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Stability Analysis of the Modified IMPES Scheme for Two–Phase Flow in Porous Media Including Dynamic Capillary Pressure
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International Conference on Computational Science, ICCS 2017, 12-14 June 2017, Zurich, Switzerland
Stability Analysis of the Modified IMPES Scheme for Two–Phase Flow inofPorous Media IMPES Including Dynamic Stability Analysis the Modified Scheme for Two–Phase Flow Capillary in Porous Pressure Media Including Dynamic Mohamed F. Pressure El-Amin1,2∗ Capillary 1
1
College of Engineering, Effat University, Jeddah 21478, Kingdom of Saudi Arabia 1,2∗ 2 Mohamed F.ofEl-Amin Mathematics Department, Faculty Science, Aswan University, Egypt [email protected] College of Engineering, Effat University, Jeddah 21478, Kingdom of Saudi Arabia 2 Mathematics Department, Faculty of Science, Aswan University, Egypt [email protected]
Abstract In this paper, the problem of two-phase flow in porous media including dynamic capillary pressure has been studied numerically. The IMplicit Pressure Explicit Saturation (IMPES) Abstract scheme has been modified to solve the governing equations. The pressure equation is treated In this paper, the problem of two-phase flow in porous media including dynamic capillary implicitly with the saturation equation to obtain the pressure, while the saturation equation pressure has been studied numerically. The IMplicit Pressure Explicit Saturation (IMPES) is solved explicitly to update the saturation at each time step. We introduce stability analysis scheme has been modified to solve the governing equations. The pressure equation is treated of the modified scheme and its stability condition has been derived. Comparison between the implicitly with the saturation equation to obtain the pressure, while the saturation equation static and the dynamic capillary pressure has been introduced to illustrate the efficiency of the is solved explicitly to update the saturation at each time step. We introduce stability analysis modified scheme. of the modified scheme and its stability condition has been derived. Comparison between the Keywords: flow, Porous media, pressure, IMPES, Stability analysis. static the dynamic capillary pressure has beencapillary introduced to illustrate the efficiency of the © 2017and The Two–phase Authors. Published by Elsevier B.V.Dynamic Peer-review under responsibility of the scientific committee of the International Conference on Computational Science modified scheme. Keywords: Two–phase flow, Porous media, Dynamic capillary pressure, IMPES, Stability analysis.
1
Introduction
All standard empirical relationships between capillary pressure and saturation (namely, the 1 static Introduction capillary pressure) were correlated by laboratory experiments under equilibrium conditions. These formulas of the static capillary pressure (see e.g. [1, 14]) have been used in most of All standard empirical relationships between capillary pressure and saturation (namely, the the mathematical models of immiscible two–phase flows in porous media. However, it was found static capillary pressure) were correlated by laboratory experiments under equilibrium condiexperimentally and theoretically that under non–equilibrium conditions the capillary pressure tions. These formulas of the static capillary pressure (see e.g. [1, 14]) have been used in most of does not correspond to the static capillary pressure. The dynamic capillary pressure–saturation the mathematical models of immiscible two–phase flows in porous media. However, it was found relationship has been obtained in the framework of a macroscopic theory of porous media flow experimentally and theoretically that under non–equilibrium conditions the capillary pressure by Hassanizadeh and Gray [6]. Moreover, many experiments reported in the literature include does not correspond to the static capillary pressure. The dynamic capillary pressure–saturation evidence of the dynamic effect, such as Stauffer [13]. So, using the static capillary pressurerelationship has been obtained in the framework of a macroscopic theory of porous media flow saturation relationship when the fluid content is in motion may be not suitable. An alternative by Hassanizadeh and Gray [6]. Moreover, many experiments reported in the literature include model of the capillary pressure-saturation relationship was proposed and referred to as dynamic evidence of the dynamic effect, such as Stauffer [13]. So, using the static capillary pressurecapillary pressure is more suitable. When gradients of fluids pressure and velocities are large, saturation relationship when the fluid content is in motion may be not suitable. An alternative ∗ Corresponding Author model of the capillary pressure-saturation relationship was proposed and referred to as dynamic capillary pressure is more suitable. When gradients of fluids pressure and velocities are large, 1 ∗ Corresponding Author 1877-0509 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the International Conference on Computational Science 10.1016/j.procs.2017.05.150
