Research Article Pressure-Transient Behavior in a...
8
Research Article Pressure-Transient Behavior in a Multilayered Polymer Flooding Reservoir Jianping Xu 1,2 and Daolun Li 3,4 1 Xi’an Petroleum University, Xi’an, Shanxi 710065, China 2 PetroChina Dagang Oilfield Company, Tianjin 300280, China 3 HeFei University of Technology, Hefei, China 4 University of Science and Technology of China, Hefei, China Correspondence should be addressed to Jianping Xu; [email protected] Received 7 May 2015; Accepted 18 May 2015 Academic Editor: Yves Grohens Copyright © 2015 J. Xu and D. Li. 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. A new well-test model is presented for unsteady flow in multizone with crossflow layers in non-Newtonian polymer flooding reservoir by utilizing the supposition of semipermeable wall and combining it with the first approximation of layered stable flow rates, and the effects of wellbore storage and skin were considered in this model and proposed the analytical solutions in Laplace space for the cases of infinite-acting and bounded systems. Finally, the stable layer flow rates are provided for commingled system and crossflow system in late-time radial flow periods. 1. Introduction Many reservoirs are formed from layers of different physical properties because of the different geological deposition ro- tary loops. Among them, if these layers do not communicate in terms of fluid flow through the formation but may be pro- duced by the same wellbore, these types of reservoir are called commingled systems; if there exits fluid that connects between these layers, they are referred to as crossflow systems. e pressure-transient behavior depends on the comprehensive properties of these multilayers. For the unstable flow of Newtonian fluids in a multilay- ered reservoir, Russell et al. [1, 2] studied pressure behavior of single-phase fluid in two layers with formation crossflow and derived the conclusion that it is similar to the flow behavior of the two layers without formation crossflow in early time. Raghavan et al. [3] studied the problem of well test in a mul- tilayered reservoir. Bourdet [4] established using steady state approximation to the presentation interlayer flow model. Gao and Deans [5] studied the behavior of multilayered reservoir with formation crossflow. Ehlig-Economides [6] has system- atically established the combination of commingled system and crossflow system unsteady flow models and provided the rule of the pressure and flow for each layer. Bidaux et al. [7], using layered pressure and flow data, conducted a study on the theory and practical application of multilayered reservoir. For the unsteady flow of non-Newtonian fluids in a mul- tilayered reservoir, van Poollen and Jargon [8] studied non- Newtonian power-law fluid unsteady flow in porous media and showed that the transient pressure response character- istics are different from that of Newtonian fluid. Ikoku and Ramey [9] studied non-Newtonian power-law fluid unsteady flow characteristics in porous medium, and the consideration of wellbore storage and skin effect is obtained in homo- geneous infinite reservoir model in Laplace space solution. Lund and Ikoku [10, 11] proposed non-Newtonian power-law fluid (polymer solution) and Newtonian fluid (oil) composite model of transient well-test analysis method. Xu et al. [12] proposed the infinite reservoir Laplace space spherical transient pressure solution and discussed the characteristics of wellbore pressure at early times and later times. e above research result is to solve the problem of single layer. Escobar et al. [13] presented equations to estimate permeability, non- Newtonian bank radius, and skin factor for the well test data in reservoirs with non-Newtonian power-law fluids. Hindawi Publishing Corporation Journal of Chemistry Volume 2015, Article ID 875068, 7 pages http://dx.doi.org/10.1155/2015/875068
Transcript of Research Article Pressure-Transient Behavior in a...
Research ArticlePressure-Transient Behavior in a Multilayered PolymerFlooding Reservoir
Jianping Xu12 and Daolun Li34
1Xirsquoan Petroleum University Xirsquoan Shanxi 710065 China2PetroChina Dagang Oilfield Company Tianjin 300280 China3HeFei University of Technology Hefei China4University of Science and Technology of China Hefei China
Correspondence should be addressed to Jianping Xu xujpingpetrochinacomcn
Received 7 May 2015 Accepted 18 May 2015
Academic Editor Yves Grohens
Copyright copy 2015 J Xu and D LiThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A new well-test model is presented for unsteady flow in multizone with crossflow layers in non-Newtonian polymer floodingreservoir by utilizing the supposition of semipermeable wall and combining it with the first approximation of layered stable flowrates and the effects of wellbore storage and skin were considered in this model and proposed the analytical solutions in Laplacespace for the cases of infinite-acting and bounded systems Finally the stable layer flow rates are provided for commingled systemand crossflow system in late-time radial flow periods
1 Introduction
Many reservoirs are formed from layers of different physicalproperties because of the different geological deposition ro-tary loops Among them if these layers do not communicatein terms of fluid flow through the formation but may be pro-duced by the samewellbore these types of reservoir are calledcommingled systems if there exits fluid that connects betweenthese layers they are referred to as crossflow systems Thepressure-transient behavior depends on the comprehensiveproperties of these multilayers
For the unstable flow of Newtonian fluids in a multilay-ered reservoir Russell et al [1 2] studied pressure behavior ofsingle-phase fluid in two layers with formation crossflow andderived the conclusion that it is similar to the flow behaviorof the two layers without formation crossflow in early timeRaghavan et al [3] studied the problem of well test in a mul-tilayered reservoir Bourdet [4] established using steady stateapproximation to the presentation interlayer flowmodel Gaoand Deans [5] studied the behavior of multilayered reservoirwith formation crossflow Ehlig-Economides [6] has system-atically established the combination of commingled systemand crossflow system unsteady flow models and provided
the rule of the pressure and flow for each layer Bidauxet al [7] using layered pressure and flow data conducted astudy on the theory and practical application of multilayeredreservoir
For the unsteady flow of non-Newtonian fluids in a mul-tilayered reservoir van Poollen and Jargon [8] studied non-Newtonian power-law fluid unsteady flow in porous mediaand showed that the transient pressure response character-istics are different from that of Newtonian fluid Ikoku andRamey [9] studied non-Newtonian power-law fluid unsteadyflow characteristics in porousmedium and the considerationof wellbore storage and skin effect is obtained in homo-geneous infinite reservoir model in Laplace space solutionLund and Ikoku [10 11] proposed non-Newtonian power-lawfluid (polymer solution) and Newtonian fluid (oil) compositemodel of transient well-test analysis method Xu et al[12] proposed the infinite reservoir Laplace space sphericaltransient pressure solution and discussed the characteristicsof wellbore pressure at early times and later times The aboveresearch result is to solve the problem of single layer Escobaret al [13] presented equations to estimate permeability non-Newtonian bank radius and skin factor for the well testdata in reservoirs with non-Newtonian power-law fluids
Hindawi Publishing CorporationJournal of ChemistryVolume 2015 Article ID 875068 7 pageshttpdxdoiorg1011552015875068
2 Journal of Chemistry
q1 S1 k1 1205931 h1
q2 S2 k2 1205932 h2
q3 S3 k3 1205933 h3
q4 S4 k4 1205934 h4
qt
1
2
1
2
3
4 1205823 = 0
1205822
1205823
Zone Layer
1205821 = 0
Figure 1 Sketch of a multilayer reservoir
Martinez et al [14] studied the transient pressure behaviorsfor a Bingham type fluid and the influence of the minimumpressure gradient Escobar et al [15 16] studied transientpressure analysis for non-Newtonian fluids in naturally frac-tured formations modelled as double-porosity model They[17] extended TDS technique to injection and fall-off testsof non-Newtonian pseudoplastic fluids van den Hoek et al[18] presented a simple and practical methodology to inferthe in situ polymer rheology from PFO (Pressure Fall-Off)tests To the problem ofmultilayered Yu et al [19] establisheda well testing model for polymer flooding and presented anumerical well testing interpretation technique to evaluateformation in crossflow double-layer reservoirs
This paper presents a well-test model and the analyticalsolution for non-Newtonian polymer-flooding unsteady flowin multizone with crossflow layers and laid the foundationtheory of field test data interpretation
2 The Model Description
The reservoir model for the 119873-layered system is shown inFigure 1 Each layer is assumed to be homogeneous andisotropic with injected polymer non-Newtonian power-lawfluids
A symmetrically located well penetrates all the layers andeach layer has a skin of arbitrary value S
119894 wellbore storage
coefficient 119862 is assumed to be constant and crossflow mayoccur in the reservoir between any two adjacent layers
Assuming that the fluid is slightly compressible the com-pression coefficient is constant the permeability porosityand thickness of each layer can be different respectively119862119905119895 119896119895B119895 ℎ119895to distinguish between layer pairs with forma-
tion crossflow with and noncommunicating layers and thereservoir is divided into 119873119885 (leN) zones Between any twoadjacent zones there is no formation crossflow Gravity andcapillary forces can be neglected Assuming weak formationcrossflow the flow is about interlayer pressure difference andhas nothing to do with the shear rate crossflow coefficient120594119895 on behalf of interlayer communicating ability Formation
crossflow is modeled as in the semipermeable-wall model of
Deans and Gao [7] which assumes that all resistance to verti-cal flow is concentrated in thewall (layer top bottom)Hencethe pressure difference between adjacent layers depends ononly radial position and time and flow within the layers isstrictly horizontal and assuming that each layer has the sameinitial pressure
The flow in each layer 119895 (119895 = 1 2 119873) is governed bythe following equation
119896119895ℎ119895nabla sdot (
1120583119895
nabla119875119895) = 120601
119895ℎ119895119862119905119895
120597119875119895
120597119905+ 120594119895minus1 (119875119895 minus119875
119895minus1)
minus 120594119895(119875119895+1 minus119875
119895)
(1)
where 120594119895is given by
120594119895=
22 [Δℎ119895119896V119895] + 119909
119895+1 + 119909119895
119895 = 1 119873 (2)
where 1205940 = 120594119899
= 0 Δℎ119895and 119896V119895 are thickness and vertical
permeability of a nonperforated zone between layers 119895 and119895 + 1 and 119909
119895= ℎ119895119896119911119895 where 119896
119911119895is the vertical permeability
for layer 119895 that is the resistance to flow per unit length at the119895th layer interface If there is no nonperforated zone betweenlayers 119895 and 119895 + 1 then the flow resistance on behalf of a 119895
layer interface unit length If on the 119895 layer and the 119895 + 1 layerbut perforated belt then (Δℎ)
119895is zero If there is no formation
crossflow between layers 119895 and 119895 + 1 then 120594119895is zero
The sand surface flow of each layer as a function of time
119902119895 (119905) =
minus2120587119896119895ℎ119895
120583119895
119903119908
120597119875119895
120597119903
1003816100381610038161003816100381610038161003816100381610038161003816119903119908
(3)
Assume that polymer solution viscosity and shear rate ofpower-law relations [4] are as follows
120583119894= 119867]119894
119899minus1 (4)
where119867 is a constant and ]119894and 119899 respectively represent the
shear rate and power-law index when 119899 tends to one fluidshowed a Newtonian fluid properties
Because of the flow in each layer 119895 changing over timethe crossflow rate is much smaller than stable flow rate so theflow rate 119902
119894can be replaced by steady flow rate to calculate the
shear rate
]119894=
119863119902119894119904
119903ℎ119894(119870119894120601119894)05 (119894 = 1 119873) (5)
where 119863 is a constant and layer stable flow rate 119902119894119904can be
obtained by the late-time in unsteady flowWith the viscosity of polymer solution into (1) for the
shear rate representation for radial flow under cylindricalcoordinate dimensionless forms are obtained by finishingafter
120581119895(
1205972119875119895119863
1205971199031198632 +
119899
119903119863
120597119875119895119863
120597119903119863
)
= 120596119895119903119863
1minus119899 120597119875119895119863
120597119905119863
minus 119903119863
1minus119899120582119895minus1 (119875119895119863 minus119875
119895minus1119863)
+ 119903119863
1minus119899120582119895(119875119895+1119863 minus119875
119895119863)
(6)
Journal of Chemistry 3
The boundary condition at the well is given by the fol-lowing equations which account for both skin and wellborestorage
119875119908119863
= 119875119895119863
(1 119905119863) minus 119904119895
120597119875119895119863
120597119903119863
100381610038161003816100381610038161003816100381610038161003816119903119863=1
1 = 119862119863
119889119875119908119863
119889119905119863
minus
119873
sum
119895=1120581119895
120597119875119895119863
120597119903119863
100381610038161003816100381610038161003816100381610038161003816119903119863=1
(7)
For infinite-outer-boundary condition
119875119895119863
(119903119863 119905119863) 997888rarr 0 (119903
119863997888rarr infin) (8)
For no-flow outer-boundary condition
120597119875119895119863
120597119903119863
= 0 (119903119863
= 119903119890119863
) (9)
For initial condition
119875119895119863
(119903119863 0) = 0 (10)
For dimensionless layer flow rates
119902119895119863
(119905119863) =
119902119895
119902= minus 120581119895
120597119875119895119863
120597119903119863
100381610038161003816100381610038161003816100381610038161003816119903119863=1
(11)
where the dimensionless variables are defined by the follow-ing
119875119895119863
=002120587
119902
119873
sum
119895=1
[
[
119896119895
(1+119899)2ℎ119895
119899(119863119902119895119904)1minus119899
1198671199031199081minus119899120601119895
(1minus119899)2]
]
(119875119895minus1198750) (12)
119905119863
= 001sum119873
119895=1 (120581119895(1+119899)2
ℎ119895
119899(119863119902119895119904)1minus119899
120601119895
(1minus119899)2)
119867sum119873
119895=1 120601119895ℎ1198951198621199051198951199031199083minus119899
119905 (13)
120581119895=
119896119895
(1+119899)2ℎ119895
119899120601119895
(119899minus1)2(119863119902119895119904)1minus119899
sum119873
119895=1 119896119895(1+119899)2
ℎ119895
119899120601119895
(119899minus1)2(119863119902119895119904)1minus119899 (14)
120596119895=
120601119895ℎ119895119862119905119895
sum119873
119895=1 120601119895ℎ119895119862119905119895 (15)
120582119895=
119867119903119908
3minus119899120594119895
sum119873
119895=1 119896119895(1+119899)2
ℎ119895
119899120601119895
(119899minus1)2(119863119902119895119904)1minus119899 (16)
119862119863
=119862
21205871199031199082 sum119873
119895=1 120601119895ℎ119895119862119905119895 (17)
119903119863
=119903
119903119908
(18)
119903119890119863
=119903119890
119903119908
(19)
