Research Article Pressure-Transient Behavior in a Multilayered Polymer Flooding Reservoir · 2018. 12. 13. · Pressure-Transient Behavior in a Multilayered Polymer Flooding Reservoir

Research ArticlePressure-Transient Behavior in a Multilayered PolymerFlooding Reservoir

Jianping Xu1,2 and Daolun Li3,4

1Xi’an Petroleum University, Xi’an, 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; [email protected]

Received 7 May 2015; Accepted 18 May 2015

Academic Editor: Yves Grohens

Copyright © 2015 J. Xu and D. Li.This 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 time.Raghavan 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 solution.Lund 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 pageshttp://dx.doi.org/10.1155/2015/875068

2 Journal of Chemistry

q1, S1, k1, 𝜑1, h1

q2, S2, k2, 𝜑2, h2

q3, S3, k3, 𝜑3, h3

q4, S4, k4, 𝜑4, h4

qt

1

2

1

2

3

4 𝜆3 = 0

𝜆2

𝜆3

Zone Layer

𝜆1 = 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 𝑁-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

𝑖, wellbore storage

coefficient 𝐶 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, porosity,and thickness of each layer can be different, respectively,𝐶𝑡𝑗, 𝑘𝑗,B𝑗, ℎ𝑗to distinguish between layer pairs with forma-

tion crossflow with and noncommunicating layers, and thereservoir is divided into 𝑁𝑍 (≤N) 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 coefficient𝜒𝑗, 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).Hence,the 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 𝑗 (𝑗 = 1, 2, . . . , 𝑁) is governed bythe following equation:

𝑘𝑗ℎ𝑗∇ ⋅ (

1𝜇𝑗

∇𝑃𝑗) = 𝜙

𝑗ℎ𝑗𝐶𝑡𝑗

𝜕𝑃𝑗

𝜕𝑡+ 𝜒𝑗−1 (𝑃𝑗 −𝑃𝑗−1)

− 𝜒𝑗(𝑃𝑗+1 −𝑃𝑗) ,

(1)

where 𝜒𝑗is given by

𝜒𝑗=

22 [Δℎ𝑗/𝑘V𝑗] + 𝑥𝑗+1 + 𝑥𝑗

, 𝑗 = 1, . . . , 𝑁, (2)

where 𝜒0 = 𝜒𝑛 = 0, Δℎ𝑗 and 𝑘V𝑗 are thickness and verticalpermeability of a nonperforated zone between layers 𝑗 and𝑗 + 1, and 𝑥

𝑗= ℎ𝑗/𝑘𝑧𝑗, where 𝑘

𝑧𝑗is the vertical permeability

for layer 𝑗 that is the resistance to flow per unit length at the𝑗th layer interface. If there is no nonperforated zone betweenlayers 𝑗 and 𝑗 + 1, then the flow resistance on behalf of a 𝑗layer interface unit length. If on the 𝑗 layer and the 𝑗 + 1 layerbut perforated belt, then (Δℎ)

𝑗is zero. If there is no formation

crossflow between layers 𝑗 and 𝑗 + 1, then 𝜒𝑗is zero.

The sand surface flow of each layer as a function of time:

𝑞𝑗 (𝑡) =

−2𝜋𝑘𝑗ℎ𝑗

𝜇𝑗

𝑟𝑤

𝜕𝑃𝑗

𝜕𝑟

𝑟𝑤

. (3)

Assume that polymer solution viscosity and shear rate ofpower-law relations [4] are as follows:

𝜇𝑖= 𝐻]𝑖

𝑛−1, (4)

where𝐻 is a constant and ]𝑖and 𝑛, respectively, represent the

shear rate and power-law index; when 𝑛 tends to one, fluidshowed a Newtonian fluid properties.

