# The Pressure Buildup Well Test Analysis considering Stress Sensitivity Effect for Deepwater Composite Gas Reservoir with High Temperature and Pressure

Authors: Yihua Gao; Ruizhong Jiang; Xiangdong Xu; Zhaobo Sun; Zhiwang Yuan; Kang Ma; Bin Jiang; Botao Kang; Guoning Chen; Chenxi Li
Journal: Geofluids
 (2021)

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## Abstract

Some deepwater gas reservoirs with high temperature and pressure have obvious stress sensitivity effect resulting in difficulty in well test interpretations. The influence of stress sensitivity effect on the pressure drawdown well test is discussed in many papers. However, the influence on the pressure buildup well test is barely discussed. For practices in oilfields, the quality of pressure data from the drawdown stage of well test is poor due to the influence of production fluctuation. Thus, the pressure data from the buildup stage is used for well test interpretations in most cases. In order to analyze the influence of stress sensitivity effect on the pressure buildup well test, this paper establishes a composite gas reservoir pressure buildup well test model considering the stress sensitivity effect and the hysteresis effect. Numerical solutions to both pressure drawdown and buildup well test models are obtained by the numerical differentiation method. The numerical solutions are verified by comparing with analytical solutions and the homogeneous gas reservoir well test solution. Then, the differences between pressure drawdown and buildup well test curves considering the stress sensitivity effect are compared. The parameter sensitivity analysis is conducted. Compared with the conventional well test curve, the pressure derivative curve of pressure drawdown well test considering the stress sensitivity effect deviates upward from the 0.5 horizontal line at the inner zone radial flow stage, while it deviates upward from theM/2 (mobility ratio/2) horizontal line at the outer zone radial flow stage. However, for the pressure buildup well test curve considering the stress sensitivity effect, the pressure derivative curve gradually descends to the 0.5 horizontal line at the inner zone radial flow stage, while it descends to the M/2 (mobility ratio/2) horizontal line at the outer zone radial flow stage. The pressure derivative curve of pressure buildup well test considering the hysteresis effect is higher than the curve without considering the hysteresis effect, because the permeability cannot be recovered to its original value in the buildup stage after considering the hysteresis effect. Meanwhile, skin factor and mobility ratio have different effects on pressure drawdown and buildup well test curves. Based on the model, a well test interpretation case from a deepwater gas reservoir with high temperature and pressure is studied. The result indicates that the accuracy of the interpretation is improved after considering the stress sensitivity effect, and the skin factor will be exaggerated without considering the stress sensitivity effect.


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## Body

## 1. Introduction

Well test is an important way to obtain physical parameters and evaluate the development performance of gas reservoirs [1–9]. It can help to understand the seepage mechanism and the development law of gas reservoirs with high temperature and pressure. Compared with the conventional gas reservoir, there is more obvious rock deformation during the development process of the gas reservoir with high temperature and pressure resulting in stronger stress sensitivity effect [10–15]. Thus, it is necessary to consider the influence of stress sensitivity effect on well test curves. Many experiments and production practices show that there is irreversible deformation in some reservoirs during the development process [16–18]. Reservoir permeability decreases with the decrease of formation pressure. However, the permeability cannot be completely recovered to its original value after the formation pressure increases to the initial value if there is irreversible deformation in reservoirs. This phenomenon is called the hysteresis effect [18].At present, many researchers have studied the influence of stress sensitivity effect on well test curves. However, the influence on the pressure drawdown well test is mainly discussed in these studies [19–30]. For practices in oilfields, the quality of pressure data from the drawdown stage of well test is poor due to the influence of production fluctuation. Thus, the pressure data from the buildup stage is used in most cases of well test interpretations for its good data quality [31–34]. For conventional well test interpretations, the superposition principle is used to obtain solutions to pressure buildup well test models from that of pressure drawdown well tests. However, for well test models considering the stress sensitivity effect, the seepage differential equations are nonlinear equations, and the superposition principle cannot be directly used for nonlinear equations to obtain the solutions to pressure buildup well test models [31]. Therefore, it is necessary to analyze the influence of stress sensitivity effect on the pressure buildup well test to guide well test interpretations. However, the studies about pressure buildup well test models considering the stress sensitivity effect are not sufficient. Zhang et al. established pressure buildup well test models considering the stress sensitivity effect for vertical fractured well and vertical well in a homogeneous reservoir [18, 31]. One of the researches mainly discussed the influence of stress sensitivity effect and hysteresis effect on the variation of fracture conductivity for vertical fractured well in a homogeneous reservoir at the buildup stage [18]. Another research only discussed the influence of stress sensitivity effect on vertical well in a homogeneous reservoir at the buildup stage [31]. There are few researches about the pressure buildup well test of composite gas reservoir considering both the stress sensitivity effect and the hysteresis effect.Thus, the paper establishes theoretical pressure drawdown and buildup well test models for the composite gas reservoir with high temperature and pressure considering both the stress sensitivity effect and the hysteresis effect. The solutions to these models are obtained by the numerical differentiation method. The numerical solutions to pressure drawdown and buildup well test models are verified by comparing with analytical solutions [23] and the homogeneous gas reservoir well test solution [31]. The differences between pressure buildup and drawdown well test curves considering the stress sensitivity effect are analyzed. The parameter sensitivity analysis is conducted. Then, a field case from a gas reservoir with high temperature and pressure is presented indicating that the accuracy of interpretation is improved after considering the stress sensitivity effect.


## 2. Model Description and Solution

### 2.1. Physical Model and Basic Assumptions

A physical model of a vertical well with fixed gas production rate in a composite gas reservoir is established shown in Figure1. The basic assumptions of the model are as follows:
(1)
An isotropic gas reservoir is bounded by impermeable layers at the bottom and top. The gas reservoir has two zones with different permeability and porosity. The radius of inner zone boundary isr1, and the outer zone boundary is infinite(2)
The vertical gas well is on production at fixed rate with the initial pressurepi followed by a shut-in stage(3)
The gas in the reservoir is highly compressible fluid with the compressibilityCg, compressibility factor Z, and viscosity μg(4)
Considering the stress sensitivity effect and the hysteresis effect, the gas reservoir permeability changes with the formation pressure resulting in irreversible deformation to the reservoir(5)
The gravity effect is negligible. The temperature in the reservoir is constant at production and shut-in stages(6)
The effect of wellbore storage is considered by a constant wellbore storage coefficient. The effective wellbore radius is used to describe the skin damageFigure 1
Physical model of vertical well producing in composite gas reservoir.


### 2.2. Mathematical Model

#### 2.2.1. Characterization of Stress Sensitivity Effect and Hysteresis Effect

The stress sensitivity coefficientγ is used to describe the change of reservoir permeability with the pressure. Some experimental studies indicate that the relationship between the reservoir permeability and the pressure change conforms to the exponential form [23, 35, 36]. The changes of reservoir permeability with the pressure at pressure drawdown and buildup stages are shown in Figure 2.Figure 2
The change of reservoir permeability with pressure.At the pressure drawdown stage, the stress sensitivity effect can be described by Equation (1):
(1)γd=1kiddkdp⇒kd=kidexp−γdpi−pd.At the pressure buildup stage, the stress sensitivity effect and the hysteresis effect can be described by Equation (2):
(2)γb=1kibdkdp⇒kb=kibexp−γbpi−pb=kidexp−γd−γbpi−pdexp−γbpi−pb,whereγd and γb are the stress sensitivity coefficients at pressure drawdown and buildup stage, 1/Pa; there will be the hysteresis effect when γb is less than γd; kid is the initial permeability of gas reservoir, m2; k is the permeability of gas reservoir, m2; p is the pressure of gas reservoir, Pa; pi is the initial pressure of gas reservoir, Pa; pd and pb are the gas reservoir pressure at pressure drawdown and buildup stages, Pa; kd and kb are the gas reservoir permeability at pressure drawdown and buildup stages, m2; kib is the maximum permeability value that can be restored after the gas reservoir pressure increases to the initial pressure, m2.The gas pseudo-pressure is generally used in gas reservoir well tests, and the definition of the gas pseudo-pressure is as Equation (3):
(3)ψ=2∫pscppμZdp,whereψ is the gas pseudo-pressure, Pa/s; μ is the gas viscosity, Pa.s; Z is the gas compressibility factor, decimal; psc is the pressure at the standard condition, Pa.Substituting Equation (3) into Equations (1) and (2), stress sensitivity equations in the pseudo-pressure form can be expressed as Equations (4) and (5):
(4)γd=1kiddkdp=1kiddkdψddψddp=1kiddkdψdpμZ,(5)γb=1kibdkdp=1kibdkdψbdψbdp=1kibdkdψbpμZ.Then, stress sensitivity coefficients in the pseudo-pressure form can be obtained as Equations (6) and (7):
(6)γd∗=γdp/μZ=1kiddkdψd⇒kd=kidexp−γd∗ψi−ψd,(7)γb∗=γbp/μZ=1kibdkdψb⇒kb=kibexp−γb∗ψi−ψb,whereγ∗d and γ∗b are stress sensitivity coefficients in the pseudo-pressure form at pressure drawdown and buildup stages, Pa/s; ψi is the initial pseudo-pressure, Pa/s; ψd and ψb are the pseudo-pressure at pressure drawdown and buildup stages, Pa/s.


#### 2.2.2. Pressure Drawdown Well Test Model

The differential equation of natural gas seepage considering the stress sensitivity effect and the hysteresis effect is as Equation (8):
(8)1r∂∂rkprpZμ∂p∂r=ϕ∂∂tpZ,wherer is the radius, m; kp is the permeability considering the stress sensitivity effect and the hysteresis effect, m2; t is time, s; ϕ is porosity, decimal.For the pressure drawdown well test, the comprehensive seepage differential equation considering the stress sensitivity effect in the inner zone of composite gas reservoir is as Equation (9):
(9)1r∂∂rki1e−γdpi−p1rp1Zμ1∂p1∂r=ϕ1∂∂tp1Z,whereki1 is the initial inner zone permeability of composite gas reservoir, m2; p1 is the inner zone pressure of composite gas reservoir, Pa; μ1 is the inner zone gas viscosity of composite gas reservoir, Pa.s; ϕ1 is the inner zone porosity of composite gas reservoir, decimal.The comprehensive seepage differential equation considering the stress sensitivity effect in the outer zone of composite gas reservoir is as Equation (10):
(10)1r∂∂rki2e−γdpi−p2rp2Zμ2∂p2∂r=ϕ2∂∂tp2Z,whereki2 is the initial outer zone permeability of composite gas reservoir, m2; p2 is the outer zone pressure of composite gas reservoir, Pa; μ2 is the outer zone gas viscosity of composite gas reservoir, Pa.s; ϕ2 is the outer zone porosity of composite gas reservoir, decimal.After substituting Equation (3) into Equations (9) and (10), gas comprehensive seepage differential equations in the pseudo-pressure form can be expressed as Equations (11) and (12):
(11)1r∂ψ1∂r+∂2ψ1∂r2+γd∗∂ψ1∂r2=ϕ1μ1Cgki1eγd∗ψi−ψ1∂ψ1∂t,(12)1r∂ψ2∂r+∂2ψ2∂r2+γd∗∂ψ2∂r2=ϕ2μ2Cgki2eγd∗ψi−ψ2∂ψ2∂t,whereψ1 is the inner zone pseudo-pressure, Pa/s; ψ2 is the outer zone pseudo-pressure of composite gas reservoir, Pa/s; Cg is the gas compressibility coefficient, Pa-1.The initial condition can be expressed as Equation (13):
(13)ψ1r,t=0=ψ2r,t=0=ψi.The inner boundary considering the skin damage and the well storage effect for the pressure drawdown well test can be expressed as Equation (14):
(14)re−γd∗ψi−ψ1∂ψ1∂rr=rwe=qscpscTπki1hTscZsc+Cμ12πki1h∂ψ1∂t,whererwe=rwexp−S is the effective well radius, m; S is the skin factor, dimensionless; qsc is the production of gas well at the standard condition, m3/s; Tsc is the temperature at the standard condition, K; T is the formation temperature, K; C is the wellbore storage coefficient, m3/Pa; h is the effective thickness of reservoir, m.The pressure and flow rate are equal in the interface between the inner zone and the outer zone as described in Equations (15) and (16):
(15)ki1e−γd∗ψi−ψ1μ1∂ψ1∂rr=r1=ki2e−γd∗ψi−ψ2μ2∂ψ2∂rr=r1,(16)ψ1r=r1,t=ψ2r=r1,t,wherer1 is the inner zone radius, m.Considering the infinite outer boundary, Equation (17) is obtained:
(17)limr⟶∞ψ2r,t=ψi.Thus, the mathematical model of composite gas reservoir pressure drawdown well test considering the stress sensitivity effect can be expressed as Equation (18):
(18)1r∂ψ1∂r+∂2ψ1∂r2+γd∗∂ψ1∂r2=ϕ1μ1Cgki1eγd∗ψi−ψ1∂ψ1∂t,1r∂ψ2∂r+∂2ψ2∂r2+γd∗∂ψ2∂r2=ϕ2μ2Cgki2eγd∗ψi−ψ2∂ψ2∂t,re−γd∗ψi−ψ1∂ψ1∂rr=rwe=qscpscTπki1hTscZsc+Cμ12πki1h∂ψ1∂t,ki1e−γd∗ψi−ψ1μ1∂ψ1∂rr=r1=ki2e−γd∗ψi−ψ2μ2∂ψ2∂rr=r1,ψ1r,t=0=ψ2r,t=0=ψi,ψ1r=r1,t=ψ2r=r1,t,limr⟶∞ψ2r,t=ψi.In order to obtain the dimensionless mathematical model, some dimensionless variables are introduced as follows:(19)ψ1D=πki1hTscψi−ψ1qscpscT,ψ2D=πki1hTscψi−ψ2qscpscT,tD=ki1tϕ1μ1Cgrw2,ω=ϕ1Cg1ϕ2Cg2,rD=rrwe=rrwexp−S,r1D=r1rwe=r1rwexp−S,γdD∗=γdμiZi2piqscpscTπki1hTsc,γbD∗=γbμiZi2piqscpscTπki1hTsc,M=ki1/μ1ki2/μ2,CD=C2πhϕ1Cgrw2,whereψ1D is the dimensionless pseudo-pressure in the inner zone; ψ2D is the dimensionless pseudo-pressure in the outer zone; tD is the dimensionless time; ω is the storage ratio between the inner zone and the outer zone; rD is the dimensionless radius; r1D is the dimensionless inner zone radius; γ∗dD and γ∗bD are dimensionless stress sensitivity coefficients at pressure drawdown and buildup stages, respectively; M is the mobility ratio between the inner zone and the outer zone; CD is the dimensionless wellbore storage coefficient.Substituting dimensionless variables into Equation (18), the dimensionless mathematical model of the pressure drawdown well test considering the stress sensitivity effect can be expressed as Equation (20):
(20)1rD∂ψ1D∂rD+∂2ψ1D∂rD2−γdD∗∂ψ1D∂rD2=eγdD∗ψ1De2S∂ψ1D∂tD,1rD∂ψ2D∂rD+∂2ψ2D∂rD2−γdD∗∂ψ2D∂rD2=ωMeγdD∗ψ2De2S∂ψ2D∂tD,CD∂ψ1D∂tD−e−γdD∗ψ1D∂ψ1D∂rDrD=1=1,ψ1DrD,tD=0=ψ2DrD,tD=0=0,e−γdD∗ψ1D∂ψ1D∂rDrD=r1D=e−γdD∗ψ2D1M∂ψ2D∂rDrD=r1D,ψ1DrD=r1D,tD=ψ2DrD=r1D,tD,limrD⟶∞ψ2DrD,tD=0.


