trolled and the surge pressure should be consideredin the design The greatest advantages of new modelare the ability to reduce nonproductive time (NPT) indrilling
Nomenclature
A Wellbore effective cross-area (m2)1198600 The outer drill pipe cross-area (m2)
1198602 Wellbore cross-area (m2)
1198603 The inner Drill pipe cross-area (m2)
119860119890 Effective cross-area of jet nozzle (m2)
119888 Wave velocity (ms)119889119904 One discrete length of wellbore (m)
14 Mathematical Problems in Engineering
119863 Wellbore effective diameter (m)1198631 The inner diameter of casing (m)
1198632 The outer diameter of casing (m)
1198633 The inner diameter of drill pipe (m)
1198634 The outer diameter of drill pipe (m)
119863119896119895 Differential equation coefficients
119864119896119895 Differential equation coefficients
119864119904 Modulus of elasticity of the drill pipe
(MPa)119864119891 Modulus of elasticity of formation (MPa)
1198642 Modulus of elasticity of casing (MPa)
119891119866 Shear stresses coefficient of gas interface
119891 Friction coefficient1198912 Wellbore friction coefficient
1198913 Drill pipe friction coefficient
119891119871 Shear stresses coefficient of drilling mud
119865119911 Pressure applied on the two-phase flow
(N)119892 Acceleration due to gravity (m2s)119892119888 Conversion factor
119892119879 Geothermal temperature gradient (Km)
119896119890 Wellbore effective roughness (m)
1198960 Roughness of outer pipe (m)
119896119894 Roughness of inner pipe (m)
119867 The depth of well (m)1198711 The length from pipe bottom to hole
bottom (m)1198712 The length from pipe bottom to well
head (m)119871119892 Flow pattern119898 Hydraulic radius (m)1198982 Wellbore hydraulic radius (m)
1198983 Drill pipe hydraulic radius (m)
119873 Parameter combiningThompson energyeffects
119875 Pressure (MPa)1198750 Initial pressure of wellbore (MPa)
1198752 Pressure within the empty wellbore
(MPa)1198753 Pressure within the drill pipe (MPa)
119875119888 Critical pressure (kPa)
119875119871 Pressure of drilling mud (MPa)
119875119903 Reduced pressure
119875119895119905 Pressure at node (119895 119905) (MPa)
119902119898 Volumetric flow rate of two phase (m3s)
119902119866 Velocity of the gas (m3s)
119902119871 Velocity of the drilling mud (m3s)
119876119871 Mud flow rate within the wellbore(m3h)
1198761 Drilling mud flow rate within hole(m3h)
1198762 Mud flow rate within the wellbore(m3h)
1198763 Mud flow rate within the drill pipe(m3h)
119876119908 Pump flow rate (m3s)
119876119895119905 Flow rate at time 119905 and nod 119895 (m3s)
119877 Constant of EOS (JKgK)
119904 The length of wellbore along wellbore(m)
119905 Time (s)119879 Temperature (K)119879119888 Critical temperature (K)
119879119903 Reduced temperature
119879119890119894 Undistributed temperature at a depth
(K)119879119890119887ℎ
Undistributed temperature of wellhead(K)
119879119891119887ℎ
Undistributed temperature at bottom-hole (K)
V Velocity of drilling mud and gas (ms)V119901 Velocity of drill pipe (ms)
V119898 Gas and drilling mud flow velocity (ms)
V119866 Gas flow velocity (ms)
V119904 Slip velocity (ms)
V119904119892 Superficial gas velocity (ms)
V119871 Drilling mud flow velocity (ms)
119882119898 Mass flow velocity (kgm3)
119911119866 Gas deviation factor
1199110 Initial calculation height of wellbore (m)
119911119887ℎ Total well depth from surface (m)
119911 Variable of distance from surface (m)
Greek Letters
120588119866 Gas density (kgm3)
120591119891 Frictional pressure gradient (Pam)
120591119908 Frictional pressure coefficient
120588119898 Gas and drilling density (kgm3)
1205880 Density under atmospheric pressure (kgm3)
120588119871 Drilling mud density (kgm3)
120590119904 Surface tension (Nm2)
120601119866 Gas void fraction
120601119871 Drilling mud holdup
120588119903 Reduced density
120583119871 Viscosity of drilling mud (Pasdots)
120579 The ground angle (rad)120583119891 Poissonrsquos ratio of formation
120583119904 Poissonrsquos ratio of drill pipe
1205832 Poissonrsquos ratio of casing
Δ119860 The change of the across areas (m2)
Subscripts
CPOP Closed pipe without pumpingDAPC Dynamic annular pressure controlEOS Equations of stateERV Relief-coupling valveMPD Managed pressure drillingOPOP Open pipe without pumpingOPWP Open pipe with pumpingR-K4 The fourth-order explicit Runge-KuttaRCD The rotating control device
Mathematical Problems in Engineering 15
Subscripts of Graph
119888 Wave velocity in two-phase flow (ms)119863 Wellbore effective diameter (m)119867 Depth of wellbore (m)119871 The length from drill pipe and drill pipe (m)119876119892 Gas influx rate at the bottomhole (m3h)
120601119866 Void fraction ()
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 51274170 andQC2012C128) Without their support this work would nothave been possible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of AirInjection in Annuli to manage pressure during CementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling Conferenceand Exhibition San Diego Calif USA March 2012
[4] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the Norh Africa Technical Conference andExhibition Cairo Egypt February 2012
[5] J Saponja A Adeleye and B Hucik ldquoManaged-pressuredrilling (MPD) field trials demonstrate technology valuerdquo inProceedings of the IADCSPE Drilling Conference Miami FlaUSA February 2006
[6] G Liu ldquoPredicting surge pressures that result from runninglinersrdquoWorld Oil vol 222 no 4 p 92 2001
[7] K Zhou J Yang B Zhong and L Tang ldquoSurge pressure inwellborerdquo Oil Drilling and Production Technology vol 12 no 2pp 1ndash10 99 1990
[8] R S Rudi Rubiandini ldquoNew formula of surge pressure fordetermining safe trip velocitiesrdquo in Proceedings of the SPE AsiaPacific Oil and Gas Conference and Exhibition pp 16ndash18 ausOctober 2000
[9] J Li Z Huang K Li and W Wu ldquoThe influence of collaron surge pressure caused by the drilling fluid viscous forceunder pumping conditionrdquoResearch Journal of Applied SciencesEngineering and Technology vol 5 no 5 pp 1781ndash1785 2040ndash7459 2013
[10] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvancedMaterials Research vol 468ndash471 pp 1736ndash1742 2012
[11] F Crespo and R Ahmed ldquoSurge-and-swab pressure predictionsfor yield-power-law drilling fluidsrdquo SPE Drilling amp Completionvol 27 no 4 pp 574ndash585 2012
[12] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[13] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquo PHDThesis vol 50 no 9 p 143 1989
[14] H Fan ldquoDevelopment of application software for dynamicsurge pressure while trippingrdquo Petroleum Drilling and Technol-ogy vol 23 no 4 pp 11ndash14 1995
[15] K Zhou J Yang B Zhong and L Tang ldquoSurge pressurein wellborerdquo Journal of Petroleum Exploration and ProductionTechnology vol 12 no 2 pp 1ndash10 1990
[16] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquoDissertation Abstracts International Bvol 50 no 9 1989
[17] Z Jiang ldquoStudies of wellbore surge pressure by a dynamicmethodrdquo Journal of Petroleum Exploration and ProductionTechnology vol 13 no 3 pp 39ndash44 97 1991
[18] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[19] D Johnston M Bracken and J Thornton ldquoAssessing pipecondition useful remaining life and maximum operating andsurge pressures when to change the pipe or control thepressurerdquoMagazine of the International Water Association vol14 no 5 pp 43ndash44 2012
[20] H Fan Y Chu and X Liu ldquoPrediction for wellbore dynamicsurge pressure while tripping a drill piperdquo Journal of ChinaUniversity of Petroleum vol 19 no 5 pp 36ndash41 1995
[21] C K Burgess J E Miller and K D Vondervort ldquoDownholesurge pressure reduction system and method of userdquo World9848 143 1998
[22] J P Allamon J E Miller and A M Macfarlane ldquoDownholeSurge Reduction Method and Apparatusrdquo 2004
[23] K S Bjoslashrkevoll R Rommetveit B Aas H Gjeraldstveit andA Merlo ldquoTransient gel breaking model for critical wellsapplications with field data verificationrdquo in Proceedings of the2003 Offshore Mediterranean Conference February 2003
[24] G H Nygaard E Johannessen J E Gravdal and F IversenldquoAutomatic coordinated control of pump rates and choke valvefor compensating pressure fluctuations during surge and swaboperationsrdquo in Proceedings of the IADCSPE Managed PressureDrilling and Underbalanced Operations Conference pp 120ndash125March 2007
[25] J V Fedevjcyk S L M Junqueira and C O R Negrao ldquoSurgeamp swab pressures in wells with cross-section changesrdquo BoletimTecnico da Producao de Petroleo vol 4 no 2 pp 221ndash234 2009
[26] S Tian G H Medley and C R Stone ldquoUnderstandingyield point effect onpressure surges critical to managing deepdifficultMPDwellsrdquoDrilling Contractor vol 65 no 4 p 46 4850 53 2009
[27] Y Sun Q Li and J Zhao ldquoNew method of predicting surgepressure apply to horizontal well based on caisson flowrdquoNaturalScience vol 2 no 12 pp 1394ndash1399 2150ndash4091 2010
[28] B T Lebele-Alawa and F E Oparadike ldquoAnalysis of the effectsof valve propagated pressure surge on pipe flowrdquo Engineeringvol 3 no 11 pp 1098ndash1101 2011
[29] X Kong Q Yuan and Y Qiu ldquoThe research of two-phasefluctuation pressure in wellbore tripping operationrdquo InnerMongolia Petrochemical Industry TE242 2011
[30] F Crespo and R Ahmed ldquoA simplified surge and swab pressuremodel for yield power law fluidsrdquo Journal of Petroleum Scienceand Engineering vol 101 pp 12ndash20 2013
16 Mathematical Problems in Engineering
[31] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
[32] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[33] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86-87 pp 127ndash136 2012
[34] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[35] M J SanchezComparison of Correlations for Predicting PressureLosses in Vertical Multiphase Annular Flow University of TulsaTulsa Okla USA 1972
[36] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[37] Y Lin X Kong Y Qiu and Q Yuan ldquoCalculation analysis ofpressure wave velocity in gas and drilling mud two-phase fluidin annulus during drilling operationsrdquo Mathematical Problemsin Engineering vol 2013 Article ID 318912 17 pages 2013
[38] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[39] H Junfang ldquoBalanced drilling and well controlrdquo PetroleumIndustry Press vol 1 article 33 1992
[40] W A Bacon Consideration of compressibility effects for applied-back- pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Va USA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Drilling mud
Gas
Drill pipe
Cross-section 2
Cross-section 1
Casing
p + dp + d
z
120579
p
Figure 1 The schematic of gas and drilling mud two-phase flowduring tripping operations
The object of the present work is to present a newcalculation model for predicting surge pressure during trip-ping operations which is applicable for gas drilling mudtwo-phase fluid In this paper in addition to the pressuretemperature and the void fraction in the wellbore the com-pressibility of the gas phase the wave velocity in real-timethe changes of wellbore parameter and three scenarios arealso taken into consideration By introducing themomentumconservation equations and the mass conservation equationsin MPD operations pressure wave velocity in real-time gas-drilling mud equations of state (EOS) and a new model forpredicting dynamic surge pressure in gas drilling mud two-phase during tripping operations is developed Numericalsolutions difference method and characteristics method areobtained to solve the model The model can be used topredict the change of the surge pressure at any well positionat different influx rate drill pipe operating speed and drillingparameter
2 The Mathematical Model
21 The Basic Equation Drilling mud contains clay cuttingsbarite other solids and so forth The solid particles aresmall and uniformly distributed therefore drilling mudis considered to be a pseudohomogeneous liquid and thenatural gas influx is considered to be the gas phase Asshown in Figure 1 the wellbore is filled with gas and drillingmud two-phase fluid The downward movement of drill pipealong wellbore filled with drilling mud can create a surge ofpressure The two-phase drilling fluid and the passagewayare all compressible and expansibleTherefore the two-phasedrilling fluid flow induced by the trip is unstable especiallyin the deep well
To establish the model about surge pressure during tripoperations the following assumptions are made
(i) the two-phase flow is treated as one-dimensional
(ii) no mass transfers between the gas and drilling mud
(iii) the flow pattern in wellbore is either bubble or slugflow
(iv) the annulus is concentric
When drilling strings are moving in the wellbore fullof drilling mud the pressure gradient equations the massconservation equations and the momentum conservationequations along the flow direction in the wellbore is com-bined to study the surge pressure in tripping operations
The mass conservation equation and momentum conser-vation equation are shown as follows
119901119860 + 120588119898119892(119860 +
120597119860
120597119911sdot119889119911
2) 119889119911 sdot sin 120579 + (119901 +
120597119901
120597119911sdot119889119911
2)120597119860
120597119911
sdot 119889119911 minus (119901119860 +120597 (119901119860)
120597119911119889119911)
minus 1205910119883119889119911 = 120588
119898(119860 +
120597119860
120597119911sdot119889119911
2) 119889119911 sdot
119889V119889119905
(1)
1
120588119898119892
120597119901
120597119911+1
119892(120597V120597119905
+ V120597V120597119911) +
119891V |V|8119898119892
= 0 (2)
The total pressure drop gradient is the sum of pressuredrop gradients due to potential energy change kinetic energyand frictional loss From (2) the equation used to calculatepressure gradient of gas drilling mud two-phase flow withinthe wellbore can be written as
119889119901
119889119911= 120588119898119892 sin 120579 minus
120591119908120587119863
119860minus 120588119898V119898
119889V119898
119889119911 (3)
Assuming the compressibility of the gas is only relatedwith the pressure in the wellbore the kinetic energy andacceleration term in the equation above can be simplified to
120588119898V119898
119889V119898
119889119911= minus
120588119898V119898V119904119892
119901
119889119901
119889119911= minus
119882119898119902119866
1198602119901
119889119901
119889119911 (4)
Substituting (3) into (4) the total pressure drop gradientalong the flow direction within the wellbore can be expressedas
119889119901
119889119911=
120588119898119892 + 120591119891
1 minus119882119898119902119866 (1198602119901)
(5)
The continuity equations can be obtained directly basedon (1)
1198882
(119875 119879119863 120601 119908 119871119892)120597V120597119911
+1
120588119898
119889119901
119889119905+ V
120597119901
120597119911= 0 (6)
where wave velocity 119888(119875 119879119863 120601 119908 119871119892) changes in real-timewhen the gas influx occurs at the bottom of drilling well
The equations of motion can be obtained directly basedon (2)
1
120588119898
120597119901
120597119911+120597V120597119905
+ 119892119889V119889119911
+119891V |V|2119863
= 0 (7)
22 Pressure Wave Velocity Only consider the deformationof wellbore caused by surge pressure the elastic constant ofdrill pipe is
120573 =2
1198631Δ119901
(1 minus 119906119904
119864119904
Δ1199011198632
1
11986322minus 11986321
1198631+1 + 119906119904
119864119904
1198632
2minus 1198632
1Δ119901
11986322minus 11986321
1
119863)
(8)
4 Mathematical Problems in Engineering
The elastic constant of well wall is
120573 =2
119864119891
(1 + 120583119891) (9)
The elastic constant between well wall and drill pipe is
120573 =2
11987722minus 1
[1198772
2
119864119891
(1 + 119906119891) +
1
119864119904
(1198772
1+ 1
11987721minus 1
minus 119906119904)] (10)
The elastic constant between casing and drill pipe is
120573 =2
11987722minus 1
[1198772
2
1198643
(1198772
3+ 1
11987721minus 1
+ 1199063) +
1
1198642
(1198772
1+ 1
11987721minus 1
minus 1199062)]
(11)
where 1198771= 11986321198631 1198772= 11986331198634 and 119877
3= 11986341198633
The density of gas and drilling mud mixture is describedas the following formula
120588119898= 120601119866120588119866+ 120601119871120588119871 (12)
The small perturbation theory is also applied to thesolution of wave velocity model According to the solvablecondition of the homogenous linear equations that thedeterminant of the equations is zero the equation of pressurewave can be expressed in the following form [31]
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(120588119892+ 119888119901120601119866120588119871
V2119904
1198882119892
)119908120601119892
1198882119892
[1 minus 119888119901120601119871]V2119904
1198882119871
119908 minus[120601119892120588119892119896 + 2119888
119901120601119892120601119871120588119871
V119904
1198882119871
119908] 22119888119901120601119892120601119871120588119871
V119904
1198882119871
119908
minus120588119871119908
1 minus 120601119866
1198882119871
119908 0 minus119896 (1 minus 120601119866) 120588119871
120588119871V2119903119896 (minus120601
119866119888119901+ 119888119903minus 119888119894+ 1198881198982) minus120601119866119896 [1 minus 120601
119871
119888119901V2119904
1198882119871
+ 119888119894
V2119904
1198882119871
]
120601119892(120588119892+ 119888V119898120588119871)119908
minus119894 (3
4
119888119863
119903120588119871120601119892V119904+4
119863119891119892119908120588119892V119892)
minus119888V119898120601119892120588119871119908 + 119894 (3
4
119888119863
119903120588119871120601119892V119904)
120588119871V2119904119896 (120601119871119888119901minus 2119888119903minus 1198881198982) minus119896(120601
119871+ 119888119903120601119892
V2119904
1198882119871
) minus119888V119898120601119892120588119871119908 + 119894 (3
4
119888119863
119903120588119871120601119892V119904)
120588119871[120601119871+ 120601119892119888V119898]119908
minus119894 (3
4
119888119863
119903120588119871120601119892V119904+4
119863119891119871120588119871V119871)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 0
(13)
where 119888119894= 03 119888
119901= 025 119888
1198982= 01 and 119888
119903= 02
The real value of wave number is determined by thepressure wave velocity 119888(119875 119879119863 120601 119908 119871119892) and pressure wavevelocity in the gas drilling mud two phase flow is
119888 (119875 119879119863 120601 119908 119871119892) =
1003816100381610038161003816119908119877+
(119896) minus 119908119877minus
(119896)1003816100381610038161003816
2 (14)
23 Physical Equations
231 Equations of State for Gas The equation of state (EOS)for gas can be expressed as follows
120588119866=
119875
(119885119866sdot 119877 sdot 119879)
(15)
where 119885119866is the compression factor of gas
The formula presented by Dranchuk has been used tosolve the compression factor under the condition of low andmedium pressure (119875 lt 35MPa) [32]
119911119866= 1 + (03051 minus
10467
119879119903
minus05783
1198793119903
)120588119903
+ (05353 minus026123
119879119903
minus06816
1198793119903
)1205882
119903
(16)
where 119879119903= 119879119879
119888 119901119903= 119901119901
119888 120588119903= 027119901
119903119885119866119879119903
The formula presented by Dranchuk and Abou-Kassemhas been adopted to solve the compression factor under thecondition of high pressure (119875 ge 35MPa) [32]
119911119866=006125119875
119903119879minus1
119903exp (minus12 (1 minus 119879minus1
119903) 2)
119884 (17)
where 119884 is given by
minus 006125119875119903119879minus1
119903exp (minus12(1 minus 119879minus1
119903)2
) +119884 + 119884
2
+ 1198843
+ 1198844
(1 minus 119884)3
= (1476119879minus1
119903minus 976119879
minus2
119903+ 458119879
minus3
119903) 1198842
minus (907119879minus1
119903minus 2422119879
minus2
119903+ 424119879
minus3
119903) 119884(218+282119879
minus1
119903
)
(18)
232 Equations of State for Drilling Mud Under differenttemperature and pressure the density of drilling mud can beobtained by the empirical formulas If 119879 lt 130
∘C the densityof drilling mud can be obtained by the following equation
120588119871= 1205880(1 + 4 times 10
minus10
119875119871minus 4 times 10
minus5
119879 minus 3 times 10minus6
1198792
) (19)
Mathematical Problems in Engineering 5
If 119879 ge 130∘C the density of drilling mud is
120588119871= 1205880(1 + 4 times 10
minus10
119875 minus 4 times 10minus5
119879 minus 3 times 10minus6
1198792
+ 04(119879 minus 130
119879)
2
)
(20)
233 Correlation of Temperature Distribution The tempera-ture of the drilling mud at different depth of the wellbore canbe determined by the relationship presented by Hasan andKabir [33]
119879 = 119879119890119894+ 119865 [1 minus 119890
(119911119887ℎ
minus119911)119860
]
times (minus119892 sin 120579119892119888119869119888119901119898
+ 119873 + 119892119879sin 120579)
+ 119890(119911119887ℎ
minus119911)119860
(119879119891119887ℎ
minus 119879119890119887ℎ)
(21)
24 Flow Pattern Analysis Based on the analysis of flowcharacteristics in the closed drilling system it can be safelyassumed that the flow pattern in wellbore is either bubbleor slug flow The pattern transition criteria for bubbly flowand slug flow given by Orkiszewski are used to judge the flowpattern in the gas-drilling mud two-phase flow [34]
For bubbly flow
119902119866
119902119898
lt 119871119861 (22)
For slug flow
119902119866
119902119898
gt 119871119861 119873
119866119881lt 119871119904 (23)
where 119902119898is the volumetric flow rate of two-phase flow and
119902119898= 119902119866+ 119902119871
The dimensionless numbers 119871119904and 119871
119861are defined as
119871119904= 50 +
36119873119866119881119902119871
119902119866
119871119861= 1071 minus
07277V2119898
119863
(24)
where
119873119866119881
= V119904(120588119871
119892120590119904
)
025
(25)
Flow parameters such as void fractionmixed density andvirtual mass force coefficient are discussed for specific flowpattern
The correlation between void fraction and drilling mudholdup is expressed as
120601119866+ 120601119871= 1 (26)
241 Bubble Flow The average density of the mixture forbubble flow is
120588119898= 120601119871120588119871+ 120601119866120588119866 (27)
The void fraction for bubble flow is determined by thefollowing equation
120601119866=1
2[
[
1 +119902119898
V119904119860minus radic(1 +
119902119898
V119904119860)
2
minus4119902119866
V119904119860]
]
(28)
The friction pressure gradient can be obtained from thefollowing formula
120591119891= 119891
120588119871V2119871
2119863 (29)
242 Slug Flow The distribution coefficient of gas in thedrilling mud phase is determined by
1198620=000252lg (103120583L)
119860138minus 0782 + 0232lgVm minus 0428lgA
(30)
The average density of the mixture for slug flow is
120588119898=119882119898+ 120588119871V119904119860
119902119898+ V119904119860
+ 1198620120588119871 (31)
The void fraction for slug flow is determined by thefollowing equation
120601119866=
119902119866
119902119866+ 119902119871
(32)
The friction pressure gradient can be obtained from thefollowing formula
120591119891= 119891
120588119871V2119898
2119863(119902119871+ V119904119860
119902119898+ V119904119860+ 1198620) (33)
25 Wellbore Characteristic Analysis Effective diameter pro-posed by Sanchez [35] is
119863 =120587 (1198632
2minus 1198632
1) 4
120587 (1198632+ 1198631) 4
= 1198632minus 1198631 (34)
The effective roughness of the wellbore can be expressedas
119896119890= 1198960
1198632
1198632+ 1198631
+ 119896119894
1198631
1198632+ 1198631
(35)
With 119863119894and 119863
119900being the diameters of inner pipe and
outer pipe respectively the 1198960and 119896
119894are the roughness of
outer pipe and the inner pipe respectively
6 Mathematical Problems in Engineering
Gas
Drilling pipe
Drilling string
Drilling mudI
IIIII II
Q1
Q2 Q3 Q2
(a) OPOP
I
IIIII II
Q1
Q2 Q3 Q2
(b) CPOP
0
I
IIIII II
Q2 Q3 Q2
L2
L1Q1
(c) OPWP
Figure 2 Different pipe end conditions in trip operations
3 Boundary Conditions and InitialConditions of the Model
When the drill pipe runs into the wellbore filled with drillingmud and gas at high velocities there are three scenarios forthe movement of the drill pipe They are closed pipe withoutpumping (CPOP) open pipe without pumping (OPOP)and open pipe with pumping (OPWP) respectively Thethree scenarios are illustrated in Figure 2 As drill pipe ismoved downward into a wellbore the drilling mud willmove upward To analyze the transient flow in the wellborecaused by the downward movement of drill pipe accordingto (6) and (7) and calculate the surge pressure at that timeit is indispensable to obtain the definite conditions of themodel for the three scenarios that is initial conditions andboundary conditions
We established coordinate systems of the hydraulic sys-tem in the wellboreThe coordinate systems have their originat the end of the drill pipe The flow rate of gas and drillingmud two-phase in open borehole annulus (comprised bydrill pipe and the casing) and drill pipe were respectivelydefined as 119876
1 1198762 and 119876
3 Respectively the corresponding
passageways filled with gas and drilling mud two-phase fluidwere marked as I III and II andthe pressure is 119875
1 1198752 and
1198753 The positive direction of passageway I was towards the
bottomhole while the positive directions of the passagewayIII and II were towards the wellhead As seen in Figure 2 119871
1
was the length from the origin to the bottomhole and 1198712was
the length from the origin to the wellhead By calculating thevelocity of two phases flow the new model for surge pressurein gas-drilling mud two-phase during the movement of drillpipe can be deduced [36]
In actual hydraulic system of the well many pipelineseries are included By assuming the length between thenodes and origin as 119871
3 the boundary consistency conditions
are provided by
119876119869+1
119894(1198713) minus 119876119869
119894(1198713) = V119901sdot Δ119860
119901119869+1
119894(1198713) = 119901119869
119894(1198713)
(119894 = 1 2 3) (36)
According to the relationship between the pressure andvelocity at the wellbore the boundary conditions can bedetermined as follows
119901119894(1198712) = 0
119876119894(1198711) = 0
(119894 = 2 3) (37)
The effective cross-area of jet nozzles is
119860119890=1
4120587
119899
sum
119894=1
1198892
119894 (38)
31 OPOP Scenario The following equation presents theinitial conditions of the flow rate and pressure for OPOPscenario
119901119894(119911) = 0
119876119894(119911) = 0
(119894 = 1 2 3) (39)
The boundary conditions of the flow rate and pressure are
1198761(0 119905) + 119876
2(0 119905) + 119876
3(0 119905) = V
119901(119905) (119860
0minus 1198603)
1199011(0 119905) = 119901
2(0 119905)
1199011(0 119905) minus 119901
3(0 119905)
=120588119898
2120583119898
(1198763(0 119905)
119860119890
+ V119901(119905))
10038161003816100381610038161003816100381610038161003816
1198763(0 119905)
119860119890
+ V119901(119905)
10038161003816100381610038161003816100381610038161003816
(40)
32 CPOP Scenario The following equation presents theinitial conditions of the flow rate and pressure for CPOPscenario
The initial conditions of the flow rate and pressure are asfollows
1199011(119911 0) = 0 119876
1(119911 0) = 0
1199012(119911 0) =
11989121199111205881198981198762
119908
811989821198602
1198762(119911 0) = 119876
119908
1199013(119911 0) =
1198913(1198712minus 119911) 120588
1198981198762
119908
811989831198603
1198763(119911 0) = 119876
119908
(41)
Theboundary conditions of the flow rate and pressure canbe written in the following form
1198761(0 119905) + 119876
2(0 119905) = V
119901(119905) 1198600
1198763(0 119905) = minusV
119901(119905) 1198603
1199011(0 119905) = 119901
2(0 119905)
(42)
33 OPWP Scenario As shown in Figure 2(c) the OPWPscenario is discussed below
Excluding atmospheric pressure the initial conditions atthe wellhead and bottom hole are
1199011(119911 0) = 0 119876
1(119911 0) = 0
1199012(119911 0) =
11989121199111205881198981198762
119908
811989821198602
1198762(119911 0) = 119876
119908
1199013(119911 0) =
1198913(1198712minus 119911) 120588
1198981198762
119908
811989831198603
1198763(119911 0) = 119876
119908
(43)
Mathematical Problems in Engineering 7
The boundary conditions of the flow rate and pressure areas follows
1198761(0 119905) + 119876
2(0 119905) = V
119901(119905) (119860
0minus 1198603) + 119876119908(0 119905)
1198763(0 119905) = minusV
119901(119905) 1198603+ 119876119908(0 119905)
1199011(0 119905) = 119901
2(0 119905)
(44)
4 Solution of the United Model
Referring to Figure 4 the variable 119909 is defined as a short pipeand the constraint is 119871
119909lt 119888119909sdot Δ119905 The variables 119910 and 119911 are
short pipe which are made up of wellbore and drill pipe andthe constraint is 119871
119910lt 119888119910sdot Δ119905 119871
119911lt 119888119911sdot Δ119905 The variables
119909 + 1 119910 + 1 and 119911 + 1 are defined as a long pipe and theconstraint is 119871
119909+ 1 lt 119888
119909+ 1 sdot Δ119905 119871
119910+ 1 lt 119888
119910+ 1 sdot Δ119905 119871
119911+ 1 lt
119888119911+ 1 sdot Δ119905 The implicit finite-difference method is applied to
calculate the short pipe and characteristicsmethod is appliedto calculate the long pipe The pressure and flow rate can beobtained at the end of drill pipe as follows
[1199012119905]2minus 120588119898[119888119910+1
119860119910+1
119876119910+1
2119905]
2
minus [1199013119905minusΔ119905
]2minus 120588119898[119888119910+1
119860119910+1
119876119905minusΔ119905
]
2
+120588119898Δ119905
8[119888119910+1
119898119910+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
2
= 0
[1199012119905]3minus 120588119898[119888119911+1
119860119911+1
119876119911+1
2119905]
3
minus [1199013119905minusΔ119905
]3minus 120588119898[119888119911+1
119860119911+1
1198763119905minusΔ119905
]
3
+120588119898Δ119905
8[119888119911+1
119898119911+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
3
= 0
[1199012119905]1minus 120588119898[119888119909+1
119860119909+1
119876119909+1
2119905]
1
minus [1199013119905minusΔ119905
]1minus 120588119898[119888119909+1
119860119909+1
1198763119905minusΔ119905
]
1
+120588119898Δ119905
8[119888119909+1
119898119909+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
1
= 0
[1199012119905+ 1199011119905]2+ 11986311[119876119910
2119905+ 119876119910
1119905]2
+ 11986312= 0
[1199012119905minus 1199011119905]2+ 11986411[119876119910
2119905minus 119876119910
1119905]2
+ 11986412= 0
[1199012119905+ 1199011119905]3+ 11986321[1198762119905+ 119876119911
1119905]2
+ 11986322= 0
[1199012119905minus 1199011119905]3+ 11986421[1198762119905minus 119876119911
1119905]2
+ 11986422= 0
[1199012119905+ 1199011119905]1+ 11986331[1198762119905minus 119876119909
1119905]1
+ 11986332= 0
[1199012119905minus 1199011119905]1+ 11986431[1198762119905+ 119876119909
1119905]1
+ 11986432= 0
1
2
3
i minus 1
i
i + 1
N minus 2
N minus 1
N
Bottomhole
Wellhead
3
3
2
1
2
3
c+y+1cz+1
Δsx+1
Δsx+1
Ly = Lz
Lx
cminusx+1
z
Figure 3 Schematic diagram of the solve
[1198761119905]1+ [1198761119905]2+ [1198761119905]2minus V119901(119905)
sdot (1198600minus 119860119894) = 0
[1199011119905]1minus [1199011119905]3minus
120588119898
2(119906119860119890)2
times [1198761119905]3+ 119860119890V119901(119905)
10038161003816100381610038161003816[1198761119905]3+ 119860119890V119901(119905)
10038161003816100381610038161003816= 0
119876119910+1
2119905minus 119876119910
2119905minus V119901(119905)
sdot Δ119860 = 0
[1199011119905]1= [1199011119905]2
11986311=1205881198981198882
119860sdotΔ119905
119871119869
11986312=120588119898
119860sdot119871119869
119871119869
(45)
The119863119896119895 119864119896119895can be expressed as follows
119863119894119905= minus (119901
119894+1119905minusΔ119905+ 119901119894119905) + 11986311(119876119894+1119905minusΔ119905
minus 119876119894119905minusΔ119905
)
119864119894119905= minus (119901
119894+1119905minusΔ119905+ 119901119894119905) + 11986411(119876119894+1119905minusΔ119905
minus 119876119894119905minusΔ119905
)
+120588119898119891119871119869
16119898(V119894+1119905minusΔ119905
+ V119894119905minusΔ119905
)1003816100381610038161003816(V119894+1119905minusΔ119905 + V
119894119905minusΔ119905)1003816100381610038161003816
(46)
where 119896 = 1 2 3 and 119895 = 1 2Obtaining the analytical solution of the mathematical
models concerned with flow pattern void fraction param-eters and pressure drop gradient are generally impossible formost practical in two-phase flow In this paper the Runge-Kutta method (R-K4) is used to discrete the new model forsurge pressure At different wellbore depths we can obtainpressure temperature gas velocity drilling mud velocity and
8 Mathematical Problems in Engineering
void fraction in MPD operations by application of the R-K4The solution of pressure drop gradient equation (3) can beseen as an initial-value problem of the ordinary differentialequation
119889119901
119889119911= 119865 (119911 119901)
119901 (1199110) = 1198750
(47)
With the initial value (1199110 1199010) and the function 119865(119911 119901)
(3)ndash(5) can be obtained
1198961= 119865 (119911
0 1199010)
1198962= 119865(119911
0+ℎ
2 1199010+ℎ
21198961)
1198963= 119865(119911
0+ℎ
2 1199010+ℎ
21198962)
1198964= 119865 (119911
0+ ℎ 119901
0+ ℎ1198963)
(48)
where ℎ is the step of depthThe pressure on the nod 119894 = 119894 + 1can be obtained by
1199011= 1199010+ Δ119901 = 119901
0+ℎ
6(1198961+ 21198962+ 21198963+ 1198964) (49)
In the present work the surge pressure calculation modelsolved by personally compiled code on VBNET (Version2010) The schematic diagram of solve is shown in Figure 3and the solution procedure for the surge pressure in thewellbore is shown in Figure 4 At initial time the wellheadtemperature wellbore structure well depths gas and drillingmud two-phase properties and so forth are known On thenode 119894 the pressure gradient wave velocity varieties in real-time temperature and the void fraction can be obtained byadopting R-K4 The density of gas can be obtained by gasEOS (15) and the density of drilling mud can be obtained byequations (19) and equations (20) Then the surge pressureat different depths of wellbore in tripping operations can besolved by (45)Theprocess is repeated until the surge pressurein the whole wellbore has been obtained [37 38]
The developed model takes full consideration of the gasinflux and wave velocity in real-time Owing to the complexconditions of wellbore in tripping operations measurementof surge pressure in the actual drilling process is very difficultIn order to verify the new model for surge pressure thepredicted surge pressure is compared with the results ofprevious experimental investigations presented by Junfang intripping operations in Figure 5 [39] The comparisons revealthat the developed new model fits well with the experimentsdata Thus the new model can be used to accurately predictsurge pressure on different scenario and gas influx rate intripping operations The value of surge pressure calculatedby adopted Burkhardtrsquos steady method is larger than theexperiments data
Table 1 Parameters of calculation well
Type Property Value
Mud Dynamic viscosity (Pasdots) 0056Density (kgm3) 1460
Gas Relative density 065Viscosity (Pasdots) 114 times 10minus5
StringElastic modulus of string (Pa) 207 times 1011
Poisson ratio of string 03Roughness (m) 154 times 10minus7
Surface condition Surface temperature (K) 298Atmosphere pressure (MPa) 0101
5 Analysis and Discussion
The drilling system described is a closed system Theschematic diagram of gas influx process in tripping opera-tions is illustrated in Figure 6 The drilling mud is pumpedfrom surface storage down the drill pipe returns from theannulus and travels back through surface processing wheredrilling solids are removed to surface storage
The well used for calculation is a gas well in XinjiangUygur Autonomous Region China The wellbore structurewell design parameters (depths and diameters) gas anddrilling mud two-phase properties (density and viscosity)and operational conditions of calculation well are displayedin Table 1 [40]
The drilling mud mixed with gas is considered as a two-phase flow medium The surge pressure in the gas-drillingmud are calculated and discussed by using the establishedmodel and well parameters As shown in Figure 7 accordingto the drill conditions such as the drill pipe velocity gasdrilling mud two-phase flow velocity the total length of drillpipe and drill pipe is 1000mwe can account for the changingrules of surge pressureThe following examples illustrate thatthe surge pressure is affected by different well parametersdrilling mud density gas influx rate and well depth in thetripping operations
51 The Effects of Various Drilling Parameters on SurgePressure Figure 7 presents the influence of acceleration ondrilling mud velocity drilling mud velocity and drill pipeoperating speed with respect to the drilling parameters Themovement of drill pipe starts at the depth of 1000m Withthe movement of the drill pipe the drilling mud also movesin the opposite direction along wellbore at the same timeThe drill pipe operating speed reaches the maximum valueat 23 seconds and stops at 30 seconds In order to make acomparison of the surge pressure in different well parametersthe analysis and compared Figures 8ndash15 based on the data ofFigure 7
Figures 8 and 9 graphically show the distributions ofvoid fraction and variations of wave velocity along the flowdirection in the wellbore When gas influx occurs at thebottom of well some gas invades into the wellbore andmigrates form the bottomhole to the wellhead along the flowdirection At low gas influx rate it is extremely obvious that
Mathematical Problems in Engineering 9
Start
Initial conditions
Grid division
Solve equation (21) for temperature
Calculate k1 minus k4Pi+1 = Pi + ΔP
zi+1 = zi + h
Solve equation (12) for 120588m
Solve equation (14) for c
Determine the time and the step
Enter the pipe velocity
Spline smoothing
Satisfy the boundary conditions and initial conditions
Δt rarr 0
Analysis conditions at the end of pipe
Solve pressure and flow at boundary points
Solve pressure and flow in pipe points
T = T + Δt
NoT lt Tmax
YesEnd
Figure 4 Solution procedure for surge pressure in MPD operations
80 16 24 32 40
00
05
10
15
20
Valu
e
t (s)
Pipe velocity (ms)This model (MPa) Experimental data (MPa)
Burkhardt data (MPa)
Figure 5 Experimental verification by comparison with previousexperiment data
the void fraction and wave velocity first slightly change in acomparatively smooth value then change sharply It is becausethe volume of gas expanded rapidly with the decreasingof pressure near the wellhead the void fraction increasessharply and at the same time the wave velocity decreasesobviously At a high gas influx rate the wave velocity tends toincrease because the void fraction in thewellhead is increasedto a high extent In conclusion the wave velocity is sensitiveto the void fraction and the void fraction is dominated by
MudMotor
Choke
z
DAPC
Casing
Drill pipeAnnulus
Drill collar
Formation
RCD0
800m
1200m
4000m
1778mm 78 mm ) times G105
2159mm times G105120601
120601 (120601
1270mm 78 mm ) times G105120601 (120601
Figure 6 Schematic diagram of gas influx process in tripping
influx rate and pressure in the wellbore especially the influxrate
As shown in Figure 9 within the range of low gas influxrate the wave velocity decreases significantly It is becausethe compressibility of the gas increases remarkably andthe medium appears high elasticity though the density ofgas-drilling mud two-phase flow changes slightly With theincrease of gas influx rate and corresponding increase of thevoid fraction in the wellbore the compressibility of the two-phase unceasingly increases which promotes themomentumand energy exchange in the interface Therefore the wavevelocity continuously decreases
10 Mathematical Problems in EngineeringVa
lue
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
minus05
minus10
Pipe acceleration (ms2)Pipe velocity (ms)Fluid velocity (ms)
Figure 7 Drill pipe velocity variation at different accelerations
0 800 1600 2400 3200 4000H (m)
100
80
60
40
20
0
minus20
G(
)
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
120601
Figure 8 Void fraction distribution at different gas influx rates
0 800 1600 2400 3200 4000H (m)
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
1500
1200
900
600
300
0
c (m
s)
Figure 9 Wave velocity variation at different gas influxes
0 10 20 30 40 50 60
25
20
15
10
05
00
minus05
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
P (M
Pa)
t (s)
Figure 10 Surge pressure variation at different gas influxes
Figure 10 illustrates the effect of gas influx on the surgepressure with the change of gas influx rate As shown inFigure 10 under different gas influx rates (0m3h 2427m3hand 8312m3h resp) the maximum surge pressure mayoccur before drill pipe reaches bottomhole With theincreases of the gas influx the corresponding surge pressuredecreases Considering the gas influx in the calculation of thesurge pressure in the wellbore the surge pressure weakensobviously As the gas influx rate decreases the surge pressureat bottom of the wellbore is reduced gradually while surgepressure can cause a significant time lag if the gas influxoccurs at the bottom of well The lag time is mainly affectedby wave velocity
Figure 11 presents a change of surge pressure in thewellbore at different scenario (OPOP CPOP and OPWPresp) According to Figure 11 surge pressure is increasedin CPOP scenario by comparing with the surge pressure inOPOP scenarios It is because the speed of the drilling mudincreases remarkably in CPOP scenario the surge pressurebecomes larger obviously Especially when the maximumsurge pressure occurs a strong increase of surge pressure isobserved at OPWP scenario If the movement of drill pipeis stop immediately the surge pressure will not disappearimmediately and the surge pressure will exist and changeslightly for short while
Figures 12 and 13 show that the surge pressure varies withwellbore diameter caused by the drill pipe operating speedaccording to Figure 7 The surge pressure decreases with theincrease of wellbore diameter It is because the drilling mudflow rate decreases with the increases of wellbore diameterBased on (1) and (2) surge pressure decreases at each positiondue to the decrease of drilling mud flow rate Surge pressurecaused by converting of potential energy into kinetic energyand then kinetic energy back into potential energy over andover again During the tripping operations especially in slimwellbore drilling process the speed of drill pipe movementshould be appropriately reduced The surge pressure in
Mathematical Problems in Engineering 11
OPWPCPOPOPOP
t (s)0 10 20 30 40 50 60
16
12
08
04
00
P (M
Pa)
Qg = 2427m3h
Figure 11 Surge pressure variation at different scenarios
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 2427m3h
Figure 12 Surge pressure variation at different wellbore diameters(gas influx 119876
119892= 2427m3h)
Figure 13 is prominently larger than that in Figure 12 Thisis because both the decrease of the wave velocity and theincrease of the compression in gas drilling mud two-phasedue to the gas influx result in the comparatively large surgepressure
Figure 14 illustrates that the surge pressure varied withthe length of drill pipe The surge pressure caused by shorterdrill pipe (whose the total length is 1000m) is smaller thanthe longer drill pipe (whose total length is 3000m) beforethe maximum surge pressure occurs This can explain thatsurge pressure is affected by the friction force of drill pipe andwellbore Compared with the longer drill pipe the frictionforce caused by shorter drill pipe is largerTherefore the surgepressure gradually decreases As a result the surge pressure isgreater at 1000m
