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International Journal of Partial Differential Equations
Volume 2014 (2014), Article ID 245086, 16 pages
Modified Method of Characteristics Combined with Finite Volume Element Methods for Incompressible Miscible Displacement Problems in Porous Media
1Department of Mathematics, Indian Institute of Space Science and Technology, Thiruvananthapuram, Kerala 695 547, India
2Department of Mathematics, BITS Pilani, Pilani, Jhunjhunu, Rajasthan 333031, India
Received 8 October 2013; Accepted 27 November 2013; Published 19 January 2014
Academic Editor: Yuncheng You
Copyright © 2014 Sarvesh Kumar and Sangita Yadav. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 element method (FVEM) for the approximation of the pressure-velocity equation. Since modified method 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 norm are derived for velocity, pressure and concentration. Numerical results are presented to substantiate the validity of the theoretical results.
A mathematical model describing miscible displacement of one incompressible fluid by another in a horizontal porous medium reservoir with boundary of unit thickness over a time period of is given by with boundary conditions and initial condition where , and are, respectively, the Darcy velocity and the pressure of the fluid mixture, is the concentration of the fluid, is the concentration of the injected fluid, is the concentration dependent viscosity of the mixture, is the 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, is the diffusion-dispersion tensor where is the molecular diffusion, and are, respectively, the longitudinal and transverse dispersion coefficients, is the tensor that projects onto direction, whose th component is given by and is the identity matrix of order . is a known function representing sources denoted as for convenience and represents the initial concentration. For physical relevance and denotes the unit exterior normal to . 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 , , , , , , , such that The authors in [1–3] have discussed mathematical theory and existence of a unique weak solution of the above system (1)–(6) under suitable assumptions on the data. The pressure-velocity 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 , and Darlow et al.  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 norm for the velocity and concentration. Recently, Kumar  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  introduced and analyzed MMOC for the approximation of convection dominated diffusion equations. The authors in [10–12] 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, , as a directional derivative.
Set (see Figure 1) The characteristic direction with respect to the operator is the unit vector The directional derivative of the concentration in the direction of is given by This implies that .
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.
Let be a given partition of the time interval with the time step size . For very small values of , the characteristic direction starting from crosses at (see Figure 2) where .
This suggests us to approximate the characteristic directional derivative at as where .
Using (16), we obtain where .
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 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 [15–18] 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.
2. Finite Volume Element Approximation
Let . 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 and , respectively, and integrate over . A use of Green’s formula and yields the following weak formulation: Find 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 be a regular, quasiuniform partition of the domain into closed triangles ; that is, . Let and . Let the trial function spaces and 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 as where . For , it is straight forward to check that where is a constant independent of . For the inequality also holds true when is in and the triangulation is quasiuniform and can be proved using the same arguments as in the proof of Lemma in [19, pp. 67].
The dual grid consists of interior quadrilaterals and boundary triangles, which are constructed by joining barycenter to the vertices. For the construction of the dual grid we refer to . 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 of . The test space 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 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 , integrating over the control volumes , applying the Gauss’s divergence theorem, and summing up over all the control volumes, we obtain where 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 such that, for , where is an approximation to obtained from (34).
Now, we introduce a dual mesh based on which will be used for the approximation of the concentration equation. For construction we refer to .
For applying the standard FVEM to approximate the concentration, we define the trial space on and the test space on as follows: where is the control volume associated with node . Again, we define a transfer function by where 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 to the concentration , we multiply (15) by , integrate over the control volumes , and apply the Gauss’s divergence theorem. Then we sum up over the control volumes to obtain the FVE approximation corresponding to the concentration equation (15) as a solution such that for , where is an approximation to to be defined later and the bilinear form is defined by being the unit outward normal to the boundary of with , , and .
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 , 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 , find such that where and is a projection of onto which will be defined in (45).
Note that in (17). we use the following notation for the exact velocity:
3. Error Estimates
3.1. Error Estimates for Velocity
For a given , the auxiliary functions are defined as follows: For a proof of existence and uniqueness of the discrete solution of (41), we refer to [20, pp. 52]. For and , the following error estimates can be obtained (see ): 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  have derived error estimates for mixed covolume method by using Raviart-Thomas projection and 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 is quasiuniform. Let and , respectively, be the solutions of (1)-(2) and (30). Then, there exists a positive constant independent of but dependent on the bounds of and such that
3.2. Error Estimates for Concentration
We write , where is the projection of defined by where The function will be chosen such that the coercivity of is assured.
We use frequently the following trace inequality [21, pp. 417]: for , where . Further, we need the following inverse inequalities (see [22, pp. 141]): Using the properties of operator, it is easy to see that, for and the following holds true: Now, using (49), we have the following lemma [17, pp. 317].