1
Mohamed F. El-Amin Stability Analysis for an IMPES Scheme . . . et al. / Procedia Computer Science 108C (2017) 2328–2332 El-Amin and Sun
non–equilibrium effects in the capillary pressure can be significant. For instant, flow in some industrial porous media, such as paper pulp drying process (Lewalle et al. [8]). Recently, many experimental studies on the dynamics capillary pressure have been introduced [11, 10]. A comprehensive review surveyed the experimental works in which non–equilibrium effects have been observed [5]. Moreover, other computational studies on dynamics capillary pressure using Darcy–scale models have been also introduced [9, 3]. In the case of two–phase flow in porous media, the capillary pressure relationship under non–equilibrium conditions is determined using future water saturation. The model of two-phase fluid flow in porous media is a coupled system of nonlinear timedependent partial differential equations. The IMplicit Pressure Explicit Saturation (IMPES) approach solves the pressure equation implicitly and updates the saturation explicitly. The instability of the IMPES method results of the decoupling between the pressure equation and the saturation equation as well as the explicit treatment of the capillary pressure [2]. Iterative IMPES splits the equation system into an equation for pressure and another saturation equation that are solved sequentially as IMPES [4]. A linear approximation of capillary function is introduced to couple the implicit saturation equation into pressure equation [7]. In this work, we modified the IMPES scheme introduced in [7] to solve the flow equation of the model with the dynamic capillary pressure.
2
Modeling and Mathematical Formulation
In this section, a mathematical model is developed to describe the two-phase flow in porous media including the dynamic capillary pressure. Consider two–phase immiscible incompressible flow in a porous medium which is governed by the Darcy’s law and the equations of mass conservation for each phase as, ∂ (φSα ) + ∇ · u α = qα , ∂t
uα = −
krα K∇pα , μα
α = w, n.
(1)
where Sα is the saturation, uα is the velocity of the phase α. w stands for the wetting phase, and n stands for the nonwetting phase. φ is the porosity of the medium, qα is the external mass flow rate. K is the absolute permeability tensor chosen as K = kI, I is the identity matrix and k is a positive real number. krα is the relative permeability, ρα is the density, and pα is the pressure, μα is the viscosity of the phase α. The fluid saturations for the wetting and non-wetting are interrelated by, Sw + Sn = 1. Now, we describe the governing equations used in [7] as, − ∇ · λt K∇pw − ∇ · λn K∇pc = qw + qn . (2) and ∂ (φSα ) − qw = −∇ · (fw ua ) . ∂t
(3)
where fw = λw /λt is the flow fraction, λα = krα /μα is the mobility. The classical capillary pressure–saturation relationship based on the thermodynamic equilibrium assumption, is commonly given as, pn − pw = ps (Sw ). For non-equilibrium conditions, the following equation for the fluids pressure difference has been suggested (Hassanizadeh and Gray [6]), ∂Sw pn − pw = ps (Sw ) − τ (Sw ) . (4) ∂t 2
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Mohamed F. El-Amin Stability Analysis for an IMPES Scheme . . . et al. / Procedia Computer Science 108C (2017) 2328–2332 El-Amin and Sun
where τ is a non-equilibrium capillarity coefficient (material property) that may be a function of saturation and other fluid–fluid properties. The dynamic capillary pressure may be written as, ∂Sw pd (Sw , S˙ w ) = ps (Sw ) − τB (Sw ) p�s . (5) ∂t
3
Solution Method
Define the time step length Δtn = tn+1 − tn , the total time interval [0, T ] may be divided into NT time steps as 0 = t0 < t1 < · · · < tNT = T . The current time step is represented by the superscript n + 1, while the current time step is represented by the superscript n. The backward Euler time discretization is used for the equation of saturation. The discretized governing equations can be given as, n+1 n n Aa (Sw = Qn+1 ) Pn+1 + Ac (Sw ) Pd Sw w ac .
