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Pressure Buildup AnalysisWith Wellbore Phase RedistributionWalter B. Fair Jr., SPE, Shell Oil Co.

AbstractThis paper presents an analysis of the effects ofwell bore phase redistribution on pressure builduptests. Wellbore phase redistribution is shown to be awell bore storage effect and is incorporatedmathematically into a new solution of the diffusivityequation. Dimensionless pressure solutions based onan infinite radial reservoir are presented for typecurve matching to analyze pressure buildup testsinfluenced by well bore phase redistribution, andexample analyses of actual field data are included.The parameters that affect phase redistribution andgas humping are documented also. This informationpermits analysis of many anomalous pressurebuildup tests which previously could not be analyzedquantitatively.IntroductionPressure buildup tests and other types of transientpressure tests have been used for many years toevaluate reservoir fluid flow characteristics and wellcompletion efficiency. The basic theory andequations for the analysis of these tests are welldocumented.! M ~ n y factors that influence thepressure response in transient flow conditions havebeen investigated - i.e., the effects of reservoirboundaries, heterogeneities, and fractures, wellborestorage of fluids, and various types of well impairments, skin effects, and completion practices.However, little information concerning the effects ofthe redistribution of gas and liquid phases in thewell bore has been presented.The phenomenon of well bore phase redistributionoccurs in a well which is shut in with gas and liquidflowing simultaneously in the tubing. As shown byStegemier and Matthews, 2 when such a well is shut inat the surface, gravity effects cause the liquid to fall0197-7520/81/0004-8206$00.25Copyright 1981 Society of Petroleum Engineers of AIMEAPRIL 1981

and the gas to rise to the surface. Because of therelative incompressibility of the liquid and theinability of the gas to expand in a closed system, thisredistribution of phases causes a net increase in thewell bore pressure. When this phenomenon occurs ina pressure buildup test, the increased pressure in thewell bore is relieved through the formation, andequilibrium between the well bore and the adjacentformation will be attained eventually. However, atearly times the pressure may increase above theformation pressure, causing an anomalous hump inthe buildup pressure which cannot be analyzed withconventional techniques. In less severe cases, thewell bore pressure may not rise sufficiently to attain amaximum buildup pressure.General analyses of well bore phase redistributionhave been presented by Stegemeier and Matthews2and by Pitzer et al. 3 Both of these investigationsdocumented the association of the pressure builduphump with phase redistribution and indicated thatthe size of the hump was correlated with the amountof gas flowing in the tubing. Stege meier and Matthews also noted an apparent correlation betweenestimated gas rise velocity and the time at which thehump occurred.Earlougher! also noted (on the basis of the shapeof the log t:..p vs. log t:..t plot of buildup test data) thatphase redistribution seems to be related to theproblem of well bore storage. Other authors haverecognized the significance of wellbore phaseredistribution; however, no complete analysis of thephenomenon has been presented and generalmethods for analyzing buildup data influenced byphase redistribution in the wellbore have not beenavailable.Mathematical Analysisof Phase RedistributionI f we consider a well where well bore phase

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redistribution occurs, it is apparent that well borestorage also must occur. I f the well bore could notstore fluids of finite compressibility, the phaseredistribution process could either (1) physically notoccur or (2) be associated with a zero pressure increase. It is also interesting to note that thetechniques presented by Stegemeier and Matthews2and Pitzer et al. 3 for minimizing wellbore phaseredistribution also minimize wellbore storage effects.

For a well where well bore storage occurs theeffects of the storage can be described by Eq. 1.4 Theeffect of the changing sand-face flow rate on thewellbore pressure also can be obtained from Eq. 2.

!bi.. = 1 - CD dPwD ..................... 1)q dtDdPwD _ 1 !bi..-- - - (1 - ).. ................ (2)dtD CD qTo describe the effect of wellbore phaseredistribution, note that not all of the pressurechange in the wellbore can be attributed to well borestorage flow rate effects, since some of the pressurechange is caused by phase redistribution.Thus, Eq. 2 can be modified by adding a termdescribing the pressure change caused by phaseredistribution, as in Eq. 3, which also can berearranged to show the sand-face flow rate dependency in Eq. 4.dPwD 1 !bi.. dp

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While a is not determined as easily, it is knownthat it will depend mainly on those factors whichcontrol the gas bubble or slug rise time in a well.Finally, to keep the dimensionless quantitiesconsistent, the dimensionless phase-redistributionpressure function is defined in Eq. 9.