Figure 15 illustrates that the surge pressure varied withthe drilling fluid density At the bottom of well the surge
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 0m3h
Figure 13 Surge pressure at different borehole diameters (gas influx119876119892= 0m3h)
t (s)0 10 20 30 40 50 60
P (M
Pa)
16
12
08
04
00
Qg = 2427m3h
L = 1000mL = 3000m
Figure 14 Surge pressure variation at different well depths
pressure increases with the increase of drilling fluid densityThe changes are significant in the maximum surge pressureThis is because the surge pressure is mainly affected by wavevelocity density and velocity of gas drilling mud two-phaseAccording to (6) and (7) the surge pressure is increased as thedrilling fluid density increasesThemaximum surge pressureoccurs before drill pipe reaches bottomhole
52 The Effects of Gas Influx Rate on Surge Pressure Figures16 17 and 18 illustrate that the surge pressure varied withthe increase of the gas influx rate Figure 17 shows the surgepressure with the gas influx rate of 2427m3h at the bottomof well Figure 18 shows the surge pressure with the gas influxrate of 8312m3h at the bottom of well Mainly affected bywave velocity density and velocity in gas-drilling mud twophase the surge pressure is greatest at the gas influx rate of0m3h Figure 16 illustrates an example of a minimum wavevelocity at the gas influx rate of 8312m3h at the bottom
12 Mathematical Problems in Engineering
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
Qg = 2427m3h
= 1500kgm3=
= 2000 kgm3=
120588m120588m
Figure 15 Surge pressure variation at different drilling fluid densi-ties
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 0m3h
Figure 16 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 0m3h)
of well As the gas influx rate increases the surge pressuredecreases When gas influx occurs the lag time is lengthenedremarkably compared with the condition of no gas influxThe longer the distance is the greater the lag time will beThe lag time is mainly determined by the propagation time ofpressure wave in the wellbore
Figures 19 and 20 illustrate the surge pressure caused bydifferent operating speeds of the drill pipe at various gasinflux rates at the bottom of well As drilling mudmixed withinflux gas not only reduces the pressure wave velocity butalso reduces the density of themixture density surge pressuremainly affected by wave velocity andmixture density reducesdrastically with the increase of gas influx rate at the bottomof well Figures 19 and 20 are compared to Figures 17 and 18
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 2427m3h
Figure 17 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 8312m3h
Figure 18 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 8312m3h)
it can be concluded that the operating speed is also a majorfactor of surge pressure
53The Effects of Drill Pipe Operating Speed on Surge PressureFigure 21 illustrates the effect of drill pipe speed on the surgepressure The change of surge pressure is smaller in gasdrilling two-phase than in single-phase drilling mud Thesurge pressure becomes smaller and closes to zero after themovement is stopped due to greater friction force existing inwell wall during drill pipe operating
6 Conclusions
With full consideration of the important factors such as wavevelocity in real-time the compressibility of the gas phasegas influx rate pressure temperature and well parametersa new surge pressure model has been proposed based on the
Mathematical Problems in Engineering 13Va
lue
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 2427m3h
Figure 19 Surge pressure and fluid velocity variation at differentpipe velocities (gas influx 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 8312m3h
Figure 20 Surge pressure and fluid velocity variation at differentgas influxes (gas influx 119876
119892= 8312m3h)
30
24
18
12
06
00
Valu
e
t (s)0 10 20 30 40 50 60
Pipe velocity (ms)Pipe velocity (ms)Surge pressure (MPa)
Surge pressure (MPa)
Qg = 2427m3h
Figure 21 Surge pressure variations at different pipe velocities (gasinflux 119876
119892= 2427m3h)
mass conservation equations the momentum conservationequations characteristic line method and the gas-drillingmud equations of state (EOS) duringMPDoperations Solvedby the difference method and fourth-order explicit Runge-Kutta method the model is used to predict surge pressurefor different drill pipe operating speed and gas influx rates inMPD operations The main conclusions can be summarizedas follows
(1) An accurate mathematical model to predict surgepressure after gas influx is of great importancebecause of the effects of surge pressure on operationalsafety and success The new model provides betterprediction of surge pressures in comparison with theexisting experiment
(2) The surge pressure tends to increase with the increaseof drill pipe operating speed and decrease of gas influxrate andwell diameter Considering that the gas influxoccurs the surge pressure weakens obviously Withthe increases in well depth the maximum allowabletripping speed gradually become smaller the surgepressure increases at the bottom of well and drillingsafety window is accordingly narrower The surgepressure can cause a significant time lag if the gasinflux occurs at the bottom of well and the timelag is mainly determined by the propagation time ofpressure wave in the wellbore
(3) The surge pressure obviously varies at different sce-narios (OPOP CPOP and OPWP resp) duringthe tripping operations Surge pressure increased inCPOP scenario than in OPOP scenario Especiallywhen the maximum surge pressure occurs a strongincrease of surge pressure is observed at OPWPscenario By running the new surge pressure modeldrilling engineers can identify potential surge prob-lems and optimize tripping operations for differentscenarios as to avoid complicated accidents to happenin wellbore during drilling operations
(4) The calculation of surge pressure in gas drillingmud two-phase is more complex than conventionalcalculation in single drillingmudDrill pipe operatingspeed in drilling operations should be strictly con-trolled and the surge pressure should be consideredin the design The greatest advantages of new modelare the ability to reduce nonproductive time (NPT) indrilling
Nomenclature
A Wellbore effective cross-area (m2)1198600 The outer drill pipe cross-area (m2)
1198602 Wellbore cross-area (m2)
1198603 The inner Drill pipe cross-area (m2)
119860119890 Effective cross-area of jet nozzle (m2)
119888 Wave velocity (ms)119889119904 One discrete length of wellbore (m)
14 Mathematical Problems in Engineering
119863 Wellbore effective diameter (m)1198631 The inner diameter of casing (m)
1198632 The outer diameter of casing (m)
1198633 The inner diameter of drill pipe (m)
1198634 The outer diameter of drill pipe (m)
119863119896119895 Differential equation coefficients
119864119896119895 Differential equation coefficients
119864119904 Modulus of elasticity of the drill pipe
(MPa)119864119891 Modulus of elasticity of formation (MPa)
1198642 Modulus of elasticity of casing (MPa)
119891119866 Shear stresses coefficient of gas interface
119891 Friction coefficient1198912 Wellbore friction coefficient
1198913 Drill pipe friction coefficient
119891119871 Shear stresses coefficient of drilling mud
119865119911 Pressure applied on the two-phase flow
(N)119892 Acceleration due to gravity (m2s)119892119888 Conversion factor
119892119879 Geothermal temperature gradient (Km)
119896119890 Wellbore effective roughness (m)
1198960 Roughness of outer pipe (m)
119896119894 Roughness of inner pipe (m)
119867 The depth of well (m)1198711 The length from pipe bottom to hole
bottom (m)1198712 The length from pipe bottom to well
head (m)119871119892 Flow pattern119898 Hydraulic radius (m)1198982 Wellbore hydraulic radius (m)
1198983 Drill pipe hydraulic radius (m)
119873 Parameter combiningThompson energyeffects
119875 Pressure (MPa)1198750 Initial pressure of wellbore (MPa)
1198752 Pressure within the empty wellbore
(MPa)1198753 Pressure within the drill pipe (MPa)
119875119888 Critical pressure (kPa)
119875119871 Pressure of drilling mud (MPa)
119875119903 Reduced pressure
119875119895119905 Pressure at node (119895 119905) (MPa)
119902119898 Volumetric flow rate of two phase (m3s)
119902119866 Velocity of the gas (m3s)
119902119871 Velocity of the drilling mud (m3s)
119876119871 Mud flow rate within the wellbore(m3h)
1198761 Drilling mud flow rate within hole(m3h)
1198762 Mud flow rate within the wellbore(m3h)
1198763 Mud flow rate within the drill pipe(m3h)
119876119908 Pump flow rate (m3s)
119876119895119905 Flow rate at time 119905 and nod 119895 (m3s)
119877 Constant of EOS (JKgK)
119904 The length of wellbore along wellbore(m)
119905 Time (s)119879 Temperature (K)119879119888 Critical temperature (K)
119879119903 Reduced temperature
119879119890119894 Undistributed temperature at a depth
(K)119879119890119887ℎ
Undistributed temperature of wellhead(K)
119879119891119887ℎ
Undistributed temperature at bottom-hole (K)
V Velocity of drilling mud and gas (ms)V119901 Velocity of drill pipe (ms)
V119898 Gas and drilling mud flow velocity (ms)
V119866 Gas flow velocity (ms)
V119904 Slip velocity (ms)
V119904119892 Superficial gas velocity (ms)
V119871 Drilling mud flow velocity (ms)
119882119898 Mass flow velocity (kgm3)
119911119866 Gas deviation factor
1199110 Initial calculation height of wellbore (m)
119911119887ℎ Total well depth from surface (m)
119911 Variable of distance from surface (m)
Greek Letters
120588119866 Gas density (kgm3)
120591119891 Frictional pressure gradient (Pam)
120591119908 Frictional pressure coefficient
120588119898 Gas and drilling density (kgm3)
1205880 Density under atmospheric pressure (kgm3)
120588119871 Drilling mud density (kgm3)
120590119904 Surface tension (Nm2)
120601119866 Gas void fraction
120601119871 Drilling mud holdup
120588119903 Reduced density
120583119871 Viscosity of drilling mud (Pasdots)
120579 The ground angle (rad)120583119891 Poissonrsquos ratio of formation
120583119904 Poissonrsquos ratio of drill pipe
1205832 Poissonrsquos ratio of casing
Δ119860 The change of the across areas (m2)
Subscripts
CPOP Closed pipe without pumpingDAPC Dynamic annular pressure controlEOS Equations of stateERV Relief-coupling valveMPD Managed pressure drillingOPOP Open pipe without pumpingOPWP Open pipe with pumpingR-K4 The fourth-order explicit Runge-KuttaRCD The rotating control device
Mathematical Problems in Engineering 15
Subscripts of Graph
119888 Wave velocity in two-phase flow (ms)119863 Wellbore effective diameter (m)119867 Depth of wellbore (m)119871 The length from drill pipe and drill pipe (m)119876119892 Gas influx rate at the bottomhole (m3h)
120601119866 Void fraction ()
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 51274170 andQC2012C128) Without their support this work would nothave been possible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of AirInjection in Annuli to manage pressure during CementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling Conferenceand Exhibition San Diego Calif USA March 2012
[4] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the Norh Africa Technical Conference andExhibition Cairo Egypt February 2012
[5] J Saponja A Adeleye and B Hucik ldquoManaged-pressuredrilling (MPD) field trials demonstrate technology valuerdquo inProceedings of the IADCSPE Drilling Conference Miami FlaUSA February 2006
[6] G Liu ldquoPredicting surge pressures that result from runninglinersrdquoWorld Oil vol 222 no 4 p 92 2001
[7] K Zhou J Yang B Zhong and L Tang ldquoSurge pressure inwellborerdquo Oil Drilling and Production Technology vol 12 no 2pp 1ndash10 99 1990
[8] R S Rudi Rubiandini ldquoNew formula of surge pressure fordetermining safe trip velocitiesrdquo in Proceedings of the SPE AsiaPacific Oil and Gas Conference and Exhibition pp 16ndash18 ausOctober 2000
[9] J Li Z Huang K Li and W Wu ldquoThe influence of collaron surge pressure caused by the drilling fluid viscous forceunder pumping conditionrdquoResearch Journal of Applied SciencesEngineering and Technology vol 5 no 5 pp 1781ndash1785 2040ndash7459 2013
[10] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvancedMaterials Research vol 468ndash471 pp 1736ndash1742 2012
[11] F Crespo and R Ahmed ldquoSurge-and-swab pressure predictionsfor yield-power-law drilling fluidsrdquo SPE Drilling amp Completionvol 27 no 4 pp 574ndash585 2012
[12] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[13] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquo PHDThesis vol 50 no 9 p 143 1989
[14] H Fan ldquoDevelopment of application software for dynamicsurge pressure while trippingrdquo Petroleum Drilling and Technol-ogy vol 23 no 4 pp 11ndash14 1995
[15] K Zhou J Yang B Zhong and L Tang ldquoSurge pressurein wellborerdquo Journal of Petroleum Exploration and ProductionTechnology vol 12 no 2 pp 1ndash10 1990
[16] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquoDissertation Abstracts International Bvol 50 no 9 1989
[17] Z Jiang ldquoStudies of wellbore surge pressure by a dynamicmethodrdquo Journal of Petroleum Exploration and ProductionTechnology vol 13 no 3 pp 39ndash44 97 1991
[18] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[19] D Johnston M Bracken and J Thornton ldquoAssessing pipecondition useful remaining life and maximum operating andsurge pressures when to change the pipe or control thepressurerdquoMagazine of the International Water Association vol14 no 5 pp 43ndash44 2012
[20] H Fan Y Chu and X Liu ldquoPrediction for wellbore dynamicsurge pressure while tripping a drill piperdquo Journal of ChinaUniversity of Petroleum vol 19 no 5 pp 36ndash41 1995
[21] C K Burgess J E Miller and K D Vondervort ldquoDownholesurge pressure reduction system and method of userdquo World9848 143 1998
[22] J P Allamon J E Miller and A M Macfarlane ldquoDownholeSurge Reduction Method and Apparatusrdquo 2004
[23] K S Bjoslashrkevoll R Rommetveit B Aas H Gjeraldstveit andA Merlo ldquoTransient gel breaking model for critical wellsapplications with field data verificationrdquo in Proceedings of the2003 Offshore Mediterranean Conference February 2003
[24] G H Nygaard E Johannessen J E Gravdal and F IversenldquoAutomatic coordinated control of pump rates and choke valvefor compensating pressure fluctuations during surge and swaboperationsrdquo in Proceedings of the IADCSPE Managed PressureDrilling and Underbalanced Operations Conference pp 120ndash125March 2007
[25] J V Fedevjcyk S L M Junqueira and C O R Negrao ldquoSurgeamp swab pressures in wells with cross-section changesrdquo BoletimTecnico da Producao de Petroleo vol 4 no 2 pp 221ndash234 2009
[26] S Tian G H Medley and C R Stone ldquoUnderstandingyield point effect onpressure surges critical to managing deepdifficultMPDwellsrdquoDrilling Contractor vol 65 no 4 p 46 4850 53 2009
[27] Y Sun Q Li and J Zhao ldquoNew method of predicting surgepressure apply to horizontal well based on caisson flowrdquoNaturalScience vol 2 no 12 pp 1394ndash1399 2150ndash4091 2010
[28] B T Lebele-Alawa and F E Oparadike ldquoAnalysis of the effectsof valve propagated pressure surge on pipe flowrdquo Engineeringvol 3 no 11 pp 1098ndash1101 2011
[29] X Kong Q Yuan and Y Qiu ldquoThe research of two-phasefluctuation pressure in wellbore tripping operationrdquo InnerMongolia Petrochemical Industry TE242 2011
[30] F Crespo and R Ahmed ldquoA simplified surge and swab pressuremodel for yield power law fluidsrdquo Journal of Petroleum Scienceand Engineering vol 101 pp 12ndash20 2013
16 Mathematical Problems in Engineering
[31] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
[32] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[33] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86-87 pp 127ndash136 2012
[34] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[35] M J SanchezComparison of Correlations for Predicting PressureLosses in Vertical Multiphase Annular Flow University of TulsaTulsa Okla USA 1972
[36] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[37] Y Lin X Kong Y Qiu and Q Yuan ldquoCalculation analysis ofpressure wave velocity in gas and drilling mud two-phase fluidin annulus during drilling operationsrdquo Mathematical Problemsin Engineering vol 2013 Article ID 318912 17 pages 2013
[38] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[39] H Junfang ldquoBalanced drilling and well controlrdquo PetroleumIndustry Press vol 1 article 33 1992
[40] W A Bacon Consideration of compressibility effects for applied-back- pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Va USA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
The elastic constant of well wall is
120573 =2
119864119891
(1 + 120583119891) (9)
The elastic constant between well wall and drill pipe is
120573 =2
11987722minus 1
[1198772
2
119864119891
(1 + 119906119891) +
1
119864119904
(1198772
1+ 1
11987721minus 1
minus 119906119904)] (10)
The elastic constant between casing and drill pipe is
120573 =2
11987722minus 1
[1198772
2
1198643
(1198772
3+ 1
11987721minus 1
+ 1199063) +
1
1198642
(1198772
1+ 1
11987721minus 1
minus 1199062)]
(11)
where 1198771= 11986321198631 1198772= 11986331198634 and 119877
3= 11986341198633
The density of gas and drilling mud mixture is describedas the following formula
120588119898= 120601119866120588119866+ 120601119871120588119871 (12)
The small perturbation theory is also applied to thesolution of wave velocity model According to the solvablecondition of the homogenous linear equations that thedeterminant of the equations is zero the equation of pressurewave can be expressed in the following form [31]
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
(120588119892+ 119888119901120601119866120588119871
V2119904
1198882119892
)119908120601119892
1198882119892
[1 minus 119888119901120601119871]V2119904
1198882119871
119908 minus[120601119892120588119892119896 + 2119888
119901120601119892120601119871120588119871
V119904
1198882119871
119908] 22119888119901120601119892120601119871120588119871
V119904
1198882119871
119908
minus120588119871119908
1 minus 120601119866
1198882119871
119908 0 minus119896 (1 minus 120601119866) 120588119871
120588119871V2119903119896 (minus120601
119866119888119901+ 119888119903minus 119888119894+ 1198881198982) minus120601119866119896 [1 minus 120601
119871
119888119901V2119904
1198882119871
+ 119888119894
V2119904
1198882119871
]
120601119892(120588119892+ 119888V119898120588119871)119908
minus119894 (3
4
119888119863
119903120588119871120601119892V119904+4
119863119891119892119908120588119892V119892)
minus119888V119898120601119892120588119871119908 + 119894 (3
4
119888119863
119903120588119871120601119892V119904)
120588119871V2119904119896 (120601119871119888119901minus 2119888119903minus 1198881198982) minus119896(120601
119871+ 119888119903120601119892
V2119904
1198882119871
) minus119888V119898120601119892120588119871119908 + 119894 (3
4
119888119863
119903120588119871120601119892V119904)
120588119871[120601119871+ 120601119892119888V119898]119908
minus119894 (3
4
119888119863
119903120588119871120601119892V119904+4
119863119891119871120588119871V119871)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
= 0
(13)
where 119888119894= 03 119888
119901= 025 119888
1198982= 01 and 119888
119903= 02
The real value of wave number is determined by thepressure wave velocity 119888(119875 119879119863 120601 119908 119871119892) and pressure wavevelocity in the gas drilling mud two phase flow is
119888 (119875 119879119863 120601 119908 119871119892) =
1003816100381610038161003816119908119877+
(119896) minus 119908119877minus
(119896)1003816100381610038161003816
2 (14)
23 Physical Equations
231 Equations of State for Gas The equation of state (EOS)for gas can be expressed as follows
120588119866=
119875
(119885119866sdot 119877 sdot 119879)
(15)
where 119885119866is the compression factor of gas
The formula presented by Dranchuk has been used tosolve the compression factor under the condition of low andmedium pressure (119875 lt 35MPa) [32]
119911119866= 1 + (03051 minus
10467
119879119903
minus05783
1198793119903
)120588119903
+ (05353 minus026123
119879119903
minus06816
1198793119903
)1205882
119903
(16)
where 119879119903= 119879119879
119888 119901119903= 119901119901
119888 120588119903= 027119901
119903119885119866119879119903
The formula presented by Dranchuk and Abou-Kassemhas been adopted to solve the compression factor under thecondition of high pressure (119875 ge 35MPa) [32]
119911119866=006125119875
119903119879minus1
119903exp (minus12 (1 minus 119879minus1
119903) 2)
119884 (17)
where 119884 is given by
minus 006125119875119903119879minus1
119903exp (minus12(1 minus 119879minus1
119903)2
) +119884 + 119884
2
+ 1198843
+ 1198844
(1 minus 119884)3
= (1476119879minus1
119903minus 976119879
minus2
119903+ 458119879
minus3
119903) 1198842
minus (907119879minus1
119903minus 2422119879
minus2
119903+ 424119879
minus3
119903) 119884(218+282119879
minus1
119903
)
(18)
232 Equations of State for Drilling Mud Under differenttemperature and pressure the density of drilling mud can beobtained by the empirical formulas If 119879 lt 130
∘C the densityof drilling mud can be obtained by the following equation
120588119871= 1205880(1 + 4 times 10
minus10
119875119871minus 4 times 10
minus5
119879 minus 3 times 10minus6
1198792
) (19)
Mathematical Problems in Engineering 5
If 119879 ge 130∘C the density of drilling mud is
120588119871= 1205880(1 + 4 times 10
minus10
119875 minus 4 times 10minus5
119879 minus 3 times 10minus6
1198792
+ 04(119879 minus 130
119879)
2
)
(20)
233 Correlation of Temperature Distribution The tempera-ture of the drilling mud at different depth of the wellbore canbe determined by the relationship presented by Hasan andKabir [33]
119879 = 119879119890119894+ 119865 [1 minus 119890
(119911119887ℎ
minus119911)119860
]
times (minus119892 sin 120579119892119888119869119888119901119898
+ 119873 + 119892119879sin 120579)
+ 119890(119911119887ℎ
minus119911)119860
(119879119891119887ℎ
minus 119879119890119887ℎ)
(21)
24 Flow Pattern Analysis Based on the analysis of flowcharacteristics in the closed drilling system it can be safelyassumed that the flow pattern in wellbore is either bubbleor slug flow The pattern transition criteria for bubbly flowand slug flow given by Orkiszewski are used to judge the flowpattern in the gas-drilling mud two-phase flow [34]
For bubbly flow
119902119866
119902119898
lt 119871119861 (22)
For slug flow
119902119866
119902119898
gt 119871119861 119873
119866119881lt 119871119904 (23)
where 119902119898is the volumetric flow rate of two-phase flow and
119902119898= 119902119866+ 119902119871
The dimensionless numbers 119871119904and 119871
119861are defined as
119871119904= 50 +
36119873119866119881119902119871
119902119866
119871119861= 1071 minus
07277V2119898
119863
(24)
where
119873119866119881
= V119904(120588119871
119892120590119904
)
025
(25)
Flow parameters such as void fractionmixed density andvirtual mass force coefficient are discussed for specific flowpattern
The correlation between void fraction and drilling mudholdup is expressed as
120601119866+ 120601119871= 1 (26)
241 Bubble Flow The average density of the mixture forbubble flow is
120588119898= 120601119871120588119871+ 120601119866120588119866 (27)
The void fraction for bubble flow is determined by thefollowing equation
120601119866=1
2[
[
1 +119902119898
V119904119860minus radic(1 +
119902119898
V119904119860)
2
minus4119902119866
V119904119860]
]
(28)
The friction pressure gradient can be obtained from thefollowing formula
120591119891= 119891
120588119871V2119871
2119863 (29)
242 Slug Flow The distribution coefficient of gas in thedrilling mud phase is determined by
1198620=000252lg (103120583L)
119860138minus 0782 + 0232lgVm minus 0428lgA
(30)
The average density of the mixture for slug flow is
120588119898=119882119898+ 120588119871V119904119860
119902119898+ V119904119860
+ 1198620120588119871 (31)
The void fraction for slug flow is determined by thefollowing equation
120601119866=
119902119866
119902119866+ 119902119871
(32)
The friction pressure gradient can be obtained from thefollowing formula
120591119891= 119891
120588119871V2119898
2119863(119902119871+ V119904119860
119902119898+ V119904119860+ 1198620) (33)
25 Wellbore Characteristic Analysis Effective diameter pro-posed by Sanchez [35] is
119863 =120587 (1198632
2minus 1198632
1) 4
120587 (1198632+ 1198631) 4
= 1198632minus 1198631 (34)
The effective roughness of the wellbore can be expressedas
119896119890= 1198960
1198632
1198632+ 1198631
+ 119896119894
1198631
1198632+ 1198631
(35)
With 119863119894and 119863
119900being the diameters of inner pipe and
outer pipe respectively the 1198960and 119896
119894are the roughness of
outer pipe and the inner pipe respectively
6 Mathematical Problems in Engineering
Gas
Drilling pipe
Drilling string
Drilling mudI
IIIII II
Q1
Q2 Q3 Q2
(a) OPOP
I
IIIII II
Q1
Q2 Q3 Q2
(b) CPOP
0
I
IIIII II
Q2 Q3 Q2
L2
L1Q1
(c) OPWP
Figure 2 Different pipe end conditions in trip operations
3 Boundary Conditions and InitialConditions of the Model
When the drill pipe runs into the wellbore filled with drillingmud and gas at high velocities there are three scenarios forthe movement of the drill pipe They are closed pipe withoutpumping (CPOP) open pipe without pumping (OPOP)and open pipe with pumping (OPWP) respectively Thethree scenarios are illustrated in Figure 2 As drill pipe ismoved downward into a wellbore the drilling mud willmove upward To analyze the transient flow in the wellborecaused by the downward movement of drill pipe accordingto (6) and (7) and calculate the surge pressure at that timeit is indispensable to obtain the definite conditions of themodel for the three scenarios that is initial conditions andboundary conditions
We established coordinate systems of the hydraulic sys-tem in the wellboreThe coordinate systems have their originat the end of the drill pipe The flow rate of gas and drillingmud two-phase in open borehole annulus (comprised bydrill pipe and the casing) and drill pipe were respectivelydefined as 119876
1 1198762 and 119876
3 Respectively the corresponding
passageways filled with gas and drilling mud two-phase fluidwere marked as I III and II andthe pressure is 119875
1 1198752 and
1198753 The positive direction of passageway I was towards the
bottomhole while the positive directions of the passagewayIII and II were towards the wellhead As seen in Figure 2 119871
1
was the length from the origin to the bottomhole and 1198712was
the length from the origin to the wellhead By calculating thevelocity of two phases flow the new model for surge pressurein gas-drilling mud two-phase during the movement of drillpipe can be deduced [36]
In actual hydraulic system of the well many pipelineseries are included By assuming the length between thenodes and origin as 119871
3 the boundary consistency conditions
are provided by
119876119869+1
119894(1198713) minus 119876119869
119894(1198713) = V119901sdot Δ119860
119901119869+1
119894(1198713) = 119901119869
119894(1198713)
(119894 = 1 2 3) (36)
According to the relationship between the pressure andvelocity at the wellbore the boundary conditions can bedetermined as follows
119901119894(1198712) = 0
119876119894(1198711) = 0
(119894 = 2 3) (37)
The effective cross-area of jet nozzles is
119860119890=1
4120587
119899
sum
119894=1
1198892
119894 (38)
31 OPOP Scenario The following equation presents theinitial conditions of the flow rate and pressure for OPOPscenario
119901119894(119911) = 0
119876119894(119911) = 0
(119894 = 1 2 3) (39)
The boundary conditions of the flow rate and pressure are
1198761(0 119905) + 119876
2(0 119905) + 119876
3(0 119905) = V
119901(119905) (119860
0minus 1198603)
1199011(0 119905) = 119901
2(0 119905)
1199011(0 119905) minus 119901
3(0 119905)
=120588119898
2120583119898
(1198763(0 119905)
119860119890
+ V119901(119905))
10038161003816100381610038161003816100381610038161003816
1198763(0 119905)
119860119890
+ V119901(119905)
10038161003816100381610038161003816100381610038161003816
(40)
32 CPOP Scenario The following equation presents theinitial conditions of the flow rate and pressure for CPOPscenario
The initial conditions of the flow rate and pressure are asfollows
1199011(119911 0) = 0 119876
1(119911 0) = 0
1199012(119911 0) =
11989121199111205881198981198762
119908
811989821198602
1198762(119911 0) = 119876
119908
1199013(119911 0) =
1198913(1198712minus 119911) 120588
1198981198762
119908
811989831198603
1198763(119911 0) = 119876
119908
(41)
Theboundary conditions of the flow rate and pressure canbe written in the following form
1198761(0 119905) + 119876
2(0 119905) = V
119901(119905) 1198600
1198763(0 119905) = minusV
119901(119905) 1198603
1199011(0 119905) = 119901
2(0 119905)
(42)
33 OPWP Scenario As shown in Figure 2(c) the OPWPscenario is discussed below
Excluding atmospheric pressure the initial conditions atthe wellhead and bottom hole are
1199011(119911 0) = 0 119876
1(119911 0) = 0
1199012(119911 0) =
11989121199111205881198981198762
119908
811989821198602
1198762(119911 0) = 119876
119908
1199013(119911 0) =
1198913(1198712minus 119911) 120588
1198981198762
119908
811989831198603
1198763(119911 0) = 119876
119908
(43)
Mathematical Problems in Engineering 7
The boundary conditions of the flow rate and pressure areas follows
1198761(0 119905) + 119876
2(0 119905) = V
119901(119905) (119860
0minus 1198603) + 119876119908(0 119905)
1198763(0 119905) = minusV
119901(119905) 1198603+ 119876119908(0 119905)
1199011(0 119905) = 119901
2(0 119905)
(44)
4 Solution of the United Model
Referring to Figure 4 the variable 119909 is defined as a short pipeand the constraint is 119871
119909lt 119888119909sdot Δ119905 The variables 119910 and 119911 are
short pipe which are made up of wellbore and drill pipe andthe constraint is 119871
119910lt 119888119910sdot Δ119905 119871
119911lt 119888119911sdot Δ119905 The variables
119909 + 1 119910 + 1 and 119911 + 1 are defined as a long pipe and theconstraint is 119871
119909+ 1 lt 119888
119909+ 1 sdot Δ119905 119871
119910+ 1 lt 119888
119910+ 1 sdot Δ119905 119871
119911+ 1 lt
119888119911+ 1 sdot Δ119905 The implicit finite-difference method is applied to
calculate the short pipe and characteristicsmethod is appliedto calculate the long pipe The pressure and flow rate can beobtained at the end of drill pipe as follows
[1199012119905]2minus 120588119898[119888119910+1
119860119910+1
119876119910+1
2119905]
2
minus [1199013119905minusΔ119905
]2minus 120588119898[119888119910+1
119860119910+1
119876119905minusΔ119905
]
2
+120588119898Δ119905
8[119888119910+1
119898119910+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
2
= 0
[1199012119905]3minus 120588119898[119888119911+1
119860119911+1
119876119911+1
2119905]
3
minus [1199013119905minusΔ119905
]3minus 120588119898[119888119911+1
119860119911+1
1198763119905minusΔ119905
]
3
+120588119898Δ119905
8[119888119911+1
119898119911+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
3
= 0
[1199012119905]1minus 120588119898[119888119909+1
119860119909+1
119876119909+1
2119905]
1
minus [1199013119905minusΔ119905
]1minus 120588119898[119888119909+1
119860119909+1
1198763119905minusΔ119905
]
1
+120588119898Δ119905
8[119888119909+1
119898119909+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
1
= 0
[1199012119905+ 1199011119905]2+ 11986311[119876119910
2119905+ 119876119910
1119905]2
+ 11986312= 0
[1199012119905minus 1199011119905]2+ 11986411[119876119910
2119905minus 119876119910
1119905]2
+ 11986412= 0
[1199012119905+ 1199011119905]3+ 11986321[1198762119905+ 119876119911
1119905]2
+ 11986322= 0
[1199012119905minus 1199011119905]3+ 11986421[1198762119905minus 119876119911
1119905]2
+ 11986422= 0
[1199012119905+ 1199011119905]1+ 11986331[1198762119905minus 119876119909
1119905]1
+ 11986332= 0
[1199012119905minus 1199011119905]1+ 11986431[1198762119905+ 119876119909
1119905]1
+ 11986432= 0
1
2
3
i minus 1
i
i + 1
N minus 2
N minus 1
N
Bottomhole
Wellhead
3
3
2
1
2
3
c+y+1cz+1
Δsx+1
Δsx+1
Ly = Lz
Lx
cminusx+1
z
Figure 3 Schematic diagram of the solve
[1198761119905]1+ [1198761119905]2+ [1198761119905]2minus V119901(119905)
sdot (1198600minus 119860119894) = 0
[1199011119905]1minus [1199011119905]3minus
120588119898
2(119906119860119890)2
times [1198761119905]3+ 119860119890V119901(119905)
10038161003816100381610038161003816[1198761119905]3+ 119860119890V119901(119905)
10038161003816100381610038161003816= 0
119876119910+1
2119905minus 119876119910
2119905minus V119901(119905)
sdot Δ119860 = 0
[1199011119905]1= [1199011119905]2
11986311=1205881198981198882
119860sdotΔ119905
119871119869
11986312=120588119898
119860sdot119871119869
119871119869
(45)
The119863119896119895 119864119896119895can be expressed as follows
119863119894119905= minus (119901
119894+1119905minusΔ119905+ 119901119894119905) + 11986311(119876119894+1119905minusΔ119905
minus 119876119894119905minusΔ119905
)
119864119894119905= minus (119901
119894+1119905minusΔ119905+ 119901119894119905) + 11986411(119876119894+1119905minusΔ119905
minus 119876119894119905minusΔ119905
)
+120588119898119891119871119869
16119898(V119894+1119905minusΔ119905
+ V119894119905minusΔ119905
)1003816100381610038161003816(V119894+1119905minusΔ119905 + V
119894119905minusΔ119905)1003816100381610038161003816
(46)
where 119896 = 1 2 3 and 119895 = 1 2Obtaining the analytical solution of the mathematical
models concerned with flow pattern void fraction param-eters and pressure drop gradient are generally impossible formost practical in two-phase flow In this paper the Runge-Kutta method (R-K4) is used to discrete the new model forsurge pressure At different wellbore depths we can obtainpressure temperature gas velocity drilling mud velocity and
8 Mathematical Problems in Engineering
void fraction in MPD operations by application of the R-K4The solution of pressure drop gradient equation (3) can beseen as an initial-value problem of the ordinary differentialequation
119889119901
119889119911= 119865 (119911 119901)
119901 (1199110) = 1198750
(47)
With the initial value (1199110 1199010) and the function 119865(119911 119901)
(3)ndash(5) can be obtained
1198961= 119865 (119911
0 1199010)
1198962= 119865(119911
0+ℎ
2 1199010+ℎ
21198961)
1198963= 119865(119911
0+ℎ
2 1199010+ℎ
21198962)
1198964= 119865 (119911
0+ ℎ 119901
0+ ℎ1198963)
(48)
where ℎ is the step of depthThe pressure on the nod 119894 = 119894 + 1can be obtained by
1199011= 1199010+ Δ119901 = 119901
0+ℎ
6(1198961+ 21198962+ 21198963+ 1198964) (49)
In the present work the surge pressure calculation modelsolved by personally compiled code on VBNET (Version2010) The schematic diagram of solve is shown in Figure 3and the solution procedure for the surge pressure in thewellbore is shown in Figure 4 At initial time the wellheadtemperature wellbore structure well depths gas and drillingmud two-phase properties and so forth are known On thenode 119894 the pressure gradient wave velocity varieties in real-time temperature and the void fraction can be obtained byadopting R-K4 The density of gas can be obtained by gasEOS (15) and the density of drilling mud can be obtained byequations (19) and equations (20) Then the surge pressureat different depths of wellbore in tripping operations can besolved by (45)Theprocess is repeated until the surge pressurein the whole wellbore has been obtained [37 38]
The developed model takes full consideration of the gasinflux and wave velocity in real-time Owing to the complexconditions of wellbore in tripping operations measurementof surge pressure in the actual drilling process is very difficultIn order to verify the new model for surge pressure thepredicted surge pressure is compared with the results ofprevious experimental investigations presented by Junfang intripping operations in Figure 5 [39] The comparisons revealthat the developed new model fits well with the experimentsdata Thus the new model can be used to accurately predictsurge pressure on different scenario and gas influx rate intripping operations The value of surge pressure calculatedby adopted Burkhardtrsquos steady method is larger than theexperiments data
Table 1 Parameters of calculation well
Type Property Value
Mud Dynamic viscosity (Pasdots) 0056Density (kgm3) 1460
Gas Relative density 065Viscosity (Pasdots) 114 times 10minus5
StringElastic modulus of string (Pa) 207 times 1011
Poisson ratio of string 03Roughness (m) 154 times 10minus7
Surface condition Surface temperature (K) 298Atmosphere pressure (MPa) 0101
5 Analysis and Discussion
The drilling system described is a closed system Theschematic diagram of gas influx process in tripping opera-tions is illustrated in Figure 6 The drilling mud is pumpedfrom surface storage down the drill pipe returns from theannulus and travels back through surface processing wheredrilling solids are removed to surface storage
The well used for calculation is a gas well in XinjiangUygur Autonomous Region China The wellbore structurewell design parameters (depths and diameters) gas anddrilling mud two-phase properties (density and viscosity)and operational conditions of calculation well are displayedin Table 1 [40]
The drilling mud mixed with gas is considered as a two-phase flow medium The surge pressure in the gas-drillingmud are calculated and discussed by using the establishedmodel and well parameters As shown in Figure 7 accordingto the drill conditions such as the drill pipe velocity gasdrilling mud two-phase flow velocity the total length of drillpipe and drill pipe is 1000mwe can account for the changingrules of surge pressureThe following examples illustrate thatthe surge pressure is affected by different well parametersdrilling mud density gas influx rate and well depth in thetripping operations
51 The Effects of Various Drilling Parameters on SurgePressure Figure 7 presents the influence of acceleration ondrilling mud velocity drilling mud velocity and drill pipeoperating speed with respect to the drilling parameters Themovement of drill pipe starts at the depth of 1000m Withthe movement of the drill pipe the drilling mud also movesin the opposite direction along wellbore at the same timeThe drill pipe operating speed reaches the maximum valueat 23 seconds and stops at 30 seconds In order to make acomparison of the surge pressure in different well parametersthe analysis and compared Figures 8ndash15 based on the data ofFigure 7
Figures 8 and 9 graphically show the distributions ofvoid fraction and variations of wave velocity along the flowdirection in the wellbore When gas influx occurs at thebottom of well some gas invades into the wellbore andmigrates form the bottomhole to the wellhead along the flowdirection At low gas influx rate it is extremely obvious that
Mathematical Problems in Engineering 9
Start
Initial conditions
Grid division
Solve equation (21) for temperature
Calculate k1 minus k4Pi+1 = Pi + ΔP
zi+1 = zi + h
Solve equation (12) for 120588m
Solve equation (14) for c
Determine the time and the step
Enter the pipe velocity
Spline smoothing
Satisfy the boundary conditions and initial conditions
Δt rarr 0
Analysis conditions at the end of pipe
Solve pressure and flow at boundary points
Solve pressure and flow in pipe points
T = T + Δt
NoT lt Tmax
YesEnd
Figure 4 Solution procedure for surge pressure in MPD operations
80 16 24 32 40
00
05
10
15
20
Valu
e
t (s)
Pipe velocity (ms)This model (MPa) Experimental data (MPa)
Burkhardt data (MPa)
Figure 5 Experimental verification by comparison with previousexperiment data
the void fraction and wave velocity first slightly change in acomparatively smooth value then change sharply It is becausethe volume of gas expanded rapidly with the decreasingof pressure near the wellhead the void fraction increasessharply and at the same time the wave velocity decreasesobviously At a high gas influx rate the wave velocity tends toincrease because the void fraction in thewellhead is increasedto a high extent In conclusion the wave velocity is sensitiveto the void fraction and the void fraction is dominated by
MudMotor
Choke
z
DAPC
Casing
Drill pipeAnnulus
Drill collar
Formation
RCD0
800m
1200m
4000m
1778mm 78 mm ) times G105
2159mm times G105120601
120601 (120601
1270mm 78 mm ) times G105120601 (120601
Figure 6 Schematic diagram of gas influx process in tripping
influx rate and pressure in the wellbore especially the influxrate
As shown in Figure 9 within the range of low gas influxrate the wave velocity decreases significantly It is becausethe compressibility of the gas increases remarkably andthe medium appears high elasticity though the density ofgas-drilling mud two-phase flow changes slightly With theincrease of gas influx rate and corresponding increase of thevoid fraction in the wellbore the compressibility of the two-phase unceasingly increases which promotes themomentumand energy exchange in the interface Therefore the wavevelocity continuously decreases
10 Mathematical Problems in EngineeringVa
lue
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
minus05
minus10
Pipe acceleration (ms2)Pipe velocity (ms)Fluid velocity (ms)
Figure 7 Drill pipe velocity variation at different accelerations
0 800 1600 2400 3200 4000H (m)
100
80
60
40
20
0
minus20
G(
)
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
120601
Figure 8 Void fraction distribution at different gas influx rates
0 800 1600 2400 3200 4000H (m)
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
1500
1200
900
600
300
0
c (m
s)
Figure 9 Wave velocity variation at different gas influxes
0 10 20 30 40 50 60
25
20
15
10
05
00
minus05
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
P (M
Pa)
t (s)
Figure 10 Surge pressure variation at different gas influxes
Figure 10 illustrates the effect of gas influx on the surgepressure with the change of gas influx rate As shown inFigure 10 under different gas influx rates (0m3h 2427m3hand 8312m3h resp) the maximum surge pressure mayoccur before drill pipe reaches bottomhole With theincreases of the gas influx the corresponding surge pressuredecreases Considering the gas influx in the calculation of thesurge pressure in the wellbore the surge pressure weakensobviously As the gas influx rate decreases the surge pressureat bottom of the wellbore is reduced gradually while surgepressure can cause a significant time lag if the gas influxoccurs at the bottom of well The lag time is mainly affectedby wave velocity
Figure 11 presents a change of surge pressure in thewellbore at different scenario (OPOP CPOP and OPWPresp) According to Figure 11 surge pressure is increasedin CPOP scenario by comparing with the surge pressure inOPOP scenarios It is because the speed of the drilling mudincreases remarkably in CPOP scenario the surge pressurebecomes larger obviously Especially when the maximumsurge pressure occurs a strong increase of surge pressure isobserved at OPWP scenario If the movement of drill pipeis stop immediately the surge pressure will not disappearimmediately and the surge pressure will exist and changeslightly for short while
Figures 12 and 13 show that the surge pressure varies withwellbore diameter caused by the drill pipe operating speedaccording to Figure 7 The surge pressure decreases with theincrease of wellbore diameter It is because the drilling mudflow rate decreases with the increases of wellbore diameterBased on (1) and (2) surge pressure decreases at each positiondue to the decrease of drilling mud flow rate Surge pressurecaused by converting of potential energy into kinetic energyand then kinetic energy back into potential energy over andover again During the tripping operations especially in slimwellbore drilling process the speed of drill pipe movementshould be appropriately reduced The surge pressure in
Mathematical Problems in Engineering 11
OPWPCPOPOPOP
t (s)0 10 20 30 40 50 60