#### 2.2.3. Pressure Buildup Well Test Model

For the pressure buildup well test, the gas productionqsc in the inner boundary condition Equation (14) is zero. The stress sensitivity coefficient of the pressure buildup stage is introduced into the mathematical model. Similarly, the dimensionless mathematical model of the pressure buildup well test considering the stress sensitivity effect and the hysteresis effect can be expressed as Equation (21):
(21)1rD∂ψ1D∂rD+∂2ψ1D∂rD2−γbD∗∂ψ1D∂rD2=R1eγbD∗ψ1De2S∂ψ1D∂tD,1rD∂ψ2D∂rD+∂2ψ2D∂rD2−γbD∗∂ψ2D∂rD2=R2ωMeγbD∗ψ2De2S∂ψ2D∂tD,R1CD∂ψ1D∂tD−e−γbD∗ψ1D∂ψ1D∂rDrD=1=0,ψ1DrD,tD=0=ψ2DrD,tD=0=0,e−γbD∗ψ1D∂ψ1D∂rDrD=r1D=e−γbD∗ψ2D1M∂ψ2D∂rDrD=r1D,ψ1DrD=r1D,tD=ψ2DrD=r1D,tD,limrD⟶∞ψ2DrD,tD=0,whereR1 is the ratio of the initial permeability and the maximum permeability that can be restored after the gas reservoir pressure increases to the initial value in the inner zone, dimensionless; R2 is the ratio of the initial permeability and the maximum permeability that can be restored after the gas reservoir pressure increases to the initial value in the outer zone, dimensionless.


### 2.3. Solutions to Mathematical Models

#### 2.3.1. Solution to Pressure Drawdown Well Test Model by Perturbation Method

Kikani and Pedrosa proposed the perturbation method to solve the mathematical model for the pressure drawdown well test considering the stress sensitivity effect [23]. First, the dimensionless pseudo-pressure is transformed as Equation (22):
(22)ψ1D=−ln1−γdD∗ξ1DγdD∗,ψ2D=−ln1−γdD∗ξ2DγdD∗.Substituting Equation (22) into Equation (20), the dimensionless pressure drawdown well test mathematical model can be transformed as Equation (23):
(23)1rD∂ξ1D∂rD+∂2ξ1D∂rD2=e−2S1−γdD∗ξ1D∂ξ1D∂tD,1rD∂ξ2D∂rD+∂2ξ2D∂rD2=ωMe−2S1−γdD∗ξ2D∂ξ2D∂tD,CD∂ξ1D∂tD−rD∂ξ1D∂rDrD=1=11−γdD∗ξ1D,ξ1DrD=r1D,tD=ξ2DrD=r1D,tD,∂ξ1D∂rDrD=r1D=1M∂ξ2D∂rDrD=r1D,ξ1DrD,tD=0=ξ2DrD,tD=0=0,limrD⟶∞ξ2DrD,tD=0.After applying zero-order approximation toξ1D and ξ2D in Equation (23) [23], the new mathematical model can be expressed as Equation (24):
(24)1rD∂ξ1D∂rD+∂2ξ1D∂rD2=e−2S∂ξ1D∂tD,1rD∂ξ2D∂rD+∂2ξ2D∂rD2=ωMe−2S∂ξ2D∂tD,CD∂ξ1D∂tD−rD∂ξ1D∂rDrD=1=1,ξ1DrD=r1D,tD=ξ2DrD=r1D,tD,∂ξ1D∂rDrD=r1D=1M∂ξ2D∂rDrD=r1D,ξ1DrD,tD=0=ξ2DrD,tD=0=0,limrD⟶∞ξ2DrD,tD=0.Equation (24) is the same as the well test model of composite gas reservoir without considering the stress sensitivity effect [37]. Based on the Laplace transform and the property of Bessel function, the solution to Equation (24) in the Laplace space can be expressed as Equation (25):
(25)ξ¯wD=1uRTPuCDRTP+βMTP,whereu is the variable in the Laplace space,
(26)RTP=1+I0βK0ββK0r1DNβK1r1Dβ−N/MβK0r1DβK1r1DNβN/MβK1r1DNβI0r1Dβ+βK0r1DNβI1r1Dβ,MTP=K1βK0β−I1βK0ββK0r1DNβK1r1Dβ−N/MβK0r1DβK1r1DNβN/MK1r1DNβI0r1Dβ+βK0r1DNβI1r1Dβ,β=uexp−S,N=ωM.Based on the numerical inversion method and the perturbation method [23], the solution to the pressure drawdown well test model considering the stress sensitivity effect for the composite gas reservoir can be expressed as Equation (27):
(27)ψwD=−ln1−γdD∗ξwDγdD∗.For the conventional well test without considering the stress sensitivity effect, the solution to the pressure buildup well test mathematical model, which is a linear equation, can be obtained by the superposition principle based on the solution to the pressure drawdown well test model. However, the superposition principle cannot be applied to the well test model considering the stress sensitivity effect, which is a nonlinear equation. Thus, it is difficult to get the solution to the pressure buildup well test model considering the stress sensitivity effect by the superposition principle.


#### 2.3.2. Solutions to Pressure Drawdown and Buildup Well Test Models by Numerical Method

The well test models considering the stress sensitivity effect and the hysteresis effect as Equations (20) and (21) are second-order nonlinear equations. The analytical solution to the pressure drawdown well test model obtained by the perturbation method is an approximate result. Thus, the numerical differentiation method is used to solve these equations.Considering dramatic pressure changes near the wellbore, the unequal-space grid division is used. TakingxD=lnrD, Equation (28) is obtained:
(28)∂ψD∂rD=e−xD∂ψD∂xD,∂2ψD∂rD2=e−2xD∂2ψD∂xD2−e−2xD∂ψD∂xD.Substituting Equation (28) into the dimensionless pressure drawdown well test model Equation (20), the model can be written as Equation (29):
(29)∂2ψ1D∂xD2−γdD∗∂ψ1D∂xD2=e2xD+γdD∗ψ1De2S∂ψ1D∂tD,∂2ψ2D∂xD2−γdD∗∂ψ2D∂xD2=ωMe2xD+γdD∗ψ2De2S∂ψ2D∂tD,CD∂ψ1D∂tD−e−γdD∗ψ1D∂ψ1D∂xDxD=0=1,ψ1DxD=lnr1D,tD=ψ2DxD=lnr1D,tD,e−γdD∗ψ1D∂ψ1D∂xDxD=lnr1D=e−γdD∗ψ2D1M∂ψ2D∂xDxD=lnr1D,ψ1DxD,tD=0=ψ2DxD,tD=0=0,limxD⟶∞ψ2DxD,tD=0.The difference in space is as Equation (30):
(30)xDi=i−1Δx1Di=1,2,3,.⋯,N,lnr1D+i−NΔx2Di=N+1,N+2,N+3,.⋯,NN,whereΔx1D=lnr1D/N−1; Δx2D=lnRD/r1D/NN−N; N is the discrete number in space of the inner zone, NN is the total discrete number in space of inner and outer zones, and RD is the dimensionless outer zone boundary radius.The logarithmic interval is taken for time as Equation (31):
(31)tDj=10−6+j−1Δt,Δt=logTD+6O−1,wherej=1,2,3,⋯O,O is the discrete number of time and TD is the dimensionless time at the end of the simulation.For the dimensionless well test mathematical model Equation (29), the finite difference discretization is carried out in space and time as Equations (30) and (31). Then, the seepage equation discretization result for the composite gas reservoir can be expressed as Equation (32):
(32)aiψDi−1j+biψDij+ciψDi+1j+diψDij2+eiψDijψDi+1j+fiψDi+1j2+gierdD∗ψDij+hiψDijerdD∗ψDij=0.Wheni=1,2,⋯,N, then
(33)ai=1Δx1D2,bi=−2Δx1D2,ci=1Δx1D2,di=−γdD∗Δx1D2,ei=2γdD∗Δx1D2,fi=−γdD∗Δx1D2,gi=e2i−1Δx1De2StDj−tDj−1ψDij−1,hi=−e2i−1Δx1De2StDj−tDj−1.Wheni=N+1,N+2,⋯.,NN, then
(34)ai=1Δx2D2,bi=−2Δx2D2,ci=1Δx2D2,di=−γdD∗Δx2D2,ei=2γdD∗Δx2D2,fi=−γdD∗Δx2D2,gi=ωMe2i−1Δx2De2StDj−tDj−1ψDij−1,hi=−ωMe2i−1Δx2De2StDj−tDj−1.The discretization result of the initial condition can be expressed as Equation (35):
(35)ψDi1=0,i=1,2,⋯,NN.The discretization result of the inner boundary condition can be expressed as Equation (36):
(36)b1ψD1j+c1ψD2j+g1eγdD∗ψDij+h1ψD1jeγdD∗ψDij=0,where(37)b1=−1Δx1D,.c1=1Δx1D,g1=1+CDψD1j−1tDj−tDj−1,h1=−CDtDj−tDj−1.The discretization result of the interface condition can be expressed as Equation (38):
(38)ψDN−ψDN−1Δx1=1MψDN+1−ψDNΔx2.The discretization result of the outer boundary condition can be expressed as Equation (39):
(39)ψDNNj=0.As for the pressure buildup test, the model is obtained after modifying the internal boundary condition and introducing the stress sensitivity coefficient of the buildup stage. Then, the seepage equation discretization result for the composite gas reservoir at the buildup stage can be expressed as Equation (40):
(40)aiψDi−1j+biψDij+ciψDi+1j+diψDij2+eiψDijψDi+1j+fiψDi+1j2+gierbD∗ψDij+hiψDijerbD∗ψDij=0.Wheni=1,2,⋯,N, then
(41)ai=1Δx1D2,bi=−2Δx1D2,ci=1Δx1D2,di=−γbD∗Δx1D2,ei=2γbD∗Δx1D2,fi=−γbD∗Δx1D2,gi=R1e2i−1Δx1De2StDj−tDj−1ψDij−1,hi=−R1e2i−1Δx1De2StDj−tDj−1.Wheni=N+1,N+2,⋯.,NN, then
(42)ai=1Δx2D2,bi=−2Δx2D2,ci=1Δx2D2,di=−γbD∗Δx2D2,ei=2γbD∗Δx2D2,fi=−γbD∗Δx2D2,gi=R2ωMe2i−1Δx2De2StDj−tDj−1ψDij−1,hi=−R2ωMe2i−1Δx2De2StDj−tDj−1.The discretization result of the inner boundary condition at the buildup stage can be expressed as Equation (43):
(43)b1ψD1j+c1ψD2j+g1eγbD∗ψDij+h1ψD1jeγbD∗ψDij=0,where(44)b1=−1Δx1D,c1=1Δx1D,g1=R1CDψD1j−1tDj−tDj−1,h1=−R1CDtDj−tDj−1.The discretization results of the interface condition and the outer zone boundary condition for the pressure buildup well test are the same as Equations (38) and (39).Numerical well test models of drawdown and buildup stages for the composite gas reservoir considering the stress sensitivity effect and the hysteresis effect are established by Equations (32)–(39) and Equations (40)–(43). The Newton-Simpson method is used to solve the nonlinear equation to obtain pressure drawdown and buildup well test solutions.


