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Three dimensional pressure transient behavior study in stress sensitive reservoirs
Journal of Petroleum Science and Engineering 152 (2017) 204–211
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Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
Three dimensional pressure transient behavior study in stress sensitive reservoirs
MARK
⁎
Mostafa Moradia, Amir Shamlooa, , Mohsen Asadbegia, Alireza Daneh Dezfulib a b
Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran Department of Mechanical Engineering, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran
A R T I C L E I N F O
A BS T RAC T
Keywords: Reservoir simulation Pressure transient Finite element method Stress sensitivity
Stress sensitivity is a phenomenon that affects reservoir rock properties, such as permeability and therefore changes the well pressure transient behavior. This paper aims to study these behaviors in stress sensitive reservoirs and evaluate the pressure loss in such reservoirs during the process of hydrocarbon production. A power model is used to correlate the changes in permeability with pore pressure. A novel semi-implicit threedimensional finite element method has been employed to numerically solve the flow problem. The numerical results have been validated by analytical results obtained in a non-sensitive reservoir. Pressure drawdown test for different scenarios has been studied. The presented numerical method could contribute to better understanding the stress-sensitivity phenomenon and its effect on reservoir performance.
1. Introduction
permeability is an exponential function of pressure (Zhang and Ambastha, 1994). In addition, the effects of permeability reduction on pressure buildup solution has been studied by Ambastha and Zhang, 1996 (Ambastha and Zhang, 1996). A skin factor relation has been recommended by Jelmert and Selseng (1997) for stress sensitive reservoirs (Jelmert and Selseng, 1997). Chin et al. (2000) developed a coupled geo-mechanics and single-phase flow model to calculate the impact of changes in rock properties such as porosity and permeability on well analysis in stress sensitive reservoirs (Chin et al., 2000). Well test curve of stress sensitive reservoirs is under the influence of permeability and compressibility alternation according to Pinzon et al. (2001). Scholes et al. (2007) proposed a model for the effects of compressive stress on permeability anisotropy in porous media (Scholes et al., 2007). Chen and Li (2008) provided a mathematical model for production prediction based on reservoir skeleton deformation (Chen and Li, 2008). The relation between permeability and geological stress has been presented by Wang et al. (2010) in fractured reservoirs (Wang et al., 2010). Zhang et al. (2010) simulated well testing for stress sensitive reservoirs with heterogeneity and nonuniform thickness (Zhang et al., 2010). A method for analyzing transient linear flow in tight oil and gas reservoirs with stress sensitive permeability and multi-phase flow has been developed by Qanbari and Clarkson, (2012, 2014). High resolution three-dimensional simulations have been presented by Shaoul et al., 2015 in a tight gas reservoir based on stress sensitive permeability data obtained from core testing (Shaoul et al., 2015). Zhao et al. (2014) simulated the flow behavior of
Detailed information about the reservoir behavior and performance is a crucial necessity for efficient reservoir management. Well pressure transient study (WPTS) is an effective technique employed to gather this information. Reservoir rock properties and fluid characteristics can be obtained by WPTS. Moreover, this study can determine reservoir domain and well completion efficiency. Therefore, WPTS helps to have an efficient reservoir management. WPT is under the influence of the stress sensitivity phenomenon. The sensitivity of reservoir rock properties to a change in stress field is called stress sensitivity. This phenomenon affects reservoir rock skeleton and changes the properties of rock such as permeability. Geological parameters change reservoir rock skeleton. Strata properties, water content and reservoir depth are among these geological parameters. Depletion throughout the reservoir lifetime can also change the rock skeleton. Numerous researchers have investigated the effects of stress sensitivity on pressure transient behavior. Pedrosa (1986) developed an analytical solution based on perturbation technique for stress sensitive problem (Pedrosa, 1986). Kilmer et al. (1987) presented a log-log relation for permeability changes with net overburden pressure in gas reservoirs (Kilmer et al., 1987). Sayers (1990) studied the permeability tensor anisotropy due to stress in fractured rocks (Sayers, 1990). Drawdown and buildup solutions for stress sensitive reservoirs have been obtained using the concept of permeability modulus by Zhang and Ambastha (1994). Permeability modulus means that
⁎
Corresponding author. E-mail address: [email protected] (A. Shamloo).
http://dx.doi.org/10.1016/j.petrol.2017.02.017 Received 16 September 2016; Received in revised form 13 December 2016; Accepted 28 February 2017 Available online 06 March 2017 0920-4105/ © 2017 Elsevier B.V. All rights reserved.
Journal of Petroleum Science and Engineering 152 (2017) 204–211
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2.1. Assumptions The reservoir is initially a homogeneous and isotropic cylinder and is bounded by impermeable walls on all outer boundaries. The fluid is single-phase and slightly compressible and fluid flow obeys Darcy's equation. The producing well is a one dimensional line which is situated at the center of the reservoir. Production rate is held constant and wellbore storage and skin effects are taken into account. Gravitational effects are neglected. It is assumed that all of the reservoir properties except permeability, are independent of stress and remain constant. Although, it should be mentioned that properties such as porosity and compressibility have some degree of dependency on stress, but their changes over the pressure range are small compared to that of permeability and therefore it is reasonable to treat them as constants. A schematic diagram of the reservoir geometry is shown in Fig. 1. 2.2. Governing Equations Flow equation for single-phase slightly compressible fluid in porous media is given by Eq. (1).
Fig. 1. Schematic diagram of the reservoir geometry.
