Modified Method of Characteristics Combined with Finite Volume Element Methods for Incompressible Miscible Displacement Problems in Porous Media

The incompressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear partial differential equations, the pressure-velocity equation and the concentration equation. In this paper, we present a mixed finite volume elementmethod (FVEM) for the approximation of the pressure-velocity equation. Sincemodifiedmethod of characteristics (MMOC) minimizes the grid orientation effect, for the approximation of the concentration equation, we apply a standard FVEM combined with MMOC. A priori error estimates in L∞(L2) norm are derived for velocity, pressure and concentration. Numerical results are presented to substantiate the validity of the theoretical results.


Introduction
A mathematical model describing miscible displacement of one incompressible fluid by another in a horizontal porous medium reservoir Ω ⊂ R 2 with boundary Ω of unit thickness over a time period of  = (0, ] is given by with boundary conditions (u) ∇ ⋅ n = 0 ∀ (, ) ∈ Ω × , and initial condition  (, 0) =  0 () ∀ ∈ Ω, where  = ( 1 ,  2 ) ∈ Ω, u(, ) = ( 1 (, ),  2 (, )) and (, ) are, respectively, the Darcy velocity and the pressure of the fluid mixture, (, ) is the concentration of the fluid, c is the concentration of the injected fluid, () is the concentration dependent viscosity of the mixture, () is the 2×2 permeability tensor of the medium, (, ) is the external source/sink term that accounts for the effect of injection and production wells, and () is the porosity of the medium.Further, (u) = (, u) is the diffusion-dispersion tensor  (u) =  () [   + |u| (   (u) +   ( −  (u)))] , (7) where   is the molecular diffusion,   and   are, respectively, the longitudinal and transverse dispersion coefficients, (u) is the tensor that projects onto u direction, whose ijth component is given by International Journal of Partial Differential Equations and  is the identity matrix of order 2.  (, , ) = (c − )  is a known function representing sources denoted as () for convenience and  0 () represents the initial concentration.
For physical relevance 0 ≤  0 () ≤ 1 and n denotes the unit exterior normal to Ω.The following compatibility condition is imposed on the data: which can be easily derived from (1)-( 2) and (4).Equation (9) indicates that, for an incompressible flow with an impermeable boundary, the amount of injected fluid and the amount of fluid produced are equal.We also assume that the functions , , , and  are bounded; that is, there exist positive constants  * , The authors in [1][2][3] have discussed mathematical theory and existence of a unique weak solution of the above system (1)-( 6) under suitable assumptions on the data.The pressurevelocity equation is elliptic type while the concentration equation is convection dominated diffusion type.Since, in the concentration equation only velocity is present, one would like to find a good approximation of the velocity.Therefore, for approximating velocity, it is natural to think of some mixed methods, which provide more accurate solution for the velocity compared to the standard finite element methods.Earlier, Douglas et al. [4,5], Ewing and Wheeler [6], and Darlow et al. [7] have discussed the mixed finite element method for approximating the velocity as well as pressure and a standard Galerkin method for the concentration equation.They have also derived optimal error estimates in  ∞ ( 2 ) norm for the velocity and concentration.Recently, Kumar [8] has discussed a mixed and discontinuous FVEM for incompressible miscible displacement problems in porous media.
Since the concentration equation is a convection dominated diffusion equation, the standard numerical methods fail to provide an accurate solution for the concentration and, therefore, suitable numerical methods have been proposed in the past for the approximation of the concentration equation.The standard numerical schemes fail to provide a physically relevant solution because most of these methods suffer from grid orientation effects.The other way to minimize the grid orientation effect is to use modified methods of characteristics (MMOC).Douglas and Russell [9] introduced and analyzed MMOC for the approximation of convection dominated diffusion equations.The authors in [10][11][12] studied MMOC combined with Galerkin finite element methods for incompressible miscible displacement problems.
The basic idea behind the modified method of characteristics for approximating the concentration equation ( 3) is to set the hyperbolic part, that is, (/) + u ⋅ ∇, as a directional derivative.Set (see Figure 1) The characteristic direction with respect to the operator (/) + u ⋅ ∇ is the unit vector The directional derivative of the concentration (, ) in the direction of s is given by This implies that (, )(/) = ()(/) + u ⋅ ∇.
Hence, (3) can be rewritten as Since ( 15) is in the form of heat equation, the behavior of the numerical solution of (15) should be better than (3) if the derivative term / is approximated properly.We choose the same time steps for pressure and concentration for simplicity.However, the analysis can be extended to the case when different time steps are chosen for velocity and concentration through minor modifications.