(6)
It is noted from above algebraic equations that the matrices Aa , Ac and Pd depend on the n n+1 is given by discretizing the approximation of capillary , Sw vector Sw . The vector Pd Sw pressure, which may be rewritten in a matrix-vector form as follows, n+1 n n n+1 − Sw n n Sw = Ps (Sw , Sw ) + τB Pf (Sw ) . Pd S w n Δt
(7)
where Ps depends on the vector Sw . Pf is resulted from the discretization of the p�s and F that n n n ) = diag (F (Sw ) · p�s (Sw )) , h = 1, 2, ..., Nc , Nc is the is a diagonal matrix defined by, Pf (Sw total number of all cells. In fact, the derivative of ps is a function of ps when the saturation at each spatial point is varies with time. At the same time, the saturation is smoothly changing along with time at each spatial point even if it is discontinuously distributed in space. Moreover, the CCFD discretization of the saturation equation is, M
− Snw Sn+1 w n + Aw (Sw ) Pn+1 = Qn+1 w w , Δtn
(8)
where M is a diagonal matrix replaces the porosity. Substituting (7) and (8) into (6), one may obtain the coupled pressure equation in the following form, −1 n+1 (9) = Qn+1 Qw Aa − τB Ac Pf M−1 Aw Pn+1 w ac − Ac Ps + τB Pf M
The upwind scheme is used in the advection term of the saturation equation that used for updating saturation. Thus, M
4
n n+1 − Snw Sn+1 w n fw (Sw + As S w , Pw ) = Qn+1 s n Δt
(10)
Stability Analysis
The dynamic capillarity parameter τB , has the main effect on the stability of this scheme. In the following analysis, the effect of saturation error on the matrices Aw , Aa and Ac is neglected and the capillary pressure is concentrated. Now we need to consider the propagation of numerical errors from time step n to time step (n + 1). Local truncation error is not considered here. n = Sn + δSn , For the nth step saturation Snw we denote for a perturbed saturation by S w w w
3
Mohamed F. El-Amin Stability Analysis for an IMPES Scheme . . . et al. / Procedia Computer Science 108C (2017) 2328–2332 El-Amin and Sun
where δSnw represents the error for Snw . Similarly, we can obtain an inexact saturation at the n+1 = Sn+1 + δSn+1 and δSn+1 is the (n + 1)th step (n + 1)th step that is expressed as S w w w w n ) − Ps (Sn ) � Pf (S n )δSn , one may get, saturation error. Moreover, since we have, Ps (S w w w w −1 n n n+1 n τB n I + (1 − Δtn )Δt HPf (Sw ) and I is the identity δSw � CδSw , C = I − τB HPf (Sw )
matrix. Consequently, the scheme (I) is stable if the following condition holds, ρ (C) < 1, ρ(C) is the spectral radius of the matrix C, i.e. all of the eigenvalues of this matrix must lie within the unit circle in the complex plane. Theoretically, this condition is computable, but practically it is too expensive to compute for guiding the choice of Δtn . We assume that a square domain is partitioned into one cell. As the results of the discretization of CCFD, all the matrices M, Aw , Aa and Ac become positive scalar numbers and so is H. The stability condition becomes, τB n n 1 − (1 − Δt n )Δt HBc /Sw < 1. (11) n 1 + τB HBc /S w
τB This means that the scheme is stable for all Δt n ≥ 0.5. It is interesting to note that this scheme n is stable naturally because τB >> Δt , τB is of order 105 − 107 . For the case of the static capillary pressure (τB = 0), a small time step size is often required to attain the stability of IMPES.
5
Results
Here we introduce an example to test the performance of the presented scheme. Consider the capillary pressure formula, ps = −Bc log(S), Bc = 50 bar. The viscosities of water and oil are 1 cP and 0.45 cP, respectively. The injection rate is 0.1 PV/year and we continue the calculation until 0.5 PVI, with the time step, 0.01 day. The porosity is taken as 0.1 while the permeability is 1 md. In this example, we run the simulation for the static τB =0 and the dynamic pressure, τB >0. Fig. 1 (left) represents the water saturation profiles after 5 days of injection for the two cases static and dynamic pressures. This figure shows that the saturation in the case of dynamic capillary pressure is greater than the saturation in the case of static capillary pressure. Moreover, comparison between static and dynamic capillary pressure are plotted in Fig. 1 (right). Similarly, we may note that the dynamic capillary pressure is greater than the static capillary pressure. These results are comparable to many experimental results such as those in Fig. 5 in Ref. [12].