16
12
08
04
00
P (M
Pa)
Qg = 2427m3h
Figure 11 Surge pressure variation at different scenarios
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 2427m3h
Figure 12 Surge pressure variation at different wellbore diameters(gas influx 119876
119892= 2427m3h)
Figure 13 is prominently larger than that in Figure 12 Thisis because both the decrease of the wave velocity and theincrease of the compression in gas drilling mud two-phasedue to the gas influx result in the comparatively large surgepressure
Figure 14 illustrates that the surge pressure varied withthe length of drill pipe The surge pressure caused by shorterdrill pipe (whose the total length is 1000m) is smaller thanthe longer drill pipe (whose total length is 3000m) beforethe maximum surge pressure occurs This can explain thatsurge pressure is affected by the friction force of drill pipe andwellbore Compared with the longer drill pipe the frictionforce caused by shorter drill pipe is largerTherefore the surgepressure gradually decreases As a result the surge pressure isgreater at 1000m
Figure 15 illustrates that the surge pressure varied withthe drilling fluid density At the bottom of well the surge
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 0m3h
Figure 13 Surge pressure at different borehole diameters (gas influx119876119892= 0m3h)
t (s)0 10 20 30 40 50 60
P (M
Pa)
16
12
08
04
00
Qg = 2427m3h
L = 1000mL = 3000m
Figure 14 Surge pressure variation at different well depths
pressure increases with the increase of drilling fluid densityThe changes are significant in the maximum surge pressureThis is because the surge pressure is mainly affected by wavevelocity density and velocity of gas drilling mud two-phaseAccording to (6) and (7) the surge pressure is increased as thedrilling fluid density increasesThemaximum surge pressureoccurs before drill pipe reaches bottomhole
52 The Effects of Gas Influx Rate on Surge Pressure Figures16 17 and 18 illustrate that the surge pressure varied withthe increase of the gas influx rate Figure 17 shows the surgepressure with the gas influx rate of 2427m3h at the bottomof well Figure 18 shows the surge pressure with the gas influxrate of 8312m3h at the bottom of well Mainly affected bywave velocity density and velocity in gas-drilling mud twophase the surge pressure is greatest at the gas influx rate of0m3h Figure 16 illustrates an example of a minimum wavevelocity at the gas influx rate of 8312m3h at the bottom
12 Mathematical Problems in Engineering
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
Qg = 2427m3h
= 1500kgm3=
= 2000 kgm3=
120588m120588m
Figure 15 Surge pressure variation at different drilling fluid densi-ties
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 0m3h
Figure 16 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 0m3h)
of well As the gas influx rate increases the surge pressuredecreases When gas influx occurs the lag time is lengthenedremarkably compared with the condition of no gas influxThe longer the distance is the greater the lag time will beThe lag time is mainly determined by the propagation time ofpressure wave in the wellbore
Figures 19 and 20 illustrate the surge pressure caused bydifferent operating speeds of the drill pipe at various gasinflux rates at the bottom of well As drilling mudmixed withinflux gas not only reduces the pressure wave velocity butalso reduces the density of themixture density surge pressuremainly affected by wave velocity andmixture density reducesdrastically with the increase of gas influx rate at the bottomof well Figures 19 and 20 are compared to Figures 17 and 18
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 2427m3h
Figure 17 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 8312m3h
Figure 18 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 8312m3h)
it can be concluded that the operating speed is also a majorfactor of surge pressure
53The Effects of Drill Pipe Operating Speed on Surge PressureFigure 21 illustrates the effect of drill pipe speed on the surgepressure The change of surge pressure is smaller in gasdrilling two-phase than in single-phase drilling mud Thesurge pressure becomes smaller and closes to zero after themovement is stopped due to greater friction force existing inwell wall during drill pipe operating
6 Conclusions
With full consideration of the important factors such as wavevelocity in real-time the compressibility of the gas phasegas influx rate pressure temperature and well parametersa new surge pressure model has been proposed based on the
Mathematical Problems in Engineering 13Va
lue
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 2427m3h
Figure 19 Surge pressure and fluid velocity variation at differentpipe velocities (gas influx 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 8312m3h
Figure 20 Surge pressure and fluid velocity variation at differentgas influxes (gas influx 119876
119892= 8312m3h)
30
24
18
12
06
00
Valu
e
t (s)0 10 20 30 40 50 60
Pipe velocity (ms)Pipe velocity (ms)Surge pressure (MPa)
Surge pressure (MPa)
Qg = 2427m3h
Figure 21 Surge pressure variations at different pipe velocities (gasinflux 119876
119892= 2427m3h)
mass conservation equations the momentum conservationequations characteristic line method and the gas-drillingmud equations of state (EOS) duringMPDoperations Solvedby the difference method and fourth-order explicit Runge-Kutta method the model is used to predict surge pressurefor different drill pipe operating speed and gas influx rates inMPD operations The main conclusions can be summarizedas follows
(1) An accurate mathematical model to predict surgepressure after gas influx is of great importancebecause of the effects of surge pressure on operationalsafety and success The new model provides betterprediction of surge pressures in comparison with theexisting experiment
(2) The surge pressure tends to increase with the increaseof drill pipe operating speed and decrease of gas influxrate andwell diameter Considering that the gas influxoccurs the surge pressure weakens obviously Withthe increases in well depth the maximum allowabletripping speed gradually become smaller the surgepressure increases at the bottom of well and drillingsafety window is accordingly narrower The surgepressure can cause a significant time lag if the gasinflux occurs at the bottom of well and the timelag is mainly determined by the propagation time ofpressure wave in the wellbore
(3) The surge pressure obviously varies at different sce-narios (OPOP CPOP and OPWP resp) duringthe tripping operations Surge pressure increased inCPOP scenario than in OPOP scenario Especiallywhen the maximum surge pressure occurs a strongincrease of surge pressure is observed at OPWPscenario By running the new surge pressure modeldrilling engineers can identify potential surge prob-lems and optimize tripping operations for differentscenarios as to avoid complicated accidents to happenin wellbore during drilling operations
(4) The calculation of surge pressure in gas drillingmud two-phase is more complex than conventionalcalculation in single drillingmudDrill pipe operatingspeed in drilling operations should be strictly con-trolled and the surge pressure should be consideredin the design The greatest advantages of new modelare the ability to reduce nonproductive time (NPT) indrilling
Nomenclature
A Wellbore effective cross-area (m2)1198600 The outer drill pipe cross-area (m2)
1198602 Wellbore cross-area (m2)
1198603 The inner Drill pipe cross-area (m2)
119860119890 Effective cross-area of jet nozzle (m2)
119888 Wave velocity (ms)119889119904 One discrete length of wellbore (m)
14 Mathematical Problems in Engineering
119863 Wellbore effective diameter (m)1198631 The inner diameter of casing (m)
1198632 The outer diameter of casing (m)
1198633 The inner diameter of drill pipe (m)
1198634 The outer diameter of drill pipe (m)
119863119896119895 Differential equation coefficients
119864119896119895 Differential equation coefficients
119864119904 Modulus of elasticity of the drill pipe
(MPa)119864119891 Modulus of elasticity of formation (MPa)
1198642 Modulus of elasticity of casing (MPa)
119891119866 Shear stresses coefficient of gas interface
119891 Friction coefficient1198912 Wellbore friction coefficient
1198913 Drill pipe friction coefficient
119891119871 Shear stresses coefficient of drilling mud
119865119911 Pressure applied on the two-phase flow
(N)119892 Acceleration due to gravity (m2s)119892119888 Conversion factor
119892119879 Geothermal temperature gradient (Km)
119896119890 Wellbore effective roughness (m)
1198960 Roughness of outer pipe (m)
119896119894 Roughness of inner pipe (m)
119867 The depth of well (m)1198711 The length from pipe bottom to hole
bottom (m)1198712 The length from pipe bottom to well
head (m)119871119892 Flow pattern119898 Hydraulic radius (m)1198982 Wellbore hydraulic radius (m)
1198983 Drill pipe hydraulic radius (m)
119873 Parameter combiningThompson energyeffects
119875 Pressure (MPa)1198750 Initial pressure of wellbore (MPa)
1198752 Pressure within the empty wellbore
(MPa)1198753 Pressure within the drill pipe (MPa)
119875119888 Critical pressure (kPa)
119875119871 Pressure of drilling mud (MPa)
119875119903 Reduced pressure
119875119895119905 Pressure at node (119895 119905) (MPa)
119902119898 Volumetric flow rate of two phase (m3s)
119902119866 Velocity of the gas (m3s)
119902119871 Velocity of the drilling mud (m3s)
119876119871 Mud flow rate within the wellbore(m3h)
1198761 Drilling mud flow rate within hole(m3h)
1198762 Mud flow rate within the wellbore(m3h)
1198763 Mud flow rate within the drill pipe(m3h)
119876119908 Pump flow rate (m3s)
119876119895119905 Flow rate at time 119905 and nod 119895 (m3s)
119877 Constant of EOS (JKgK)
119904 The length of wellbore along wellbore(m)
119905 Time (s)119879 Temperature (K)119879119888 Critical temperature (K)
119879119903 Reduced temperature
119879119890119894 Undistributed temperature at a depth
(K)119879119890119887ℎ
Undistributed temperature of wellhead(K)
119879119891119887ℎ
Undistributed temperature at bottom-hole (K)
V Velocity of drilling mud and gas (ms)V119901 Velocity of drill pipe (ms)
V119898 Gas and drilling mud flow velocity (ms)
V119866 Gas flow velocity (ms)
V119904 Slip velocity (ms)
V119904119892 Superficial gas velocity (ms)
V119871 Drilling mud flow velocity (ms)
119882119898 Mass flow velocity (kgm3)
119911119866 Gas deviation factor
1199110 Initial calculation height of wellbore (m)
119911119887ℎ Total well depth from surface (m)
119911 Variable of distance from surface (m)
Greek Letters
120588119866 Gas density (kgm3)
120591119891 Frictional pressure gradient (Pam)
120591119908 Frictional pressure coefficient
120588119898 Gas and drilling density (kgm3)
1205880 Density under atmospheric pressure (kgm3)
120588119871 Drilling mud density (kgm3)
120590119904 Surface tension (Nm2)
120601119866 Gas void fraction
120601119871 Drilling mud holdup
120588119903 Reduced density
120583119871 Viscosity of drilling mud (Pasdots)
120579 The ground angle (rad)120583119891 Poissonrsquos ratio of formation
120583119904 Poissonrsquos ratio of drill pipe
1205832 Poissonrsquos ratio of casing
Δ119860 The change of the across areas (m2)
Subscripts
CPOP Closed pipe without pumpingDAPC Dynamic annular pressure controlEOS Equations of stateERV Relief-coupling valveMPD Managed pressure drillingOPOP Open pipe without pumpingOPWP Open pipe with pumpingR-K4 The fourth-order explicit Runge-KuttaRCD The rotating control device
Mathematical Problems in Engineering 15
Subscripts of Graph
119888 Wave velocity in two-phase flow (ms)119863 Wellbore effective diameter (m)119867 Depth of wellbore (m)119871 The length from drill pipe and drill pipe (m)119876119892 Gas influx rate at the bottomhole (m3h)
120601119866 Void fraction ()
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 51274170 andQC2012C128) Without their support this work would nothave been possible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of AirInjection in Annuli to manage pressure during CementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling Conferenceand Exhibition San Diego Calif USA March 2012
[4] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the Norh Africa Technical Conference andExhibition Cairo Egypt February 2012
[5] J Saponja A Adeleye and B Hucik ldquoManaged-pressuredrilling (MPD) field trials demonstrate technology valuerdquo inProceedings of the IADCSPE Drilling Conference Miami FlaUSA February 2006
[6] G Liu ldquoPredicting surge pressures that result from runninglinersrdquoWorld Oil vol 222 no 4 p 92 2001
[7] K Zhou J Yang B Zhong and L Tang ldquoSurge pressure inwellborerdquo Oil Drilling and Production Technology vol 12 no 2pp 1ndash10 99 1990
[8] R S Rudi Rubiandini ldquoNew formula of surge pressure fordetermining safe trip velocitiesrdquo in Proceedings of the SPE AsiaPacific Oil and Gas Conference and Exhibition pp 16ndash18 ausOctober 2000
[9] J Li Z Huang K Li and W Wu ldquoThe influence of collaron surge pressure caused by the drilling fluid viscous forceunder pumping conditionrdquoResearch Journal of Applied SciencesEngineering and Technology vol 5 no 5 pp 1781ndash1785 2040ndash7459 2013
[10] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvancedMaterials Research vol 468ndash471 pp 1736ndash1742 2012
[11] F Crespo and R Ahmed ldquoSurge-and-swab pressure predictionsfor yield-power-law drilling fluidsrdquo SPE Drilling amp Completionvol 27 no 4 pp 574ndash585 2012
[12] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[13] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquo PHDThesis vol 50 no 9 p 143 1989
[14] H Fan ldquoDevelopment of application software for dynamicsurge pressure while trippingrdquo Petroleum Drilling and Technol-ogy vol 23 no 4 pp 11ndash14 1995
[15] K Zhou J Yang B Zhong and L Tang ldquoSurge pressurein wellborerdquo Journal of Petroleum Exploration and ProductionTechnology vol 12 no 2 pp 1ndash10 1990
[16] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquoDissertation Abstracts International Bvol 50 no 9 1989
[17] Z Jiang ldquoStudies of wellbore surge pressure by a dynamicmethodrdquo Journal of Petroleum Exploration and ProductionTechnology vol 13 no 3 pp 39ndash44 97 1991
[18] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[19] D Johnston M Bracken and J Thornton ldquoAssessing pipecondition useful remaining life and maximum operating andsurge pressures when to change the pipe or control thepressurerdquoMagazine of the International Water Association vol14 no 5 pp 43ndash44 2012
[20] H Fan Y Chu and X Liu ldquoPrediction for wellbore dynamicsurge pressure while tripping a drill piperdquo Journal of ChinaUniversity of Petroleum vol 19 no 5 pp 36ndash41 1995
[21] C K Burgess J E Miller and K D Vondervort ldquoDownholesurge pressure reduction system and method of userdquo World9848 143 1998
[22] J P Allamon J E Miller and A M Macfarlane ldquoDownholeSurge Reduction Method and Apparatusrdquo 2004
[23] K S Bjoslashrkevoll R Rommetveit B Aas H Gjeraldstveit andA Merlo ldquoTransient gel breaking model for critical wellsapplications with field data verificationrdquo in Proceedings of the2003 Offshore Mediterranean Conference February 2003
[24] G H Nygaard E Johannessen J E Gravdal and F IversenldquoAutomatic coordinated control of pump rates and choke valvefor compensating pressure fluctuations during surge and swaboperationsrdquo in Proceedings of the IADCSPE Managed PressureDrilling and Underbalanced Operations Conference pp 120ndash125March 2007
[25] J V Fedevjcyk S L M Junqueira and C O R Negrao ldquoSurgeamp swab pressures in wells with cross-section changesrdquo BoletimTecnico da Producao de Petroleo vol 4 no 2 pp 221ndash234 2009
[26] S Tian G H Medley and C R Stone ldquoUnderstandingyield point effect onpressure surges critical to managing deepdifficultMPDwellsrdquoDrilling Contractor vol 65 no 4 p 46 4850 53 2009
[27] Y Sun Q Li and J Zhao ldquoNew method of predicting surgepressure apply to horizontal well based on caisson flowrdquoNaturalScience vol 2 no 12 pp 1394ndash1399 2150ndash4091 2010
[28] B T Lebele-Alawa and F E Oparadike ldquoAnalysis of the effectsof valve propagated pressure surge on pipe flowrdquo Engineeringvol 3 no 11 pp 1098ndash1101 2011
[29] X Kong Q Yuan and Y Qiu ldquoThe research of two-phasefluctuation pressure in wellbore tripping operationrdquo InnerMongolia Petrochemical Industry TE242 2011
[30] F Crespo and R Ahmed ldquoA simplified surge and swab pressuremodel for yield power law fluidsrdquo Journal of Petroleum Scienceand Engineering vol 101 pp 12ndash20 2013
16 Mathematical Problems in Engineering
[31] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
[32] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[33] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86-87 pp 127ndash136 2012
[34] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[35] M J SanchezComparison of Correlations for Predicting PressureLosses in Vertical Multiphase Annular Flow University of TulsaTulsa Okla USA 1972
[36] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[37] Y Lin X Kong Y Qiu and Q Yuan ldquoCalculation analysis ofpressure wave velocity in gas and drilling mud two-phase fluidin annulus during drilling operationsrdquo Mathematical Problemsin Engineering vol 2013 Article ID 318912 17 pages 2013
[38] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[39] H Junfang ldquoBalanced drilling and well controlrdquo PetroleumIndustry Press vol 1 article 33 1992
[40] W A Bacon Consideration of compressibility effects for applied-back- pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Va USA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
If 119879 ge 130∘C the density of drilling mud is
120588119871= 1205880(1 + 4 times 10
minus10
119875 minus 4 times 10minus5
119879 minus 3 times 10minus6
1198792
+ 04(119879 minus 130
119879)
2
)
(20)
233 Correlation of Temperature Distribution The tempera-ture of the drilling mud at different depth of the wellbore canbe determined by the relationship presented by Hasan andKabir [33]
119879 = 119879119890119894+ 119865 [1 minus 119890
(119911119887ℎ
minus119911)119860
]
times (minus119892 sin 120579119892119888119869119888119901119898
+ 119873 + 119892119879sin 120579)
+ 119890(119911119887ℎ
minus119911)119860
(119879119891119887ℎ
minus 119879119890119887ℎ)
(21)
24 Flow Pattern Analysis Based on the analysis of flowcharacteristics in the closed drilling system it can be safelyassumed that the flow pattern in wellbore is either bubbleor slug flow The pattern transition criteria for bubbly flowand slug flow given by Orkiszewski are used to judge the flowpattern in the gas-drilling mud two-phase flow [34]
For bubbly flow
119902119866
119902119898
lt 119871119861 (22)
For slug flow
119902119866
119902119898
gt 119871119861 119873
119866119881lt 119871119904 (23)
where 119902119898is the volumetric flow rate of two-phase flow and
119902119898= 119902119866+ 119902119871
The dimensionless numbers 119871119904and 119871
119861are defined as
119871119904= 50 +
36119873119866119881119902119871
119902119866
119871119861= 1071 minus
07277V2119898
119863
(24)
where
119873119866119881
= V119904(120588119871
119892120590119904
)
025
(25)
Flow parameters such as void fractionmixed density andvirtual mass force coefficient are discussed for specific flowpattern
The correlation between void fraction and drilling mudholdup is expressed as
120601119866+ 120601119871= 1 (26)
241 Bubble Flow The average density of the mixture forbubble flow is
120588119898= 120601119871120588119871+ 120601119866120588119866 (27)
The void fraction for bubble flow is determined by thefollowing equation
120601119866=1
2[
[
1 +119902119898
V119904119860minus radic(1 +
119902119898
V119904119860)
2
minus4119902119866
V119904119860]
]
(28)
The friction pressure gradient can be obtained from thefollowing formula
120591119891= 119891
120588119871V2119871
2119863 (29)
242 Slug Flow The distribution coefficient of gas in thedrilling mud phase is determined by
1198620=000252lg (103120583L)
119860138minus 0782 + 0232lgVm minus 0428lgA
(30)
The average density of the mixture for slug flow is
120588119898=119882119898+ 120588119871V119904119860
119902119898+ V119904119860
+ 1198620120588119871 (31)
The void fraction for slug flow is determined by thefollowing equation
120601119866=
119902119866
119902119866+ 119902119871
(32)
The friction pressure gradient can be obtained from thefollowing formula
120591119891= 119891
120588119871V2119898
2119863(119902119871+ V119904119860
119902119898+ V119904119860+ 1198620) (33)
25 Wellbore Characteristic Analysis Effective diameter pro-posed by Sanchez [35] is
119863 =120587 (1198632
2minus 1198632
1) 4
120587 (1198632+ 1198631) 4
= 1198632minus 1198631 (34)
The effective roughness of the wellbore can be expressedas
119896119890= 1198960
1198632
1198632+ 1198631
+ 119896119894
1198631
1198632+ 1198631
(35)
With 119863119894and 119863
119900being the diameters of inner pipe and
outer pipe respectively the 1198960and 119896
119894are the roughness of
outer pipe and the inner pipe respectively
6 Mathematical Problems in Engineering
Gas
Drilling pipe
Drilling string
Drilling mudI
IIIII II
Q1
Q2 Q3 Q2
(a) OPOP
I
IIIII II
Q1
Q2 Q3 Q2
(b) CPOP
0
I
IIIII II
Q2 Q3 Q2
L2
L1Q1
(c) OPWP
Figure 2 Different pipe end conditions in trip operations
3 Boundary Conditions and InitialConditions of the Model
When the drill pipe runs into the wellbore filled with drillingmud and gas at high velocities there are three scenarios forthe movement of the drill pipe They are closed pipe withoutpumping (CPOP) open pipe without pumping (OPOP)and open pipe with pumping (OPWP) respectively Thethree scenarios are illustrated in Figure 2 As drill pipe ismoved downward into a wellbore the drilling mud willmove upward To analyze the transient flow in the wellborecaused by the downward movement of drill pipe accordingto (6) and (7) and calculate the surge pressure at that timeit is indispensable to obtain the definite conditions of themodel for the three scenarios that is initial conditions andboundary conditions
We established coordinate systems of the hydraulic sys-tem in the wellboreThe coordinate systems have their originat the end of the drill pipe The flow rate of gas and drillingmud two-phase in open borehole annulus (comprised bydrill pipe and the casing) and drill pipe were respectivelydefined as 119876
1 1198762 and 119876
3 Respectively the corresponding
passageways filled with gas and drilling mud two-phase fluidwere marked as I III and II andthe pressure is 119875
1 1198752 and
1198753 The positive direction of passageway I was towards the
bottomhole while the positive directions of the passagewayIII and II were towards the wellhead As seen in Figure 2 119871
1
was the length from the origin to the bottomhole and 1198712was
the length from the origin to the wellhead By calculating thevelocity of two phases flow the new model for surge pressurein gas-drilling mud two-phase during the movement of drillpipe can be deduced [36]
In actual hydraulic system of the well many pipelineseries are included By assuming the length between thenodes and origin as 119871
3 the boundary consistency conditions
are provided by
119876119869+1
119894(1198713) minus 119876119869
119894(1198713) = V119901sdot Δ119860
119901119869+1
119894(1198713) = 119901119869
119894(1198713)
(119894 = 1 2 3) (36)
According to the relationship between the pressure andvelocity at the wellbore the boundary conditions can bedetermined as follows
119901119894(1198712) = 0
119876119894(1198711) = 0
(119894 = 2 3) (37)
The effective cross-area of jet nozzles is
119860119890=1
4120587
119899
sum
119894=1
1198892
119894 (38)
31 OPOP Scenario The following equation presents theinitial conditions of the flow rate and pressure for OPOPscenario
119901119894(119911) = 0
119876119894(119911) = 0
(119894 = 1 2 3) (39)
The boundary conditions of the flow rate and pressure are
1198761(0 119905) + 119876
2(0 119905) + 119876
3(0 119905) = V
119901(119905) (119860
0minus 1198603)
1199011(0 119905) = 119901
2(0 119905)
1199011(0 119905) minus 119901
3(0 119905)
=120588119898
2120583119898
(1198763(0 119905)
119860119890
+ V119901(119905))
10038161003816100381610038161003816100381610038161003816
1198763(0 119905)
119860119890
+ V119901(119905)
10038161003816100381610038161003816100381610038161003816
(40)
32 CPOP Scenario The following equation presents theinitial conditions of the flow rate and pressure for CPOPscenario
The initial conditions of the flow rate and pressure are asfollows
1199011(119911 0) = 0 119876
1(119911 0) = 0
1199012(119911 0) =
11989121199111205881198981198762
119908
811989821198602
1198762(119911 0) = 119876
119908
1199013(119911 0) =
1198913(1198712minus 119911) 120588
1198981198762
119908
811989831198603
1198763(119911 0) = 119876
119908
(41)
Theboundary conditions of the flow rate and pressure canbe written in the following form
1198761(0 119905) + 119876
2(0 119905) = V
119901(119905) 1198600
1198763(0 119905) = minusV
119901(119905) 1198603
1199011(0 119905) = 119901
2(0 119905)
(42)
33 OPWP Scenario As shown in Figure 2(c) the OPWPscenario is discussed below
Excluding atmospheric pressure the initial conditions atthe wellhead and bottom hole are
1199011(119911 0) = 0 119876
1(119911 0) = 0
1199012(119911 0) =
11989121199111205881198981198762
119908
811989821198602
1198762(119911 0) = 119876
119908
1199013(119911 0) =
1198913(1198712minus 119911) 120588
1198981198762
119908
811989831198603
1198763(119911 0) = 119876
119908
(43)
Mathematical Problems in Engineering 7
The boundary conditions of the flow rate and pressure areas follows
1198761(0 119905) + 119876
2(0 119905) = V
119901(119905) (119860
0minus 1198603) + 119876119908(0 119905)
1198763(0 119905) = minusV
119901(119905) 1198603+ 119876119908(0 119905)
1199011(0 119905) = 119901
2(0 119905)
(44)
4 Solution of the United Model
Referring to Figure 4 the variable 119909 is defined as a short pipeand the constraint is 119871
119909lt 119888119909sdot Δ119905 The variables 119910 and 119911 are
short pipe which are made up of wellbore and drill pipe andthe constraint is 119871
119910lt 119888119910sdot Δ119905 119871
119911lt 119888119911sdot Δ119905 The variables
119909 + 1 119910 + 1 and 119911 + 1 are defined as a long pipe and theconstraint is 119871
119909+ 1 lt 119888
119909+ 1 sdot Δ119905 119871
119910+ 1 lt 119888
119910+ 1 sdot Δ119905 119871
119911+ 1 lt
119888119911+ 1 sdot Δ119905 The implicit finite-difference method is applied to
calculate the short pipe and characteristicsmethod is appliedto calculate the long pipe The pressure and flow rate can beobtained at the end of drill pipe as follows
[1199012119905]2minus 120588119898[119888119910+1
119860119910+1
119876119910+1
2119905]
2
minus [1199013119905minusΔ119905
]2minus 120588119898[119888119910+1
119860119910+1
119876119905minusΔ119905
]
2
+120588119898Δ119905
8[119888119910+1
119898119910+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
2
= 0
[1199012119905]3minus 120588119898[119888119911+1
119860119911+1
119876119911+1
2119905]
3
minus [1199013119905minusΔ119905
]3minus 120588119898[119888119911+1
119860119911+1
1198763119905minusΔ119905
]
3
+120588119898Δ119905
8[119888119911+1
119898119911+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
3
= 0
[1199012119905]1minus 120588119898[119888119909+1
119860119909+1
119876119909+1
2119905]
1
minus [1199013119905minusΔ119905
]1minus 120588119898[119888119909+1
119860119909+1
1198763119905minusΔ119905
]
1
+120588119898Δ119905
8[119888119909+1
119898119909+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
1
= 0
[1199012119905+ 1199011119905]2+ 11986311[119876119910
2119905+ 119876119910
1119905]2
+ 11986312= 0
[1199012119905minus 1199011119905]2+ 11986411[119876119910
2119905minus 119876119910
1119905]2
+ 11986412= 0
[1199012119905+ 1199011119905]3+ 11986321[1198762119905+ 119876119911
1119905]2
+ 11986322= 0
[1199012119905minus 1199011119905]3+ 11986421[1198762119905minus 119876119911
1119905]2
+ 11986422= 0
[1199012119905+ 1199011119905]1+ 11986331[1198762119905minus 119876119909
1119905]1
+ 11986332= 0
[1199012119905minus 1199011119905]1+ 11986431[1198762119905+ 119876119909
1119905]1
+ 11986432= 0
1
2
3
i minus 1
i
i + 1
N minus 2
N minus 1
N
Bottomhole
Wellhead
3
3
2
1
2
3
c+y+1cz+1
Δsx+1
Δsx+1
Ly = Lz
Lx
cminusx+1
z
Figure 3 Schematic diagram of the solve
[1198761119905]1+ [1198761119905]2+ [1198761119905]2minus V119901(119905)
sdot (1198600minus 119860119894) = 0
[1199011119905]1minus [1199011119905]3minus
120588119898
2(119906119860119890)2
times [1198761119905]3+ 119860119890V119901(119905)
10038161003816100381610038161003816[1198761119905]3+ 119860119890V119901(119905)
10038161003816100381610038161003816= 0
119876119910+1
2119905minus 119876119910
2119905minus V119901(119905)
sdot Δ119860 = 0
[1199011119905]1= [1199011119905]2
11986311=1205881198981198882
119860sdotΔ119905
119871119869
11986312=120588119898
119860sdot119871119869
119871119869
(45)
The119863119896119895 119864119896119895can be expressed as follows
119863119894119905= minus (119901
119894+1119905minusΔ119905+ 119901119894119905) + 11986311(119876119894+1119905minusΔ119905
minus 119876119894119905minusΔ119905
)
119864119894119905= minus (119901
119894+1119905minusΔ119905+ 119901119894119905) + 11986411(119876119894+1119905minusΔ119905
minus 119876119894119905minusΔ119905
)
+120588119898119891119871119869
16119898(V119894+1119905minusΔ119905
+ V119894119905minusΔ119905
)1003816100381610038161003816(V119894+1119905minusΔ119905 + V
119894119905minusΔ119905)1003816100381610038161003816
(46)
where 119896 = 1 2 3 and 119895 = 1 2Obtaining the analytical solution of the mathematical
models concerned with flow pattern void fraction param-eters and pressure drop gradient are generally impossible formost practical in two-phase flow In this paper the Runge-Kutta method (R-K4) is used to discrete the new model forsurge pressure At different wellbore depths we can obtainpressure temperature gas velocity drilling mud velocity and
8 Mathematical Problems in Engineering
void fraction in MPD operations by application of the R-K4The solution of pressure drop gradient equation (3) can beseen as an initial-value problem of the ordinary differentialequation
119889119901
119889119911= 119865 (119911 119901)
119901 (1199110) = 1198750
(47)
With the initial value (1199110 1199010) and the function 119865(119911 119901)
(3)ndash(5) can be obtained
1198961= 119865 (119911
0 1199010)
1198962= 119865(119911
0+ℎ
2 1199010+ℎ
21198961)
1198963= 119865(119911
0+ℎ
2 1199010+ℎ
21198962)
1198964= 119865 (119911
0+ ℎ 119901
0+ ℎ1198963)
(48)
where ℎ is the step of depthThe pressure on the nod 119894 = 119894 + 1can be obtained by
1199011= 1199010+ Δ119901 = 119901
0+ℎ
6(1198961+ 21198962+ 21198963+ 1198964) (49)
In the present work the surge pressure calculation modelsolved by personally compiled code on VBNET (Version2010) The schematic diagram of solve is shown in Figure 3and the solution procedure for the surge pressure in thewellbore is shown in Figure 4 At initial time the wellheadtemperature wellbore structure well depths gas and drillingmud two-phase properties and so forth are known On thenode 119894 the pressure gradient wave velocity varieties in real-time temperature and the void fraction can be obtained byadopting R-K4 The density of gas can be obtained by gasEOS (15) and the density of drilling mud can be obtained byequations (19) and equations (20) Then the surge pressureat different depths of wellbore in tripping operations can besolved by (45)Theprocess is repeated until the surge pressurein the whole wellbore has been obtained [37 38]
The developed model takes full consideration of the gasinflux and wave velocity in real-time Owing to the complexconditions of wellbore in tripping operations measurementof surge pressure in the actual drilling process is very difficultIn order to verify the new model for surge pressure thepredicted surge pressure is compared with the results ofprevious experimental investigations presented by Junfang intripping operations in Figure 5 [39] The comparisons revealthat the developed new model fits well with the experimentsdata Thus the new model can be used to accurately predictsurge pressure on different scenario and gas influx rate intripping operations The value of surge pressure calculatedby adopted Burkhardtrsquos steady method is larger than theexperiments data
Table 1 Parameters of calculation well
Type Property Value
Mud Dynamic viscosity (Pasdots) 0056Density (kgm3) 1460
Gas Relative density 065Viscosity (Pasdots) 114 times 10minus5
StringElastic modulus of string (Pa) 207 times 1011
Poisson ratio of string 03Roughness (m) 154 times 10minus7
Surface condition Surface temperature (K) 298Atmosphere pressure (MPa) 0101
5 Analysis and Discussion
The drilling system described is a closed system Theschematic diagram of gas influx process in tripping opera-tions is illustrated in Figure 6 The drilling mud is pumpedfrom surface storage down the drill pipe returns from theannulus and travels back through surface processing wheredrilling solids are removed to surface storage
The well used for calculation is a gas well in XinjiangUygur Autonomous Region China The wellbore structurewell design parameters (depths and diameters) gas anddrilling mud two-phase properties (density and viscosity)and operational conditions of calculation well are displayedin Table 1 [40]
The drilling mud mixed with gas is considered as a two-phase flow medium The surge pressure in the gas-drillingmud are calculated and discussed by using the establishedmodel and well parameters As shown in Figure 7 accordingto the drill conditions such as the drill pipe velocity gasdrilling mud two-phase flow velocity the total length of drillpipe and drill pipe is 1000mwe can account for the changingrules of surge pressureThe following examples illustrate thatthe surge pressure is affected by different well parametersdrilling mud density gas influx rate and well depth in thetripping operations
51 The Effects of Various Drilling Parameters on SurgePressure Figure 7 presents the influence of acceleration ondrilling mud velocity drilling mud velocity and drill pipeoperating speed with respect to the drilling parameters Themovement of drill pipe starts at the depth of 1000m Withthe movement of the drill pipe the drilling mud also movesin the opposite direction along wellbore at the same timeThe drill pipe operating speed reaches the maximum valueat 23 seconds and stops at 30 seconds In order to make acomparison of the surge pressure in different well parametersthe analysis and compared Figures 8ndash15 based on the data ofFigure 7
Figures 8 and 9 graphically show the distributions ofvoid fraction and variations of wave velocity along the flowdirection in the wellbore When gas influx occurs at thebottom of well some gas invades into the wellbore andmigrates form the bottomhole to the wellhead along the flowdirection At low gas influx rate it is extremely obvious that
Mathematical Problems in Engineering 9
Start
Initial conditions
Grid division
Solve equation (21) for temperature
Calculate k1 minus k4Pi+1 = Pi + ΔP
zi+1 = zi + h
Solve equation (12) for 120588m
Solve equation (14) for c
Determine the time and the step
Enter the pipe velocity
Spline smoothing
Satisfy the boundary conditions and initial conditions
Δt rarr 0
Analysis conditions at the end of pipe
Solve pressure and flow at boundary points
Solve pressure and flow in pipe points
T = T + Δt
NoT lt Tmax
YesEnd
Figure 4 Solution procedure for surge pressure in MPD operations
80 16 24 32 40
00
05
10
15
20
Valu
e
t (s)
Pipe velocity (ms)This model (MPa) Experimental data (MPa)
Burkhardt data (MPa)
Figure 5 Experimental verification by comparison with previousexperiment data
the void fraction and wave velocity first slightly change in acomparatively smooth value then change sharply It is becausethe volume of gas expanded rapidly with the decreasingof pressure near the wellhead the void fraction increasessharply and at the same time the wave velocity decreasesobviously At a high gas influx rate the wave velocity tends toincrease because the void fraction in thewellhead is increasedto a high extent In conclusion the wave velocity is sensitiveto the void fraction and the void fraction is dominated by
MudMotor
Choke
z
DAPC
Casing
Drill pipeAnnulus
Drill collar
Formation
RCD0
800m
1200m
4000m
1778mm 78 mm ) times G105
2159mm times G105120601
120601 (120601
1270mm 78 mm ) times G105120601 (120601
Figure 6 Schematic diagram of gas influx process in tripping
influx rate and pressure in the wellbore especially the influxrate
As shown in Figure 9 within the range of low gas influxrate the wave velocity decreases significantly It is becausethe compressibility of the gas increases remarkably andthe medium appears high elasticity though the density ofgas-drilling mud two-phase flow changes slightly With theincrease of gas influx rate and corresponding increase of thevoid fraction in the wellbore the compressibility of the two-phase unceasingly increases which promotes themomentumand energy exchange in the interface Therefore the wavevelocity continuously decreases
10 Mathematical Problems in EngineeringVa
lue
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
minus05
minus10
Pipe acceleration (ms2)Pipe velocity (ms)Fluid velocity (ms)
Figure 7 Drill pipe velocity variation at different accelerations
0 800 1600 2400 3200 4000H (m)
100
80
60
40
20
0
minus20
G(
)
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
120601
Figure 8 Void fraction distribution at different gas influx rates
0 800 1600 2400 3200 4000H (m)
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
1500
1200
900
600
300
0
c (m
s)
Figure 9 Wave velocity variation at different gas influxes
0 10 20 30 40 50 60
25
20
15
10
05
00
minus05
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
P (M
Pa)
t (s)
Figure 10 Surge pressure variation at different gas influxes
Figure 10 illustrates the effect of gas influx on the surgepressure with the change of gas influx rate As shown inFigure 10 under different gas influx rates (0m3h 2427m3hand 8312m3h resp) the maximum surge pressure mayoccur before drill pipe reaches bottomhole With theincreases of the gas influx the corresponding surge pressuredecreases Considering the gas influx in the calculation of thesurge pressure in the wellbore the surge pressure weakensobviously As the gas influx rate decreases the surge pressureat bottom of the wellbore is reduced gradually while surgepressure can cause a significant time lag if the gas influxoccurs at the bottom of well The lag time is mainly affectedby wave velocity
Figure 11 presents a change of surge pressure in thewellbore at different scenario (OPOP CPOP and OPWPresp) According to Figure 11 surge pressure is increasedin CPOP scenario by comparing with the surge pressure inOPOP scenarios It is because the speed of the drilling mudincreases remarkably in CPOP scenario the surge pressurebecomes larger obviously Especially when the maximumsurge pressure occurs a strong increase of surge pressure isobserved at OPWP scenario If the movement of drill pipeis stop immediately the surge pressure will not disappearimmediately and the surge pressure will exist and changeslightly for short while
Figures 12 and 13 show that the surge pressure varies withwellbore diameter caused by the drill pipe operating speedaccording to Figure 7 The surge pressure decreases with theincrease of wellbore diameter It is because the drilling mudflow rate decreases with the increases of wellbore diameterBased on (1) and (2) surge pressure decreases at each positiondue to the decrease of drilling mud flow rate Surge pressurecaused by converting of potential energy into kinetic energyand then kinetic energy back into potential energy over andover again During the tripping operations especially in slimwellbore drilling process the speed of drill pipe movementshould be appropriately reduced The surge pressure in
Mathematical Problems in Engineering 11
OPWPCPOPOPOP
t (s)0 10 20 30 40 50 60
16
12
08
04
00
P (M
Pa)
Qg = 2427m3h
Figure 11 Surge pressure variation at different scenarios
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 2427m3h
Figure 12 Surge pressure variation at different wellbore diameters(gas influx 119876
119892= 2427m3h)
Figure 13 is prominently larger than that in Figure 12 Thisis because both the decrease of the wave velocity and theincrease of the compression in gas drilling mud two-phasedue to the gas influx result in the comparatively large surgepressure
Figure 14 illustrates that the surge pressure varied withthe length of drill pipe The surge pressure caused by shorterdrill pipe (whose the total length is 1000m) is smaller thanthe longer drill pipe (whose total length is 3000m) beforethe maximum surge pressure occurs This can explain thatsurge pressure is affected by the friction force of drill pipe andwellbore Compared with the longer drill pipe the frictionforce caused by shorter drill pipe is largerTherefore the surgepressure gradually decreases As a result the surge pressure isgreater at 1000m
Figure 15 illustrates that the surge pressure varied withthe drilling fluid density At the bottom of well the surge
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 0m3h
Figure 13 Surge pressure at different borehole diameters (gas influx119876119892= 0m3h)
t (s)0 10 20 30 40 50 60
P (M
Pa)
16
12
08
04
00
Qg = 2427m3h
L = 1000mL = 3000m
Figure 14 Surge pressure variation at different well depths
pressure increases with the increase of drilling fluid densityThe changes are significant in the maximum surge pressureThis is because the surge pressure is mainly affected by wavevelocity density and velocity of gas drilling mud two-phaseAccording to (6) and (7) the surge pressure is increased as thedrilling fluid density increasesThemaximum surge pressureoccurs before drill pipe reaches bottomhole
52 The Effects of Gas Influx Rate on Surge Pressure Figures16 17 and 18 illustrate that the surge pressure varied withthe increase of the gas influx rate Figure 17 shows the surgepressure with the gas influx rate of 2427m3h at the bottomof well Figure 18 shows the surge pressure with the gas influxrate of 8312m3h at the bottom of well Mainly affected bywave velocity density and velocity in gas-drilling mud twophase the surge pressure is greatest at the gas influx rate of0m3h Figure 16 illustrates an example of a minimum wavevelocity at the gas influx rate of 8312m3h at the bottom
12 Mathematical Problems in Engineering
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
Qg = 2427m3h
= 1500kgm3=
= 2000 kgm3=
120588m120588m
Figure 15 Surge pressure variation at different drilling fluid densi-ties
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 0m3h
Figure 16 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 0m3h)
of well As the gas influx rate increases the surge pressuredecreases When gas influx occurs the lag time is lengthenedremarkably compared with the condition of no gas influxThe longer the distance is the greater the lag time will beThe lag time is mainly determined by the propagation time ofpressure wave in the wellbore
Figures 19 and 20 illustrate the surge pressure caused bydifferent operating speeds of the drill pipe at various gasinflux rates at the bottom of well As drilling mudmixed withinflux gas not only reduces the pressure wave velocity butalso reduces the density of themixture density surge pressuremainly affected by wave velocity andmixture density reducesdrastically with the increase of gas influx rate at the bottomof well Figures 19 and 20 are compared to Figures 17 and 18
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 2427m3h