wherekhp

PD = 141.2C _ kh CD - 141.2

0.000264 klID= 2 ' ~ C t r wand

0.000264 kaaD= 2 ~ C t r wIn SPE preferred SI units, replace 11141.2 with7.271- x 10 -6 and 0.000264 with 3.6 x 10 -6 .Determination of DimensionlessWellbore PressuresTo obtain dimensionless pressure solutions for use inthe analysis of pressure buildup tests, it is necessaryto incorporate the effects of wellbore phaseredistribution into the diffusivity equation. Forradial flow in an infinite, homogeneous, isotropicreservoir of a fluid of small compressibility, thisproblem is stated in dimensionless variables asfollows. The diffusivity equation is

a2pD 1 apD apD--2 + - - -=- .............. 10)arD rD arD aIDThe boundary conditions are

PD (rD'O) =0......................... 11)lim PD(rD,ID ) =0................... 12)rD - ex>_ ( aPD ) =1_cD(dPWD _ dPD).arD rD=1 dID dID......................... (13)

P D = [PD _s (aPD ) ] _ ............. 14)warD rD-ISeveral authors 5 have shown that this problemalso can be written as a convolution integral to account for well bore storage. This approach leads to

Eq.15.

APRIL I'lXI

. [dPwD(tD) _ dPD(tD) ]1 .... 15)dID dID j

Eq. 15 can be solved for (PwD)' the Laplacetransform of the desired pressure function. Thisresults in Eq. 16. (s denotes the Laplace transformvariable.)

[s (PD) +S][1 + CDS2 (PD)] (p D) = .w s[1 +CDS(S(PD) +S)]................................ (16)

Note that the solution still is entirely general, sinceno constraints have been placed on either P D or PD'except that these functions exist and are Laplacetransformable. Thus, if PD represents any type ofreservoir condition, the required pressure solutionsfor those conditions can be determined in principle.This statement also applies to the phaseredistribution pressure.In this work, Eq. 9 is used for the phaseredistribution effect. Its Laplace transform is:

CD CD (PD) = - - . . . . . . . . . . . . . 17)S S+ 1IaDThe required expression for (p D) has beenpresented by Van Everdingen and Hurst4 as Eq. 18,where Ko and Klare modified Bessel functions.

Ko(Ys) (PD) = s312 KI (Ys)' ................. (18)I t also has been shown that at long times thissimplifies to the line source solution in Eq. 19, since

YsK I (Ys)-1 whens-Oor/D-oo.1(PD) = -Ko(Ys) .................. 19)s

A further long-time approximation for (PD)can be obtained by noting that as ID -oo,s-O, andYsKo(Ys) - - [In( 2 +1'],

where 1'=0.577 215 664901 52 .. . denotes Euler'sconstant. This gives Eq. 20.(PD) = - [ I n ( ~ ) +1'] ............. (20)Combining the definition of (PD) in Eq. 17with the various forms of (PD) from Eqs. 18, 19,and 20 yields required expressions for (PwD) as

follows.Cylindrical Source Well.

[ Ko(Ys) ] [ 2 ( 1 1 )]sK I (Ys) + S 1+ CDCDs - s+ 1IaD{ [ K (Ys) ]J+ CDs 0 Ys +SsK I ( s)...................... (21)

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C00 - 1 00

Fig. 1 - Type curves with phase redistribution (CaD = 0,CD =100).

"wo' O O - ~ - - - - - , - - - - : = c r - - - - - , - - - - - , - - - - - - - - ,

]()Co 1000

O > , ' - ; - o ' - - - - ' ' ~ , ----"-':-'--,-, --",,'----'-,"'-----',,'

Fig. 2 - Type curves with phase redistribution (CaD = 20,CD =1,000).

OliO", - - " ~ ' - - - - ' - ; " ' - - , - , --':,,'-----","'-----',,'Fig. 3 - Type curves with phase redistribution (CaD = 20,CD = 10,000)."wa'00,----___ - -,___ ----__ -,___ ,---------,

O \ O ' - ; - , - - - , , ~ , - - - , - ' : - O . - - " - - - - - ' : " ' - - - - - " , " ' - - - - - - ' ' ' '

Fig. 4 - Type curves with phase redistribution (CaD =100,CD = 1,000).262

Line Source Well.

\ (s[1 + CDs[KO (Vs) +SlJ)............. (22)Line Source Well Long-Time Approximation.

[ Vs 2 1 1 J[S-ln( -) 1'][1 + CDCDs (- - )]2 s s+ l IaD\ (s[1 +CDs[S-ln( ) -I'll). . ........ (23)

The long-time approximating form of the PwDfunction can be derived from Eqs. 21,22, or 23 bynoting that

S 2 ( ~ - 1 )-0 as s-O( tD- oo ).s s+ l IaDThus, these equations reduce to the well bore storageequation given by Agarwal et al. 5 which furtherapproaches Eq. 24.