6
Conclusion
In the current work, we investigated the problem of two-phase flow in porous media including the dynamic capillary pressure has been studied. The IMPES scheme is considered to solving the problem under consideration. The saturation equation is used in two different locations of the scheme. The first location was when substituting the time derivative of the saturation in the capillary pressure term in the pressure equation. In the second location, is the saturation equation which is explicitly with the upwinding advection to update the saturation. The stability condition of the scheme has been determined. We conclude from this study that the dynamic capillary pressure IMPES scheme is naturally stable. 4
2331
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Mohamed F. El-Amin Stability Analysis for an IMPES Scheme . . . et al. / Procedia Computer Science 108C (2017) 2328–2332 El-Amin and Sun
"
!
!
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Figure 1: (Left) saturation profiles after seven days of injection at the static and the dynamic capillary pressures; (right) comparison between the static and the dynamic capillary pressures
References [1] R.H. Brooks and A.T. Corey. Hydraulic properties of porous media. Hydrology, 1, 1964. [2] K. H. Coats. Impes stability: selection of stable time steps. SPE-84924, SPE ReservoirSimulation Symposium, Houston, TX., 2001. [3] D. B. Das, M. Mirzaei, and N. Widdows. Non–uniqueness in capillary pressure–saturation–relative permeability relationships for two-phase flow in porous media: interplay between intensity and distribution of random micro–heterogeneities. Vadose Zone J., pages 6786–6803, 2006. [4] M. F. El-Amin, J. Kou, and S. Sun. Convergence analysis of the nonlinear iterative method for two-phase flow in porous media associated with nanoparticle injection. Int. J. Numer. Meth. Heat Fluid Flow, page 1, 2017. [5] S. M. Hassanizadeh, S. Celia, and H. Dahle. Dynamic effects in the capillary pressuresaturation relationship and their impacts on unsaturated flow. Vadose Zone J., pages 38–57, 2002. [6] S. M. Hassanizadeh and W. G. Gray. Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv. Water Res., 13:196–186, 1990. [7] J. Kou and S. Sun. On iterative impes formulation for two phase flow with capillarity in heterogeneous porous media. Int. J. Numer. Anal. Model. Ser. B, 1(1):20–40, 2010. [8] J. Lewalle, K. Singh, and J. Bambacht. Analysis of the wet pressing of paper pulp. Int. J. Multiphase Flow, 20:415–437, 1994. [9] S. Manthey, Hassanizadeh S. M., and R. Helmig. Macro–scale dynamic effects in homogeneous and heterogeneous porous media. Trans. Porous Media, pages 121–145, 2005. [10] D. M. O’Carroll, T. J. Phelan, and L. M. Abriola. Exploring dynamic effects in capillary pressure in multistep outflow experiments. Water Res. Res., page W11419, 2005. [11] O. Oung, S. M. Hassanizadeh, and A. Bezuijen. Two-phase flow experiments in a geocentrifuge and the significance of dynamic capillary pressure effect. J. Porous Media, pages 247–257, 2005. [12] G. C. Roach, D. M. O’Carroll, T. A. Newson, T. Sakaki, and T. H. Illangasekare. Experimental investigation of dynamic effects in capillary pressure: Grain size dependency and upscaling. Water Resour. Res., page W08544, 2010. [13] F. Stauffer. Time dependence of the relations between capillary pressure, water content and conductivity during drainage of porous media. On Scale Effects in Porous Media, IAHR, Thessaloniki, Greece, 1978. [14] M. T. van Genuchten. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sc. Soc. Amer. J., 44:892–898, 1980.