Figure 17 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 8312m3h
Figure 18 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 8312m3h)
it can be concluded that the operating speed is also a majorfactor of surge pressure
53The Effects of Drill Pipe Operating Speed on Surge PressureFigure 21 illustrates the effect of drill pipe speed on the surgepressure The change of surge pressure is smaller in gasdrilling two-phase than in single-phase drilling mud Thesurge pressure becomes smaller and closes to zero after themovement is stopped due to greater friction force existing inwell wall during drill pipe operating
6 Conclusions
With full consideration of the important factors such as wavevelocity in real-time the compressibility of the gas phasegas influx rate pressure temperature and well parametersa new surge pressure model has been proposed based on the
Mathematical Problems in Engineering 13Va
lue
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 2427m3h
Figure 19 Surge pressure and fluid velocity variation at differentpipe velocities (gas influx 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 8312m3h
Figure 20 Surge pressure and fluid velocity variation at differentgas influxes (gas influx 119876
119892= 8312m3h)
30
24
18
12
06
00
Valu
e
t (s)0 10 20 30 40 50 60
Pipe velocity (ms)Pipe velocity (ms)Surge pressure (MPa)
Surge pressure (MPa)
Qg = 2427m3h
Figure 21 Surge pressure variations at different pipe velocities (gasinflux 119876
119892= 2427m3h)
mass conservation equations the momentum conservationequations characteristic line method and the gas-drillingmud equations of state (EOS) duringMPDoperations Solvedby the difference method and fourth-order explicit Runge-Kutta method the model is used to predict surge pressurefor different drill pipe operating speed and gas influx rates inMPD operations The main conclusions can be summarizedas follows
(1) An accurate mathematical model to predict surgepressure after gas influx is of great importancebecause of the effects of surge pressure on operationalsafety and success The new model provides betterprediction of surge pressures in comparison with theexisting experiment
(2) The surge pressure tends to increase with the increaseof drill pipe operating speed and decrease of gas influxrate andwell diameter Considering that the gas influxoccurs the surge pressure weakens obviously Withthe increases in well depth the maximum allowabletripping speed gradually become smaller the surgepressure increases at the bottom of well and drillingsafety window is accordingly narrower The surgepressure can cause a significant time lag if the gasinflux occurs at the bottom of well and the timelag is mainly determined by the propagation time ofpressure wave in the wellbore
(3) The surge pressure obviously varies at different sce-narios (OPOP CPOP and OPWP resp) duringthe tripping operations Surge pressure increased inCPOP scenario than in OPOP scenario Especiallywhen the maximum surge pressure occurs a strongincrease of surge pressure is observed at OPWPscenario By running the new surge pressure modeldrilling engineers can identify potential surge prob-lems and optimize tripping operations for differentscenarios as to avoid complicated accidents to happenin wellbore during drilling operations
(4) The calculation of surge pressure in gas drillingmud two-phase is more complex than conventionalcalculation in single drillingmudDrill pipe operatingspeed in drilling operations should be strictly con-trolled and the surge pressure should be consideredin the design The greatest advantages of new modelare the ability to reduce nonproductive time (NPT) indrilling
Nomenclature
A Wellbore effective cross-area (m2)1198600 The outer drill pipe cross-area (m2)
1198602 Wellbore cross-area (m2)
1198603 The inner Drill pipe cross-area (m2)
119860119890 Effective cross-area of jet nozzle (m2)
119888 Wave velocity (ms)119889119904 One discrete length of wellbore (m)
14 Mathematical Problems in Engineering
119863 Wellbore effective diameter (m)1198631 The inner diameter of casing (m)
1198632 The outer diameter of casing (m)
1198633 The inner diameter of drill pipe (m)
1198634 The outer diameter of drill pipe (m)
119863119896119895 Differential equation coefficients
119864119896119895 Differential equation coefficients
119864119904 Modulus of elasticity of the drill pipe
(MPa)119864119891 Modulus of elasticity of formation (MPa)
1198642 Modulus of elasticity of casing (MPa)
119891119866 Shear stresses coefficient of gas interface
119891 Friction coefficient1198912 Wellbore friction coefficient
1198913 Drill pipe friction coefficient
119891119871 Shear stresses coefficient of drilling mud
119865119911 Pressure applied on the two-phase flow
(N)119892 Acceleration due to gravity (m2s)119892119888 Conversion factor
119892119879 Geothermal temperature gradient (Km)
119896119890 Wellbore effective roughness (m)
1198960 Roughness of outer pipe (m)
119896119894 Roughness of inner pipe (m)
119867 The depth of well (m)1198711 The length from pipe bottom to hole
bottom (m)1198712 The length from pipe bottom to well
head (m)119871119892 Flow pattern119898 Hydraulic radius (m)1198982 Wellbore hydraulic radius (m)
1198983 Drill pipe hydraulic radius (m)
119873 Parameter combiningThompson energyeffects
119875 Pressure (MPa)1198750 Initial pressure of wellbore (MPa)
1198752 Pressure within the empty wellbore
(MPa)1198753 Pressure within the drill pipe (MPa)
119875119888 Critical pressure (kPa)
119875119871 Pressure of drilling mud (MPa)
119875119903 Reduced pressure
119875119895119905 Pressure at node (119895 119905) (MPa)
119902119898 Volumetric flow rate of two phase (m3s)
119902119866 Velocity of the gas (m3s)
119902119871 Velocity of the drilling mud (m3s)
119876119871 Mud flow rate within the wellbore(m3h)
1198761 Drilling mud flow rate within hole(m3h)
1198762 Mud flow rate within the wellbore(m3h)
1198763 Mud flow rate within the drill pipe(m3h)
119876119908 Pump flow rate (m3s)
119876119895119905 Flow rate at time 119905 and nod 119895 (m3s)
119877 Constant of EOS (JKgK)
119904 The length of wellbore along wellbore(m)
119905 Time (s)119879 Temperature (K)119879119888 Critical temperature (K)
119879119903 Reduced temperature
119879119890119894 Undistributed temperature at a depth
(K)119879119890119887ℎ
Undistributed temperature of wellhead(K)
119879119891119887ℎ
Undistributed temperature at bottom-hole (K)
V Velocity of drilling mud and gas (ms)V119901 Velocity of drill pipe (ms)
V119898 Gas and drilling mud flow velocity (ms)
V119866 Gas flow velocity (ms)
V119904 Slip velocity (ms)
V119904119892 Superficial gas velocity (ms)
V119871 Drilling mud flow velocity (ms)
119882119898 Mass flow velocity (kgm3)
119911119866 Gas deviation factor
1199110 Initial calculation height of wellbore (m)
119911119887ℎ Total well depth from surface (m)
119911 Variable of distance from surface (m)
Greek Letters
120588119866 Gas density (kgm3)
120591119891 Frictional pressure gradient (Pam)
120591119908 Frictional pressure coefficient
120588119898 Gas and drilling density (kgm3)
1205880 Density under atmospheric pressure (kgm3)
120588119871 Drilling mud density (kgm3)
120590119904 Surface tension (Nm2)
120601119866 Gas void fraction
120601119871 Drilling mud holdup
120588119903 Reduced density
120583119871 Viscosity of drilling mud (Pasdots)
120579 The ground angle (rad)120583119891 Poissonrsquos ratio of formation
120583119904 Poissonrsquos ratio of drill pipe
1205832 Poissonrsquos ratio of casing
Δ119860 The change of the across areas (m2)
Subscripts
CPOP Closed pipe without pumpingDAPC Dynamic annular pressure controlEOS Equations of stateERV Relief-coupling valveMPD Managed pressure drillingOPOP Open pipe without pumpingOPWP Open pipe with pumpingR-K4 The fourth-order explicit Runge-KuttaRCD The rotating control device
Mathematical Problems in Engineering 15
Subscripts of Graph
119888 Wave velocity in two-phase flow (ms)119863 Wellbore effective diameter (m)119867 Depth of wellbore (m)119871 The length from drill pipe and drill pipe (m)119876119892 Gas influx rate at the bottomhole (m3h)
120601119866 Void fraction ()
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 51274170 andQC2012C128) Without their support this work would nothave been possible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of AirInjection in Annuli to manage pressure during CementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling Conferenceand Exhibition San Diego Calif USA March 2012
[4] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the Norh Africa Technical Conference andExhibition Cairo Egypt February 2012
[5] J Saponja A Adeleye and B Hucik ldquoManaged-pressuredrilling (MPD) field trials demonstrate technology valuerdquo inProceedings of the IADCSPE Drilling Conference Miami FlaUSA February 2006
[6] G Liu ldquoPredicting surge pressures that result from runninglinersrdquoWorld Oil vol 222 no 4 p 92 2001
[7] K Zhou J Yang B Zhong and L Tang ldquoSurge pressure inwellborerdquo Oil Drilling and Production Technology vol 12 no 2pp 1ndash10 99 1990
[8] R S Rudi Rubiandini ldquoNew formula of surge pressure fordetermining safe trip velocitiesrdquo in Proceedings of the SPE AsiaPacific Oil and Gas Conference and Exhibition pp 16ndash18 ausOctober 2000
[9] J Li Z Huang K Li and W Wu ldquoThe influence of collaron surge pressure caused by the drilling fluid viscous forceunder pumping conditionrdquoResearch Journal of Applied SciencesEngineering and Technology vol 5 no 5 pp 1781ndash1785 2040ndash7459 2013
[10] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvancedMaterials Research vol 468ndash471 pp 1736ndash1742 2012
[11] F Crespo and R Ahmed ldquoSurge-and-swab pressure predictionsfor yield-power-law drilling fluidsrdquo SPE Drilling amp Completionvol 27 no 4 pp 574ndash585 2012
[12] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[13] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquo PHDThesis vol 50 no 9 p 143 1989
[14] H Fan ldquoDevelopment of application software for dynamicsurge pressure while trippingrdquo Petroleum Drilling and Technol-ogy vol 23 no 4 pp 11ndash14 1995
[15] K Zhou J Yang B Zhong and L Tang ldquoSurge pressurein wellborerdquo Journal of Petroleum Exploration and ProductionTechnology vol 12 no 2 pp 1ndash10 1990
[16] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquoDissertation Abstracts International Bvol 50 no 9 1989
[17] Z Jiang ldquoStudies of wellbore surge pressure by a dynamicmethodrdquo Journal of Petroleum Exploration and ProductionTechnology vol 13 no 3 pp 39ndash44 97 1991
[18] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[19] D Johnston M Bracken and J Thornton ldquoAssessing pipecondition useful remaining life and maximum operating andsurge pressures when to change the pipe or control thepressurerdquoMagazine of the International Water Association vol14 no 5 pp 43ndash44 2012
[20] H Fan Y Chu and X Liu ldquoPrediction for wellbore dynamicsurge pressure while tripping a drill piperdquo Journal of ChinaUniversity of Petroleum vol 19 no 5 pp 36ndash41 1995
[21] C K Burgess J E Miller and K D Vondervort ldquoDownholesurge pressure reduction system and method of userdquo World9848 143 1998
[22] J P Allamon J E Miller and A M Macfarlane ldquoDownholeSurge Reduction Method and Apparatusrdquo 2004
[23] K S Bjoslashrkevoll R Rommetveit B Aas H Gjeraldstveit andA Merlo ldquoTransient gel breaking model for critical wellsapplications with field data verificationrdquo in Proceedings of the2003 Offshore Mediterranean Conference February 2003
[24] G H Nygaard E Johannessen J E Gravdal and F IversenldquoAutomatic coordinated control of pump rates and choke valvefor compensating pressure fluctuations during surge and swaboperationsrdquo in Proceedings of the IADCSPE Managed PressureDrilling and Underbalanced Operations Conference pp 120ndash125March 2007
[25] J V Fedevjcyk S L M Junqueira and C O R Negrao ldquoSurgeamp swab pressures in wells with cross-section changesrdquo BoletimTecnico da Producao de Petroleo vol 4 no 2 pp 221ndash234 2009
[26] S Tian G H Medley and C R Stone ldquoUnderstandingyield point effect onpressure surges critical to managing deepdifficultMPDwellsrdquoDrilling Contractor vol 65 no 4 p 46 4850 53 2009
[27] Y Sun Q Li and J Zhao ldquoNew method of predicting surgepressure apply to horizontal well based on caisson flowrdquoNaturalScience vol 2 no 12 pp 1394ndash1399 2150ndash4091 2010
[28] B T Lebele-Alawa and F E Oparadike ldquoAnalysis of the effectsof valve propagated pressure surge on pipe flowrdquo Engineeringvol 3 no 11 pp 1098ndash1101 2011
[29] X Kong Q Yuan and Y Qiu ldquoThe research of two-phasefluctuation pressure in wellbore tripping operationrdquo InnerMongolia Petrochemical Industry TE242 2011
[30] F Crespo and R Ahmed ldquoA simplified surge and swab pressuremodel for yield power law fluidsrdquo Journal of Petroleum Scienceand Engineering vol 101 pp 12ndash20 2013
16 Mathematical Problems in Engineering
[31] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
[32] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[33] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86-87 pp 127ndash136 2012
[34] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[35] M J SanchezComparison of Correlations for Predicting PressureLosses in Vertical Multiphase Annular Flow University of TulsaTulsa Okla USA 1972
[36] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[37] Y Lin X Kong Y Qiu and Q Yuan ldquoCalculation analysis ofpressure wave velocity in gas and drilling mud two-phase fluidin annulus during drilling operationsrdquo Mathematical Problemsin Engineering vol 2013 Article ID 318912 17 pages 2013
[38] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[39] H Junfang ldquoBalanced drilling and well controlrdquo PetroleumIndustry Press vol 1 article 33 1992
[40] W A Bacon Consideration of compressibility effects for applied-back- pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Va USA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
Gas
Drilling pipe
Drilling string
Drilling mudI
IIIII II
Q1
Q2 Q3 Q2
(a) OPOP
I
IIIII II
Q1
Q2 Q3 Q2
(b) CPOP
0
I
IIIII II
Q2 Q3 Q2
L2
L1Q1
(c) OPWP
Figure 2 Different pipe end conditions in trip operations
3 Boundary Conditions and InitialConditions of the Model
When the drill pipe runs into the wellbore filled with drillingmud and gas at high velocities there are three scenarios forthe movement of the drill pipe They are closed pipe withoutpumping (CPOP) open pipe without pumping (OPOP)and open pipe with pumping (OPWP) respectively Thethree scenarios are illustrated in Figure 2 As drill pipe ismoved downward into a wellbore the drilling mud willmove upward To analyze the transient flow in the wellborecaused by the downward movement of drill pipe accordingto (6) and (7) and calculate the surge pressure at that timeit is indispensable to obtain the definite conditions of themodel for the three scenarios that is initial conditions andboundary conditions
We established coordinate systems of the hydraulic sys-tem in the wellboreThe coordinate systems have their originat the end of the drill pipe The flow rate of gas and drillingmud two-phase in open borehole annulus (comprised bydrill pipe and the casing) and drill pipe were respectivelydefined as 119876
1 1198762 and 119876
3 Respectively the corresponding
passageways filled with gas and drilling mud two-phase fluidwere marked as I III and II andthe pressure is 119875
1 1198752 and
1198753 The positive direction of passageway I was towards the
bottomhole while the positive directions of the passagewayIII and II were towards the wellhead As seen in Figure 2 119871
1
was the length from the origin to the bottomhole and 1198712was
the length from the origin to the wellhead By calculating thevelocity of two phases flow the new model for surge pressurein gas-drilling mud two-phase during the movement of drillpipe can be deduced [36]
In actual hydraulic system of the well many pipelineseries are included By assuming the length between thenodes and origin as 119871
3 the boundary consistency conditions
are provided by
119876119869+1
119894(1198713) minus 119876119869
119894(1198713) = V119901sdot Δ119860
119901119869+1
119894(1198713) = 119901119869
119894(1198713)
(119894 = 1 2 3) (36)
According to the relationship between the pressure andvelocity at the wellbore the boundary conditions can bedetermined as follows
119901119894(1198712) = 0
119876119894(1198711) = 0
(119894 = 2 3) (37)
The effective cross-area of jet nozzles is
119860119890=1
4120587
119899
sum
119894=1
1198892
119894 (38)
31 OPOP Scenario The following equation presents theinitial conditions of the flow rate and pressure for OPOPscenario
119901119894(119911) = 0
119876119894(119911) = 0
(119894 = 1 2 3) (39)
The boundary conditions of the flow rate and pressure are
1198761(0 119905) + 119876
2(0 119905) + 119876
3(0 119905) = V
119901(119905) (119860
0minus 1198603)
1199011(0 119905) = 119901
2(0 119905)
1199011(0 119905) minus 119901
3(0 119905)
=120588119898
2120583119898
(1198763(0 119905)
119860119890
+ V119901(119905))
10038161003816100381610038161003816100381610038161003816
1198763(0 119905)
119860119890
+ V119901(119905)
10038161003816100381610038161003816100381610038161003816
(40)
32 CPOP Scenario The following equation presents theinitial conditions of the flow rate and pressure for CPOPscenario
The initial conditions of the flow rate and pressure are asfollows
1199011(119911 0) = 0 119876
1(119911 0) = 0
1199012(119911 0) =
11989121199111205881198981198762
119908
811989821198602
1198762(119911 0) = 119876
119908
1199013(119911 0) =
1198913(1198712minus 119911) 120588
1198981198762
119908
811989831198603
1198763(119911 0) = 119876
119908
(41)
Theboundary conditions of the flow rate and pressure canbe written in the following form
1198761(0 119905) + 119876
2(0 119905) = V
119901(119905) 1198600
1198763(0 119905) = minusV
119901(119905) 1198603
1199011(0 119905) = 119901
2(0 119905)
(42)
33 OPWP Scenario As shown in Figure 2(c) the OPWPscenario is discussed below
Excluding atmospheric pressure the initial conditions atthe wellhead and bottom hole are
1199011(119911 0) = 0 119876
1(119911 0) = 0
1199012(119911 0) =
11989121199111205881198981198762
119908
811989821198602
1198762(119911 0) = 119876
119908
1199013(119911 0) =
1198913(1198712minus 119911) 120588
1198981198762
119908
811989831198603
1198763(119911 0) = 119876
119908
(43)
Mathematical Problems in Engineering 7
The boundary conditions of the flow rate and pressure areas follows
1198761(0 119905) + 119876
2(0 119905) = V
119901(119905) (119860
0minus 1198603) + 119876119908(0 119905)
1198763(0 119905) = minusV
119901(119905) 1198603+ 119876119908(0 119905)
1199011(0 119905) = 119901
2(0 119905)
(44)
4 Solution of the United Model
Referring to Figure 4 the variable 119909 is defined as a short pipeand the constraint is 119871
119909lt 119888119909sdot Δ119905 The variables 119910 and 119911 are
short pipe which are made up of wellbore and drill pipe andthe constraint is 119871
119910lt 119888119910sdot Δ119905 119871
119911lt 119888119911sdot Δ119905 The variables
119909 + 1 119910 + 1 and 119911 + 1 are defined as a long pipe and theconstraint is 119871
119909+ 1 lt 119888
119909+ 1 sdot Δ119905 119871
119910+ 1 lt 119888
119910+ 1 sdot Δ119905 119871
119911+ 1 lt
119888119911+ 1 sdot Δ119905 The implicit finite-difference method is applied to
calculate the short pipe and characteristicsmethod is appliedto calculate the long pipe The pressure and flow rate can beobtained at the end of drill pipe as follows
[1199012119905]2minus 120588119898[119888119910+1
119860119910+1
119876119910+1
2119905]
2
minus [1199013119905minusΔ119905
]2minus 120588119898[119888119910+1
119860119910+1
119876119905minusΔ119905
]
2
+120588119898Δ119905
8[119888119910+1
119898119910+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
2
= 0
[1199012119905]3minus 120588119898[119888119911+1
119860119911+1
119876119911+1
2119905]
3
minus [1199013119905minusΔ119905
]3minus 120588119898[119888119911+1
119860119911+1
1198763119905minusΔ119905
]
3
+120588119898Δ119905
8[119888119911+1
119898119911+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
3
= 0
[1199012119905]1minus 120588119898[119888119909+1
119860119909+1
119876119909+1
2119905]
1
minus [1199013119905minusΔ119905
]1minus 120588119898[119888119909+1
119860119909+1
1198763119905minusΔ119905
]
1
+120588119898Δ119905
8[119888119909+1
119898119909+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
1
= 0
[1199012119905+ 1199011119905]2+ 11986311[119876119910
2119905+ 119876119910
1119905]2
+ 11986312= 0
[1199012119905minus 1199011119905]2+ 11986411[119876119910
2119905minus 119876119910
1119905]2
+ 11986412= 0
[1199012119905+ 1199011119905]3+ 11986321[1198762119905+ 119876119911
1119905]2
+ 11986322= 0
[1199012119905minus 1199011119905]3+ 11986421[1198762119905minus 119876119911
1119905]2
+ 11986422= 0
[1199012119905+ 1199011119905]1+ 11986331[1198762119905minus 119876119909
1119905]1
+ 11986332= 0
[1199012119905minus 1199011119905]1+ 11986431[1198762119905+ 119876119909
1119905]1
+ 11986432= 0
1
2
3
i minus 1
i
i + 1
N minus 2
N minus 1
N
Bottomhole
Wellhead
3
3
2
1
2
3
c+y+1cz+1
Δsx+1
Δsx+1
Ly = Lz
Lx
cminusx+1
z
Figure 3 Schematic diagram of the solve
[1198761119905]1+ [1198761119905]2+ [1198761119905]2minus V119901(119905)
sdot (1198600minus 119860119894) = 0
[1199011119905]1minus [1199011119905]3minus
120588119898
2(119906119860119890)2
times [1198761119905]3+ 119860119890V119901(119905)
10038161003816100381610038161003816[1198761119905]3+ 119860119890V119901(119905)
10038161003816100381610038161003816= 0
119876119910+1
2119905minus 119876119910
2119905minus V119901(119905)
sdot Δ119860 = 0
[1199011119905]1= [1199011119905]2
11986311=1205881198981198882
119860sdotΔ119905
119871119869
11986312=120588119898
119860sdot119871119869
119871119869
(45)
The119863119896119895 119864119896119895can be expressed as follows
119863119894119905= minus (119901
119894+1119905minusΔ119905+ 119901119894119905) + 11986311(119876119894+1119905minusΔ119905
minus 119876119894119905minusΔ119905
)
119864119894119905= minus (119901
119894+1119905minusΔ119905+ 119901119894119905) + 11986411(119876119894+1119905minusΔ119905
minus 119876119894119905minusΔ119905
)
+120588119898119891119871119869
16119898(V119894+1119905minusΔ119905
+ V119894119905minusΔ119905
)1003816100381610038161003816(V119894+1119905minusΔ119905 + V
119894119905minusΔ119905)1003816100381610038161003816
(46)
where 119896 = 1 2 3 and 119895 = 1 2Obtaining the analytical solution of the mathematical
models concerned with flow pattern void fraction param-eters and pressure drop gradient are generally impossible formost practical in two-phase flow In this paper the Runge-Kutta method (R-K4) is used to discrete the new model forsurge pressure At different wellbore depths we can obtainpressure temperature gas velocity drilling mud velocity and
8 Mathematical Problems in Engineering
void fraction in MPD operations by application of the R-K4The solution of pressure drop gradient equation (3) can beseen as an initial-value problem of the ordinary differentialequation
119889119901
119889119911= 119865 (119911 119901)
119901 (1199110) = 1198750
(47)
With the initial value (1199110 1199010) and the function 119865(119911 119901)
(3)ndash(5) can be obtained
1198961= 119865 (119911
0 1199010)
1198962= 119865(119911
0+ℎ
2 1199010+ℎ
21198961)
1198963= 119865(119911
0+ℎ
2 1199010+ℎ
21198962)
1198964= 119865 (119911
0+ ℎ 119901
0+ ℎ1198963)
(48)
where ℎ is the step of depthThe pressure on the nod 119894 = 119894 + 1can be obtained by
1199011= 1199010+ Δ119901 = 119901
0+ℎ
6(1198961+ 21198962+ 21198963+ 1198964) (49)
In the present work the surge pressure calculation modelsolved by personally compiled code on VBNET (Version2010) The schematic diagram of solve is shown in Figure 3and the solution procedure for the surge pressure in thewellbore is shown in Figure 4 At initial time the wellheadtemperature wellbore structure well depths gas and drillingmud two-phase properties and so forth are known On thenode 119894 the pressure gradient wave velocity varieties in real-time temperature and the void fraction can be obtained byadopting R-K4 The density of gas can be obtained by gasEOS (15) and the density of drilling mud can be obtained byequations (19) and equations (20) Then the surge pressureat different depths of wellbore in tripping operations can besolved by (45)Theprocess is repeated until the surge pressurein the whole wellbore has been obtained [37 38]
The developed model takes full consideration of the gasinflux and wave velocity in real-time Owing to the complexconditions of wellbore in tripping operations measurementof surge pressure in the actual drilling process is very difficultIn order to verify the new model for surge pressure thepredicted surge pressure is compared with the results ofprevious experimental investigations presented by Junfang intripping operations in Figure 5 [39] The comparisons revealthat the developed new model fits well with the experimentsdata Thus the new model can be used to accurately predictsurge pressure on different scenario and gas influx rate intripping operations The value of surge pressure calculatedby adopted Burkhardtrsquos steady method is larger than theexperiments data
Table 1 Parameters of calculation well
Type Property Value
Mud Dynamic viscosity (Pasdots) 0056Density (kgm3) 1460
Gas Relative density 065Viscosity (Pasdots) 114 times 10minus5
StringElastic modulus of string (Pa) 207 times 1011
Poisson ratio of string 03Roughness (m) 154 times 10minus7
Surface condition Surface temperature (K) 298Atmosphere pressure (MPa) 0101
5 Analysis and Discussion
The drilling system described is a closed system Theschematic diagram of gas influx process in tripping opera-tions is illustrated in Figure 6 The drilling mud is pumpedfrom surface storage down the drill pipe returns from theannulus and travels back through surface processing wheredrilling solids are removed to surface storage
The well used for calculation is a gas well in XinjiangUygur Autonomous Region China The wellbore structurewell design parameters (depths and diameters) gas anddrilling mud two-phase properties (density and viscosity)and operational conditions of calculation well are displayedin Table 1 [40]
The drilling mud mixed with gas is considered as a two-phase flow medium The surge pressure in the gas-drillingmud are calculated and discussed by using the establishedmodel and well parameters As shown in Figure 7 accordingto the drill conditions such as the drill pipe velocity gasdrilling mud two-phase flow velocity the total length of drillpipe and drill pipe is 1000mwe can account for the changingrules of surge pressureThe following examples illustrate thatthe surge pressure is affected by different well parametersdrilling mud density gas influx rate and well depth in thetripping operations
51 The Effects of Various Drilling Parameters on SurgePressure Figure 7 presents the influence of acceleration ondrilling mud velocity drilling mud velocity and drill pipeoperating speed with respect to the drilling parameters Themovement of drill pipe starts at the depth of 1000m Withthe movement of the drill pipe the drilling mud also movesin the opposite direction along wellbore at the same timeThe drill pipe operating speed reaches the maximum valueat 23 seconds and stops at 30 seconds In order to make acomparison of the surge pressure in different well parametersthe analysis and compared Figures 8ndash15 based on the data ofFigure 7
Figures 8 and 9 graphically show the distributions ofvoid fraction and variations of wave velocity along the flowdirection in the wellbore When gas influx occurs at thebottom of well some gas invades into the wellbore andmigrates form the bottomhole to the wellhead along the flowdirection At low gas influx rate it is extremely obvious that
Mathematical Problems in Engineering 9
Start
Initial conditions
Grid division
Solve equation (21) for temperature
Calculate k1 minus k4Pi+1 = Pi + ΔP
zi+1 = zi + h
Solve equation (12) for 120588m
Solve equation (14) for c
Determine the time and the step
Enter the pipe velocity
Spline smoothing
Satisfy the boundary conditions and initial conditions
Δt rarr 0
Analysis conditions at the end of pipe
Solve pressure and flow at boundary points
Solve pressure and flow in pipe points
T = T + Δt
NoT lt Tmax
YesEnd
Figure 4 Solution procedure for surge pressure in MPD operations
80 16 24 32 40
00
05
10
15
20
Valu
e
t (s)
Pipe velocity (ms)This model (MPa) Experimental data (MPa)
Burkhardt data (MPa)
Figure 5 Experimental verification by comparison with previousexperiment data
the void fraction and wave velocity first slightly change in acomparatively smooth value then change sharply It is becausethe volume of gas expanded rapidly with the decreasingof pressure near the wellhead the void fraction increasessharply and at the same time the wave velocity decreasesobviously At a high gas influx rate the wave velocity tends toincrease because the void fraction in thewellhead is increasedto a high extent In conclusion the wave velocity is sensitiveto the void fraction and the void fraction is dominated by
MudMotor
Choke
z
DAPC
Casing
Drill pipeAnnulus
Drill collar
Formation
RCD0
800m
1200m
4000m
1778mm 78 mm ) times G105
2159mm times G105120601
120601 (120601
1270mm 78 mm ) times G105120601 (120601
Figure 6 Schematic diagram of gas influx process in tripping
influx rate and pressure in the wellbore especially the influxrate
As shown in Figure 9 within the range of low gas influxrate the wave velocity decreases significantly It is becausethe compressibility of the gas increases remarkably andthe medium appears high elasticity though the density ofgas-drilling mud two-phase flow changes slightly With theincrease of gas influx rate and corresponding increase of thevoid fraction in the wellbore the compressibility of the two-phase unceasingly increases which promotes themomentumand energy exchange in the interface Therefore the wavevelocity continuously decreases
10 Mathematical Problems in EngineeringVa
lue
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
minus05
minus10
Pipe acceleration (ms2)Pipe velocity (ms)Fluid velocity (ms)
Figure 7 Drill pipe velocity variation at different accelerations
0 800 1600 2400 3200 4000H (m)
100
80
60
40
20
0
minus20
G(
)
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
120601
Figure 8 Void fraction distribution at different gas influx rates
0 800 1600 2400 3200 4000H (m)
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
1500
1200
900
600
300
0
c (m
s)
Figure 9 Wave velocity variation at different gas influxes
0 10 20 30 40 50 60
25
20
15
10
05
00
minus05
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
P (M
Pa)
t (s)
Figure 10 Surge pressure variation at different gas influxes
Figure 10 illustrates the effect of gas influx on the surgepressure with the change of gas influx rate As shown inFigure 10 under different gas influx rates (0m3h 2427m3hand 8312m3h resp) the maximum surge pressure mayoccur before drill pipe reaches bottomhole With theincreases of the gas influx the corresponding surge pressuredecreases Considering the gas influx in the calculation of thesurge pressure in the wellbore the surge pressure weakensobviously As the gas influx rate decreases the surge pressureat bottom of the wellbore is reduced gradually while surgepressure can cause a significant time lag if the gas influxoccurs at the bottom of well The lag time is mainly affectedby wave velocity
Figure 11 presents a change of surge pressure in thewellbore at different scenario (OPOP CPOP and OPWPresp) According to Figure 11 surge pressure is increasedin CPOP scenario by comparing with the surge pressure inOPOP scenarios It is because the speed of the drilling mudincreases remarkably in CPOP scenario the surge pressurebecomes larger obviously Especially when the maximumsurge pressure occurs a strong increase of surge pressure isobserved at OPWP scenario If the movement of drill pipeis stop immediately the surge pressure will not disappearimmediately and the surge pressure will exist and changeslightly for short while
Figures 12 and 13 show that the surge pressure varies withwellbore diameter caused by the drill pipe operating speedaccording to Figure 7 The surge pressure decreases with theincrease of wellbore diameter It is because the drilling mudflow rate decreases with the increases of wellbore diameterBased on (1) and (2) surge pressure decreases at each positiondue to the decrease of drilling mud flow rate Surge pressurecaused by converting of potential energy into kinetic energyand then kinetic energy back into potential energy over andover again During the tripping operations especially in slimwellbore drilling process the speed of drill pipe movementshould be appropriately reduced The surge pressure in
Mathematical Problems in Engineering 11
OPWPCPOPOPOP
t (s)0 10 20 30 40 50 60
16
12
08
04
00
P (M
Pa)
Qg = 2427m3h
Figure 11 Surge pressure variation at different scenarios
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 2427m3h
Figure 12 Surge pressure variation at different wellbore diameters(gas influx 119876
119892= 2427m3h)
Figure 13 is prominently larger than that in Figure 12 Thisis because both the decrease of the wave velocity and theincrease of the compression in gas drilling mud two-phasedue to the gas influx result in the comparatively large surgepressure
Figure 14 illustrates that the surge pressure varied withthe length of drill pipe The surge pressure caused by shorterdrill pipe (whose the total length is 1000m) is smaller thanthe longer drill pipe (whose total length is 3000m) beforethe maximum surge pressure occurs This can explain thatsurge pressure is affected by the friction force of drill pipe andwellbore Compared with the longer drill pipe the frictionforce caused by shorter drill pipe is largerTherefore the surgepressure gradually decreases As a result the surge pressure isgreater at 1000m
Figure 15 illustrates that the surge pressure varied withthe drilling fluid density At the bottom of well the surge
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 0m3h
Figure 13 Surge pressure at different borehole diameters (gas influx119876119892= 0m3h)
t (s)0 10 20 30 40 50 60
P (M
Pa)
16
12
08
04
00
Qg = 2427m3h
L = 1000mL = 3000m
Figure 14 Surge pressure variation at different well depths
pressure increases with the increase of drilling fluid densityThe changes are significant in the maximum surge pressureThis is because the surge pressure is mainly affected by wavevelocity density and velocity of gas drilling mud two-phaseAccording to (6) and (7) the surge pressure is increased as thedrilling fluid density increasesThemaximum surge pressureoccurs before drill pipe reaches bottomhole
52 The Effects of Gas Influx Rate on Surge Pressure Figures16 17 and 18 illustrate that the surge pressure varied withthe increase of the gas influx rate Figure 17 shows the surgepressure with the gas influx rate of 2427m3h at the bottomof well Figure 18 shows the surge pressure with the gas influxrate of 8312m3h at the bottom of well Mainly affected bywave velocity density and velocity in gas-drilling mud twophase the surge pressure is greatest at the gas influx rate of0m3h Figure 16 illustrates an example of a minimum wavevelocity at the gas influx rate of 8312m3h at the bottom
12 Mathematical Problems in Engineering
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
Qg = 2427m3h
= 1500kgm3=
= 2000 kgm3=
120588m120588m
Figure 15 Surge pressure variation at different drilling fluid densi-ties
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 0m3h
Figure 16 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 0m3h)
of well As the gas influx rate increases the surge pressuredecreases When gas influx occurs the lag time is lengthenedremarkably compared with the condition of no gas influxThe longer the distance is the greater the lag time will beThe lag time is mainly determined by the propagation time ofpressure wave in the wellbore
Figures 19 and 20 illustrate the surge pressure caused bydifferent operating speeds of the drill pipe at various gasinflux rates at the bottom of well As drilling mudmixed withinflux gas not only reduces the pressure wave velocity butalso reduces the density of themixture density surge pressuremainly affected by wave velocity andmixture density reducesdrastically with the increase of gas influx rate at the bottomof well Figures 19 and 20 are compared to Figures 17 and 18
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 2427m3h
Figure 17 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 8312m3h
Figure 18 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 8312m3h)
it can be concluded that the operating speed is also a majorfactor of surge pressure
53The Effects of Drill Pipe Operating Speed on Surge PressureFigure 21 illustrates the effect of drill pipe speed on the surgepressure The change of surge pressure is smaller in gasdrilling two-phase than in single-phase drilling mud Thesurge pressure becomes smaller and closes to zero after themovement is stopped due to greater friction force existing inwell wall during drill pipe operating
6 Conclusions
With full consideration of the important factors such as wavevelocity in real-time the compressibility of the gas phasegas influx rate pressure temperature and well parametersa new surge pressure model has been proposed based on the
Mathematical Problems in Engineering 13Va
lue
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 2427m3h
Figure 19 Surge pressure and fluid velocity variation at differentpipe velocities (gas influx 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 8312m3h
Figure 20 Surge pressure and fluid velocity variation at differentgas influxes (gas influx 119876
119892= 8312m3h)
30
24
18
12
06
00
Valu
e
t (s)0 10 20 30 40 50 60
Pipe velocity (ms)Pipe velocity (ms)Surge pressure (MPa)
Surge pressure (MPa)
Qg = 2427m3h
Figure 21 Surge pressure variations at different pipe velocities (gasinflux 119876
119892= 2427m3h)
mass conservation equations the momentum conservationequations characteristic line method and the gas-drillingmud equations of state (EOS) duringMPDoperations Solvedby the difference method and fourth-order explicit Runge-Kutta method the model is used to predict surge pressurefor different drill pipe operating speed and gas influx rates inMPD operations The main conclusions can be summarizedas follows
(1) An accurate mathematical model to predict surgepressure after gas influx is of great importancebecause of the effects of surge pressure on operationalsafety and success The new model provides betterprediction of surge pressures in comparison with theexisting experiment
(2) The surge pressure tends to increase with the increaseof drill pipe operating speed and decrease of gas influxrate andwell diameter Considering that the gas influxoccurs the surge pressure weakens obviously Withthe increases in well depth the maximum allowabletripping speed gradually become smaller the surgepressure increases at the bottom of well and drillingsafety window is accordingly narrower The surgepressure can cause a significant time lag if the gasinflux occurs at the bottom of well and the timelag is mainly determined by the propagation time ofpressure wave in the wellbore
(3) The surge pressure obviously varies at different sce-narios (OPOP CPOP and OPWP resp) duringthe tripping operations Surge pressure increased inCPOP scenario than in OPOP scenario Especiallywhen the maximum surge pressure occurs a strongincrease of surge pressure is observed at OPWPscenario By running the new surge pressure modeldrilling engineers can identify potential surge prob-lems and optimize tripping operations for differentscenarios as to avoid complicated accidents to happenin wellbore during drilling operations
(4) The calculation of surge pressure in gas drillingmud two-phase is more complex than conventionalcalculation in single drillingmudDrill pipe operatingspeed in drilling operations should be strictly con-trolled and the surge pressure should be consideredin the design The greatest advantages of new modelare the ability to reduce nonproductive time (NPT) indrilling
Nomenclature
A Wellbore effective cross-area (m2)1198600 The outer drill pipe cross-area (m2)
1198602 Wellbore cross-area (m2)
1198603 The inner Drill pipe cross-area (m2)
119860119890 Effective cross-area of jet nozzle (m2)
119888 Wave velocity (ms)119889119904 One discrete length of wellbore (m)
14 Mathematical Problems in Engineering
119863 Wellbore effective diameter (m)1198631 The inner diameter of casing (m)
1198632 The outer diameter of casing (m)
1198633 The inner diameter of drill pipe (m)
1198634 The outer diameter of drill pipe (m)
119863119896119895 Differential equation coefficients
119864119896119895 Differential equation coefficients
119864119904 Modulus of elasticity of the drill pipe
(MPa)119864119891 Modulus of elasticity of formation (MPa)
1198642 Modulus of elasticity of casing (MPa)
119891119866 Shear stresses coefficient of gas interface
119891 Friction coefficient1198912 Wellbore friction coefficient
1198913 Drill pipe friction coefficient
119891119871 Shear stresses coefficient of drilling mud
119865119911 Pressure applied on the two-phase flow
(N)119892 Acceleration due to gravity (m2s)119892119888 Conversion factor
119892119879 Geothermal temperature gradient (Km)
119896119890 Wellbore effective roughness (m)
1198960 Roughness of outer pipe (m)
119896119894 Roughness of inner pipe (m)
119867 The depth of well (m)1198711 The length from pipe bottom to hole
bottom (m)1198712 The length from pipe bottom to well
head (m)119871119892 Flow pattern119898 Hydraulic radius (m)1198982 Wellbore hydraulic radius (m)
1198983 Drill pipe hydraulic radius (m)
119873 Parameter combiningThompson energyeffects
119875 Pressure (MPa)1198750 Initial pressure of wellbore (MPa)
1198752 Pressure within the empty wellbore
(MPa)1198753 Pressure within the drill pipe (MPa)
119875119888 Critical pressure (kPa)
119875119871 Pressure of drilling mud (MPa)
119875119903 Reduced pressure
119875119895119905 Pressure at node (119895 119905) (MPa)
119902119898 Volumetric flow rate of two phase (m3s)
119902119866 Velocity of the gas (m3s)
119902119871 Velocity of the drilling mud (m3s)
119876119871 Mud flow rate within the wellbore(m3h)
1198761 Drilling mud flow rate within hole(m3h)
1198762 Mud flow rate within the wellbore(m3h)
1198763 Mud flow rate within the drill pipe(m3h)
119876119908 Pump flow rate (m3s)
119876119895119905 Flow rate at time 119905 and nod 119895 (m3s)
119877 Constant of EOS (JKgK)
119904 The length of wellbore along wellbore(m)
119905 Time (s)119879 Temperature (K)119879119888 Critical temperature (K)
119879119903 Reduced temperature
119879119890119894 Undistributed temperature at a depth
(K)119879119890119887ℎ
Undistributed temperature of wellhead(K)
119879119891119887ℎ
Undistributed temperature at bottom-hole (K)
V Velocity of drilling mud and gas (ms)V119901 Velocity of drill pipe (ms)
V119898 Gas and drilling mud flow velocity (ms)
V119866 Gas flow velocity (ms)
V119904 Slip velocity (ms)
V119904119892 Superficial gas velocity (ms)
V119871 Drilling mud flow velocity (ms)
119882119898 Mass flow velocity (kgm3)
119911119866 Gas deviation factor
1199110 Initial calculation height of wellbore (m)
119911119887ℎ Total well depth from surface (m)
119911 Variable of distance from surface (m)
Greek Letters
120588119866 Gas density (kgm3)
120591119891 Frictional pressure gradient (Pam)
120591119908 Frictional pressure coefficient
120588119898 Gas and drilling density (kgm3)
1205880 Density under atmospheric pressure (kgm3)