3 The Solution of the Model
The above equation is given by Laplace transform on timeSet
119897 =2
3 minus 119899
119898 =1 minus 119899
3 minus 119899
(20)
The basic control equation solution is as follows
119875119895119863
=
119873
sum
119896=1119903119863
(1minus119899)2[119860119895
119896119870119898
(119897120590119896119903119863
(3minus119899)2)
+119861119895
119896119868119898
(119897120590119896119903119863
(3minus119899)2)]
(21)
where the subscript on119860 indexes the layer and the superscriptindexes 120590119860 and119861 are respectively the first and the two classof 119898 order modified Bessel function 119896
1198951205902 is eigenvalue of
real symmetric three diagonal positive definitematrices [1198861015840119895119896]
where
1198861015840
119895119896= minus120582
119895minus1 119896 = 119895 minus 1 119895 gt 1 120596119895119911 minus 120582119895minus1 minus120582
119895 119896 = 119895
minus 120582119895 119896 = 119895 + 1 119895 lt119873 0 119896 = 119895 minus 1 119895 or 119895 + 1
(22)
According to the boundary conditions a function relation-ship between 119860
119895
119896 and 119861119895
119896 is as follows
1198602119896=
minus1198861111988612
1198601119896= 12057221198961198601119896
1198603119896=
minus (119886211198601119896+ 119886221198602
119896)
11988623= 12057231198961198601119896
119860119873
119896=
minus (119886119873minus1119873minus2119860119873minus2
119896+ 119886119873minus1119873minus1119860119873minus1
119896)
119886119873minus1119873
= 120572119873
1198961198601119896
119861119895
119896= 120572119895
1198961198611119896 (119895 = 1 119873)
(23)
Then1198732 values for120572119895
119896 can be computed directly from theabove recursion formula Let zone 119894 (119894 = 1 119873119885) contain119898119894layer the zone 119894 with a specific layer 119895 dimensionless
pressure
119875119895119863
=
119898119894
sum
119896119894=1119903119863
(1minus119899)2[119860119895
119896119894119870119898
(119897120590119896119894
119903119863
(3minus119899)2)
+119861119895
119896119894119868119898
(119897120590119896119894
119903119863
(3minus119899)2)]
(24)
where each layer index 119896119894 in the sum refers to the same zone
119894 as Layer 119895
4 Journal of Chemistry
Finally the coefficients for each zone can be expressed asmultiples of the coefficients1198601
119896119894 1198611119896119894 of the uppermost layer
in zone 119894 with (12)
119875119895119863
=
119898119894
sum
119896119894=1
119903119863
(1minus119899)2[1198601119896119894120572119895
119896119894119870119898
(119897120590119896119894
119903119863
(3minus119899)2)
+1198611119896119894120572119895
119896119894119868119898
(119897120590119896119894
119903119863
(3minus119899)2)]
(25)
External boundary condition implies the relationshipbetween 1198601
119896119894 and 1198611
119896119894 is as follows
1198611119896119894 = 1198871198961198941198601119896119894 (26)
For infinite-outer-boundary condition
119887119896119894 = 0 (27)
For no-flow outer-boundary conditions
119887119896119894 =
119870119897(119897120590119896119894
119903119890119863
(3minus119899)2)
119868minus119897
(119897120590119896119894
119903119890119863(3minus119899)2)
(28)
According to the boundary conditions the layer 119895 minus 1 andlayer 119895 which in the same zone have the following relations
119898119894
sum
119896119894=11198601119896119894 (120572119895minus1119896119894 119870119898
(119897120590119896119894
) + 119887119896119894119868119898
(119897120590119896119894
)
+ 119904119895minus1120590119896
119894
[(119897120590119896119894
) minus 119887119896119894119868minus119897
(119897120590119896119894
)]
minus 120572119895
119896119894 119870119898
(119897120590119896119894
) + 119887120581119894119868119898
(119897120590119896119894
)
+ 119904119895120590119896119894
[119870119897(119897120590119896119894
) minus 119887120581119894119868minus119897
(119897120590119896119894
)]) = 0
(29)
The layer 119895 minus 1 of zone 119894 minus 1 and the layer 119895 of zone 119894 minus 1 havethe following relations
119898119894minus1
sum
119896119894minus1=1
(1198601119896119894minus1120572119895minus1119896119894minus1 119870119898
(119897120590119896119894minus1
) + 119887119896119894minus1119868119898
(119897120590119896119894minus1
) + 119904119895minus1120590119896
119894minus1[119870119897(119897120590119896119894minus1
) minus 119887119896119894minus1119868minus119897
(119897120590119896119894minus1
)])
minus
119898119894
sum
119896119894=1
(1198601119896119894120572119895
119896119894 119870119898
(119897120590119896119894
) + 119887120581119894119868119898
(119897120590119896119894
) + 119904119895120590119896119894
[119870119897(119897120590119896119894
) minus 119887120581119894119868minus119897
(119897120590119896119894
)]) = 0
(30)
The wellbore pressure is as follows
119875119908119863
=1
(1198621198631199112 + 1119875
119908119863119862119863=0)
(31)
Equation (31) is a general solution and it is by no wellborestorage pressure solution conversion into thewellbore storagepressure solutions Therefore we will solve the 119862
119863= 0 cases
of equationsEquations (29)-(30) are linear equations with coefficient
1198601119896119894 which can be solved by numerical methodThe wellbore
pressure without wellbore storage is given as
119875119908119863119862119863=0 =
119898119894
sum
119896119894=1
(11986011198961198941205721119896119894 119870119898
(119897120590119896119894
) + 119887119896119894119868119898
(119897120590119896119894
)
+ 1199041120590119896119894
[119870119897(119897120590119896119894
) minus 119887119896119894119868minus119897
(119897120590119896119894
)])
(32)
In addition layer flow rate is given as follows
119902119895119863
= (1minus119862119863119875119908119863
1199112)
sdot 120581119895
119898119894
sum
119896119894=1
1198601119896119894120572119895
119896119894120590119896119894
119870119897(119897120590119896119894
) minus 119887119896119894119868minus119897
(119897120590119896119894
)
(33)
Distribution of radial pressure on each layer becomes
119875119895119863
= (1minus119862119863119875119908119863
1199112) 119875119895119863119862119863=0 = (1minus119862
119863119875119908119863
1199112)
sdot
119898119894
sum
119896119894=1
(1198601119896119894120572119895
119896119894 119870119898
(119897120590119896119894
119903119863
(3minus119899)2)
+ 119887119896119894119868119898
(119897120590119896119894
119903119863
(3minus119899)2))
(34)
For the case of double-layer reservoirs with formation cross-flow assume that choosing parameters is as follows
119862119863
= 1 1199041 = 1199042 = 1 120582 = 1119890 minus 5 120581 = 095120596 = 01 and 119899 = 01 03 05 07 09 through the numericalinversion of the Laplace transform of Stehfest [20] Pressureand flow in the real space are obtainedThe wellbore pressurecurve is shown in Figure 1 The wellbore pressure curve isshown in Figure 2 It can be seen that the pressure derivativecurve is unit slope straight line in the early-time namelylinear unit slope represents pure effect of wellbore storagein the medium term the curve is concave interporosityflow transition characteristics in the late stage the pressurederivative curve is approximately straight line up and theslope is related to the power law index The slope is 119898
119871=
(1minus119899)(3minus119899) And it can be seen that the pressure derivativerises and increases with the reduction of power-law index 119899
steepened
Journal of Chemistry 5
108107106105104
103
103
102
102
101
101
100
100
10minus1
10minus1
10minus2
10minus2
CD = 1
s1 = 1s2 = 1
120582 = 1e minus 5120581 = 095
120596 = 01
tD
n
n = [01 03 05 07 09]
PwDdpwD
dln
t D
Figure 2 Dimensionless pressure for two-layer system
00
02
04
06
08
10
10810710610510410310210110010minus110minus2
tD
n
q1D
CD = 1
s1 = 1s2 = 1
120582 = 1e minus 5120581 = 095
120596 = 01
n = [01 03 05 07 09]
Figure 3 Dimensionless flow-rate for the first layer
In Figures 3 and 4 flow curve can be seen for the firstlayer with high quasi capacity coefficient the dimensionlessflow rate increases with the time increasing and finally tendsto be stable flow rate 120581 for the second layer with low quasicapacity coefficient the dimensionless flow rate increaseswith the time increasing In a certain period due to highpermeability layer crossflow pressure tends to be balancedand the crossflow that is more and more weak finally tendsto be stable flow rate 1 minus 120581
4 The Late Stable Flow Model
For both cases stable layer flow-rates were discussed includ-ing (1) without formation crossflow and (2) with formationcrossflow We only discussed the behavior in late time soignore the effect of wellbore storage effect
000
005
010
015
020
10810710610510410310210110010minus2 10minus110minus4 10minus3
tD
n
q2D
CD = 1
s1 = 1s2 = 1
120582 = 1e minus 5120581 = 095
120596 = 01
n = [01 03 05 07 09]
Figure 4 Dimensionless flow-rate for the second layer
(1) Without Formation Crossflow
for the infinite-out-boundary the stable flow rate forlayer 119895 is as follows
119902119895119904
=
119902120581119895
119897120596119895
119898
sum119873
119896=1 120581119896119897120596119896119898 (35)
For the no-flow-boundary the stable flow rate forlayer 119895 is as follows
119902119895119904
=
119902120601119895ℎ119895119862119905119895
sum119873
119895=1 120601119895ℎ119895119862119905119895 (36)
(2) With Formation Crossflow
Regarding the eigenvalues of matrix 1198861015840
119895119896 there is a
value to meet 1205902
= 119911 and the rest is independentof 119911 and 120596
119895and depends only on 1205811 1205812 120581119899 and
1205821 1205822 120582119899 Assume that 12059012= 119911 Stable layer flow
rate through the determinant value can be expressedas
119902119895119904
= 119902120581119895lim119911rarr 0
[
[
det 11986211989511989611
det 119862119895119896
120572119895
11205901 119870119897 (1198971205901)
minus 1198871119868minus119897
(1198971205901)]
]
+ 119902120581119895
119873
sum
119896119894=2lim119911rarr 0
[
det 119862119894119896119896119894119896119894
det 119862119894119896
sdot 120572119895
119896119894120590119896119894
119870119897(119897120590119896) minus 119887119896119894119868minus119897
(119897120590119896)]
(37)
where det119862119895119896 is determinant of matrix 119862
119895119896
6 Journal of Chemistry
The matrix element is as follows
119862119895119896
= 120572119895
119896119870119898
(119897120590119896) + 119887119896119868119898
(119897120590119896)
+ 119904119896120590119896[119870119897(119897120590119896) minus 119887119896119868minus119897
(119897120590119896)]
minus 120572119895+1119896119870119898
(119897120590119896+1) + 119887
119896119868119898
(119897120590119896+1)
+ 119904119896+1120590119896+1 [119870
119897(119897120590119896+1) minus 119887
119896119868minus119897
(119897120590119896+1)]
(38)
Thematrix 119862119895119896119894119894is used in matrix in 119862
119895119896 column 119894
replacement for (0 0 119860 1119911)119879 where superscript
119879 is the vector transpose
5 Conclusions
Establishing the multilayer reservoir well-test model forpolymer flooding gives the formation and wellbore transientpressure and the layer flow-rate finallyThe expressions of latetime stable flow rate in each layer are given
The unsteady well-test model provides theoreticalmethod for polymer flooding well test analysis of multilayerreservoir the experimental data and the theoretical curvefitting can be used to determine the permeability skin factorand effective interlayer vertical permeability and otherimportant parameters
Nomenclature
1198861015840
119895119896 Matrix elements defined in (22)
119860119895
119896 119861119895
119896 Coefficient for 119895th layer 119896th root defined in(23)
1198601119896119894 1198611119896119894 Coefficient for 119895th layer 119896th root in zone 119868
defined in (25)119887119896119894 Coefficient for outer boundary condition
defined in (27) or (28)119862 Wellbore-storage coefficient (cm3kPa)119862119905119895 Total compressibility in layer 119868 (kPaminus1)
119862119895119896 Matrix elements defined in (38)
119863 Constant defined in (5)ℎ119895 Formation thickness in layer 119895 (cm)
119867 Constant defined in (4)119868119898(sdot) 119870119898(sdot) Modified 119898 order Bessel function of the first
and second kindΔℎ119895 Thickness of a nonperforated zone between
layers 119895 and 119895 + 1 (cm)119870119895 Horizontal permeability in layer 119895 120583m2
119870V119895 Vertical permeability of a tight zone betweenLayers 119895 and 119895 minus 1 (120583m2)
119870119911119895 Vertical permeability in Layer 119895 (120583m2)
119897 119898 Constant defined in (20)119898119894 Number of layers in Zone 119894
119899 Power-law index119873 Number of layers in reservoir system119873119885 Number of Zones in reservoir system119875119894 Reservoir pressure in layer 119868 (kPa)
119875119895
119894 Reservoir pressure in layer 119895 in Zone 119894
(kPa)119875119895119863 Dimensionless reservoir pressure in
layer 119895 defined in (12)1198750 Initial reservoir pressure (kPa)
119875119908119863
Dimensionless bottomhole pressure inLaplace space
119875119908119863119862119863=0 Dimensionless bottomhole pressure
without wellbore storage in Laplacespace
119876 Surface production rate (cm3s)119902119895 Flow rate for layer 119895 (cm3s)
119902119895119904 Stable flow rate for layer 119895 (cm3s)
119903 Radial distance (cm)119903119863 Dimensionless radius defined in (18)
119903119890 Reservoir outer radius (cm)
119903119890119863 Dimensionless outer reservoir radius
defined in (19)119903119908 Wellbore radius (cm)
119878119895 Wellbore skin factor for layer 119895
119905 Time (s)Δ119905 Elapsed time after rate change (s)119905119863 Dimensionless time referenced to
producing layer]119894 Shear rate in layer 119868 (sminus1)
120594119895 Crossflow coefficient between layers 119895
and 119895 + 1 defined in (2)119909119895 Resistance to flow per unit length at the
119895th layer interface119885 Laplace space variable 119894
120572119895
119896 Coefficient for layer 119895 Root 119896 definedin (23)
120572119895
119896119894 Coefficient for layer 119895 Root 119896 defined
in Zone 119868 defined in (25)120581119895 Coefficient for layer 119895 defined in (14)
120582119895 Dimensionless semipermeability
between layers 119895 and 119895 + 1 defined in(16)
119872 Dynamic viscosity (mpasdots)Φ119895 Porosity fraction for layer 119895
120596119895 Coefficient for layer 119895 defined in (15)
]119895 Shear rate for layer 119895 defined in (5)
det119862119895119896 Determinant of matrix 119862
119895119896
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The project was supported by the National Key Science andTechnology Project (2011ZX05009-006) the Major StateBasic Research Development Program of China (973 Pro-gram) (no 2011CB707305) and the China ScholarshipCouncil CAS Strategic Priority Research Program(XDB10030402)
Journal of Chemistry 7
References
[1] D G Russell and M Prats ldquoPerformance of layered reser-voirs with crossflowmdashsingle-compressible-fluid caserdquo Society ofPetroleum Engineers Journal vol 2 no 1 pp 53ndash67 1962
[2] D Russell and M Prats ldquoThe practical aspects of interlayercrossflowrdquo Journal of Petroleum Technology vol 14 no 06 pp589ndash594 2013
[3] R Raghavan R Prijambodo and A C Reynolds ldquoWell testanalysis for well producing layered Reservoirs with crossflowrdquoSociety of Petroleum Engineers Journal vol 25 no 3 pp 380ndash396 1985