Lemma 3. There exists a positive constant independent of such that where
Also note that, by the usual interpolation theory, the operator 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.
Lemma 4 (see [24, pp. 240]). The operator has the following properties.(i)With , the norms and are equivalent on ; that is, there exist positive constants and , independent of , such that (ii) is stable with respect to the norm; that is, there exists a positive constant independent of such that
Lemma 5 (see [25, pp. 1871]). Assume that . Then one has Moreover, for ,
Remark 6. Note that (55) also holds true for .
Since is uniformly positive definite, we obtain following from (56): Choose such that, for , and hence Now, we derive the error bound in and norms for . Let be the continuous interpolant onto satisfying the following approximation properties. For with , we have  Moreover, if , then
Lemma 7. There exists a positive constant independent of such that provided , for a.e. .
Proof. The coercivity and boundedness of with (45) yield
where depends on the bound of given in (11). Combine the estimates (63) and (59) and use the triangle inequality to complete the proof.
For deriving the error bounds for , we need the following lemma.
Lemma 8. Let . There exists a positive constant such that, for and , Using (55) (see also Remark 6), we find that To bound , first we use the fact that is linear on each triangle to obtain Now use (3), (49), and (50) to obtain where denotes the average value of on triangle .
Based on the analysis in [25, pp. 1873], we estimate as follows. Note that an appeal to the continuity of with (49) yields where and is a function such that, for any edge of a triangle , and is the midpoint of . Since , we use trace inequalities (47) and (61) to arrive at Substitute the estimates of and in (65) to complete the rest of the proof.
Lemma 9. There exists a positive constant independent of such that provided , and for a.e.
Proof. To obtain optimal error estimates for , we now appeal to Aubin-Nitsche duality argument. Let be a solution of the following adjoint problem: which satisfies the elliptic regularity condition Multiply the above equation by and integrate over . An integration by parts and a use of (45) yield For , use (61) to find that The bound for follows from Lemma 8 and hence Substitute (75) and (76) in (74) to find that Now choose in (77). Then use elliptic regularity condition (73) with (59) to obtain and this completes the proof.
Lemma 10. There exists a positive constant such that ,
Proof. Note that
where 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 and ,
A use of the trace inequalities (47) and (84) yields
Now, using Taylor series expansion, we find that
Let . Since is linear on each triangle, seminorms and are identical.
Substitute (86) in (85) and use the estimates for to arrive at and this completes the rest of the proof.
Now, we prove our main theorem.
Theorem 11. Let and be the solutions of (3) and (39) at , respectively, and let . Further assume that . Then, for sufficiently small , there exists a positive constant independent of but dependent on the bounds of and such that
Proof. Write . Since the estimates of are known, it is enough to estimate .
Multiplying (3) by and subtracting the resulting equation from (39) at , we obtain Choose in (89) and use the definition of to obtain To estimate , , and , we use the Cauchy-Schwarz inequality, boundedness of , and (54) to obtain To bound , we use Lemma 10 to obtain Since hence, and, similarly, Hence, Using the Cauchy-Schwarz inequality and (54), we obtain Let , so that where approximates the characteristic unit vector . Now using the same arguments given in , we have the following bound: Hence, is bounded as follows: To bound , we proceed as follows: Now can be written as Following the proof lines of Theorem in , it can be shown that, for , there exists a positive constant independent of and such that A use of (103) yields It is easy to show that Using (52), (104), (105), and the Cauchy-Schwarz inequality, we obtain can be bounded as follows: A use of Cauchy-Schwarz inequality yields In order to bound , we use the following result.
If and with , for a nonzero function such that and are bounded, then where is a positive constant independent of and ; for a proof, we refer to [9, pp. 875]. Using (109), we obtain This implies that Now, using (106), (111), and (101), we obtain the following bound for : Using the Cauchy-Schwarz inequality, we obtain We use (103) and (109) to bound and , respectively. For this, we need that and its first derivative are bounded.
First let us make an induction hypothesis. Assume that there is a constant say with such that where is the projection of at defined in (41). To bound , we use inverse inequality and (114): where we have used the assumption that .
Using (103), we have Using (109), we have Now, using (52), (116), inverse inequality (48), and (117), we obtain the following bound for : To bound , we use maximum norm estimate of (see (79)): And, hence, using (54), is bounded as follows: Substitute the estimates of in (90) and use nonsingular property of and a kick-back argument with the Young’s inequality to obtain Using (43) and (44), we obtain Substitute (122) in (121) to obtain Taking summation over , we obtain Now, using discrete Gronwall’s, equivalence of the norms (53), and the estimates of , we obtain Now, it remains to show the induction hypothesis (114). Using (23) and (24), we have Using , we have Now using (42) and (125), we obtain for small