PwD """PD +S ........................ 24)The short-time approximation also can be obtained from Eq. 21 by noting that the well borestorage factor obtained by letting CD = 0 reduces to

1. (PwD ) = -2- ' ................ (25)storage s CDAlso, sinces2 [ l Is - [ l I (s+ 1)/(aD)J]-I /aD at larges, . (PwD) must approach

1 CD. (PwD) """ --2 + --2' .............. (26)CDs aDsand PwD approaches

tDPWD=-CaDwhere1 1 CD-= -+ - . . . . . . . . . . . . . . . . . . . . 27)CaD CD aD

Note that Eq. 27 indicates that a representation verysimilar to well bore storage will exist at short times.This is consistent with Earlougher's 1 earlier comments.To obtain dimensionless pressures for use inanalyzing pressure buildup tests with wellbore phaseredistribution, Eqs. 21, 22, or 23 must be inverted.Since these expressions are too complicated foranalytical inversion, the inverse Laplace transforms

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were calculated numerically using an inversiontechnique presented by Stedhfest6 adapted for use onthe TI-59 programmable calculator. The type curvesshown in Figs. 1 through 6 indicate that the pressurefunctions may show a tendency toward a dampedoscillation. According to Stehfest, 6 such anoscillation may render the numerical techniqueuseless unless certain conditions on wavelength ofoscillation are met. However, it can be shown thatthe functions obtained in this work do not oscillate,since the Laplace transform can be written as the sumof three terms. Two of the terms representmonotonic functions, while the inverse transform ofthe remaining term has a single maximum. Thus,Stehfest's criteria of functional "smoothness" is meton each term and, by virtue of the linearity propertyof the Laplace transform and of the numericaltechnique, it is valid to use the numerical methodwith these functions.Eq. 21 was programmed and inverted for severalvalues of the wellbore-storage coefficient CD andskin factor S. Results and a comparison with datapreviously reported by Agarwal et al. 5 are shown inTable 1. The excellent agreement indicates that thenumerical technique is well suited to the calculatorprecision. Eq. 22 also was programmed and invertedfor several values of CD and S, again with closeagreement to the Agarwal et al. 5 results for the line

TABLE 1 - COMPARISON OF CALCULATED PwoWITH REF. 5 (CYLINDRICAL SOURCE WELL, Co = )

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264

TABLE 2 - COMPARISON OF CALCULATED PwDWITH REF. 5 (LINE SOURCE WELL, C",D = 0)

to PwOCD =100, S=O

100 0.79381,000 3.263910,000 4.9563100,000 6.15481,000,000 7.311610,000,000 8.4635

Pwo,Ref. 50.79383.26404.95646.15487.31168.4635

Co = 10,000, S=O100 0.00998 0.009981,000 0.0984 0.098410,000 0.8925 0.8925100,000 4.6771 4.67721,000,000 7.2307 7.230910,000,000 8.4550 8.4550

CD = 100, S=O100 0.97761,000 8.121210,000 24.242100,000 26.1341,000,000 27.31010,000,000 28.463

0.97768.121224.24126.13427.31028.463

CD =10,000, S =20100 0.01000 0.010001,000 0.0998 0.099810,000 0.9797 0.9797100,000 8.2698 8.26981,000,000 26.286 26.28610,000,000 28.434 28.434

TABLE 3 - COMPARISON OF PwD CALCULATIONS(C"'D =0 )

Pwo, Pwo,Cylinder Line Pwo,to Source Source Approx.

Co =100,S=0100 0.7975 0.7938 0.79291,000 3.2680 3.2639 3.263410,000 4.9566 4.9563 4.9563100,000 6.1548 6.1548 6.1548

1,000,000 7.3116 7.3116 7.311610,000,000 8.4635 8.4635 8.4635

Co =10,000,S=0100 0.00998 0.00998 0.009981,000 0.0984 0.0984 0.098410,000 0.8925 0.8925 0.8925100,000 4.6771 4.6771 4.67711,000,000 7.2308 7.2307 7.230710,000,000 8.4550 8.4550 8.4550

CD =100,S=0100 0.9777 0.9776 0.97761,000 8.1220 8.1212 8.121110,000 24.242 24.242 24.242100,000 26.134 26.134 26.1341,000,000 27.310 27.310 27.31010,000,000 28.463 28.463 28.463

CD = 0,000, S = 0100 0.01000 0.01000 0.010001,000 0.0998 0.0998 0.099810,000 0.9797 0.9797 0.9797100,000 8.2698 8.2698 8.26981,000,000 26.286 26.286 26.28610,000,000 28.434 28.434 28.434

TABLE 4 - COMPARISON OF PwD CALCULATIONS (5 =0)Pwo, Pwo,Cylinder Line Pwo,

to Source Source Approx.CD =100, C",o = 10, aD =1 000100 1.5541 1.5468 1.54501,000 5.0013 4.9962 4.995710,000 5.0199 5.0196 5.0196100,000 6.1600 6.1599 6.15991,000,000 7.3121 7.3120 7.312110,000,000 8.4635 8.4635 8.4635