## 2.1. Physical Model and Basic Assumptions

A physical model of a vertical well with fixed gas production rate in a composite gas reservoir is established shown in Figure1. The basic assumptions of the model are as follows:
(1)
An isotropic gas reservoir is bounded by impermeable layers at the bottom and top. The gas reservoir has two zones with different permeability and porosity. The radius of inner zone boundary isr1, and the outer zone boundary is infinite(2)
The vertical gas well is on production at fixed rate with the initial pressurepi followed by a shut-in stage(3)
The gas in the reservoir is highly compressible fluid with the compressibilityCg, compressibility factor Z, and viscosity μg(4)
Considering the stress sensitivity effect and the hysteresis effect, the gas reservoir permeability changes with the formation pressure resulting in irreversible deformation to the reservoir(5)
The gravity effect is negligible. The temperature in the reservoir is constant at production and shut-in stages(6)
The effect of wellbore storage is considered by a constant wellbore storage coefficient. The effective wellbore radius is used to describe the skin damageFigure 1
Physical model of vertical well producing in composite gas reservoir.


## 2.2. Mathematical Model

### 2.2.1. Characterization of Stress Sensitivity Effect and Hysteresis Effect

The stress sensitivity coefficientγ is used to describe the change of reservoir permeability with the pressure. Some experimental studies indicate that the relationship between the reservoir permeability and the pressure change conforms to the exponential form [23, 35, 36]. The changes of reservoir permeability with the pressure at pressure drawdown and buildup stages are shown in Figure 2.Figure 2
The change of reservoir permeability with pressure.At the pressure drawdown stage, the stress sensitivity effect can be described by Equation (1):
(1)γd=1kiddkdp⇒kd=kidexp−γdpi−pd.At the pressure buildup stage, the stress sensitivity effect and the hysteresis effect can be described by Equation (2):
(2)γb=1kibdkdp⇒kb=kibexp−γbpi−pb=kidexp−γd−γbpi−pdexp−γbpi−pb,whereγd and γb are the stress sensitivity coefficients at pressure drawdown and buildup stage, 1/Pa; there will be the hysteresis effect when γb is less than γd; kid is the initial permeability of gas reservoir, m2; k is the permeability of gas reservoir, m2; p is the pressure of gas reservoir, Pa; pi is the initial pressure of gas reservoir, Pa; pd and pb are the gas reservoir pressure at pressure drawdown and buildup stages, Pa; kd and kb are the gas reservoir permeability at pressure drawdown and buildup stages, m2; kib is the maximum permeability value that can be restored after the gas reservoir pressure increases to the initial pressure, m2.The gas pseudo-pressure is generally used in gas reservoir well tests, and the definition of the gas pseudo-pressure is as Equation (3):
(3)ψ=2∫pscppμZdp,whereψ is the gas pseudo-pressure, Pa/s; μ is the gas viscosity, Pa.s; Z is the gas compressibility factor, decimal; psc is the pressure at the standard condition, Pa.Substituting Equation (3) into Equations (1) and (2), stress sensitivity equations in the pseudo-pressure form can be expressed as Equations (4) and (5):
(4)γd=1kiddkdp=1kiddkdψddψddp=1kiddkdψdpμZ,(5)γb=1kibdkdp=1kibdkdψbdψbdp=1kibdkdψbpμZ.Then, stress sensitivity coefficients in the pseudo-pressure form can be obtained as Equations (6) and (7):
(6)γd∗=γdp/μZ=1kiddkdψd⇒kd=kidexp−γd∗ψi−ψd,(7)γb∗=γbp/μZ=1kibdkdψb⇒kb=kibexp−γb∗ψi−ψb,whereγ∗d and γ∗b are stress sensitivity coefficients in the pseudo-pressure form at pressure drawdown and buildup stages, Pa/s; ψi is the initial pseudo-pressure, Pa/s; ψd and ψb are the pseudo-pressure at pressure drawdown and buildup stages, Pa/s.


### 2.2.2. Pressure Drawdown Well Test Model

The differential equation of natural gas seepage considering the stress sensitivity effect and the hysteresis effect is as Equation (8):
(8)1r∂∂rkprpZμ∂p∂r=ϕ∂∂tpZ,wherer is the radius, m; kp is the permeability considering the stress sensitivity effect and the hysteresis effect, m2; t is time, s; ϕ is porosity, decimal.For the pressure drawdown well test, the comprehensive seepage differential equation considering the stress sensitivity effect in the inner zone of composite gas reservoir is as Equation (9):
(9)1r∂∂rki1e−γdpi−p1rp1Zμ1∂p1∂r=ϕ1∂∂tp1Z,whereki1 is the initial inner zone permeability of composite gas reservoir, m2; p1 is the inner zone pressure of composite gas reservoir, Pa; μ1 is the inner zone gas viscosity of composite gas reservoir, Pa.s; ϕ1 is the inner zone porosity of composite gas reservoir, decimal.The comprehensive seepage differential equation considering the stress sensitivity effect in the outer zone of composite gas reservoir is as Equation (10):
(10)1r∂∂rki2e−γdpi−p2rp2Zμ2∂p2∂r=ϕ2∂∂tp2Z,whereki2 is the initial outer zone permeability of composite gas reservoir, m2; p2 is the outer zone pressure of composite gas reservoir, Pa; μ2 is the outer zone gas viscosity of composite gas reservoir, Pa.s; ϕ2 is the outer zone porosity of composite gas reservoir, decimal.After substituting Equation (3) into Equations (9) and (10), gas comprehensive seepage differential equations in the pseudo-pressure form can be expressed as Equations (11) and (12):
(11)1r∂ψ1∂r+∂2ψ1∂r2+γd∗∂ψ1∂r2=ϕ1μ1Cgki1eγd∗ψi−ψ1∂ψ1∂t,(12)1r∂ψ2∂r+∂2ψ2∂r2+γd∗∂ψ2∂r2=ϕ2μ2Cgki2eγd∗ψi−ψ2∂ψ2∂t,whereψ1 is the inner zone pseudo-pressure, Pa/s; ψ2 is the outer zone pseudo-pressure of composite gas reservoir, Pa/s; Cg is the gas compressibility coefficient, Pa-1.The initial condition can be expressed as Equation (13):
(13)ψ1r,t=0=ψ2r,t=0=ψi.The inner boundary considering the skin damage and the well storage effect for the pressure drawdown well test can be expressed as Equation (14):
(14)re−γd∗ψi−ψ1∂ψ1∂rr=rwe=qscpscTπki1hTscZsc+Cμ12πki1h∂ψ1∂t,whererwe=rwexp−S is the effective well radius, m; S is the skin factor, dimensionless; qsc is the production of gas well at the standard condition, m3/s; Tsc is the temperature at the standard condition, K; T is the formation temperature, K; C is the wellbore storage coefficient, m3/Pa; h is the effective thickness of reservoir, m.The pressure and flow rate are equal in the interface between the inner zone and the outer zone as described in Equations (15) and (16):
(15)ki1e−γd∗ψi−ψ1μ1∂ψ1∂rr=r1=ki2e−γd∗ψi−ψ2μ2∂ψ2∂rr=r1,(16)ψ1r=r1,t=ψ2r=r1,t,wherer1 is the inner zone radius, m.Considering the infinite outer boundary, Equation (17) is obtained:
(17)limr⟶∞ψ2r,t=ψi.Thus, the mathematical model of composite gas reservoir pressure drawdown well test considering the stress sensitivity effect can be expressed as Equation (18):
(18)1r∂ψ1∂r+∂2ψ1∂r2+γd∗∂ψ1∂r2=ϕ1μ1Cgki1eγd∗ψi−ψ1∂ψ1∂t,1r∂ψ2∂r+∂2ψ2∂r2+γd∗∂ψ2∂r2=ϕ2μ2Cgki2eγd∗ψi−ψ2∂ψ2∂t,re−γd∗ψi−ψ1∂ψ1∂rr=rwe=qscpscTπki1hTscZsc+Cμ12πki1h∂ψ1∂t,ki1e−γd∗ψi−ψ1μ1∂ψ1∂rr=r1=ki2e−γd∗ψi−ψ2μ2∂ψ2∂rr=r1,ψ1r,t=0=ψ2r,t=0=ψi,ψ1r=r1,t=ψ2r=r1,t,limr⟶∞ψ2r,t=ψi.In order to obtain the dimensionless mathematical model, some dimensionless variables are introduced as follows:(19)ψ1D=πki1hTscψi−ψ1qscpscT,ψ2D=πki1hTscψi−ψ2qscpscT,tD=ki1tϕ1μ1Cgrw2,ω=ϕ1Cg1ϕ2Cg2,rD=rrwe=rrwexp−S,r1D=r1rwe=r1rwexp−S,γdD∗=γdμiZi2piqscpscTπki1hTsc,γbD∗=γbμiZi2piqscpscTπki1hTsc,M=ki1/μ1ki2/μ2,CD=C2πhϕ1Cgrw2,whereψ1D is the dimensionless pseudo-pressure in the inner zone; ψ2D is the dimensionless pseudo-pressure in the outer zone; tD is the dimensionless time; ω is the storage ratio between the inner zone and the outer zone; rD is the dimensionless radius; r1D is the dimensionless inner zone radius; γ∗dD and γ∗bD are dimensionless stress sensitivity coefficients at pressure drawdown and buildup stages, respectively; M is the mobility ratio between the inner zone and the outer zone; CD is the dimensionless wellbore storage coefficient.Substituting dimensionless variables into Equation (18), the dimensionless mathematical model of the pressure drawdown well test considering the stress sensitivity effect can be expressed as Equation (20):
(20)1rD∂ψ1D∂rD+∂2ψ1D∂rD2−γdD∗∂ψ1D∂rD2=eγdD∗ψ1De2S∂ψ1D∂tD,1rD∂ψ2D∂rD+∂2ψ2D∂rD2−γdD∗∂ψ2D∂rD2=ωMeγdD∗ψ2De2S∂ψ2D∂tD,CD∂ψ1D∂tD−e−γdD∗ψ1D∂ψ1D∂rDrD=1=1,ψ1DrD,tD=0=ψ2DrD,tD=0=0,e−γdD∗ψ1D∂ψ1D∂rDrD=r1D=e−γdD∗ψ2D1M∂ψ2D∂rDrD=r1D,ψ1DrD=r1D,tD=ψ2DrD=r1D,tD,limrD⟶∞ψ2DrD,tD=0.


### 2.2.3. Pressure Buildup Well Test Model

For the pressure buildup well test, the gas productionqsc in the inner boundary condition Equation (14) is zero. The stress sensitivity coefficient of the pressure buildup stage is introduced into the mathematical model. Similarly, the dimensionless mathematical model of the pressure buildup well test considering the stress sensitivity effect and the hysteresis effect can be expressed as Equation (21):
(21)1rD∂ψ1D∂rD+∂2ψ1D∂rD2−γbD∗∂ψ1D∂rD2=R1eγbD∗ψ1De2S∂ψ1D∂tD,1rD∂ψ2D∂rD+∂2ψ2D∂rD2−γbD∗∂ψ2D∂rD2=R2ωMeγbD∗ψ2De2S∂ψ2D∂tD,R1CD∂ψ1D∂tD−e−γbD∗ψ1D∂ψ1D∂rDrD=1=0,ψ1DrD,tD=0=ψ2DrD,tD=0=0,e−γbD∗ψ1D∂ψ1D∂rDrD=r1D=e−γbD∗ψ2D1M∂ψ2D∂rDrD=r1D,ψ1DrD=r1D,tD=ψ2DrD=r1D,tD,limrD⟶∞ψ2DrD,tD=0,whereR1 is the ratio of the initial permeability and the maximum permeability that can be restored after the gas reservoir pressure increases to the initial value in the inner zone, dimensionless; R2 is the ratio of the initial permeability and the maximum permeability that can be restored after the gas reservoir pressure increases to the initial value in the outer zone, dimensionless.


