Numerical Treatment of Two-Phase Flow in Porous Media Including Specific Interfacial Area

Transcription

1 Procedia Computer cience Volume 51, 2015, Pages ICC 2015 International Conference On Computational cience Numerical Treatment of To-Phase Flo in Porous Media Including pecific Interfacial Area M. F. El-Amin 1, R. Meftah 1,2, A. alama 1, and un 1 1 Computational Transport Phenomena Laboratory CTPL, Division of Physical ciences and Engineering PE, King Abdullah University of cience and Technology KAUT, Thual , Jeddah, Kingdom of audi Arabia 2 University Pierre Marie Curie, Paris, France mohamed.elamin@kaust.edu.sa Abstract In this ork, e present a numerical treatment for the model of to-phase flo in porous media including specific interfacial area. For numerical discretization e use the cell-centered finite difference CCFD method based on the shifting-matrices method hich can reduce the time-consuming operations. A ne iterative implicit algorithm has been developed to solve the problem under consideration. All advection and advection-like terms that appear in saturation equation and interfacial area equation are treated using upind schemes. elected simulation results such as p c a n surface, capillary pressure, saturation and specific interfacial area ith various values of model parameters have been introduced. The simulation results sho a good agreement ith those in the literature using either pore netork modeling or Darcy scale modeling. Keyords: Interfacial area, To-phase flo, Porous media, Capillary pressure, CCFD method, hiftingmatrices method 1 Introduction The model of flo and transport in porous media, Darcy s la, as originally proposed empirically for one-dimensional isothermal flo of an incompressible fluid in a rigid, homogeneous and isotropic porous medium. The extended Darcy s La is used for highly complex situations like non-isothermal, multiphase and multicomponent flo and transport, ithout introducing any additional driving forces. The resisting force, hich balances the driving force, is assumed to be linearly proportional to the relative fluid velocity ith respect to the solid. This results in a linear relationship beteen the flo velocity and driving forces. While these assumptions are reasonable for single-phase flo, one may expect many other factors to affect the balance of forces in the case of multiphase flo. Hassanizadeh and Gray [7, 8] developed a unified model Corresponding Author election and peer-revie under responsibility of the cientific Programme Committee of ICC 2015 c The Authors. Published by Elsevier B.V doi: /j.procs

2 Numerical Treatment of To-Phase Flo... El-Amin, Meftah, alama and un based on rational thermodynamics leading to additional driving forces. The macroscale effects of interfacial forces ere included and the momentum balance equations are derived for both phases and interfaces. They found that the driving forces for the flo of a phase ere the gradients of Gibbs free energy of the phase plus gravity. In the case of single-phase flo, the gradient of Gibbs free energy reduces to the gradient of pressure. Moreover, capillary pressure is not a hysteretic function of saturation only, but uniquely defined by a capillary pressure-saturationspecific interfacial area surface. In Hassanizadeh and Gray [7], the equations constitute a very complex model that includes many different effects. Hoever, some of these effects are being of second order, hereas several of the involved constitutive relationships have not been investigated so far. In Hassanizadeh and Gray [8], the momentum balance equations ere simplified. These studies ere folloed by a number of other studies e.g. [9, 10]. Most of these orks ere done using pore netork modeling and/or experiments and very fe attempts ere done to simulate the macroscale multiphase flo model including the specific interfacial area such as Ref. [17]. The IMplicit Pressure Explicit aturation IMPE approach solves the pressure equation implicitly and updates the saturation explicitly. In IMPE method, the time step size should be very small, in particular, for highly heterogeneous permeable media. Because the decoupling beteen the pressure equation and the saturation equation, the IMPE method is conditionally stable. Iterative IMPE splits the equation system into a pressure and a saturation equation that are solved sequentially as IMPE [14, 15, 13]. The objective of this ork is to develop an accurate numerical simulator for to-phase flo in porous media including specific interfacial area using appropriate numerical methods. The cell-centered finite difference CCFD method together ith the shifting-matrices approach are used for spacial discretization, hile the upind scheme is used to treat the advection terms in the differential system. 2 Modeling and Mathematical Formulation Hassanizadeh and Gray [7] have used a thermodynamic approach using the concept of Gibbs free energy. They concluded that the movement of each phase or interface is governed by its gradient in Gibbs free energy. No, let us consider the flo of to immiscible and compressible fluids including interfacial area. Assuming that the porosity is constant; the interfacial mass density is constant; the gravity does not have effect on interfacial movement; all material properties are neglected; mass exchange beteen phases and interfaces has negligible effect on mass balance of phases; and the cross-coupling terms are negligible. Therefore, the governing equations describing the above system can be ritten as follos [16], φ α + v α = q α, α =, n 1 v α = K k rα μ α p α ρ α g, α =, n 2 a n + a n v n = E n 3 v n = K n a n 4 here φ is porosity, α, v α and q α denote, respectively, to the saturation, the Darcy s velocity and the source term of the α phase. α stands for the etting phase or the nonetting phase n. K is the intrinsic permeability, μ α is the dynamic viscosity, k rα denotes the relative 1250