⎞ ⎛k ∂P ∇. ⎜ ∇P⎟ = ϕc +q ∂t ⎝μ ⎠
a horizontal well using the perturbation technique (Zhao et al., 2014, 2016). Almost all previous studies have utilized an exponential model to correlate the permeability changes with pore pressure. New experimental researches demonstrate that a power model better describes stress sensitive behavior than the classical exponential model (Ren and Guo, 2014 ). Stress-sensitivity models correlate rock permeability with the reservoir pore pressure. These models are based on many experimental measurements from different rock types. Curve-fitting techniques are used to implement these experimental data into the mathematical equation which govern the flow inside the reservoir rock. One of the widely used curve-fitting techniques is the exponential model. In the present work WPT behavior in three-dimensional stress sensitive reservoirs using a power model and the Galerkin variational method with finite element discretization is simulated. It should be mentioned that poroelastic effects are not considered in this work and experimental relations are used to model the effect of stress sensitivity on rock skeleton. This work aims to study the effects of stresssensitivity on rock permeability reduction and the wellbore pressure in different scenarios. Pressure drawdown test under several situations, such as different degrees of stress sensitivity and well penetration ratios is investigated. All of the reservoir rock properties except permeability are assumed to remain unchanged. Moreover, it is assumed that fluid properties are independent of stress and remain constant. This paper is organized as follows: First, the mathematical formulation of the problem and a description of the numerical method is presented. Results and discussion section consists of four different cases. The first case is a WPT analysis without the effects of formation damage, partial well completion, wellbore storage and stress sensitivity, and is carried out in order to evaluate the accuracy of the method. In the second case, the influence of skin factor and wellbore storage on WPT is investigated. In the third case, the effects of stress sensitivity is added to the second case. And lastly, the combined effects of stress sensitivity and partial well completion on WPT is studied in the fourth case.
c = cr + cf
(1)
Where P is the pore pressure, k is the reservoir rock permeability, ϕ is the porosity, cf is the fluid compressibility, cr is the rock compressibility, c is the total compressibility and q is the volumetric flow rate. In Eq. (1), permeability is a function of pore pressure. Several models have been developed to express permeability in terms of pore pressure. In this work, the relation between pressure and permeability as proposed by Ren et al. (2014) is modelled using a power model as shown in Eq. (2).
⎛ P − P ⎞−γ k = ⎜ ob ⎟ ki ⎝ Pob − Pi ⎠
(2)
Where Pob is the overburden pressure and γ is the dimensionless stress sensitivity coefficient, which depends on reservoir rock type. The stress sensitivity coefficient can be determined by experimental core testing. The wellbore pressure decreases due to formation damage. Eq. (3) is used to model the effect of formation damage on the wellbore pressure.
Pw = Pw′ −ΔPs ΔPs =
qsf Bμ kH
(3)
S = qsf s
Where P′w is the wellbore pressure without formation damage, Pw is the corrected wellbore pressure, ΔPs is the pressure loss due to formation damage, S is the skin factor, s is the skin factor per unit volumetric flow rate and H is the reservoir thickness. Whenever there is a change of production rate, part of the production due to the wellbore fluid volume changes. This is known as the wellbore storage effect and is modelled by Eq. (4).
qsf = q +
Cs dPw B dt
(4)
Where CS is the wellbore storage coefficient, q is the production rate, qsf is the reservoir flow rate and B is the formation volume factor. Combining Eqs. (1), (3) and (4), fluid flow equation for wellbore nodes becomes:
2. Mathematical Formulation In the following subsections, the assumptions used in solving the flow problem are described. Moreover, mathematical formulae governing the problem are established. The numerical method and computational grid used to solve the flow equations are also presented.
⎧ ⎛ ⎛ ⎪ ∂P C dP ⎞ C dP ⎞⎤⎫ 1 ⎪ ⎡ ∇.⎨ = ϕc w + ⎜q + s w ⎟ k ∇⎢Pw+s⎜qsf + s w ⎟⎥⎬ ⎪ ⎪ ⎥ ⎢ ⎝ ⎝ ∂t B dt ⎠ B dt ⎠⎦⎭ μ ⎩ ⎣ 205
(5)
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25 20
Van Everdingen & Hurst Present Work
15
10
PWD
S=0 CD =0 reD =1000 5
Fig. 2. Unstructured adaptive grid around the wellbore.
For nodes without source term, qsf , fluid flow equation is reduced to Eq. (6).
1 ∂P ∇. (k ∇P ) = ϕc μ ∂t
2
10
3
4
10
5
10
6
10
Fig. 3. Comparison of numerical result for a pressure drawdown test with Van Everdingen and Hurst's analytical solution for closed outer boundary reservoir.
2.3. Numerical method
the following dimensionless variables are defined.
In this work, the standard finite element method is used to solve the problem. Although, this method is numerically more expensive than the conventional finite difference method, but since it can be used with unstructured gridding, it is suitable for three-dimensional problems with complex geometry. Another advantage is that mesh refinement can be limited to the area around the wellbore, which is not possible in finite difference method (Karimi-Fard and Firoozabadi, 2003). Many variants of the finite element method have been developed to solve the flow problem in porous media. The Galerkin variational method is suitable for single-phase problems and the accuracy of results obtained with this method is acceptable (Chen, 2005). Since permeability is a function of pressure, the governing partial differential equation is nonlinear and cannot be directly solved using the finite element method. Two approaches can be taken to overcome this issue. One approach is to use the Newton-Raphson method as used by Ren and Gou, 2014 to iteratively solve the nonlinear problem. Another approach that has been used in this work is to linearize the problem by letting the nonlinearities to lag one time step behind. Applying the finite element method to Eqs. (5) and (6), leads to a system of linear equations in the form of Eq. (7). The details of this procedure has been explained in the appendix. Eq. (7) is then solved using the Gauss elimination method to find the nodal pressures at the next time step. It should be mentioned that time step is not constant and gradually increases as time elapses.
Dimensionless wellbore pressurePwD =
Dimensionless timetD =
2πkih(Pob − Pi ) qscBμ
kit ϕμct rw2
Dimensionless outer boundary radiusreD =
re rw
Dimensionless wellbore storage coefficientCD =
Cs 2πϕct hrw2
Where subscripts i ,w and e denote the initial value, wellbore and outer boundary, respectively. 3. Computational grid generation Appropriate grid generation is a requirement for obtaining accurate flow solutions in porous media. To generate an appropriate computational grid, geological features such as well positions have to be taken into account. Such features, reflect their effects on grid resolution. By increasing the stability and accuracy of the solution, optimum grid resolution can enhance flow transport prediction (Bahrainian and Daneh Dezfuli, 2014). In this work, the three dimensional grid generation has been carried out using the Gmsh software (Geuzaine and Remacle, 2009). Gmsh is a program that utilizes Delaunay-based triangulation for grid generation. Since the rate of change of pressure is very rapid near the wellbore and very slow far from the wellbore, an
(7)
Table 1 Rock and fluid properties.