## 2.2.1. Characterization of Stress Sensitivity Effect and Hysteresis Effect

The stress sensitivity coefficientγ is used to describe the change of reservoir permeability with the pressure. Some experimental studies indicate that the relationship between the reservoir permeability and the pressure change conforms to the exponential form [23, 35, 36]. The changes of reservoir permeability with the pressure at pressure drawdown and buildup stages are shown in Figure 2.Figure 2
The change of reservoir permeability with pressure.At the pressure drawdown stage, the stress sensitivity effect can be described by Equation (1):
(1)γd=1kiddkdp⇒kd=kidexp−γdpi−pd.At the pressure buildup stage, the stress sensitivity effect and the hysteresis effect can be described by Equation (2):
(2)γb=1kibdkdp⇒kb=kibexp−γbpi−pb=kidexp−γd−γbpi−pdexp−γbpi−pb,whereγd and γb are the stress sensitivity coefficients at pressure drawdown and buildup stage, 1/Pa; there will be the hysteresis effect when γb is less than γd; kid is the initial permeability of gas reservoir, m2; k is the permeability of gas reservoir, m2; p is the pressure of gas reservoir, Pa; pi is the initial pressure of gas reservoir, Pa; pd and pb are the gas reservoir pressure at pressure drawdown and buildup stages, Pa; kd and kb are the gas reservoir permeability at pressure drawdown and buildup stages, m2; kib is the maximum permeability value that can be restored after the gas reservoir pressure increases to the initial pressure, m2.The gas pseudo-pressure is generally used in gas reservoir well tests, and the definition of the gas pseudo-pressure is as Equation (3):
(3)ψ=2∫pscppμZdp,whereψ is the gas pseudo-pressure, Pa/s; μ is the gas viscosity, Pa.s; Z is the gas compressibility factor, decimal; psc is the pressure at the standard condition, Pa.Substituting Equation (3) into Equations (1) and (2), stress sensitivity equations in the pseudo-pressure form can be expressed as Equations (4) and (5):
(4)γd=1kiddkdp=1kiddkdψddψddp=1kiddkdψdpμZ,(5)γb=1kibdkdp=1kibdkdψbdψbdp=1kibdkdψbpμZ.Then, stress sensitivity coefficients in the pseudo-pressure form can be obtained as Equations (6) and (7):
(6)γd∗=γdp/μZ=1kiddkdψd⇒kd=kidexp−γd∗ψi−ψd,(7)γb∗=γbp/μZ=1kibdkdψb⇒kb=kibexp−γb∗ψi−ψb,whereγ∗d and γ∗b are stress sensitivity coefficients in the pseudo-pressure form at pressure drawdown and buildup stages, Pa/s; ψi is the initial pseudo-pressure, Pa/s; ψd and ψb are the pseudo-pressure at pressure drawdown and buildup stages, Pa/s.


## 2.2.2. Pressure Drawdown Well Test Model

The differential equation of natural gas seepage considering the stress sensitivity effect and the hysteresis effect is as Equation (8):
(8)1r∂∂rkprpZμ∂p∂r=ϕ∂∂tpZ,wherer is the radius, m; kp is the permeability considering the stress sensitivity effect and the hysteresis effect, m2; t is time, s; ϕ is porosity, decimal.For the pressure drawdown well test, the comprehensive seepage differential equation considering the stress sensitivity effect in the inner zone of composite gas reservoir is as Equation (9):
(9)1r∂∂rki1e−γdpi−p1rp1Zμ1∂p1∂r=ϕ1∂∂tp1Z,whereki1 is the initial inner zone permeability of composite gas reservoir, m2; p1 is the inner zone pressure of composite gas reservoir, Pa; μ1 is the inner zone gas viscosity of composite gas reservoir, Pa.s; ϕ1 is the inner zone porosity of composite gas reservoir, decimal.The comprehensive seepage differential equation considering the stress sensitivity effect in the outer zone of composite gas reservoir is as Equation (10):
(10)1r∂∂rki2e−γdpi−p2rp2Zμ2∂p2∂r=ϕ2∂∂tp2Z,whereki2 is the initial outer zone permeability of composite gas reservoir, m2; p2 is the outer zone pressure of composite gas reservoir, Pa; μ2 is the outer zone gas viscosity of composite gas reservoir, Pa.s; ϕ2 is the outer zone porosity of composite gas reservoir, decimal.After substituting Equation (3) into Equations (9) and (10), gas comprehensive seepage differential equations in the pseudo-pressure form can be expressed as Equations (11) and (12):
(11)1r∂ψ1∂r+∂2ψ1∂r2+γd∗∂ψ1∂r2=ϕ1μ1Cgki1eγd∗ψi−ψ1∂ψ1∂t,(12)1r∂ψ2∂r+∂2ψ2∂r2+γd∗∂ψ2∂r2=ϕ2μ2Cgki2eγd∗ψi−ψ2∂ψ2∂t,whereψ1 is the inner zone pseudo-pressure, Pa/s; ψ2 is the outer zone pseudo-pressure of composite gas reservoir, Pa/s; Cg is the gas compressibility coefficient, Pa-1.The initial condition can be expressed as Equation (13):
(13)ψ1r,t=0=ψ2r,t=0=ψi.The inner boundary considering the skin damage and the well storage effect for the pressure drawdown well test can be expressed as Equation (14):
(14)re−γd∗ψi−ψ1∂ψ1∂rr=rwe=qscpscTπki1hTscZsc+Cμ12πki1h∂ψ1∂t,whererwe=rwexp−S is the effective well radius, m; S is the skin factor, dimensionless; qsc is the production of gas well at the standard condition, m3/s; Tsc is the temperature at the standard condition, K; T is the formation temperature, K; C is the wellbore storage coefficient, m3/Pa; h is the effective thickness of reservoir, m.The pressure and flow rate are equal in the interface between the inner zone and the outer zone as described in Equations (15) and (16):
(15)ki1e−γd∗ψi−ψ1μ1∂ψ1∂rr=r1=ki2e−γd∗ψi−ψ2μ2∂ψ2∂rr=r1,(16)ψ1r=r1,t=ψ2r=r1,t,wherer1 is the inner zone radius, m.Considering the infinite outer boundary, Equation (17) is obtained:
(17)limr⟶∞ψ2r,t=ψi.Thus, the mathematical model of composite gas reservoir pressure drawdown well test considering the stress sensitivity effect can be expressed as Equation (18):
(18)1r∂ψ1∂r+∂2ψ1∂r2+γd∗∂ψ1∂r2=ϕ1μ1Cgki1eγd∗ψi−ψ1∂ψ1∂t,1r∂ψ2∂r+∂2ψ2∂r2+γd∗∂ψ2∂r2=ϕ2μ2Cgki2eγd∗ψi−ψ2∂ψ2∂t,re−γd∗ψi−ψ1∂ψ1∂rr=rwe=qscpscTπki1hTscZsc+Cμ12πki1h∂ψ1∂t,ki1e−γd∗ψi−ψ1μ1∂ψ1∂rr=r1=ki2e−γd∗ψi−ψ2μ2∂ψ2∂rr=r1,ψ1r,t=0=ψ2r,t=0=ψi,ψ1r=r1,t=ψ2r=r1,t,limr⟶∞ψ2r,t=ψi.In order to obtain the dimensionless mathematical model, some dimensionless variables are introduced as follows:(19)ψ1D=πki1hTscψi−ψ1qscpscT,ψ2D=πki1hTscψi−ψ2qscpscT,tD=ki1tϕ1μ1Cgrw2,ω=ϕ1Cg1ϕ2Cg2,rD=rrwe=rrwexp−S,r1D=r1rwe=r1rwexp−S,γdD∗=γdμiZi2piqscpscTπki1hTsc,γbD∗=γbμiZi2piqscpscTπki1hTsc,M=ki1/μ1ki2/μ2,CD=C2πhϕ1Cgrw2,whereψ1D is the dimensionless pseudo-pressure in the inner zone; ψ2D is the dimensionless pseudo-pressure in the outer zone; tD is the dimensionless time; ω is the storage ratio between the inner zone and the outer zone; rD is the dimensionless radius; r1D is the dimensionless inner zone radius; γ∗dD and γ∗bD are dimensionless stress sensitivity coefficients at pressure drawdown and buildup stages, respectively; M is the mobility ratio between the inner zone and the outer zone; CD is the dimensionless wellbore storage coefficient.Substituting dimensionless variables into Equation (18), the dimensionless mathematical model of the pressure drawdown well test considering the stress sensitivity effect can be expressed as Equation (20):
(20)1rD∂ψ1D∂rD+∂2ψ1D∂rD2−γdD∗∂ψ1D∂rD2=eγdD∗ψ1De2S∂ψ1D∂tD,1rD∂ψ2D∂rD+∂2ψ2D∂rD2−γdD∗∂ψ2D∂rD2=ωMeγdD∗ψ2De2S∂ψ2D∂tD,CD∂ψ1D∂tD−e−γdD∗ψ1D∂ψ1D∂rDrD=1=1,ψ1DrD,tD=0=ψ2DrD,tD=0=0,e−γdD∗ψ1D∂ψ1D∂rDrD=r1D=e−γdD∗ψ2D1M∂ψ2D∂rDrD=r1D,ψ1DrD=r1D,tD=ψ2DrD=r1D,tD,limrD⟶∞ψ2DrD,tD=0.


## 2.2.3. Pressure Buildup Well Test Model

For the pressure buildup well test, the gas productionqsc in the inner boundary condition Equation (14) is zero. The stress sensitivity coefficient of the pressure buildup stage is introduced into the mathematical model. Similarly, the dimensionless mathematical model of the pressure buildup well test considering the stress sensitivity effect and the hysteresis effect can be expressed as Equation (21):
(21)1rD∂ψ1D∂rD+∂2ψ1D∂rD2−γbD∗∂ψ1D∂rD2=R1eγbD∗ψ1De2S∂ψ1D∂tD,1rD∂ψ2D∂rD+∂2ψ2D∂rD2−γbD∗∂ψ2D∂rD2=R2ωMeγbD∗ψ2De2S∂ψ2D∂tD,R1CD∂ψ1D∂tD−e−γbD∗ψ1D∂ψ1D∂rDrD=1=0,ψ1DrD,tD=0=ψ2DrD,tD=0=0,e−γbD∗ψ1D∂ψ1D∂rDrD=r1D=e−γbD∗ψ2D1M∂ψ2D∂rDrD=r1D,ψ1DrD=r1D,tD=ψ2DrD=r1D,tD,limrD⟶∞ψ2DrD,tD=0,whereR1 is the ratio of the initial permeability and the maximum permeability that can be restored after the gas reservoir pressure increases to the initial value in the inner zone, dimensionless; R2 is the ratio of the initial permeability and the maximum permeability that can be restored after the gas reservoir pressure increases to the initial value in the outer zone, dimensionless.


## 2.3. Solutions to Mathematical Models

### 2.3.1. Solution to Pressure Drawdown Well Test Model by Perturbation Method

Kikani and Pedrosa proposed the perturbation method to solve the mathematical model for the pressure drawdown well test considering the stress sensitivity effect [23]. First, the dimensionless pseudo-pressure is transformed as Equation (22):
(22)ψ1D=−ln1−γdD∗ξ1DγdD∗,ψ2D=−ln1−γdD∗ξ2DγdD∗.Substituting Equation (22) into Equation (20), the dimensionless pressure drawdown well test mathematical model can be transformed as Equation (23):
(23)1rD∂ξ1D∂rD+∂2ξ1D∂rD2=e−2S1−γdD∗ξ1D∂ξ1D∂tD,1rD∂ξ2D∂rD+∂2ξ2D∂rD2=ωMe−2S1−γdD∗ξ2D∂ξ2D∂tD,CD∂ξ1D∂tD−rD∂ξ1D∂rDrD=1=11−γdD∗ξ1D,ξ1DrD=r1D,tD=ξ2DrD=r1D,tD,∂ξ1D∂rDrD=r1D=1M∂ξ2D∂rDrD=r1D,ξ1DrD,tD=0=ξ2DrD,tD=0=0,limrD⟶∞ξ2DrD,tD=0.After applying zero-order approximation toξ1D and ξ2D in Equation (23) [23], the new mathematical model can be expressed as Equation (24):
(24)1rD∂ξ1D∂rD+∂2ξ1D∂rD2=e−2S∂ξ1D∂tD,1rD∂ξ2D∂rD+∂2ξ2D∂rD2=ωMe−2S∂ξ2D∂tD,CD∂ξ1D∂tD−rD∂ξ1D∂rDrD=1=1,ξ1DrD=r1D,tD=ξ2DrD=r1D,tD,∂ξ1D∂rDrD=r1D=1M∂ξ2D∂rDrD=r1D,ξ1DrD,tD=0=ξ2DrD,tD=0=0,limrD⟶∞ξ2DrD,tD=0.Equation (24) is the same as the well test model of composite gas reservoir without considering the stress sensitivity effect [37]. Based on the Laplace transform and the property of Bessel function, the solution to Equation (24) in the Laplace space can be expressed as Equation (25):
(25)ξ¯wD=1uRTPuCDRTP+βMTP,whereu is the variable in the Laplace space,
(26)RTP=1+I0βK0ββK0r1DNβK1r1Dβ−N/MβK0r1DβK1r1DNβN/MβK1r1DNβI0r1Dβ+βK0r1DNβI1r1Dβ,MTP=K1βK0β−I1βK0ββK0r1DNβK1r1Dβ−N/MβK0r1DβK1r1DNβN/MK1r1DNβI0r1Dβ+βK0r1DNβI1r1Dβ,β=uexp−S,N=ωM.Based on the numerical inversion method and the perturbation method [23], the solution to the pressure drawdown well test model considering the stress sensitivity effect for the composite gas reservoir can be expressed as Equation (27):
(27)ψwD=−ln1−γdD∗ξwDγdD∗.For the conventional well test without considering the stress sensitivity effect, the solution to the pressure buildup well test mathematical model, which is a linear equation, can be obtained by the superposition principle based on the solution to the pressure drawdown well test model. However, the superposition principle cannot be applied to the well test model considering the stress sensitivity effect, which is a nonlinear equation. Thus, it is difficult to get the solution to the pressure buildup well test model considering the stress sensitivity effect by the superposition principle.