Because of the flow in each layer 𝑗 changing over time,the crossflow rate is much smaller than stable flow rate, so theflow rate 𝑞

𝑖can be replaced by steady flow rate to calculate the

shear rate:

]𝑖=

𝐷𝑞𝑖𝑠

𝑟ℎ𝑖(𝐾𝑖𝜙𝑖)0.5 , (𝑖 = 1, . . . , 𝑁) , (5)

where 𝐷 is a constant and layer stable flow rate 𝑞𝑖𝑠can be

obtained by the late-time in unsteady flow.With the viscosity of polymer solution into (1) for the

shear rate representation, for radial flow, under cylindricalcoordinate, dimensionless forms are obtained by finishingafter

𝜅𝑗(

𝜕2𝑃𝑗𝐷

𝜕𝑟𝐷2 +

𝑛

𝑟𝐷

𝜕𝑃𝑗𝐷

𝜕𝑟𝐷

)

= 𝜔𝑗𝑟𝐷

1−𝑛 𝜕𝑃𝑗𝐷

𝜕𝑡𝐷

− 𝑟𝐷

1−𝑛𝜆𝑗−1 (𝑃𝑗𝐷 −𝑃𝑗−1𝐷)

+ 𝑟𝐷

1−𝑛𝜆𝑗(𝑃𝑗+1𝐷 −𝑃𝑗𝐷) .

(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:

𝑃𝑤𝐷

= 𝑃𝑗𝐷

(1, 𝑡𝐷) − 𝑠𝑗

𝜕𝑃𝑗𝐷

𝜕𝑟𝐷

𝑟𝐷=1

,

1 = 𝐶𝐷

𝑑𝑃𝑤𝐷

𝑑𝑡𝐷

−

𝑁

∑

𝑗=1𝜅𝑗

𝜕𝑃𝑗𝐷

𝜕𝑟𝐷

𝑟𝐷=1

.

(7)

For infinite-outer-boundary condition,

𝑃𝑗𝐷

(𝑟𝐷, 𝑡𝐷) → 0 (𝑟

𝐷→ ∞) . (8)

For no-flow outer-boundary condition,

𝜕𝑃𝑗𝐷

𝜕𝑟𝐷

= 0 (𝑟𝐷

= 𝑟𝑒𝐷

) . (9)

For initial condition,

𝑃𝑗𝐷

(𝑟𝐷, 0) = 0. (10)

For dimensionless layer flow rates,

𝑞𝑗𝐷

(𝑡𝐷) =

𝑞𝑗

𝑞= − 𝜅𝑗

𝜕𝑃𝑗𝐷

𝜕𝑟𝐷

𝑟𝐷=1

, (11)

where the dimensionless variables are defined by the follow-ing:

𝑃𝑗𝐷

=0.02𝜋

𝑞

𝑁

∑

𝑗=1

[

[

𝑘𝑗

(1+𝑛)/2ℎ𝑗

𝑛(𝐷𝑞𝑗𝑠)1−𝑛

𝐻𝑟𝑤1−𝑛𝜙𝑗

(1−𝑛)/2]

]

(𝑃𝑗−𝑃0) , (12)

𝑡𝐷

= 0.01∑𝑁

𝑗=1 (𝜅𝑗(1+𝑛)/2

ℎ𝑗

𝑛(𝐷𝑞𝑗𝑠)1−𝑛

/𝜙𝑗

(1−𝑛)/2)

𝐻∑𝑁

𝑗=1 𝜙𝑗ℎ𝑗𝐶𝑡𝑗𝑟𝑤3−𝑛

𝑡, (13)

𝜅𝑗=

𝑘𝑗

(1+𝑛)/2ℎ𝑗

𝑛𝜙𝑗

(𝑛−1)/2(𝐷𝑞𝑗𝑠)1−𝑛

∑𝑁

𝑗=1 𝑘𝑗(1+𝑛)/2

ℎ𝑗

𝑛𝜙𝑗

(𝑛−1)/2(𝐷𝑞𝑗𝑠)1−𝑛 , (14)

𝜔𝑗=

𝜙𝑗ℎ𝑗𝐶𝑡𝑗

∑𝑁

𝑗=1 𝜙𝑗ℎ𝑗𝐶𝑡𝑗, (15)

𝜆𝑗=

𝐻𝑟𝑤

3−𝑛𝜒𝑗

∑𝑁

𝑗=1 𝑘𝑗(1+𝑛)/2

ℎ𝑗

𝑛𝜙𝑗

(𝑛−1)/2(𝐷𝑞𝑗𝑠)1−𝑛 , (16)

𝐶𝐷

=𝐶

2𝜋𝑟𝑤2 ∑𝑁

𝑗=1 𝜙𝑗ℎ𝑗𝐶𝑡𝑗, (17)

𝑟𝐷

=𝑟

𝑟𝑤

, (18)

𝑟𝑒𝐷

=𝑟𝑒

𝑟𝑤

. (19)

3. The Solution of the Model

The above equation is given by Laplace transform on time.Set

𝑙 =2

3 − 𝑛,

𝑚 =1 − 𝑛3 − 𝑛

.