120588119871 Drilling mud density (kgm3)
120590119904 Surface tension (Nm2)
120601119866 Gas void fraction
120601119871 Drilling mud holdup
120588119903 Reduced density
120583119871 Viscosity of drilling mud (Pasdots)
120579 The ground angle (rad)120583119891 Poissonrsquos ratio of formation
120583119904 Poissonrsquos ratio of drill pipe
1205832 Poissonrsquos ratio of casing
Δ119860 The change of the across areas (m2)
Subscripts
CPOP Closed pipe without pumpingDAPC Dynamic annular pressure controlEOS Equations of stateERV Relief-coupling valveMPD Managed pressure drillingOPOP Open pipe without pumpingOPWP Open pipe with pumpingR-K4 The fourth-order explicit Runge-KuttaRCD The rotating control device
Mathematical Problems in Engineering 15
Subscripts of Graph
119888 Wave velocity in two-phase flow (ms)119863 Wellbore effective diameter (m)119867 Depth of wellbore (m)119871 The length from drill pipe and drill pipe (m)119876119892 Gas influx rate at the bottomhole (m3h)
120601119866 Void fraction ()
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 51274170 andQC2012C128) Without their support this work would nothave been possible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of AirInjection in Annuli to manage pressure during CementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling Conferenceand Exhibition San Diego Calif USA March 2012
[4] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the Norh Africa Technical Conference andExhibition Cairo Egypt February 2012
[5] J Saponja A Adeleye and B Hucik ldquoManaged-pressuredrilling (MPD) field trials demonstrate technology valuerdquo inProceedings of the IADCSPE Drilling Conference Miami FlaUSA February 2006
[6] G Liu ldquoPredicting surge pressures that result from runninglinersrdquoWorld Oil vol 222 no 4 p 92 2001
[7] K Zhou J Yang B Zhong and L Tang ldquoSurge pressure inwellborerdquo Oil Drilling and Production Technology vol 12 no 2pp 1ndash10 99 1990
[8] R S Rudi Rubiandini ldquoNew formula of surge pressure fordetermining safe trip velocitiesrdquo in Proceedings of the SPE AsiaPacific Oil and Gas Conference and Exhibition pp 16ndash18 ausOctober 2000
[9] J Li Z Huang K Li and W Wu ldquoThe influence of collaron surge pressure caused by the drilling fluid viscous forceunder pumping conditionrdquoResearch Journal of Applied SciencesEngineering and Technology vol 5 no 5 pp 1781ndash1785 2040ndash7459 2013
[10] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvancedMaterials Research vol 468ndash471 pp 1736ndash1742 2012
[11] F Crespo and R Ahmed ldquoSurge-and-swab pressure predictionsfor yield-power-law drilling fluidsrdquo SPE Drilling amp Completionvol 27 no 4 pp 574ndash585 2012
[12] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[13] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquo PHDThesis vol 50 no 9 p 143 1989
[14] H Fan ldquoDevelopment of application software for dynamicsurge pressure while trippingrdquo Petroleum Drilling and Technol-ogy vol 23 no 4 pp 11ndash14 1995
[15] K Zhou J Yang B Zhong and L Tang ldquoSurge pressurein wellborerdquo Journal of Petroleum Exploration and ProductionTechnology vol 12 no 2 pp 1ndash10 1990
[16] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquoDissertation Abstracts International Bvol 50 no 9 1989
[17] Z Jiang ldquoStudies of wellbore surge pressure by a dynamicmethodrdquo Journal of Petroleum Exploration and ProductionTechnology vol 13 no 3 pp 39ndash44 97 1991
[18] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[19] D Johnston M Bracken and J Thornton ldquoAssessing pipecondition useful remaining life and maximum operating andsurge pressures when to change the pipe or control thepressurerdquoMagazine of the International Water Association vol14 no 5 pp 43ndash44 2012
[20] H Fan Y Chu and X Liu ldquoPrediction for wellbore dynamicsurge pressure while tripping a drill piperdquo Journal of ChinaUniversity of Petroleum vol 19 no 5 pp 36ndash41 1995
[21] C K Burgess J E Miller and K D Vondervort ldquoDownholesurge pressure reduction system and method of userdquo World9848 143 1998
[22] J P Allamon J E Miller and A M Macfarlane ldquoDownholeSurge Reduction Method and Apparatusrdquo 2004
[23] K S Bjoslashrkevoll R Rommetveit B Aas H Gjeraldstveit andA Merlo ldquoTransient gel breaking model for critical wellsapplications with field data verificationrdquo in Proceedings of the2003 Offshore Mediterranean Conference February 2003
[24] G H Nygaard E Johannessen J E Gravdal and F IversenldquoAutomatic coordinated control of pump rates and choke valvefor compensating pressure fluctuations during surge and swaboperationsrdquo in Proceedings of the IADCSPE Managed PressureDrilling and Underbalanced Operations Conference pp 120ndash125March 2007
[25] J V Fedevjcyk S L M Junqueira and C O R Negrao ldquoSurgeamp swab pressures in wells with cross-section changesrdquo BoletimTecnico da Producao de Petroleo vol 4 no 2 pp 221ndash234 2009
[26] S Tian G H Medley and C R Stone ldquoUnderstandingyield point effect onpressure surges critical to managing deepdifficultMPDwellsrdquoDrilling Contractor vol 65 no 4 p 46 4850 53 2009
[27] Y Sun Q Li and J Zhao ldquoNew method of predicting surgepressure apply to horizontal well based on caisson flowrdquoNaturalScience vol 2 no 12 pp 1394ndash1399 2150ndash4091 2010
[28] B T Lebele-Alawa and F E Oparadike ldquoAnalysis of the effectsof valve propagated pressure surge on pipe flowrdquo Engineeringvol 3 no 11 pp 1098ndash1101 2011
[29] X Kong Q Yuan and Y Qiu ldquoThe research of two-phasefluctuation pressure in wellbore tripping operationrdquo InnerMongolia Petrochemical Industry TE242 2011
[30] F Crespo and R Ahmed ldquoA simplified surge and swab pressuremodel for yield power law fluidsrdquo Journal of Petroleum Scienceand Engineering vol 101 pp 12ndash20 2013
16 Mathematical Problems in Engineering
[31] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
[32] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[33] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86-87 pp 127ndash136 2012
[34] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[35] M J SanchezComparison of Correlations for Predicting PressureLosses in Vertical Multiphase Annular Flow University of TulsaTulsa Okla USA 1972
[36] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[37] Y Lin X Kong Y Qiu and Q Yuan ldquoCalculation analysis ofpressure wave velocity in gas and drilling mud two-phase fluidin annulus during drilling operationsrdquo Mathematical Problemsin Engineering vol 2013 Article ID 318912 17 pages 2013
[38] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[39] H Junfang ldquoBalanced drilling and well controlrdquo PetroleumIndustry Press vol 1 article 33 1992
[40] W A Bacon Consideration of compressibility effects for applied-back- pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Va USA 2011
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Mathematical Problems in Engineering 7
The boundary conditions of the flow rate and pressure areas follows
1198761(0 119905) + 119876
2(0 119905) = V
119901(119905) (119860
0minus 1198603) + 119876119908(0 119905)
1198763(0 119905) = minusV
119901(119905) 1198603+ 119876119908(0 119905)
1199011(0 119905) = 119901
2(0 119905)
(44)
4 Solution of the United Model
Referring to Figure 4 the variable 119909 is defined as a short pipeand the constraint is 119871
119909lt 119888119909sdot Δ119905 The variables 119910 and 119911 are
short pipe which are made up of wellbore and drill pipe andthe constraint is 119871
119910lt 119888119910sdot Δ119905 119871
119911lt 119888119911sdot Δ119905 The variables
119909 + 1 119910 + 1 and 119911 + 1 are defined as a long pipe and theconstraint is 119871
119909+ 1 lt 119888
119909+ 1 sdot Δ119905 119871
119910+ 1 lt 119888
119910+ 1 sdot Δ119905 119871
119911+ 1 lt
119888119911+ 1 sdot Δ119905 The implicit finite-difference method is applied to
calculate the short pipe and characteristicsmethod is appliedto calculate the long pipe The pressure and flow rate can beobtained at the end of drill pipe as follows
[1199012119905]2minus 120588119898[119888119910+1
119860119910+1
119876119910+1
2119905]
2
minus [1199013119905minusΔ119905
]2minus 120588119898[119888119910+1
119860119910+1
119876119905minusΔ119905
]
2
+120588119898Δ119905
8[119888119910+1
119898119910+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
2
= 0
[1199012119905]3minus 120588119898[119888119911+1
119860119911+1
119876119911+1
2119905]
3
minus [1199013119905minusΔ119905
]3minus 120588119898[119888119911+1
119860119911+1
1198763119905minusΔ119905
]
3
+120588119898Δ119905
8[119888119911+1
119898119911+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
3
= 0
[1199012119905]1minus 120588119898[119888119909+1
119860119909+1
119876119909+1
2119905]
1
minus [1199013119905minusΔ119905
]1minus 120588119898[119888119909+1
119860119909+1
1198763119905minusΔ119905
]
1
+120588119898Δ119905
8[119888119909+1
119898119909+1
1198913119905minusΔ119905
V3119905minusΔ119905
1003816100381610038161003816V3119905minusΔ1199051003816100381610038161003816]
1
= 0
[1199012119905+ 1199011119905]2+ 11986311[119876119910
2119905+ 119876119910
1119905]2
+ 11986312= 0
[1199012119905minus 1199011119905]2+ 11986411[119876119910
2119905minus 119876119910
1119905]2
+ 11986412= 0
[1199012119905+ 1199011119905]3+ 11986321[1198762119905+ 119876119911
1119905]2
+ 11986322= 0
[1199012119905minus 1199011119905]3+ 11986421[1198762119905minus 119876119911
1119905]2
+ 11986422= 0
[1199012119905+ 1199011119905]1+ 11986331[1198762119905minus 119876119909
1119905]1
+ 11986332= 0
[1199012119905minus 1199011119905]1+ 11986431[1198762119905+ 119876119909
1119905]1
+ 11986432= 0
1
2
3
i minus 1
i
i + 1
N minus 2
N minus 1
N
Bottomhole
Wellhead
3
3
2
1
2
3
c+y+1cz+1
Δsx+1
Δsx+1
Ly = Lz
Lx
cminusx+1
z
Figure 3 Schematic diagram of the solve
[1198761119905]1+ [1198761119905]2+ [1198761119905]2minus V119901(119905)
sdot (1198600minus 119860119894) = 0
[1199011119905]1minus [1199011119905]3minus
120588119898
2(119906119860119890)2
times [1198761119905]3+ 119860119890V119901(119905)
10038161003816100381610038161003816[1198761119905]3+ 119860119890V119901(119905)
10038161003816100381610038161003816= 0
119876119910+1
2119905minus 119876119910
2119905minus V119901(119905)
sdot Δ119860 = 0
[1199011119905]1= [1199011119905]2
11986311=1205881198981198882
119860sdotΔ119905
119871119869
11986312=120588119898
119860sdot119871119869
119871119869
(45)
The119863119896119895 119864119896119895can be expressed as follows
119863119894119905= minus (119901
119894+1119905minusΔ119905+ 119901119894119905) + 11986311(119876119894+1119905minusΔ119905
minus 119876119894119905minusΔ119905
)
119864119894119905= minus (119901
119894+1119905minusΔ119905+ 119901119894119905) + 11986411(119876119894+1119905minusΔ119905
minus 119876119894119905minusΔ119905
)
+120588119898119891119871119869
16119898(V119894+1119905minusΔ119905
+ V119894119905minusΔ119905
)1003816100381610038161003816(V119894+1119905minusΔ119905 + V
119894119905minusΔ119905)1003816100381610038161003816
(46)
where 119896 = 1 2 3 and 119895 = 1 2Obtaining the analytical solution of the mathematical
models concerned with flow pattern void fraction param-eters and pressure drop gradient are generally impossible formost practical in two-phase flow In this paper the Runge-Kutta method (R-K4) is used to discrete the new model forsurge pressure At different wellbore depths we can obtainpressure temperature gas velocity drilling mud velocity and
8 Mathematical Problems in Engineering
void fraction in MPD operations by application of the R-K4The solution of pressure drop gradient equation (3) can beseen as an initial-value problem of the ordinary differentialequation
119889119901
119889119911= 119865 (119911 119901)
119901 (1199110) = 1198750
(47)
With the initial value (1199110 1199010) and the function 119865(119911 119901)
(3)ndash(5) can be obtained
1198961= 119865 (119911
0 1199010)
1198962= 119865(119911
0+ℎ
2 1199010+ℎ
21198961)
1198963= 119865(119911
0+ℎ
2 1199010+ℎ
21198962)
1198964= 119865 (119911
0+ ℎ 119901
0+ ℎ1198963)
(48)
where ℎ is the step of depthThe pressure on the nod 119894 = 119894 + 1can be obtained by
1199011= 1199010+ Δ119901 = 119901
0+ℎ
6(1198961+ 21198962+ 21198963+ 1198964) (49)
In the present work the surge pressure calculation modelsolved by personally compiled code on VBNET (Version2010) The schematic diagram of solve is shown in Figure 3and the solution procedure for the surge pressure in thewellbore is shown in Figure 4 At initial time the wellheadtemperature wellbore structure well depths gas and drillingmud two-phase properties and so forth are known On thenode 119894 the pressure gradient wave velocity varieties in real-time temperature and the void fraction can be obtained byadopting R-K4 The density of gas can be obtained by gasEOS (15) and the density of drilling mud can be obtained byequations (19) and equations (20) Then the surge pressureat different depths of wellbore in tripping operations can besolved by (45)Theprocess is repeated until the surge pressurein the whole wellbore has been obtained [37 38]
The developed model takes full consideration of the gasinflux and wave velocity in real-time Owing to the complexconditions of wellbore in tripping operations measurementof surge pressure in the actual drilling process is very difficultIn order to verify the new model for surge pressure thepredicted surge pressure is compared with the results ofprevious experimental investigations presented by Junfang intripping operations in Figure 5 [39] The comparisons revealthat the developed new model fits well with the experimentsdata Thus the new model can be used to accurately predictsurge pressure on different scenario and gas influx rate intripping operations The value of surge pressure calculatedby adopted Burkhardtrsquos steady method is larger than theexperiments data
Table 1 Parameters of calculation well
Type Property Value
Mud Dynamic viscosity (Pasdots) 0056Density (kgm3) 1460
Gas Relative density 065Viscosity (Pasdots) 114 times 10minus5
StringElastic modulus of string (Pa) 207 times 1011
Poisson ratio of string 03Roughness (m) 154 times 10minus7
Surface condition Surface temperature (K) 298Atmosphere pressure (MPa) 0101
5 Analysis and Discussion
The drilling system described is a closed system Theschematic diagram of gas influx process in tripping opera-tions is illustrated in Figure 6 The drilling mud is pumpedfrom surface storage down the drill pipe returns from theannulus and travels back through surface processing wheredrilling solids are removed to surface storage
The well used for calculation is a gas well in XinjiangUygur Autonomous Region China The wellbore structurewell design parameters (depths and diameters) gas anddrilling mud two-phase properties (density and viscosity)and operational conditions of calculation well are displayedin Table 1 [40]
The drilling mud mixed with gas is considered as a two-phase flow medium The surge pressure in the gas-drillingmud are calculated and discussed by using the establishedmodel and well parameters As shown in Figure 7 accordingto the drill conditions such as the drill pipe velocity gasdrilling mud two-phase flow velocity the total length of drillpipe and drill pipe is 1000mwe can account for the changingrules of surge pressureThe following examples illustrate thatthe surge pressure is affected by different well parametersdrilling mud density gas influx rate and well depth in thetripping operations
51 The Effects of Various Drilling Parameters on SurgePressure Figure 7 presents the influence of acceleration ondrilling mud velocity drilling mud velocity and drill pipeoperating speed with respect to the drilling parameters Themovement of drill pipe starts at the depth of 1000m Withthe movement of the drill pipe the drilling mud also movesin the opposite direction along wellbore at the same timeThe drill pipe operating speed reaches the maximum valueat 23 seconds and stops at 30 seconds In order to make acomparison of the surge pressure in different well parametersthe analysis and compared Figures 8ndash15 based on the data ofFigure 7
Figures 8 and 9 graphically show the distributions ofvoid fraction and variations of wave velocity along the flowdirection in the wellbore When gas influx occurs at thebottom of well some gas invades into the wellbore andmigrates form the bottomhole to the wellhead along the flowdirection At low gas influx rate it is extremely obvious that
Mathematical Problems in Engineering 9
Start
Initial conditions
Grid division
Solve equation (21) for temperature
Calculate k1 minus k4Pi+1 = Pi + ΔP
zi+1 = zi + h
Solve equation (12) for 120588m
Solve equation (14) for c
Determine the time and the step
Enter the pipe velocity
Spline smoothing
Satisfy the boundary conditions and initial conditions
Δt rarr 0
Analysis conditions at the end of pipe
Solve pressure and flow at boundary points
Solve pressure and flow in pipe points
T = T + Δt
NoT lt Tmax
YesEnd
Figure 4 Solution procedure for surge pressure in MPD operations
80 16 24 32 40
00
05
10
15
20
Valu
e
t (s)
Pipe velocity (ms)This model (MPa) Experimental data (MPa)
Burkhardt data (MPa)
Figure 5 Experimental verification by comparison with previousexperiment data
the void fraction and wave velocity first slightly change in acomparatively smooth value then change sharply It is becausethe volume of gas expanded rapidly with the decreasingof pressure near the wellhead the void fraction increasessharply and at the same time the wave velocity decreasesobviously At a high gas influx rate the wave velocity tends toincrease because the void fraction in thewellhead is increasedto a high extent In conclusion the wave velocity is sensitiveto the void fraction and the void fraction is dominated by
MudMotor
Choke
z
DAPC
Casing
Drill pipeAnnulus
Drill collar
Formation
RCD0
800m
1200m
4000m
1778mm 78 mm ) times G105
2159mm times G105120601
120601 (120601
1270mm 78 mm ) times G105120601 (120601
Figure 6 Schematic diagram of gas influx process in tripping
influx rate and pressure in the wellbore especially the influxrate
As shown in Figure 9 within the range of low gas influxrate the wave velocity decreases significantly It is becausethe compressibility of the gas increases remarkably andthe medium appears high elasticity though the density ofgas-drilling mud two-phase flow changes slightly With theincrease of gas influx rate and corresponding increase of thevoid fraction in the wellbore the compressibility of the two-phase unceasingly increases which promotes themomentumand energy exchange in the interface Therefore the wavevelocity continuously decreases
10 Mathematical Problems in EngineeringVa
lue
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
minus05
minus10
Pipe acceleration (ms2)Pipe velocity (ms)Fluid velocity (ms)
Figure 7 Drill pipe velocity variation at different accelerations
0 800 1600 2400 3200 4000H (m)
100
80
60
40
20
0
minus20
G(
)
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
120601
Figure 8 Void fraction distribution at different gas influx rates
0 800 1600 2400 3200 4000H (m)
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
1500
1200
900
600
300
0
c (m
s)
Figure 9 Wave velocity variation at different gas influxes
0 10 20 30 40 50 60
25
20
15
10
05
00
minus05
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
P (M
Pa)
t (s)
Figure 10 Surge pressure variation at different gas influxes
Figure 10 illustrates the effect of gas influx on the surgepressure with the change of gas influx rate As shown inFigure 10 under different gas influx rates (0m3h 2427m3hand 8312m3h resp) the maximum surge pressure mayoccur before drill pipe reaches bottomhole With theincreases of the gas influx the corresponding surge pressuredecreases Considering the gas influx in the calculation of thesurge pressure in the wellbore the surge pressure weakensobviously As the gas influx rate decreases the surge pressureat bottom of the wellbore is reduced gradually while surgepressure can cause a significant time lag if the gas influxoccurs at the bottom of well The lag time is mainly affectedby wave velocity
Figure 11 presents a change of surge pressure in thewellbore at different scenario (OPOP CPOP and OPWPresp) According to Figure 11 surge pressure is increasedin CPOP scenario by comparing with the surge pressure inOPOP scenarios It is because the speed of the drilling mudincreases remarkably in CPOP scenario the surge pressurebecomes larger obviously Especially when the maximumsurge pressure occurs a strong increase of surge pressure isobserved at OPWP scenario If the movement of drill pipeis stop immediately the surge pressure will not disappearimmediately and the surge pressure will exist and changeslightly for short while
Figures 12 and 13 show that the surge pressure varies withwellbore diameter caused by the drill pipe operating speedaccording to Figure 7 The surge pressure decreases with theincrease of wellbore diameter It is because the drilling mudflow rate decreases with the increases of wellbore diameterBased on (1) and (2) surge pressure decreases at each positiondue to the decrease of drilling mud flow rate Surge pressurecaused by converting of potential energy into kinetic energyand then kinetic energy back into potential energy over andover again During the tripping operations especially in slimwellbore drilling process the speed of drill pipe movementshould be appropriately reduced The surge pressure in
Mathematical Problems in Engineering 11
OPWPCPOPOPOP
t (s)0 10 20 30 40 50 60
16
12
08
04
00
P (M
Pa)
Qg = 2427m3h
Figure 11 Surge pressure variation at different scenarios
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 2427m3h
Figure 12 Surge pressure variation at different wellbore diameters(gas influx 119876
119892= 2427m3h)
Figure 13 is prominently larger than that in Figure 12 Thisis because both the decrease of the wave velocity and theincrease of the compression in gas drilling mud two-phasedue to the gas influx result in the comparatively large surgepressure
Figure 14 illustrates that the surge pressure varied withthe length of drill pipe The surge pressure caused by shorterdrill pipe (whose the total length is 1000m) is smaller thanthe longer drill pipe (whose total length is 3000m) beforethe maximum surge pressure occurs This can explain thatsurge pressure is affected by the friction force of drill pipe andwellbore Compared with the longer drill pipe the frictionforce caused by shorter drill pipe is largerTherefore the surgepressure gradually decreases As a result the surge pressure isgreater at 1000m
Figure 15 illustrates that the surge pressure varied withthe drilling fluid density At the bottom of well the surge
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 0m3h
Figure 13 Surge pressure at different borehole diameters (gas influx119876119892= 0m3h)
t (s)0 10 20 30 40 50 60
P (M
Pa)
16
12
08
04
00
Qg = 2427m3h
L = 1000mL = 3000m
Figure 14 Surge pressure variation at different well depths
pressure increases with the increase of drilling fluid densityThe changes are significant in the maximum surge pressureThis is because the surge pressure is mainly affected by wavevelocity density and velocity of gas drilling mud two-phaseAccording to (6) and (7) the surge pressure is increased as thedrilling fluid density increasesThemaximum surge pressureoccurs before drill pipe reaches bottomhole
52 The Effects of Gas Influx Rate on Surge Pressure Figures16 17 and 18 illustrate that the surge pressure varied withthe increase of the gas influx rate Figure 17 shows the surgepressure with the gas influx rate of 2427m3h at the bottomof well Figure 18 shows the surge pressure with the gas influxrate of 8312m3h at the bottom of well Mainly affected bywave velocity density and velocity in gas-drilling mud twophase the surge pressure is greatest at the gas influx rate of0m3h Figure 16 illustrates an example of a minimum wavevelocity at the gas influx rate of 8312m3h at the bottom
12 Mathematical Problems in Engineering
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
Qg = 2427m3h
= 1500kgm3=
= 2000 kgm3=
120588m120588m
Figure 15 Surge pressure variation at different drilling fluid densi-ties
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 0m3h
Figure 16 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 0m3h)
of well As the gas influx rate increases the surge pressuredecreases When gas influx occurs the lag time is lengthenedremarkably compared with the condition of no gas influxThe longer the distance is the greater the lag time will beThe lag time is mainly determined by the propagation time ofpressure wave in the wellbore
Figures 19 and 20 illustrate the surge pressure caused bydifferent operating speeds of the drill pipe at various gasinflux rates at the bottom of well As drilling mudmixed withinflux gas not only reduces the pressure wave velocity butalso reduces the density of themixture density surge pressuremainly affected by wave velocity andmixture density reducesdrastically with the increase of gas influx rate at the bottomof well Figures 19 and 20 are compared to Figures 17 and 18
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 2427m3h
Figure 17 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 8312m3h
Figure 18 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 8312m3h)
it can be concluded that the operating speed is also a majorfactor of surge pressure
53The Effects of Drill Pipe Operating Speed on Surge PressureFigure 21 illustrates the effect of drill pipe speed on the surgepressure The change of surge pressure is smaller in gasdrilling two-phase than in single-phase drilling mud Thesurge pressure becomes smaller and closes to zero after themovement is stopped due to greater friction force existing inwell wall during drill pipe operating
6 Conclusions
With full consideration of the important factors such as wavevelocity in real-time the compressibility of the gas phasegas influx rate pressure temperature and well parametersa new surge pressure model has been proposed based on the
Mathematical Problems in Engineering 13Va
lue
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 2427m3h
Figure 19 Surge pressure and fluid velocity variation at differentpipe velocities (gas influx 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 8312m3h
Figure 20 Surge pressure and fluid velocity variation at differentgas influxes (gas influx 119876
119892= 8312m3h)
30
24
18
12
06
00
Valu
e
t (s)0 10 20 30 40 50 60
Pipe velocity (ms)Pipe velocity (ms)Surge pressure (MPa)
Surge pressure (MPa)
Qg = 2427m3h
Figure 21 Surge pressure variations at different pipe velocities (gasinflux 119876
119892= 2427m3h)
mass conservation equations the momentum conservationequations characteristic line method and the gas-drillingmud equations of state (EOS) duringMPDoperations Solvedby the difference method and fourth-order explicit Runge-Kutta method the model is used to predict surge pressurefor different drill pipe operating speed and gas influx rates inMPD operations The main conclusions can be summarizedas follows
(1) An accurate mathematical model to predict surgepressure after gas influx is of great importancebecause of the effects of surge pressure on operationalsafety and success The new model provides betterprediction of surge pressures in comparison with theexisting experiment
(2) The surge pressure tends to increase with the increaseof drill pipe operating speed and decrease of gas influxrate andwell diameter Considering that the gas influxoccurs the surge pressure weakens obviously Withthe increases in well depth the maximum allowabletripping speed gradually become smaller the surgepressure increases at the bottom of well and drillingsafety window is accordingly narrower The surgepressure can cause a significant time lag if the gasinflux occurs at the bottom of well and the timelag is mainly determined by the propagation time ofpressure wave in the wellbore
(3) The surge pressure obviously varies at different sce-narios (OPOP CPOP and OPWP resp) duringthe tripping operations Surge pressure increased inCPOP scenario than in OPOP scenario Especiallywhen the maximum surge pressure occurs a strongincrease of surge pressure is observed at OPWPscenario By running the new surge pressure modeldrilling engineers can identify potential surge prob-lems and optimize tripping operations for differentscenarios as to avoid complicated accidents to happenin wellbore during drilling operations
(4) The calculation of surge pressure in gas drillingmud two-phase is more complex than conventionalcalculation in single drillingmudDrill pipe operatingspeed in drilling operations should be strictly con-trolled and the surge pressure should be consideredin the design The greatest advantages of new modelare the ability to reduce nonproductive time (NPT) indrilling
Nomenclature
A Wellbore effective cross-area (m2)1198600 The outer drill pipe cross-area (m2)
1198602 Wellbore cross-area (m2)
1198603 The inner Drill pipe cross-area (m2)
119860119890 Effective cross-area of jet nozzle (m2)
119888 Wave velocity (ms)119889119904 One discrete length of wellbore (m)
14 Mathematical Problems in Engineering
119863 Wellbore effective diameter (m)1198631 The inner diameter of casing (m)
1198632 The outer diameter of casing (m)
1198633 The inner diameter of drill pipe (m)
1198634 The outer diameter of drill pipe (m)
119863119896119895 Differential equation coefficients
119864119896119895 Differential equation coefficients
119864119904 Modulus of elasticity of the drill pipe
(MPa)119864119891 Modulus of elasticity of formation (MPa)
1198642 Modulus of elasticity of casing (MPa)
119891119866 Shear stresses coefficient of gas interface
119891 Friction coefficient1198912 Wellbore friction coefficient
1198913 Drill pipe friction coefficient
119891119871 Shear stresses coefficient of drilling mud
119865119911 Pressure applied on the two-phase flow
(N)119892 Acceleration due to gravity (m2s)119892119888 Conversion factor
119892119879 Geothermal temperature gradient (Km)
119896119890 Wellbore effective roughness (m)
1198960 Roughness of outer pipe (m)
119896119894 Roughness of inner pipe (m)
119867 The depth of well (m)1198711 The length from pipe bottom to hole
bottom (m)1198712 The length from pipe bottom to well
head (m)119871119892 Flow pattern119898 Hydraulic radius (m)1198982 Wellbore hydraulic radius (m)
1198983 Drill pipe hydraulic radius (m)
119873 Parameter combiningThompson energyeffects
119875 Pressure (MPa)1198750 Initial pressure of wellbore (MPa)
1198752 Pressure within the empty wellbore
(MPa)1198753 Pressure within the drill pipe (MPa)
119875119888 Critical pressure (kPa)
119875119871 Pressure of drilling mud (MPa)
119875119903 Reduced pressure
119875119895119905 Pressure at node (119895 119905) (MPa)
119902119898 Volumetric flow rate of two phase (m3s)
119902119866 Velocity of the gas (m3s)
119902119871 Velocity of the drilling mud (m3s)
119876119871 Mud flow rate within the wellbore(m3h)
1198761 Drilling mud flow rate within hole(m3h)
1198762 Mud flow rate within the wellbore(m3h)
1198763 Mud flow rate within the drill pipe(m3h)
119876119908 Pump flow rate (m3s)
119876119895119905 Flow rate at time 119905 and nod 119895 (m3s)
119877 Constant of EOS (JKgK)
119904 The length of wellbore along wellbore(m)
119905 Time (s)119879 Temperature (K)119879119888 Critical temperature (K)
119879119903 Reduced temperature
119879119890119894 Undistributed temperature at a depth
(K)119879119890119887ℎ
Undistributed temperature of wellhead(K)
119879119891119887ℎ
Undistributed temperature at bottom-hole (K)
V Velocity of drilling mud and gas (ms)V119901 Velocity of drill pipe (ms)
V119898 Gas and drilling mud flow velocity (ms)
V119866 Gas flow velocity (ms)
V119904 Slip velocity (ms)
V119904119892 Superficial gas velocity (ms)
V119871 Drilling mud flow velocity (ms)
119882119898 Mass flow velocity (kgm3)
119911119866 Gas deviation factor
1199110 Initial calculation height of wellbore (m)
119911119887ℎ Total well depth from surface (m)
119911 Variable of distance from surface (m)
Greek Letters
120588119866 Gas density (kgm3)
120591119891 Frictional pressure gradient (Pam)
120591119908 Frictional pressure coefficient
120588119898 Gas and drilling density (kgm3)
1205880 Density under atmospheric pressure (kgm3)
120588119871 Drilling mud density (kgm3)
120590119904 Surface tension (Nm2)
120601119866 Gas void fraction
120601119871 Drilling mud holdup
120588119903 Reduced density
120583119871 Viscosity of drilling mud (Pasdots)
120579 The ground angle (rad)120583119891 Poissonrsquos ratio of formation
120583119904 Poissonrsquos ratio of drill pipe
1205832 Poissonrsquos ratio of casing
Δ119860 The change of the across areas (m2)
Subscripts
CPOP Closed pipe without pumpingDAPC Dynamic annular pressure controlEOS Equations of stateERV Relief-coupling valveMPD Managed pressure drillingOPOP Open pipe without pumpingOPWP Open pipe with pumpingR-K4 The fourth-order explicit Runge-KuttaRCD The rotating control device
Mathematical Problems in Engineering 15
Subscripts of Graph
119888 Wave velocity in two-phase flow (ms)119863 Wellbore effective diameter (m)119867 Depth of wellbore (m)119871 The length from drill pipe and drill pipe (m)119876119892 Gas influx rate at the bottomhole (m3h)
120601119866 Void fraction ()
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 51274170 andQC2012C128) Without their support this work would nothave been possible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of AirInjection in Annuli to manage pressure during CementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling Conferenceand Exhibition San Diego Calif USA March 2012
[4] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the Norh Africa Technical Conference andExhibition Cairo Egypt February 2012
[5] J Saponja A Adeleye and B Hucik ldquoManaged-pressuredrilling (MPD) field trials demonstrate technology valuerdquo inProceedings of the IADCSPE Drilling Conference Miami FlaUSA February 2006
[6] G Liu ldquoPredicting surge pressures that result from runninglinersrdquoWorld Oil vol 222 no 4 p 92 2001
[7] K Zhou J Yang B Zhong and L Tang ldquoSurge pressure inwellborerdquo Oil Drilling and Production Technology vol 12 no 2pp 1ndash10 99 1990
[8] R S Rudi Rubiandini ldquoNew formula of surge pressure fordetermining safe trip velocitiesrdquo in Proceedings of the SPE AsiaPacific Oil and Gas Conference and Exhibition pp 16ndash18 ausOctober 2000
[9] J Li Z Huang K Li and W Wu ldquoThe influence of collaron surge pressure caused by the drilling fluid viscous forceunder pumping conditionrdquoResearch Journal of Applied SciencesEngineering and Technology vol 5 no 5 pp 1781ndash1785 2040ndash7459 2013
[10] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvancedMaterials Research vol 468ndash471 pp 1736ndash1742 2012
[11] F Crespo and R Ahmed ldquoSurge-and-swab pressure predictionsfor yield-power-law drilling fluidsrdquo SPE Drilling amp Completionvol 27 no 4 pp 574ndash585 2012
[12] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[13] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquo PHDThesis vol 50 no 9 p 143 1989
[14] H Fan ldquoDevelopment of application software for dynamicsurge pressure while trippingrdquo Petroleum Drilling and Technol-ogy vol 23 no 4 pp 11ndash14 1995
[15] K Zhou J Yang B Zhong and L Tang ldquoSurge pressurein wellborerdquo Journal of Petroleum Exploration and ProductionTechnology vol 12 no 2 pp 1ndash10 1990
[16] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquoDissertation Abstracts International Bvol 50 no 9 1989
[17] Z Jiang ldquoStudies of wellbore surge pressure by a dynamicmethodrdquo Journal of Petroleum Exploration and ProductionTechnology vol 13 no 3 pp 39ndash44 97 1991
[18] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[19] D Johnston M Bracken and J Thornton ldquoAssessing pipecondition useful remaining life and maximum operating andsurge pressures when to change the pipe or control thepressurerdquoMagazine of the International Water Association vol14 no 5 pp 43ndash44 2012
[20] H Fan Y Chu and X Liu ldquoPrediction for wellbore dynamicsurge pressure while tripping a drill piperdquo Journal of ChinaUniversity of Petroleum vol 19 no 5 pp 36ndash41 1995
[21] C K Burgess J E Miller and K D Vondervort ldquoDownholesurge pressure reduction system and method of userdquo World9848 143 1998
[22] J P Allamon J E Miller and A M Macfarlane ldquoDownholeSurge Reduction Method and Apparatusrdquo 2004
[23] K S Bjoslashrkevoll R Rommetveit B Aas H Gjeraldstveit andA Merlo ldquoTransient gel breaking model for critical wellsapplications with field data verificationrdquo in Proceedings of the2003 Offshore Mediterranean Conference February 2003
[24] G H Nygaard E Johannessen J E Gravdal and F IversenldquoAutomatic coordinated control of pump rates and choke valvefor compensating pressure fluctuations during surge and swaboperationsrdquo in Proceedings of the IADCSPE Managed PressureDrilling and Underbalanced Operations Conference pp 120ndash125March 2007
[25] J V Fedevjcyk S L M Junqueira and C O R Negrao ldquoSurgeamp swab pressures in wells with cross-section changesrdquo BoletimTecnico da Producao de Petroleo vol 4 no 2 pp 221ndash234 2009
[26] S Tian G H Medley and C R Stone ldquoUnderstandingyield point effect onpressure surges critical to managing deepdifficultMPDwellsrdquoDrilling Contractor vol 65 no 4 p 46 4850 53 2009
[27] Y Sun Q Li and J Zhao ldquoNew method of predicting surgepressure apply to horizontal well based on caisson flowrdquoNaturalScience vol 2 no 12 pp 1394ndash1399 2150ndash4091 2010
[28] B T Lebele-Alawa and F E Oparadike ldquoAnalysis of the effectsof valve propagated pressure surge on pipe flowrdquo Engineeringvol 3 no 11 pp 1098ndash1101 2011
[29] X Kong Q Yuan and Y Qiu ldquoThe research of two-phasefluctuation pressure in wellbore tripping operationrdquo InnerMongolia Petrochemical Industry TE242 2011
[30] F Crespo and R Ahmed ldquoA simplified surge and swab pressuremodel for yield power law fluidsrdquo Journal of Petroleum Scienceand Engineering vol 101 pp 12ndash20 2013
16 Mathematical Problems in Engineering
[31] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
[32] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[33] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86-87 pp 127ndash136 2012
[34] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[35] M J SanchezComparison of Correlations for Predicting PressureLosses in Vertical Multiphase Annular Flow University of TulsaTulsa Okla USA 1972
[36] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[37] Y Lin X Kong Y Qiu and Q Yuan ldquoCalculation analysis ofpressure wave velocity in gas and drilling mud two-phase fluidin annulus during drilling operationsrdquo Mathematical Problemsin Engineering vol 2013 Article ID 318912 17 pages 2013
[38] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[39] H Junfang ldquoBalanced drilling and well controlrdquo PetroleumIndustry Press vol 1 article 33 1992
[40] W A Bacon Consideration of compressibility effects for applied-back- pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Va USA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
void fraction in MPD operations by application of the R-K4The solution of pressure drop gradient equation (3) can beseen as an initial-value problem of the ordinary differentialequation
119889119901
119889119911= 119865 (119911 119901)
119901 (1199110) = 1198750
(47)
With the initial value (1199110 1199010) and the function 119865(119911 119901)
(3)ndash(5) can be obtained
1198961= 119865 (119911
0 1199010)
1198962= 119865(119911
0+ℎ
2 1199010+ℎ
21198961)
1198963= 119865(119911
0+ℎ
2 1199010+ℎ
21198962)
1198964= 119865 (119911
0+ ℎ 119901
0+ ℎ1198963)
(48)
where ℎ is the step of depthThe pressure on the nod 119894 = 119894 + 1can be obtained by
1199011= 1199010+ Δ119901 = 119901
0+ℎ
6(1198961+ 21198962+ 21198963+ 1198964) (49)
In the present work the surge pressure calculation modelsolved by personally compiled code on VBNET (Version2010) The schematic diagram of solve is shown in Figure 3and the solution procedure for the surge pressure in thewellbore is shown in Figure 4 At initial time the wellheadtemperature wellbore structure well depths gas and drillingmud two-phase properties and so forth are known On thenode 119894 the pressure gradient wave velocity varieties in real-time temperature and the void fraction can be obtained byadopting R-K4 The density of gas can be obtained by gasEOS (15) and the density of drilling mud can be obtained byequations (19) and equations (20) Then the surge pressureat different depths of wellbore in tripping operations can besolved by (45)Theprocess is repeated until the surge pressurein the whole wellbore has been obtained [37 38]
The developed model takes full consideration of the gasinflux and wave velocity in real-time Owing to the complexconditions of wellbore in tripping operations measurementof surge pressure in the actual drilling process is very difficultIn order to verify the new model for surge pressure thepredicted surge pressure is compared with the results ofprevious experimental investigations presented by Junfang intripping operations in Figure 5 [39] The comparisons revealthat the developed new model fits well with the experimentsdata Thus the new model can be used to accurately predictsurge pressure on different scenario and gas influx rate intripping operations The value of surge pressure calculatedby adopted Burkhardtrsquos steady method is larger than theexperiments data
Table 1 Parameters of calculation well
Type Property Value
Mud Dynamic viscosity (Pasdots) 0056Density (kgm3) 1460
Gas Relative density 065Viscosity (Pasdots) 114 times 10minus5
StringElastic modulus of string (Pa) 207 times 1011
Poisson ratio of string 03Roughness (m) 154 times 10minus7
Surface condition Surface temperature (K) 298Atmosphere pressure (MPa) 0101
5 Analysis and Discussion
The drilling system described is a closed system Theschematic diagram of gas influx process in tripping opera-tions is illustrated in Figure 6 The drilling mud is pumpedfrom surface storage down the drill pipe returns from theannulus and travels back through surface processing wheredrilling solids are removed to surface storage
The well used for calculation is a gas well in XinjiangUygur Autonomous Region China The wellbore structurewell design parameters (depths and diameters) gas anddrilling mud two-phase properties (density and viscosity)and operational conditions of calculation well are displayedin Table 1 [40]
The drilling mud mixed with gas is considered as a two-phase flow medium The surge pressure in the gas-drillingmud are calculated and discussed by using the establishedmodel and well parameters As shown in Figure 7 accordingto the drill conditions such as the drill pipe velocity gasdrilling mud two-phase flow velocity the total length of drillpipe and drill pipe is 1000mwe can account for the changingrules of surge pressureThe following examples illustrate thatthe surge pressure is affected by different well parametersdrilling mud density gas influx rate and well depth in thetripping operations
51 The Effects of Various Drilling Parameters on SurgePressure Figure 7 presents the influence of acceleration ondrilling mud velocity drilling mud velocity and drill pipeoperating speed with respect to the drilling parameters Themovement of drill pipe starts at the depth of 1000m Withthe movement of the drill pipe the drilling mud also movesin the opposite direction along wellbore at the same timeThe drill pipe operating speed reaches the maximum valueat 23 seconds and stops at 30 seconds In order to make acomparison of the surge pressure in different well parametersthe analysis and compared Figures 8ndash15 based on the data ofFigure 7