[4] D Bourdet ldquoPressure behavior of Layered reservoirs withcrossflowrdquo inProceedings of the SPECalifornia RegionalMeetingSPE-13628-MS Bakersfield Calif USA March 1985
[5] C-T Gao and H A Deans ldquoPressure transients and crossflowcaused by diffusivities inmultiplayer reservoirsrdquo SPE FormationEvaluation vol 3 no 2 pp 438ndash448 1988
[6] C A Ehlig-Economides ldquoA new test for determination ofindividual layer properties in a multilayered reservoirrdquo SPEFormation Evaluation vol 2 no 3 pp 261ndash283 1987
[7] P Bidaux T M Whittle P J Coveney and A C GringartenldquoAnalysis of pressure and rate transient data from wells inmultilayered reservoirs theory and applicationrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE24679 Washington DC USA October 1992
[8] H van Poollen and J Jargon ldquoSteady-state and unsteady-stateflow of Non-Newtonian fluids through porous mediardquo Societyof Petroleum Engineers Journal vol 9 no 1 pp 80ndash88 2013
[9] C U Ikoku and H J Ramey Jr ldquoWellbore storage and skineffects during the transient flow of non-Newtonian power-lawfluids in porous mediardquo Society of Petroleum Engineers Journalvol 20 no 1 pp 25ndash38 1980
[10] O Lund and C U Ikoku ldquoPressure transient behavior of non-newtoniannewtonian fluid composite reservoirsrdquo Society ofPetroleum Engineers Journal vol 21 no 2 pp 271ndash280 1981
[11] R Prijambodo R Raghavan and A C Reynolds Well TestAnalysis for Wells Producing Layered Reservoirs With CrossflowSociety of Petroleum Engineers 1985
[12] J Xu LWang andK Zhu ldquoPressure behavior for spherical flowin a reservoir of non-Newtonian power-law fluidsrdquo PetroleumScience and Technology vol 21 no 1-2 pp 133ndash143 2003
[13] F-H Escobar J-A Martınez and M Montealegre-MaderoldquoPressure and pressure derivative analysis for a well in aradial composite reservoir with a non-NewtonianNewtonianinterfacerdquo Ciencia Tecnologia y Futuro vol 4 no 2 pp 33ndash422010
[14] J A Martinez F H Escobar and M M Montealegre ldquoVerticalwell pressure and pressure derivative analysis for binghamfluidsin a homogeneous reservoirsrdquo Dyna vol 78 no 166 pp 21ndash282011
[15] F H Escobar A P Zambrano D V Giraldo and J H CantilloldquoPressure and pressure derivative analysis for non-Newtonianpseudoplastic fluids in double-porosity formationsrdquo CTampFmdashCiencia Tecnologıa y Futuro vol 4 no 3 pp 47ndash59 2011
[16] F H Escobar A Martınez and D M Silva ldquoConventionalpressure analysis for naturally-fractured reservoirs with non-Newtonian pseudoplastic fluidsrdquo Fuentes Journal vol 11 no 1pp 27ndash34 2013
[17] F H Escobar J M Ascencio and D F Real ldquoInjection and fall-off tests transient analysis of non-newtonian fluidsrdquo Journal ofEngineering and Applied Sciences vol 8 no 9 pp 703ndash707 2013
[18] P van den Hoek H Mahani T Sorop et al ldquoApplicationof injection fall-off analysis in polymer floodingrdquo in Proceed-ings of the SPE EuropecEAGE Annual Conference Society ofPetroleum Engineers Copenhagen Denmark June 2012
[19] H Y Yu H Guo Y W He et al ldquoNumerical well testing in-terpretation model and applications in crossflow double-layerreservoirs by polymer floodingrdquo The Scientific World Journalvol 2014 Article ID 890874 11 pages 2014
[20] H Stehfest ldquoAlgorithm 368 numerical inversion of Laplacetransforms [D5]rdquo Communications of the ACM vol 13 no 1pp 47ndash49 1970
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
2 Journal of Chemistry
q1 S1 k1 1205931 h1
q2 S2 k2 1205932 h2
q3 S3 k3 1205933 h3
q4 S4 k4 1205934 h4
qt
1
2
1
2
3
4 1205823 = 0
1205822
1205823
Zone Layer
1205821 = 0
Figure 1 Sketch of a multilayer reservoir
Martinez et al [14] studied the transient pressure behaviorsfor a Bingham type fluid and the influence of the minimumpressure gradient Escobar et al [15 16] studied transientpressure analysis for non-Newtonian fluids in naturally frac-tured formations modelled as double-porosity model They[17] extended TDS technique to injection and fall-off testsof non-Newtonian pseudoplastic fluids van den Hoek et al[18] presented a simple and practical methodology to inferthe in situ polymer rheology from PFO (Pressure Fall-Off)tests To the problem ofmultilayered Yu et al [19] establisheda well testing model for polymer flooding and presented anumerical well testing interpretation technique to evaluateformation in crossflow double-layer reservoirs
This paper presents a well-test model and the analyticalsolution for non-Newtonian polymer-flooding unsteady flowin multizone with crossflow layers and laid the foundationtheory of field test data interpretation
2 The Model Description
The reservoir model for the 119873-layered system is shown inFigure 1 Each layer is assumed to be homogeneous andisotropic with injected polymer non-Newtonian power-lawfluids
A symmetrically located well penetrates all the layers andeach layer has a skin of arbitrary value S
119894 wellbore storage
coefficient 119862 is assumed to be constant and crossflow mayoccur in the reservoir between any two adjacent layers
Assuming that the fluid is slightly compressible the com-pression coefficient is constant the permeability porosityand thickness of each layer can be different respectively119862119905119895 119896119895B119895 ℎ119895to distinguish between layer pairs with forma-
tion crossflow with and noncommunicating layers and thereservoir is divided into 119873119885 (leN) zones Between any twoadjacent zones there is no formation crossflow Gravity andcapillary forces can be neglected Assuming weak formationcrossflow the flow is about interlayer pressure difference andhas nothing to do with the shear rate crossflow coefficient120594119895 on behalf of interlayer communicating ability Formation
crossflow is modeled as in the semipermeable-wall model of
Deans and Gao [7] which assumes that all resistance to verti-cal flow is concentrated in thewall (layer top bottom)Hencethe pressure difference between adjacent layers depends ononly radial position and time and flow within the layers isstrictly horizontal and assuming that each layer has the sameinitial pressure
The flow in each layer 119895 (119895 = 1 2 119873) is governed bythe following equation
119896119895ℎ119895nabla sdot (
1120583119895
nabla119875119895) = 120601
119895ℎ119895119862119905119895
120597119875119895
120597119905+ 120594119895minus1 (119875119895 minus119875
119895minus1)
minus 120594119895(119875119895+1 minus119875
119895)
(1)
where 120594119895is given by
120594119895=
22 [Δℎ119895119896V119895] + 119909
119895+1 + 119909119895
119895 = 1 119873 (2)
where 1205940 = 120594119899
= 0 Δℎ119895and 119896V119895 are thickness and vertical
permeability of a nonperforated zone between layers 119895 and119895 + 1 and 119909
119895= ℎ119895119896119911119895 where 119896
119911119895is the vertical permeability
for layer 119895 that is the resistance to flow per unit length at the119895th layer interface If there is no nonperforated zone betweenlayers 119895 and 119895 + 1 then the flow resistance on behalf of a 119895
layer interface unit length If on the 119895 layer and the 119895 + 1 layerbut perforated belt then (Δℎ)
119895is zero If there is no formation
crossflow between layers 119895 and 119895 + 1 then 120594119895is zero
The sand surface flow of each layer as a function of time
119902119895 (119905) =
minus2120587119896119895ℎ119895
120583119895
119903119908
120597119875119895
120597119903
1003816100381610038161003816100381610038161003816100381610038161003816119903119908
(3)
Assume that polymer solution viscosity and shear rate ofpower-law relations [4] are as follows
120583119894= 119867]119894
119899minus1 (4)
where119867 is a constant and ]119894and 119899 respectively represent the
shear rate and power-law index when 119899 tends to one fluidshowed a Newtonian fluid properties
Because of the flow in each layer 119895 changing over timethe crossflow rate is much smaller than stable flow rate so theflow rate 119902
119894can be replaced by steady flow rate to calculate the
shear rate
]119894=
119863119902119894119904
119903ℎ119894(119870119894120601119894)05 (119894 = 1 119873) (5)
where 119863 is a constant and layer stable flow rate 119902119894119904can be
obtained by the late-time in unsteady flowWith the viscosity of polymer solution into (1) for the
shear rate representation for radial flow under cylindricalcoordinate dimensionless forms are obtained by finishingafter
120581119895(
1205972119875119895119863
1205971199031198632 +
119899
119903119863
120597119875119895119863
120597119903119863
)
= 120596119895119903119863
1minus119899 120597119875119895119863
120597119905119863
minus 119903119863
1minus119899120582119895minus1 (119875119895119863 minus119875
119895minus1119863)
+ 119903119863
1minus119899120582119895(119875119895+1119863 minus119875
119895119863)
(6)
Journal of Chemistry 3
The boundary condition at the well is given by the fol-lowing equations which account for both skin and wellborestorage
119875119908119863
= 119875119895119863
(1 119905119863) minus 119904119895
120597119875119895119863
120597119903119863
100381610038161003816100381610038161003816100381610038161003816119903119863=1
1 = 119862119863
119889119875119908119863
119889119905119863
minus
119873
sum
119895=1120581119895
120597119875119895119863
120597119903119863
100381610038161003816100381610038161003816100381610038161003816119903119863=1
(7)
For infinite-outer-boundary condition
119875119895119863
(119903119863 119905119863) 997888rarr 0 (119903
119863997888rarr infin) (8)
For no-flow outer-boundary condition
120597119875119895119863
120597119903119863
= 0 (119903119863
= 119903119890119863
) (9)
For initial condition
119875119895119863
(119903119863 0) = 0 (10)
For dimensionless layer flow rates
119902119895119863
(119905119863) =
119902119895
119902= minus 120581119895
120597119875119895119863
120597119903119863
100381610038161003816100381610038161003816100381610038161003816119903119863=1
(11)
where the dimensionless variables are defined by the follow-ing
119875119895119863
=002120587
119902
119873
sum
119895=1
[
[
119896119895
(1+119899)2ℎ119895
119899(119863119902119895119904)1minus119899
1198671199031199081minus119899120601119895
(1minus119899)2]
]
(119875119895minus1198750) (12)
119905119863
= 001sum119873
119895=1 (120581119895(1+119899)2
ℎ119895
119899(119863119902119895119904)1minus119899
120601119895
(1minus119899)2)
119867sum119873
119895=1 120601119895ℎ1198951198621199051198951199031199083minus119899
119905 (13)
120581119895=
119896119895
(1+119899)2ℎ119895
119899120601119895
(119899minus1)2(119863119902119895119904)1minus119899
sum119873
119895=1 119896119895(1+119899)2
ℎ119895
119899120601119895
(119899minus1)2(119863119902119895119904)1minus119899 (14)
120596119895=
120601119895ℎ119895119862119905119895
sum119873
119895=1 120601119895ℎ119895119862119905119895 (15)
120582119895=
119867119903119908
3minus119899120594119895
sum119873
119895=1 119896119895(1+119899)2
ℎ119895
119899120601119895
(119899minus1)2(119863119902119895119904)1minus119899 (16)
119862119863
=119862
21205871199031199082 sum119873
119895=1 120601119895ℎ119895119862119905119895 (17)
119903119863
=119903
119903119908
(18)
119903119890119863
=119903119890
119903119908
(19)
3 The Solution of the Model
The above equation is given by Laplace transform on timeSet
119897 =2
3 minus 119899
119898 =1 minus 119899
3 minus 119899
(20)
The basic control equation solution is as follows
119875119895119863
=
119873
sum
119896=1119903119863
(1minus119899)2[119860119895
119896119870119898
(119897120590119896119903119863
(3minus119899)2)
+119861119895
119896119868119898
(119897120590119896119903119863
(3minus119899)2)]
(21)
where the subscript on119860 indexes the layer and the superscriptindexes 120590119860 and119861 are respectively the first and the two classof 119898 order modified Bessel function 119896
1198951205902 is eigenvalue of
real symmetric three diagonal positive definitematrices [1198861015840119895119896]
where
1198861015840
119895119896= minus120582
119895minus1 119896 = 119895 minus 1 119895 gt 1 120596119895119911 minus 120582119895minus1 minus120582
119895 119896 = 119895
minus 120582119895 119896 = 119895 + 1 119895 lt119873 0 119896 = 119895 minus 1 119895 or 119895 + 1
(22)
According to the boundary conditions a function relation-ship between 119860
119895
119896 and 119861119895
119896 is as follows
1198602119896=
minus1198861111988612
1198601119896= 12057221198961198601119896
1198603119896=
minus (119886211198601119896+ 119886221198602
119896)
11988623= 12057231198961198601119896
119860119873
119896=
minus (119886119873minus1119873minus2119860119873minus2
119896+ 119886119873minus1119873minus1119860119873minus1
119896)
119886119873minus1119873
= 120572119873
1198961198601119896
119861119895
119896= 120572119895
1198961198611119896 (119895 = 1 119873)
(23)
Then1198732 values for120572119895
119896 can be computed directly from theabove recursion formula Let zone 119894 (119894 = 1 119873119885) contain119898119894layer the zone 119894 with a specific layer 119895 dimensionless
pressure
119875119895119863
=
119898119894
sum
119896119894=1119903119863
(1minus119899)2[119860119895
119896119894119870119898
(119897120590119896119894
119903119863
(3minus119899)2)
+119861119895
119896119894119868119898
(119897120590119896119894
119903119863
(3minus119899)2)]
(24)
where each layer index 119896119894 in the sum refers to the same zone
119894 as Layer 119895
4 Journal of Chemistry
Finally the coefficients for each zone can be expressed asmultiples of the coefficients1198601
119896119894 1198611119896119894 of the uppermost layer
in zone 119894 with (12)
119875119895119863
=
119898119894
sum
119896119894=1
119903119863
(1minus119899)2[1198601119896119894120572119895
119896119894119870119898
(119897120590119896119894
119903119863
(3minus119899)2)
+1198611119896119894120572119895
119896119894119868119898
(119897120590119896119894
119903119863
(3minus119899)2)]
(25)
External boundary condition implies the relationshipbetween 1198601
119896119894 and 1198611
119896119894 is as follows
1198611119896119894 = 1198871198961198941198601119896119894 (26)
For infinite-outer-boundary condition
119887119896119894 = 0 (27)
For no-flow outer-boundary conditions
119887119896119894 =
119870119897(119897120590119896119894
119903119890119863
(3minus119899)2)
119868minus119897
(119897120590119896119894
119903119890119863(3minus119899)2)
(28)
According to the boundary conditions the layer 119895 minus 1 andlayer 119895 which in the same zone have the following relations
119898119894
sum
119896119894=11198601119896119894 (120572119895minus1119896119894 119870119898
(119897120590119896119894
) + 119887119896119894119868119898
(119897120590119896119894
)
+ 119904119895minus1120590119896
119894
[(119897120590119896119894
) minus 119887119896119894119868minus119897
(119897120590119896119894
)]
minus 120572119895
119896119894 119870119898
(119897120590119896119894
) + 119887120581119894119868119898
(119897120590119896119894
)
+ 119904119895120590119896119894
[119870119897(119897120590119896119894
) minus 119887120581119894119868minus119897
(119897120590119896119894
)]) = 0
(29)
The layer 119895 minus 1 of zone 119894 minus 1 and the layer 119895 of zone 119894 minus 1 havethe following relations
119898119894minus1
sum
119896119894minus1=1
(1198601119896119894minus1120572119895minus1119896119894minus1 119870119898
(119897120590119896119894minus1
) + 119887119896119894minus1119868119898
(119897120590119896119894minus1
) + 119904119895minus1120590119896
119894minus1[119870119897(119897120590119896119894minus1
