A stabilized mixed discontinuous Galerkin method for the incompressible miscible displacement problem

A new fully discrete stabilized discontinuous Galerkin method is proposed to solve the incompressible miscible displacement problem. For the pressure equation, we develop a mixed, stabilized, discontinuous Galerkin formulation. We can obtain the optimal priori estimates for both concentration and pressure.

We consider the problem of miscible displacement which has considerable and practical importance in petroleum engineering. This problem can be considered as the result of advective-diffusive equation for concentrations and the Darcy flow equation. The more popular approach in application so far has been based on the mixed formulation. In a previous work, Douglas and Roberts [1] presented a mixed finite element (MFE) method for the compressible miscible displacement problem. For the Darcy flow, Masud and Hughes [2] introduced a stabilized finite element formulation in which an appropriately weighted residual of the Darcy law is added to the standard mixed formulation. Recently, discontinuous Galerkin for miscible displacement has been investigated by numerical experiments and was reported to exhibit good numerical performance [3,4]. In Hughes-Masud-Wan [5], the method of [2] was extended to the discontinuous Galerkin framework for the Darcy flow. A family of mixed finite element discretizations of the Darcy flow equations using totally discontinuous elements was introduced in [6]. In [7] primal semi-discrete discontinuous Galerkin methods with interior penalty are proposed to solve the coupled system of flow and reactive transport in porous media, which arises from many applications including miscible displacement and acid-stimulated flow. In [8], stable Crank-Nicolson discretization was given for incompressible miscible displacement problem.

The discontinuous Galerkin (DG) method was introduced by Reed and Hill [9], and extended by Cockburn and Shu [10-12] to conservation law and system of conservation laws,respectively. Due to localizability of the discontinuous Galerkin method, it is easy to construct higher order element to obtain higher order accuracy and to derive highly parallel algorithms. Because of these advantages, the discontinuous Galerkin method has become a very active area of research [4-7,13-18]. Most of the literature concerning discontinuous Galerkin methods can be found in [13].

In this paper, we analyze a fully discrete finite element method with the stabilized mixed discontinuous Galerkin methods for the incompressible miscible displacement problem in porous media. For the pressure equation, we develop a mixed, stabilized, discontinuous Galerkin formulation. To some extent, we develop a more general stabilized formulation and because of the proper choose of the parameters γ and β, this paper includes the methods of [2,6] and [5]. All the schemes are stable for any combination of discontinuous discrete concentration, velocity and pressure spaces. Based on our results, we can assert that the mixed stabilized discontinuous Galerkin formulation of the incompressible miscible displacement problem is mathematically viable, and we also believe it may be practically useful. It generalizes and encompasses all the successful elements described in [2,6] and [5]. Optimal error estimate are obtained for the concentration, velocity and pressure.

An outline of the remainder of the paper follows: In Section 2, we describe the modeling equations. The DG schemes for the concentration and some of their properties are introduced in Section 3. Stabilized mixed DG methods are introduced for the velocity and pressure in Section 4. In Section 5, we propose the numerical approximation scheme of incompressible miscible displacement problems with a fully discrete in time, combined with a mixed, stabilized and discontinuous Galerkin method. The boundedness and stability of the finite element formulation are studied in Section 6. Error estimates for the incompressible miscible displacement problem are obtained in Section 7.

Throughout the paper, we denote by C a generic positive constant that is independent of h and Δt, but might depend on the partial differential equation solution; we denote by ε a fixed positive constant that can be chosen arbitrarily small.

Miscible displacement of one incompressible fluid by another in a porous medium Ω ∈ R^d(d = 2, 3) over time interval J = (0, T] is modeled by the system concentration equation:

Pressure equation:

The initial conditions

The no-flow boundary conditions

Dispersion/diffusion tensor

where the unknowns are p (the pressure in the fluid mixture), u (the Darcy velocity of the mixture, i.e., the volume of fluid flowing cross a unit across-section per unit time) and c (the concentration of the interested species, i.e., the amount of the species per unit volume of the fluid mixture). ϕ = ϕ(x) is the porosity of the medium, uniformly bounded above and below by positive numbers. The E(u) is the tensor that projects onto the u direction, whose (i,j) component is ; d_m is the molecular diffusivity and assumed to be strictly positive; d_l and d_t are the longitudinal and the transverse dispersivities, respectively, and are assumed to be nonnegative. The imposed external total flow rate q is sum of sources (injection) and sinks (extraction) and is assumed to be bounded. Concentration c* in the source term is the injected concentration c_w if q ≥ 0 and is the resident concentration c if q < 0. Here, we assume that the a(c) is a globally Lipschitz continuous function of c, and is uniformly symmetric positive definite and bounded.