Figures 8 and 9 graphically show the distributions ofvoid fraction and variations of wave velocity along the flowdirection in the wellbore When gas influx occurs at thebottom of well some gas invades into the wellbore andmigrates form the bottomhole to the wellhead along the flowdirection At low gas influx rate it is extremely obvious that
Mathematical Problems in Engineering 9
Start
Initial conditions
Grid division
Solve equation (21) for temperature
Calculate k1 minus k4Pi+1 = Pi + ΔP
zi+1 = zi + h
Solve equation (12) for 120588m
Solve equation (14) for c
Determine the time and the step
Enter the pipe velocity
Spline smoothing
Satisfy the boundary conditions and initial conditions
Δt rarr 0
Analysis conditions at the end of pipe
Solve pressure and flow at boundary points
Solve pressure and flow in pipe points
T = T + Δt
NoT lt Tmax
YesEnd
Figure 4 Solution procedure for surge pressure in MPD operations
80 16 24 32 40
00
05
10
15
20
Valu
e
t (s)
Pipe velocity (ms)This model (MPa) Experimental data (MPa)
Burkhardt data (MPa)
Figure 5 Experimental verification by comparison with previousexperiment data
the void fraction and wave velocity first slightly change in acomparatively smooth value then change sharply It is becausethe volume of gas expanded rapidly with the decreasingof pressure near the wellhead the void fraction increasessharply and at the same time the wave velocity decreasesobviously At a high gas influx rate the wave velocity tends toincrease because the void fraction in thewellhead is increasedto a high extent In conclusion the wave velocity is sensitiveto the void fraction and the void fraction is dominated by
MudMotor
Choke
z
DAPC
Casing
Drill pipeAnnulus
Drill collar
Formation
RCD0
800m
1200m
4000m
1778mm 78 mm ) times G105
2159mm times G105120601
120601 (120601
1270mm 78 mm ) times G105120601 (120601
Figure 6 Schematic diagram of gas influx process in tripping
influx rate and pressure in the wellbore especially the influxrate
As shown in Figure 9 within the range of low gas influxrate the wave velocity decreases significantly It is becausethe compressibility of the gas increases remarkably andthe medium appears high elasticity though the density ofgas-drilling mud two-phase flow changes slightly With theincrease of gas influx rate and corresponding increase of thevoid fraction in the wellbore the compressibility of the two-phase unceasingly increases which promotes themomentumand energy exchange in the interface Therefore the wavevelocity continuously decreases
10 Mathematical Problems in EngineeringVa
lue
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
minus05
minus10
Pipe acceleration (ms2)Pipe velocity (ms)Fluid velocity (ms)
Figure 7 Drill pipe velocity variation at different accelerations
0 800 1600 2400 3200 4000H (m)
100
80
60
40
20
0
minus20
G(
)
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
120601
Figure 8 Void fraction distribution at different gas influx rates
0 800 1600 2400 3200 4000H (m)
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
1500
1200
900
600
300
0
c (m
s)
Figure 9 Wave velocity variation at different gas influxes
0 10 20 30 40 50 60
25
20
15
10
05
00
minus05
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
P (M
Pa)
t (s)
Figure 10 Surge pressure variation at different gas influxes
Figure 10 illustrates the effect of gas influx on the surgepressure with the change of gas influx rate As shown inFigure 10 under different gas influx rates (0m3h 2427m3hand 8312m3h resp) the maximum surge pressure mayoccur before drill pipe reaches bottomhole With theincreases of the gas influx the corresponding surge pressuredecreases Considering the gas influx in the calculation of thesurge pressure in the wellbore the surge pressure weakensobviously As the gas influx rate decreases the surge pressureat bottom of the wellbore is reduced gradually while surgepressure can cause a significant time lag if the gas influxoccurs at the bottom of well The lag time is mainly affectedby wave velocity
Figure 11 presents a change of surge pressure in thewellbore at different scenario (OPOP CPOP and OPWPresp) According to Figure 11 surge pressure is increasedin CPOP scenario by comparing with the surge pressure inOPOP scenarios It is because the speed of the drilling mudincreases remarkably in CPOP scenario the surge pressurebecomes larger obviously Especially when the maximumsurge pressure occurs a strong increase of surge pressure isobserved at OPWP scenario If the movement of drill pipeis stop immediately the surge pressure will not disappearimmediately and the surge pressure will exist and changeslightly for short while
Figures 12 and 13 show that the surge pressure varies withwellbore diameter caused by the drill pipe operating speedaccording to Figure 7 The surge pressure decreases with theincrease of wellbore diameter It is because the drilling mudflow rate decreases with the increases of wellbore diameterBased on (1) and (2) surge pressure decreases at each positiondue to the decrease of drilling mud flow rate Surge pressurecaused by converting of potential energy into kinetic energyand then kinetic energy back into potential energy over andover again During the tripping operations especially in slimwellbore drilling process the speed of drill pipe movementshould be appropriately reduced The surge pressure in
Mathematical Problems in Engineering 11
OPWPCPOPOPOP
t (s)0 10 20 30 40 50 60
16
12
08
04
00
P (M
Pa)
Qg = 2427m3h
Figure 11 Surge pressure variation at different scenarios
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 2427m3h
Figure 12 Surge pressure variation at different wellbore diameters(gas influx 119876
119892= 2427m3h)
Figure 13 is prominently larger than that in Figure 12 Thisis because both the decrease of the wave velocity and theincrease of the compression in gas drilling mud two-phasedue to the gas influx result in the comparatively large surgepressure
Figure 14 illustrates that the surge pressure varied withthe length of drill pipe The surge pressure caused by shorterdrill pipe (whose the total length is 1000m) is smaller thanthe longer drill pipe (whose total length is 3000m) beforethe maximum surge pressure occurs This can explain thatsurge pressure is affected by the friction force of drill pipe andwellbore Compared with the longer drill pipe the frictionforce caused by shorter drill pipe is largerTherefore the surgepressure gradually decreases As a result the surge pressure isgreater at 1000m
Figure 15 illustrates that the surge pressure varied withthe drilling fluid density At the bottom of well the surge
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 0m3h
Figure 13 Surge pressure at different borehole diameters (gas influx119876119892= 0m3h)
t (s)0 10 20 30 40 50 60
P (M
Pa)
16
12
08
04
00
Qg = 2427m3h
L = 1000mL = 3000m
Figure 14 Surge pressure variation at different well depths
pressure increases with the increase of drilling fluid densityThe changes are significant in the maximum surge pressureThis is because the surge pressure is mainly affected by wavevelocity density and velocity of gas drilling mud two-phaseAccording to (6) and (7) the surge pressure is increased as thedrilling fluid density increasesThemaximum surge pressureoccurs before drill pipe reaches bottomhole
52 The Effects of Gas Influx Rate on Surge Pressure Figures16 17 and 18 illustrate that the surge pressure varied withthe increase of the gas influx rate Figure 17 shows the surgepressure with the gas influx rate of 2427m3h at the bottomof well Figure 18 shows the surge pressure with the gas influxrate of 8312m3h at the bottom of well Mainly affected bywave velocity density and velocity in gas-drilling mud twophase the surge pressure is greatest at the gas influx rate of0m3h Figure 16 illustrates an example of a minimum wavevelocity at the gas influx rate of 8312m3h at the bottom
12 Mathematical Problems in Engineering
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
Qg = 2427m3h
= 1500kgm3=
= 2000 kgm3=
120588m120588m
Figure 15 Surge pressure variation at different drilling fluid densi-ties
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 0m3h
Figure 16 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 0m3h)
of well As the gas influx rate increases the surge pressuredecreases When gas influx occurs the lag time is lengthenedremarkably compared with the condition of no gas influxThe longer the distance is the greater the lag time will beThe lag time is mainly determined by the propagation time ofpressure wave in the wellbore
Figures 19 and 20 illustrate the surge pressure caused bydifferent operating speeds of the drill pipe at various gasinflux rates at the bottom of well As drilling mudmixed withinflux gas not only reduces the pressure wave velocity butalso reduces the density of themixture density surge pressuremainly affected by wave velocity andmixture density reducesdrastically with the increase of gas influx rate at the bottomof well Figures 19 and 20 are compared to Figures 17 and 18
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 2427m3h
Figure 17 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 8312m3h
Figure 18 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 8312m3h)
it can be concluded that the operating speed is also a majorfactor of surge pressure
53The Effects of Drill Pipe Operating Speed on Surge PressureFigure 21 illustrates the effect of drill pipe speed on the surgepressure The change of surge pressure is smaller in gasdrilling two-phase than in single-phase drilling mud Thesurge pressure becomes smaller and closes to zero after themovement is stopped due to greater friction force existing inwell wall during drill pipe operating
6 Conclusions
With full consideration of the important factors such as wavevelocity in real-time the compressibility of the gas phasegas influx rate pressure temperature and well parametersa new surge pressure model has been proposed based on the
Mathematical Problems in Engineering 13Va
lue
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 2427m3h
Figure 19 Surge pressure and fluid velocity variation at differentpipe velocities (gas influx 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 8312m3h
Figure 20 Surge pressure and fluid velocity variation at differentgas influxes (gas influx 119876
119892= 8312m3h)
30
24
18
12
06
00
Valu
e
t (s)0 10 20 30 40 50 60
Pipe velocity (ms)Pipe velocity (ms)Surge pressure (MPa)
Surge pressure (MPa)
Qg = 2427m3h
Figure 21 Surge pressure variations at different pipe velocities (gasinflux 119876
119892= 2427m3h)
mass conservation equations the momentum conservationequations characteristic line method and the gas-drillingmud equations of state (EOS) duringMPDoperations Solvedby the difference method and fourth-order explicit Runge-Kutta method the model is used to predict surge pressurefor different drill pipe operating speed and gas influx rates inMPD operations The main conclusions can be summarizedas follows
(1) An accurate mathematical model to predict surgepressure after gas influx is of great importancebecause of the effects of surge pressure on operationalsafety and success The new model provides betterprediction of surge pressures in comparison with theexisting experiment
(2) The surge pressure tends to increase with the increaseof drill pipe operating speed and decrease of gas influxrate andwell diameter Considering that the gas influxoccurs the surge pressure weakens obviously Withthe increases in well depth the maximum allowabletripping speed gradually become smaller the surgepressure increases at the bottom of well and drillingsafety window is accordingly narrower The surgepressure can cause a significant time lag if the gasinflux occurs at the bottom of well and the timelag is mainly determined by the propagation time ofpressure wave in the wellbore
(3) The surge pressure obviously varies at different sce-narios (OPOP CPOP and OPWP resp) duringthe tripping operations Surge pressure increased inCPOP scenario than in OPOP scenario Especiallywhen the maximum surge pressure occurs a strongincrease of surge pressure is observed at OPWPscenario By running the new surge pressure modeldrilling engineers can identify potential surge prob-lems and optimize tripping operations for differentscenarios as to avoid complicated accidents to happenin wellbore during drilling operations
(4) The calculation of surge pressure in gas drillingmud two-phase is more complex than conventionalcalculation in single drillingmudDrill pipe operatingspeed in drilling operations should be strictly con-trolled and the surge pressure should be consideredin the design The greatest advantages of new modelare the ability to reduce nonproductive time (NPT) indrilling
Nomenclature
A Wellbore effective cross-area (m2)1198600 The outer drill pipe cross-area (m2)
1198602 Wellbore cross-area (m2)
1198603 The inner Drill pipe cross-area (m2)
119860119890 Effective cross-area of jet nozzle (m2)
119888 Wave velocity (ms)119889119904 One discrete length of wellbore (m)
14 Mathematical Problems in Engineering
119863 Wellbore effective diameter (m)1198631 The inner diameter of casing (m)
1198632 The outer diameter of casing (m)
1198633 The inner diameter of drill pipe (m)
1198634 The outer diameter of drill pipe (m)
119863119896119895 Differential equation coefficients
119864119896119895 Differential equation coefficients
119864119904 Modulus of elasticity of the drill pipe
(MPa)119864119891 Modulus of elasticity of formation (MPa)
1198642 Modulus of elasticity of casing (MPa)
119891119866 Shear stresses coefficient of gas interface
119891 Friction coefficient1198912 Wellbore friction coefficient
1198913 Drill pipe friction coefficient
119891119871 Shear stresses coefficient of drilling mud
119865119911 Pressure applied on the two-phase flow
(N)119892 Acceleration due to gravity (m2s)119892119888 Conversion factor
119892119879 Geothermal temperature gradient (Km)
119896119890 Wellbore effective roughness (m)
1198960 Roughness of outer pipe (m)
119896119894 Roughness of inner pipe (m)
119867 The depth of well (m)1198711 The length from pipe bottom to hole
bottom (m)1198712 The length from pipe bottom to well
head (m)119871119892 Flow pattern119898 Hydraulic radius (m)1198982 Wellbore hydraulic radius (m)
1198983 Drill pipe hydraulic radius (m)
119873 Parameter combiningThompson energyeffects
119875 Pressure (MPa)1198750 Initial pressure of wellbore (MPa)
1198752 Pressure within the empty wellbore
(MPa)1198753 Pressure within the drill pipe (MPa)
119875119888 Critical pressure (kPa)
119875119871 Pressure of drilling mud (MPa)
119875119903 Reduced pressure
119875119895119905 Pressure at node (119895 119905) (MPa)
119902119898 Volumetric flow rate of two phase (m3s)
119902119866 Velocity of the gas (m3s)
119902119871 Velocity of the drilling mud (m3s)
119876119871 Mud flow rate within the wellbore(m3h)
1198761 Drilling mud flow rate within hole(m3h)
1198762 Mud flow rate within the wellbore(m3h)
1198763 Mud flow rate within the drill pipe(m3h)
119876119908 Pump flow rate (m3s)
119876119895119905 Flow rate at time 119905 and nod 119895 (m3s)
119877 Constant of EOS (JKgK)
119904 The length of wellbore along wellbore(m)
119905 Time (s)119879 Temperature (K)119879119888 Critical temperature (K)
119879119903 Reduced temperature
119879119890119894 Undistributed temperature at a depth
(K)119879119890119887ℎ
Undistributed temperature of wellhead(K)
119879119891119887ℎ
Undistributed temperature at bottom-hole (K)
V Velocity of drilling mud and gas (ms)V119901 Velocity of drill pipe (ms)
V119898 Gas and drilling mud flow velocity (ms)
V119866 Gas flow velocity (ms)
V119904 Slip velocity (ms)
V119904119892 Superficial gas velocity (ms)
V119871 Drilling mud flow velocity (ms)
119882119898 Mass flow velocity (kgm3)
119911119866 Gas deviation factor
1199110 Initial calculation height of wellbore (m)
119911119887ℎ Total well depth from surface (m)
119911 Variable of distance from surface (m)
Greek Letters
120588119866 Gas density (kgm3)
120591119891 Frictional pressure gradient (Pam)
120591119908 Frictional pressure coefficient
120588119898 Gas and drilling density (kgm3)
1205880 Density under atmospheric pressure (kgm3)
120588119871 Drilling mud density (kgm3)
120590119904 Surface tension (Nm2)
120601119866 Gas void fraction
120601119871 Drilling mud holdup
120588119903 Reduced density
120583119871 Viscosity of drilling mud (Pasdots)
120579 The ground angle (rad)120583119891 Poissonrsquos ratio of formation
120583119904 Poissonrsquos ratio of drill pipe
1205832 Poissonrsquos ratio of casing
Δ119860 The change of the across areas (m2)
Subscripts
CPOP Closed pipe without pumpingDAPC Dynamic annular pressure controlEOS Equations of stateERV Relief-coupling valveMPD Managed pressure drillingOPOP Open pipe without pumpingOPWP Open pipe with pumpingR-K4 The fourth-order explicit Runge-KuttaRCD The rotating control device
Mathematical Problems in Engineering 15
Subscripts of Graph
119888 Wave velocity in two-phase flow (ms)119863 Wellbore effective diameter (m)119867 Depth of wellbore (m)119871 The length from drill pipe and drill pipe (m)119876119892 Gas influx rate at the bottomhole (m3h)
120601119866 Void fraction ()
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 51274170 andQC2012C128) Without their support this work would nothave been possible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of AirInjection in Annuli to manage pressure during CementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling Conferenceand Exhibition San Diego Calif USA March 2012
[4] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the Norh Africa Technical Conference andExhibition Cairo Egypt February 2012
[5] J Saponja A Adeleye and B Hucik ldquoManaged-pressuredrilling (MPD) field trials demonstrate technology valuerdquo inProceedings of the IADCSPE Drilling Conference Miami FlaUSA February 2006
[6] G Liu ldquoPredicting surge pressures that result from runninglinersrdquoWorld Oil vol 222 no 4 p 92 2001
[7] K Zhou J Yang B Zhong and L Tang ldquoSurge pressure inwellborerdquo Oil Drilling and Production Technology vol 12 no 2pp 1ndash10 99 1990
[8] R S Rudi Rubiandini ldquoNew formula of surge pressure fordetermining safe trip velocitiesrdquo in Proceedings of the SPE AsiaPacific Oil and Gas Conference and Exhibition pp 16ndash18 ausOctober 2000
[9] J Li Z Huang K Li and W Wu ldquoThe influence of collaron surge pressure caused by the drilling fluid viscous forceunder pumping conditionrdquoResearch Journal of Applied SciencesEngineering and Technology vol 5 no 5 pp 1781ndash1785 2040ndash7459 2013
[10] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvancedMaterials Research vol 468ndash471 pp 1736ndash1742 2012
[11] F Crespo and R Ahmed ldquoSurge-and-swab pressure predictionsfor yield-power-law drilling fluidsrdquo SPE Drilling amp Completionvol 27 no 4 pp 574ndash585 2012
[12] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[13] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquo PHDThesis vol 50 no 9 p 143 1989
[14] H Fan ldquoDevelopment of application software for dynamicsurge pressure while trippingrdquo Petroleum Drilling and Technol-ogy vol 23 no 4 pp 11ndash14 1995
[15] K Zhou J Yang B Zhong and L Tang ldquoSurge pressurein wellborerdquo Journal of Petroleum Exploration and ProductionTechnology vol 12 no 2 pp 1ndash10 1990
[16] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquoDissertation Abstracts International Bvol 50 no 9 1989
[17] Z Jiang ldquoStudies of wellbore surge pressure by a dynamicmethodrdquo Journal of Petroleum Exploration and ProductionTechnology vol 13 no 3 pp 39ndash44 97 1991
[18] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[19] D Johnston M Bracken and J Thornton ldquoAssessing pipecondition useful remaining life and maximum operating andsurge pressures when to change the pipe or control thepressurerdquoMagazine of the International Water Association vol14 no 5 pp 43ndash44 2012
[20] H Fan Y Chu and X Liu ldquoPrediction for wellbore dynamicsurge pressure while tripping a drill piperdquo Journal of ChinaUniversity of Petroleum vol 19 no 5 pp 36ndash41 1995
[21] C K Burgess J E Miller and K D Vondervort ldquoDownholesurge pressure reduction system and method of userdquo World9848 143 1998
[22] J P Allamon J E Miller and A M Macfarlane ldquoDownholeSurge Reduction Method and Apparatusrdquo 2004
[23] K S Bjoslashrkevoll R Rommetveit B Aas H Gjeraldstveit andA Merlo ldquoTransient gel breaking model for critical wellsapplications with field data verificationrdquo in Proceedings of the2003 Offshore Mediterranean Conference February 2003
[24] G H Nygaard E Johannessen J E Gravdal and F IversenldquoAutomatic coordinated control of pump rates and choke valvefor compensating pressure fluctuations during surge and swaboperationsrdquo in Proceedings of the IADCSPE Managed PressureDrilling and Underbalanced Operations Conference pp 120ndash125March 2007
[25] J V Fedevjcyk S L M Junqueira and C O R Negrao ldquoSurgeamp swab pressures in wells with cross-section changesrdquo BoletimTecnico da Producao de Petroleo vol 4 no 2 pp 221ndash234 2009
[26] S Tian G H Medley and C R Stone ldquoUnderstandingyield point effect onpressure surges critical to managing deepdifficultMPDwellsrdquoDrilling Contractor vol 65 no 4 p 46 4850 53 2009
[27] Y Sun Q Li and J Zhao ldquoNew method of predicting surgepressure apply to horizontal well based on caisson flowrdquoNaturalScience vol 2 no 12 pp 1394ndash1399 2150ndash4091 2010
[28] B T Lebele-Alawa and F E Oparadike ldquoAnalysis of the effectsof valve propagated pressure surge on pipe flowrdquo Engineeringvol 3 no 11 pp 1098ndash1101 2011
[29] X Kong Q Yuan and Y Qiu ldquoThe research of two-phasefluctuation pressure in wellbore tripping operationrdquo InnerMongolia Petrochemical Industry TE242 2011
[30] F Crespo and R Ahmed ldquoA simplified surge and swab pressuremodel for yield power law fluidsrdquo Journal of Petroleum Scienceand Engineering vol 101 pp 12ndash20 2013
16 Mathematical Problems in Engineering
[31] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
[32] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[33] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86-87 pp 127ndash136 2012
[34] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[35] M J SanchezComparison of Correlations for Predicting PressureLosses in Vertical Multiphase Annular Flow University of TulsaTulsa Okla USA 1972
[36] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[37] Y Lin X Kong Y Qiu and Q Yuan ldquoCalculation analysis ofpressure wave velocity in gas and drilling mud two-phase fluidin annulus during drilling operationsrdquo Mathematical Problemsin Engineering vol 2013 Article ID 318912 17 pages 2013
[38] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[39] H Junfang ldquoBalanced drilling and well controlrdquo PetroleumIndustry Press vol 1 article 33 1992
[40] W A Bacon Consideration of compressibility effects for applied-back- pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Va USA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Start
Initial conditions
Grid division
Solve equation (21) for temperature
Calculate k1 minus k4Pi+1 = Pi + ΔP
zi+1 = zi + h
Solve equation (12) for 120588m
Solve equation (14) for c
Determine the time and the step
Enter the pipe velocity
Spline smoothing
Satisfy the boundary conditions and initial conditions
Δt rarr 0
Analysis conditions at the end of pipe
Solve pressure and flow at boundary points
Solve pressure and flow in pipe points
T = T + Δt
NoT lt Tmax
YesEnd
Figure 4 Solution procedure for surge pressure in MPD operations
80 16 24 32 40
00
05
10
15
20
Valu
e
t (s)
Pipe velocity (ms)This model (MPa) Experimental data (MPa)
Burkhardt data (MPa)
Figure 5 Experimental verification by comparison with previousexperiment data
the void fraction and wave velocity first slightly change in acomparatively smooth value then change sharply It is becausethe volume of gas expanded rapidly with the decreasingof pressure near the wellhead the void fraction increasessharply and at the same time the wave velocity decreasesobviously At a high gas influx rate the wave velocity tends toincrease because the void fraction in thewellhead is increasedto a high extent In conclusion the wave velocity is sensitiveto the void fraction and the void fraction is dominated by
MudMotor
Choke
z
DAPC
Casing
Drill pipeAnnulus
Drill collar
Formation
RCD0
800m
1200m
4000m
1778mm 78 mm ) times G105
2159mm times G105120601
120601 (120601
1270mm 78 mm ) times G105120601 (120601
Figure 6 Schematic diagram of gas influx process in tripping
influx rate and pressure in the wellbore especially the influxrate
As shown in Figure 9 within the range of low gas influxrate the wave velocity decreases significantly It is becausethe compressibility of the gas increases remarkably andthe medium appears high elasticity though the density ofgas-drilling mud two-phase flow changes slightly With theincrease of gas influx rate and corresponding increase of thevoid fraction in the wellbore the compressibility of the two-phase unceasingly increases which promotes themomentumand energy exchange in the interface Therefore the wavevelocity continuously decreases
10 Mathematical Problems in EngineeringVa
lue
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
minus05
minus10
Pipe acceleration (ms2)Pipe velocity (ms)Fluid velocity (ms)
Figure 7 Drill pipe velocity variation at different accelerations
0 800 1600 2400 3200 4000H (m)
100
80
60
40
20
0
minus20
G(
)
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
120601
Figure 8 Void fraction distribution at different gas influx rates
0 800 1600 2400 3200 4000H (m)
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
1500
1200
900
600
300
0
c (m
s)
Figure 9 Wave velocity variation at different gas influxes
0 10 20 30 40 50 60
25
20
15
10
05
00
minus05
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
P (M
Pa)
t (s)
Figure 10 Surge pressure variation at different gas influxes
Figure 10 illustrates the effect of gas influx on the surgepressure with the change of gas influx rate As shown inFigure 10 under different gas influx rates (0m3h 2427m3hand 8312m3h resp) the maximum surge pressure mayoccur before drill pipe reaches bottomhole With theincreases of the gas influx the corresponding surge pressuredecreases Considering the gas influx in the calculation of thesurge pressure in the wellbore the surge pressure weakensobviously As the gas influx rate decreases the surge pressureat bottom of the wellbore is reduced gradually while surgepressure can cause a significant time lag if the gas influxoccurs at the bottom of well The lag time is mainly affectedby wave velocity
Figure 11 presents a change of surge pressure in thewellbore at different scenario (OPOP CPOP and OPWPresp) According to Figure 11 surge pressure is increasedin CPOP scenario by comparing with the surge pressure inOPOP scenarios It is because the speed of the drilling mudincreases remarkably in CPOP scenario the surge pressurebecomes larger obviously Especially when the maximumsurge pressure occurs a strong increase of surge pressure isobserved at OPWP scenario If the movement of drill pipeis stop immediately the surge pressure will not disappearimmediately and the surge pressure will exist and changeslightly for short while
Figures 12 and 13 show that the surge pressure varies withwellbore diameter caused by the drill pipe operating speedaccording to Figure 7 The surge pressure decreases with theincrease of wellbore diameter It is because the drilling mudflow rate decreases with the increases of wellbore diameterBased on (1) and (2) surge pressure decreases at each positiondue to the decrease of drilling mud flow rate Surge pressurecaused by converting of potential energy into kinetic energyand then kinetic energy back into potential energy over andover again During the tripping operations especially in slimwellbore drilling process the speed of drill pipe movementshould be appropriately reduced The surge pressure in
Mathematical Problems in Engineering 11
OPWPCPOPOPOP
t (s)0 10 20 30 40 50 60
16
12
08
04
00
P (M
Pa)
Qg = 2427m3h
Figure 11 Surge pressure variation at different scenarios
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 2427m3h
Figure 12 Surge pressure variation at different wellbore diameters(gas influx 119876
119892= 2427m3h)
Figure 13 is prominently larger than that in Figure 12 Thisis because both the decrease of the wave velocity and theincrease of the compression in gas drilling mud two-phasedue to the gas influx result in the comparatively large surgepressure
Figure 14 illustrates that the surge pressure varied withthe length of drill pipe The surge pressure caused by shorterdrill pipe (whose the total length is 1000m) is smaller thanthe longer drill pipe (whose total length is 3000m) beforethe maximum surge pressure occurs This can explain thatsurge pressure is affected by the friction force of drill pipe andwellbore Compared with the longer drill pipe the frictionforce caused by shorter drill pipe is largerTherefore the surgepressure gradually decreases As a result the surge pressure isgreater at 1000m
Figure 15 illustrates that the surge pressure varied withthe drilling fluid density At the bottom of well the surge
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 0m3h
Figure 13 Surge pressure at different borehole diameters (gas influx119876119892= 0m3h)
t (s)0 10 20 30 40 50 60
P (M
Pa)
16
12
08
04
00
Qg = 2427m3h
L = 1000mL = 3000m
Figure 14 Surge pressure variation at different well depths
pressure increases with the increase of drilling fluid densityThe changes are significant in the maximum surge pressureThis is because the surge pressure is mainly affected by wavevelocity density and velocity of gas drilling mud two-phaseAccording to (6) and (7) the surge pressure is increased as thedrilling fluid density increasesThemaximum surge pressureoccurs before drill pipe reaches bottomhole
52 The Effects of Gas Influx Rate on Surge Pressure Figures16 17 and 18 illustrate that the surge pressure varied withthe increase of the gas influx rate Figure 17 shows the surgepressure with the gas influx rate of 2427m3h at the bottomof well Figure 18 shows the surge pressure with the gas influxrate of 8312m3h at the bottom of well Mainly affected bywave velocity density and velocity in gas-drilling mud twophase the surge pressure is greatest at the gas influx rate of0m3h Figure 16 illustrates an example of a minimum wavevelocity at the gas influx rate of 8312m3h at the bottom
12 Mathematical Problems in Engineering
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
Qg = 2427m3h
= 1500kgm3=
= 2000 kgm3=
120588m120588m
Figure 15 Surge pressure variation at different drilling fluid densi-ties
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 0m3h
Figure 16 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 0m3h)
of well As the gas influx rate increases the surge pressuredecreases When gas influx occurs the lag time is lengthenedremarkably compared with the condition of no gas influxThe longer the distance is the greater the lag time will beThe lag time is mainly determined by the propagation time ofpressure wave in the wellbore
Figures 19 and 20 illustrate the surge pressure caused bydifferent operating speeds of the drill pipe at various gasinflux rates at the bottom of well As drilling mudmixed withinflux gas not only reduces the pressure wave velocity butalso reduces the density of themixture density surge pressuremainly affected by wave velocity andmixture density reducesdrastically with the increase of gas influx rate at the bottomof well Figures 19 and 20 are compared to Figures 17 and 18
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 2427m3h
Figure 17 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 8312m3h
Figure 18 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 8312m3h)
it can be concluded that the operating speed is also a majorfactor of surge pressure
53The Effects of Drill Pipe Operating Speed on Surge PressureFigure 21 illustrates the effect of drill pipe speed on the surgepressure The change of surge pressure is smaller in gasdrilling two-phase than in single-phase drilling mud Thesurge pressure becomes smaller and closes to zero after themovement is stopped due to greater friction force existing inwell wall during drill pipe operating
6 Conclusions
With full consideration of the important factors such as wavevelocity in real-time the compressibility of the gas phasegas influx rate pressure temperature and well parametersa new surge pressure model has been proposed based on the
Mathematical Problems in Engineering 13Va
lue
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 2427m3h
Figure 19 Surge pressure and fluid velocity variation at differentpipe velocities (gas influx 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 8312m3h
Figure 20 Surge pressure and fluid velocity variation at differentgas influxes (gas influx 119876
119892= 8312m3h)
30
24
18
12
06
00
Valu
e
t (s)0 10 20 30 40 50 60
Pipe velocity (ms)Pipe velocity (ms)Surge pressure (MPa)
Surge pressure (MPa)
Qg = 2427m3h
Figure 21 Surge pressure variations at different pipe velocities (gasinflux 119876
119892= 2427m3h)
mass conservation equations the momentum conservationequations characteristic line method and the gas-drillingmud equations of state (EOS) duringMPDoperations Solvedby the difference method and fourth-order explicit Runge-Kutta method the model is used to predict surge pressurefor different drill pipe operating speed and gas influx rates inMPD operations The main conclusions can be summarizedas follows
(1) An accurate mathematical model to predict surgepressure after gas influx is of great importancebecause of the effects of surge pressure on operationalsafety and success The new model provides betterprediction of surge pressures in comparison with theexisting experiment
(2) The surge pressure tends to increase with the increaseof drill pipe operating speed and decrease of gas influxrate andwell diameter Considering that the gas influxoccurs the surge pressure weakens obviously Withthe increases in well depth the maximum allowabletripping speed gradually become smaller the surgepressure increases at the bottom of well and drillingsafety window is accordingly narrower The surgepressure can cause a significant time lag if the gasinflux occurs at the bottom of well and the timelag is mainly determined by the propagation time ofpressure wave in the wellbore
(3) The surge pressure obviously varies at different sce-narios (OPOP CPOP and OPWP resp) duringthe tripping operations Surge pressure increased inCPOP scenario than in OPOP scenario Especiallywhen the maximum surge pressure occurs a strongincrease of surge pressure is observed at OPWPscenario By running the new surge pressure modeldrilling engineers can identify potential surge prob-lems and optimize tripping operations for differentscenarios as to avoid complicated accidents to happenin wellbore during drilling operations
(4) The calculation of surge pressure in gas drillingmud two-phase is more complex than conventionalcalculation in single drillingmudDrill pipe operatingspeed in drilling operations should be strictly con-trolled and the surge pressure should be consideredin the design The greatest advantages of new modelare the ability to reduce nonproductive time (NPT) indrilling
Nomenclature
A Wellbore effective cross-area (m2)1198600 The outer drill pipe cross-area (m2)
1198602 Wellbore cross-area (m2)
1198603 The inner Drill pipe cross-area (m2)
119860119890 Effective cross-area of jet nozzle (m2)
119888 Wave velocity (ms)119889119904 One discrete length of wellbore (m)
14 Mathematical Problems in Engineering
119863 Wellbore effective diameter (m)1198631 The inner diameter of casing (m)
1198632 The outer diameter of casing (m)
1198633 The inner diameter of drill pipe (m)
1198634 The outer diameter of drill pipe (m)
119863119896119895 Differential equation coefficients
119864119896119895 Differential equation coefficients
119864119904 Modulus of elasticity of the drill pipe
(MPa)119864119891 Modulus of elasticity of formation (MPa)
1198642 Modulus of elasticity of casing (MPa)
119891119866 Shear stresses coefficient of gas interface
119891 Friction coefficient1198912 Wellbore friction coefficient
1198913 Drill pipe friction coefficient
119891119871 Shear stresses coefficient of drilling mud
119865119911 Pressure applied on the two-phase flow
(N)119892 Acceleration due to gravity (m2s)119892119888 Conversion factor
119892119879 Geothermal temperature gradient (Km)
119896119890 Wellbore effective roughness (m)
1198960 Roughness of outer pipe (m)
119896119894 Roughness of inner pipe (m)
119867 The depth of well (m)1198711 The length from pipe bottom to hole
bottom (m)1198712 The length from pipe bottom to well
head (m)119871119892 Flow pattern119898 Hydraulic radius (m)1198982 Wellbore hydraulic radius (m)
1198983 Drill pipe hydraulic radius (m)
119873 Parameter combiningThompson energyeffects
119875 Pressure (MPa)1198750 Initial pressure of wellbore (MPa)
1198752 Pressure within the empty wellbore
(MPa)1198753 Pressure within the drill pipe (MPa)
119875119888 Critical pressure (kPa)
119875119871 Pressure of drilling mud (MPa)
119875119903 Reduced pressure
119875119895119905 Pressure at node (119895 119905) (MPa)
119902119898 Volumetric flow rate of two phase (m3s)
119902119866 Velocity of the gas (m3s)
119902119871 Velocity of the drilling mud (m3s)
119876119871 Mud flow rate within the wellbore(m3h)
1198761 Drilling mud flow rate within hole(m3h)
1198762 Mud flow rate within the wellbore(m3h)
1198763 Mud flow rate within the drill pipe(m3h)
119876119908 Pump flow rate (m3s)
119876119895119905 Flow rate at time 119905 and nod 119895 (m3s)
119877 Constant of EOS (JKgK)
119904 The length of wellbore along wellbore(m)
119905 Time (s)119879 Temperature (K)119879119888 Critical temperature (K)
119879119903 Reduced temperature
119879119890119894 Undistributed temperature at a depth
(K)119879119890119887ℎ
Undistributed temperature of wellhead(K)
119879119891119887ℎ
Undistributed temperature at bottom-hole (K)
V Velocity of drilling mud and gas (ms)V119901 Velocity of drill pipe (ms)
V119898 Gas and drilling mud flow velocity (ms)
V119866 Gas flow velocity (ms)
V119904 Slip velocity (ms)
V119904119892 Superficial gas velocity (ms)
V119871 Drilling mud flow velocity (ms)
119882119898 Mass flow velocity (kgm3)
119911119866 Gas deviation factor
1199110 Initial calculation height of wellbore (m)
119911119887ℎ Total well depth from surface (m)
119911 Variable of distance from surface (m)
Greek Letters
120588119866 Gas density (kgm3)
120591119891 Frictional pressure gradient (Pam)
120591119908 Frictional pressure coefficient
120588119898 Gas and drilling density (kgm3)
1205880 Density under atmospheric pressure (kgm3)
120588119871 Drilling mud density (kgm3)
120590119904 Surface tension (Nm2)
120601119866 Gas void fraction
120601119871 Drilling mud holdup
120588119903 Reduced density
120583119871 Viscosity of drilling mud (Pasdots)
120579 The ground angle (rad)120583119891 Poissonrsquos ratio of formation
120583119904 Poissonrsquos ratio of drill pipe
1205832 Poissonrsquos ratio of casing
Δ119860 The change of the across areas (m2)
Subscripts
CPOP Closed pipe without pumpingDAPC Dynamic annular pressure controlEOS Equations of stateERV Relief-coupling valveMPD Managed pressure drillingOPOP Open pipe without pumpingOPWP Open pipe with pumpingR-K4 The fourth-order explicit Runge-KuttaRCD The rotating control device
Mathematical Problems in Engineering 15
Subscripts of Graph
119888 Wave velocity in two-phase flow (ms)119863 Wellbore effective diameter (m)119867 Depth of wellbore (m)119871 The length from drill pipe and drill pipe (m)119876119892 Gas influx rate at the bottomhole (m3h)
120601119866 Void fraction ()
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 51274170 andQC2012C128) Without their support this work would nothave been possible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of AirInjection in Annuli to manage pressure during CementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling Conferenceand Exhibition San Diego Calif USA March 2012
[4] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the Norh Africa Technical Conference andExhibition Cairo Egypt February 2012
[5] J Saponja A Adeleye and B Hucik ldquoManaged-pressuredrilling (MPD) field trials demonstrate technology valuerdquo inProceedings of the IADCSPE Drilling Conference Miami FlaUSA February 2006
[6] G Liu ldquoPredicting surge pressures that result from runninglinersrdquoWorld Oil vol 222 no 4 p 92 2001
[7] K Zhou J Yang B Zhong and L Tang ldquoSurge pressure inwellborerdquo Oil Drilling and Production Technology vol 12 no 2pp 1ndash10 99 1990
[8] R S Rudi Rubiandini ldquoNew formula of surge pressure fordetermining safe trip velocitiesrdquo in Proceedings of the SPE AsiaPacific Oil and Gas Conference and Exhibition pp 16ndash18 ausOctober 2000
[9] J Li Z Huang K Li and W Wu ldquoThe influence of collaron surge pressure caused by the drilling fluid viscous forceunder pumping conditionrdquoResearch Journal of Applied SciencesEngineering and Technology vol 5 no 5 pp 1781ndash1785 2040ndash7459 2013
[10] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvancedMaterials Research vol 468ndash471 pp 1736ndash1742 2012
[11] F Crespo and R Ahmed ldquoSurge-and-swab pressure predictionsfor yield-power-law drilling fluidsrdquo SPE Drilling amp Completionvol 27 no 4 pp 574ndash585 2012
[12] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[13] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquo PHDThesis vol 50 no 9 p 143 1989
[14] H Fan ldquoDevelopment of application software for dynamicsurge pressure while trippingrdquo Petroleum Drilling and Technol-ogy vol 23 no 4 pp 11ndash14 1995
[15] K Zhou J Yang B Zhong and L Tang ldquoSurge pressurein wellborerdquo Journal of Petroleum Exploration and ProductionTechnology vol 12 no 2 pp 1ndash10 1990
[16] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquoDissertation Abstracts International Bvol 50 no 9 1989
[17] Z Jiang ldquoStudies of wellbore surge pressure by a dynamicmethodrdquo Journal of Petroleum Exploration and ProductionTechnology vol 13 no 3 pp 39ndash44 97 1991
[18] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[19] D Johnston M Bracken and J Thornton ldquoAssessing pipecondition useful remaining life and maximum operating andsurge pressures when to change the pipe or control thepressurerdquoMagazine of the International Water Association vol14 no 5 pp 43ndash44 2012
[20] H Fan Y Chu and X Liu ldquoPrediction for wellbore dynamicsurge pressure while tripping a drill piperdquo Journal of ChinaUniversity of Petroleum vol 19 no 5 pp 36ndash41 1995
[21] C K Burgess J E Miller and K D Vondervort ldquoDownholesurge pressure reduction system and method of userdquo World9848 143 1998
[22] J P Allamon J E Miller and A M Macfarlane ldquoDownholeSurge Reduction Method and Apparatusrdquo 2004
[23] K S Bjoslashrkevoll R Rommetveit B Aas H Gjeraldstveit andA Merlo ldquoTransient gel breaking model for critical wellsapplications with field data verificationrdquo in Proceedings of the2003 Offshore Mediterranean Conference February 2003
[24] G H Nygaard E Johannessen J E Gravdal and F IversenldquoAutomatic coordinated control of pump rates and choke valvefor compensating pressure fluctuations during surge and swaboperationsrdquo in Proceedings of the IADCSPE Managed PressureDrilling and Underbalanced Operations Conference pp 120ndash125March 2007
[25] J V Fedevjcyk S L M Junqueira and C O R Negrao ldquoSurgeamp swab pressures in wells with cross-section changesrdquo BoletimTecnico da Producao de Petroleo vol 4 no 2 pp 221ndash234 2009