) minus 119887119896119894minus1119868minus119897
(119897120590119896119894minus1
)])
minus
119898119894
sum
119896119894=1
(1198601119896119894120572119895
119896119894 119870119898
(119897120590119896119894
) + 119887120581119894119868119898
(119897120590119896119894
) + 119904119895120590119896119894
[119870119897(119897120590119896119894
) minus 119887120581119894119868minus119897
(119897120590119896119894
)]) = 0
(30)
The wellbore pressure is as follows
119875119908119863
=1
(1198621198631199112 + 1119875
119908119863119862119863=0)
(31)
Equation (31) is a general solution and it is by no wellborestorage pressure solution conversion into thewellbore storagepressure solutions Therefore we will solve the 119862
119863= 0 cases
of equationsEquations (29)-(30) are linear equations with coefficient
1198601119896119894 which can be solved by numerical methodThe wellbore
pressure without wellbore storage is given as
119875119908119863119862119863=0 =
119898119894
sum
119896119894=1
(11986011198961198941205721119896119894 119870119898
(119897120590119896119894
) + 119887119896119894119868119898
(119897120590119896119894
)
+ 1199041120590119896119894
[119870119897(119897120590119896119894
) minus 119887119896119894119868minus119897
(119897120590119896119894
)])
(32)
In addition layer flow rate is given as follows
119902119895119863
= (1minus119862119863119875119908119863
1199112)
sdot 120581119895
119898119894
sum
119896119894=1
1198601119896119894120572119895
119896119894120590119896119894
119870119897(119897120590119896119894
) minus 119887119896119894119868minus119897
(119897120590119896119894
)
(33)
Distribution of radial pressure on each layer becomes
119875119895119863
= (1minus119862119863119875119908119863
1199112) 119875119895119863119862119863=0 = (1minus119862
119863119875119908119863
1199112)
sdot
119898119894
sum
119896119894=1
(1198601119896119894120572119895
119896119894 119870119898
(119897120590119896119894
119903119863
(3minus119899)2)
+ 119887119896119894119868119898
(119897120590119896119894
119903119863
(3minus119899)2))
(34)
For the case of double-layer reservoirs with formation cross-flow assume that choosing parameters is as follows
119862119863
= 1 1199041 = 1199042 = 1 120582 = 1119890 minus 5 120581 = 095120596 = 01 and 119899 = 01 03 05 07 09 through the numericalinversion of the Laplace transform of Stehfest [20] Pressureand flow in the real space are obtainedThe wellbore pressurecurve is shown in Figure 1 The wellbore pressure curve isshown in Figure 2 It can be seen that the pressure derivativecurve is unit slope straight line in the early-time namelylinear unit slope represents pure effect of wellbore storagein the medium term the curve is concave interporosityflow transition characteristics in the late stage the pressurederivative curve is approximately straight line up and theslope is related to the power law index The slope is 119898
119871=
(1minus119899)(3minus119899) And it can be seen that the pressure derivativerises and increases with the reduction of power-law index 119899
steepened
Journal of Chemistry 5
108107106105104
103
103
102
102
101
101
100
100
10minus1
10minus1
10minus2
10minus2
CD = 1
s1 = 1s2 = 1
120582 = 1e minus 5120581 = 095
120596 = 01
tD
n
n = [01 03 05 07 09]
PwDdpwD
dln
t D
Figure 2 Dimensionless pressure for two-layer system
00
02
04
06
08
10
10810710610510410310210110010minus110minus2
tD
n
q1D
CD = 1
s1 = 1s2 = 1
120582 = 1e minus 5120581 = 095
120596 = 01
n = [01 03 05 07 09]
Figure 3 Dimensionless flow-rate for the first layer
In Figures 3 and 4 flow curve can be seen for the firstlayer with high quasi capacity coefficient the dimensionlessflow rate increases with the time increasing and finally tendsto be stable flow rate 120581 for the second layer with low quasicapacity coefficient the dimensionless flow rate increaseswith the time increasing In a certain period due to highpermeability layer crossflow pressure tends to be balancedand the crossflow that is more and more weak finally tendsto be stable flow rate 1 minus 120581
4 The Late Stable Flow Model
For both cases stable layer flow-rates were discussed includ-ing (1) without formation crossflow and (2) with formationcrossflow We only discussed the behavior in late time soignore the effect of wellbore storage effect
000
005
010
015
020
10810710610510410310210110010minus2 10minus110minus4 10minus3
tD
n
q2D
CD = 1
s1 = 1s2 = 1
120582 = 1e minus 5120581 = 095
120596 = 01
n = [01 03 05 07 09]
Figure 4 Dimensionless flow-rate for the second layer
(1) Without Formation Crossflow
for the infinite-out-boundary the stable flow rate forlayer 119895 is as follows
119902119895119904
=
119902120581119895
119897120596119895
119898
sum119873
119896=1 120581119896119897120596119896119898 (35)
For the no-flow-boundary the stable flow rate forlayer 119895 is as follows
119902119895119904
=
119902120601119895ℎ119895119862119905119895
sum119873
119895=1 120601119895ℎ119895119862119905119895 (36)
(2) With Formation Crossflow
Regarding the eigenvalues of matrix 1198861015840
119895119896 there is a
value to meet 1205902
= 119911 and the rest is independentof 119911 and 120596
119895and depends only on 1205811 1205812 120581119899 and
1205821 1205822 120582119899 Assume that 12059012= 119911 Stable layer flow
rate through the determinant value can be expressedas
119902119895119904
= 119902120581119895lim119911rarr 0
[
[
det 11986211989511989611
det 119862119895119896
120572119895
11205901 119870119897 (1198971205901)
minus 1198871119868minus119897
(1198971205901)]
]
+ 119902120581119895
119873
sum
119896119894=2lim119911rarr 0
[
det 119862119894119896119896119894119896119894
det 119862119894119896
sdot 120572119895
119896119894120590119896119894
119870119897(119897120590119896) minus 119887119896119894119868minus119897
(119897120590119896)]
(37)
where det119862119895119896 is determinant of matrix 119862
119895119896
6 Journal of Chemistry
The matrix element is as follows
119862119895119896
= 120572119895
119896119870119898
(119897120590119896) + 119887119896119868119898
(119897120590119896)
+ 119904119896120590119896[119870119897(119897120590119896) minus 119887119896119868minus119897
(119897120590119896)]
minus 120572119895+1119896119870119898
(119897120590119896+1) + 119887
119896119868119898
(119897120590119896+1)
+ 119904119896+1120590119896+1 [119870
119897(119897120590119896+1) minus 119887
119896119868minus119897
(119897120590119896+1)]
(38)
Thematrix 119862119895119896119894119894is used in matrix in 119862
119895119896 column 119894
replacement for (0 0 119860 1119911)119879 where superscript
119879 is the vector transpose
5 Conclusions
Establishing the multilayer reservoir well-test model forpolymer flooding gives the formation and wellbore transientpressure and the layer flow-rate finallyThe expressions of latetime stable flow rate in each layer are given
The unsteady well-test model provides theoreticalmethod for polymer flooding well test analysis of multilayerreservoir the experimental data and the theoretical curvefitting can be used to determine the permeability skin factorand effective interlayer vertical permeability and otherimportant parameters
Nomenclature
1198861015840
119895119896 Matrix elements defined in (22)
119860119895
119896 119861119895
119896 Coefficient for 119895th layer 119896th root defined in(23)
1198601119896119894 1198611119896119894 Coefficient for 119895th layer 119896th root in zone 119868
defined in (25)119887119896119894 Coefficient for outer boundary condition
defined in (27) or (28)119862 Wellbore-storage coefficient (cm3kPa)119862119905119895 Total compressibility in layer 119868 (kPaminus1)
119862119895119896 Matrix elements defined in (38)
119863 Constant defined in (5)ℎ119895 Formation thickness in layer 119895 (cm)
119867 Constant defined in (4)119868119898(sdot) 119870119898(sdot) Modified 119898 order Bessel function of the first
and second kindΔℎ119895 Thickness of a nonperforated zone between
layers 119895 and 119895 + 1 (cm)119870119895 Horizontal permeability in layer 119895 120583m2
119870V119895 Vertical permeability of a tight zone betweenLayers 119895 and 119895 minus 1 (120583m2)
119870119911119895 Vertical permeability in Layer 119895 (120583m2)
119897 119898 Constant defined in (20)119898119894 Number of layers in Zone 119894
119899 Power-law index119873 Number of layers in reservoir system119873119885 Number of Zones in reservoir system119875119894 Reservoir pressure in layer 119868 (kPa)
119875119895
119894 Reservoir pressure in layer 119895 in Zone 119894
(kPa)119875119895119863 Dimensionless reservoir pressure in
layer 119895 defined in (12)1198750 Initial reservoir pressure (kPa)
119875119908119863
Dimensionless bottomhole pressure inLaplace space
119875119908119863119862119863=0 Dimensionless bottomhole pressure
without wellbore storage in Laplacespace
119876 Surface production rate (cm3s)119902119895 Flow rate for layer 119895 (cm3s)
119902119895119904 Stable flow rate for layer 119895 (cm3s)
119903 Radial distance (cm)119903119863 Dimensionless radius defined in (18)
119903119890 Reservoir outer radius (cm)
119903119890119863 Dimensionless outer reservoir radius
defined in (19)119903119908 Wellbore radius (cm)
119878119895 Wellbore skin factor for layer 119895
119905 Time (s)Δ119905 Elapsed time after rate change (s)119905119863 Dimensionless time referenced to
producing layer]119894 Shear rate in layer 119868 (sminus1)
120594119895 Crossflow coefficient between layers 119895
and 119895 + 1 defined in (2)119909119895 Resistance to flow per unit length at the
119895th layer interface119885 Laplace space variable 119894
120572119895
119896 Coefficient for layer 119895 Root 119896 definedin (23)
120572119895
119896119894 Coefficient for layer 119895 Root 119896 defined
in Zone 119868 defined in (25)120581119895 Coefficient for layer 119895 defined in (14)
120582119895 Dimensionless semipermeability
between layers 119895 and 119895 + 1 defined in(16)
119872 Dynamic viscosity (mpasdots)Φ119895 Porosity fraction for layer 119895
120596119895 Coefficient for layer 119895 defined in (15)
]119895 Shear rate for layer 119895 defined in (5)
det119862119895119896 Determinant of matrix 119862
119895119896
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The project was supported by the National Key Science andTechnology Project (2011ZX05009-006) the Major StateBasic Research Development Program of China (973 Pro-gram) (no 2011CB707305) and the China ScholarshipCouncil CAS Strategic Priority Research Program(XDB10030402)
Journal of Chemistry 7
References
[1] D G Russell and M Prats ldquoPerformance of layered reser-voirs with crossflowmdashsingle-compressible-fluid caserdquo Society ofPetroleum Engineers Journal vol 2 no 1 pp 53ndash67 1962
[2] D Russell and M Prats ldquoThe practical aspects of interlayercrossflowrdquo Journal of Petroleum Technology vol 14 no 06 pp589ndash594 2013
[3] R Raghavan R Prijambodo and A C Reynolds ldquoWell testanalysis for well producing layered Reservoirs with crossflowrdquoSociety of Petroleum Engineers Journal vol 25 no 3 pp 380ndash396 1985
[4] D Bourdet ldquoPressure behavior of Layered reservoirs withcrossflowrdquo inProceedings of the SPECalifornia RegionalMeetingSPE-13628-MS Bakersfield Calif USA March 1985
[5] C-T Gao and H A Deans ldquoPressure transients and crossflowcaused by diffusivities inmultiplayer reservoirsrdquo SPE FormationEvaluation vol 3 no 2 pp 438ndash448 1988
[6] C A Ehlig-Economides ldquoA new test for determination ofindividual layer properties in a multilayered reservoirrdquo SPEFormation Evaluation vol 2 no 3 pp 261ndash283 1987
[7] P Bidaux T M Whittle P J Coveney and A C GringartenldquoAnalysis of pressure and rate transient data from wells inmultilayered reservoirs theory and applicationrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE24679 Washington DC USA October 1992
[8] H van Poollen and J Jargon ldquoSteady-state and unsteady-stateflow of Non-Newtonian fluids through porous mediardquo Societyof Petroleum Engineers Journal vol 9 no 1 pp 80ndash88 2013
[9] C U Ikoku and H J Ramey Jr ldquoWellbore storage and skineffects during the transient flow of non-Newtonian power-lawfluids in porous mediardquo Society of Petroleum Engineers Journalvol 20 no 1 pp 25ndash38 1980
[10] O Lund and C U Ikoku ldquoPressure transient behavior of non-newtoniannewtonian fluid composite reservoirsrdquo Society ofPetroleum Engineers Journal vol 21 no 2 pp 271ndash280 1981
[11] R Prijambodo R Raghavan and A C Reynolds Well TestAnalysis for Wells Producing Layered Reservoirs With CrossflowSociety of Petroleum Engineers 1985
[12] J Xu LWang andK Zhu ldquoPressure behavior for spherical flowin a reservoir of non-Newtonian power-law fluidsrdquo PetroleumScience and Technology vol 21 no 1-2 pp 133ndash143 2003
[13] F-H Escobar J-A Martınez and M Montealegre-MaderoldquoPressure and pressure derivative analysis for a well in aradial composite reservoir with a non-NewtonianNewtonianinterfacerdquo Ciencia Tecnologia y Futuro vol 4 no 2 pp 33ndash422010
[14] J A Martinez F H Escobar and M M Montealegre ldquoVerticalwell pressure and pressure derivative analysis for binghamfluidsin a homogeneous reservoirsrdquo Dyna vol 78 no 166 pp 21ndash282011
[15] F H Escobar A P Zambrano D V Giraldo and J H CantilloldquoPressure and pressure derivative analysis for non-Newtonianpseudoplastic fluids in double-porosity formationsrdquo CTampFmdashCiencia Tecnologıa y Futuro vol 4 no 3 pp 47ndash59 2011
[16] F H Escobar A Martınez and D M Silva ldquoConventionalpressure analysis for naturally-fractured reservoirs with non-Newtonian pseudoplastic fluidsrdquo Fuentes Journal vol 11 no 1pp 27ndash34 2013
[17] F H Escobar J M Ascencio and D F Real ldquoInjection and fall-off tests transient analysis of non-newtonian fluidsrdquo Journal ofEngineering and Applied Sciences vol 8 no 9 pp 703ndash707 2013
[18] P van den Hoek H Mahani T Sorop et al ldquoApplicationof injection fall-off analysis in polymer floodingrdquo in Proceed-ings of the SPE EuropecEAGE Annual Conference Society ofPetroleum Engineers Copenhagen Denmark June 2012
[19] H Y Yu H Guo Y W He et al ldquoNumerical well testing in-terpretation model and applications in crossflow double-layerreservoirs by polymer floodingrdquo The Scientific World Journalvol 2014 Article ID 890874 11 pages 2014
[20] H Stehfest ldquoAlgorithm 368 numerical inversion of Laplacetransforms [D5]rdquo Communications of the ACM vol 13 no 1pp 47ndash49 1970
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
Journal of Chemistry 3
The boundary condition at the well is given by the fol-lowing equations which account for both skin and wellborestorage
119875119908119863
= 119875119895119863
(1 119905119863) minus 119904119895
120597119875119895119863
120597119903119863
100381610038161003816100381610038161003816100381610038161003816119903119863=1
1 = 119862119863
119889119875119908119863
119889119905119863
minus
119873
sum
119895=1120581119895
120597119875119895119863
120597119903119863
100381610038161003816100381610038161003816100381610038161003816119903119863=1
(7)
For infinite-outer-boundary condition
119875119895119863
(119903119863 119905119863) 997888rarr 0 (119903
119863997888rarr infin) (8)
For no-flow outer-boundary condition
120597119875119895119863
120597119903119863
= 0 (119903119863
= 119903119890119863
) (9)
For initial condition
119875119895119863
(119903119863 0) = 0 (10)
For dimensionless layer flow rates
119902119895119863
(119905119863) =