### 2.3.2. Solutions to Pressure Drawdown and Buildup Well Test Models by Numerical Method

The well test models considering the stress sensitivity effect and the hysteresis effect as Equations (20) and (21) are second-order nonlinear equations. The analytical solution to the pressure drawdown well test model obtained by the perturbation method is an approximate result. Thus, the numerical differentiation method is used to solve these equations.Considering dramatic pressure changes near the wellbore, the unequal-space grid division is used. TakingxD=lnrD, Equation (28) is obtained:
(28)∂ψD∂rD=e−xD∂ψD∂xD,∂2ψD∂rD2=e−2xD∂2ψD∂xD2−e−2xD∂ψD∂xD.Substituting Equation (28) into the dimensionless pressure drawdown well test model Equation (20), the model can be written as Equation (29):
(29)∂2ψ1D∂xD2−γdD∗∂ψ1D∂xD2=e2xD+γdD∗ψ1De2S∂ψ1D∂tD,∂2ψ2D∂xD2−γdD∗∂ψ2D∂xD2=ωMe2xD+γdD∗ψ2De2S∂ψ2D∂tD,CD∂ψ1D∂tD−e−γdD∗ψ1D∂ψ1D∂xDxD=0=1,ψ1DxD=lnr1D,tD=ψ2DxD=lnr1D,tD,e−γdD∗ψ1D∂ψ1D∂xDxD=lnr1D=e−γdD∗ψ2D1M∂ψ2D∂xDxD=lnr1D,ψ1DxD,tD=0=ψ2DxD,tD=0=0,limxD⟶∞ψ2DxD,tD=0.The difference in space is as Equation (30):
(30)xDi=i−1Δx1Di=1,2,3,.⋯,N,lnr1D+i−NΔx2Di=N+1,N+2,N+3,.⋯,NN,whereΔx1D=lnr1D/N−1; Δx2D=lnRD/r1D/NN−N; N is the discrete number in space of the inner zone, NN is the total discrete number in space of inner and outer zones, and RD is the dimensionless outer zone boundary radius.The logarithmic interval is taken for time as Equation (31):
(31)tDj=10−6+j−1Δt,Δt=logTD+6O−1,wherej=1,2,3,⋯O,O is the discrete number of time and TD is the dimensionless time at the end of the simulation.For the dimensionless well test mathematical model Equation (29), the finite difference discretization is carried out in space and time as Equations (30) and (31). Then, the seepage equation discretization result for the composite gas reservoir can be expressed as Equation (32):
(32)aiψDi−1j+biψDij+ciψDi+1j+diψDij2+eiψDijψDi+1j+fiψDi+1j2+gierdD∗ψDij+hiψDijerdD∗ψDij=0.Wheni=1,2,⋯,N, then
(33)ai=1Δx1D2,bi=−2Δx1D2,ci=1Δx1D2,di=−γdD∗Δx1D2,ei=2γdD∗Δx1D2,fi=−γdD∗Δx1D2,gi=e2i−1Δx1De2StDj−tDj−1ψDij−1,hi=−e2i−1Δx1De2StDj−tDj−1.Wheni=N+1,N+2,⋯.,NN, then
(34)ai=1Δx2D2,bi=−2Δx2D2,ci=1Δx2D2,di=−γdD∗Δx2D2,ei=2γdD∗Δx2D2,fi=−γdD∗Δx2D2,gi=ωMe2i−1Δx2De2StDj−tDj−1ψDij−1,hi=−ωMe2i−1Δx2De2StDj−tDj−1.The discretization result of the initial condition can be expressed as Equation (35):
(35)ψDi1=0,i=1,2,⋯,NN.The discretization result of the inner boundary condition can be expressed as Equation (36):
(36)b1ψD1j+c1ψD2j+g1eγdD∗ψDij+h1ψD1jeγdD∗ψDij=0,where(37)b1=−1Δx1D,.c1=1Δx1D,g1=1+CDψD1j−1tDj−tDj−1,h1=−CDtDj−tDj−1.The discretization result of the interface condition can be expressed as Equation (38):
(38)ψDN−ψDN−1Δx1=1MψDN+1−ψDNΔx2.The discretization result of the outer boundary condition can be expressed as Equation (39):
(39)ψDNNj=0.As for the pressure buildup test, the model is obtained after modifying the internal boundary condition and introducing the stress sensitivity coefficient of the buildup stage. Then, the seepage equation discretization result for the composite gas reservoir at the buildup stage can be expressed as Equation (40):
(40)aiψDi−1j+biψDij+ciψDi+1j+diψDij2+eiψDijψDi+1j+fiψDi+1j2+gierbD∗ψDij+hiψDijerbD∗ψDij=0.Wheni=1,2,⋯,N, then
(41)ai=1Δx1D2,bi=−2Δx1D2,ci=1Δx1D2,di=−γbD∗Δx1D2,ei=2γbD∗Δx1D2,fi=−γbD∗Δx1D2,gi=R1e2i−1Δx1De2StDj−tDj−1ψDij−1,hi=−R1e2i−1Δx1De2StDj−tDj−1.Wheni=N+1,N+2,⋯.,NN, then
(42)ai=1Δx2D2,bi=−2Δx2D2,ci=1Δx2D2,di=−γbD∗Δx2D2,ei=2γbD∗Δx2D2,fi=−γbD∗Δx2D2,gi=R2ωMe2i−1Δx2De2StDj−tDj−1ψDij−1,hi=−R2ωMe2i−1Δx2De2StDj−tDj−1.The discretization result of the inner boundary condition at the buildup stage can be expressed as Equation (43):
(43)b1ψD1j+c1ψD2j+g1eγbD∗ψDij+h1ψD1jeγbD∗ψDij=0,where(44)b1=−1Δx1D,c1=1Δx1D,g1=R1CDψD1j−1tDj−tDj−1,h1=−R1CDtDj−tDj−1.The discretization results of the interface condition and the outer zone boundary condition for the pressure buildup well test are the same as Equations (38) and (39).Numerical well test models of drawdown and buildup stages for the composite gas reservoir considering the stress sensitivity effect and the hysteresis effect are established by Equations (32)–(39) and Equations (40)–(43). The Newton-Simpson method is used to solve the nonlinear equation to obtain pressure drawdown and buildup well test solutions.


## 2.3.1. Solution to Pressure Drawdown Well Test Model by Perturbation Method

Kikani and Pedrosa proposed the perturbation method to solve the mathematical model for the pressure drawdown well test considering the stress sensitivity effect [23]. First, the dimensionless pseudo-pressure is transformed as Equation (22):
(22)ψ1D=−ln1−γdD∗ξ1DγdD∗,ψ2D=−ln1−γdD∗ξ2DγdD∗.Substituting Equation (22) into Equation (20), the dimensionless pressure drawdown well test mathematical model can be transformed as Equation (23):
(23)1rD∂ξ1D∂rD+∂2ξ1D∂rD2=e−2S1−γdD∗ξ1D∂ξ1D∂tD,1rD∂ξ2D∂rD+∂2ξ2D∂rD2=ωMe−2S1−γdD∗ξ2D∂ξ2D∂tD,CD∂ξ1D∂tD−rD∂ξ1D∂rDrD=1=11−γdD∗ξ1D,ξ1DrD=r1D,tD=ξ2DrD=r1D,tD,∂ξ1D∂rDrD=r1D=1M∂ξ2D∂rDrD=r1D,ξ1DrD,tD=0=ξ2DrD,tD=0=0,limrD⟶∞ξ2DrD,tD=0.After applying zero-order approximation toξ1D and ξ2D in Equation (23) [23], the new mathematical model can be expressed as Equation (24):
(24)1rD∂ξ1D∂rD+∂2ξ1D∂rD2=e−2S∂ξ1D∂tD,1rD∂ξ2D∂rD+∂2ξ2D∂rD2=ωMe−2S∂ξ2D∂tD,CD∂ξ1D∂tD−rD∂ξ1D∂rDrD=1=1,ξ1DrD=r1D,tD=ξ2DrD=r1D,tD,∂ξ1D∂rDrD=r1D=1M∂ξ2D∂rDrD=r1D,ξ1DrD,tD=0=ξ2DrD,tD=0=0,limrD⟶∞ξ2DrD,tD=0.Equation (24) is the same as the well test model of composite gas reservoir without considering the stress sensitivity effect [37]. Based on the Laplace transform and the property of Bessel function, the solution to Equation (24) in the Laplace space can be expressed as Equation (25):
(25)ξ¯wD=1uRTPuCDRTP+βMTP,whereu is the variable in the Laplace space,
(26)RTP=1+I0βK0ββK0r1DNβK1r1Dβ−N/MβK0r1DβK1r1DNβN/MβK1r1DNβI0r1Dβ+βK0r1DNβI1r1Dβ,MTP=K1βK0β−I1βK0ββK0r1DNβK1r1Dβ−N/MβK0r1DβK1r1DNβN/MK1r1DNβI0r1Dβ+βK0r1DNβI1r1Dβ,β=uexp−S,N=ωM.Based on the numerical inversion method and the perturbation method [23], the solution to the pressure drawdown well test model considering the stress sensitivity effect for the composite gas reservoir can be expressed as Equation (27):
(27)ψwD=−ln1−γdD∗ξwDγdD∗.For the conventional well test without considering the stress sensitivity effect, the solution to the pressure buildup well test mathematical model, which is a linear equation, can be obtained by the superposition principle based on the solution to the pressure drawdown well test model. However, the superposition principle cannot be applied to the well test model considering the stress sensitivity effect, which is a nonlinear equation. Thus, it is difficult to get the solution to the pressure buildup well test model considering the stress sensitivity effect by the superposition principle.


## 2.3.2. Solutions to Pressure Drawdown and Buildup Well Test Models by Numerical Method

The well test models considering the stress sensitivity effect and the hysteresis effect as Equations (20) and (21) are second-order nonlinear equations. The analytical solution to the pressure drawdown well test model obtained by the perturbation method is an approximate result. Thus, the numerical differentiation method is used to solve these equations.Considering dramatic pressure changes near the wellbore, the unequal-space grid division is used. TakingxD=lnrD, Equation (28) is obtained:
(28)∂ψD∂rD=e−xD∂ψD∂xD,∂2ψD∂rD2=e−2xD∂2ψD∂xD2−e−2xD∂ψD∂xD.Substituting Equation (28) into the dimensionless pressure drawdown well test model Equation (20), the model can be written as Equation (29):
(29)∂2ψ1D∂xD2−γdD∗∂ψ1D∂xD2=e2xD+γdD∗ψ1De2S∂ψ1D∂tD,∂2ψ2D∂xD2−γdD∗∂ψ2D∂xD2=ωMe2xD+γdD∗ψ2De2S∂ψ2D∂tD,CD∂ψ1D∂tD−e−γdD∗ψ1D∂ψ1D∂xDxD=0=1,ψ1DxD=lnr1D,tD=ψ2DxD=lnr1D,tD,e−γdD∗ψ1D∂ψ1D∂xDxD=lnr1D=e−γdD∗ψ2D1M∂ψ2D∂xDxD=lnr1D,ψ1DxD,tD=0=ψ2DxD,tD=0=0,limxD⟶∞ψ2DxD,tD=0.The difference in space is as Equation (30):
(30)xDi=i−1Δx1Di=1,2,3,.⋯,N,lnr1D+i−NΔx2Di=N+1,N+2,N+3,.⋯,NN,whereΔx1D=lnr1D/N−1; Δx2D=lnRD/r1D/NN−N; N is the discrete number in space of the inner zone, NN is the total discrete number in space of inner and outer zones, and RD is the dimensionless outer zone boundary radius.The logarithmic interval is taken for time as Equation (31):
(31)tDj=10−6+j−1Δt,Δt=logTD+6O−1,wherej=1,2,3,⋯O,O is the discrete number of time and TD is the dimensionless time at the end of the simulation.For the dimensionless well test mathematical model Equation (29), the finite difference discretization is carried out in space and time as Equations (30) and (31). Then, the seepage equation discretization result for the composite gas reservoir can be expressed as Equation (32):
(32)aiψDi−1j+biψDij+ciψDi+1j+diψDij2+eiψDijψDi+1j+fiψDi+1j2+gierdD∗ψDij+hiψDijerdD∗ψDij=0.Wheni=1,2,⋯,N, then
(33)ai=1Δx1D2,bi=−2Δx1D2,ci=1Δx1D2,di=−γdD∗Δx1D2,ei=2γdD∗Δx1D2,fi=−γdD∗Δx1D2,gi=e2i−1Δx1De2StDj−tDj−1ψDij−1,hi=−e2i−1Δx1De2StDj−tDj−1.Wheni=N+1,N+2,⋯.,NN, then
(34)ai=1Δx2D2,bi=−2Δx2D2,ci=1Δx2D2,di=−γdD∗Δx2D2,ei=2γdD∗Δx2D2,fi=−γdD∗Δx2D2,gi=ωMe2i−1Δx2De2StDj−tDj−1ψDij−1,hi=−ωMe2i−1Δx2De2StDj−tDj−1.The discretization result of the initial condition can be expressed as Equation (35):
(35)ψDi1=0,i=1,2,⋯,NN.The discretization result of the inner boundary condition can be expressed as Equation (36):
(36)b1ψD1j+c1ψD2j+g1eγdD∗ψDij+h1ψD1jeγdD∗ψDij=0,where(37)b1=−1Δx1D,.c1=1Δx1D,g1=1+CDψD1j−1tDj−tDj−1,h1=−CDtDj−tDj−1.The discretization result of the interface condition can be expressed as Equation (38):
(38)ψDN−ψDN−1Δx1=1MψDN+1−ψDNΔx2.The discretization result of the outer boundary condition can be expressed as Equation (39):
(39)ψDNNj=0.As for the pressure buildup test, the model is obtained after modifying the internal boundary condition and introducing the stress sensitivity coefficient of the buildup stage. Then, the seepage equation discretization result for the composite gas reservoir at the buildup stage can be expressed as Equation (40):
(40)aiψDi−1j+biψDij+ciψDi+1j+diψDij2+eiψDijψDi+1j+fiψDi+1j2+gierbD∗ψDij+hiψDijerbD∗ψDij=0.Wheni=1,2,⋯,N, then
(41)ai=1Δx1D2,bi=−2Δx1D2,ci=1Δx1D2,di=−γbD∗Δx1D2,ei=2γbD∗Δx1D2,fi=−γbD∗Δx1D2,gi=R1e2i−1Δx1De2StDj−tDj−1ψDij−1,hi=−R1e2i−1Δx1De2StDj−tDj−1.Wheni=N+1,N+2,⋯.,NN, then
(42)ai=1Δx2D2,bi=−2Δx2D2,ci=1Δx2D2,di=−γbD∗Δx2D2,ei=2γbD∗Δx2D2,fi=−γbD∗Δx2D2,gi=R2ωMe2i−1Δx2De2StDj−tDj−1ψDij−1,hi=−R2ωMe2i−1Δx2De2StDj−tDj−1.The discretization result of the inner boundary condition at the buildup stage can be expressed as Equation (43):
(43)b1ψD1j+c1ψD2j+g1eγbD∗ψDij+h1ψD1jeγbD∗ψDij=0,where(44)b1=−1Δx1D,c1=1Δx1D,g1=R1CDψD1j−1tDj−tDj−1,h1=−R1CDtDj−tDj−1.The discretization results of the interface condition and the outer zone boundary condition for the pressure buildup well test are the same as Equations (38) and (39).Numerical well test models of drawdown and buildup stages for the composite gas reservoir considering the stress sensitivity effect and the hysteresis effect are established by Equations (32)–(39) and Equations (40)–(43). The Newton-Simpson method is used to solve the nonlinear equation to obtain pressure drawdown and buildup well test solutions.