International Journal of Partial Differential Equations This suggests us to approximate the characteristic directional derivative at  =  +1 as where  +1 = (,  +1 ).
Using (16), we obtain where c  = ( x ,   ).Compared to the conforming finite element methods (FEM), the finite volume methods are conservative in nature, and, hence, they preserve the physical conservative properties.In a mixed FVEM, one uses two different kinds of grids: a primal grid and a dual grid (see Figures 3 and 4).Mixed FVEM can also be thought of as a Petrov-Galerkin method.The analysis of these methods is based on the tools borrowed from the mixed FEM.Using a transfer operator which maps the trial space to the test space, the mixed covolume methods can be put in the framework of mixed FEM.This transfer operator plays a vital role in deriving the optimal error estimates.Earlier, Chou et al. [13,14] have discussed and analyzed mixed covolume or FVEM for the second-order linear elliptic problems in two-dimensional domains.The standard FVEM can also be considered as a Petrov-Galerkin finite element method in which the trial space is chosen as  0 piecewise linear polynomials on the finite element partition of the domain and the test space, as piecewise constants over the control volumes are to be defined in Section 2. For more details on finite volume methods, we refer to [15][16][17][18] and references there in.In this paper, we present a mixed FVEM for approximating the pressure-velocity equations ( 1)-( 2) and (4) and a standard FVEM combined with MMOC for the approximation of the concentration equations ( 15) and ( 5)-( 6).This paper is organized as follows.In Section 2, FVE approximation procedure is discussed.A priori error estimates for velocity, pressure, and concentration are presented in Section 3. Finally in Section 4, the numerical procedure is discussed and some numerical experiments are presented.

Finite Volume Element Approximation
Let  = {k ∈ (div; Ω) : k ⋅ n = 0 on Ω}.Note that (1)-( 2) with boundary condition (4) has a solution for pressure, which is only unique up to an additive constant.The nonuniqueness of ( 1)-( 2) may be avoided by considering the following quotient space: Multiply ( 1) and (2) by k ∈  and  ∈ , respectively, and integrate over Ω.A use of Green's formula and k ⋅ n = 0 on Ω yields the following weak formulation: Find (u, ) :  →  ×  satisfying We use a mixed FVEM for the simultaneous approximation of velocity and pressure in (1)-( 2) and a standard FVEM for the approximation of the concentration in (15).For this purpose, we introduce three kinds of grids: one primal grid and two dual grids.
Let T ℎ = {} be a regular, quasiuniform partition of the domain Ω into closed triangles ; that is, Ω = ∪ ∈T ℎ .Let ℎ  = diam() and ℎ = max ∈T ℎ ℎ  .Let the trial function spaces  ℎ and  ℎ associated with the approximation of velocity and pressure, respectively, be the lowest order Raviart-Thomas space for triangles defined by Let us define the discrete norm for where where  is a constant independent of ℎ.For k h ∈  ℎ the inequality also holds true when Ω is in R 2 and the triangulation T ℎ is quasiuniform and can be proved using the same arguments as in the proof of Lemma 4 in [19, pp. 67].
The dual grid T * ℎ consists of interior quadrilaterals and boundary triangles, which are constructed by joining barycenter to the vertices.For the construction of the dual grid T * ℎ we refer to [14].In general, let  *  denote the dual element corresponding to the midside node .The union of all the dual elements/control volume elements forms a partition T * ℎ of Ω.The test space  ℎ is defined by where  *  denotes the dual element corresponding to the midside node .For connecting our trial and test spaces, we define a transfer operator  ℎ :  ℎ →  ℎ by where   is the total number of midside nodes and  *  's are the scalar characteristic functions corresponding to the control volume  *   defined by Multiplying (1) by  ℎ k h ∈  ℎ , integrating over the control volumes  *  ∈ T * ℎ , applying the Gauss's divergence theorem, and summing up over all the control volumes, we obtain where n  *   denotes the outward normal vector to the boundary of  *   .Set Then, the mixed FVE approximation corresponding to (1)-( 2) can be written as follows: find (u h ,  ℎ ) :  →  ℎ ×  ℎ such that, for  ∈ (0, ], where  ℎ is an approximation to  obtained from (34).Now, we introduce a dual mesh V * ℎ based on T ℎ which will be used for the approximation of the concentration equation.For construction we refer to [18].