119902119895
119902= minus 120581119895
120597119875119895119863
120597119903119863
100381610038161003816100381610038161003816100381610038161003816119903119863=1
(11)
where the dimensionless variables are defined by the follow-ing
119875119895119863
=002120587
119902
119873
sum
119895=1
[
[
119896119895
(1+119899)2ℎ119895
119899(119863119902119895119904)1minus119899
1198671199031199081minus119899120601119895
(1minus119899)2]
]
(119875119895minus1198750) (12)
119905119863
= 001sum119873
119895=1 (120581119895(1+119899)2
ℎ119895
119899(119863119902119895119904)1minus119899
120601119895
(1minus119899)2)
119867sum119873
119895=1 120601119895ℎ1198951198621199051198951199031199083minus119899
119905 (13)
120581119895=
119896119895
(1+119899)2ℎ119895
119899120601119895
(119899minus1)2(119863119902119895119904)1minus119899
sum119873
119895=1 119896119895(1+119899)2
ℎ119895
119899120601119895
(119899minus1)2(119863119902119895119904)1minus119899 (14)
120596119895=
120601119895ℎ119895119862119905119895
sum119873
119895=1 120601119895ℎ119895119862119905119895 (15)
120582119895=
119867119903119908
3minus119899120594119895
sum119873
119895=1 119896119895(1+119899)2
ℎ119895
119899120601119895
(119899minus1)2(119863119902119895119904)1minus119899 (16)
119862119863
=119862
21205871199031199082 sum119873
119895=1 120601119895ℎ119895119862119905119895 (17)
119903119863
=119903
119903119908
(18)
119903119890119863
=119903119890
119903119908
(19)
3 The Solution of the Model
The above equation is given by Laplace transform on timeSet
119897 =2
3 minus 119899
119898 =1 minus 119899
3 minus 119899
(20)
The basic control equation solution is as follows
119875119895119863
=
119873
sum
119896=1119903119863
(1minus119899)2[119860119895
119896119870119898
(119897120590119896119903119863
(3minus119899)2)
+119861119895
119896119868119898
(119897120590119896119903119863
(3minus119899)2)]
(21)
where the subscript on119860 indexes the layer and the superscriptindexes 120590119860 and119861 are respectively the first and the two classof 119898 order modified Bessel function 119896
1198951205902 is eigenvalue of
real symmetric three diagonal positive definitematrices [1198861015840119895119896]
where
1198861015840
119895119896= minus120582
119895minus1 119896 = 119895 minus 1 119895 gt 1 120596119895119911 minus 120582119895minus1 minus120582
119895 119896 = 119895
minus 120582119895 119896 = 119895 + 1 119895 lt119873 0 119896 = 119895 minus 1 119895 or 119895 + 1
(22)
According to the boundary conditions a function relation-ship between 119860
119895
119896 and 119861119895
119896 is as follows
1198602119896=
minus1198861111988612
1198601119896= 12057221198961198601119896
1198603119896=
minus (119886211198601119896+ 119886221198602
119896)
11988623= 12057231198961198601119896
119860119873
119896=
minus (119886119873minus1119873minus2119860119873minus2
119896+ 119886119873minus1119873minus1119860119873minus1
119896)
119886119873minus1119873
= 120572119873
1198961198601119896
119861119895
119896= 120572119895
1198961198611119896 (119895 = 1 119873)
(23)
Then1198732 values for120572119895
119896 can be computed directly from theabove recursion formula Let zone 119894 (119894 = 1 119873119885) contain119898119894layer the zone 119894 with a specific layer 119895 dimensionless
pressure
119875119895119863
=
119898119894
sum
119896119894=1119903119863
(1minus119899)2[119860119895
119896119894119870119898
(119897120590119896119894
119903119863
(3minus119899)2)
+119861119895
119896119894119868119898
(119897120590119896119894
119903119863
(3minus119899)2)]
(24)
where each layer index 119896119894 in the sum refers to the same zone
119894 as Layer 119895
4 Journal of Chemistry
Finally the coefficients for each zone can be expressed asmultiples of the coefficients1198601
119896119894 1198611119896119894 of the uppermost layer
in zone 119894 with (12)
119875119895119863
=
119898119894
sum
119896119894=1
119903119863
(1minus119899)2[1198601119896119894120572119895
119896119894119870119898
(119897120590119896119894
119903119863
(3minus119899)2)
+1198611119896119894120572119895
119896119894119868119898
(119897120590119896119894
119903119863
(3minus119899)2)]
(25)
External boundary condition implies the relationshipbetween 1198601
119896119894 and 1198611
119896119894 is as follows
1198611119896119894 = 1198871198961198941198601119896119894 (26)
For infinite-outer-boundary condition
119887119896119894 = 0 (27)
For no-flow outer-boundary conditions
119887119896119894 =
119870119897(119897120590119896119894
119903119890119863
(3minus119899)2)
119868minus119897
(119897120590119896119894
119903119890119863(3minus119899)2)
(28)
According to the boundary conditions the layer 119895 minus 1 andlayer 119895 which in the same zone have the following relations
119898119894
sum
119896119894=11198601119896119894 (120572119895minus1119896119894 119870119898
(119897120590119896119894
) + 119887119896119894119868119898
(119897120590119896119894
)
+ 119904119895minus1120590119896
119894
[(119897120590119896119894
) minus 119887119896119894119868minus119897
(119897120590119896119894
)]
minus 120572119895
119896119894 119870119898
(119897120590119896119894
) + 119887120581119894119868119898
(119897120590119896119894
)
+ 119904119895120590119896119894
[119870119897(119897120590119896119894
) minus 119887120581119894119868minus119897
(119897120590119896119894
)]) = 0
(29)
The layer 119895 minus 1 of zone 119894 minus 1 and the layer 119895 of zone 119894 minus 1 havethe following relations
119898119894minus1
sum
119896119894minus1=1
(1198601119896119894minus1120572119895minus1119896119894minus1 119870119898
(119897120590119896119894minus1
) + 119887119896119894minus1119868119898
(119897120590119896119894minus1
) + 119904119895minus1120590119896
119894minus1[119870119897(119897120590119896119894minus1
) minus 119887119896119894minus1119868minus119897
(119897120590119896119894minus1
)])
minus
119898119894
sum
119896119894=1
(1198601119896119894120572119895
119896119894 119870119898
(119897120590119896119894
) + 119887120581119894119868119898
(119897120590119896119894
) + 119904119895120590119896119894
[119870119897(119897120590119896119894
) minus 119887120581119894119868minus119897
(119897120590119896119894
)]) = 0
(30)
The wellbore pressure is as follows
119875119908119863
=1
(1198621198631199112 + 1119875
119908119863119862119863=0)
(31)
Equation (31) is a general solution and it is by no wellborestorage pressure solution conversion into thewellbore storagepressure solutions Therefore we will solve the 119862
119863= 0 cases
of equationsEquations (29)-(30) are linear equations with coefficient
1198601119896119894 which can be solved by numerical methodThe wellbore
pressure without wellbore storage is given as
119875119908119863119862119863=0 =
119898119894
sum
119896119894=1
(11986011198961198941205721119896119894 119870119898
(119897120590119896119894
) + 119887119896119894119868119898
(119897120590119896119894
)
+ 1199041120590119896119894
[119870119897(119897120590119896119894
) minus 119887119896119894119868minus119897
(119897120590119896119894
)])
(32)
In addition layer flow rate is given as follows
119902119895119863
= (1minus119862119863119875119908119863
1199112)
sdot 120581119895
119898119894
sum
119896119894=1
1198601119896119894120572119895
119896119894120590119896119894
119870119897(119897120590119896119894
) minus 119887119896119894119868minus119897
(119897120590119896119894
)
(33)
Distribution of radial pressure on each layer becomes
119875119895119863
= (1minus119862119863119875119908119863
1199112) 119875119895119863119862119863=0 = (1minus119862
119863119875119908119863
1199112)
sdot
119898119894
sum
119896119894=1
(1198601119896119894120572119895
119896119894 119870119898
(119897120590119896119894
119903119863
(3minus119899)2)
+ 119887119896119894119868119898
(119897120590119896119894
119903119863
(3minus119899)2))
(34)
For the case of double-layer reservoirs with formation cross-flow assume that choosing parameters is as follows
119862119863
= 1 1199041 = 1199042 = 1 120582 = 1119890 minus 5 120581 = 095120596 = 01 and 119899 = 01 03 05 07 09 through the numericalinversion of the Laplace transform of Stehfest [20] Pressureand flow in the real space are obtainedThe wellbore pressurecurve is shown in Figure 1 The wellbore pressure curve isshown in Figure 2 It can be seen that the pressure derivativecurve is unit slope straight line in the early-time namelylinear unit slope represents pure effect of wellbore storagein the medium term the curve is concave interporosityflow transition characteristics in the late stage the pressurederivative curve is approximately straight line up and theslope is related to the power law index The slope is 119898
119871=
(1minus119899)(3minus119899) And it can be seen that the pressure derivativerises and increases with the reduction of power-law index 119899
steepened
Journal of Chemistry 5
108107106105104
103
103
102
102
101
101
100
100
10minus1
10minus1
10minus2
10minus2
CD = 1
s1 = 1s2 = 1
120582 = 1e minus 5120581 = 095
120596 = 01
tD
n
n = [01 03 05 07 09]
PwDdpwD
dln
t D
Figure 2 Dimensionless pressure for two-layer system
00
02
04
06
08
10
10810710610510410310210110010minus110minus2
tD
n
q1D
CD = 1
s1 = 1s2 = 1
120582 = 1e minus 5120581 = 095
120596 = 01
n = [01 03 05 07 09]
Figure 3 Dimensionless flow-rate for the first layer
In Figures 3 and 4 flow curve can be seen for the firstlayer with high quasi capacity coefficient the dimensionlessflow rate increases with the time increasing and finally tendsto be stable flow rate 120581 for the second layer with low quasicapacity coefficient the dimensionless flow rate increaseswith the time increasing In a certain period due to highpermeability layer crossflow pressure tends to be balancedand the crossflow that is more and more weak finally tendsto be stable flow rate 1 minus 120581
4 The Late Stable Flow Model
For both cases stable layer flow-rates were discussed includ-ing (1) without formation crossflow and (2) with formationcrossflow We only discussed the behavior in late time soignore the effect of wellbore storage effect
000
005
010
015
020
10810710610510410310210110010minus2 10minus110minus4 10minus3
tD
n
q2D
CD = 1
s1 = 1s2 = 1
120582 = 1e minus 5120581 = 095
120596 = 01
n = [01 03 05 07 09]
Figure 4 Dimensionless flow-rate for the second layer
(1) Without Formation Crossflow
for the infinite-out-boundary the stable flow rate forlayer 119895 is as follows
119902119895119904
=
119902120581119895
119897120596119895
119898
sum119873
119896=1 120581119896119897120596119896119898 (35)
For the no-flow-boundary the stable flow rate forlayer 119895 is as follows
119902119895119904
=
119902120601119895ℎ119895119862119905119895
sum119873
119895=1 120601119895ℎ119895119862119905119895 (36)
(2) With Formation Crossflow
Regarding the eigenvalues of matrix 1198861015840
119895119896 there is a
value to meet 1205902
= 119911 and the rest is independentof 119911 and 120596
119895and depends only on 1205811 1205812 120581119899 and
1205821 1205822 120582119899 Assume that 12059012= 119911 Stable layer flow
rate through the determinant value can be expressedas
119902119895119904
= 119902120581119895lim119911rarr 0
[
[
det 11986211989511989611
det 119862119895119896
120572119895
11205901 119870119897 (1198971205901)
minus 1198871119868minus119897
(1198971205901)]
]
+ 119902120581119895
119873
sum
119896119894=2lim119911rarr 0
[
det 119862119894119896119896119894119896119894
det 119862119894119896
sdot 120572119895
119896119894120590119896119894
119870119897(119897120590119896) minus 119887119896119894119868minus119897
(119897120590119896)]
(37)
where det119862119895119896 is determinant of matrix 119862
119895119896
6 Journal of Chemistry
The matrix element is as follows
119862119895119896
= 120572119895
119896119870119898
(119897120590119896) + 119887119896119868119898
(119897120590119896)
+ 119904119896120590119896[119870119897(119897120590119896) minus 119887119896119868minus119897
(119897120590119896)]
minus 120572119895+1119896119870119898
(119897120590119896+1) + 119887
119896119868119898
(119897120590119896+1)
+ 119904119896+1120590119896+1 [119870
119897(119897120590119896+1) minus 119887
119896119868minus119897
(119897120590119896+1)]
(38)
Thematrix 119862119895119896119894119894is used in matrix in 119862
119895119896 column 119894
replacement for (0 0 119860 1119911)119879 where superscript
119879 is the vector transpose
5 Conclusions
Establishing the multilayer reservoir well-test model forpolymer flooding gives the formation and wellbore transientpressure and the layer flow-rate finallyThe expressions of latetime stable flow rate in each layer are given
The unsteady well-test model provides theoreticalmethod for polymer flooding well test analysis of multilayerreservoir the experimental data and the theoretical curvefitting can be used to determine the permeability skin factorand effective interlayer vertical permeability and otherimportant parameters
Nomenclature
1198861015840
119895119896 Matrix elements defined in (22)
119860119895
119896 119861119895
119896 Coefficient for 119895th layer 119896th root defined in(23)
1198601119896119894 1198611119896119894 Coefficient for 119895th layer 119896th root in zone 119868
defined in (25)119887119896119894 Coefficient for outer boundary condition
defined in (27) or (28)119862 Wellbore-storage coefficient (cm3kPa)119862119905119895 Total compressibility in layer 119868 (kPaminus1)
119862119895119896 Matrix elements defined in (38)
119863 Constant defined in (5)ℎ119895 Formation thickness in layer 119895 (cm)
119867 Constant defined in (4)119868119898(sdot) 119870119898(sdot) Modified 119898 order Bessel function of the first
and second kindΔℎ119895 Thickness of a nonperforated zone between
layers 119895 and 119895 + 1 (cm)119870119895 Horizontal permeability in layer 119895 120583m2
119870V119895 Vertical permeability of a tight zone betweenLayers 119895 and 119895 minus 1 (120583m2)
119870119911119895 Vertical permeability in Layer 119895 (120583m2)
119897 119898 Constant defined in (20)119898119894 Number of layers in Zone 119894
119899 Power-law index119873 Number of layers in reservoir system119873119885 Number of Zones in reservoir system119875119894 Reservoir pressure in layer 119868 (kPa)
119875119895
119894 Reservoir pressure in layer 119895 in Zone 119894
(kPa)119875119895119863 Dimensionless reservoir pressure in
layer 119895 defined in (12)1198750 Initial reservoir pressure (kPa)
119875119908119863
Dimensionless bottomhole pressure inLaplace space
119875119908119863119862119863=0 Dimensionless bottomhole pressure
without wellbore storage in Laplacespace
119876 Surface production rate (cm3s)119902119895 Flow rate for layer 119895 (cm3s)
119902119895119904 Stable flow rate for layer 119895 (cm3s)
119903 Radial distance (cm)119903119863 Dimensionless radius defined in (18)
119903119890 Reservoir outer radius (cm)
119903119890119863 Dimensionless outer reservoir radius
defined in (19)119903119908 Wellbore radius (cm)
119878119895 Wellbore skin factor for layer 119895
119905 Time (s)Δ119905 Elapsed time after rate change (s)119905119863 Dimensionless time referenced to
producing layer]119894 Shear rate in layer 119868 (sminus1)
120594119895 Crossflow coefficient between layers 119895
and 119895 + 1 defined in (2)119909119895 Resistance to flow per unit length at the
119895th layer interface119885 Laplace space variable 119894
120572119895
119896 Coefficient for layer 119895 Root 119896 definedin (23)