3.1 Notation

Let T_h = (K) be a sequence of finite element partitions of Ω. Let Γ_I denote the set of all interior edges, Γ_B the set of the edges e on ∂Ω, and Γ_h = Γ_B + Γ_I. K⁺, K^- be two adjacent elements of T_h; let x be an arbitrary point of the set e = ∂K⁺ ∩ ∂K^-, which is assumed to have a nonzero (d - 1) dimensional measure; and let n⁺, n^- be the corresponding outward unit normals at that point. Let (u, p) be a function smooth inside each element K^± and let us denote by (u^±, p^±) the traces of (u, p) on e from the interior of K^±. Then we define the mean values {{·}} and jumps [[·]] at x ∈ {e} as

For e ∈ Γ_B, the obvious definitions is {{p}} = p, [[u]] = u·n, with n denoting the outward unit normal vector on ∂Ω. we define the set 〈K, K'〉 as

For s ≥ 0, we define

The usual Sobolev norm on Ω is denoted by ||·||_{m, Ω} [19]. The broken norms are defined, for a positive number m, as

The discontinuous finite element space is taken to be

where P_r(K) denotes the space of polynomials of (total) degree less than or equal to r (r ≥ 0) on K. Note that we present error estimators in this paper for the local space P_r, but the results also apply to the local space Q_r (the tensor product of the polynomial spaces of degree less than or equal to r in each spatial dimension) because P_r(K) ⊂ Q_r(K).

The cut-off operator is defined as

where M is a large positive constant. By a straightforward argument, we can show that the cut-off operator is uniformly Lipschitz continuous in the following sense.

Lemma 3.1 [7]  (Property of operator ) The cut-off operator defined as in Equation 3.4 is uniformly Lipschitz continuous with a Lipschitz constant one, that is

We shall also use the following inverse inequalities, which can be derived using the method in [20]. Let K ∈ T_h, v ∈ P_r(K) and h_K is the diameter of K. Then there exists a constant C independent of v and h_K, such that

3.2 Discontinuous Galerkin schemes

Let ∇_h · v and ∇_hv be the functions whose restriction to each element are equal to ∇ · v, ∇v, respectively. We introduce the bilinear form B(c, w; u) and the linear functional L(w; u, c)

with

here c₁₁ > 0 is a constant independent of the meshsize.

We now define the weak formulation on which our mixed discontinuous method is based

Let N be a positive integer, and t_m = mΔt for m = 0, 1, ..., N. The approximation of c_t at t = tⁿ⁺¹ can be discreted by the forward difference. The DG schemes for approximating concentration are as follows. We seek c_h ∈ W^1,∞(0, T; D_k-1(T_h)) satisfying

where with the DG velocity u_h defined below

4.1 Elimination for the flux variable u

Letting α(c) = a(c)^-1. For the velocity and pressure, we define the following forms

The discrete problem for the velocity and pressure can be written as: find u_h ∈ (D_l-2(T_h))^d, (l ≥ 2), p_h ∈ D_l-1(T_h) such as

In order to eliminate the flux variable, we first recall a useful identity, that holds for vectors u and scalars ψ piecewise smooth on T_h:

Using (4.4) we have

Substituting (4.5) in the first equation of (4.3) we obtain

We introduce the lift operator R:L¹(∪∂K) → (D_l-2(T_h))^d defined by

From (4.6) and (4.7) we have

We also introduce the L²-projection π onto (D_l-2(T_h))^d

Equation 4.8 gives now

Noting that ∇_hD_l-1(T_h) ⊂ (D_l-2(T_h))^d, we have π∇_hp_h ≡ ∇_hp_h for all p_h ∈ D_l-1(T_h). The Equation 4.10 gives

Using (4.5) and the lifting operator R defined in (4.7) we have

Substituting (4.12) in the second equation of (4.3) and using (4.11) we have

For future reference, it is convenient to rewrite (4.13) as follows

where

4.2 Stabilization of formulation (4.3)

We write first (4.3) in the equivalent form: find (u_h, p_h) ∈ (D_l-2(T_h))^d × D_l-1(T_h) such that

where

In a sense, (4.16) can be seen as a Darcy problem. The usual way to stabilized it is to introduce penalty terms on the jumps of p and/or on the jumps of u. In [2], Masud and Hughes introduced a stabilized finite element formulation in which an appropriately weighted residual of the Darcy law is added to the standard mixed formulation. In Hughes-Masud-Wan [5], the method was extend within the discontinuous Galerkin framework. A family of mixed finite element discretizations of the Darcy flow equations using totally discontinuous elements was introduced in [6]. In this paper, we consider the following stabilized formulation which includes the methods of [2,6] and [5].

The stabilized formulation of (4.16) is

where

where γ and β are chosen as the following (i) γ = 1, β = 1. (ii) γ = 0, β = 1, δ could assume either the value +1 or the value -1. The definition of θ will be given in the following content.

By combining (3.8) with (4.18), we have the stabilized DG for the approximating (2.1)-(2.5): seek c_h ∈ W^1,∞(0, T; D_k-1(T_h)) =: W_h, p_h ∈ W^1,∞(0, T; D_l-1(T_h)) =: Q_h and u_h ∈ (W^1,∞(0, T; D_l-2(T_h)))^d =: V_h satisfying

We define the "stability norm" by

where

From [6], we can state the following results.