3 Numerical Treatment of To-Phase Flo... El-Amin, Meftah, alama and un permeability, ρ α the density and p α the pressure of the phase α. a n [m 1 ] is the specific interfacial area. v n [m s 1 ] is the interfacial velocity. K n [m 3 s 1 ] is the interfacial permeability. E n [m 1 s 1 ] is the production rate of specific interfacial area. For the production rate of specific interfacial area Joekar-Niasar et al. [11] have defined it as, E n = e n 5 To estimate the production rate of specific interfacial area they neglect the advective flux term of Eq.3. Also the term is calculated as the rate of change of specific interfacial area ith time. They defined a linear relationship beteen E n and /. In their ork, they also defined e n as being a linear function of : e n = e 1 + e 2. The coefficients e 1 and e 2 depend on etting-phase and non-etting phase viscosity. On the other hand, in order to close the above system of equations three relations have been added. The first relation is the the sum of etting-phase and non-etting phase saturations equals one. The second relation is the difference beteen non-etting phase and etting-phase pressure hich is equal to the capillary pressure. The last relation indicates that interfacial specific area is a function of the etting-phase saturation and the capillary pressure. + n =1 6 p c = p n p 7 a n = a n,p c 8 For the last relation of closure one can use a bi-quadratic relationship depends on and p c see Ref. [12]. Instead of the above complicated formula, e ill use the folloing flexible form Ref. [10], a n,p c =α 1 1 α2 p c α 3 9 here α 1,α 2 and α 3 are constants. No, let us define the potential phase α pressure as Φ α = p α ρ α g, and the potential capillary pressure as Φ c = p c ρ n ρ g. Also, the total velocity is defined as v t = v + v n, the total mobility λ t = λ + λ n. Adding the saturation equation of etting phase to the saturation equation of nonetting phase 1 ith the aid of 6, one may obtain, v t λ t K Φ λ n K Φ c,a n =Q t 10 here Q t = q + q n. Also, the etting phase velocity may be defined as, v = λ K Φ, Therefore, saturation equation of the etting phase may be ritten as, φ λ K Φ φ v = q. 11 From 3, 5 and 11, e can rite the equation of specific interfacial area as, φ a n + φ v n a n +e n v =e n q

4 Numerical Treatment of To-Phase Flo... El-Amin, Meftah, alama and un The saturation of the etting phase at the beginning of the flo displacing process is initially defined by, = 0 at t =0. 13 Also, the initial specific interfacial area is zero, i.e., a n = a 0 n at t =0. 14 The general boundary conditions considered in this study are summarized as follo, p or p n =p D on Γ D, 15 v t n = q N on Γ N, 16 v n n = N on Γ N, 17 here Γ D is the Dirichlet boundary and Γ N is the Neumann boundary. n is the outard unit normal vector to Γ N, p D is the pressure on Γ D and q N the imposed inflo rate on Γ N, respectively. The saturations on the boundary are subject to, and the specific interfacial area on the boundary is subject to, 3 Iterative Implicit cheme or n = N on Γ N, 18 a n = a N n on Γ N. 19 Define the time step length = n t n, the total time interval [0,T] may be divided into N time steps as 0 = t 0 <t 1 < <t N = T. The current time step is represented by the superscript n + 1. The backard Euler time discretization is used for the equations 10-12, to obtain, λ t K Φ λ n K Φc,a n = Q t, 20 and, φ n λ K Φ = q, 21 φ a n a n n + φ v n a n e n λ K Φ = e n q. 22 The system is fully implicit, coupled and highly nonlinear, so it can not be solved directly. Hence, iterative methods are often employed to solve such kind of complicated systems. No, let us introduce the iterative Implicit Pressure Explicit aturation-interfacial Area IMPEA formulation for the equations that is given as, λ t K Φ +1 λ n K Φc +1, ã +1 n = Q t, 23 φ +1 n Δt λ K Φ +1 = q,