Table 2 Input data in cases 4.2 and 4.3.
Rock properties
Porosity, ϕ Compressibility, cr
2πkih(Pi − Pw ) qscBμ
Dimensionless overburden pressurePobD =
In order to present the numerical results in a dimensionless form,
Initial permeability, ki
10
tD
(6)
→ ⎯n [A]P = [B]
7
10
2.4×10−15m 2 0.1
7.5×10−10Pa−1
Fluid properties Viscosity, μ
1.81×10−3Pa. s
Compressibility, cf
1.6×10−9Pa−1
206
Wellbore radius, rw Outer boundary radius, re Reservoir thickness, H Initial pressure, Pi
6.9×107Pa
Production rate, q
2×10−4m3 /s
0.11m 110m 9. 9m
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20
102
reD = 1000
P
101
S=5
'
P
.t
'
CD =1 2 0
=7.15 =14.3 =21.5 =35.8
PWD, PWD.tD
10
CD =1 3 0
h/H = 1
CD =1 0
PWD
15
γ=1.2,P γ=1.2,P γ=1.2,P γ=1.2,P γ=0
S=10
S=2 S=0
0
10
5
0 0 10
r = 1000 S =2 C = 10 h/H = 1 -1
1
10
2
10
3
4
10
10
5
10
6
10
10
7
10
10-1
100
101
102
103
104
105
106
tD/CD
tD Fig. 4. Pressure transient behavior for different values of skin factor and wellbore storage coefficient in a closed outer boundary reservoir.
Fig. 5. Effect of dimensionless net overburden pressure on the pressure curve for a closed outer boundary reservoir.
adaptive grid is used. Fig. 2 shows the generated adaptive grid. The grid is finest near the wellbore and becomes coarser at outer boundaries of the reservoir.
4.3. Case 3 In this case, permeability is a function of pressure and is computed for each computational node using Eq. (2). As mentioned before, permeability is calculated explicitly one time step ahead of pressure. Although this adds some error to the solution, but extrapolation
4. Results and discussion 4.1. Case 1
102
In order to validate the numerical results and evaluate the accuracy of the solution, the pressure drawdown solution is compared to the analytical solution obtained by van Everdingen and Hurst (1949). In this case, formation damage, wellbore storage, and stress sensitivity effects are not considered. Fluid and rock properties used in all cases can be found in Table 1. As shown in Fig. 3, the comparison demonstrates that the numerical results are accurate. Transient flow exists in linear portion of the curve and once the pressure distribution reaches the outer boundary of the reservoir, i.e. the starting point of the nonlinear portion, the flow becomes pseudo-steady and the wellbore pressure falls at the same rate as the reservoir pressure.
γ=1.2,P γ=0.8,P γ=0.5,P γ=0.3,P γ=0
=14.3 =14.3 =14.3 =14.3
P
101
'
.t
PWD,P 'WD.tD
P
100
4.2. Case 2 r =1000 S=2 C =10 h/H=1
This case study is performed to illustrate the effects of wellbore storage and formation damage. At the beginning of the production process, immediately after the well is opened, unloading of fluid inside the wellbore alters the well pressure curve. The same thing happens when the well is shut in which is also called “afterflow”. Wellbore storage effects describes this unloading of wellbore fluid and afterflow. Rock and fluid properties are shown in Table 1 and geometrical parameters and initial conditions are given in Table 2. Fig. 4 shows the effects of formation damage (S) and wellbore storage (CD ) on the WPT behavior. It is evident that wellbore storage only affects the pressure behavior at early stages of production. An increase in wellbore storage coefficient results in an increase in dimensionless pore pressure and causes the pressure curve to be affected for a longer period of time. Formation damage decreases the pressure near the wellbore and therefore increases the dimensionless well pressure. This decrease in pressure is due to the lower rock permeability around the wellbore. This change in permeability has nothing to do with stress-sensitivity phenomenon and is caused by drilling operations.
10-1 -1 10
100
101
102
103
104
105
106
tD/CD Fig. 6. Effect of stress sensitivity coefficient on the pressure curve for a closed outer boundary reservoir.
Table 3 Input data in case 4.4. Wellbore radius, rw Outer boundary radius, re Reservoir thickness, H Initial pressure, Pi Dimensionless overburden pressure, PobD Stress sensitivity coefficient, γ
207
0. 11m 11m 9. 9m
6.9×107Pa 14.3 1.2
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103
(Eq. (7)) in nth time step. By doing so, permeability, k , can be calculated using Eq. (2) based on the average value of approximate value of pressure at n+1 time step, P′n+1, and P n . For n ≥ 2 , P′n+1 is calculated by Eq. (8).
h/H= 0.125 h/H= 0.25 h/H= 0.5 h/H= 1
102
⎛ Δt n +1 ⎞ Δt n +1 n −1 n P′n +1 = ⎜1+ P n ⎟P − Δt ⎠ Δt n ⎝
P '
PWD, P'WD.tD
P
.t
The effects of net overburden pressure on the WPT behavior is shown in Fig. 5. It can be seen in Fig. 5 that as the initial net overburden pressure decreases, the dimensionless pressure increases. This is due to the fact that the decrease in permeability at lower net overburden pressures is more dramatic. As a result, wellbore pressure is more affected at lower initial net overburden pressures. Fig. 6 demonstrates the influence of stress sensitivity coefficient, γ , on the WPT curve. Stress-sensitivity coefficient is a measure of the degree of dependency of rock permeability on pore pressure. In other words, for higher values of γ , rock permeability changes more with pressure. The stress sensitivity coefficient is a reservoir rock property and can be obtained experimentally. As shown in this figure, the dimensionless pressure and pressure derivative increase with increasing the stress sensitivity coefficient. In the wellbore storage period, i.e. the log-log linear portion, stress sensitivity has no considerable effect on the pressure curve. But, once the flow reaches the radial flow period, the stress sensitivity effects become more significant. Higher stress sensitivity coefficient results in a higher pressure loss. This higher pressure loss implies that for the same amount of hydrocarbon extracted from the reservoir, the natural energy of the
101
100
r =100 S=2 C =10
-1
10
10-1
100
101
102
103
104
105
(8)
106
tD/CD Fig. 7. Pressure curve for different well penetration ratios.
techniques can be used to improve the accuracy of this method. Using the extrapolation technique, the unknown variable, P, can be approximated in n+1 time step before solving the linear system of equations
Fig. 8. reD=100 , h /H = 0.125, H /rw=90 , Pi=10000psi a, ki=2.4md b. a Pressure distribution for stress sensitive reservoir, b pressure distribution for non-sensitive reservoir, c permeability distribution for stress sensitive reservoir. a 1psi = 6.9×103Pa , b 1md =0. 987×10−12m 2 .