## 3. Type Curve of Composite Gas Reservoir Well Test

### 3.1. Model Verification

#### 3.1.1. Verification of Pressure Drawdown Well Test Model

When dimensionless stress sensitivity coefficientsγdD∗ and γbD∗ are taken as zero, the model is the same as the conventional pressure drawdown well test model of composite gas reservoir [37]. In order to verify the model in the paper, the type curve obtained by the numerical method without considering the stress sensitivity effect is compared with the conventional composite gas reservoir well test curve obtained by the analytical solution shown in Figure 3.Figure 3
Comparison between the numerical solution and the analytical solution to the pressure drawdown well test model without considering the stress sensitivity effect.As shown in Figure3, the type curves of pressure drawdown well test obtained by the numerical solution and the analytical solution are the same when dimensionless stress sensitivity coefficients are set to zero. The curves show common characteristics. The pressure derivative curve is a 0.5 horizontal line at the inner zone radial flow stage, while it is a M/2 (mobility ratio/2) horizontal line at the outer zone radial flow stage. Thus, the numerical solution in the paper is reliable.In order to further verify the model, the type curves of pressure drawdown well test considering the stress sensitivity effect obtained by the numerical method and the perturbation method [23] are compared shown in Figure 4.Figure 4
Comparison between the numerical solution and the perturbation solution to the pressure drawdown well test considering the stress sensitivity effect.It can be seen from Figure4 that the numerical solution is consistent with the solution obtained by the perturbation method, which verifies the reliability of the numerical solution.


#### 3.1.2. Verification of Pressure Buildup Well Test Model

In order to verify the pressure buildup well test model of composite gas reservoir in the paper, the model is simplified into a homogeneous gas reservoir model considering the stress sensitivity effect by settingM=1, ω=1, and γ∗bD=γ∗dD. With those parameter settings, the model does not reflect the influence of the hysteresis effect and the differences of permeability and porosity between the inner zone and the outer zone. Then, the result of the model is compared with the research of Zhang and He [31] shown in Figure 5.Figure 5
Comparison of the homogeneous gas reservoir pressure buildup well test curve obtained from the paper and the research of Zhang and He.It can be seen from Figure5 that the well test curve obtained from the simplified model in the paper is consistent with the curve from the research of Zhang and He [31]. Thus, the numerical solution to the pressure buildup well test model in the paper is reliable.


### 3.2. Type Curve Analysis

Figures6 and 7 are type curves for pressure drawdown and buildup well tests in a composite gas reservoir. The corresponding parameter values are CD=1, S=1, M=0.5, ω=1, r1D=500, γdD∗=0 or 0.04, and γbD∗=0 or 0.04. And when γdD∗=γbD∗=0, the well test model is suitable for the conventional composite gas reservoir; when γdD∗=γbD∗=0.04, the model is suitable for the composite gas reservoir considering the stress sensitivity effect.Figure 6
Well test curves for the pressure drawdown well test in composite gas reservoir.Figure 7
Well test curves for the pressure buildup well test in composite gas reservoir.It can be seen from Figures6 and 7 that type curves of the composite gas reservoir well test considering the stress sensitivity effect can be divided into five basic flow regimes including ① the wellbore storage stage, ② the transition stage dominated by the skin factor, ③ the inner zone radial flow stage, ④ the transition stage between the inner zone and outer zone radial flow stages, and ⑤ the outer zone radial flow stage. The characteristics of the wellbore storage stage are not affected by the stress sensitivity effect. The transition stage and the radial flow stage are significantly affected by the stress sensitivity effect. For the pressure drawdown well test shown in Figure 6, the pseudo-pressure curve has a significant uplift compared with the conventional well test curve indicating a larger pressure drawdown at the same dimensionless time tD after considering the stress sensitivity effect; the pseudo-pressure derivative curve at the inner zone radial flow stage gradually deviates upward from the 0.5 horizontal line indicating the permeability decreases with the increase of the pressure drawdown at the inner zone radial flow stage; the pseudo-pressure derivative curve at the outer zone radial flow stage gradually deviates upward from the M/2 (mobility ratio/2) horizontal line indicating the permeability decreases with the increase of the pressure drawdown at the outer zone radial flow stage.For the pressure buildup well test shown in Figure7, the pseudo-pressure curve has a significant uplift compared with the conventional well test curve after considering the stress sensitivity effect; the pseudo-pressure derivative curve at the inner zone radial flow stage gradually descends towards the 0.5 horizontal line indicating the permeability increases with the increase of the formation pressure during the buildup stage, while the pseudo-pressure derivative curve at the outer zone radial flow stage gradually descends towards the M/2 horizontal line. The pseudo-pressure derivative curve is slightly higher than the M/2 horizontal line at the outer zone radial flow stage indicating that the formation pressure is close to the initial pressure.


### 3.3. Parameter Sensitivity Analysis

#### 3.3.1. The Effect of Stress Sensitivity Coefficient

The type curves of pressure drawdown and buildup well tests considering the stress sensitivity effect under different stress sensitivity coefficients are shown in Figures8 and 9. The corresponding parameter values are CD=1, S=1, r1D=500, M=0.5, ω=1, and γdD∗=γbD∗=0, 0.02, 0.04, and 0.06.Figure 8
Well test curves of the pressure drawdown well test under different stress sensitivity coefficients.Figure 9
Well test curves of the pressure buildup well test under different stress sensitivity coefficients.As shown in Figure8, for the pressure drawdown well test, as stress sensitivity coefficients γdD∗ and γbD∗ increase, the pseudo-pressure curve gradually rises, and the pseudo-pressure derivative curve deviates upward from the 0.5 horizontal line at the inner zone radial flow stage, while it deviates upward from the M/2 (mobility ratio/2) horizontal line at the outer zone radial flow stage. The changes indicate that the pressure drawdown becomes larger, and the permeability becomes lower at the same dimensionless time tD as stress sensitivity coefficients increase.For the pressure buildup well test shown in Figure9, as stress sensitivity coefficients γdD∗ and γbD∗ increase, the pseudo-pressure and pseudo-pressure derivative curves gradually rise, which means that the speed of formation pressure buildup is quicker as stress sensitivity coefficients γdD∗ and γbD∗ increase. The pseudo-pressure derivative curve at the inner zone radial flow stage gradually descends towards the 0.5 horizontal line, while the pseudo-pressure derivative curve at the outer zone radial flow stage gradually descends towards the M/2 (mobility ratio/2) horizontal line in all cases.


#### 3.3.2. The Effect of Hysteresis Effect

The type curves of pressure drawdown and buildup well tests considering the hysteresis effect are shown in Figures10 and 11. The corresponding parameter values are CD=1, S=1, r1D=500, M=0.5, ω=1, γdD∗=0.1, and γbD∗=0.1, 0.05, and 0.01.Figure 10
Well test curves of the pressure drawdown well test considering the hysteresis effect.Figure 11
Well test curves of the pressure buildup well test considering the hysteresis effect.As shown in Figure10, for the pressure drawdown well test with one production stage followed by one shut-in stage, the stress hysteresis effect has no effect on the curve of pressure drawdown well test due to the same stress sensitivity coefficient γdD∗ at the pressure drawdown stage.For the pressure buildup well test curve shown in Figure11, as the stress sensitivity coefficient γbD∗ of buildup stage decreases, and the pseudo-pressure derivative curve gradually rises at both inner zone and outer zone radial flow stages. Moreover, the transition stage between inner zone and outer zone radial flow stages appears later. Because the lower stress sensitivity coefficient of buildup stage γbD∗ is, the slower of permeability recovery speed is. The final permeability recovery value is also smaller at the pressure buildup stage.


#### 3.3.3. The Effect of Skin Factor

The type curves of pressure drawdown and buildup well tests considering the stress sensitivity effect under different skin factors are shown in Figures12 and 13. The corresponding parameter values are CD=1, γdD∗=γbD∗=0.04, M=0.5, ω=1, r1D=500, and S=−1, 1, 3, and 5.Figure 12
Well test curves of the pressure drawdown well test under different skin factors.Figure 13
Well test curves of the pressure buildup well test under different skin factors.As shown in Figures12 and 13, the pseudo-pressure curves of pressure drawdown and buildup well tests both have an uplift as the skin factor S increases indicating a larger pressure drawdown or buildup at the same dimensionless time tD. At the inner zone or outer zone radial flow stages, the pseudo-pressure derivative curves of pressure drawdown well test obviously deviate upward as the skin factor increases, but the pseudo-pressure derivative curves at the radial flow stage of pressure buildup well test are not affected by the skin factor. This feature is different from that of the conventional composite gas reservoir well test curve. For conventional composite gas reservoir well test type curves under different skin factors, the pseudo-pressure derivative curves for both pressure drawdown and buildup well tests are 0.5 horizontal lines at inner zone radial flow stages. Those are M/2 horizontal lines at the outer zone radial flow stages.


#### 3.3.4. The Effect of Inner Radius

The type curves of pressure drawdown and buildup well tests considering the stress sensitivity effect under different inner radii are shown in Figures14 and 15. The corresponding parameter values are CD=1, S=1, γdD∗=γbD∗=0.04, M=0.5, ω=1, and r1D=200, 300, 400, and 500.Figure 14
Well test curves of the pressure drawdown well test under different inner zone radii.Figure 15
Well test curves of the pressure buildup well test under different inner zone radii.As shown in Figures14 and 15, the radius of inner zone mainly affects the duration of inner zone radial flow. The larger inner zone radius leads to a longer duration of inner zone radial flow and a later appearance of the transition stage between inner zone and outer zone radial flow stages.


#### 3.3.5. The Effect of Mobility Ratio

The type curves of pressure drawdown and buildup well tests considering the stress sensitivity effect under different mobility ratios are shown in Figures16 and 17. The corresponding parameter values are CD=1, S=1, γdD∗=γbD∗=0.04, ω=1, r1D=500, and M=0.2, 0.6, 1.0, 1.5, and 2.Figure 16
Well test curves of the pressure drawdown well test under different mobility ratios.Figure 17
Well test curves of the pressure buildup well test under different mobility ratios.As shown in Figures16 and 17, for the pressure drawdown well test, the mobility ratio mainly affects characteristics of the transition stage and the outer zone radial flow stage. A larger mobility ratio leads to a higher pseudo-pressure and pseudo-pressure derivative curve at the outer zone radial flow stage indicating a larger pressure drawdown and a lower permeability in the outer zone at the same dimensionless time. For the pressure buildup well test, the mobility ratio has an effect on both inner zone and outer zone radial flow stages. A larger mobility ratio leads to a higher pseudo-pressure and pseudo-pressure derivative curve in both two flow regimes. The feature is different from the pressure buildup well test curve of conventional composite gas reservoir. The mobility ratio has no effect on the pseudo-pressure derivative of both pressure drawdown and buildup well tests at the inner zone radial flow stage for the conventional well test curve.