CD =100, C",o = 10, aD =100100 5.6832 5.65751,000 4.5588 4.558110,000 5.0123 5.0121100,000 6.1599 6.15991,000,000 7.3121 7.312010,000,000 8.4635 8.4635

5.65194.55835.01206.15987.31218.4635

CD = 0,000, C",o =100, aD = ,000100 9.5031 9.5025 9.50231,000 62.1794 62.1759 62.175310,000 82.9727 82.9691 82.9689100,000 25.4042 25.4045 25.40441,000,000 7.8197 7.8195 7.819610,000,000 8.5059 8.5058 8.5059

CD = 10,000, C",o =100, aD = 100100 63.0504 63.0466 63.04561,000 97.4584 97.4528 97.452010,000 81.5483 81.5449 81.5449100,000 25.1634 25.1638 25.16361,000,000 7.8191 7.8188 7.819010,000,000 8.5059 8.5059 8.5059

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source well solution, as shown in Table 2. Finally,Eq. 23 was inverted and, again, the results shown inTables 3 and 4 are in close agreement with previousresults. Eq. 23 therefore was used in the remainder ofthis study.

Fig. 7. Note that at long times the new curvescoincide with the storage curves, while at early timesthe apparent wellbore storage effect is obvious. Atintermediate times, the phase redistribution effectcauses the curves to trend away from the apparentstorage behavior to the true storage behavior. Atlarge values of CcpD' the "gas hump" is apparent,while at small values of C cpD the phase redistributioneffect is much diminished. A potential problem inpressure data interpretation also is shown at intermediate values of CcpD' since the curve forCcpD =JO resembles the storage curve with CD = 100while the true CD is 1,000. An attempt to type-curvematch phase-redistribution data to a storage curve

To facilitate use of the dimensionless pressures inbuildup analysis, CaD from Eq. 27 was used as avariable rather than CiD' which is more difficult todetermine. Results of the inversion are shown inTables 5 and 6, and log PwD vs. log tD type curvesare presented in Figs. 1 through 6. Accuracy isO.1070.A comparison of the phase-redistribution typecurves with well bore-storage type curves is shown in

TABLE 5 - DIMENSIONLESS WELLBORE PRESSURES WITH PHASE REDISTRIBUTION

APRIL 1981

to1002005001,000

2,0005,00010,00020,00050,000100,000200,000500,0001,000,0002,000,0005,000,00010,000,000

1002005001,0002,0005,00010,00020,00050,000100,000200,000500,0001,000,0002,000,0005,000,00010,000,000

1002005001,0002,0005,00010,00020,00050,000100,000200,000500,0001,000,0002,000,0005,000,00010,000,000

CaD = 0, Co = 00S=O S=10 S=20

C",o = 11.4961.9002.6933.3783.9674.5704.9625.3315.8046.1556.5056.9647.3127.6598.1178.463

3.3744.9685.8284.9914.4474.6885.0135.3555.8136.1606.5076.9657.3127.6598.1178.464

1.8812.7164.8367.50010.7413.9014.8115.2715.7816.1516.5016.9617.3117.6618.1218.46

C",o = 104.1146.87710.8812.6213.4114.3514.8915.3015.7916.1516.5016.9617.3117.6618.1218.46

C",o = 1003.898 4.7206.594 8.93611.05 19.0513.30 29.5812.77 36.818.531 28.396.104 18.035.646 15.665.912 15.906.206 16.206.530 16.526.974 16.977.317 17.327.661 17.668.118 18.128.464 18.46

1.9262.8385.3038.78113.9221.2424.2725.1925.7626.1326.4926.9627.3127.6628.1228.46

4.1967.15011.9915.2118.3122.5924.5125.2325.7726.1426.5026.9627.3127.6628.1228.46

4.8109.26320.7134.5248.6047.4731.9825.8225.8926.1926.5226.9727.3127.6628.1228.46

CaD =20, Co =1,000S=O S=10 S=20

1.0551.1251.3151.6132.1313.2204.1874.9875.6896.1006.4766.9537.3057.6558.1158.463

3.8716.1488.5978.6987.8046.3715.6225.4675.8086.1516.5006.9627.3107.6588.1168.463

Cpo = 11.0861.1841.4541.8922.7244.9087.70911.2314.8615.9316.4116.9317.3017.6518.1118.46

C",o = 03.9566.3879.38710.3110.6311.3512.3413.7015.3316.0116.4416.9417.3017.6518.1118.46

C",o = 1004.762 4.8619.137 9.45720.31 21.7833.73 38.1547.43 59.2445.83 76.1025.87 64.9111.41 41.917.067 20.586.678 16.876.742 16.737.055 17.047.356 17.357.680 17.688.125 18.128.468 18.47