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Fig. 9. reD=100 , h /H = 0.5, H /rw=90 , Pi=10000psi a, ki=2.4md b. a Pressure distribution for stress sensitive reservoir, b pressure distribution for non-sensitive reservoir, c permeability distribution for stress sensitive reservoir.
at later times, the difference turns out to be more substantial. Fig. 8(c) shows the permeability distribution for a stress sensitive reservoir. Permeability changes very rapidly near the producing well. But, the permeability distribution is much smoother at other regions of the reservoir. In this case, similar to case of Section 4.3, in the stress-sensitive reservoir loss of pressure is higher than the conventional reservoir. Furthermore, partial well completion exacerbates the situation and causes an additional loss in pressure near the wellbore. As a result, reservoir energy depletion intensifies and could lead to a decrease in hydrocarbon recovery percentage.
reservoir in the stress-sensitive case will be lower. This will lead to a reduction in the percentage of hydrocarbon that can be recovered without the aid of external sources of energy (i.e. water or gas injection).
4.4. Case 4 In this section, stress-sensitivity is studied in a reservoir with partial well completion. Geometrical parameters and initial conditions are presented in Table 3, and rock and fluid properties used in this section are shown in Table 1. Partial well completion causes a positive pseudo skin effect to occur near the producing well and consequently alters the pressure transient behavior. Fig. 7 shows the effects of partial completion for different well penetration ratios. In early times, wellbore storage is dominant and the pressure curve is a unit slope line. Afterward, spherical flow occurs as a result of partial completion. Consequently, the wellbore pressure drop is much greater than case 1. After that, pressure distribution reaches the lower and upper boundaries of the reservoir and radial flow starts to develop. It can be seen in the pressure derivative curve that the radial flow portion of the curve is smaller for lower well penetration ratios. Finally the reservoir reaches the pseudo-steady state and the pressure drops equally at any point within the reservoir.
5. Conclusion This paper presents a novel semi-implicit three-dimensional numerical method based on Galerkin method with finite element discretization, which can be used to compute the well pressure transient response in stress sensitive reservoirs. A power model has been implemented to quantify the stress sensitive phenomenon and correlate rock permeability with reservoir pore pressure. The effects of various parameters has been investigated on the pressure drawdown test in stress sensitive reservoirs. Moreover, pressure and permeability distribution for a partially penetrating well in a stress sensitive reservoir has been studied. In early stages of production, stresssensitivity has no significant effect on pressure and pressure derivative curves. After the wellbore storage period, dimensionless pressure and pressure derivative values are higher for stress-sensitive reservoirs in
Figs. 8 and 9 present the combined effects of partial completion and stress sensitivity. At early stages of production, there isn’t substantial difference between the sensitive and non-sensitive cases. Nevertheless, 209
Journal of Petroleum Science and Engineering 152 (2017) 204–211
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Acknowledgements
comparison to conventional reservoirs. Also, stress sensitivity effects are more significant for lower overburden pressures and higher stress sensitivity coefficients. In future studies, more detailed scenarios such as the effects of stress sensitivity in a field case can be investigated.
The authors would like to express their sincere gratitude to National Iranian South Oil Company (NISOC).
Appendix A In the finite element method, the unknown variable is approximated as the sum of its value at all nodes. Using the definition of shape function, N , the pressure can be written in terms of nodal values and shape functions as follows. n (e )
P=
∑ PN i i
(A.1)
i =1
Integrating Eq. (5) over the volume of element e and using Green's formula, results in the weak form of Eq. (5). Eq. (A.2) is for wellbore nodes and Eq. (A.3) is for nodes without source term.
−
1 μ
∭e k∇P.∇NdV = ϕc ∭e
−
1 μ
∭e ∇.{k∇} = ϕc ∭e
∂P NdV ∂t
∂P NdV + ∂t
(A.2)
∭e qdV
(A.3)
In this paper, Crank-Nicholson method is used for temporal discretization. Eq. (A.1) is combined with Eqs. (A.2) and (A.3), and is rewritten in matrix form as follows.
[A]{P}n = [B]
(A.4)
[A] and [B] matrices are given by Eqs. (A.5) and (A.6), respectively. [A] =
Δt [T ] + [M ] + μCs[Wq ] + μSCs[WT ] 2
(A.5)
⎞ ⎛ Δt [B] = − Δt (Bμ + S[T ]){q} + ⎜[M ]{P}n −1 − [T ] + μCS[Wq ] + μSCs[WT ]⎟{P}n −1 ⎠ ⎝ 2
(A.6)
[T] and [M ] matrices can be calculated using Eqs. (A.7) and (A.8).
[T ]ij = −
k μ
[M ]ij = ϕc
∭e ∇Ni.∇NjdV
(A.7)
∭e NN i j dV
(A.8)
[WT ] and [Wq ] are defined as follows. ⎧ ⎪[WT ]i, j = [T ]i, j ⎨ ⎪[W ] ⎩ T i, j = 0
if j is a wellbore node if j is not a wellbore node
i = 1, 2, …, n (A.9)
Where n is the number of all nodes.
⎧ [Wq ] = 0| i, j = 1, 2, …, n ⎪ i, j ⎨ ⎪[W ] = 1 if i is a wellbore node ⎩ q i, i
(A.10)
As mentioned earlier, matrix [T], which is the source of nonlinearity is calculated at n − 1 time step. Eq. (A.4) is used to find the nodal pressures at n th time step. After that, [T ] is recalculated explicitly using Eqs. (A.7) and (2). For further details about the numerical method refer to Zienkiewicz and Taylor (2005) and Zhang et al. (2016).