#### 3.3.6. The Effect of Storage Ratio

The type curves of pressure drawdown and buildup well tests considering the stress sensitivity effect under different storage ratios are shown in Figures18 and 19. The corresponding parameter values are CD=1, S=1, γdD∗=γbD∗=0.04, r1D=500, M=1, and ω=0.1, 0.5, 1, 5, and 10.Figure 18
Well test curves of the pressure drawdown well test under different storage ratios.Figure 19
Well test curves of the pressure buildup test under different storage ratios.As shown in Figures18 and 19, when the storage ratio ω is greater than 1, the curve shows a small hump at the transition stage between inner zone and outer zone radial flow stages indicating that the inner zone storage capacity is greater than that of the outer zone. When the storage ratio ω is less than 1, the curve shows a small concave.


## 3.1. Model Verification

### 3.1.1. Verification of Pressure Drawdown Well Test Model

When dimensionless stress sensitivity coefficientsγdD∗ and γbD∗ are taken as zero, the model is the same as the conventional pressure drawdown well test model of composite gas reservoir [37]. In order to verify the model in the paper, the type curve obtained by the numerical method without considering the stress sensitivity effect is compared with the conventional composite gas reservoir well test curve obtained by the analytical solution shown in Figure 3.Figure 3
Comparison between the numerical solution and the analytical solution to the pressure drawdown well test model without considering the stress sensitivity effect.As shown in Figure3, the type curves of pressure drawdown well test obtained by the numerical solution and the analytical solution are the same when dimensionless stress sensitivity coefficients are set to zero. The curves show common characteristics. The pressure derivative curve is a 0.5 horizontal line at the inner zone radial flow stage, while it is a M/2 (mobility ratio/2) horizontal line at the outer zone radial flow stage. Thus, the numerical solution in the paper is reliable.In order to further verify the model, the type curves of pressure drawdown well test considering the stress sensitivity effect obtained by the numerical method and the perturbation method [23] are compared shown in Figure 4.Figure 4
Comparison between the numerical solution and the perturbation solution to the pressure drawdown well test considering the stress sensitivity effect.It can be seen from Figure4 that the numerical solution is consistent with the solution obtained by the perturbation method, which verifies the reliability of the numerical solution.


### 3.1.2. Verification of Pressure Buildup Well Test Model

In order to verify the pressure buildup well test model of composite gas reservoir in the paper, the model is simplified into a homogeneous gas reservoir model considering the stress sensitivity effect by settingM=1, ω=1, and γ∗bD=γ∗dD. With those parameter settings, the model does not reflect the influence of the hysteresis effect and the differences of permeability and porosity between the inner zone and the outer zone. Then, the result of the model is compared with the research of Zhang and He [31] shown in Figure 5.Figure 5
Comparison of the homogeneous gas reservoir pressure buildup well test curve obtained from the paper and the research of Zhang and He.It can be seen from Figure5 that the well test curve obtained from the simplified model in the paper is consistent with the curve from the research of Zhang and He [31]. Thus, the numerical solution to the pressure buildup well test model in the paper is reliable.


## 3.1.1. Verification of Pressure Drawdown Well Test Model

When dimensionless stress sensitivity coefficientsγdD∗ and γbD∗ are taken as zero, the model is the same as the conventional pressure drawdown well test model of composite gas reservoir [37]. In order to verify the model in the paper, the type curve obtained by the numerical method without considering the stress sensitivity effect is compared with the conventional composite gas reservoir well test curve obtained by the analytical solution shown in Figure 3.Figure 3
Comparison between the numerical solution and the analytical solution to the pressure drawdown well test model without considering the stress sensitivity effect.As shown in Figure3, the type curves of pressure drawdown well test obtained by the numerical solution and the analytical solution are the same when dimensionless stress sensitivity coefficients are set to zero. The curves show common characteristics. The pressure derivative curve is a 0.5 horizontal line at the inner zone radial flow stage, while it is a M/2 (mobility ratio/2) horizontal line at the outer zone radial flow stage. Thus, the numerical solution in the paper is reliable.In order to further verify the model, the type curves of pressure drawdown well test considering the stress sensitivity effect obtained by the numerical method and the perturbation method [23] are compared shown in Figure 4.Figure 4
Comparison between the numerical solution and the perturbation solution to the pressure drawdown well test considering the stress sensitivity effect.It can be seen from Figure4 that the numerical solution is consistent with the solution obtained by the perturbation method, which verifies the reliability of the numerical solution.


## 3.1.2. Verification of Pressure Buildup Well Test Model

In order to verify the pressure buildup well test model of composite gas reservoir in the paper, the model is simplified into a homogeneous gas reservoir model considering the stress sensitivity effect by settingM=1, ω=1, and γ∗bD=γ∗dD. With those parameter settings, the model does not reflect the influence of the hysteresis effect and the differences of permeability and porosity between the inner zone and the outer zone. Then, the result of the model is compared with the research of Zhang and He [31] shown in Figure 5.Figure 5
Comparison of the homogeneous gas reservoir pressure buildup well test curve obtained from the paper and the research of Zhang and He.It can be seen from Figure5 that the well test curve obtained from the simplified model in the paper is consistent with the curve from the research of Zhang and He [31]. Thus, the numerical solution to the pressure buildup well test model in the paper is reliable.


## 3.2. Type Curve Analysis

Figures6 and 7 are type curves for pressure drawdown and buildup well tests in a composite gas reservoir. The corresponding parameter values are CD=1, S=1, M=0.5, ω=1, r1D=500, γdD∗=0 or 0.04, and γbD∗=0 or 0.04. And when γdD∗=γbD∗=0, the well test model is suitable for the conventional composite gas reservoir; when γdD∗=γbD∗=0.04, the model is suitable for the composite gas reservoir considering the stress sensitivity effect.Figure 6
Well test curves for the pressure drawdown well test in composite gas reservoir.Figure 7
Well test curves for the pressure buildup well test in composite gas reservoir.It can be seen from Figures6 and 7 that type curves of the composite gas reservoir well test considering the stress sensitivity effect can be divided into five basic flow regimes including ① the wellbore storage stage, ② the transition stage dominated by the skin factor, ③ the inner zone radial flow stage, ④ the transition stage between the inner zone and outer zone radial flow stages, and ⑤ the outer zone radial flow stage. The characteristics of the wellbore storage stage are not affected by the stress sensitivity effect. The transition stage and the radial flow stage are significantly affected by the stress sensitivity effect. For the pressure drawdown well test shown in Figure 6, the pseudo-pressure curve has a significant uplift compared with the conventional well test curve indicating a larger pressure drawdown at the same dimensionless time tD after considering the stress sensitivity effect; the pseudo-pressure derivative curve at the inner zone radial flow stage gradually deviates upward from the 0.5 horizontal line indicating the permeability decreases with the increase of the pressure drawdown at the inner zone radial flow stage; the pseudo-pressure derivative curve at the outer zone radial flow stage gradually deviates upward from the M/2 (mobility ratio/2) horizontal line indicating the permeability decreases with the increase of the pressure drawdown at the outer zone radial flow stage.For the pressure buildup well test shown in Figure7, the pseudo-pressure curve has a significant uplift compared with the conventional well test curve after considering the stress sensitivity effect; the pseudo-pressure derivative curve at the inner zone radial flow stage gradually descends towards the 0.5 horizontal line indicating the permeability increases with the increase of the formation pressure during the buildup stage, while the pseudo-pressure derivative curve at the outer zone radial flow stage gradually descends towards the M/2 horizontal line. The pseudo-pressure derivative curve is slightly higher than the M/2 horizontal line at the outer zone radial flow stage indicating that the formation pressure is close to the initial pressure.


## 3.3. Parameter Sensitivity Analysis

### 3.3.1. The Effect of Stress Sensitivity Coefficient

The type curves of pressure drawdown and buildup well tests considering the stress sensitivity effect under different stress sensitivity coefficients are shown in Figures8 and 9. The corresponding parameter values are CD=1, S=1, r1D=500, M=0.5, ω=1, and γdD∗=γbD∗=0, 0.02, 0.04, and 0.06.Figure 8
Well test curves of the pressure drawdown well test under different stress sensitivity coefficients.Figure 9
Well test curves of the pressure buildup well test under different stress sensitivity coefficients.As shown in Figure8, for the pressure drawdown well test, as stress sensitivity coefficients γdD∗ and γbD∗ increase, the pseudo-pressure curve gradually rises, and the pseudo-pressure derivative curve deviates upward from the 0.5 horizontal line at the inner zone radial flow stage, while it deviates upward from the M/2 (mobility ratio/2) horizontal line at the outer zone radial flow stage. The changes indicate that the pressure drawdown becomes larger, and the permeability becomes lower at the same dimensionless time tD as stress sensitivity coefficients increase.For the pressure buildup well test shown in Figure9, as stress sensitivity coefficients γdD∗ and γbD∗ increase, the pseudo-pressure and pseudo-pressure derivative curves gradually rise, which means that the speed of formation pressure buildup is quicker as stress sensitivity coefficients γdD∗ and γbD∗ increase. The pseudo-pressure derivative curve at the inner zone radial flow stage gradually descends towards the 0.5 horizontal line, while the pseudo-pressure derivative curve at the outer zone radial flow stage gradually descends towards the M/2 (mobility ratio/2) horizontal line in all cases.


### 3.3.2. The Effect of Hysteresis Effect

The type curves of pressure drawdown and buildup well tests considering the hysteresis effect are shown in Figures10 and 11. The corresponding parameter values are CD=1, S=1, r1D=500, M=0.5, ω=1, γdD∗=0.1, and γbD∗=0.1, 0.05, and 0.01.Figure 10
Well test curves of the pressure drawdown well test considering the hysteresis effect.Figure 11
Well test curves of the pressure buildup well test considering the hysteresis effect.As shown in Figure10, for the pressure drawdown well test with one production stage followed by one shut-in stage, the stress hysteresis effect has no effect on the curve of pressure drawdown well test due to the same stress sensitivity coefficient γdD∗ at the pressure drawdown stage.For the pressure buildup well test curve shown in Figure11, as the stress sensitivity coefficient γbD∗ of buildup stage decreases, and the pseudo-pressure derivative curve gradually rises at both inner zone and outer zone radial flow stages. Moreover, the transition stage between inner zone and outer zone radial flow stages appears later. Because the lower stress sensitivity coefficient of buildup stage γbD∗ is, the slower of permeability recovery speed is. The final permeability recovery value is also smaller at the pressure buildup stage.


### 3.3.3. The Effect of Skin Factor

The type curves of pressure drawdown and buildup well tests considering the stress sensitivity effect under different skin factors are shown in Figures12 and 13. The corresponding parameter values are CD=1, γdD∗=γbD∗=0.04, M=0.5, ω=1, r1D=500, and S=−1, 1, 3, and 5.Figure 12
Well test curves of the pressure drawdown well test under different skin factors.Figure 13
Well test curves of the pressure buildup well test under different skin factors.As shown in Figures12 and 13, the pseudo-pressure curves of pressure drawdown and buildup well tests both have an uplift as the skin factor S increases indicating a larger pressure drawdown or buildup at the same dimensionless time tD. At the inner zone or outer zone radial flow stages, the pseudo-pressure derivative curves of pressure drawdown well test obviously deviate upward as the skin factor increases, but the pseudo-pressure derivative curves at the radial flow stage of pressure buildup well test are not affected by the skin factor. This feature is different from that of the conventional composite gas reservoir well test curve. For conventional composite gas reservoir well test type curves under different skin factors, the pseudo-pressure derivative curves for both pressure drawdown and buildup well tests are 0.5 horizontal lines at inner zone radial flow stages. Those are M/2 horizontal lines at the outer zone radial flow stages.


### 3.3.4. The Effect of Inner Radius

The type curves of pressure drawdown and buildup well tests considering the stress sensitivity effect under different inner radii are shown in Figures14 and 15. The corresponding parameter values are CD=1, S=1, γdD∗=γbD∗=0.04, M=0.5, ω=1, and r1D=200, 300, 400, and 500.Figure 14
Well test curves of the pressure drawdown well test under different inner zone radii.Figure 15
Well test curves of the pressure buildup well test under different inner zone radii.As shown in Figures14 and 15, the radius of inner zone mainly affects the duration of inner zone radial flow. The larger inner zone radius leads to a longer duration of inner zone radial flow and a later appearance of the transition stage between inner zone and outer zone radial flow stages.