1.0891.1911.4741.9382.8375.3278.86514.1621.9325.3026.3326.9127.2827.6528.1128.46

3.9646.4149.49510.5711.1912.7214.9218.2523.2625.5526.3726.9227.2927.6528.1128.46

4.8719.49221.9838.8261.4083.8479.9462.6537.6328.2326.7327.0227.3427.6728.1228.47

CaD =20, Co =10,000S=O S=10 S=20

0.99971.0131.0341.0701.1431.3591.6982.2983.6234.8825.9566.8047.2377.6218.1018.456

3.9286.3039.1169.7949.7219.3948.9748.3597.3416.7286.5806.9327.2907.6458.1118.460

Cpo = 11.0031.0181.0461.0921.1841.4571.8992.7444.9807.90011.6815.8017.0417.5618.0818.45

C",o = 103.9376.3289.1999.97310.0710.1610.3110.6211.5012.7014.3416.3517.1317.5918.0918.45

C",o = 1004.865 4.8769.477 9.51021.93 22.0938.72 39.2261.30 62.8084.81 90.0184.01 94.9171.18 90.4545.70 77.4725.68 61.2412.93 41.308.225 21.897.820 18.057.888 17.878.204 18.198.506 18.50

1.0031.0191.0481.0961.1911.4741.9402.8445.3558.94814.3922.6026.3227.4728.0628.43

3.9386.3319.2109.99910.1210.3010.5911.1612.7515.0618.5924.0326.6027.5028.0728.44

4.8769.51422.1139.2963.0390.9397.0994.9287.2976.6360.9438.4229.4427.8828.1728.49

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could give a reasonable type-curve match, but any I f the producing time is large compared to the shut-inestimates of reservoir parameters might be greatly in time M D , then tD+MD=:: tD , so that Eq. 28 sim-error. Fortunately, this problem can be resolved by plifies tocomparing the estimates of the true and apparent khstorage constants as in the following examples. Ww(tD+tJ.tD ) -Pwj1= (tJ.tD )Analysis of Pressure Buildup Tests 141.2 qBp.Generally, to analyze pressure buildup test data, the ................................ (29)superposition principle is applied to any dimen-sionless pressure functions yielding Thus, a log (pw -Pwj ) vs. log (tJ.t) plot of the

kh buildup data can be type-curve matched according toW (t +tJ.t ) -P 11 = standard procedures, providing that P wD ( tD + J.tD)141.2 qBp. w D D w=::PwD (tD)' I f this assumption is not valid, the morePwD (tD) -PwD (t D + J.tD ) +PwD (tJ.tD ) . . (28) general superposition in Eq. 28 must be used;

TABLE 6 - DIMENSIONLESS WELLBORE PRESSURES WITH PHASE REDISTRIBUTIONCaD =100, CD =1,000 CaD = 00, CD =10,000 CaD = ,000, CD =10,000

tD S= O S=10 S=20 S= O S=10 S=20 S= O S=10 S=20Cq,D =1 Cq,D =1 Cpo =1

100 0.6748 0.6902 0.6917 0.6366 0.6381 0.6383 0.09583 0.09603 0.09605200 0.9841 1.024 1.029 0.8772 0.8809 0.8814 0.1839 0.1846 0.1846500 1.326 1.450 1.467 1.029 1.040 1.041 0.4085 0.4115 0.41191,000 1.630 1.898 1.941 1.072 1.093 1.096 0.6813 0.6905 0.69182,000 2.143 2.730 2.841 1.145 1.185 1.191 0.9997 1.025 1.0295,000 3.225 4.912 5.330 1.360 1.457 1.475 1.369 1.453 1.46810,000 4.189 7.712 8.868 1.699 1.900 1.940 1.715 1.906 1.94420,000 4.988 11.23 14.16 2.299 2.744 2.844 2.312 2.750 2.84850,000 5.689 14.86 21.93 3.623 4.980 5.355 3.630 4.985 5.358100,000 6.100 15.93 25.30 4.882 7.900 8.949 4.885 7.904 8.951200,000 6.476 16.41 26.33 5.956 11.68 14.39 5.957 11.68 14.39500,000 6.953 16.93 26.91 6.804 15.80 22.60 6.804 15.80 22.601,000,000 7.305 17.30 27.28 7.237 17.04 26.32 7.237 17.04 26.322,000,000 7.655 17.65 27.64 7.621 17.56 27.47 7.621 17.56 27.475,000,000 8.115 18.11 28.11 8.102 18.08 28.06 8.102 18.08 28.0610,000,000 8.463 18.46 28.46 8.456 18.45 28.43 8.456 18.45 28.43