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Contents lists available at ScienceDirect
Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
Three dimensional pressure transient behavior study in stress sensitive reservoirs
MARK
⁎
Mostafa Moradia, Amir Shamlooa, , Mohsen Asadbegia, Alireza Daneh Dezfulib a b
Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran Department of Mechanical Engineering, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran
A R T I C L E I N F O
A BS T RAC T
Keywords: Reservoir simulation Pressure transient Finite element method Stress sensitivity
Stress sensitivity is a phenomenon that affects reservoir rock properties, such as permeability and therefore changes the well pressure transient behavior. This paper aims to study these behaviors in stress sensitive reservoirs and evaluate the pressure loss in such reservoirs during the process of hydrocarbon production. A power model is used to correlate the changes in permeability with pore pressure. A novel semi-implicit threedimensional finite element method has been employed to numerically solve the flow problem. The numerical results have been validated by analytical results obtained in a non-sensitive reservoir. Pressure drawdown test for different scenarios has been studied. The presented numerical method could contribute to better understanding the stress-sensitivity phenomenon and its effect on reservoir performance.
1. Introduction
permeability is an exponential function of pressure (Zhang and Ambastha, 1994). In addition, the effects of permeability reduction on pressure buildup solution has been studied by Ambastha and Zhang, 1996 (Ambastha and Zhang, 1996). A skin factor relation has been recommended by Jelmert and Selseng (1997) for stress sensitive reservoirs (Jelmert and Selseng, 1997). Chin et al. (2000) developed a coupled geo-mechanics and single-phase flow model to calculate the impact of changes in rock properties such as porosity and permeability on well analysis in stress sensitive reservoirs (Chin et al., 2000). Well test curve of stress sensitive reservoirs is under the influence of permeability and compressibility alternation according to Pinzon et al. (2001). Scholes et al. (2007) proposed a model for the effects of compressive stress on permeability anisotropy in porous media (Scholes et al., 2007). Chen and Li (2008) provided a mathematical model for production prediction based on reservoir skeleton deformation (Chen and Li, 2008). The relation between permeability and geological stress has been presented by Wang et al. (2010) in fractured reservoirs (Wang et al., 2010). Zhang et al. (2010) simulated well testing for stress sensitive reservoirs with heterogeneity and nonuniform thickness (Zhang et al., 2010). A method for analyzing transient linear flow in tight oil and gas reservoirs with stress sensitive permeability and multi-phase flow has been developed by Qanbari and Clarkson, (2012, 2014). High resolution three-dimensional simulations have been presented by Shaoul et al., 2015 in a tight gas reservoir based on stress sensitive permeability data obtained from core testing (Shaoul et al., 2015). Zhao et al. (2014) simulated the flow behavior of
Detailed information about the reservoir behavior and performance is a crucial necessity for efficient reservoir management. Well pressure transient study (WPTS) is an effective technique employed to gather this information. Reservoir rock properties and fluid characteristics can be obtained by WPTS. Moreover, this study can determine reservoir domain and well completion efficiency. Therefore, WPTS helps to have an efficient reservoir management. WPT is under the influence of the stress sensitivity phenomenon. The sensitivity of reservoir rock properties to a change in stress field is called stress sensitivity. This phenomenon affects reservoir rock skeleton and changes the properties of rock such as permeability. Geological parameters change reservoir rock skeleton. Strata properties, water content and reservoir depth are among these geological parameters. Depletion throughout the reservoir lifetime can also change the rock skeleton. Numerous researchers have investigated the effects of stress sensitivity on pressure transient behavior. Pedrosa (1986) developed an analytical solution based on perturbation technique for stress sensitive problem (Pedrosa, 1986). Kilmer et al. (1987) presented a log-log relation for permeability changes with net overburden pressure in gas reservoirs (Kilmer et al., 1987). Sayers (1990) studied the permeability tensor anisotropy due to stress in fractured rocks (Sayers, 1990). Drawdown and buildup solutions for stress sensitive reservoirs have been obtained using the concept of permeability modulus by Zhang and Ambastha (1994). Permeability modulus means that
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Corresponding author. E-mail address: [email protected] (A. Shamloo).
http://dx.doi.org/10.1016/j.petrol.2017.02.017 Received 16 September 2016; Received in revised form 13 December 2016; Accepted 28 February 2017 Available online 06 March 2017 0920-4105/ © 2017 Elsevier B.V. All rights reserved.
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2.1. Assumptions The reservoir is initially a homogeneous and isotropic cylinder and is bounded by impermeable walls on all outer boundaries. The fluid is single-phase and slightly compressible and fluid flow obeys Darcy's equation. The producing well is a one dimensional line which is situated at the center of the reservoir. Production rate is held constant and wellbore storage and skin effects are taken into account. Gravitational effects are neglected. It is assumed that all of the reservoir properties except permeability, are independent of stress and remain constant. Although, it should be mentioned that properties such as porosity and compressibility have some degree of dependency on stress, but their changes over the pressure range are small compared to that of permeability and therefore it is reasonable to treat them as constants. A schematic diagram of the reservoir geometry is shown in Fig. 1. 2.2. Governing Equations Flow equation for single-phase slightly compressible fluid in porous media is given by Eq. (1).
Fig. 1. Schematic diagram of the reservoir geometry.