### 3.3.5. The Effect of Mobility Ratio

The type curves of pressure drawdown and buildup well tests considering the stress sensitivity effect under different mobility ratios are shown in Figures16 and 17. The corresponding parameter values are CD=1, S=1, γdD∗=γbD∗=0.04, ω=1, r1D=500, and M=0.2, 0.6, 1.0, 1.5, and 2.Figure 16
Well test curves of the pressure drawdown well test under different mobility ratios.Figure 17
Well test curves of the pressure buildup well test under different mobility ratios.As shown in Figures16 and 17, for the pressure drawdown well test, the mobility ratio mainly affects characteristics of the transition stage and the outer zone radial flow stage. A larger mobility ratio leads to a higher pseudo-pressure and pseudo-pressure derivative curve at the outer zone radial flow stage indicating a larger pressure drawdown and a lower permeability in the outer zone at the same dimensionless time. For the pressure buildup well test, the mobility ratio has an effect on both inner zone and outer zone radial flow stages. A larger mobility ratio leads to a higher pseudo-pressure and pseudo-pressure derivative curve in both two flow regimes. The feature is different from the pressure buildup well test curve of conventional composite gas reservoir. The mobility ratio has no effect on the pseudo-pressure derivative of both pressure drawdown and buildup well tests at the inner zone radial flow stage for the conventional well test curve.


### 3.3.6. The Effect of Storage Ratio

The type curves of pressure drawdown and buildup well tests considering the stress sensitivity effect under different storage ratios are shown in Figures18 and 19. The corresponding parameter values are CD=1, S=1, γdD∗=γbD∗=0.04, r1D=500, M=1, and ω=0.1, 0.5, 1, 5, and 10.Figure 18
Well test curves of the pressure drawdown well test under different storage ratios.Figure 19
Well test curves of the pressure buildup test under different storage ratios.As shown in Figures18 and 19, when the storage ratio ω is greater than 1, the curve shows a small hump at the transition stage between inner zone and outer zone radial flow stages indicating that the inner zone storage capacity is greater than that of the outer zone. When the storage ratio ω is less than 1, the curve shows a small concave.


## 3.3.1. The Effect of Stress Sensitivity Coefficient

The type curves of pressure drawdown and buildup well tests considering the stress sensitivity effect under different stress sensitivity coefficients are shown in Figures8 and 9. The corresponding parameter values are CD=1, S=1, r1D=500, M=0.5, ω=1, and γdD∗=γbD∗=0, 0.02, 0.04, and 0.06.Figure 8
Well test curves of the pressure drawdown well test under different stress sensitivity coefficients.Figure 9
Well test curves of the pressure buildup well test under different stress sensitivity coefficients.As shown in Figure8, for the pressure drawdown well test, as stress sensitivity coefficients γdD∗ and γbD∗ increase, the pseudo-pressure curve gradually rises, and the pseudo-pressure derivative curve deviates upward from the 0.5 horizontal line at the inner zone radial flow stage, while it deviates upward from the M/2 (mobility ratio/2) horizontal line at the outer zone radial flow stage. The changes indicate that the pressure drawdown becomes larger, and the permeability becomes lower at the same dimensionless time tD as stress sensitivity coefficients increase.For the pressure buildup well test shown in Figure9, as stress sensitivity coefficients γdD∗ and γbD∗ increase, the pseudo-pressure and pseudo-pressure derivative curves gradually rise, which means that the speed of formation pressure buildup is quicker as stress sensitivity coefficients γdD∗ and γbD∗ increase. The pseudo-pressure derivative curve at the inner zone radial flow stage gradually descends towards the 0.5 horizontal line, while the pseudo-pressure derivative curve at the outer zone radial flow stage gradually descends towards the M/2 (mobility ratio/2) horizontal line in all cases.


## 3.3.2. The Effect of Hysteresis Effect

The type curves of pressure drawdown and buildup well tests considering the hysteresis effect are shown in Figures10 and 11. The corresponding parameter values are CD=1, S=1, r1D=500, M=0.5, ω=1, γdD∗=0.1, and γbD∗=0.1, 0.05, and 0.01.Figure 10
Well test curves of the pressure drawdown well test considering the hysteresis effect.Figure 11
Well test curves of the pressure buildup well test considering the hysteresis effect.As shown in Figure10, for the pressure drawdown well test with one production stage followed by one shut-in stage, the stress hysteresis effect has no effect on the curve of pressure drawdown well test due to the same stress sensitivity coefficient γdD∗ at the pressure drawdown stage.For the pressure buildup well test curve shown in Figure11, as the stress sensitivity coefficient γbD∗ of buildup stage decreases, and the pseudo-pressure derivative curve gradually rises at both inner zone and outer zone radial flow stages. Moreover, the transition stage between inner zone and outer zone radial flow stages appears later. Because the lower stress sensitivity coefficient of buildup stage γbD∗ is, the slower of permeability recovery speed is. The final permeability recovery value is also smaller at the pressure buildup stage.


## 3.3.3. The Effect of Skin Factor

The type curves of pressure drawdown and buildup well tests considering the stress sensitivity effect under different skin factors are shown in Figures12 and 13. The corresponding parameter values are CD=1, γdD∗=γbD∗=0.04, M=0.5, ω=1, r1D=500, and S=−1, 1, 3, and 5.Figure 12
Well test curves of the pressure drawdown well test under different skin factors.Figure 13
Well test curves of the pressure buildup well test under different skin factors.As shown in Figures12 and 13, the pseudo-pressure curves of pressure drawdown and buildup well tests both have an uplift as the skin factor S increases indicating a larger pressure drawdown or buildup at the same dimensionless time tD. At the inner zone or outer zone radial flow stages, the pseudo-pressure derivative curves of pressure drawdown well test obviously deviate upward as the skin factor increases, but the pseudo-pressure derivative curves at the radial flow stage of pressure buildup well test are not affected by the skin factor. This feature is different from that of the conventional composite gas reservoir well test curve. For conventional composite gas reservoir well test type curves under different skin factors, the pseudo-pressure derivative curves for both pressure drawdown and buildup well tests are 0.5 horizontal lines at inner zone radial flow stages. Those are M/2 horizontal lines at the outer zone radial flow stages.


## 3.3.4. The Effect of Inner Radius

The type curves of pressure drawdown and buildup well tests considering the stress sensitivity effect under different inner radii are shown in Figures14 and 15. The corresponding parameter values are CD=1, S=1, γdD∗=γbD∗=0.04, M=0.5, ω=1, and r1D=200, 300, 400, and 500.Figure 14
Well test curves of the pressure drawdown well test under different inner zone radii.Figure 15
Well test curves of the pressure buildup well test under different inner zone radii.As shown in Figures14 and 15, the radius of inner zone mainly affects the duration of inner zone radial flow. The larger inner zone radius leads to a longer duration of inner zone radial flow and a later appearance of the transition stage between inner zone and outer zone radial flow stages.


## 3.3.5. The Effect of Mobility Ratio

The type curves of pressure drawdown and buildup well tests considering the stress sensitivity effect under different mobility ratios are shown in Figures16 and 17. The corresponding parameter values are CD=1, S=1, γdD∗=γbD∗=0.04, ω=1, r1D=500, and M=0.2, 0.6, 1.0, 1.5, and 2.Figure 16
Well test curves of the pressure drawdown well test under different mobility ratios.Figure 17
Well test curves of the pressure buildup well test under different mobility ratios.As shown in Figures16 and 17, for the pressure drawdown well test, the mobility ratio mainly affects characteristics of the transition stage and the outer zone radial flow stage. A larger mobility ratio leads to a higher pseudo-pressure and pseudo-pressure derivative curve at the outer zone radial flow stage indicating a larger pressure drawdown and a lower permeability in the outer zone at the same dimensionless time. For the pressure buildup well test, the mobility ratio has an effect on both inner zone and outer zone radial flow stages. A larger mobility ratio leads to a higher pseudo-pressure and pseudo-pressure derivative curve in both two flow regimes. The feature is different from the pressure buildup well test curve of conventional composite gas reservoir. The mobility ratio has no effect on the pseudo-pressure derivative of both pressure drawdown and buildup well tests at the inner zone radial flow stage for the conventional well test curve.


## 3.3.6. The Effect of Storage Ratio

The type curves of pressure drawdown and buildup well tests considering the stress sensitivity effect under different storage ratios are shown in Figures18 and 19. The corresponding parameter values are CD=1, S=1, γdD∗=γbD∗=0.04, r1D=500, M=1, and ω=0.1, 0.5, 1, 5, and 10.Figure 18
Well test curves of the pressure drawdown well test under different storage ratios.Figure 19
Well test curves of the pressure buildup test under different storage ratios.As shown in Figures18 and 19, when the storage ratio ω is greater than 1, the curve shows a small hump at the transition stage between inner zone and outer zone radial flow stages indicating that the inner zone storage capacity is greater than that of the outer zone. When the storage ratio ω is less than 1, the curve shows a small concave.


## 4. Case Study

Taking a case from a deepwater gas reservoir with high temperature and pressure as an example, the basic parameter of the well is shown in Table1. The pressure history of the well is shown in Figure 20.Table 1
The basic parameters of gas well test in the deepwater gas reservoir.
ParametersValuesWell typeVertical wellInitial pressure (MPa)72.91Reservoir temperature (°C)167.68Effective thickness (m)9.1Wellbore radius (m)0.127Porosity (decimal)0.2197Rock compressibility (MPa-1)0.000501Relative density of natural gas (decimal)0.764Figure 20
The pressure history of the well in the deepwater gas reservoir.Taking the second buildup stage to conduct the well test interpretation, interpretation results are shown in Figure21 and Table 2. It can be seen that there is no obvious boundary response in the well test curve; thus, the well test model is chosen as the infinite model with a constant well storage coefficient and skin factor. The pressure derivative curve of the pressure buildup well test has obvious downward trend indicating the influence of the stress sensitivity effect. Therefore, a well test model considering the stress sensitivity effect is used in the well test interpretation of this case. Besides, the results of well test interpretation with and without considering the stress sensitivity effect are compared shown in Figure 21 and Table 2.Figure 21
Comparison of interpretation results with and without considering the stress sensitivity effect. (a, b) Well test interpretation results in the double logarithmic curve. (a) The interpretation result without considering the stress sensitivity effect. (b) The interpretation result considering the stress sensitivity effect. (c, d) Well test interpretation results in the semilogarithm curve. (c) The interpretation result without considering the stress sensitivity effect. (d) The interpretation result considering the stress sensitivity effect.
(a)(b)(c)(d)Table 2
Comparison of interpretation results with and without considering the stress sensitivity effect.
ParametersConsidering the stress sensitivity effectWithout considering the stress sensitivity effectVariation amplitude of the interpretation parameters (%)Well storage coefficient (m3/MPa)0.2800.2831.06Skin factor3.505.4656.0Permeability (mD)5.546.5017.3Stress sensitivity coefficient (MPa-1)0.012----Mobility ratio1.01.0--Storage ratio1.01.0--It can be seen from Table2 and Figure 21 that better fitting results of pressure curves can be obtained after considering the stress sensitivity effect. The skin factor will be exaggerated without considering the stress sensitivity effect. Therefore, the accuracy of the well test interpretation for the deepwater gas reservoir with high temperature and pressure is improved after considering the stress sensitivity effect.


## 5. Conclusion

The paper establishes a composite gas reservoir pressure buildup well test model considering the stress sensitivity and the hysteresis effect to guide well test interpretations for high temperature and pressure composite gas reservoirs in deepwater. Based on the research, the following conclusions are obtained:(1)
The solutions to composite gas reservoir pressure drawdown and buildup well test models considering the stress sensitivity and the hysteresis effect are obtained by the numerical differentiation method. Then, the solution to pressure drawdown well test model is verified by comparing with the analytical solution obtained by the perturbation method and the analytical solution to the conventional well test model. The solution to the pressure buildup well test model is verified by comparing with the homogeneous gas reservoir well test solution from the research of Zhang and He after settingM=1, ω=1, and γ∗bD=γ∗dD(2)
The differences between pressure drawdown and buildup well test curves considering the stress sensitivity effect and the hysteresis effect are compared. The main flow regimes affected by the stress sensitivity effect are the inner zone and outer zone radial flow stages. The pressure derivative curve of pressure drawdown well test considering the stress sensitivity effect deviates upward from the 0.5 horizontal line at the inner zone radial flow stage, while it deviates upward from theM/2 (mobility ratio/2) horizontal line at the outer zone radial flow stage. However, for the pressure buildup well test curve considering the stress sensitivity effect, the pressure derivative curve gradually descends to the 0.5 horizontal line at the inner zone radial flow stage, while it descends to the M/2 (mobility ratio/2) horizontal line at the outer zone radial flow stage. The pressure derivative curve considering the hysteresis effect is higher than the curve without considering the hysteresis effect for the pressure buildup well test, because the permeability cannot be recovered to its original value in the buildup stage after considering the hysteresis effect(3)
The effects of different parameters such as stress sensitivity coefficient, hysteresis effect coefficient, skin factor, inner radius, mobility ratio, and storage ratio on type curves are analyzed. The stress sensitivity coefficient, skin factor, and mobility ratio have different effects on pressure drawdown and buildup well test curves after considering the stress sensitivity effect, which are different from conventional well test curves without considering the stress sensitivity effect(4)
A well test interpretation case from the deepwater gas reservoir with high temperature and pressure is studied. The result indicates that the accuracy of interpretation is improved after considering the stress sensitivity effect. The skin factor will be exaggerated without considering the stress sensitivity effect



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*Source: 5054246-2021-12-15.xml*