C",D =10 C"'D =10 C"'D =10100 0.9370 0.9566 0.9585 0.9502 0.9522 0.9524 0.09935 0.09955 0.09957200 1.769 1.832 1.839 1.808 1.815 1.815 0.1976 0.1982 0.1983500 3.760 4.039 4.076 3.917 3.946 3.950 0.4857 0.4891 0.48951,000 5.847 6.646 6.768 6.270 6.356 6.368 0.9454 0.9570 0.95862,000 7.474 9.457 9.823 8.505 8.734 8.770 1.794 1.833 1.8395,000 7.018 11.70 12.92 9.482 10.14 10.26 3.859 4.046 4.07810,000 5.876 12.66 15.17 9.102 10.36 10.62 6.107 6.668 6.77620,000 5.514 13.86 18.42 8.457 10.67 11.19 8.049 9.523 9.84650,000 5.811 15.36 23.31 7.393 11.53 12.78 8.097 11.91 13.00100,000 6.152 16.01 25.56 6.750 12.72 15.08 7.081 13.07 15.35200,000 6.500 16.44 26.37 6.585 14.36 18.61 6.659 14.54 18.79500,000 6.962 16.94 26.92 6.933 16.35 24.03 6.938 16.38 24.091,000,000 7.310 17.30 27.29 7.290 17.13 26.60 7.290 17.14 26.612,000,000 7.658 17.65 27.65 7.646 17.59 27.50 7.646 17.59 27.505,000,000 8.116 18.11 28.11 8.111 18.09 28.07 8.111 18.09 28.0710,000,000 8.463 18.46 28.46 8.460 18.45 28.44 8.460 18.45 28.44Cq,D =100 Cq,D =100 C

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",(PSI) . r - ~ ~ - r ~ ~ ~ , - - - ~ - , ~ ~ ~ ~ ~ ~ - ,

' -MATCHPOINT>,10 . ' 0 6900oil lOO.''wO'lOl

. TYPE CURVECaD 400 S- 0Co 750 Cl2lo- 10

Fig. 8 - Pressure buildup data, Example 1.

however, since there is apparently no drawdownequivalent to wellbore phase redistribution in abuildup test, the appropriate PwD functions must beused for PwD (t D) and PwD (t D + :.tD). An exampleof such superposition would be the case of a shortdrawdown with storage followed by a buildup withphase redistribution. The superposition of this ratehistory would be as in Eq. 30, where phaseredistribution effects are in the third term only.

kh141.2 qBp. [Pw (tD +t:.tD ) -Pwj l =PwD (tD'CD,S) -PwD (tD + :.tD,CD,S)+ PwD (t:.tD,CD,S,PD)' ............... (30)

In SPE preferred SI units, 1/141.2 is replaced by7.27r x 10 - 6 in Eqs. 28 through 30.The examples provided in the following sectionillustrate the analysis of bottomhole pressure-buildupsurveys which are influenced by well bore phaseredistribution; however, note that not all surveys areanalyzed as easily. The tests documented here aretaken in gas-lifted oil wells in southern Louisiana,and one factor which makes these tests amenable toanalysis is that little free gas enters the wellbore aftershut-in. Most of the gas in the tubing string whichcontributes to the phase redistribution processoriginates from the annulus through the gas-liftvalves. Thus, the true wellbore storage coefficientCD is controlled by the rising liquid level of theinflowing fluid which remains essentially constant.In other cases, it is not obvious that the density ofthe inflowing fluid remains constant, since theflowing gas1oil ratio may change as the well is shutin; this would cause a changing storage coefficient.In addition, the compression of the gas near thesurface may not be accounted for correctly, whichalso will cause a variable wellbore storage. AlthoughAPRIL 1981

TABLE 7 - PRESSURE BUILDUP DATA FOR FIELD EXAMPLE 1q = 212 BID (33.71 m3 /d)J1. = 4cp(4x10- 3 Pas)B = 1.1 RB/STB(1.1 res m3 /stocktank m3 )h = 10 ft (3.05 m)Aw = 0.00387 bbl/ft (0.002 02 m2 ) = 0.28c t = 60x10- 6 psi- 1 (8.70x10- 6 kPa- 1 )rw 2 = 0.083 sq ft (0.007 71 m2 )PI = 0.330 psi/ft (7.46 kPa/m), measured

t,.t(hours)o0.250.300.75124678910

/Jwpsi (kPa)296 (2041)449 (3096)520 (3585)574 (3958)597 (4116)588 (4054)576 (3971)576 (3971)576 (3971)578 (3985)578 (3985)

at long times these affects will be negligible, at shorttimes they will cause the pressure buildup to deviatefrom the unit slope log P wD vs. log t:.t D line. In wellswhich flow significant quantities of free gas, thiseffect may be pronounced. 1-3In addition, the problems of buildup analysiswithout phase redistribution also exist in the analysisof data with phase redistribution. Multiple stringers,mechanical problems, and other effects may makeanalysis difficult, if not impossible.Another observation of interest in the analysis ofbuildup surveys is a discrepancy between storageconstants calculated from pressure data and fromwell completion data. I have found that theseestimates of the storage constant rarely agree. Thetype curves presented in this work offer one possibleexplanation for this common discrepancy, since atsmall times it is apparent that the phaseredistribution effect greatly controls the apparentstorage constant CaD calculated from the pressuredata. Several types of storage behavior are observedin practice.