(20)

The basic control equation solution is as follows:

𝑃𝑗𝐷

=

𝑁

∑

𝑘=1𝑟𝐷

(1−𝑛)/2[𝐴𝑗

𝑘𝐾𝑚

(𝑙𝜎𝑘𝑟𝐷

(3−𝑛)/2)

+𝐵𝑗

𝑘𝐼𝑚

(𝑙𝜎𝑘𝑟𝐷

(3−𝑛)/2)] ,

(21)

where the subscript on𝐴 indexes the layer and the superscriptindexes 𝜎,𝐴, and𝐵 are, respectively, the first and the two classof 𝑚 order modified Bessel function. 𝑘

𝑗𝜎2 is eigenvalue of

real symmetric three diagonal positive definitematrices [𝑎𝑗𝑘],

where

𝑎

𝑗𝑘= {−𝜆

𝑗−1, 𝑘 = 𝑗 − 1, 𝑗 > 1; 𝜔𝑗𝑧 − 𝜆𝑗−1 −𝜆𝑗, 𝑘 = 𝑗;

− 𝜆𝑗, 𝑘 = 𝑗 + 1, 𝑗


Finally, the coefficients for each zone can be expressed asmultiples of the coefficients,𝐴1

𝑘𝑖 ,𝐵1𝑘𝑖 , of the uppermost layer

in zone 𝑖 with (12):

𝑃𝑗𝐷

=

𝑚𝑖

∑

𝑘𝑖=1

𝑟𝐷

(1−𝑛)/2[𝐴1𝑘𝑖𝛼𝑗

𝑘𝑖𝐾𝑚

(𝑙𝜎𝑘𝑖

𝑟𝐷

(3−𝑛)/2)

+𝐵1𝑘𝑖𝛼𝑗

𝑘𝑖𝐼𝑚

(𝑙𝜎𝑘𝑖

𝑟𝐷

(3−𝑛)/2)] .

(25)

External boundary condition implies the relationshipbetween 𝐴1

𝑘𝑖 and 𝐵1

𝑘𝑖 is as follows:

𝐵1𝑘𝑖 = 𝑏𝑘𝑖𝐴1𝑘𝑖 . (26)

For infinite-outer-boundary condition,

𝑏𝑘𝑖 = 0. (27)

For no-flow outer-boundary conditions,

𝑏𝑘𝑖 =

𝐾𝑙(𝑙𝜎𝑘𝑖

𝑟𝑒𝐷

(3−𝑛)/2)

𝐼−𝑙

(𝑙𝜎𝑘𝑖

𝑟𝑒𝐷(3−𝑛)/2)

. (28)

According to the boundary conditions, the layer 𝑗 − 1 andlayer 𝑗 which in the same zone have the following relations:

𝑚𝑖

∑

𝑘𝑖=1𝐴1𝑘𝑖 (𝛼𝑗−1𝑘𝑖 {𝐾𝑚

(𝑙𝜎𝑘𝑖

) + 𝑏𝑘𝑖𝐼𝑚

(𝑙𝜎𝑘𝑖

)

+ 𝑠𝑗−1𝜎𝑘

𝑖

[(𝑙𝜎𝑘𝑖

) − 𝑏𝑘𝑖𝐼−𝑙

(𝑙𝜎𝑘𝑖

)]}

− 𝛼𝑗

𝑘𝑖 {𝐾𝑚

(𝑙𝜎𝑘𝑖

) + 𝑏𝜅𝑖𝐼𝑚

(𝑙𝜎𝑘𝑖

)

+ 𝑠𝑗𝜎𝑘𝑖

[𝐾𝑙(𝑙𝜎𝑘𝑖

) − 𝑏𝜅𝑖𝐼−𝑙

(𝑙𝜎𝑘𝑖

)]}) = 0.