[26] S Tian G H Medley and C R Stone ldquoUnderstandingyield point effect onpressure surges critical to managing deepdifficultMPDwellsrdquoDrilling Contractor vol 65 no 4 p 46 4850 53 2009
[27] Y Sun Q Li and J Zhao ldquoNew method of predicting surgepressure apply to horizontal well based on caisson flowrdquoNaturalScience vol 2 no 12 pp 1394ndash1399 2150ndash4091 2010
[28] B T Lebele-Alawa and F E Oparadike ldquoAnalysis of the effectsof valve propagated pressure surge on pipe flowrdquo Engineeringvol 3 no 11 pp 1098ndash1101 2011
[29] X Kong Q Yuan and Y Qiu ldquoThe research of two-phasefluctuation pressure in wellbore tripping operationrdquo InnerMongolia Petrochemical Industry TE242 2011
[30] F Crespo and R Ahmed ldquoA simplified surge and swab pressuremodel for yield power law fluidsrdquo Journal of Petroleum Scienceand Engineering vol 101 pp 12ndash20 2013
16 Mathematical Problems in Engineering
[31] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
[32] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[33] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86-87 pp 127ndash136 2012
[34] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[35] M J SanchezComparison of Correlations for Predicting PressureLosses in Vertical Multiphase Annular Flow University of TulsaTulsa Okla USA 1972
[36] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[37] Y Lin X Kong Y Qiu and Q Yuan ldquoCalculation analysis ofpressure wave velocity in gas and drilling mud two-phase fluidin annulus during drilling operationsrdquo Mathematical Problemsin Engineering vol 2013 Article ID 318912 17 pages 2013
[38] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[39] H Junfang ldquoBalanced drilling and well controlrdquo PetroleumIndustry Press vol 1 article 33 1992
[40] W A Bacon Consideration of compressibility effects for applied-back- pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Va USA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in EngineeringVa
lue
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
minus05
minus10
Pipe acceleration (ms2)Pipe velocity (ms)Fluid velocity (ms)
Figure 7 Drill pipe velocity variation at different accelerations
0 800 1600 2400 3200 4000H (m)
100
80
60
40
20
0
minus20
G(
)
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
120601
Figure 8 Void fraction distribution at different gas influx rates
0 800 1600 2400 3200 4000H (m)
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
1500
1200
900
600
300
0
c (m
s)
Figure 9 Wave velocity variation at different gas influxes
0 10 20 30 40 50 60
25
20
15
10
05
00
minus05
Qg = 0m3h
Qg = 8312m3hQg = 2427m3h
P (M
Pa)
t (s)
Figure 10 Surge pressure variation at different gas influxes
Figure 10 illustrates the effect of gas influx on the surgepressure with the change of gas influx rate As shown inFigure 10 under different gas influx rates (0m3h 2427m3hand 8312m3h resp) the maximum surge pressure mayoccur before drill pipe reaches bottomhole With theincreases of the gas influx the corresponding surge pressuredecreases Considering the gas influx in the calculation of thesurge pressure in the wellbore the surge pressure weakensobviously As the gas influx rate decreases the surge pressureat bottom of the wellbore is reduced gradually while surgepressure can cause a significant time lag if the gas influxoccurs at the bottom of well The lag time is mainly affectedby wave velocity
Figure 11 presents a change of surge pressure in thewellbore at different scenario (OPOP CPOP and OPWPresp) According to Figure 11 surge pressure is increasedin CPOP scenario by comparing with the surge pressure inOPOP scenarios It is because the speed of the drilling mudincreases remarkably in CPOP scenario the surge pressurebecomes larger obviously Especially when the maximumsurge pressure occurs a strong increase of surge pressure isobserved at OPWP scenario If the movement of drill pipeis stop immediately the surge pressure will not disappearimmediately and the surge pressure will exist and changeslightly for short while
Figures 12 and 13 show that the surge pressure varies withwellbore diameter caused by the drill pipe operating speedaccording to Figure 7 The surge pressure decreases with theincrease of wellbore diameter It is because the drilling mudflow rate decreases with the increases of wellbore diameterBased on (1) and (2) surge pressure decreases at each positiondue to the decrease of drilling mud flow rate Surge pressurecaused by converting of potential energy into kinetic energyand then kinetic energy back into potential energy over andover again During the tripping operations especially in slimwellbore drilling process the speed of drill pipe movementshould be appropriately reduced The surge pressure in
Mathematical Problems in Engineering 11
OPWPCPOPOPOP
t (s)0 10 20 30 40 50 60
16
12
08
04
00
P (M
Pa)
Qg = 2427m3h
Figure 11 Surge pressure variation at different scenarios
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 2427m3h
Figure 12 Surge pressure variation at different wellbore diameters(gas influx 119876
119892= 2427m3h)
Figure 13 is prominently larger than that in Figure 12 Thisis because both the decrease of the wave velocity and theincrease of the compression in gas drilling mud two-phasedue to the gas influx result in the comparatively large surgepressure
Figure 14 illustrates that the surge pressure varied withthe length of drill pipe The surge pressure caused by shorterdrill pipe (whose the total length is 1000m) is smaller thanthe longer drill pipe (whose total length is 3000m) beforethe maximum surge pressure occurs This can explain thatsurge pressure is affected by the friction force of drill pipe andwellbore Compared with the longer drill pipe the frictionforce caused by shorter drill pipe is largerTherefore the surgepressure gradually decreases As a result the surge pressure isgreater at 1000m
Figure 15 illustrates that the surge pressure varied withthe drilling fluid density At the bottom of well the surge
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 0m3h
Figure 13 Surge pressure at different borehole diameters (gas influx119876119892= 0m3h)
t (s)0 10 20 30 40 50 60
P (M
Pa)
16
12
08
04
00
Qg = 2427m3h
L = 1000mL = 3000m
Figure 14 Surge pressure variation at different well depths
pressure increases with the increase of drilling fluid densityThe changes are significant in the maximum surge pressureThis is because the surge pressure is mainly affected by wavevelocity density and velocity of gas drilling mud two-phaseAccording to (6) and (7) the surge pressure is increased as thedrilling fluid density increasesThemaximum surge pressureoccurs before drill pipe reaches bottomhole
52 The Effects of Gas Influx Rate on Surge Pressure Figures16 17 and 18 illustrate that the surge pressure varied withthe increase of the gas influx rate Figure 17 shows the surgepressure with the gas influx rate of 2427m3h at the bottomof well Figure 18 shows the surge pressure with the gas influxrate of 8312m3h at the bottom of well Mainly affected bywave velocity density and velocity in gas-drilling mud twophase the surge pressure is greatest at the gas influx rate of0m3h Figure 16 illustrates an example of a minimum wavevelocity at the gas influx rate of 8312m3h at the bottom
12 Mathematical Problems in Engineering
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
Qg = 2427m3h
= 1500kgm3=
= 2000 kgm3=
120588m120588m
Figure 15 Surge pressure variation at different drilling fluid densi-ties
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 0m3h
Figure 16 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 0m3h)
of well As the gas influx rate increases the surge pressuredecreases When gas influx occurs the lag time is lengthenedremarkably compared with the condition of no gas influxThe longer the distance is the greater the lag time will beThe lag time is mainly determined by the propagation time ofpressure wave in the wellbore
Figures 19 and 20 illustrate the surge pressure caused bydifferent operating speeds of the drill pipe at various gasinflux rates at the bottom of well As drilling mudmixed withinflux gas not only reduces the pressure wave velocity butalso reduces the density of themixture density surge pressuremainly affected by wave velocity andmixture density reducesdrastically with the increase of gas influx rate at the bottomof well Figures 19 and 20 are compared to Figures 17 and 18
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 2427m3h
Figure 17 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 8312m3h
Figure 18 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 8312m3h)
it can be concluded that the operating speed is also a majorfactor of surge pressure
53The Effects of Drill Pipe Operating Speed on Surge PressureFigure 21 illustrates the effect of drill pipe speed on the surgepressure The change of surge pressure is smaller in gasdrilling two-phase than in single-phase drilling mud Thesurge pressure becomes smaller and closes to zero after themovement is stopped due to greater friction force existing inwell wall during drill pipe operating
6 Conclusions
With full consideration of the important factors such as wavevelocity in real-time the compressibility of the gas phasegas influx rate pressure temperature and well parametersa new surge pressure model has been proposed based on the
Mathematical Problems in Engineering 13Va
lue
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 2427m3h
Figure 19 Surge pressure and fluid velocity variation at differentpipe velocities (gas influx 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 8312m3h
Figure 20 Surge pressure and fluid velocity variation at differentgas influxes (gas influx 119876
119892= 8312m3h)
30
24
18
12
06
00
Valu
e
t (s)0 10 20 30 40 50 60
Pipe velocity (ms)Pipe velocity (ms)Surge pressure (MPa)
Surge pressure (MPa)
Qg = 2427m3h
Figure 21 Surge pressure variations at different pipe velocities (gasinflux 119876
119892= 2427m3h)
mass conservation equations the momentum conservationequations characteristic line method and the gas-drillingmud equations of state (EOS) duringMPDoperations Solvedby the difference method and fourth-order explicit Runge-Kutta method the model is used to predict surge pressurefor different drill pipe operating speed and gas influx rates inMPD operations The main conclusions can be summarizedas follows
(1) An accurate mathematical model to predict surgepressure after gas influx is of great importancebecause of the effects of surge pressure on operationalsafety and success The new model provides betterprediction of surge pressures in comparison with theexisting experiment
(2) The surge pressure tends to increase with the increaseof drill pipe operating speed and decrease of gas influxrate andwell diameter Considering that the gas influxoccurs the surge pressure weakens obviously Withthe increases in well depth the maximum allowabletripping speed gradually become smaller the surgepressure increases at the bottom of well and drillingsafety window is accordingly narrower The surgepressure can cause a significant time lag if the gasinflux occurs at the bottom of well and the timelag is mainly determined by the propagation time ofpressure wave in the wellbore
(3) The surge pressure obviously varies at different sce-narios (OPOP CPOP and OPWP resp) duringthe tripping operations Surge pressure increased inCPOP scenario than in OPOP scenario Especiallywhen the maximum surge pressure occurs a strongincrease of surge pressure is observed at OPWPscenario By running the new surge pressure modeldrilling engineers can identify potential surge prob-lems and optimize tripping operations for differentscenarios as to avoid complicated accidents to happenin wellbore during drilling operations
(4) The calculation of surge pressure in gas drillingmud two-phase is more complex than conventionalcalculation in single drillingmudDrill pipe operatingspeed in drilling operations should be strictly con-trolled and the surge pressure should be consideredin the design The greatest advantages of new modelare the ability to reduce nonproductive time (NPT) indrilling
Nomenclature
A Wellbore effective cross-area (m2)1198600 The outer drill pipe cross-area (m2)
1198602 Wellbore cross-area (m2)
1198603 The inner Drill pipe cross-area (m2)
119860119890 Effective cross-area of jet nozzle (m2)
119888 Wave velocity (ms)119889119904 One discrete length of wellbore (m)
14 Mathematical Problems in Engineering
119863 Wellbore effective diameter (m)1198631 The inner diameter of casing (m)
1198632 The outer diameter of casing (m)
1198633 The inner diameter of drill pipe (m)
1198634 The outer diameter of drill pipe (m)
119863119896119895 Differential equation coefficients
119864119896119895 Differential equation coefficients
119864119904 Modulus of elasticity of the drill pipe
(MPa)119864119891 Modulus of elasticity of formation (MPa)
1198642 Modulus of elasticity of casing (MPa)
119891119866 Shear stresses coefficient of gas interface
119891 Friction coefficient1198912 Wellbore friction coefficient
1198913 Drill pipe friction coefficient
119891119871 Shear stresses coefficient of drilling mud
119865119911 Pressure applied on the two-phase flow
(N)119892 Acceleration due to gravity (m2s)119892119888 Conversion factor
119892119879 Geothermal temperature gradient (Km)
119896119890 Wellbore effective roughness (m)
1198960 Roughness of outer pipe (m)
119896119894 Roughness of inner pipe (m)
119867 The depth of well (m)1198711 The length from pipe bottom to hole
bottom (m)1198712 The length from pipe bottom to well
head (m)119871119892 Flow pattern119898 Hydraulic radius (m)1198982 Wellbore hydraulic radius (m)
1198983 Drill pipe hydraulic radius (m)
119873 Parameter combiningThompson energyeffects
119875 Pressure (MPa)1198750 Initial pressure of wellbore (MPa)
1198752 Pressure within the empty wellbore
(MPa)1198753 Pressure within the drill pipe (MPa)
119875119888 Critical pressure (kPa)
119875119871 Pressure of drilling mud (MPa)
119875119903 Reduced pressure
119875119895119905 Pressure at node (119895 119905) (MPa)
119902119898 Volumetric flow rate of two phase (m3s)
119902119866 Velocity of the gas (m3s)
119902119871 Velocity of the drilling mud (m3s)
119876119871 Mud flow rate within the wellbore(m3h)
1198761 Drilling mud flow rate within hole(m3h)
1198762 Mud flow rate within the wellbore(m3h)
1198763 Mud flow rate within the drill pipe(m3h)
119876119908 Pump flow rate (m3s)
119876119895119905 Flow rate at time 119905 and nod 119895 (m3s)
119877 Constant of EOS (JKgK)
119904 The length of wellbore along wellbore(m)
119905 Time (s)119879 Temperature (K)119879119888 Critical temperature (K)
119879119903 Reduced temperature
119879119890119894 Undistributed temperature at a depth
(K)119879119890119887ℎ
Undistributed temperature of wellhead(K)
119879119891119887ℎ
Undistributed temperature at bottom-hole (K)
V Velocity of drilling mud and gas (ms)V119901 Velocity of drill pipe (ms)
V119898 Gas and drilling mud flow velocity (ms)
V119866 Gas flow velocity (ms)
V119904 Slip velocity (ms)
V119904119892 Superficial gas velocity (ms)
V119871 Drilling mud flow velocity (ms)
119882119898 Mass flow velocity (kgm3)
119911119866 Gas deviation factor
1199110 Initial calculation height of wellbore (m)
119911119887ℎ Total well depth from surface (m)
119911 Variable of distance from surface (m)
Greek Letters
120588119866 Gas density (kgm3)
120591119891 Frictional pressure gradient (Pam)
120591119908 Frictional pressure coefficient
120588119898 Gas and drilling density (kgm3)
1205880 Density under atmospheric pressure (kgm3)
120588119871 Drilling mud density (kgm3)
120590119904 Surface tension (Nm2)
120601119866 Gas void fraction
120601119871 Drilling mud holdup
120588119903 Reduced density
120583119871 Viscosity of drilling mud (Pasdots)
120579 The ground angle (rad)120583119891 Poissonrsquos ratio of formation
120583119904 Poissonrsquos ratio of drill pipe
1205832 Poissonrsquos ratio of casing
Δ119860 The change of the across areas (m2)
Subscripts
CPOP Closed pipe without pumpingDAPC Dynamic annular pressure controlEOS Equations of stateERV Relief-coupling valveMPD Managed pressure drillingOPOP Open pipe without pumpingOPWP Open pipe with pumpingR-K4 The fourth-order explicit Runge-KuttaRCD The rotating control device
Mathematical Problems in Engineering 15
Subscripts of Graph
119888 Wave velocity in two-phase flow (ms)119863 Wellbore effective diameter (m)119867 Depth of wellbore (m)119871 The length from drill pipe and drill pipe (m)119876119892 Gas influx rate at the bottomhole (m3h)
120601119866 Void fraction ()
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 51274170 andQC2012C128) Without their support this work would nothave been possible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of AirInjection in Annuli to manage pressure during CementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling Conferenceand Exhibition San Diego Calif USA March 2012
[4] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the Norh Africa Technical Conference andExhibition Cairo Egypt February 2012
[5] J Saponja A Adeleye and B Hucik ldquoManaged-pressuredrilling (MPD) field trials demonstrate technology valuerdquo inProceedings of the IADCSPE Drilling Conference Miami FlaUSA February 2006
[6] G Liu ldquoPredicting surge pressures that result from runninglinersrdquoWorld Oil vol 222 no 4 p 92 2001
[7] K Zhou J Yang B Zhong and L Tang ldquoSurge pressure inwellborerdquo Oil Drilling and Production Technology vol 12 no 2pp 1ndash10 99 1990
[8] R S Rudi Rubiandini ldquoNew formula of surge pressure fordetermining safe trip velocitiesrdquo in Proceedings of the SPE AsiaPacific Oil and Gas Conference and Exhibition pp 16ndash18 ausOctober 2000
[9] J Li Z Huang K Li and W Wu ldquoThe influence of collaron surge pressure caused by the drilling fluid viscous forceunder pumping conditionrdquoResearch Journal of Applied SciencesEngineering and Technology vol 5 no 5 pp 1781ndash1785 2040ndash7459 2013
[10] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvancedMaterials Research vol 468ndash471 pp 1736ndash1742 2012
[11] F Crespo and R Ahmed ldquoSurge-and-swab pressure predictionsfor yield-power-law drilling fluidsrdquo SPE Drilling amp Completionvol 27 no 4 pp 574ndash585 2012
[12] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[13] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquo PHDThesis vol 50 no 9 p 143 1989
[14] H Fan ldquoDevelopment of application software for dynamicsurge pressure while trippingrdquo Petroleum Drilling and Technol-ogy vol 23 no 4 pp 11ndash14 1995
[15] K Zhou J Yang B Zhong and L Tang ldquoSurge pressurein wellborerdquo Journal of Petroleum Exploration and ProductionTechnology vol 12 no 2 pp 1ndash10 1990
[16] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquoDissertation Abstracts International Bvol 50 no 9 1989
[17] Z Jiang ldquoStudies of wellbore surge pressure by a dynamicmethodrdquo Journal of Petroleum Exploration and ProductionTechnology vol 13 no 3 pp 39ndash44 97 1991
[18] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[19] D Johnston M Bracken and J Thornton ldquoAssessing pipecondition useful remaining life and maximum operating andsurge pressures when to change the pipe or control thepressurerdquoMagazine of the International Water Association vol14 no 5 pp 43ndash44 2012
[20] H Fan Y Chu and X Liu ldquoPrediction for wellbore dynamicsurge pressure while tripping a drill piperdquo Journal of ChinaUniversity of Petroleum vol 19 no 5 pp 36ndash41 1995
[21] C K Burgess J E Miller and K D Vondervort ldquoDownholesurge pressure reduction system and method of userdquo World9848 143 1998
[22] J P Allamon J E Miller and A M Macfarlane ldquoDownholeSurge Reduction Method and Apparatusrdquo 2004
[23] K S Bjoslashrkevoll R Rommetveit B Aas H Gjeraldstveit andA Merlo ldquoTransient gel breaking model for critical wellsapplications with field data verificationrdquo in Proceedings of the2003 Offshore Mediterranean Conference February 2003
[24] G H Nygaard E Johannessen J E Gravdal and F IversenldquoAutomatic coordinated control of pump rates and choke valvefor compensating pressure fluctuations during surge and swaboperationsrdquo in Proceedings of the IADCSPE Managed PressureDrilling and Underbalanced Operations Conference pp 120ndash125March 2007
[25] J V Fedevjcyk S L M Junqueira and C O R Negrao ldquoSurgeamp swab pressures in wells with cross-section changesrdquo BoletimTecnico da Producao de Petroleo vol 4 no 2 pp 221ndash234 2009
[26] S Tian G H Medley and C R Stone ldquoUnderstandingyield point effect onpressure surges critical to managing deepdifficultMPDwellsrdquoDrilling Contractor vol 65 no 4 p 46 4850 53 2009
[27] Y Sun Q Li and J Zhao ldquoNew method of predicting surgepressure apply to horizontal well based on caisson flowrdquoNaturalScience vol 2 no 12 pp 1394ndash1399 2150ndash4091 2010
[28] B T Lebele-Alawa and F E Oparadike ldquoAnalysis of the effectsof valve propagated pressure surge on pipe flowrdquo Engineeringvol 3 no 11 pp 1098ndash1101 2011
[29] X Kong Q Yuan and Y Qiu ldquoThe research of two-phasefluctuation pressure in wellbore tripping operationrdquo InnerMongolia Petrochemical Industry TE242 2011
[30] F Crespo and R Ahmed ldquoA simplified surge and swab pressuremodel for yield power law fluidsrdquo Journal of Petroleum Scienceand Engineering vol 101 pp 12ndash20 2013
16 Mathematical Problems in Engineering
[31] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
[32] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[33] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86-87 pp 127ndash136 2012
[34] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[35] M J SanchezComparison of Correlations for Predicting PressureLosses in Vertical Multiphase Annular Flow University of TulsaTulsa Okla USA 1972
[36] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[37] Y Lin X Kong Y Qiu and Q Yuan ldquoCalculation analysis ofpressure wave velocity in gas and drilling mud two-phase fluidin annulus during drilling operationsrdquo Mathematical Problemsin Engineering vol 2013 Article ID 318912 17 pages 2013
[38] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[39] H Junfang ldquoBalanced drilling and well controlrdquo PetroleumIndustry Press vol 1 article 33 1992
[40] W A Bacon Consideration of compressibility effects for applied-back- pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Va USA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
OPWPCPOPOPOP
t (s)0 10 20 30 40 50 60
16
12
08
04
00
P (M
Pa)
Qg = 2427m3h
Figure 11 Surge pressure variation at different scenarios
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 2427m3h
Figure 12 Surge pressure variation at different wellbore diameters(gas influx 119876
119892= 2427m3h)
Figure 13 is prominently larger than that in Figure 12 Thisis because both the decrease of the wave velocity and theincrease of the compression in gas drilling mud two-phasedue to the gas influx result in the comparatively large surgepressure
Figure 14 illustrates that the surge pressure varied withthe length of drill pipe The surge pressure caused by shorterdrill pipe (whose the total length is 1000m) is smaller thanthe longer drill pipe (whose total length is 3000m) beforethe maximum surge pressure occurs This can explain thatsurge pressure is affected by the friction force of drill pipe andwellbore Compared with the longer drill pipe the frictionforce caused by shorter drill pipe is largerTherefore the surgepressure gradually decreases As a result the surge pressure isgreater at 1000m
Figure 15 illustrates that the surge pressure varied withthe drilling fluid density At the bottom of well the surge
t (s)0 10 20 30 40 50 60
25
20
15
10
05
00
P (M
Pa)
D = 02540mD = 02159m
Qg = 0m3h
Figure 13 Surge pressure at different borehole diameters (gas influx119876119892= 0m3h)
t (s)0 10 20 30 40 50 60
P (M
Pa)
16
12
08
04
00
Qg = 2427m3h
L = 1000mL = 3000m
Figure 14 Surge pressure variation at different well depths
pressure increases with the increase of drilling fluid densityThe changes are significant in the maximum surge pressureThis is because the surge pressure is mainly affected by wavevelocity density and velocity of gas drilling mud two-phaseAccording to (6) and (7) the surge pressure is increased as thedrilling fluid density increasesThemaximum surge pressureoccurs before drill pipe reaches bottomhole
52 The Effects of Gas Influx Rate on Surge Pressure Figures16 17 and 18 illustrate that the surge pressure varied withthe increase of the gas influx rate Figure 17 shows the surgepressure with the gas influx rate of 2427m3h at the bottomof well Figure 18 shows the surge pressure with the gas influxrate of 8312m3h at the bottom of well Mainly affected bywave velocity density and velocity in gas-drilling mud twophase the surge pressure is greatest at the gas influx rate of0m3h Figure 16 illustrates an example of a minimum wavevelocity at the gas influx rate of 8312m3h at the bottom
12 Mathematical Problems in Engineering
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
Qg = 2427m3h
= 1500kgm3=
= 2000 kgm3=
120588m120588m
Figure 15 Surge pressure variation at different drilling fluid densi-ties
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 0m3h
Figure 16 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 0m3h)
of well As the gas influx rate increases the surge pressuredecreases When gas influx occurs the lag time is lengthenedremarkably compared with the condition of no gas influxThe longer the distance is the greater the lag time will beThe lag time is mainly determined by the propagation time ofpressure wave in the wellbore
Figures 19 and 20 illustrate the surge pressure caused bydifferent operating speeds of the drill pipe at various gasinflux rates at the bottom of well As drilling mudmixed withinflux gas not only reduces the pressure wave velocity butalso reduces the density of themixture density surge pressuremainly affected by wave velocity andmixture density reducesdrastically with the increase of gas influx rate at the bottomof well Figures 19 and 20 are compared to Figures 17 and 18
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 2427m3h
Figure 17 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 8312m3h
Figure 18 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 8312m3h)
it can be concluded that the operating speed is also a majorfactor of surge pressure
53The Effects of Drill Pipe Operating Speed on Surge PressureFigure 21 illustrates the effect of drill pipe speed on the surgepressure The change of surge pressure is smaller in gasdrilling two-phase than in single-phase drilling mud Thesurge pressure becomes smaller and closes to zero after themovement is stopped due to greater friction force existing inwell wall during drill pipe operating
6 Conclusions
With full consideration of the important factors such as wavevelocity in real-time the compressibility of the gas phasegas influx rate pressure temperature and well parametersa new surge pressure model has been proposed based on the
Mathematical Problems in Engineering 13Va
lue
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 2427m3h
Figure 19 Surge pressure and fluid velocity variation at differentpipe velocities (gas influx 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 8312m3h
Figure 20 Surge pressure and fluid velocity variation at differentgas influxes (gas influx 119876
119892= 8312m3h)
30
24
18
12
06
00
Valu
e
t (s)0 10 20 30 40 50 60
Pipe velocity (ms)Pipe velocity (ms)Surge pressure (MPa)
Surge pressure (MPa)
Qg = 2427m3h
Figure 21 Surge pressure variations at different pipe velocities (gasinflux 119876
119892= 2427m3h)
mass conservation equations the momentum conservationequations characteristic line method and the gas-drillingmud equations of state (EOS) duringMPDoperations Solvedby the difference method and fourth-order explicit Runge-Kutta method the model is used to predict surge pressurefor different drill pipe operating speed and gas influx rates inMPD operations The main conclusions can be summarizedas follows
(1) An accurate mathematical model to predict surgepressure after gas influx is of great importancebecause of the effects of surge pressure on operationalsafety and success The new model provides betterprediction of surge pressures in comparison with theexisting experiment
(2) The surge pressure tends to increase with the increaseof drill pipe operating speed and decrease of gas influxrate andwell diameter Considering that the gas influxoccurs the surge pressure weakens obviously Withthe increases in well depth the maximum allowabletripping speed gradually become smaller the surgepressure increases at the bottom of well and drillingsafety window is accordingly narrower The surgepressure can cause a significant time lag if the gasinflux occurs at the bottom of well and the timelag is mainly determined by the propagation time ofpressure wave in the wellbore
(3) The surge pressure obviously varies at different sce-narios (OPOP CPOP and OPWP resp) duringthe tripping operations Surge pressure increased inCPOP scenario than in OPOP scenario Especiallywhen the maximum surge pressure occurs a strongincrease of surge pressure is observed at OPWPscenario By running the new surge pressure modeldrilling engineers can identify potential surge prob-lems and optimize tripping operations for differentscenarios as to avoid complicated accidents to happenin wellbore during drilling operations
(4) The calculation of surge pressure in gas drillingmud two-phase is more complex than conventionalcalculation in single drillingmudDrill pipe operatingspeed in drilling operations should be strictly con-trolled and the surge pressure should be consideredin the design The greatest advantages of new modelare the ability to reduce nonproductive time (NPT) indrilling
Nomenclature
A Wellbore effective cross-area (m2)1198600 The outer drill pipe cross-area (m2)
1198602 Wellbore cross-area (m2)
1198603 The inner Drill pipe cross-area (m2)
119860119890 Effective cross-area of jet nozzle (m2)
119888 Wave velocity (ms)119889119904 One discrete length of wellbore (m)
14 Mathematical Problems in Engineering
119863 Wellbore effective diameter (m)1198631 The inner diameter of casing (m)
1198632 The outer diameter of casing (m)
1198633 The inner diameter of drill pipe (m)
1198634 The outer diameter of drill pipe (m)
119863119896119895 Differential equation coefficients
119864119896119895 Differential equation coefficients
119864119904 Modulus of elasticity of the drill pipe
(MPa)119864119891 Modulus of elasticity of formation (MPa)
1198642 Modulus of elasticity of casing (MPa)
119891119866 Shear stresses coefficient of gas interface
119891 Friction coefficient1198912 Wellbore friction coefficient
1198913 Drill pipe friction coefficient
119891119871 Shear stresses coefficient of drilling mud
119865119911 Pressure applied on the two-phase flow
(N)119892 Acceleration due to gravity (m2s)119892119888 Conversion factor
119892119879 Geothermal temperature gradient (Km)
119896119890 Wellbore effective roughness (m)
1198960 Roughness of outer pipe (m)
119896119894 Roughness of inner pipe (m)
119867 The depth of well (m)1198711 The length from pipe bottom to hole
bottom (m)1198712 The length from pipe bottom to well
head (m)119871119892 Flow pattern119898 Hydraulic radius (m)1198982 Wellbore hydraulic radius (m)
1198983 Drill pipe hydraulic radius (m)
119873 Parameter combiningThompson energyeffects
119875 Pressure (MPa)1198750 Initial pressure of wellbore (MPa)
1198752 Pressure within the empty wellbore
(MPa)1198753 Pressure within the drill pipe (MPa)
119875119888 Critical pressure (kPa)
119875119871 Pressure of drilling mud (MPa)
119875119903 Reduced pressure
119875119895119905 Pressure at node (119895 119905) (MPa)
119902119898 Volumetric flow rate of two phase (m3s)
119902119866 Velocity of the gas (m3s)
119902119871 Velocity of the drilling mud (m3s)
119876119871 Mud flow rate within the wellbore(m3h)
1198761 Drilling mud flow rate within hole(m3h)
1198762 Mud flow rate within the wellbore(m3h)
1198763 Mud flow rate within the drill pipe(m3h)
119876119908 Pump flow rate (m3s)
119876119895119905 Flow rate at time 119905 and nod 119895 (m3s)
119877 Constant of EOS (JKgK)
119904 The length of wellbore along wellbore(m)
119905 Time (s)119879 Temperature (K)119879119888 Critical temperature (K)
119879119903 Reduced temperature
119879119890119894 Undistributed temperature at a depth
(K)119879119890119887ℎ
Undistributed temperature of wellhead(K)
119879119891119887ℎ
Undistributed temperature at bottom-hole (K)
V Velocity of drilling mud and gas (ms)V119901 Velocity of drill pipe (ms)
V119898 Gas and drilling mud flow velocity (ms)
V119866 Gas flow velocity (ms)
V119904 Slip velocity (ms)
V119904119892 Superficial gas velocity (ms)
V119871 Drilling mud flow velocity (ms)
119882119898 Mass flow velocity (kgm3)
119911119866 Gas deviation factor
1199110 Initial calculation height of wellbore (m)
119911119887ℎ Total well depth from surface (m)
119911 Variable of distance from surface (m)
Greek Letters
120588119866 Gas density (kgm3)
120591119891 Frictional pressure gradient (Pam)
120591119908 Frictional pressure coefficient
120588119898 Gas and drilling density (kgm3)
1205880 Density under atmospheric pressure (kgm3)
120588119871 Drilling mud density (kgm3)
120590119904 Surface tension (Nm2)
120601119866 Gas void fraction
120601119871 Drilling mud holdup
120588119903 Reduced density
120583119871 Viscosity of drilling mud (Pasdots)
120579 The ground angle (rad)120583119891 Poissonrsquos ratio of formation
120583119904 Poissonrsquos ratio of drill pipe
1205832 Poissonrsquos ratio of casing
Δ119860 The change of the across areas (m2)
Subscripts
CPOP Closed pipe without pumpingDAPC Dynamic annular pressure controlEOS Equations of stateERV Relief-coupling valveMPD Managed pressure drillingOPOP Open pipe without pumpingOPWP Open pipe with pumpingR-K4 The fourth-order explicit Runge-KuttaRCD The rotating control device
Mathematical Problems in Engineering 15
Subscripts of Graph
119888 Wave velocity in two-phase flow (ms)119863 Wellbore effective diameter (m)119867 Depth of wellbore (m)119871 The length from drill pipe and drill pipe (m)119876119892 Gas influx rate at the bottomhole (m3h)
120601119866 Void fraction ()
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 51274170 andQC2012C128) Without their support this work would nothave been possible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of AirInjection in Annuli to manage pressure during CementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling Conferenceand Exhibition San Diego Calif USA March 2012
[4] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the Norh Africa Technical Conference andExhibition Cairo Egypt February 2012
[5] J Saponja A Adeleye and B Hucik ldquoManaged-pressuredrilling (MPD) field trials demonstrate technology valuerdquo inProceedings of the IADCSPE Drilling Conference Miami FlaUSA February 2006
[6] G Liu ldquoPredicting surge pressures that result from runninglinersrdquoWorld Oil vol 222 no 4 p 92 2001
[7] K Zhou J Yang B Zhong and L Tang ldquoSurge pressure inwellborerdquo Oil Drilling and Production Technology vol 12 no 2pp 1ndash10 99 1990
[8] R S Rudi Rubiandini ldquoNew formula of surge pressure fordetermining safe trip velocitiesrdquo in Proceedings of the SPE AsiaPacific Oil and Gas Conference and Exhibition pp 16ndash18 ausOctober 2000
[9] J Li Z Huang K Li and W Wu ldquoThe influence of collaron surge pressure caused by the drilling fluid viscous forceunder pumping conditionrdquoResearch Journal of Applied SciencesEngineering and Technology vol 5 no 5 pp 1781ndash1785 2040ndash7459 2013
[10] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvancedMaterials Research vol 468ndash471 pp 1736ndash1742 2012
[11] F Crespo and R Ahmed ldquoSurge-and-swab pressure predictionsfor yield-power-law drilling fluidsrdquo SPE Drilling amp Completionvol 27 no 4 pp 574ndash585 2012
[12] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[13] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquo PHDThesis vol 50 no 9 p 143 1989
[14] H Fan ldquoDevelopment of application software for dynamicsurge pressure while trippingrdquo Petroleum Drilling and Technol-ogy vol 23 no 4 pp 11ndash14 1995
[15] K Zhou J Yang B Zhong and L Tang ldquoSurge pressurein wellborerdquo Journal of Petroleum Exploration and ProductionTechnology vol 12 no 2 pp 1ndash10 1990
[16] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquoDissertation Abstracts International Bvol 50 no 9 1989
[17] Z Jiang ldquoStudies of wellbore surge pressure by a dynamicmethodrdquo Journal of Petroleum Exploration and ProductionTechnology vol 13 no 3 pp 39ndash44 97 1991
[18] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[19] D Johnston M Bracken and J Thornton ldquoAssessing pipecondition useful remaining life and maximum operating andsurge pressures when to change the pipe or control thepressurerdquoMagazine of the International Water Association vol14 no 5 pp 43ndash44 2012
[20] H Fan Y Chu and X Liu ldquoPrediction for wellbore dynamicsurge pressure while tripping a drill piperdquo Journal of ChinaUniversity of Petroleum vol 19 no 5 pp 36ndash41 1995
[21] C K Burgess J E Miller and K D Vondervort ldquoDownholesurge pressure reduction system and method of userdquo World9848 143 1998
[22] J P Allamon J E Miller and A M Macfarlane ldquoDownholeSurge Reduction Method and Apparatusrdquo 2004
[23] K S Bjoslashrkevoll R Rommetveit B Aas H Gjeraldstveit andA Merlo ldquoTransient gel breaking model for critical wellsapplications with field data verificationrdquo in Proceedings of the2003 Offshore Mediterranean Conference February 2003
[24] G H Nygaard E Johannessen J E Gravdal and F IversenldquoAutomatic coordinated control of pump rates and choke valvefor compensating pressure fluctuations during surge and swaboperationsrdquo in Proceedings of the IADCSPE Managed PressureDrilling and Underbalanced Operations Conference pp 120ndash125March 2007
[25] J V Fedevjcyk S L M Junqueira and C O R Negrao ldquoSurgeamp swab pressures in wells with cross-section changesrdquo BoletimTecnico da Producao de Petroleo vol 4 no 2 pp 221ndash234 2009
[26] S Tian G H Medley and C R Stone ldquoUnderstandingyield point effect onpressure surges critical to managing deepdifficultMPDwellsrdquoDrilling Contractor vol 65 no 4 p 46 4850 53 2009
[27] Y Sun Q Li and J Zhao ldquoNew method of predicting surgepressure apply to horizontal well based on caisson flowrdquoNaturalScience vol 2 no 12 pp 1394ndash1399 2150ndash4091 2010
[28] B T Lebele-Alawa and F E Oparadike ldquoAnalysis of the effectsof valve propagated pressure surge on pipe flowrdquo Engineeringvol 3 no 11 pp 1098ndash1101 2011
[29] X Kong Q Yuan and Y Qiu ldquoThe research of two-phasefluctuation pressure in wellbore tripping operationrdquo InnerMongolia Petrochemical Industry TE242 2011
[30] F Crespo and R Ahmed ldquoA simplified surge and swab pressuremodel for yield power law fluidsrdquo Journal of Petroleum Scienceand Engineering vol 101 pp 12ndash20 2013
16 Mathematical Problems in Engineering
[31] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
[32] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[33] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86-87 pp 127ndash136 2012
[34] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[35] M J SanchezComparison of Correlations for Predicting PressureLosses in Vertical Multiphase Annular Flow University of TulsaTulsa Okla USA 1972
[36] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[37] Y Lin X Kong Y Qiu and Q Yuan ldquoCalculation analysis ofpressure wave velocity in gas and drilling mud two-phase fluidin annulus during drilling operationsrdquo Mathematical Problemsin Engineering vol 2013 Article ID 318912 17 pages 2013
[38] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[39] H Junfang ldquoBalanced drilling and well controlrdquo PetroleumIndustry Press vol 1 article 33 1992
[40] W A Bacon Consideration of compressibility effects for applied-back- pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Va USA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
t (s)0 10 20 30 40 50 60
15
12
09
06
03
00
P (M
Pa)
Qg = 2427m3h
= 1500kgm3=
= 2000 kgm3=
120588m120588m
Figure 15 Surge pressure variation at different drilling fluid densi-ties
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 0m3h
Figure 16 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 0m3h)
of well As the gas influx rate increases the surge pressuredecreases When gas influx occurs the lag time is lengthenedremarkably compared with the condition of no gas influxThe longer the distance is the greater the lag time will beThe lag time is mainly determined by the propagation time ofpressure wave in the wellbore
Figures 19 and 20 illustrate the surge pressure caused bydifferent operating speeds of the drill pipe at various gasinflux rates at the bottom of well As drilling mudmixed withinflux gas not only reduces the pressure wave velocity butalso reduces the density of themixture density surge pressuremainly affected by wave velocity andmixture density reducesdrastically with the increase of gas influx rate at the bottomof well Figures 19 and 20 are compared to Figures 17 and 18
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 2427m3h
Figure 17 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
30
15
00
minus15
Qg = 8312m3h
Figure 18 Surge pressure variation at different pipe velocities (gasinflux 119876
119892= 8312m3h)
it can be concluded that the operating speed is also a majorfactor of surge pressure
53The Effects of Drill Pipe Operating Speed on Surge PressureFigure 21 illustrates the effect of drill pipe speed on the surgepressure The change of surge pressure is smaller in gasdrilling two-phase than in single-phase drilling mud Thesurge pressure becomes smaller and closes to zero after themovement is stopped due to greater friction force existing inwell wall during drill pipe operating
6 Conclusions
With full consideration of the important factors such as wavevelocity in real-time the compressibility of the gas phasegas influx rate pressure temperature and well parametersa new surge pressure model has been proposed based on the
Mathematical Problems in Engineering 13Va
lue
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 2427m3h
Figure 19 Surge pressure and fluid velocity variation at differentpipe velocities (gas influx 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 8312m3h
Figure 20 Surge pressure and fluid velocity variation at differentgas influxes (gas influx 119876
119892= 8312m3h)
30
24
18
12
06
00
Valu
e
t (s)0 10 20 30 40 50 60
Pipe velocity (ms)Pipe velocity (ms)Surge pressure (MPa)
Surge pressure (MPa)
Qg = 2427m3h
Figure 21 Surge pressure variations at different pipe velocities (gasinflux 119876
119892= 2427m3h)
mass conservation equations the momentum conservationequations characteristic line method and the gas-drillingmud equations of state (EOS) duringMPDoperations Solvedby the difference method and fourth-order explicit Runge-Kutta method the model is used to predict surge pressurefor different drill pipe operating speed and gas influx rates inMPD operations The main conclusions can be summarizedas follows
(1) An accurate mathematical model to predict surgepressure after gas influx is of great importancebecause of the effects of surge pressure on operationalsafety and success The new model provides betterprediction of surge pressures in comparison with theexisting experiment