1. When Cad ::= CD ' the well exhibits a true storagebehavior.2. When CaD < CD' the buildup usually is controlled by phase redistribution.3. When CaD >CD , mechanical problems,multiple l a y : ~ r s , or an enlarged wellbore usually canexplain the discrepancy.Note that these observations are based on experience and apply only to pressure tests in unfractured reservoirs where the storage is caused by arising fluid level. Note also that the correct value forthe fluid density or gradient must be used incalculating the true storage constant, since an error inthe gradient will cause directly a corresponding errorin the calculated storage constant. For this reason, it

is recommended that the fluid gradient under flowingand static conditions be measured in conjunction267

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. "IPSI )

"00'.,-----,---------,-----,-------,-------,, = ~ - - f ~ - + - - - ~ I - - - - - - - - - - - - - - - - j - ~ - - - - - - - j

T ~ P f C L J R V f fORl d t ) l OC I) 1()()

OOOOOQ(3)_____L---"-"-i'-Cjl!O 10

"wo l8"" t."W lOOPS''I) 1]0"" . ' 1HR

Fig. 9 - Pressure buildup data, Example 2.

with the buildup data. In gas-lifted oil wells, thisgradient must be measured below the point of gas-liftgas entry. Any differences in the flowing and staticgradients which cannot be attributed to frictionaleffects generally will give an indication of the flow offree gas from the reservoir. The gradients used in thefollowing examples were measured in conjunctionwith the pressure surveys.Note also that the type curves presented are notmeant to replace semilog analysis methods or the useof previously published type curves. I f it is possibleto analyze well test data using semilog methods,greater accuracy will be obtained in nearly all cases,mainly due to the similarity of the shape of the typecurves which makes type-curve matching difficult.When such simple analyses are not possible,however, the type curves presented in this work maypermit approximate analysis which would not bepossible otherwise.Example AnalysesExample 1 is an actual set of pressure buildup datameasured in a gas-lifted oil well in southeastLouisiana. The basic data are shown in Table 7 and alog-log plot of the pressure data is shown in Fig. 8.

From the data plot in Fig. 8, a point on the unitslope straight line is estimated to be !::t.p = 153 psi(lOSS kPa) at !::t.t = 0.1 hour. The wellbore storagecoefficient is calculated as in Eq. 30 and the apparentstorage coefficient as in Eq. 31. The gradient used inEq. 30 is calculated from flowing- and static-pressuresurveys measured in conjunction with the builduptest.AwC= - = 0.01173 bbllpsiPI(0.000 270 5 m 3IkPa). . ............... (30)

268

TABLE 8 - PRESSURE BUILDUP DATA FOR FIELD EXAMPLE 2q = 14BID(2.23m 3 /d)Ii = 4 cp (5 x 10- 4 Pa s)B = 1.05 RB/STB (1.05 res m 3 /stocktank m3 )h = 20 ft (6.10 m)

Aw = 0.00387 bbl/ft (0.002 02 m2) = 0.28

Ct = 150x10- 6 psi- 1 (2.176x10- 5 kPa- 1 )rw2 = 0.085 ft2 (0.007 90 m2)PI = 0.420 psi/ft (9.50 kPa/m), measured

M(hours)o0.250.500.7512346810121416.5

C4Jwpsi (kPa)102 (703)190 (1310)254 (1751)278 (1917)306 (2110)292 (2013)284 (1958)273 (1882)276 (1903)276 (1903)276 (1903)278 (1917)281 (1937)

qB!::t.tCa = 2 - - =0.00635 bbllpsi4!::t.p(0.000 1464 m 3 IkPa). . .............. (31)

Since Ca < C, we can conclude that phaseredistribution effects are significant. These valuesyield CD =752 and CaD =407.The data then are matched to the type curves forCaD =400 and CD =750 as indicated, with a matchpoint chosen as CD = 10, S = 0, t D = 6,800 at !::t.t = 1hour, and PwD = 1.02 at !::t.pw = 100 psi (689 kPa).From the standard definitions of tD and PwD' thepermeability is calculated as follows.FrompwD match: k= 134 md.From tD match: k= 144 md.Example 2 consists of data taken in a well insoutheast Louisiana producing at low rates and highwater cuts from a shaly sand. Pressure buildup datais given in Table 8 and the log !::t.pw vs. log !::t.t plot isshown in Fig. 9. From the static and flowing surveystaken in conjunction with the pressure buildup, C iscalculated as shown and Ca also is estimated from anextrapolation of the short-time data.