(29)

The layer 𝑗 − 1 of zone 𝑖 − 1 and the layer 𝑗 of zone 𝑖 − 1 havethe following relations:

𝑚𝑖−1

∑

𝑘𝑖−1=1

(𝐴1𝑘𝑖−1𝛼𝑗−1𝑘𝑖−1 {𝐾𝑚

(𝑙𝜎𝑘𝑖−1

) + 𝑏𝑘𝑖−1𝐼𝑚


) + 𝑠𝑗−1𝜎𝑘

𝑖−1[𝐾𝑙(𝑙𝜎𝑘𝑖−1

) − 𝑏𝑘𝑖−1𝐼−𝑙


)]})

−

𝑚𝑖

∑

𝑘𝑖=1

(𝐴1𝑘𝑖𝛼𝑗

𝑘𝑖 {𝐾𝑚

(𝑙𝜎𝑘𝑖

) + 𝑏𝜅𝑖𝐼𝑚

(𝑙𝜎𝑘𝑖

) + 𝑠𝑗𝜎𝑘𝑖


) − 𝑏𝜅𝑖𝐼−𝑙

(𝑙𝜎𝑘𝑖

)]}) = 0.

(30)

The wellbore pressure is as follows:

𝑃𝑤𝐷

=1

(𝐶𝐷𝑧2 + 1/𝑃

𝑤𝐷𝐶𝐷=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 𝐶

𝐷= 0 cases

of equations.Equations (29)-(30) are linear equations with coefficient

𝐴1𝑘𝑖 which can be solved by numerical method.The wellbore

pressure without wellbore storage is given as

𝑃𝑤𝐷𝐶𝐷=0 =

𝑚𝑖

∑

𝑘𝑖=1

(𝐴1𝑘𝑖𝛼1𝑘𝑖 {𝐾𝑚

(𝑙𝜎𝑘𝑖

) + 𝑏𝑘𝑖𝐼𝑚

(𝑙𝜎𝑘𝑖

)

+ 𝑠1𝜎𝑘𝑖



(𝑙𝜎𝑘𝑖

)]}) .

(32)

In addition, layer flow rate is given as follows:

𝑞𝑗𝐷

= (1−𝐶𝐷𝑃𝑤𝐷

𝑧2)

⋅ 𝜅𝑗

𝑚𝑖

∑

𝑘𝑖=1

𝐴1𝑘𝑖𝛼𝑗

𝑘𝑖𝜎𝑘𝑖

{𝐾𝑙(𝑙𝜎𝑘𝑖


(𝑙𝜎𝑘𝑖

)} .

(33)

Distribution of radial pressure on each layer becomes

𝑃𝑗𝐷

= (1−𝐶𝐷𝑃𝑤𝐷

𝑧2) 𝑃𝑗𝐷𝐶𝐷=0 = (1−𝐶𝐷𝑃𝑤𝐷𝑧

2)

⋅

𝑚𝑖

∑

𝑘𝑖=1

(𝐴1𝑘𝑖𝛼𝑗

𝑘𝑖 {𝐾𝑚

(𝑙𝜎𝑘𝑖

𝑟𝐷

(3−𝑛)/2)

+ 𝑏𝑘𝑖𝐼𝑚

(𝑙𝜎𝑘𝑖

𝑟𝐷

(3−𝑛)/2)}) .

(34)

For the case of double-layer reservoirs with formation cross-flow, assume that choosing parameters is as follows.

𝐶𝐷

= 1, 𝑠1 = 𝑠2 = 1, 𝜆 = 1𝑒 − 5, 𝜅 = 0.95,𝜔 = 0.1, and 𝑛 = 0.1, 0.3, 0.5, 0.7, 0.9, through the numericalinversion of the Laplace transform of Stehfest [20]. Pressureand flow in the real space are obtained.The 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; namely,linear unit slope represents pure effect of wellbore storage;in 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 𝑚

𝐿=

(1−𝑛)/(3−𝑛). And it can be seen that the pressure derivativerises and increases with the reduction of power-law index 𝑛steepened.


108107106105104

103

103

102

102

101

101

100

100

10−1

10−1

10−2

10−2

CD = 1

s1 = 1s2 = 1

𝜆 = 1e − 5𝜅 = 0.95

𝜔 = 0.1

tD

n

n = [0.1, 0.3, 0.5, 0.7, 0.9]

PwD,dpwD

/dln

t D

Figure 2: Dimensionless pressure for two-layer system.