120572119895
119896119894 Coefficient for layer 119895 Root 119896 defined
in Zone 119868 defined in (25)120581119895 Coefficient for layer 119895 defined in (14)
120582119895 Dimensionless semipermeability
between layers 119895 and 119895 + 1 defined in(16)
119872 Dynamic viscosity (mpasdots)Φ119895 Porosity fraction for layer 119895
120596119895 Coefficient for layer 119895 defined in (15)
]119895 Shear rate for layer 119895 defined in (5)
det119862119895119896 Determinant of matrix 119862
119895119896
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The project was supported by the National Key Science andTechnology Project (2011ZX05009-006) the Major StateBasic Research Development Program of China (973 Pro-gram) (no 2011CB707305) and the China ScholarshipCouncil CAS Strategic Priority Research Program(XDB10030402)
Journal of Chemistry 7
References
[1] D G Russell and M Prats ldquoPerformance of layered reser-voirs with crossflowmdashsingle-compressible-fluid caserdquo Society ofPetroleum Engineers Journal vol 2 no 1 pp 53ndash67 1962
[2] D Russell and M Prats ldquoThe practical aspects of interlayercrossflowrdquo Journal of Petroleum Technology vol 14 no 06 pp589ndash594 2013
[3] R Raghavan R Prijambodo and A C Reynolds ldquoWell testanalysis for well producing layered Reservoirs with crossflowrdquoSociety of Petroleum Engineers Journal vol 25 no 3 pp 380ndash396 1985
[4] D Bourdet ldquoPressure behavior of Layered reservoirs withcrossflowrdquo inProceedings of the SPECalifornia RegionalMeetingSPE-13628-MS Bakersfield Calif USA March 1985
[5] C-T Gao and H A Deans ldquoPressure transients and crossflowcaused by diffusivities inmultiplayer reservoirsrdquo SPE FormationEvaluation vol 3 no 2 pp 438ndash448 1988
[6] C A Ehlig-Economides ldquoA new test for determination ofindividual layer properties in a multilayered reservoirrdquo SPEFormation Evaluation vol 2 no 3 pp 261ndash283 1987
[7] P Bidaux T M Whittle P J Coveney and A C GringartenldquoAnalysis of pressure and rate transient data from wells inmultilayered reservoirs theory and applicationrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE24679 Washington DC USA October 1992
[8] H van Poollen and J Jargon ldquoSteady-state and unsteady-stateflow of Non-Newtonian fluids through porous mediardquo Societyof Petroleum Engineers Journal vol 9 no 1 pp 80ndash88 2013
[9] C U Ikoku and H J Ramey Jr ldquoWellbore storage and skineffects during the transient flow of non-Newtonian power-lawfluids in porous mediardquo Society of Petroleum Engineers Journalvol 20 no 1 pp 25ndash38 1980
[10] O Lund and C U Ikoku ldquoPressure transient behavior of non-newtoniannewtonian fluid composite reservoirsrdquo Society ofPetroleum Engineers Journal vol 21 no 2 pp 271ndash280 1981
[11] R Prijambodo R Raghavan and A C Reynolds Well TestAnalysis for Wells Producing Layered Reservoirs With CrossflowSociety of Petroleum Engineers 1985
[12] J Xu LWang andK Zhu ldquoPressure behavior for spherical flowin a reservoir of non-Newtonian power-law fluidsrdquo PetroleumScience and Technology vol 21 no 1-2 pp 133ndash143 2003
[13] F-H Escobar J-A Martınez and M Montealegre-MaderoldquoPressure and pressure derivative analysis for a well in aradial composite reservoir with a non-NewtonianNewtonianinterfacerdquo Ciencia Tecnologia y Futuro vol 4 no 2 pp 33ndash422010
[14] J A Martinez F H Escobar and M M Montealegre ldquoVerticalwell pressure and pressure derivative analysis for binghamfluidsin a homogeneous reservoirsrdquo Dyna vol 78 no 166 pp 21ndash282011
[15] F H Escobar A P Zambrano D V Giraldo and J H CantilloldquoPressure and pressure derivative analysis for non-Newtonianpseudoplastic fluids in double-porosity formationsrdquo CTampFmdashCiencia Tecnologıa y Futuro vol 4 no 3 pp 47ndash59 2011
[16] F H Escobar A Martınez and D M Silva ldquoConventionalpressure analysis for naturally-fractured reservoirs with non-Newtonian pseudoplastic fluidsrdquo Fuentes Journal vol 11 no 1pp 27ndash34 2013
[17] F H Escobar J M Ascencio and D F Real ldquoInjection and fall-off tests transient analysis of non-newtonian fluidsrdquo Journal ofEngineering and Applied Sciences vol 8 no 9 pp 703ndash707 2013
[18] P van den Hoek H Mahani T Sorop et al ldquoApplicationof injection fall-off analysis in polymer floodingrdquo in Proceed-ings of the SPE EuropecEAGE Annual Conference Society ofPetroleum Engineers Copenhagen Denmark June 2012
[19] H Y Yu H Guo Y W He et al ldquoNumerical well testing in-terpretation model and applications in crossflow double-layerreservoirs by polymer floodingrdquo The Scientific World Journalvol 2014 Article ID 890874 11 pages 2014
[20] H Stehfest ldquoAlgorithm 368 numerical inversion of Laplacetransforms [D5]rdquo Communications of the ACM vol 13 no 1pp 47ndash49 1970
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
4 Journal of Chemistry
Finally the coefficients for each zone can be expressed asmultiples of the coefficients1198601
119896119894 1198611119896119894 of the uppermost layer
in zone 119894 with (12)
119875119895119863
=
119898119894
sum
119896119894=1
119903119863
(1minus119899)2[1198601119896119894120572119895
119896119894119870119898
(119897120590119896119894
119903119863
(3minus119899)2)
+1198611119896119894120572119895
119896119894119868119898
(119897120590119896119894
119903119863
(3minus119899)2)]
(25)
External boundary condition implies the relationshipbetween 1198601
119896119894 and 1198611
119896119894 is as follows
1198611119896119894 = 1198871198961198941198601119896119894 (26)
For infinite-outer-boundary condition
119887119896119894 = 0 (27)
For no-flow outer-boundary conditions
119887119896119894 =
119870119897(119897120590119896119894
119903119890119863
(3minus119899)2)
119868minus119897
(119897120590119896119894
119903119890119863(3minus119899)2)
(28)
According to the boundary conditions the layer 119895 minus 1 andlayer 119895 which in the same zone have the following relations
119898119894
sum
119896119894=11198601119896119894 (120572119895minus1119896119894 119870119898
(119897120590119896119894
) + 119887119896119894119868119898
(119897120590119896119894
)
+ 119904119895minus1120590119896
119894
[(119897120590119896119894
) minus 119887119896119894119868minus119897
(119897120590119896119894
)]
minus 120572119895
119896119894 119870119898
(119897120590119896119894
) + 119887120581119894119868119898
(119897120590119896119894
)
+ 119904119895120590119896119894
[119870119897(119897120590119896119894
) minus 119887120581119894119868minus119897
(119897120590119896119894
)]) = 0
(29)
The layer 119895 minus 1 of zone 119894 minus 1 and the layer 119895 of zone 119894 minus 1 havethe following relations
119898119894minus1
sum
119896119894minus1=1
(1198601119896119894minus1120572119895minus1119896119894minus1 119870119898
(119897120590119896119894minus1
) + 119887119896119894minus1119868119898
(119897120590119896119894minus1
) + 119904119895minus1120590119896
119894minus1[119870119897(119897120590119896119894minus1
) minus 119887119896119894minus1119868minus119897
(119897120590119896119894minus1
)])
minus
119898119894
sum
119896119894=1
(1198601119896119894120572119895
119896119894 119870119898
(119897120590119896119894
) + 119887120581119894119868119898
(119897120590119896119894
) + 119904119895120590119896119894
[119870119897(119897120590119896119894
) minus 119887120581119894119868minus119897
(119897120590119896119894
)]) = 0
(30)
The wellbore pressure is as follows
119875119908119863
=1
(1198621198631199112 + 1119875
119908119863119862119863=0)
(31)
Equation (31) is a general solution and it is by no wellborestorage pressure solution conversion into thewellbore storagepressure solutions Therefore we will solve the 119862
119863= 0 cases
of equationsEquations (29)-(30) are linear equations with coefficient
1198601119896119894 which can be solved by numerical methodThe wellbore
pressure without wellbore storage is given as
119875119908119863119862119863=0 =
119898119894
sum
119896119894=1
(11986011198961198941205721119896119894 119870119898
(119897120590119896119894
) + 119887119896119894119868119898
(119897120590119896119894
)
+ 1199041120590119896119894
[119870119897(119897120590119896119894
) minus 119887119896119894119868minus119897
(119897120590119896119894
)])
(32)
In addition layer flow rate is given as follows
119902119895119863
= (1minus119862119863119875119908119863
1199112)
sdot 120581119895
119898119894
sum
119896119894=1
1198601119896119894120572119895
119896119894120590119896119894
119870119897(119897120590119896119894
) minus 119887119896119894119868minus119897
(119897120590119896119894
)
(33)
Distribution of radial pressure on each layer becomes
119875119895119863
= (1minus119862119863119875119908119863
1199112) 119875119895119863119862119863=0 = (1minus119862
119863119875119908119863
1199112)
sdot
119898119894
sum
119896119894=1
(1198601119896119894120572119895
119896119894 119870119898
(119897120590119896119894
119903119863
(3minus119899)2)
+ 119887119896119894119868119898
(119897120590119896119894
119903119863
(3minus119899)2))
(34)
For the case of double-layer reservoirs with formation cross-flow assume that choosing parameters is as follows
119862119863
= 1 1199041 = 1199042 = 1 120582 = 1119890 minus 5 120581 = 095120596 = 01 and 119899 = 01 03 05 07 09 through the numericalinversion of the Laplace transform of Stehfest [20] Pressureand flow in the real space are obtainedThe wellbore pressurecurve is shown in Figure 1 The wellbore pressure curve isshown in Figure 2 It can be seen that the pressure derivativecurve is unit slope straight line in the early-time namelylinear unit slope represents pure effect of wellbore storagein the medium term the curve is concave interporosityflow transition characteristics in the late stage the pressurederivative curve is approximately straight line up and theslope is related to the power law index The slope is 119898
119871=
(1minus119899)(3minus119899) And it can be seen that the pressure derivativerises and increases with the reduction of power-law index 119899
steepened
Journal of Chemistry 5
108107106105104
103
103
102
102
101
101
100
100
10minus1
10minus1
10minus2
10minus2
CD = 1
s1 = 1s2 = 1
120582 = 1e minus 5120581 = 095
120596 = 01
tD
n
n = [01 03 05 07 09]
PwDdpwD
dln
t D
Figure 2 Dimensionless pressure for two-layer system
00
02
04
06
08
10
10810710610510410310210110010minus110minus2
tD
n
q1D
CD = 1
s1 = 1s2 = 1
120582 = 1e minus 5120581 = 095
120596 = 01
n = [01 03 05 07 09]
Figure 3 Dimensionless flow-rate for the first layer
In Figures 3 and 4 flow curve can be seen for the firstlayer with high quasi capacity coefficient the dimensionlessflow rate increases with the time increasing and finally tendsto be stable flow rate 120581 for the second layer with low quasicapacity coefficient the dimensionless flow rate increaseswith the time increasing In a certain period due to highpermeability layer crossflow pressure tends to be balancedand the crossflow that is more and more weak finally tendsto be stable flow rate 1 minus 120581
4 The Late Stable Flow Model
For both cases stable layer flow-rates were discussed includ-ing (1) without formation crossflow and (2) with formationcrossflow We only discussed the behavior in late time soignore the effect of wellbore storage effect
000
005
010
015
020
10810710610510410310210110010minus2 10minus110minus4 10minus3
tD
n
q2D
CD = 1
s1 = 1s2 = 1
120582 = 1e minus 5120581 = 095
120596 = 01
n = [01 03 05 07 09]
Figure 4 Dimensionless flow-rate for the second layer
(1) Without Formation Crossflow
for the infinite-out-boundary the stable flow rate forlayer 119895 is as follows
119902119895119904
=
119902120581119895
119897120596119895
119898
sum119873
119896=1 120581119896119897120596119896119898 (35)
For the no-flow-boundary the stable flow rate forlayer 119895 is as follows
119902119895119904
=
119902120601119895ℎ119895119862119905119895
sum119873
119895=1 120601119895ℎ119895119862119905119895 (36)
(2) With Formation Crossflow
Regarding the eigenvalues of matrix 1198861015840
119895119896 there is a
value to meet 1205902
= 119911 and the rest is independentof 119911 and 120596
119895and depends only on 1205811 1205812 120581119899 and
1205821 1205822 120582119899 Assume that 12059012= 119911 Stable layer flow
rate through the determinant value can be expressedas
119902119895119904
= 119902120581119895lim119911rarr 0
[
[
det 11986211989511989611
det 119862119895119896
120572119895
11205901 119870119897 (1198971205901)
minus 1198871119868minus119897
(1198971205901)]
]
+ 119902120581119895
119873
sum
119896119894=2lim119911rarr 0
[
det 119862119894119896119896119894119896119894
det 119862119894119896
sdot 120572119895
119896119894120590119896119894
119870119897(119897120590119896) minus 119887119896119894119868minus119897
(119897120590119896)]
(37)
where det119862119895119896 is determinant of matrix 119862
119895119896
6 Journal of Chemistry
The matrix element is as follows
119862119895119896
= 120572119895
119896119870119898
(119897120590119896) + 119887119896119868119898
(119897120590119896)
+ 119904119896120590119896[119870119897(119897120590119896) minus 119887119896119868minus119897
(119897120590119896)]
minus 120572119895+1119896119870119898
(119897120590119896+1) + 119887
119896119868119898
(119897120590119896+1)
+ 119904119896+1120590119896+1 [119870
119897(119897120590119896+1) minus 119887
119896119868minus119897
(119897120590119896+1)]
(38)
Thematrix 119862119895119896119894119894is used in matrix in 119862
119895119896 column 119894
replacement for (0 0 119860 1119911)119879 where superscript
119879 is the vector transpose
5 Conclusions
Establishing the multilayer reservoir well-test model forpolymer flooding gives the formation and wellbore transientpressure and the layer flow-rate finallyThe expressions of latetime stable flow rate in each layer are given
The unsteady well-test model provides theoreticalmethod for polymer flooding well test analysis of multilayerreservoir the experimental data and the theoretical curvefitting can be used to determine the permeability skin factorand effective interlayer vertical permeability and otherimportant parameters
Nomenclature
1198861015840
119895119896 Matrix elements defined in (22)
119860119895
119896 119861119895
119896 Coefficient for 119895th layer 119896th root defined in(23)
1198601119896119894 1198611119896119894 Coefficient for 119895th layer 119896th root in zone 119868
defined in (25)119887119896119894 Coefficient for outer boundary condition
defined in (27) or (28)119862 Wellbore-storage coefficient (cm3kPa)119862119905119895 Total compressibility in layer 119868 (kPaminus1)
119862119895119896 Matrix elements defined in (38)
119863 Constant defined in (5)ℎ119895 Formation thickness in layer 119895 (cm)
119867 Constant defined in (4)119868119898(sdot) 119870119898(sdot) Modified 119898 order Bessel function of the first
and second kindΔℎ119895 Thickness of a nonperforated zone between
layers 119895 and 119895 + 1 (cm)119870119895 Horizontal permeability in layer 119895 120583m2
119870V119895 Vertical permeability of a tight zone betweenLayers 119895 and 119895 minus 1 (120583m2)
119870119911119895 Vertical permeability in Layer 119895 (120583m2)
119897 119898 Constant defined in (20)119898119894 Number of layers in Zone 119894
119899 Power-law index119873 Number of layers in reservoir system119873119885 Number of Zones in reservoir system119875119894 Reservoir pressure in layer 119868 (kPa)
119875119895
119894 Reservoir pressure in layer 119895 in Zone 119894