5 Numerical Treatment of To-Phase Flo... El-Amin, Meftah, alama and un φ ã+1 n a n n + φ v n a n e n = e n q. λ K Φ +1 The superscripts k and k +1 represent the iterative steps ithin the current time step. For each iteration, the variables λ,λ n and λ t are calculated using the saturation from the previous iteration. Hoever, e may consider the saturation and the specific interfacial area at the current iteration step instead of the previous iteration to obtain the capillary potential Φ c. We may linearize the capillary potential Φ c as follos, Φ c +1, ã +1 n [ +1 = Φc ] + Φc,a n a n + Φ c,a n,a [ã+1 n n ] a n The changes of saturation and interfacial area in a time step are often very small, and hence the linear approximation is reasonable. Moreover, the relaxation approach is often applied to control the convergence of nonlinear iterative solvers. In this algorithm e use the folloing to relaxation relationships for saturation and interfacial area as, and, +1 a +1 n = = a n here θ s,θ a 0, 1] are relaxation factors. 4 patial Discretization + θ s +1 + θ a ã+1 n 25 26, 27 a n, 28 No, let us apply the CCFD scheme to the system of equations to obtain the iterative discretization. The discretization form of the pressure equation may be given as, A t Φ +1 + A c Pc +1, ã +1 n = Q ct. 29 It is noted from above algebraic equations that the matrices A and A c depend on the vector at the previous iteration step, hile the vector P c depends on both vectors +1 and a +1 n at the current iteration step, hich may be ritten as, in a matrix-vector form as follos, P c +1, ã +1 n [ +1 = Φc ],a n + P a + Ps,a n,a [ã+1 n n ] a n here the matrices P s and P a are diagonal resulted from the discretization of Φc and Φc a n, respectively. In fact, the derivatives of Φ c are vieed as functions of Φ c hen the saturation and the interfacial area at each spatial point vary ith time. At the same time, the saturation and the interfacial area are smoothly changing along ith time at each spatial point even they are discontinuously distributed in space. Also, the CCFD discretization of the saturation equation 24 may lead to, +1 n M 30 + A Φ +1 = Q,

6 Numerical Treatment of To-Phase Flo... El-Amin, Meftah, alama and un here M is a diagonal matrix replaces the porosity that appears in the saturation equation and also it is a function of cell area. It is orth mentioning that the above equation of saturation is coupled ith the pressure equation to be solved implicitly together, hoever it is not used to update the saturation. imilarly, the CCFD discretization of the interfacial area equation 25 can be given as, M ã+1 n a n n E + M [ A n,a a n a n A Φ +1 + A n,s a n = E Q, ] + here E is a diagonal matrix. The above equation of saturation is coupled ith the pressure equation to be solved implicitly together, hoever it is not used to update the saturation. ubstituting from into 29, the coupled pressure equation becomes, here and A t A t,a n Q t P s P a [ P s,a n Φ +1 = At A c P a,a n,a n = Q ct n = Q t [ Ps 32,a n. 33,a n M 1 + M 1 E ] 34 A A c + Pa [ An,a a n a n,a n M 1 + P a { Φc + a n n a a + A n,s a n,a n ] + M 1 E ] } Q. After updating the velocity u +1, e can update the saturation, e consider the folloing explicit scheme, φ +1 n + u +1 = q. 36 No, e can update all variables that are functions in. Then, after updating n +1,the interfacial area can be updated by the folloing explicit scheme, φ a+1 n a n n + φ +1 n a n 35 + e +1 n u +1 = e +1 n q. 37 The upind CCFD discretization of the above to equations 36 and 37 are, and, Mã+1 n a n n +1 n M + MA + A u,a n a n,a n = Q s, 38 + EA u,a n = EQ a, 39 respectively. The hifting Matrices method [19] is used in the implementation of the above discretized system. 1254