0.0

0.2

0.4

0.6

0.8

1.0

10810710610510410310210110010−110−2

tD

n

q1D

CD = 1

s1 = 1s2 = 1

𝜆 = 1e − 5𝜅 = 0.95

𝜔 = 0.1

n = [0.1, 0.3, 0.5, 0.7, 0.9]

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 𝜅; 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 balanced,and the crossflow that is more and more weak, finally, tendsto be stable flow rate 1 − 𝜅.

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.

0.00

0.05

0.10

0.15

0.20

10810710610510410310210110010−2 10−110−4 10−3

tD

n

q2D

CD = 1

s1 = 1s2 = 1

𝜆 = 1e − 5𝜅 = 0.95

𝜔 = 0.1

n = [0.1, 0.3, 0.5, 0.7, 0.9]

Figure 4: Dimensionless flow-rate for the second layer.

(1) Without Formation Crossflow.

for the infinite-out-boundary, the stable flow rate, forlayer 𝑗, is as follows:

𝑞𝑗𝑠

=

𝑞𝜅𝑗

𝑙𝜔𝑗

𝑚

∑𝑁

𝑘=1 𝜅𝑘𝑙𝜔𝑘𝑚. (35)

For the no-flow-boundary, the stable flow rate, forlayer 𝑗, is as follows:

𝑞𝑗𝑠

=

𝑞𝜙𝑗ℎ𝑗𝐶𝑡𝑗

∑𝑁

𝑗=1 𝜙𝑗ℎ𝑗𝐶𝑡𝑗. (36)

(2) With Formation Crossflow.

Regarding the eigenvalues of matrix {𝑎𝑗𝑘}, there is a

value to meet 𝜎2 = 𝑧, and the rest is independentof 𝑧 and 𝜔

𝑗and depends only on 𝜅1, 𝜅2, . . . , 𝜅𝑛 and

𝜆1, 𝜆2, . . . , 𝜆𝑛. Assume that 𝜎12= 𝑧. Stable layer flow

rate through the determinant value can be expressedas

𝑞𝑗𝑠

= 𝑞𝜅𝑗lim𝑧→ 0

[

[

det {𝐶𝑗𝑘}1,1

det {𝐶𝑗𝑘}

𝛼𝑗

1𝜎1 {𝐾𝑙 (𝑙𝜎1)

− 𝑏1𝐼−𝑙

(𝑙𝜎1)}]

]

+ 𝑞𝜅𝑗

𝑁

∑

𝑘𝑖=2lim𝑧→ 0

[

det {𝐶𝑖𝑘}𝑘𝑖,𝑘𝑖

det {𝐶𝑖𝑘}

⋅ 𝛼𝑗

𝑘𝑖𝜎𝑘𝑖

{𝐾𝑙(𝑙𝜎𝑘) − 𝑏𝑘𝑖𝐼−𝑙

(𝑙𝜎𝑘)}] ,

(37)

where det{𝐶𝑗𝑘} is determinant of matrix {𝐶

𝑗𝑘}.


The matrix element is as follows:

𝐶𝑗𝑘

= 𝛼𝑗

𝑘{𝐾𝑚

(𝑙𝜎𝑘) + 𝑏𝑘𝐼𝑚

(𝑙𝜎𝑘)

+ 𝑠𝑘𝜎𝑘[𝐾𝑙(𝑙𝜎𝑘) − 𝑏𝑘𝐼−𝑙

(𝑙𝜎𝑘)]}

− 𝛼𝑗+1𝑘{𝐾𝑚

(𝑙𝜎𝑘+1) + 𝑏

𝑘𝐼𝑚

(𝑙𝜎𝑘+1)

+ 𝑠𝑘+1𝜎𝑘+1 [𝐾𝑙 (𝑙𝜎𝑘+1) − 𝑏

𝑘𝐼−𝑙

(𝑙𝜎𝑘+1)]} .