5
ScienceDirect This space is Computer reservedScience for the Procedia header, do not use it Procedia 108C (2017) 2328–2332 This space is reserved for the Procedia header, do not use it
International Conference on Computational Science, ICCS 2017, 12-14 June 2017, Zurich, Switzerland
Stability Analysis of the Modified IMPES Scheme for Two–Phase Flow inofPorous Media IMPES Including Dynamic Stability Analysis the Modified Scheme for Two–Phase Flow Capillary in Porous Pressure Media Including Dynamic Mohamed F. Pressure El-Amin1,2∗ Capillary 1
1
College of Engineering, Effat University, Jeddah 21478, Kingdom of Saudi Arabia 1,2∗ 2 Mohamed F.ofEl-Amin Mathematics Department, Faculty Science, Aswan University, Egypt [email protected] College of Engineering, Effat University, Jeddah 21478, Kingdom of Saudi Arabia 2 Mathematics Department, Faculty of Science, Aswan University, Egypt [email protected]
Abstract In this paper, the problem of two-phase flow in porous media including dynamic capillary pressure has been studied numerically. The IMplicit Pressure Explicit Saturation (IMPES) Abstract scheme has been modified to solve the governing equations. The pressure equation is treated In this paper, the problem of two-phase flow in porous media including dynamic capillary implicitly with the saturation equation to obtain the pressure, while the saturation equation pressure has been studied numerically. The IMplicit Pressure Explicit Saturation (IMPES) is solved explicitly to update the saturation at each time step. We introduce stability analysis scheme has been modified to solve the governing equations. The pressure equation is treated of the modified scheme and its stability condition has been derived. Comparison between the implicitly with the saturation equation to obtain the pressure, while the saturation equation static and the dynamic capillary pressure has been introduced to illustrate the efficiency of the is solved explicitly to update the saturation at each time step. We introduce stability analysis modified scheme. of the modified scheme and its stability condition has been derived. Comparison between the Keywords: flow, Porous media, pressure, IMPES, Stability analysis. static the dynamic capillary pressure has beencapillary introduced to illustrate the efficiency of the © 2017and The Two–phase Authors. Published by Elsevier B.V.Dynamic Peer-review under responsibility of the scientific committee of the International Conference on Computational Science modified scheme. Keywords: Two–phase flow, Porous media, Dynamic capillary pressure, IMPES, Stability analysis.
1
Introduction
All standard empirical relationships between capillary pressure and saturation (namely, the 1 static Introduction capillary pressure) were correlated by laboratory experiments under equilibrium conditions. These formulas of the static capillary pressure (see e.g. [1, 14]) have been used in most of All standard empirical relationships between capillary pressure and saturation (namely, the the mathematical models of immiscible two–phase flows in porous media. However, it was found static capillary pressure) were correlated by laboratory experiments under equilibrium condiexperimentally and theoretically that under non–equilibrium conditions the capillary pressure tions. These formulas of the static capillary pressure (see e.g. [1, 14]) have been used in most of does not correspond to the static capillary pressure. The dynamic capillary pressure–saturation the mathematical models of immiscible two–phase flows in porous media. However, it was found relationship has been obtained in the framework of a macroscopic theory of porous media flow experimentally and theoretically that under non–equilibrium conditions the capillary pressure by Hassanizadeh and Gray [6]. Moreover, many experiments reported in the literature include does not correspond to the static capillary pressure. The dynamic capillary pressure–saturation evidence of the dynamic effect, such as Stauffer [13]. So, using the static capillary pressurerelationship has been obtained in the framework of a macroscopic theory of porous media flow saturation relationship when the fluid content is in motion may be not suitable. An alternative by Hassanizadeh and Gray [6]. Moreover, many experiments reported in the literature include model of the capillary pressure-saturation relationship was proposed and referred to as dynamic evidence of the dynamic effect, such as Stauffer [13]. So, using the static capillary pressurecapillary pressure is more suitable. When gradients of fluids pressure and velocities are large, saturation relationship when the fluid content is in motion may be not suitable. An alternative ∗ Corresponding Author model of the capillary pressure-saturation relationship was proposed and referred to as dynamic capillary pressure is more suitable. When gradients of fluids pressure and velocities are large, 1 ∗ Corresponding Author 1877-0509 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the International Conference on Computational Science 10.1016/j.procs.2017.05.150
1
Mohamed F. El-Amin Stability Analysis for an IMPES Scheme . . . et al. / Procedia Computer Science 108C (2017) 2328–2332 El-Amin and Sun
non–equilibrium effects in the capillary pressure can be significant. For instant, flow in some industrial porous media, such as paper pulp drying process (Lewalle et al. [8]). Recently, many experimental studies on the dynamics capillary pressure have been introduced [11, 10]. A comprehensive review surveyed the experimental works in which non–equilibrium effects have been observed [5]. Moreover, other computational studies on dynamics capillary pressure using Darcy–scale models have been also introduced [9, 3]. In the case of two–phase flow in porous media, the capillary pressure relationship under non–equilibrium conditions is determined using future water saturation. The model of two-phase fluid flow in porous media is a coupled system of nonlinear timedependent partial differential equations. The IMplicit Pressure Explicit Saturation (IMPES) approach solves the pressure equation implicitly and updates the saturation explicitly. The instability of the IMPES method results of the decoupling between the pressure equation and the saturation equation as well as the explicit treatment of the capillary pressure [2]. Iterative IMPES splits the equation system into an equation for pressure and another saturation equation that are solved sequentially as IMPES [4]. A linear approximation of capillary function is introduced to couple the implicit saturation equation into pressure equation [7]. In this work, we modified the IMPES scheme introduced in [7] to solve the flow equation of the model with the dynamic capillary pressure.