For applying the standard FVEM to approximate the concentration, we define the trial space  ℎ on T ℎ and the test space  ℎ on V * ℎ as follows: where  *  is the control volume associated with node .Again, we define a transfer function Π * ℎ :  ℎ →  ℎ by where  ℎ is the total number of vertices and   's are the characteristic functions corresponding to the control volume  *   given by For obtaining a finite volume approximation  ℎ to the concentration , we multiply (15) by Π * ℎ  ℎ ∈  ℎ , integrate over the control volumes   *  ∈ V * ℎ , and apply the Gauss's divergence theorem.Then we sum up over the control volumes to obtain the FVE approximation  ℎ corresponding to the concentration equation ( 15) as a solution  ℎ :  →  ℎ such that for  ∈ (0, ], where  0,ℎ is an approximation to  0 to be defined later and the bilinear form  ℎ (k; ⋅, ⋅) is defined by n   being the unit outward normal to the boundary of  *   with k ∈ ,  ∈  1 (Ω), and  ℎ ∈  ℎ .
Remark 1.The three grids are introduced each for the pressure, velocity, and concentration variables.This is to balance the number of unknowns and the equations in the coupled systems (30) and (34).
To approximate the concentration at any time, say  +1 , we use the approximation to the velocity at the previous time step.The fully discrete scheme corresponding to (30) and ( 34) is defined as follows.For  = 0, 1 . . ., find where ĉ ℎ =  ℎ ( x,   ) =  ℎ ( − (u n h /)Δ,   ) and  ℎ  is a projection of  onto  ℎ which will be defined in (45).
Note that in (17).we use the following notation for the exact velocity:

Error Estimates
3.1.Error Estimates for Velocity.For a given , the auxiliary functions (ũ ℎ , pℎ ) : [0, ] →  ℎ ×  ℎ are defined as follows: For a proof of existence and uniqueness of the discrete solution of (41), we refer to [20, pp. 52].For u h and ũℎ , the following error estimates can be obtained (see [14]): Now, since concentration depends on the velocity and vice versa, to derive the error estimates for the concentration, we also need error estimates for the velocity.For elliptic problems, the authors in [14] have derived error estimates for mixed covolume method by using Raviart-Thomas projection and  2 projection.In the similar way, for a given , the following theorem can be shown but the proof is long so we omit it here.
Theorem 2. Assume that the triangulation T ℎ is quasiuniform.Let (u, ) and (u h ,  ℎ ), respectively, be the solutions of (1)-( 2) and (30).Then, there exists a positive constant  independent of ℎ but dependent on the bounds of  −1 and  such that (44)

Error Estimates for Concentration. We write 𝑐 − 𝑐
where The function  will be chosen such that the coercivity of (u; ⋅, ⋅) is assured.We use frequently the following trace inequality [21, pp.417]: for  ∈  1 (), where ‖‖ 2  = ∫   2 .Further, we need the following inverse inequalities (see [22, where Also note that, by the usual interpolation theory, the operator Π * ℎ has the following approximation property [23, pp. 466]: We also recall two well-known results (Lemmas 4 and 5), which will be used in the the proof of Lemmas 7-9.
Combine the estimates (63) and (59) and use the triangle inequality to complete the proof.For deriving the  2 error bounds for  −  ℎ , we need the following lemma.Lemma 8. Let   (u; ,  ℎ ) = (u; ,  ℎ ) −  ℎ (u; ,  ℎ ).There exists a positive constant  such that, for  ∈ To bound  1 , first we use the fact that  ℎ  is linear on each triangle  to obtain where (∇ ⋅ (u))  denotes the average value of ∇ ⋅ (u) on triangle .
Based on the analysis in [25, pp.1873], we estimate  2 as follows.Note that an appeal to the continuity of ∇ ⋅ n with (49) yields where  = (u) and   is a function such that, for any edge of a triangle  ∈ T ℎ , and   is the midpoint of .Since |() −   | ≤ ℎ  ‖‖ 1,∞ , we use trace inequalities (47) and ( 61) to arrive at where   = ∑ 3 =1 ∫  *  ∩ ((u) − (u h ))∇ ℎ  ⋅ n     and   = (  ); see Figure 5.For each triangle ,   can be written as Using the Cauchy-Schwarz inequality and (79), we obtain It has been proved in [26, pp. 332] that the matrix  is uniformly Lipschitz; that is, there exists a constant  such that for u and k ∈ ( 2 (Ω)) 2 , A use of the trace inequalities (47) and (84) yields Let and this completes the rest of the proof.Now, we prove our main theorem.