(kPa)119875119895119863 Dimensionless reservoir pressure in
layer 119895 defined in (12)1198750 Initial reservoir pressure (kPa)
119875119908119863
Dimensionless bottomhole pressure inLaplace space
119875119908119863119862119863=0 Dimensionless bottomhole pressure
without wellbore storage in Laplacespace
119876 Surface production rate (cm3s)119902119895 Flow rate for layer 119895 (cm3s)
119902119895119904 Stable flow rate for layer 119895 (cm3s)
119903 Radial distance (cm)119903119863 Dimensionless radius defined in (18)
119903119890 Reservoir outer radius (cm)
119903119890119863 Dimensionless outer reservoir radius
defined in (19)119903119908 Wellbore radius (cm)
119878119895 Wellbore skin factor for layer 119895
119905 Time (s)Δ119905 Elapsed time after rate change (s)119905119863 Dimensionless time referenced to
producing layer]119894 Shear rate in layer 119868 (sminus1)
120594119895 Crossflow coefficient between layers 119895
and 119895 + 1 defined in (2)119909119895 Resistance to flow per unit length at the
119895th layer interface119885 Laplace space variable 119894
120572119895
119896 Coefficient for layer 119895 Root 119896 definedin (23)
120572119895
119896119894 Coefficient for layer 119895 Root 119896 defined
in Zone 119868 defined in (25)120581119895 Coefficient for layer 119895 defined in (14)
120582119895 Dimensionless semipermeability
between layers 119895 and 119895 + 1 defined in(16)
119872 Dynamic viscosity (mpasdots)Φ119895 Porosity fraction for layer 119895
120596119895 Coefficient for layer 119895 defined in (15)
]119895 Shear rate for layer 119895 defined in (5)
det119862119895119896 Determinant of matrix 119862
119895119896
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The project was supported by the National Key Science andTechnology Project (2011ZX05009-006) the Major StateBasic Research Development Program of China (973 Pro-gram) (no 2011CB707305) and the China ScholarshipCouncil CAS Strategic Priority Research Program(XDB10030402)
Journal of Chemistry 7
References
[1] D G Russell and M Prats ldquoPerformance of layered reser-voirs with crossflowmdashsingle-compressible-fluid caserdquo Society ofPetroleum Engineers Journal vol 2 no 1 pp 53ndash67 1962
[2] D Russell and M Prats ldquoThe practical aspects of interlayercrossflowrdquo Journal of Petroleum Technology vol 14 no 06 pp589ndash594 2013
[3] R Raghavan R Prijambodo and A C Reynolds ldquoWell testanalysis for well producing layered Reservoirs with crossflowrdquoSociety of Petroleum Engineers Journal vol 25 no 3 pp 380ndash396 1985
[4] D Bourdet ldquoPressure behavior of Layered reservoirs withcrossflowrdquo inProceedings of the SPECalifornia RegionalMeetingSPE-13628-MS Bakersfield Calif USA March 1985
[5] C-T Gao and H A Deans ldquoPressure transients and crossflowcaused by diffusivities inmultiplayer reservoirsrdquo SPE FormationEvaluation vol 3 no 2 pp 438ndash448 1988
[6] C A Ehlig-Economides ldquoA new test for determination ofindividual layer properties in a multilayered reservoirrdquo SPEFormation Evaluation vol 2 no 3 pp 261ndash283 1987
[7] P Bidaux T M Whittle P J Coveney and A C GringartenldquoAnalysis of pressure and rate transient data from wells inmultilayered reservoirs theory and applicationrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE24679 Washington DC USA October 1992
[8] H van Poollen and J Jargon ldquoSteady-state and unsteady-stateflow of Non-Newtonian fluids through porous mediardquo Societyof Petroleum Engineers Journal vol 9 no 1 pp 80ndash88 2013
[9] C U Ikoku and H J Ramey Jr ldquoWellbore storage and skineffects during the transient flow of non-Newtonian power-lawfluids in porous mediardquo Society of Petroleum Engineers Journalvol 20 no 1 pp 25ndash38 1980
[10] O Lund and C U Ikoku ldquoPressure transient behavior of non-newtoniannewtonian fluid composite reservoirsrdquo Society ofPetroleum Engineers Journal vol 21 no 2 pp 271ndash280 1981
[11] R Prijambodo R Raghavan and A C Reynolds Well TestAnalysis for Wells Producing Layered Reservoirs With CrossflowSociety of Petroleum Engineers 1985
[12] J Xu LWang andK Zhu ldquoPressure behavior for spherical flowin a reservoir of non-Newtonian power-law fluidsrdquo PetroleumScience and Technology vol 21 no 1-2 pp 133ndash143 2003
[13] F-H Escobar J-A Martınez and M Montealegre-MaderoldquoPressure and pressure derivative analysis for a well in aradial composite reservoir with a non-NewtonianNewtonianinterfacerdquo Ciencia Tecnologia y Futuro vol 4 no 2 pp 33ndash422010
[14] J A Martinez F H Escobar and M M Montealegre ldquoVerticalwell pressure and pressure derivative analysis for binghamfluidsin a homogeneous reservoirsrdquo Dyna vol 78 no 166 pp 21ndash282011
[15] F H Escobar A P Zambrano D V Giraldo and J H CantilloldquoPressure and pressure derivative analysis for non-Newtonianpseudoplastic fluids in double-porosity formationsrdquo CTampFmdashCiencia Tecnologıa y Futuro vol 4 no 3 pp 47ndash59 2011
[16] F H Escobar A Martınez and D M Silva ldquoConventionalpressure analysis for naturally-fractured reservoirs with non-Newtonian pseudoplastic fluidsrdquo Fuentes Journal vol 11 no 1pp 27ndash34 2013
[17] F H Escobar J M Ascencio and D F Real ldquoInjection and fall-off tests transient analysis of non-newtonian fluidsrdquo Journal ofEngineering and Applied Sciences vol 8 no 9 pp 703ndash707 2013
[18] P van den Hoek H Mahani T Sorop et al ldquoApplicationof injection fall-off analysis in polymer floodingrdquo in Proceed-ings of the SPE EuropecEAGE Annual Conference Society ofPetroleum Engineers Copenhagen Denmark June 2012
[19] H Y Yu H Guo Y W He et al ldquoNumerical well testing in-terpretation model and applications in crossflow double-layerreservoirs by polymer floodingrdquo The Scientific World Journalvol 2014 Article ID 890874 11 pages 2014
[20] H Stehfest ldquoAlgorithm 368 numerical inversion of Laplacetransforms [D5]rdquo Communications of the ACM vol 13 no 1pp 47ndash49 1970
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
Journal of Chemistry 5
108107106105104
103
103
102
102
101
101
100
100
10minus1
10minus1
10minus2
10minus2
CD = 1
s1 = 1s2 = 1
120582 = 1e minus 5120581 = 095
120596 = 01
tD
n
n = [01 03 05 07 09]
PwDdpwD
dln
t D
Figure 2 Dimensionless pressure for two-layer system
00
02
04
06
08
10
10810710610510410310210110010minus110minus2
tD
n
q1D
CD = 1
s1 = 1s2 = 1
120582 = 1e minus 5120581 = 095
120596 = 01
n = [01 03 05 07 09]
Figure 3 Dimensionless flow-rate for the first layer
In Figures 3 and 4 flow curve can be seen for the firstlayer with high quasi capacity coefficient the dimensionlessflow rate increases with the time increasing and finally tendsto be stable flow rate 120581 for the second layer with low quasicapacity coefficient the dimensionless flow rate increaseswith the time increasing In a certain period due to highpermeability layer crossflow pressure tends to be balancedand the crossflow that is more and more weak finally tendsto be stable flow rate 1 minus 120581
4 The Late Stable Flow Model
For both cases stable layer flow-rates were discussed includ-ing (1) without formation crossflow and (2) with formationcrossflow We only discussed the behavior in late time soignore the effect of wellbore storage effect
000
005
010
015
020
10810710610510410310210110010minus2 10minus110minus4 10minus3
tD
n
q2D
CD = 1
s1 = 1s2 = 1
120582 = 1e minus 5120581 = 095
120596 = 01
n = [01 03 05 07 09]
Figure 4 Dimensionless flow-rate for the second layer
(1) Without Formation Crossflow
for the infinite-out-boundary the stable flow rate forlayer 119895 is as follows
119902119895119904
=
119902120581119895
119897120596119895
119898
sum119873
119896=1 120581119896119897120596119896119898 (35)
For the no-flow-boundary the stable flow rate forlayer 119895 is as follows
119902119895119904
=
119902120601119895ℎ119895119862119905119895
sum119873
119895=1 120601119895ℎ119895119862119905119895 (36)
(2) With Formation Crossflow
Regarding the eigenvalues of matrix 1198861015840
119895119896 there is a
value to meet 1205902
= 119911 and the rest is independentof 119911 and 120596
119895and depends only on 1205811 1205812 120581119899 and
1205821 1205822 120582119899 Assume that 12059012= 119911 Stable layer flow
rate through the determinant value can be expressedas
119902119895119904
= 119902120581119895lim119911rarr 0
[
[
det 11986211989511989611
det 119862119895119896
120572119895
11205901 119870119897 (1198971205901)
minus 1198871119868minus119897
(1198971205901)]
]
+ 119902120581119895
119873
sum
119896119894=2lim119911rarr 0
[
det 119862119894119896119896119894119896119894
det 119862119894119896
sdot 120572119895
119896119894120590119896119894
119870119897(119897120590119896) minus 119887119896119894119868minus119897
(119897120590119896)]
(37)
where det119862119895119896 is determinant of matrix 119862
119895119896
6 Journal of Chemistry
The matrix element is as follows
119862119895119896
= 120572119895
119896119870119898
(119897120590119896) + 119887119896119868119898
(119897120590119896)
+ 119904119896120590119896[119870119897(119897120590119896) minus 119887119896119868minus119897
(119897120590119896)]
minus 120572119895+1119896119870119898
(119897120590119896+1) + 119887
119896119868119898
(119897120590119896+1)
+ 119904119896+1120590119896+1 [119870
119897(119897120590119896+1) minus 119887
119896119868minus119897
(119897120590119896+1)]
(38)
Thematrix 119862119895119896119894119894is used in matrix in 119862
119895119896 column 119894
replacement for (0 0 119860 1119911)119879 where superscript
119879 is the vector transpose
5 Conclusions
Establishing the multilayer reservoir well-test model forpolymer flooding gives the formation and wellbore transientpressure and the layer flow-rate finallyThe expressions of latetime stable flow rate in each layer are given
The unsteady well-test model provides theoreticalmethod for polymer flooding well test analysis of multilayerreservoir the experimental data and the theoretical curvefitting can be used to determine the permeability skin factorand effective interlayer vertical permeability and otherimportant parameters
Nomenclature
1198861015840
119895119896 Matrix elements defined in (22)
119860119895
119896 119861119895
119896 Coefficient for 119895th layer 119896th root defined in(23)
1198601119896119894 1198611119896119894 Coefficient for 119895th layer 119896th root in zone 119868
defined in (25)119887119896119894 Coefficient for outer boundary condition
defined in (27) or (28)119862 Wellbore-storage coefficient (cm3kPa)119862119905119895 Total compressibility in layer 119868 (kPaminus1)
119862119895119896 Matrix elements defined in (38)
119863 Constant defined in (5)ℎ119895 Formation thickness in layer 119895 (cm)
119867 Constant defined in (4)119868119898(sdot) 119870119898(sdot) Modified 119898 order Bessel function of the first
and second kindΔℎ119895 Thickness of a nonperforated zone between
layers 119895 and 119895 + 1 (cm)119870119895 Horizontal permeability in layer 119895 120583m2
119870V119895 Vertical permeability of a tight zone betweenLayers 119895 and 119895 minus 1 (120583m2)
119870119911119895 Vertical permeability in Layer 119895 (120583m2)
119897 119898 Constant defined in (20)119898119894 Number of layers in Zone 119894
119899 Power-law index119873 Number of layers in reservoir system119873119885 Number of Zones in reservoir system119875119894 Reservoir pressure in layer 119868 (kPa)
119875119895
119894 Reservoir pressure in layer 119895 in Zone 119894
(kPa)119875119895119863 Dimensionless reservoir pressure in
layer 119895 defined in (12)1198750 Initial reservoir pressure (kPa)
119875119908119863
Dimensionless bottomhole pressure inLaplace space
119875119908119863119862119863=0 Dimensionless bottomhole pressure
without wellbore storage in Laplacespace
119876 Surface production rate (cm3s)119902119895 Flow rate for layer 119895 (cm3s)
119902119895119904 Stable flow rate for layer 119895 (cm3s)
119903 Radial distance (cm)119903119863 Dimensionless radius defined in (18)
119903119890 Reservoir outer radius (cm)
119903119890119863 Dimensionless outer reservoir radius
defined in (19)119903119908 Wellbore radius (cm)
119878119895 Wellbore skin factor for layer 119895
119905 Time (s)Δ119905 Elapsed time after rate change (s)119905119863 Dimensionless time referenced to
producing layer]119894 Shear rate in layer 119868 (sminus1)
120594119895 Crossflow coefficient between layers 119895
and 119895 + 1 defined in (2)119909119895 Resistance to flow per unit length at the
119895th layer interface119885 Laplace space variable 119894
120572119895
119896 Coefficient for layer 119895 Root 119896 definedin (23)
120572119895
119896119894 Coefficient for layer 119895 Root 119896 defined
in Zone 119868 defined in (25)120581119895 Coefficient for layer 119895 defined in (14)
120582119895 Dimensionless semipermeability
between layers 119895 and 119895 + 1 defined in(16)
119872 Dynamic viscosity (mpasdots)Φ119895 Porosity fraction for layer 119895
120596119895 Coefficient for layer 119895 defined in (15)
]119895 Shear rate for layer 119895 defined in (5)
det119862119895119896 Determinant of matrix 119862
119895119896
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The project was supported by the National Key Science andTechnology Project (2011ZX05009-006) the Major StateBasic Research Development Program of China (973 Pro-gram) (no 2011CB707305) and the China ScholarshipCouncil CAS Strategic Priority Research Program(XDB10030402)
Journal of Chemistry 7
References
[1] D G Russell and M Prats ldquoPerformance of layered reser-voirs with crossflowmdashsingle-compressible-fluid caserdquo Society ofPetroleum Engineers Journal vol 2 no 1 pp 53ndash67 1962
[2] D Russell and M Prats ldquoThe practical aspects of interlayercrossflowrdquo Journal of Petroleum Technology vol 14 no 06 pp589ndash594 2013
[3] R Raghavan R Prijambodo and A C Reynolds ldquoWell testanalysis for well producing layered Reservoirs with crossflowrdquoSociety of Petroleum Engineers Journal vol 25 no 3 pp 380ndash396 1985
[4] D Bourdet ldquoPressure behavior of Layered reservoirs withcrossflowrdquo inProceedings of the SPECalifornia RegionalMeetingSPE-13628-MS Bakersfield Calif USA March 1985
[5] C-T Gao and H A Deans ldquoPressure transients and crossflowcaused by diffusivities inmultiplayer reservoirsrdquo SPE FormationEvaluation vol 3 no 2 pp 438ndash448 1988
[6] C A Ehlig-Economides ldquoA new test for determination ofindividual layer properties in a multilayered reservoirrdquo SPEFormation Evaluation vol 2 no 3 pp 261ndash283 1987
[7] P Bidaux T M Whittle P J Coveney and A C GringartenldquoAnalysis of pressure and rate transient data from wells inmultilayered reservoirs theory and applicationrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE24679 Washington DC USA October 1992
[8] H van Poollen and J Jargon ldquoSteady-state and unsteady-stateflow of Non-Newtonian fluids through porous mediardquo Societyof Petroleum Engineers Journal vol 9 no 1 pp 80ndash88 2013
[9] C U Ikoku and H J Ramey Jr ldquoWellbore storage and skineffects during the transient flow of non-Newtonian power-lawfluids in porous mediardquo Society of Petroleum Engineers Journalvol 20 no 1 pp 25ndash38 1980
[10] O Lund and C U Ikoku ldquoPressure transient behavior of non-newtoniannewtonian fluid composite reservoirsrdquo Society ofPetroleum Engineers Journal vol 21 no 2 pp 271ndash280 1981
[11] R Prijambodo R Raghavan and A C Reynolds Well TestAnalysis for Wells Producing Layered Reservoirs With CrossflowSociety of Petroleum Engineers 1985