2
Modeling and Mathematical Formulation
In this section, a mathematical model is developed to describe the two-phase flow in porous media including the dynamic capillary pressure. Consider two–phase immiscible incompressible flow in a porous medium which is governed by the Darcy’s law and the equations of mass conservation for each phase as, ∂ (φSα ) + ∇ · u α = qα , ∂t
uα = −
krα K∇pα , μα
α = w, n.
(1)
where Sα is the saturation, uα is the velocity of the phase α. w stands for the wetting phase, and n stands for the nonwetting phase. φ is the porosity of the medium, qα is the external mass flow rate. K is the absolute permeability tensor chosen as K = kI, I is the identity matrix and k is a positive real number. krα is the relative permeability, ρα is the density, and pα is the pressure, μα is the viscosity of the phase α. The fluid saturations for the wetting and non-wetting are interrelated by, Sw + Sn = 1. Now, we describe the governing equations used in [7] as, − ∇ · λt K∇pw − ∇ · λn K∇pc = qw + qn . (2) and ∂ (φSα ) − qw = −∇ · (fw ua ) . ∂t
(3)
where fw = λw /λt is the flow fraction, λα = krα /μα is the mobility. The classical capillary pressure–saturation relationship based on the thermodynamic equilibrium assumption, is commonly given as, pn − pw = ps (Sw ). For non-equilibrium conditions, the following equation for the fluids pressure difference has been suggested (Hassanizadeh and Gray [6]), ∂Sw pn − pw = ps (Sw ) − τ (Sw ) . (4) ∂t 2
2329
2330
Mohamed F. El-Amin Stability Analysis for an IMPES Scheme . . . et al. / Procedia Computer Science 108C (2017) 2328–2332 El-Amin and Sun
where τ is a non-equilibrium capillarity coefficient (material property) that may be a function of saturation and other fluid–fluid properties. The dynamic capillary pressure may be written as, ∂Sw pd (Sw , S˙ w ) = ps (Sw ) − τB (Sw ) p�s . (5) ∂t
3
Solution Method
Define the time step length Δtn = tn+1 − tn , the total time interval [0, T ] may be divided into NT time steps as 0 = t0 < t1 < · · · < tNT = T . The current time step is represented by the superscript n + 1, while the current time step is represented by the superscript n. The backward Euler time discretization is used for the equation of saturation. The discretized governing equations can be given as, n+1 n n Aa (Sw = Qn+1 ) Pn+1 + Ac (Sw ) Pd Sw w ac .