(2) The surge pressure tends to increase with the increaseof drill pipe operating speed and decrease of gas influxrate andwell diameter Considering that the gas influxoccurs the surge pressure weakens obviously Withthe increases in well depth the maximum allowabletripping speed gradually become smaller the surgepressure increases at the bottom of well and drillingsafety window is accordingly narrower The surgepressure can cause a significant time lag if the gasinflux occurs at the bottom of well and the timelag is mainly determined by the propagation time ofpressure wave in the wellbore
(3) The surge pressure obviously varies at different sce-narios (OPOP CPOP and OPWP resp) duringthe tripping operations Surge pressure increased inCPOP scenario than in OPOP scenario Especiallywhen the maximum surge pressure occurs a strongincrease of surge pressure is observed at OPWPscenario By running the new surge pressure modeldrilling engineers can identify potential surge prob-lems and optimize tripping operations for differentscenarios as to avoid complicated accidents to happenin wellbore during drilling operations
(4) The calculation of surge pressure in gas drillingmud two-phase is more complex than conventionalcalculation in single drillingmudDrill pipe operatingspeed in drilling operations should be strictly con-trolled and the surge pressure should be consideredin the design The greatest advantages of new modelare the ability to reduce nonproductive time (NPT) indrilling
Nomenclature
A Wellbore effective cross-area (m2)1198600 The outer drill pipe cross-area (m2)
1198602 Wellbore cross-area (m2)
1198603 The inner Drill pipe cross-area (m2)
119860119890 Effective cross-area of jet nozzle (m2)
119888 Wave velocity (ms)119889119904 One discrete length of wellbore (m)
14 Mathematical Problems in Engineering
119863 Wellbore effective diameter (m)1198631 The inner diameter of casing (m)
1198632 The outer diameter of casing (m)
1198633 The inner diameter of drill pipe (m)
1198634 The outer diameter of drill pipe (m)
119863119896119895 Differential equation coefficients
119864119896119895 Differential equation coefficients
119864119904 Modulus of elasticity of the drill pipe
(MPa)119864119891 Modulus of elasticity of formation (MPa)
1198642 Modulus of elasticity of casing (MPa)
119891119866 Shear stresses coefficient of gas interface
119891 Friction coefficient1198912 Wellbore friction coefficient
1198913 Drill pipe friction coefficient
119891119871 Shear stresses coefficient of drilling mud
119865119911 Pressure applied on the two-phase flow
(N)119892 Acceleration due to gravity (m2s)119892119888 Conversion factor
119892119879 Geothermal temperature gradient (Km)
119896119890 Wellbore effective roughness (m)
1198960 Roughness of outer pipe (m)
119896119894 Roughness of inner pipe (m)
119867 The depth of well (m)1198711 The length from pipe bottom to hole
bottom (m)1198712 The length from pipe bottom to well
head (m)119871119892 Flow pattern119898 Hydraulic radius (m)1198982 Wellbore hydraulic radius (m)
1198983 Drill pipe hydraulic radius (m)
119873 Parameter combiningThompson energyeffects
119875 Pressure (MPa)1198750 Initial pressure of wellbore (MPa)
1198752 Pressure within the empty wellbore
(MPa)1198753 Pressure within the drill pipe (MPa)
119875119888 Critical pressure (kPa)
119875119871 Pressure of drilling mud (MPa)
119875119903 Reduced pressure
119875119895119905 Pressure at node (119895 119905) (MPa)
119902119898 Volumetric flow rate of two phase (m3s)
119902119866 Velocity of the gas (m3s)
119902119871 Velocity of the drilling mud (m3s)
119876119871 Mud flow rate within the wellbore(m3h)
1198761 Drilling mud flow rate within hole(m3h)
1198762 Mud flow rate within the wellbore(m3h)
1198763 Mud flow rate within the drill pipe(m3h)
119876119908 Pump flow rate (m3s)
119876119895119905 Flow rate at time 119905 and nod 119895 (m3s)
119877 Constant of EOS (JKgK)
119904 The length of wellbore along wellbore(m)
119905 Time (s)119879 Temperature (K)119879119888 Critical temperature (K)
119879119903 Reduced temperature
119879119890119894 Undistributed temperature at a depth
(K)119879119890119887ℎ
Undistributed temperature of wellhead(K)
119879119891119887ℎ
Undistributed temperature at bottom-hole (K)
V Velocity of drilling mud and gas (ms)V119901 Velocity of drill pipe (ms)
V119898 Gas and drilling mud flow velocity (ms)
V119866 Gas flow velocity (ms)
V119904 Slip velocity (ms)
V119904119892 Superficial gas velocity (ms)
V119871 Drilling mud flow velocity (ms)
119882119898 Mass flow velocity (kgm3)
119911119866 Gas deviation factor
1199110 Initial calculation height of wellbore (m)
119911119887ℎ Total well depth from surface (m)
119911 Variable of distance from surface (m)
Greek Letters
120588119866 Gas density (kgm3)
120591119891 Frictional pressure gradient (Pam)
120591119908 Frictional pressure coefficient
120588119898 Gas and drilling density (kgm3)
1205880 Density under atmospheric pressure (kgm3)
120588119871 Drilling mud density (kgm3)
120590119904 Surface tension (Nm2)
120601119866 Gas void fraction
120601119871 Drilling mud holdup
120588119903 Reduced density
120583119871 Viscosity of drilling mud (Pasdots)
120579 The ground angle (rad)120583119891 Poissonrsquos ratio of formation
120583119904 Poissonrsquos ratio of drill pipe
1205832 Poissonrsquos ratio of casing
Δ119860 The change of the across areas (m2)
Subscripts
CPOP Closed pipe without pumpingDAPC Dynamic annular pressure controlEOS Equations of stateERV Relief-coupling valveMPD Managed pressure drillingOPOP Open pipe without pumpingOPWP Open pipe with pumpingR-K4 The fourth-order explicit Runge-KuttaRCD The rotating control device
Mathematical Problems in Engineering 15
Subscripts of Graph
119888 Wave velocity in two-phase flow (ms)119863 Wellbore effective diameter (m)119867 Depth of wellbore (m)119871 The length from drill pipe and drill pipe (m)119876119892 Gas influx rate at the bottomhole (m3h)
120601119866 Void fraction ()
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 51274170 andQC2012C128) Without their support this work would nothave been possible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of AirInjection in Annuli to manage pressure during CementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling Conferenceand Exhibition San Diego Calif USA March 2012
[4] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the Norh Africa Technical Conference andExhibition Cairo Egypt February 2012
[5] J Saponja A Adeleye and B Hucik ldquoManaged-pressuredrilling (MPD) field trials demonstrate technology valuerdquo inProceedings of the IADCSPE Drilling Conference Miami FlaUSA February 2006
[6] G Liu ldquoPredicting surge pressures that result from runninglinersrdquoWorld Oil vol 222 no 4 p 92 2001
[7] K Zhou J Yang B Zhong and L Tang ldquoSurge pressure inwellborerdquo Oil Drilling and Production Technology vol 12 no 2pp 1ndash10 99 1990
[8] R S Rudi Rubiandini ldquoNew formula of surge pressure fordetermining safe trip velocitiesrdquo in Proceedings of the SPE AsiaPacific Oil and Gas Conference and Exhibition pp 16ndash18 ausOctober 2000
[9] J Li Z Huang K Li and W Wu ldquoThe influence of collaron surge pressure caused by the drilling fluid viscous forceunder pumping conditionrdquoResearch Journal of Applied SciencesEngineering and Technology vol 5 no 5 pp 1781ndash1785 2040ndash7459 2013
[10] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvancedMaterials Research vol 468ndash471 pp 1736ndash1742 2012
[11] F Crespo and R Ahmed ldquoSurge-and-swab pressure predictionsfor yield-power-law drilling fluidsrdquo SPE Drilling amp Completionvol 27 no 4 pp 574ndash585 2012
[12] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[13] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquo PHDThesis vol 50 no 9 p 143 1989
[14] H Fan ldquoDevelopment of application software for dynamicsurge pressure while trippingrdquo Petroleum Drilling and Technol-ogy vol 23 no 4 pp 11ndash14 1995
[15] K Zhou J Yang B Zhong and L Tang ldquoSurge pressurein wellborerdquo Journal of Petroleum Exploration and ProductionTechnology vol 12 no 2 pp 1ndash10 1990
[16] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquoDissertation Abstracts International Bvol 50 no 9 1989
[17] Z Jiang ldquoStudies of wellbore surge pressure by a dynamicmethodrdquo Journal of Petroleum Exploration and ProductionTechnology vol 13 no 3 pp 39ndash44 97 1991
[18] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[19] D Johnston M Bracken and J Thornton ldquoAssessing pipecondition useful remaining life and maximum operating andsurge pressures when to change the pipe or control thepressurerdquoMagazine of the International Water Association vol14 no 5 pp 43ndash44 2012
[20] H Fan Y Chu and X Liu ldquoPrediction for wellbore dynamicsurge pressure while tripping a drill piperdquo Journal of ChinaUniversity of Petroleum vol 19 no 5 pp 36ndash41 1995
[21] C K Burgess J E Miller and K D Vondervort ldquoDownholesurge pressure reduction system and method of userdquo World9848 143 1998
[22] J P Allamon J E Miller and A M Macfarlane ldquoDownholeSurge Reduction Method and Apparatusrdquo 2004
[23] K S Bjoslashrkevoll R Rommetveit B Aas H Gjeraldstveit andA Merlo ldquoTransient gel breaking model for critical wellsapplications with field data verificationrdquo in Proceedings of the2003 Offshore Mediterranean Conference February 2003
[24] G H Nygaard E Johannessen J E Gravdal and F IversenldquoAutomatic coordinated control of pump rates and choke valvefor compensating pressure fluctuations during surge and swaboperationsrdquo in Proceedings of the IADCSPE Managed PressureDrilling and Underbalanced Operations Conference pp 120ndash125March 2007
[25] J V Fedevjcyk S L M Junqueira and C O R Negrao ldquoSurgeamp swab pressures in wells with cross-section changesrdquo BoletimTecnico da Producao de Petroleo vol 4 no 2 pp 221ndash234 2009
[26] S Tian G H Medley and C R Stone ldquoUnderstandingyield point effect onpressure surges critical to managing deepdifficultMPDwellsrdquoDrilling Contractor vol 65 no 4 p 46 4850 53 2009
[27] Y Sun Q Li and J Zhao ldquoNew method of predicting surgepressure apply to horizontal well based on caisson flowrdquoNaturalScience vol 2 no 12 pp 1394ndash1399 2150ndash4091 2010
[28] B T Lebele-Alawa and F E Oparadike ldquoAnalysis of the effectsof valve propagated pressure surge on pipe flowrdquo Engineeringvol 3 no 11 pp 1098ndash1101 2011
[29] X Kong Q Yuan and Y Qiu ldquoThe research of two-phasefluctuation pressure in wellbore tripping operationrdquo InnerMongolia Petrochemical Industry TE242 2011
[30] F Crespo and R Ahmed ldquoA simplified surge and swab pressuremodel for yield power law fluidsrdquo Journal of Petroleum Scienceand Engineering vol 101 pp 12ndash20 2013
16 Mathematical Problems in Engineering
[31] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
[32] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[33] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86-87 pp 127ndash136 2012
[34] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[35] M J SanchezComparison of Correlations for Predicting PressureLosses in Vertical Multiphase Annular Flow University of TulsaTulsa Okla USA 1972
[36] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[37] Y Lin X Kong Y Qiu and Q Yuan ldquoCalculation analysis ofpressure wave velocity in gas and drilling mud two-phase fluidin annulus during drilling operationsrdquo Mathematical Problemsin Engineering vol 2013 Article ID 318912 17 pages 2013
[38] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[39] H Junfang ldquoBalanced drilling and well controlrdquo PetroleumIndustry Press vol 1 article 33 1992
[40] W A Bacon Consideration of compressibility effects for applied-back- pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Va USA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13Va
lue
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 2427m3h
Figure 19 Surge pressure and fluid velocity variation at differentpipe velocities (gas influx 119876
119892= 2427m3h)
Valu
e
t (s)0 10 20 30 40 50 60
Pipe acceleration (ms2)Pipe velocity (ms)
Fluid velocity (ms)Surge pressure (MPa)
3
2
1
0
minus1
minus2
Qg = 8312m3h
Figure 20 Surge pressure and fluid velocity variation at differentgas influxes (gas influx 119876
119892= 8312m3h)
30
24
18
12
06
00
Valu
e
t (s)0 10 20 30 40 50 60
Pipe velocity (ms)Pipe velocity (ms)Surge pressure (MPa)
Surge pressure (MPa)
Qg = 2427m3h
Figure 21 Surge pressure variations at different pipe velocities (gasinflux 119876
119892= 2427m3h)
mass conservation equations the momentum conservationequations characteristic line method and the gas-drillingmud equations of state (EOS) duringMPDoperations Solvedby the difference method and fourth-order explicit Runge-Kutta method the model is used to predict surge pressurefor different drill pipe operating speed and gas influx rates inMPD operations The main conclusions can be summarizedas follows
(1) An accurate mathematical model to predict surgepressure after gas influx is of great importancebecause of the effects of surge pressure on operationalsafety and success The new model provides betterprediction of surge pressures in comparison with theexisting experiment
(2) The surge pressure tends to increase with the increaseof drill pipe operating speed and decrease of gas influxrate andwell diameter Considering that the gas influxoccurs the surge pressure weakens obviously Withthe increases in well depth the maximum allowabletripping speed gradually become smaller the surgepressure increases at the bottom of well and drillingsafety window is accordingly narrower The surgepressure can cause a significant time lag if the gasinflux occurs at the bottom of well and the timelag is mainly determined by the propagation time ofpressure wave in the wellbore
(3) The surge pressure obviously varies at different sce-narios (OPOP CPOP and OPWP resp) duringthe tripping operations Surge pressure increased inCPOP scenario than in OPOP scenario Especiallywhen the maximum surge pressure occurs a strongincrease of surge pressure is observed at OPWPscenario By running the new surge pressure modeldrilling engineers can identify potential surge prob-lems and optimize tripping operations for differentscenarios as to avoid complicated accidents to happenin wellbore during drilling operations
(4) The calculation of surge pressure in gas drillingmud two-phase is more complex than conventionalcalculation in single drillingmudDrill pipe operatingspeed in drilling operations should be strictly con-trolled and the surge pressure should be consideredin the design The greatest advantages of new modelare the ability to reduce nonproductive time (NPT) indrilling
Nomenclature
A Wellbore effective cross-area (m2)1198600 The outer drill pipe cross-area (m2)
1198602 Wellbore cross-area (m2)
1198603 The inner Drill pipe cross-area (m2)
119860119890 Effective cross-area of jet nozzle (m2)
119888 Wave velocity (ms)119889119904 One discrete length of wellbore (m)
14 Mathematical Problems in Engineering
119863 Wellbore effective diameter (m)1198631 The inner diameter of casing (m)
1198632 The outer diameter of casing (m)
1198633 The inner diameter of drill pipe (m)
1198634 The outer diameter of drill pipe (m)
119863119896119895 Differential equation coefficients
119864119896119895 Differential equation coefficients
119864119904 Modulus of elasticity of the drill pipe
(MPa)119864119891 Modulus of elasticity of formation (MPa)
1198642 Modulus of elasticity of casing (MPa)
119891119866 Shear stresses coefficient of gas interface
119891 Friction coefficient1198912 Wellbore friction coefficient
1198913 Drill pipe friction coefficient
119891119871 Shear stresses coefficient of drilling mud
119865119911 Pressure applied on the two-phase flow
(N)119892 Acceleration due to gravity (m2s)119892119888 Conversion factor
119892119879 Geothermal temperature gradient (Km)
119896119890 Wellbore effective roughness (m)
1198960 Roughness of outer pipe (m)
119896119894 Roughness of inner pipe (m)
119867 The depth of well (m)1198711 The length from pipe bottom to hole
bottom (m)1198712 The length from pipe bottom to well
head (m)119871119892 Flow pattern119898 Hydraulic radius (m)1198982 Wellbore hydraulic radius (m)
1198983 Drill pipe hydraulic radius (m)
119873 Parameter combiningThompson energyeffects
119875 Pressure (MPa)1198750 Initial pressure of wellbore (MPa)
1198752 Pressure within the empty wellbore
(MPa)1198753 Pressure within the drill pipe (MPa)
119875119888 Critical pressure (kPa)
119875119871 Pressure of drilling mud (MPa)
119875119903 Reduced pressure
119875119895119905 Pressure at node (119895 119905) (MPa)
119902119898 Volumetric flow rate of two phase (m3s)
119902119866 Velocity of the gas (m3s)
119902119871 Velocity of the drilling mud (m3s)
119876119871 Mud flow rate within the wellbore(m3h)
1198761 Drilling mud flow rate within hole(m3h)
1198762 Mud flow rate within the wellbore(m3h)
1198763 Mud flow rate within the drill pipe(m3h)
119876119908 Pump flow rate (m3s)
119876119895119905 Flow rate at time 119905 and nod 119895 (m3s)
119877 Constant of EOS (JKgK)
119904 The length of wellbore along wellbore(m)
119905 Time (s)119879 Temperature (K)119879119888 Critical temperature (K)
119879119903 Reduced temperature
119879119890119894 Undistributed temperature at a depth
(K)119879119890119887ℎ
Undistributed temperature of wellhead(K)
119879119891119887ℎ
Undistributed temperature at bottom-hole (K)
V Velocity of drilling mud and gas (ms)V119901 Velocity of drill pipe (ms)
V119898 Gas and drilling mud flow velocity (ms)
V119866 Gas flow velocity (ms)
V119904 Slip velocity (ms)
V119904119892 Superficial gas velocity (ms)
V119871 Drilling mud flow velocity (ms)
119882119898 Mass flow velocity (kgm3)
119911119866 Gas deviation factor
1199110 Initial calculation height of wellbore (m)
119911119887ℎ Total well depth from surface (m)
119911 Variable of distance from surface (m)
Greek Letters
120588119866 Gas density (kgm3)
120591119891 Frictional pressure gradient (Pam)
120591119908 Frictional pressure coefficient
120588119898 Gas and drilling density (kgm3)
1205880 Density under atmospheric pressure (kgm3)
120588119871 Drilling mud density (kgm3)
120590119904 Surface tension (Nm2)
120601119866 Gas void fraction
120601119871 Drilling mud holdup
120588119903 Reduced density
120583119871 Viscosity of drilling mud (Pasdots)
120579 The ground angle (rad)120583119891 Poissonrsquos ratio of formation
120583119904 Poissonrsquos ratio of drill pipe
1205832 Poissonrsquos ratio of casing
Δ119860 The change of the across areas (m2)
Subscripts
CPOP Closed pipe without pumpingDAPC Dynamic annular pressure controlEOS Equations of stateERV Relief-coupling valveMPD Managed pressure drillingOPOP Open pipe without pumpingOPWP Open pipe with pumpingR-K4 The fourth-order explicit Runge-KuttaRCD The rotating control device
Mathematical Problems in Engineering 15
Subscripts of Graph
119888 Wave velocity in two-phase flow (ms)119863 Wellbore effective diameter (m)119867 Depth of wellbore (m)119871 The length from drill pipe and drill pipe (m)119876119892 Gas influx rate at the bottomhole (m3h)
120601119866 Void fraction ()
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 51274170 andQC2012C128) Without their support this work would nothave been possible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of AirInjection in Annuli to manage pressure during CementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling Conferenceand Exhibition San Diego Calif USA March 2012
[4] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the Norh Africa Technical Conference andExhibition Cairo Egypt February 2012
[5] J Saponja A Adeleye and B Hucik ldquoManaged-pressuredrilling (MPD) field trials demonstrate technology valuerdquo inProceedings of the IADCSPE Drilling Conference Miami FlaUSA February 2006
[6] G Liu ldquoPredicting surge pressures that result from runninglinersrdquoWorld Oil vol 222 no 4 p 92 2001
[7] K Zhou J Yang B Zhong and L Tang ldquoSurge pressure inwellborerdquo Oil Drilling and Production Technology vol 12 no 2pp 1ndash10 99 1990
[8] R S Rudi Rubiandini ldquoNew formula of surge pressure fordetermining safe trip velocitiesrdquo in Proceedings of the SPE AsiaPacific Oil and Gas Conference and Exhibition pp 16ndash18 ausOctober 2000
[9] J Li Z Huang K Li and W Wu ldquoThe influence of collaron surge pressure caused by the drilling fluid viscous forceunder pumping conditionrdquoResearch Journal of Applied SciencesEngineering and Technology vol 5 no 5 pp 1781ndash1785 2040ndash7459 2013
[10] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvancedMaterials Research vol 468ndash471 pp 1736ndash1742 2012
[11] F Crespo and R Ahmed ldquoSurge-and-swab pressure predictionsfor yield-power-law drilling fluidsrdquo SPE Drilling amp Completionvol 27 no 4 pp 574ndash585 2012
[12] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[13] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquo PHDThesis vol 50 no 9 p 143 1989
[14] H Fan ldquoDevelopment of application software for dynamicsurge pressure while trippingrdquo Petroleum Drilling and Technol-ogy vol 23 no 4 pp 11ndash14 1995
[15] K Zhou J Yang B Zhong and L Tang ldquoSurge pressurein wellborerdquo Journal of Petroleum Exploration and ProductionTechnology vol 12 no 2 pp 1ndash10 1990
[16] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquoDissertation Abstracts International Bvol 50 no 9 1989
[17] Z Jiang ldquoStudies of wellbore surge pressure by a dynamicmethodrdquo Journal of Petroleum Exploration and ProductionTechnology vol 13 no 3 pp 39ndash44 97 1991
[18] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[19] D Johnston M Bracken and J Thornton ldquoAssessing pipecondition useful remaining life and maximum operating andsurge pressures when to change the pipe or control thepressurerdquoMagazine of the International Water Association vol14 no 5 pp 43ndash44 2012
[20] H Fan Y Chu and X Liu ldquoPrediction for wellbore dynamicsurge pressure while tripping a drill piperdquo Journal of ChinaUniversity of Petroleum vol 19 no 5 pp 36ndash41 1995
[21] C K Burgess J E Miller and K D Vondervort ldquoDownholesurge pressure reduction system and method of userdquo World9848 143 1998
[22] J P Allamon J E Miller and A M Macfarlane ldquoDownholeSurge Reduction Method and Apparatusrdquo 2004
[23] K S Bjoslashrkevoll R Rommetveit B Aas H Gjeraldstveit andA Merlo ldquoTransient gel breaking model for critical wellsapplications with field data verificationrdquo in Proceedings of the2003 Offshore Mediterranean Conference February 2003
[24] G H Nygaard E Johannessen J E Gravdal and F IversenldquoAutomatic coordinated control of pump rates and choke valvefor compensating pressure fluctuations during surge and swaboperationsrdquo in Proceedings of the IADCSPE Managed PressureDrilling and Underbalanced Operations Conference pp 120ndash125March 2007
[25] J V Fedevjcyk S L M Junqueira and C O R Negrao ldquoSurgeamp swab pressures in wells with cross-section changesrdquo BoletimTecnico da Producao de Petroleo vol 4 no 2 pp 221ndash234 2009
[26] S Tian G H Medley and C R Stone ldquoUnderstandingyield point effect onpressure surges critical to managing deepdifficultMPDwellsrdquoDrilling Contractor vol 65 no 4 p 46 4850 53 2009
[27] Y Sun Q Li and J Zhao ldquoNew method of predicting surgepressure apply to horizontal well based on caisson flowrdquoNaturalScience vol 2 no 12 pp 1394ndash1399 2150ndash4091 2010
[28] B T Lebele-Alawa and F E Oparadike ldquoAnalysis of the effectsof valve propagated pressure surge on pipe flowrdquo Engineeringvol 3 no 11 pp 1098ndash1101 2011
[29] X Kong Q Yuan and Y Qiu ldquoThe research of two-phasefluctuation pressure in wellbore tripping operationrdquo InnerMongolia Petrochemical Industry TE242 2011
[30] F Crespo and R Ahmed ldquoA simplified surge and swab pressuremodel for yield power law fluidsrdquo Journal of Petroleum Scienceand Engineering vol 101 pp 12ndash20 2013
16 Mathematical Problems in Engineering
[31] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
[32] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[33] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86-87 pp 127ndash136 2012
[34] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[35] M J SanchezComparison of Correlations for Predicting PressureLosses in Vertical Multiphase Annular Flow University of TulsaTulsa Okla USA 1972
[36] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[37] Y Lin X Kong Y Qiu and Q Yuan ldquoCalculation analysis ofpressure wave velocity in gas and drilling mud two-phase fluidin annulus during drilling operationsrdquo Mathematical Problemsin Engineering vol 2013 Article ID 318912 17 pages 2013
[38] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[39] H Junfang ldquoBalanced drilling and well controlrdquo PetroleumIndustry Press vol 1 article 33 1992
[40] W A Bacon Consideration of compressibility effects for applied-back- pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Va USA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
119863 Wellbore effective diameter (m)1198631 The inner diameter of casing (m)
1198632 The outer diameter of casing (m)
1198633 The inner diameter of drill pipe (m)
1198634 The outer diameter of drill pipe (m)
119863119896119895 Differential equation coefficients
119864119896119895 Differential equation coefficients
119864119904 Modulus of elasticity of the drill pipe
(MPa)119864119891 Modulus of elasticity of formation (MPa)
1198642 Modulus of elasticity of casing (MPa)
119891119866 Shear stresses coefficient of gas interface
119891 Friction coefficient1198912 Wellbore friction coefficient
1198913 Drill pipe friction coefficient
119891119871 Shear stresses coefficient of drilling mud
119865119911 Pressure applied on the two-phase flow
(N)119892 Acceleration due to gravity (m2s)119892119888 Conversion factor
119892119879 Geothermal temperature gradient (Km)
119896119890 Wellbore effective roughness (m)
1198960 Roughness of outer pipe (m)
119896119894 Roughness of inner pipe (m)
119867 The depth of well (m)1198711 The length from pipe bottom to hole
bottom (m)1198712 The length from pipe bottom to well
head (m)119871119892 Flow pattern119898 Hydraulic radius (m)1198982 Wellbore hydraulic radius (m)
1198983 Drill pipe hydraulic radius (m)
119873 Parameter combiningThompson energyeffects
119875 Pressure (MPa)1198750 Initial pressure of wellbore (MPa)
1198752 Pressure within the empty wellbore
(MPa)1198753 Pressure within the drill pipe (MPa)
119875119888 Critical pressure (kPa)
119875119871 Pressure of drilling mud (MPa)
119875119903 Reduced pressure
119875119895119905 Pressure at node (119895 119905) (MPa)
119902119898 Volumetric flow rate of two phase (m3s)
119902119866 Velocity of the gas (m3s)
119902119871 Velocity of the drilling mud (m3s)
119876119871 Mud flow rate within the wellbore(m3h)
1198761 Drilling mud flow rate within hole(m3h)
1198762 Mud flow rate within the wellbore(m3h)
1198763 Mud flow rate within the drill pipe(m3h)
119876119908 Pump flow rate (m3s)
119876119895119905 Flow rate at time 119905 and nod 119895 (m3s)
119877 Constant of EOS (JKgK)
119904 The length of wellbore along wellbore(m)
119905 Time (s)119879 Temperature (K)119879119888 Critical temperature (K)
119879119903 Reduced temperature
119879119890119894 Undistributed temperature at a depth
(K)119879119890119887ℎ
Undistributed temperature of wellhead(K)
119879119891119887ℎ
Undistributed temperature at bottom-hole (K)
V Velocity of drilling mud and gas (ms)V119901 Velocity of drill pipe (ms)
V119898 Gas and drilling mud flow velocity (ms)
V119866 Gas flow velocity (ms)
V119904 Slip velocity (ms)
V119904119892 Superficial gas velocity (ms)
V119871 Drilling mud flow velocity (ms)
119882119898 Mass flow velocity (kgm3)
119911119866 Gas deviation factor
1199110 Initial calculation height of wellbore (m)
119911119887ℎ Total well depth from surface (m)
119911 Variable of distance from surface (m)
Greek Letters
120588119866 Gas density (kgm3)
120591119891 Frictional pressure gradient (Pam)
120591119908 Frictional pressure coefficient
120588119898 Gas and drilling density (kgm3)
1205880 Density under atmospheric pressure (kgm3)
120588119871 Drilling mud density (kgm3)
120590119904 Surface tension (Nm2)
120601119866 Gas void fraction
120601119871 Drilling mud holdup
120588119903 Reduced density
120583119871 Viscosity of drilling mud (Pasdots)
120579 The ground angle (rad)120583119891 Poissonrsquos ratio of formation
120583119904 Poissonrsquos ratio of drill pipe
1205832 Poissonrsquos ratio of casing
Δ119860 The change of the across areas (m2)
Subscripts
CPOP Closed pipe without pumpingDAPC Dynamic annular pressure controlEOS Equations of stateERV Relief-coupling valveMPD Managed pressure drillingOPOP Open pipe without pumpingOPWP Open pipe with pumpingR-K4 The fourth-order explicit Runge-KuttaRCD The rotating control device
Mathematical Problems in Engineering 15
Subscripts of Graph
119888 Wave velocity in two-phase flow (ms)119863 Wellbore effective diameter (m)119867 Depth of wellbore (m)119871 The length from drill pipe and drill pipe (m)119876119892 Gas influx rate at the bottomhole (m3h)
120601119866 Void fraction ()
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 51274170 andQC2012C128) Without their support this work would nothave been possible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of AirInjection in Annuli to manage pressure during CementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling Conferenceand Exhibition San Diego Calif USA March 2012
[4] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the Norh Africa Technical Conference andExhibition Cairo Egypt February 2012
[5] J Saponja A Adeleye and B Hucik ldquoManaged-pressuredrilling (MPD) field trials demonstrate technology valuerdquo inProceedings of the IADCSPE Drilling Conference Miami FlaUSA February 2006
[6] G Liu ldquoPredicting surge pressures that result from runninglinersrdquoWorld Oil vol 222 no 4 p 92 2001
[7] K Zhou J Yang B Zhong and L Tang ldquoSurge pressure inwellborerdquo Oil Drilling and Production Technology vol 12 no 2pp 1ndash10 99 1990
[8] R S Rudi Rubiandini ldquoNew formula of surge pressure fordetermining safe trip velocitiesrdquo in Proceedings of the SPE AsiaPacific Oil and Gas Conference and Exhibition pp 16ndash18 ausOctober 2000
[9] J Li Z Huang K Li and W Wu ldquoThe influence of collaron surge pressure caused by the drilling fluid viscous forceunder pumping conditionrdquoResearch Journal of Applied SciencesEngineering and Technology vol 5 no 5 pp 1781ndash1785 2040ndash7459 2013
[10] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvancedMaterials Research vol 468ndash471 pp 1736ndash1742 2012
[11] F Crespo and R Ahmed ldquoSurge-and-swab pressure predictionsfor yield-power-law drilling fluidsrdquo SPE Drilling amp Completionvol 27 no 4 pp 574ndash585 2012
[12] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[13] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquo PHDThesis vol 50 no 9 p 143 1989
[14] H Fan ldquoDevelopment of application software for dynamicsurge pressure while trippingrdquo Petroleum Drilling and Technol-ogy vol 23 no 4 pp 11ndash14 1995
[15] K Zhou J Yang B Zhong and L Tang ldquoSurge pressurein wellborerdquo Journal of Petroleum Exploration and ProductionTechnology vol 12 no 2 pp 1ndash10 1990
[16] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquoDissertation Abstracts International Bvol 50 no 9 1989
[17] Z Jiang ldquoStudies of wellbore surge pressure by a dynamicmethodrdquo Journal of Petroleum Exploration and ProductionTechnology vol 13 no 3 pp 39ndash44 97 1991
[18] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[19] D Johnston M Bracken and J Thornton ldquoAssessing pipecondition useful remaining life and maximum operating andsurge pressures when to change the pipe or control thepressurerdquoMagazine of the International Water Association vol14 no 5 pp 43ndash44 2012
[20] H Fan Y Chu and X Liu ldquoPrediction for wellbore dynamicsurge pressure while tripping a drill piperdquo Journal of ChinaUniversity of Petroleum vol 19 no 5 pp 36ndash41 1995
[21] C K Burgess J E Miller and K D Vondervort ldquoDownholesurge pressure reduction system and method of userdquo World9848 143 1998
[22] J P Allamon J E Miller and A M Macfarlane ldquoDownholeSurge Reduction Method and Apparatusrdquo 2004
[23] K S Bjoslashrkevoll R Rommetveit B Aas H Gjeraldstveit andA Merlo ldquoTransient gel breaking model for critical wellsapplications with field data verificationrdquo in Proceedings of the2003 Offshore Mediterranean Conference February 2003
[24] G H Nygaard E Johannessen J E Gravdal and F IversenldquoAutomatic coordinated control of pump rates and choke valvefor compensating pressure fluctuations during surge and swaboperationsrdquo in Proceedings of the IADCSPE Managed PressureDrilling and Underbalanced Operations Conference pp 120ndash125March 2007
[25] J V Fedevjcyk S L M Junqueira and C O R Negrao ldquoSurgeamp swab pressures in wells with cross-section changesrdquo BoletimTecnico da Producao de Petroleo vol 4 no 2 pp 221ndash234 2009
[26] S Tian G H Medley and C R Stone ldquoUnderstandingyield point effect onpressure surges critical to managing deepdifficultMPDwellsrdquoDrilling Contractor vol 65 no 4 p 46 4850 53 2009
[27] Y Sun Q Li and J Zhao ldquoNew method of predicting surgepressure apply to horizontal well based on caisson flowrdquoNaturalScience vol 2 no 12 pp 1394ndash1399 2150ndash4091 2010
[28] B T Lebele-Alawa and F E Oparadike ldquoAnalysis of the effectsof valve propagated pressure surge on pipe flowrdquo Engineeringvol 3 no 11 pp 1098ndash1101 2011
[29] X Kong Q Yuan and Y Qiu ldquoThe research of two-phasefluctuation pressure in wellbore tripping operationrdquo InnerMongolia Petrochemical Industry TE242 2011
[30] F Crespo and R Ahmed ldquoA simplified surge and swab pressuremodel for yield power law fluidsrdquo Journal of Petroleum Scienceand Engineering vol 101 pp 12ndash20 2013
16 Mathematical Problems in Engineering
[31] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
[32] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[33] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86-87 pp 127ndash136 2012
[34] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[35] M J SanchezComparison of Correlations for Predicting PressureLosses in Vertical Multiphase Annular Flow University of TulsaTulsa Okla USA 1972
[36] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[37] Y Lin X Kong Y Qiu and Q Yuan ldquoCalculation analysis ofpressure wave velocity in gas and drilling mud two-phase fluidin annulus during drilling operationsrdquo Mathematical Problemsin Engineering vol 2013 Article ID 318912 17 pages 2013
[38] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[39] H Junfang ldquoBalanced drilling and well controlrdquo PetroleumIndustry Press vol 1 article 33 1992
[40] W A Bacon Consideration of compressibility effects for applied-back- pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Va USA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
Subscripts of Graph
119888 Wave velocity in two-phase flow (ms)119863 Wellbore effective diameter (m)119867 Depth of wellbore (m)119871 The length from drill pipe and drill pipe (m)119876119892 Gas influx rate at the bottomhole (m3h)
120601119866 Void fraction ()
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research work was cofinanced by the National NaturalScience Foundation of China (nos 51074135 51274170 andQC2012C128) Without their support this work would nothave been possible
References
[1] Y Bu F Li Z Wang and J Li ldquoPreliminary study of AirInjection in Annuli to manage pressure during CementingrdquoProcedia Engineering vol 18 pp 329ndash334 2011
[2] W Guo F Honghai and L Gang ldquoDesign and calculation ofa MPD model with constant bottom hole pressurerdquo PetroleumExploration and Development vol 38 no 1 pp 103ndash108 2011
[3] S Saeed andR Lovorn ldquoAutomated drilling systems forMPDmdashthe realityrdquo in Proceedings of the IADCSPE Drilling Conferenceand Exhibition San Diego Calif USA March 2012
[4] P Vieira F Torres R A Qamar and G E Marin ldquoDownholepressure uncertainties related to deep wells drilling are safelyand precisely ascertained using automated MPD technologyrdquoin Proceedings of the Norh Africa Technical Conference andExhibition Cairo Egypt February 2012
[5] J Saponja A Adeleye and B Hucik ldquoManaged-pressuredrilling (MPD) field trials demonstrate technology valuerdquo inProceedings of the IADCSPE Drilling Conference Miami FlaUSA February 2006
[6] G Liu ldquoPredicting surge pressures that result from runninglinersrdquoWorld Oil vol 222 no 4 p 92 2001
[7] K Zhou J Yang B Zhong and L Tang ldquoSurge pressure inwellborerdquo Oil Drilling and Production Technology vol 12 no 2pp 1ndash10 99 1990
[8] R S Rudi Rubiandini ldquoNew formula of surge pressure fordetermining safe trip velocitiesrdquo in Proceedings of the SPE AsiaPacific Oil and Gas Conference and Exhibition pp 16ndash18 ausOctober 2000
[9] J Li Z Huang K Li and W Wu ldquoThe influence of collaron surge pressure caused by the drilling fluid viscous forceunder pumping conditionrdquoResearch Journal of Applied SciencesEngineering and Technology vol 5 no 5 pp 1781ndash1785 2040ndash7459 2013
[10] SWang C Qiang and K Bo ldquoFluctuating pressure calculationduring the progress of trip in managed pressure drillingrdquoAdvancedMaterials Research vol 468ndash471 pp 1736ndash1742 2012
[11] F Crespo and R Ahmed ldquoSurge-and-swab pressure predictionsfor yield-power-law drilling fluidsrdquo SPE Drilling amp Completionvol 27 no 4 pp 574ndash585 2012
[12] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[13] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquo PHDThesis vol 50 no 9 p 143 1989
[14] H Fan ldquoDevelopment of application software for dynamicsurge pressure while trippingrdquo Petroleum Drilling and Technol-ogy vol 23 no 4 pp 11ndash14 1995
[15] K Zhou J Yang B Zhong and L Tang ldquoSurge pressurein wellborerdquo Journal of Petroleum Exploration and ProductionTechnology vol 12 no 2 pp 1ndash10 1990
[16] G A Chukwu ldquoSurge and swab pressure computed for couetteflow of power-law fluidsrdquoDissertation Abstracts International Bvol 50 no 9 1989
[17] Z Jiang ldquoStudies of wellbore surge pressure by a dynamicmethodrdquo Journal of Petroleum Exploration and ProductionTechnology vol 13 no 3 pp 39ndash44 97 1991
[18] H Wang and X Liu ldquoStudy on steady surge pressure of cassonfluid in concentric wellbore of directional wellsrdquo Drilling FluidCompletion Fluid vol 11 no 6 pp 35ndash40 44 79 1994
[19] D Johnston M Bracken and J Thornton ldquoAssessing pipecondition useful remaining life and maximum operating andsurge pressures when to change the pipe or control thepressurerdquoMagazine of the International Water Association vol14 no 5 pp 43ndash44 2012
[20] H Fan Y Chu and X Liu ldquoPrediction for wellbore dynamicsurge pressure while tripping a drill piperdquo Journal of ChinaUniversity of Petroleum vol 19 no 5 pp 36ndash41 1995
[21] C K Burgess J E Miller and K D Vondervort ldquoDownholesurge pressure reduction system and method of userdquo World9848 143 1998
[22] J P Allamon J E Miller and A M Macfarlane ldquoDownholeSurge Reduction Method and Apparatusrdquo 2004
[23] K S Bjoslashrkevoll R Rommetveit B Aas H Gjeraldstveit andA Merlo ldquoTransient gel breaking model for critical wellsapplications with field data verificationrdquo in Proceedings of the2003 Offshore Mediterranean Conference February 2003
[24] G H Nygaard E Johannessen J E Gravdal and F IversenldquoAutomatic coordinated control of pump rates and choke valvefor compensating pressure fluctuations during surge and swaboperationsrdquo in Proceedings of the IADCSPE Managed PressureDrilling and Underbalanced Operations Conference pp 120ndash125March 2007
[25] J V Fedevjcyk S L M Junqueira and C O R Negrao ldquoSurgeamp swab pressures in wells with cross-section changesrdquo BoletimTecnico da Producao de Petroleo vol 4 no 2 pp 221ndash234 2009
[26] S Tian G H Medley and C R Stone ldquoUnderstandingyield point effect onpressure surges critical to managing deepdifficultMPDwellsrdquoDrilling Contractor vol 65 no 4 p 46 4850 53 2009
[27] Y Sun Q Li and J Zhao ldquoNew method of predicting surgepressure apply to horizontal well based on caisson flowrdquoNaturalScience vol 2 no 12 pp 1394ndash1399 2150ndash4091 2010
[28] B T Lebele-Alawa and F E Oparadike ldquoAnalysis of the effectsof valve propagated pressure surge on pipe flowrdquo Engineeringvol 3 no 11 pp 1098ndash1101 2011
[29] X Kong Q Yuan and Y Qiu ldquoThe research of two-phasefluctuation pressure in wellbore tripping operationrdquo InnerMongolia Petrochemical Industry TE242 2011
[30] F Crespo and R Ahmed ldquoA simplified surge and swab pressuremodel for yield power law fluidsrdquo Journal of Petroleum Scienceand Engineering vol 101 pp 12ndash20 2013
16 Mathematical Problems in Engineering
[31] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
[32] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[33] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86-87 pp 127ndash136 2012
[34] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[35] M J SanchezComparison of Correlations for Predicting PressureLosses in Vertical Multiphase Annular Flow University of TulsaTulsa Okla USA 1972
[36] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[37] Y Lin X Kong Y Qiu and Q Yuan ldquoCalculation analysis ofpressure wave velocity in gas and drilling mud two-phase fluidin annulus during drilling operationsrdquo Mathematical Problemsin Engineering vol 2013 Article ID 318912 17 pages 2013
[38] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[39] H Junfang ldquoBalanced drilling and well controlrdquo PetroleumIndustry Press vol 1 article 33 1992
[40] W A Bacon Consideration of compressibility effects for applied-back- pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Va USA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Mathematical Problems in Engineering
[31] F Huang M Takahashi and L Guo ldquoPressure wave propa-gation in air-water bubbly and slug flowrdquo Progress in NuclearEnergy vol 47 no 1ndash4 pp 648ndash655 2005
[32] P M Dranchuk and J H Abou-Kassem ldquoCalculation of Zfactors natural gases using equitation of staterdquo Journal ofCanadian Petroleum Technology vol 14 no 3 pp 34ndash36 1975
[33] A R Hasan and C S Kabir ldquoWellbore heat-transfer modelingand applicationsrdquo Journal of Petroleum Science and Engineeringvol 86-87 pp 127ndash136 2012
[34] J Orkiszewski ldquoPredicting two-phase pressure drops in verticalpiperdquo Journal of PetroleumTechnology vol 6 no 6 pp 829ndash8381967
[35] M J SanchezComparison of Correlations for Predicting PressureLosses in Vertical Multiphase Annular Flow University of TulsaTulsa Okla USA 1972
[36] H Li YMeng G Li et al ldquoPropagation ofmeasurement-while-drilling mud pulse during high temperature deep well drillingoperationsrdquo Mathematical Problems in Engineering vol 2013Article ID 243670 12 pages 2013
[37] Y Lin X Kong Y Qiu and Q Yuan ldquoCalculation analysis ofpressure wave velocity in gas and drilling mud two-phase fluidin annulus during drilling operationsrdquo Mathematical Problemsin Engineering vol 2013 Article ID 318912 17 pages 2013
[38] H Zhu Y Lin D Zeng D Zhang and F Wang ldquoCalculationanalysis of sustained casing pressure in gas wellsrdquo PetroleumScience vol 9 no 1 pp 66ndash74 2012
[39] H Junfang ldquoBalanced drilling and well controlrdquo PetroleumIndustry Press vol 1 article 33 1992
[40] W A Bacon Consideration of compressibility effects for applied-back- pressure dynamic well control response to a gas kick inmanaged pressure drilling operations [MS thesis] University ofTexas Arlington Va USA 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