⎞ ⎛k ∂P ∇. ⎜ ∇P⎟ = ϕc +q ∂t ⎝μ ⎠
a horizontal well using the perturbation technique (Zhao et al., 2014, 2016). Almost all previous studies have utilized an exponential model to correlate the permeability changes with pore pressure. New experimental researches demonstrate that a power model better describes stress sensitive behavior than the classical exponential model (Ren and Guo, 2014 ). Stress-sensitivity models correlate rock permeability with the reservoir pore pressure. These models are based on many experimental measurements from different rock types. Curve-fitting techniques are used to implement these experimental data into the mathematical equation which govern the flow inside the reservoir rock. One of the widely used curve-fitting techniques is the exponential model. In the present work WPT behavior in three-dimensional stress sensitive reservoirs using a power model and the Galerkin variational method with finite element discretization is simulated. It should be mentioned that poroelastic effects are not considered in this work and experimental relations are used to model the effect of stress sensitivity on rock skeleton. This work aims to study the effects of stresssensitivity on rock permeability reduction and the wellbore pressure in different scenarios. Pressure drawdown test under several situations, such as different degrees of stress sensitivity and well penetration ratios is investigated. All of the reservoir rock properties except permeability are assumed to remain unchanged. Moreover, it is assumed that fluid properties are independent of stress and remain constant. This paper is organized as follows: First, the mathematical formulation of the problem and a description of the numerical method is presented. Results and discussion section consists of four different cases. The first case is a WPT analysis without the effects of formation damage, partial well completion, wellbore storage and stress sensitivity, and is carried out in order to evaluate the accuracy of the method. In the second case, the influence of skin factor and wellbore storage on WPT is investigated. In the third case, the effects of stress sensitivity is added to the second case. And lastly, the combined effects of stress sensitivity and partial well completion on WPT is studied in the fourth case.
c = cr + cf
(1)
Where P is the pore pressure, k is the reservoir rock permeability, ϕ is the porosity, cf is the fluid compressibility, cr is the rock compressibility, c is the total compressibility and q is the volumetric flow rate. In Eq. (1), permeability is a function of pore pressure. Several models have been developed to express permeability in terms of pore pressure. In this work, the relation between pressure and permeability as proposed by Ren et al. (2014) is modelled using a power model as shown in Eq. (2).
⎛ P − P ⎞−γ k = ⎜ ob ⎟ ki ⎝ Pob − Pi ⎠
(2)
Where Pob is the overburden pressure and γ is the dimensionless stress sensitivity coefficient, which depends on reservoir rock type. The stress sensitivity coefficient can be determined by experimental core testing. The wellbore pressure decreases due to formation damage. Eq. (3) is used to model the effect of formation damage on the wellbore pressure.
Pw = Pw′ −ΔPs ΔPs =
qsf Bμ kH
(3)
S = qsf s
Where P′w is the wellbore pressure without formation damage, Pw is the corrected wellbore pressure, ΔPs is the pressure loss due to formation damage, S is the skin factor, s is the skin factor per unit volumetric flow rate and H is the reservoir thickness. Whenever there is a change of production rate, part of the production due to the wellbore fluid volume changes. This is known as the wellbore storage effect and is modelled by Eq. (4).
qsf = q +
Cs dPw B dt
(4)
Where CS is the wellbore storage coefficient, q is the production rate, qsf is the reservoir flow rate and B is the formation volume factor. Combining Eqs. (1), (3) and (4), fluid flow equation for wellbore nodes becomes:
2. Mathematical Formulation In the following subsections, the assumptions used in solving the flow problem are described. Moreover, mathematical formulae governing the problem are established. The numerical method and computational grid used to solve the flow equations are also presented.
⎧ ⎛ ⎛ ⎪ ∂P C dP ⎞ C dP ⎞⎤⎫ 1 ⎪ ⎡ ∇.⎨ = ϕc w + ⎜q + s w ⎟ k ∇⎢Pw+s⎜qsf + s w ⎟⎥⎬ ⎪ ⎪ ⎥ ⎢ ⎝ ⎝ ∂t B dt ⎠ B dt ⎠⎦⎭ μ ⎩ ⎣ 205
(5)
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25 20
Van Everdingen & Hurst Present Work
15
10
PWD
S=0 CD =0 reD =1000 5
Fig. 2. Unstructured adaptive grid around the wellbore.
For nodes without source term, qsf , fluid flow equation is reduced to Eq. (6).
1 ∂P ∇. (k ∇P ) = ϕc μ ∂t
2
10
3
4
10
5
10
6
10
Fig. 3. Comparison of numerical result for a pressure drawdown test with Van Everdingen and Hurst's analytical solution for closed outer boundary reservoir.
2.3. Numerical method
the following dimensionless variables are defined.
In this work, the standard finite element method is used to solve the problem. Although, this method is numerically more expensive than the conventional finite difference method, but since it can be used with unstructured gridding, it is suitable for three-dimensional problems with complex geometry. Another advantage is that mesh refinement can be limited to the area around the wellbore, which is not possible in finite difference method (Karimi-Fard and Firoozabadi, 2003). Many variants of the finite element method have been developed to solve the flow problem in porous media. The Galerkin variational method is suitable for single-phase problems and the accuracy of results obtained with this method is acceptable (Chen, 2005). Since permeability is a function of pressure, the governing partial differential equation is nonlinear and cannot be directly solved using the finite element method. Two approaches can be taken to overcome this issue. One approach is to use the Newton-Raphson method as used by Ren and Gou, 2014 to iteratively solve the nonlinear problem. Another approach that has been used in this work is to linearize the problem by letting the nonlinearities to lag one time step behind. Applying the finite element method to Eqs. (5) and (6), leads to a system of linear equations in the form of Eq. (7). The details of this procedure has been explained in the appendix. Eq. (7) is then solved using the Gauss elimination method to find the nodal pressures at the next time step. It should be mentioned that time step is not constant and gradually increases as time elapses.
Dimensionless wellbore pressurePwD =
Dimensionless timetD =
2πkih(Pob − Pi ) qscBμ
kit ϕμct rw2
Dimensionless outer boundary radiusreD =
re rw
Dimensionless wellbore storage coefficientCD =
Cs 2πϕct hrw2
Where subscripts i ,w and e denote the initial value, wellbore and outer boundary, respectively. 3. Computational grid generation Appropriate grid generation is a requirement for obtaining accurate flow solutions in porous media. To generate an appropriate computational grid, geological features such as well positions have to be taken into account. Such features, reflect their effects on grid resolution. By increasing the stability and accuracy of the solution, optimum grid resolution can enhance flow transport prediction (Bahrainian and Daneh Dezfuli, 2014). In this work, the three dimensional grid generation has been carried out using the Gmsh software (Geuzaine and Remacle, 2009). Gmsh is a program that utilizes Delaunay-based triangulation for grid generation. Since the rate of change of pressure is very rapid near the wellbore and very slow far from the wellbore, an
(7)
Table 1 Rock and fluid properties.