(6)
It is noted from above algebraic equations that the matrices Aa , Ac and Pd depend on the n n+1 is given by discretizing the approximation of capillary , Sw vector Sw . The vector Pd Sw pressure, which may be rewritten in a matrix-vector form as follows, n+1 n n n+1 − Sw n n Sw = Ps (Sw , Sw ) + τB Pf (Sw ) . Pd S w n Δt
(7)
where Ps depends on the vector Sw . Pf is resulted from the discretization of the p�s and F that n n n ) = diag (F (Sw ) · p�s (Sw )) , h = 1, 2, ..., Nc , Nc is the is a diagonal matrix defined by, Pf (Sw total number of all cells. In fact, the derivative of ps is a function of ps when the saturation at each spatial point is varies with time. At the same time, the saturation is smoothly changing along with time at each spatial point even if it is discontinuously distributed in space. Moreover, the CCFD discretization of the saturation equation is, M
− Snw Sn+1 w n + Aw (Sw ) Pn+1 = Qn+1 w w , Δtn
(8)
where M is a diagonal matrix replaces the porosity. Substituting (7) and (8) into (6), one may obtain the coupled pressure equation in the following form, −1 n+1 (9) = Qn+1 Qw Aa − τB Ac Pf M−1 Aw Pn+1 w ac − Ac Ps + τB Pf M
The upwind scheme is used in the advection term of the saturation equation that used for updating saturation. Thus, M
4
n n+1 − Snw Sn+1 w n fw (Sw + As S w , Pw ) = Qn+1 s n Δt
(10)
Stability Analysis
The dynamic capillarity parameter τB , has the main effect on the stability of this scheme. In the following analysis, the effect of saturation error on the matrices Aw , Aa and Ac is neglected and the capillary pressure is concentrated. Now we need to consider the propagation of numerical errors from time step n to time step (n + 1). Local truncation error is not considered here. n = Sn + δSn , For the nth step saturation Snw we denote for a perturbed saturation by S w w w
3
Mohamed F. El-Amin Stability Analysis for an IMPES Scheme . . . et al. / Procedia Computer Science 108C (2017) 2328–2332 El-Amin and Sun
where δSnw represents the error for Snw . Similarly, we can obtain an inexact saturation at the n+1 = Sn+1 + δSn+1 and δSn+1 is the (n + 1)th step (n + 1)th step that is expressed as S w w w w n ) − Ps (Sn ) � Pf (S n )δSn , one may get, saturation error. Moreover, since we have, Ps (S w w w w −1 n n n+1 n τB n I + (1 − Δtn )Δt HPf (Sw ) and I is the identity δSw � CδSw , C = I − τB HPf (Sw )
matrix. Consequently, the scheme (I) is stable if the following condition holds, ρ (C) < 1, ρ(C) is the spectral radius of the matrix C, i.e. all of the eigenvalues of this matrix must lie within the unit circle in the complex plane. Theoretically, this condition is computable, but practically it is too expensive to compute for guiding the choice of Δtn . We assume that a square domain is partitioned into one cell. As the results of the discretization of CCFD, all the matrices M, Aw , Aa and Ac become positive scalar numbers and so is H. The stability condition becomes, τB n n 1 − (1 − Δt n )Δt HBc /Sw < 1. (11) n 1 + τB HBc /S w
τB This means that the scheme is stable for all Δt n ≥ 0.5. It is interesting to note that this scheme n is stable naturally because τB >> Δt , τB is of order 105 − 107 . For the case of the static capillary pressure (τB = 0), a small time step size is often required to attain the stability of IMPES.
5
Results
Here we introduce an example to test the performance of the presented scheme. Consider the capillary pressure formula, ps = −Bc log(S), Bc = 50 bar. The viscosities of water and oil are 1 cP and 0.45 cP, respectively. The injection rate is 0.1 PV/year and we continue the calculation until 0.5 PVI, with the time step, 0.01 day. The porosity is taken as 0.1 while the permeability is 1 md. In this example, we run the simulation for the static τB =0 and the dynamic pressure, τB >0. Fig. 1 (left) represents the water saturation profiles after 5 days of injection for the two cases static and dynamic pressures. This figure shows that the saturation in the case of dynamic capillary pressure is greater than the saturation in the case of static capillary pressure. Moreover, comparison between static and dynamic capillary pressure are plotted in Fig. 1 (right). Similarly, we may note that the dynamic capillary pressure is greater than the static capillary pressure. These results are comparable to many experimental results such as those in Fig. 5 in Ref. [12].
6
Conclusion
In the current work, we investigated the problem of two-phase flow in porous media including the dynamic capillary pressure has been studied. The IMPES scheme is considered to solving the problem under consideration. The saturation equation is used in two different locations of the scheme. The first location was when substituting the time derivative of the saturation in the capillary pressure term in the pressure equation. In the second location, is the saturation equation which is explicitly with the upwinding advection to update the saturation. The stability condition of the scheme has been determined. We conclude from this study that the dynamic capillary pressure IMPES scheme is naturally stable. 4
2331
2332
Mohamed F. El-Amin Stability Analysis for an IMPES Scheme . . . et al. / Procedia Computer Science 108C (2017) 2328–2332 El-Amin and Sun
"
!
!
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Figure 1: (Left) saturation profiles after seven days of injection at the static and the dynamic capillary pressures; (right) comparison between the static and the dynamic capillary pressures
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