7 Numerical Treatment of To-Phase Flo... El-Amin, Meftah, alama and un 5 Numerical Experiments This section analyzes some results of saturation, capillary pressure and specific interfacial area. For example, e choose fluid parameters densities, viscosities, and diffusion constants of ater and air; for the matrix parameters, e take typical soil parameters, here k =10 10 m 2 and φ =0.3. We take a rough estimate of k n =10 5 m 3 s 1. We set the interfacial permeability K n to 10 5 m 3 /s. The ratio beteen non-etting phase and etting-phase viscosity is μ n /μ =0.1. The coefficients for the production rate of specific interfacial area e use the values from Ref. [10]. The initial condition for the specific interfacial a n is taken as 5790 [1/m] and at the boundary conditions are given as: 1000 [1/m] on the est boundary; 5790 [1/m] on the east boundary; and 0 [1/m] on the north and south boundary. Also, the boundary conditions of the interface velocity v n,isconsidered0.001 [m/year] on the est boundary and 0[m/year] on the east, north and south boundaries. The parameters in the capillary pressure formula are set to α 1 = , α 2 =1.18 and α 3 = 0.8. Figure 1 represents relation beteen specific interfacial area, saturation and capillary pressure. It is interesting to note that this 3D surface is comparable to previous experimental and computational studies, [1, 10, 18]. Figure 2a shos the variation of capillary pressure ith saturation. From this figure e may conclude that the capillary pressure increases ith decreasing etting-phase saturation. The variations of the specific interfacial area ith etting-phase saturation are plotted in Fig. 2b. According to this figure, one can observe that the specific interfacial area decreases ith increasing the etting-phase saturation. In another example, e choose counter-current imbibition model [3, 4, 5, 6, 2]. In the case of considering the specific interfacial area, the capillary pressure formula p c,a n isvery sensitive to its parameters. The values of these parameters may be modified based on the type of porous media and other physical and computational aspects. o, here e investigate the sensitivity of the capillary pressure parameters, namely, α 1,α 2 and α 3. In Fig. 3a, the specific interfacial area is plotted against the capillary pressure ith different values of α 1,atα 2 =1.18 and α 3 = 0.8. This figure shos that the specific interfacial area could has same values for different capillary pressure ranges based on the values of the parameter α 1. Fig. 3b shos the specific interfacial area profiles against the capillary pressure ith various values of the parameter α 2,atα 1 = and α 3 = 0.8. It is interesting to note from Fig. 3b that the behavior of the interfacial area profiles ith small values of α 2 is totally different from those ith bigger values of α 2. The specific interfacial area decreases as the parameter α 2 decreases and opposite is true for the capillary pressure. The specific interfacial area is plotted in Fig. 3c against the capillary pressure ith different values of α 3,atα 1 = and α 2 =1.18. It can be seen from this figure that as the parameter α 3 increases the interfacial area slightly increases ith bigger values of the capillary pressure. Also, Fig. Fig. 3c illustrates the capillary pressure variation against the ater saturation at different values of α 1,atα 2 =1.18 and α 3 = 0.8. It is clear from this figure that as the parameter α 1 increases the capillary pressure increases. Moreover, it is interesting to note from the same figure that the capillary pressure has reasonable physical values of α 1 o10 5. The profiles of the capillary pressure are plotted in Fig. 3d against the ater saturation ith different values of the parameter α 2 =0.01 : 2.5, at α 1 = and α 3 = 0.8. It is clear here that hen α 2 > 1, the behavior of the P c curve deviates from the standard behavior. Fig. 3e shos the capillary pressure against the ater saturation ith different values of the α 3,ithα 1 = and α 2 =1.18. From Fig. 3f, it can be noted that the behavior of the P c curve deviates from the standard behavior at α 3 >

8 Numerical Treatment of To-Phase Flo... El-Amin, Meftah, alama and un Figure 1: Distribution of p c a n surface Figure 2: Variation of a capillary pressure ith saturation, and b Variationofspecific interfacial area ith saturation References [1] J. T. Cheng, L. J. Pyrak-Nolte, D. D. Nolte, and N. J. Giordano. Linking pressure and saturation through interfacial areas in porous media. Geophysical Research Letters, 318,

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