Table 2 Input data in cases 4.2 and 4.3.
Rock properties
Porosity, ϕ Compressibility, cr
2πkih(Pi − Pw ) qscBμ
Dimensionless overburden pressurePobD =
In order to present the numerical results in a dimensionless form,
Initial permeability, ki
10
tD
(6)
→ ⎯n [A]P = [B]
7
10
2.4×10−15m 2 0.1
7.5×10−10Pa−1
Fluid properties Viscosity, μ
1.81×10−3Pa. s
Compressibility, cf
1.6×10−9Pa−1
206
Wellbore radius, rw Outer boundary radius, re Reservoir thickness, H Initial pressure, Pi
6.9×107Pa
Production rate, q
2×10−4m3 /s
0.11m 110m 9. 9m
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102
reD = 1000
P
101
S=5
'
P
.t
'
CD =1 2 0
=7.15 =14.3 =21.5 =35.8
PWD, PWD.tD
10
CD =1 3 0
h/H = 1
CD =1 0
PWD
15
γ=1.2,P γ=1.2,P γ=1.2,P γ=1.2,P γ=0
S=10
S=2 S=0
0
10
5
0 0 10
r = 1000 S =2 C = 10 h/H = 1 -1
1
10
2
10
3
4
10
10
5
10
6
10
10
7
10
10-1
100
101
102
103
104
105
106
tD/CD
tD Fig. 4. Pressure transient behavior for different values of skin factor and wellbore storage coefficient in a closed outer boundary reservoir.
Fig. 5. Effect of dimensionless net overburden pressure on the pressure curve for a closed outer boundary reservoir.
adaptive grid is used. Fig. 2 shows the generated adaptive grid. The grid is finest near the wellbore and becomes coarser at outer boundaries of the reservoir.
4.3. Case 3 In this case, permeability is a function of pressure and is computed for each computational node using Eq. (2). As mentioned before, permeability is calculated explicitly one time step ahead of pressure. Although this adds some error to the solution, but extrapolation
4. Results and discussion 4.1. Case 1
102
In order to validate the numerical results and evaluate the accuracy of the solution, the pressure drawdown solution is compared to the analytical solution obtained by van Everdingen and Hurst (1949). In this case, formation damage, wellbore storage, and stress sensitivity effects are not considered. Fluid and rock properties used in all cases can be found in Table 1. As shown in Fig. 3, the comparison demonstrates that the numerical results are accurate. Transient flow exists in linear portion of the curve and once the pressure distribution reaches the outer boundary of the reservoir, i.e. the starting point of the nonlinear portion, the flow becomes pseudo-steady and the wellbore pressure falls at the same rate as the reservoir pressure.
γ=1.2,P γ=0.8,P γ=0.5,P γ=0.3,P γ=0
=14.3 =14.3 =14.3 =14.3
P
101
'
.t
PWD,P 'WD.tD
P
100
4.2. Case 2 r =1000 S=2 C =10 h/H=1
This case study is performed to illustrate the effects of wellbore storage and formation damage. At the beginning of the production process, immediately after the well is opened, unloading of fluid inside the wellbore alters the well pressure curve. The same thing happens when the well is shut in which is also called “afterflow”. Wellbore storage effects describes this unloading of wellbore fluid and afterflow. Rock and fluid properties are shown in Table 1 and geometrical parameters and initial conditions are given in Table 2. Fig. 4 shows the effects of formation damage (S) and wellbore storage (CD ) on the WPT behavior. It is evident that wellbore storage only affects the pressure behavior at early stages of production. An increase in wellbore storage coefficient results in an increase in dimensionless pore pressure and causes the pressure curve to be affected for a longer period of time. Formation damage decreases the pressure near the wellbore and therefore increases the dimensionless well pressure. This decrease in pressure is due to the lower rock permeability around the wellbore. This change in permeability has nothing to do with stress-sensitivity phenomenon and is caused by drilling operations.
10-1 -1 10
100
101
102
103
104
105
106
tD/CD Fig. 6. Effect of stress sensitivity coefficient on the pressure curve for a closed outer boundary reservoir.
Table 3 Input data in case 4.4. Wellbore radius, rw Outer boundary radius, re Reservoir thickness, H Initial pressure, Pi Dimensionless overburden pressure, PobD Stress sensitivity coefficient, γ
207
0. 11m 11m 9. 9m
6.9×107Pa 14.3 1.2
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103
(Eq. (7)) in nth time step. By doing so, permeability, k , can be calculated using Eq. (2) based on the average value of approximate value of pressure at n+1 time step, P′n+1, and P n . For n ≥ 2 , P′n+1 is calculated by Eq. (8).
h/H= 0.125 h/H= 0.25 h/H= 0.5 h/H= 1
102
⎛ Δt n +1 ⎞ Δt n +1 n −1 n P′n +1 = ⎜1+ P n ⎟P − Δt ⎠ Δt n ⎝
P '
PWD, P'WD.tD
P
.t
The effects of net overburden pressure on the WPT behavior is shown in Fig. 5. It can be seen in Fig. 5 that as the initial net overburden pressure decreases, the dimensionless pressure increases. This is due to the fact that the decrease in permeability at lower net overburden pressures is more dramatic. As a result, wellbore pressure is more affected at lower initial net overburden pressures. Fig. 6 demonstrates the influence of stress sensitivity coefficient, γ , on the WPT curve. Stress-sensitivity coefficient is a measure of the degree of dependency of rock permeability on pore pressure. In other words, for higher values of γ , rock permeability changes more with pressure. The stress sensitivity coefficient is a reservoir rock property and can be obtained experimentally. As shown in this figure, the dimensionless pressure and pressure derivative increase with increasing the stress sensitivity coefficient. In the wellbore storage period, i.e. the log-log linear portion, stress sensitivity has no considerable effect on the pressure curve. But, once the flow reaches the radial flow period, the stress sensitivity effects become more significant. Higher stress sensitivity coefficient results in a higher pressure loss. This higher pressure loss implies that for the same amount of hydrocarbon extracted from the reservoir, the natural energy of the
101
100
r =100 S=2 C =10
-1
10
10-1
100
101
102
103
104
105
(8)
106
tD/CD Fig. 7. Pressure curve for different well penetration ratios.
techniques can be used to improve the accuracy of this method. Using the extrapolation technique, the unknown variable, P, can be approximated in n+1 time step before solving the linear system of equations
Fig. 8. reD=100 , h /H = 0.125, H /rw=90 , Pi=10000psi a, ki=2.4md b. a Pressure distribution for stress sensitive reservoir, b pressure distribution for non-sensitive reservoir, c permeability distribution for stress sensitive reservoir. a 1psi = 6.9×103Pa , b 1md =0. 987×10−12m 2 .