Aw 3C= - =0.00921 bbllpsi (0.000 212 m IkPa).PIqB!::t.tCa = - - = 0.00130 bbllpsi24!::t.p

(0.000 030 m 3IkPa).Since Ca < C, phase redistribution effects arebelieved to be significant, so CD and CaD arecalculated to be

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CD = 115 and CaD = 16.Th e data are matched on. the type curves forCD = 100, CaD = 20 and the best match is estimatedas shown in Fig. 9 to be Cq,D = 10, S = 5, PwD = 2.80at Llp w = 100 psi (689 kPa), and tD = 320 at Llt= 1hour. From the standard definitions of P wD and tD'the permeability is calculated as follows.

FromPwD match: k= 1.45 md.From tD match: k=2.16 md.Fig. 9 also seems to indicate that the last two datapoints may be close to the semi og straight line. Usingsemilog analysis techniques, the permeability an dskin are estimated to be 1.46 md and 3.4, respectively. Since the true straight line may not have beenreached and only two points are used to determine

the semi og str aight line, these es timates are inadequate agreement with the estimates obtained fromtype-curve matching.Summary and ConclusionsIt has been recognized for some time that wellborephase redistribution can cause anomalous pressurebuildup behavior in oil an d gas wells. General aspectsof the phenomenon have been presentedpreviously2,3; however, no technique for the analysisof such tests has been available.

The work presented in this paper provides ananalysis of the wellbore phase redistribution problemand, with an assumed behavior based upon physicalarguments, provides a general method for theanalysis and description of such anomalous pressurebuildup tests. I t has been shown that the wellborephase redistribution problem is a complex wellborestorage phenomenon, and mathematical methodspreviously applied to wellbore storage problems havebeen extended to solve this more general problem.

In the analysis of buildup surveys, I have foundthat the observed storage constant often does notagree with that calculated from the well completionproperties. One possible explanation for this ob-servation lies in the apparent storage observed to beassociated with phase redistribution. Even though ahump may not be observed, phase redistributioneffects may cause an inobvious distortion in the dataplot. Analysis of such data by other type curvetechniques may yield totally meaningless results. Inview of this, it is recommended that the true andapparent storage coefficients always be calculatedand checked for consistency before proceeding withdetailed analysis of a buildup survey.

The main assumption of this work is the exponential form used in representing the phaseredistribution pressure function. I have found thatthis form apparently represents phase redistributionin a gas-lifted oil well very well; however, nomeaningful experimental data are available tosubstantiate this completely. Such data would beuseful either in verifying this function or inproposing a new function for the phase redistributionpressure. This data could be collected by measuringthe pressure in a well shut in simultaneously at thesurface and at the bottom of the tubing string usingAPRIL 1981

equipment described in Ref. 3. Laboratory experiments also could measure this pressure. This dataand further analytical work is definitely needed todetermine the range of well conditions over which theassumed form is applicable and to extend the basictechnique to other conditions.In this work, only positive values of the skin effectfactor have been considered. It would pose no majorproblem to calculate dimensionless wellborepressures for negative skin factors by the techniquedescribed by Agarwal, et at. 5 Note, however, thatsuch an approach places a great emphasis on theaccuracy of the various functions used at small timesand these functions are inherently more difficult toevaluate with great preCISIOn. In the phaseredistribution problem, such an approach wouldrequire the evaluation of the dimensionless pressuresat extremely small dimensionless times, dimensionless storage coefficients, and dimensionless phaseredistribution time parameters.

Although the numerical work presented here isbased upon an infinite, homogeneous, radialreservoir model, the basic concepts are much moregeneral. In particular, it is possible to apply thetechniques used in this study to other reservoirmodels and thereby to obtain techniques for theanalysis of data in fractured systems as well as otherpractical situations.NomencJature

A =wB=

C=

cross-sectional area of the well bore ,bbl/ft (m2)formation volume factors, RSB/STB(res m 3 Istock-tank m3)wellbore storage coefficient, bbl/psi(m 3 /kPa)apparent storage coefficient, bbl/psi

(m 3 /kPa)apparent dimensionless storage coefficient,5.6146CaC D= 2a 27r(j>c th rw

= ( Cq,D + _1 ) _ 1aD CD

(CaD = p ( j > ~ ~ r w2)CeD = effective dimensionless storage coef

ficient defined in Eq. 5c ( compressibi lity, psi - 1(kPa - 1 )CD dimensionless wellbore storage coef-ficient,

(CD = C )2p(j>c(hrw 2phase redistribution pressure parameter,psi (kPa)

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dimensionless phase redistributionpressure parameter,khC

CD = 141.2 qBJl( _ 7.27rx 1O-6khC