(38)

Thematrix {𝐶𝑗𝑘}𝑖,𝑖is used in matrix in {𝐶

𝑗𝑘} column 𝑖

replacement for (0, 0, . . . , 𝐴, 1/𝑧)𝑇, where superscript𝑇 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 finally.The 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

𝑎

𝑗𝑘: Matrix elements defined in (22)

𝐴𝑗

𝑘, 𝐵𝑗

𝑘: Coefficient for 𝑗th layer, 𝑘th root defined in(23)

𝐴1𝑘𝑖 ,𝐵1𝑘𝑖 : Coefficient for 𝑗th layer, 𝑘th root, in zone 𝐼,

defined in (25)𝑏𝑘𝑖 : Coefficient for outer boundary condition

defined in (27) or (28)𝐶: Wellbore-storage coefficient (cm3/kPa)𝐶𝑡𝑗: Total compressibility in layer 𝐼 (kPa−1)

𝐶𝑗𝑘: Matrix elements defined in (38)

𝐷: Constant defined in (5)ℎ𝑗: Formation thickness in layer 𝑗 (cm)

𝐻 : Constant defined in (4)𝐼𝑚(⋅), 𝐾𝑚(⋅): Modified 𝑚 order Bessel function of the first

and second kindΔℎ𝑗: Thickness of a nonperforated zone between

layers 𝑗 and 𝑗 + 1 (cm)𝐾𝑗: Horizontal permeability in layer 𝑗, 𝜇m2

𝐾V𝑗: Vertical permeability of a tight zone betweenLayers 𝑗 and 𝑗 − 1 (𝜇m2)

𝐾𝑧𝑗: Vertical permeability in Layer 𝑗 (𝜇m2)

𝑙, 𝑚: Constant defined in (20)𝑚𝑖: Number of layers in Zone 𝑖

𝑛: Power-law index𝑁: Number of layers in reservoir system𝑁𝑍: Number of Zones in reservoir system𝑃𝑖: Reservoir pressure in layer 𝐼 (kPa)

𝑃𝑗

𝑖: Reservoir pressure in layer 𝑗 in Zone 𝑖(kPa)

𝑃𝑗𝐷: Dimensionless reservoir pressure in

layer 𝑗 defined in (12)𝑃0: Initial reservoir pressure (kPa)

𝑃𝑤𝐷

: Dimensionless bottomhole pressure inLaplace space

𝑃𝑤𝐷𝐶𝐷=0: Dimensionless bottomhole pressure

without wellbore storage in Laplacespace

𝑄: Surface production rate (cm3/s)𝑞𝑗: Flow rate for layer 𝑗 (cm3/s)

𝑞𝑗𝑠: Stable flow rate for layer 𝑗 (cm3/s)

𝑟: Radial distance (cm)𝑟𝐷: Dimensionless radius, defined in (18)

𝑟𝑒: Reservoir outer radius (cm)

𝑟𝑒𝐷: Dimensionless outer reservoir radius,

defined in (19)𝑟𝑤: Wellbore radius (cm)

𝑆𝑗: Wellbore skin factor for layer 𝑗

𝑡: Time (s)Δ𝑡: Elapsed time after rate change (s)𝑡𝐷: Dimensionless time referenced to

producing layer]𝑖: Shear rate in layer 𝐼 (s−1)

𝜒𝑗: Crossflow coefficient between layers 𝑗

and 𝑗 + 1 defined in (2)𝑥𝑗: Resistance to flow per unit length at the

𝑗th layer interface𝑍: Laplace space variable 𝑖𝛼𝑗

𝑘: Coefficient for layer 𝑗, Root 𝑘, definedin (23)

𝛼𝑗

𝑘𝑖 : Coefficient for layer 𝑗, Root 𝑘, defined

in Zone 𝐼, defined in (25)𝜅𝑗: Coefficient for layer 𝑗, defined in (14)

𝜆𝑗: Dimensionless semipermeability

between layers 𝑗 and 𝑗 + 1, defined in(16)

𝑀: Dynamic viscosity (mpa⋅s)Φ𝑗: Porosity fraction for layer 𝑗

𝜔𝑗: Coefficient for layer 𝑗, defined in (15)

]𝑗: Shear rate for layer 𝑗 defined in (5)

det{𝐶𝑗𝑘}: Determinant of matrix {𝐶

𝑗𝑘}.

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).


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Research Article Pressure-Transient Behavior in a Multilayered Polymer Flooding Reservoir · 2018. 12. 13. · Pressure-Transient Behavior in a Multilayered Polymer Flooding Reservoir