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Fig. 9. reD=100 , h /H = 0.5, H /rw=90 , Pi=10000psi a, ki=2.4md b. a Pressure distribution for stress sensitive reservoir, b pressure distribution for non-sensitive reservoir, c permeability distribution for stress sensitive reservoir.
at later times, the difference turns out to be more substantial. Fig. 8(c) shows the permeability distribution for a stress sensitive reservoir. Permeability changes very rapidly near the producing well. But, the permeability distribution is much smoother at other regions of the reservoir. In this case, similar to case of Section 4.3, in the stress-sensitive reservoir loss of pressure is higher than the conventional reservoir. Furthermore, partial well completion exacerbates the situation and causes an additional loss in pressure near the wellbore. As a result, reservoir energy depletion intensifies and could lead to a decrease in hydrocarbon recovery percentage.
reservoir in the stress-sensitive case will be lower. This will lead to a reduction in the percentage of hydrocarbon that can be recovered without the aid of external sources of energy (i.e. water or gas injection).
4.4. Case 4 In this section, stress-sensitivity is studied in a reservoir with partial well completion. Geometrical parameters and initial conditions are presented in Table 3, and rock and fluid properties used in this section are shown in Table 1. Partial well completion causes a positive pseudo skin effect to occur near the producing well and consequently alters the pressure transient behavior. Fig. 7 shows the effects of partial completion for different well penetration ratios. In early times, wellbore storage is dominant and the pressure curve is a unit slope line. Afterward, spherical flow occurs as a result of partial completion. Consequently, the wellbore pressure drop is much greater than case 1. After that, pressure distribution reaches the lower and upper boundaries of the reservoir and radial flow starts to develop. It can be seen in the pressure derivative curve that the radial flow portion of the curve is smaller for lower well penetration ratios. Finally the reservoir reaches the pseudo-steady state and the pressure drops equally at any point within the reservoir.
5. Conclusion This paper presents a novel semi-implicit three-dimensional numerical method based on Galerkin method with finite element discretization, which can be used to compute the well pressure transient response in stress sensitive reservoirs. A power model has been implemented to quantify the stress sensitive phenomenon and correlate rock permeability with reservoir pore pressure. The effects of various parameters has been investigated on the pressure drawdown test in stress sensitive reservoirs. Moreover, pressure and permeability distribution for a partially penetrating well in a stress sensitive reservoir has been studied. In early stages of production, stresssensitivity has no significant effect on pressure and pressure derivative curves. After the wellbore storage period, dimensionless pressure and pressure derivative values are higher for stress-sensitive reservoirs in
Figs. 8 and 9 present the combined effects of partial completion and stress sensitivity. At early stages of production, there isn’t substantial difference between the sensitive and non-sensitive cases. Nevertheless, 209
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Acknowledgements
comparison to conventional reservoirs. Also, stress sensitivity effects are more significant for lower overburden pressures and higher stress sensitivity coefficients. In future studies, more detailed scenarios such as the effects of stress sensitivity in a field case can be investigated.
The authors would like to express their sincere gratitude to National Iranian South Oil Company (NISOC).
Appendix A In the finite element method, the unknown variable is approximated as the sum of its value at all nodes. Using the definition of shape function, N , the pressure can be written in terms of nodal values and shape functions as follows. n (e )
P=
∑ PN i i
(A.1)
i =1
Integrating Eq. (5) over the volume of element e and using Green's formula, results in the weak form of Eq. (5). Eq. (A.2) is for wellbore nodes and Eq. (A.3) is for nodes without source term.
−
1 μ
∭e k∇P.∇NdV = ϕc ∭e
−
1 μ
∭e ∇.{k∇} = ϕc ∭e
∂P NdV ∂t
∂P NdV + ∂t
(A.2)
∭e qdV
(A.3)
In this paper, Crank-Nicholson method is used for temporal discretization. Eq. (A.1) is combined with Eqs. (A.2) and (A.3), and is rewritten in matrix form as follows.
[A]{P}n = [B]
(A.4)
[A] and [B] matrices are given by Eqs. (A.5) and (A.6), respectively. [A] =
Δt [T ] + [M ] + μCs[Wq ] + μSCs[WT ] 2
(A.5)
⎞ ⎛ Δt [B] = − Δt (Bμ + S[T ]){q} + ⎜[M ]{P}n −1 − [T ] + μCS[Wq ] + μSCs[WT ]⎟{P}n −1 ⎠ ⎝ 2
(A.6)
[T] and [M ] matrices can be calculated using Eqs. (A.7) and (A.8).
[T ]ij = −
k μ
[M ]ij = ϕc
∭e ∇Ni.∇NjdV
(A.7)
∭e NN i j dV
(A.8)
[WT ] and [Wq ] are defined as follows. ⎧ ⎪[WT ]i, j = [T ]i, j ⎨ ⎪[W ] ⎩ T i, j = 0
if j is a wellbore node if j is not a wellbore node
i = 1, 2, …, n (A.9)
Where n is the number of all nodes.
⎧ [Wq ] = 0| i, j = 1, 2, …, n ⎪ i, j ⎨ ⎪[W ] = 1 if i is a wellbore node ⎩ q i, i
(A.10)
As mentioned earlier, matrix [T], which is the source of nonlinearity is calculated at n − 1 time step. Eq. (A.4) is used to find the nodal pressures at n th time step. After that, [T ] is recalculated explicitly using Eqs. (A.7) and (2). For further details about the numerical method refer to Zienkiewicz and Taylor (2005) and Zhang et al. (2016).
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