[12] J Xu LWang andK Zhu ldquoPressure behavior for spherical flowin a reservoir of non-Newtonian power-law fluidsrdquo PetroleumScience and Technology vol 21 no 1-2 pp 133ndash143 2003
[13] F-H Escobar J-A Martınez and M Montealegre-MaderoldquoPressure and pressure derivative analysis for a well in aradial composite reservoir with a non-NewtonianNewtonianinterfacerdquo Ciencia Tecnologia y Futuro vol 4 no 2 pp 33ndash422010
[14] J A Martinez F H Escobar and M M Montealegre ldquoVerticalwell pressure and pressure derivative analysis for binghamfluidsin a homogeneous reservoirsrdquo Dyna vol 78 no 166 pp 21ndash282011
[15] F H Escobar A P Zambrano D V Giraldo and J H CantilloldquoPressure and pressure derivative analysis for non-Newtonianpseudoplastic fluids in double-porosity formationsrdquo CTampFmdashCiencia Tecnologıa y Futuro vol 4 no 3 pp 47ndash59 2011
[16] F H Escobar A Martınez and D M Silva ldquoConventionalpressure analysis for naturally-fractured reservoirs with non-Newtonian pseudoplastic fluidsrdquo Fuentes Journal vol 11 no 1pp 27ndash34 2013
[17] F H Escobar J M Ascencio and D F Real ldquoInjection and fall-off tests transient analysis of non-newtonian fluidsrdquo Journal ofEngineering and Applied Sciences vol 8 no 9 pp 703ndash707 2013
[18] P van den Hoek H Mahani T Sorop et al ldquoApplicationof injection fall-off analysis in polymer floodingrdquo in Proceed-ings of the SPE EuropecEAGE Annual Conference Society ofPetroleum Engineers Copenhagen Denmark June 2012
[19] H Y Yu H Guo Y W He et al ldquoNumerical well testing in-terpretation model and applications in crossflow double-layerreservoirs by polymer floodingrdquo The Scientific World Journalvol 2014 Article ID 890874 11 pages 2014
[20] H Stehfest ldquoAlgorithm 368 numerical inversion of Laplacetransforms [D5]rdquo Communications of the ACM vol 13 no 1pp 47ndash49 1970
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
6 Journal of Chemistry
The matrix element is as follows
119862119895119896
= 120572119895
119896119870119898
(119897120590119896) + 119887119896119868119898
(119897120590119896)
+ 119904119896120590119896[119870119897(119897120590119896) minus 119887119896119868minus119897
(119897120590119896)]
minus 120572119895+1119896119870119898
(119897120590119896+1) + 119887
119896119868119898
(119897120590119896+1)
+ 119904119896+1120590119896+1 [119870
119897(119897120590119896+1) minus 119887
119896119868minus119897
(119897120590119896+1)]
(38)
Thematrix 119862119895119896119894119894is used in matrix in 119862
119895119896 column 119894
replacement for (0 0 119860 1119911)119879 where superscript
119879 is the vector transpose
5 Conclusions
Establishing the multilayer reservoir well-test model forpolymer flooding gives the formation and wellbore transientpressure and the layer flow-rate finallyThe expressions of latetime stable flow rate in each layer are given
The unsteady well-test model provides theoreticalmethod for polymer flooding well test analysis of multilayerreservoir the experimental data and the theoretical curvefitting can be used to determine the permeability skin factorand effective interlayer vertical permeability and otherimportant parameters
Nomenclature
1198861015840
119895119896 Matrix elements defined in (22)
119860119895
119896 119861119895
119896 Coefficient for 119895th layer 119896th root defined in(23)
1198601119896119894 1198611119896119894 Coefficient for 119895th layer 119896th root in zone 119868
defined in (25)119887119896119894 Coefficient for outer boundary condition
defined in (27) or (28)119862 Wellbore-storage coefficient (cm3kPa)119862119905119895 Total compressibility in layer 119868 (kPaminus1)
119862119895119896 Matrix elements defined in (38)
119863 Constant defined in (5)ℎ119895 Formation thickness in layer 119895 (cm)
119867 Constant defined in (4)119868119898(sdot) 119870119898(sdot) Modified 119898 order Bessel function of the first
and second kindΔℎ119895 Thickness of a nonperforated zone between
layers 119895 and 119895 + 1 (cm)119870119895 Horizontal permeability in layer 119895 120583m2
119870V119895 Vertical permeability of a tight zone betweenLayers 119895 and 119895 minus 1 (120583m2)
119870119911119895 Vertical permeability in Layer 119895 (120583m2)
119897 119898 Constant defined in (20)119898119894 Number of layers in Zone 119894
119899 Power-law index119873 Number of layers in reservoir system119873119885 Number of Zones in reservoir system119875119894 Reservoir pressure in layer 119868 (kPa)
119875119895
119894 Reservoir pressure in layer 119895 in Zone 119894
(kPa)119875119895119863 Dimensionless reservoir pressure in
layer 119895 defined in (12)1198750 Initial reservoir pressure (kPa)
119875119908119863
Dimensionless bottomhole pressure inLaplace space
119875119908119863119862119863=0 Dimensionless bottomhole pressure
without wellbore storage in Laplacespace
119876 Surface production rate (cm3s)119902119895 Flow rate for layer 119895 (cm3s)
119902119895119904 Stable flow rate for layer 119895 (cm3s)
119903 Radial distance (cm)119903119863 Dimensionless radius defined in (18)
119903119890 Reservoir outer radius (cm)
119903119890119863 Dimensionless outer reservoir radius
defined in (19)119903119908 Wellbore radius (cm)
119878119895 Wellbore skin factor for layer 119895
119905 Time (s)Δ119905 Elapsed time after rate change (s)119905119863 Dimensionless time referenced to
producing layer]119894 Shear rate in layer 119868 (sminus1)
120594119895 Crossflow coefficient between layers 119895
and 119895 + 1 defined in (2)119909119895 Resistance to flow per unit length at the
119895th layer interface119885 Laplace space variable 119894
120572119895
119896 Coefficient for layer 119895 Root 119896 definedin (23)
120572119895
119896119894 Coefficient for layer 119895 Root 119896 defined
in Zone 119868 defined in (25)120581119895 Coefficient for layer 119895 defined in (14)
120582119895 Dimensionless semipermeability
between layers 119895 and 119895 + 1 defined in(16)
119872 Dynamic viscosity (mpasdots)Φ119895 Porosity fraction for layer 119895
120596119895 Coefficient for layer 119895 defined in (15)
]119895 Shear rate for layer 119895 defined in (5)
det119862119895119896 Determinant of matrix 119862
119895119896
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The project was supported by the National Key Science andTechnology Project (2011ZX05009-006) the Major StateBasic Research Development Program of China (973 Pro-gram) (no 2011CB707305) and the China ScholarshipCouncil CAS Strategic Priority Research Program(XDB10030402)
Journal of Chemistry 7
References
[1] D G Russell and M Prats ldquoPerformance of layered reser-voirs with crossflowmdashsingle-compressible-fluid caserdquo Society ofPetroleum Engineers Journal vol 2 no 1 pp 53ndash67 1962
[2] D Russell and M Prats ldquoThe practical aspects of interlayercrossflowrdquo Journal of Petroleum Technology vol 14 no 06 pp589ndash594 2013
[3] R Raghavan R Prijambodo and A C Reynolds ldquoWell testanalysis for well producing layered Reservoirs with crossflowrdquoSociety of Petroleum Engineers Journal vol 25 no 3 pp 380ndash396 1985
[4] D Bourdet ldquoPressure behavior of Layered reservoirs withcrossflowrdquo inProceedings of the SPECalifornia RegionalMeetingSPE-13628-MS Bakersfield Calif USA March 1985
[5] C-T Gao and H A Deans ldquoPressure transients and crossflowcaused by diffusivities inmultiplayer reservoirsrdquo SPE FormationEvaluation vol 3 no 2 pp 438ndash448 1988
[6] C A Ehlig-Economides ldquoA new test for determination ofindividual layer properties in a multilayered reservoirrdquo SPEFormation Evaluation vol 2 no 3 pp 261ndash283 1987
[7] P Bidaux T M Whittle P J Coveney and A C GringartenldquoAnalysis of pressure and rate transient data from wells inmultilayered reservoirs theory and applicationrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE24679 Washington DC USA October 1992
[8] H van Poollen and J Jargon ldquoSteady-state and unsteady-stateflow of Non-Newtonian fluids through porous mediardquo Societyof Petroleum Engineers Journal vol 9 no 1 pp 80ndash88 2013
[9] C U Ikoku and H J Ramey Jr ldquoWellbore storage and skineffects during the transient flow of non-Newtonian power-lawfluids in porous mediardquo Society of Petroleum Engineers Journalvol 20 no 1 pp 25ndash38 1980
[10] O Lund and C U Ikoku ldquoPressure transient behavior of non-newtoniannewtonian fluid composite reservoirsrdquo Society ofPetroleum Engineers Journal vol 21 no 2 pp 271ndash280 1981
[11] R Prijambodo R Raghavan and A C Reynolds Well TestAnalysis for Wells Producing Layered Reservoirs With CrossflowSociety of Petroleum Engineers 1985
[12] J Xu LWang andK Zhu ldquoPressure behavior for spherical flowin a reservoir of non-Newtonian power-law fluidsrdquo PetroleumScience and Technology vol 21 no 1-2 pp 133ndash143 2003
[13] F-H Escobar J-A Martınez and M Montealegre-MaderoldquoPressure and pressure derivative analysis for a well in aradial composite reservoir with a non-NewtonianNewtonianinterfacerdquo Ciencia Tecnologia y Futuro vol 4 no 2 pp 33ndash422010
[14] J A Martinez F H Escobar and M M Montealegre ldquoVerticalwell pressure and pressure derivative analysis for binghamfluidsin a homogeneous reservoirsrdquo Dyna vol 78 no 166 pp 21ndash282011
[15] F H Escobar A P Zambrano D V Giraldo and J H CantilloldquoPressure and pressure derivative analysis for non-Newtonianpseudoplastic fluids in double-porosity formationsrdquo CTampFmdashCiencia Tecnologıa y Futuro vol 4 no 3 pp 47ndash59 2011
[16] F H Escobar A Martınez and D M Silva ldquoConventionalpressure analysis for naturally-fractured reservoirs with non-Newtonian pseudoplastic fluidsrdquo Fuentes Journal vol 11 no 1pp 27ndash34 2013
[17] F H Escobar J M Ascencio and D F Real ldquoInjection and fall-off tests transient analysis of non-newtonian fluidsrdquo Journal ofEngineering and Applied Sciences vol 8 no 9 pp 703ndash707 2013
[18] P van den Hoek H Mahani T Sorop et al ldquoApplicationof injection fall-off analysis in polymer floodingrdquo in Proceed-ings of the SPE EuropecEAGE Annual Conference Society ofPetroleum Engineers Copenhagen Denmark June 2012
[19] H Y Yu H Guo Y W He et al ldquoNumerical well testing in-terpretation model and applications in crossflow double-layerreservoirs by polymer floodingrdquo The Scientific World Journalvol 2014 Article ID 890874 11 pages 2014
[20] H Stehfest ldquoAlgorithm 368 numerical inversion of Laplacetransforms [D5]rdquo Communications of the ACM vol 13 no 1pp 47ndash49 1970
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
Journal of Chemistry 7
References
[1] D G Russell and M Prats ldquoPerformance of layered reser-voirs with crossflowmdashsingle-compressible-fluid caserdquo Society ofPetroleum Engineers Journal vol 2 no 1 pp 53ndash67 1962
[2] D Russell and M Prats ldquoThe practical aspects of interlayercrossflowrdquo Journal of Petroleum Technology vol 14 no 06 pp589ndash594 2013
[3] R Raghavan R Prijambodo and A C Reynolds ldquoWell testanalysis for well producing layered Reservoirs with crossflowrdquoSociety of Petroleum Engineers Journal vol 25 no 3 pp 380ndash396 1985
[4] D Bourdet ldquoPressure behavior of Layered reservoirs withcrossflowrdquo inProceedings of the SPECalifornia RegionalMeetingSPE-13628-MS Bakersfield Calif USA March 1985
[5] C-T Gao and H A Deans ldquoPressure transients and crossflowcaused by diffusivities inmultiplayer reservoirsrdquo SPE FormationEvaluation vol 3 no 2 pp 438ndash448 1988
[6] C A Ehlig-Economides ldquoA new test for determination ofindividual layer properties in a multilayered reservoirrdquo SPEFormation Evaluation vol 2 no 3 pp 261ndash283 1987
[7] P Bidaux T M Whittle P J Coveney and A C GringartenldquoAnalysis of pressure and rate transient data from wells inmultilayered reservoirs theory and applicationrdquo in Proceedingsof the SPE Annual Technical Conference and Exhibition SPE24679 Washington DC USA October 1992
[8] H van Poollen and J Jargon ldquoSteady-state and unsteady-stateflow of Non-Newtonian fluids through porous mediardquo Societyof Petroleum Engineers Journal vol 9 no 1 pp 80ndash88 2013
[9] C U Ikoku and H J Ramey Jr ldquoWellbore storage and skineffects during the transient flow of non-Newtonian power-lawfluids in porous mediardquo Society of Petroleum Engineers Journalvol 20 no 1 pp 25ndash38 1980
[10] O Lund and C U Ikoku ldquoPressure transient behavior of non-newtoniannewtonian fluid composite reservoirsrdquo Society ofPetroleum Engineers Journal vol 21 no 2 pp 271ndash280 1981
[11] R Prijambodo R Raghavan and A C Reynolds Well TestAnalysis for Wells Producing Layered Reservoirs With CrossflowSociety of Petroleum Engineers 1985
[12] J Xu LWang andK Zhu ldquoPressure behavior for spherical flowin a reservoir of non-Newtonian power-law fluidsrdquo PetroleumScience and Technology vol 21 no 1-2 pp 133ndash143 2003
[13] F-H Escobar J-A Martınez and M Montealegre-MaderoldquoPressure and pressure derivative analysis for a well in aradial composite reservoir with a non-NewtonianNewtonianinterfacerdquo Ciencia Tecnologia y Futuro vol 4 no 2 pp 33ndash422010
[14] J A Martinez F H Escobar and M M Montealegre ldquoVerticalwell pressure and pressure derivative analysis for binghamfluidsin a homogeneous reservoirsrdquo Dyna vol 78 no 166 pp 21ndash282011
[15] F H Escobar A P Zambrano D V Giraldo and J H CantilloldquoPressure and pressure derivative analysis for non-Newtonianpseudoplastic fluids in double-porosity formationsrdquo CTampFmdashCiencia Tecnologıa y Futuro vol 4 no 3 pp 47ndash59 2011
[16] F H Escobar A Martınez and D M Silva ldquoConventionalpressure analysis for naturally-fractured reservoirs with non-Newtonian pseudoplastic fluidsrdquo Fuentes Journal vol 11 no 1pp 27ndash34 2013
[17] F H Escobar J M Ascencio and D F Real ldquoInjection and fall-off tests transient analysis of non-newtonian fluidsrdquo Journal ofEngineering and Applied Sciences vol 8 no 9 pp 703ndash707 2013
[18] P van den Hoek H Mahani T Sorop et al ldquoApplicationof injection fall-off analysis in polymer floodingrdquo in Proceed-ings of the SPE EuropecEAGE Annual Conference Society ofPetroleum Engineers Copenhagen Denmark June 2012
[19] H Y Yu H Guo Y W He et al ldquoNumerical well testing in-terpretation model and applications in crossflow double-layerreservoirs by polymer floodingrdquo The Scientific World Journalvol 2014 Article ID 890874 11 pages 2014
[20] H Stehfest ldquoAlgorithm 368 numerical inversion of Laplacetransforms [D5]rdquo Communications of the ACM vol 13 no 1pp 47ndash49 1970
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Inorganic ChemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Carbohydrate Chemistry
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
Physical Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom
Analytical Methods in Chemistry
Journal of
Volume 2014
Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
SpectroscopyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Medicinal ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chromatography Research International
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Theoretical ChemistryJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Spectroscopy
Analytical ChemistryInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Quantum Chemistry
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Organic Chemistry International
ElectrochemistryInternational Journal of
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CatalystsJournal of