Theorem 11.Let   and   ℎ be the solutions of (3) and (39) at  =   , respectively, and let  ℎ (0) =  0,ℎ =  ℎ (0).Further assume that Δ = (ℎ).Then, for sufficiently small ℎ, there exists a positive constant () independent of ℎ but dependent on the bounds of  −1 and  such that Choose  ℎ =  +1 in (89) and use the definition of  ℎ to obtain Let where  approximates the characteristic unit vector s.Now using the same arguments given in [12] To bound  6 , we proceed as follows: Now  1 can be written as Following the proof lines of Theorem 4.1 in [12], it can be shown that, for  ∈ Using ( 52), ( 104), (105), and the Cauchy-Schwarz inequality, we obtain 2 can be bounded as follows: A use of Cauchy-Schwarz inequality yields In order to bound  2 , we use the following result.
If  ∈  2 (Ω) and η () = ( x ) with x =  − r()Δ, for a nonzero function r() such that r and ∇ ⋅ r are bounded, then where  is a positive constant independent of ℎ and Δ; for a proof, we refer to [9, pp. 875 Now, it remains to show the induction hypothesis (114).Using ( 23) and ( 24), we have Using ‖∇ ⋅ (u n h −ũ n h )‖ = 0, we have Now using ( 42) and (125), we obtain for small ℎ Here, we have used Δ = (ℎ) and ℎ log (1/ℎ) → 0 as ℎ → 0 and this proves our induction hypothesis (114).Now combine the estimates of  and  and use triangle inequality to complete the rest of the proof.
Using (43) and Theorem 11, we obtain the following error estimates for velocity as well as pressure.
To solve the pressure equations, that is, (131) and (132), we use the mixed FVEM, and for the concentration equation (133), we use the standard FVEM.
For the test problems, we have taken data from [28].The spatial domain is Ω = (0, 1000) × (0, 1000) ft 2 , the time period is [0, 3600] days, and viscosity of oil is (0) = 1.0 cp.The injection well is located at the upper right corner (1000, 1000) with the injection rate  + = 30ft 2 /day and injection concentration  = 1.0.The production well is located at the lower left corner with the production rate  − = 30 ft 2 /day and the initial concentration is (, 0) = 0.For time discretization, we take Δ  = 360 days and Δ  = 120 days; that is, we divide each pressure time interval into three subintervals.
Test 1.We assume that the porous medium is homogeneous and isotropic.The permeability is  = 80.The porosity of the medium is  = .1 and the mobility ratio between the resident and injected fluid is  = 1.Furthermore, we assume that the molecular diffusion is   = 1 and the dispersion coefficients are zero.In our numerical simulation, we divide the spatial domain into 20 equal subdivisions along both  and  axis.For time discretization, we take Δ  = 360 days and Δ  = 120 days; that is, we divide each pressure time interval into three subintervals.
The surface and contour plots for the concentration at  = 3 and  = 10 years are presented in Figures 6 and 7, respectively.Since only molecular diffusion is present and viscosity is also independent of the velocity, Figure 6 shows that the velocity is radial and the contour plots for the concentration are circular until the invading fluid reaches the production well.Figure 7 shows that when these plots are reached at production well, the invading fluid continues to fill the whole domain until  = 1.
Test 2. In this test we consider the numerical simulation of a miscible displacement problem with discontinuous permeability.Here, the data is the same as given in Test 1 except the permeability of the medium ().We take  = 80 on the subdomain Ω  := (0, 1000)×(0, 500) and  = 20 on the subdomain Ω  := (0, 1000) × (500, 1000).The contour and surface plot at  = 3 and  = 10 years are given in Figures 8 and  9, respectively.Figures 8 and 9 show that when the injecting fluid reaches the lower half domain, it starts moving much faster in the horizontal direction on this domain compared to the low permeability domain, that is, upper half domain.We observe that one should put the production well in a low permeability zone to increase the area swept by the injected fluid.
Order of Convergence.In order to verify our theoretical results we also compute the order of convergence for the concentration for this particular test problem.We compute the order of convergence in  2 norm.To discretize the time interval [0, ], we take uniform time step Δ = 360 days for pressure and concentration equations.The computed order of convergence is given in Figure 10.Note that the computed order of convergence matches with the theoretical order of convergence derived in Theorem 11.
Note.This paper has been presented in the International Conference on Numerical Analysis and Applications which was held at Bulgaria during June 15-20, 2012.Moreover, some of the results without proof have been published in the proceeding of that conference, kindly see [29].

Figure 2 :
Figure 2: An illustration of the definition x .