Lemma 6.1 [6]There exist two positive constants C₁ and C₂, depending only on the minimum angle of the decomposition and on the polynomial degree

Lemma 6.2 [6]There exists two positive constants C₁ and C₂, depending only on the minimum angle of the decomposition such that

Lemma 6.3 [6]Let be a Hilbert spaces, and λ and μ positive constants. Then, for every ξ and η in we have

Theorem 6.1 (Stability) For δ = 1, problem (4.18) is stable for all θ ∈ (0,1).

Proof Consider first the case γ = 1, β = 1. From the definition of A_stab(·,·;·,·;·), we have

We remark that (6.4) can be rewritten as

and the stability in the norm (5.2) follows from .

Consider now the case γ = 0, β =1. Using the equivalent expressions (4.11) and (4.12) for the first and second equation of (4.3), respectively, the problem (4.18) for γ = 0 can be rewritten as: find u_h ∈ (D_l-2(T_h))^d, p_h ∈ D_l-1(T_h) such that

From the first equation in (6.6) and (4.9) we have

Substituting the expression (6.7) in the second equation of (6.6) for δ = 1, we have

Denote by B_1h(·,·) the bilinear form (6.8), we have

and the stability in the norm (5.3) follows from Lemma 6.1. This completes the proof.   □

Theorem 6.2 For δ = -1, problem (4.18) is stable for all θ < 0.

Proof Consider first the case γ = 1, β = 1. The problem (4.18) for δ = -1 reads

Using the arithmetic-geometric mean inequality, we have

and since θ < 0 the result follows.

Consider now the case γ = 0, β = 1. From (6.7) the second equation of (6.6) for δ = -1 can be written as

We remark that formulation (6.12) can be rewritten as

where A_BO(p_h, ψ) is introduced by Baumann and Oden [14], and given by

Denote by B_2h(·,·) the bilinear form (6.13), we have

and since θ < 0 the result follows again from Lemma 6.3 and 6.1.   □

Theorem 6.3 (Consistency) If p,c and u are the solution of (2.1)-(2.5) and are essentially bounded, then

provided that the constant M for the cut-off operator is sufficiently large.

To summarize, for all the bilinear forms in (6.4), (6.10), (6.8) or (6.13) we have: ∃C > 0 such that

and ∃C > 0 such that

where (6.17) clearly holds for every θ ∈ (0,1) for the case ((6.4), (6.8)), and for every θ < 0 for the case ((6.10), (6.13)). On the other hand, since ∇_hD_l-1(T_h) ⊂ (D_l-2(T_h))^d holds, boundedness of the bilinear form in (6.8) and (6.13) follows directly from the boundedness of the bilinear forms A_BR and A_BO, as proved in [13], thanks to the equivalence of the norms (6.1) and (6.2). Thus, we have: ∃C > 0 such that

Let be an interpolation of the exact solution (u, p, c) such that

Let us define interpolation errors, finite element solution errors and auxiliary errors

It was proven in [18] that

hold for all t ∈ J with the constant C independent only on bounds for the coefficient α(c), but not on c itself.

Theorem 7.1 (Error estimate for the velocity and pressure) Let (u, p, c) be the solution to (2.1)-(2.5), and assume p ∈ L²(0, T; H^l(T_h)), u ∈ (L²(0, T; H^l-1(T_h)))^d and c ∈ L²(0, T; H^k(T_h)). We further assume that p, ∇p, c and ∇c are essentially bounded. If the constant M for the cut-off operator is sufficiently large, then there exists a constant C independent of h such that

Proof For the sake of brevity we will assume in the following content. Consider the case γ = 1, β = 1. From the second equation of (5.1) and (6.16) we have

That is

Choosing v = ξ₁, ψ = η₁ and splitting e_p according e_p = η₁- η₂, from (7.1) and we obtain

Let us first consider the left side of error equation (7.5)

We know that (7.2) and quasi-regularity that are bounded in L^∞(Ω). So the right side of the error equation (7.5) can be bounded from below. Noting that , we have

The second and the third terms of the right side of the error equation (7.5) can be bounded using Cauchy-Schwartz inequality and approximation results,

The fourth term can be bounded in a similar way as that for the first term

The last two terms can be bounded as follows

Substituting all these inequalities into Equation 7.5, we have

Using the facts , and we have

The theorem follows from the triangle inequality.

Now consider the case γ = 0, β = 1. The bilinear form (6.8) from the second equation of (5.1) for reads

where

Replacing (6.8) with p_h = p and subtracting it from (7.13) we finally obtain

Choosing ψ = η₁, we have

Let us first estimate the left side of (7.15). From (6.17) and using the fact , we have

The first and the second terms of the right side of (7.15) can be bounded using Lemma 6.1 and (3.5)

Note that are bounded in L^∞(Ω) and , we have

Substituting all these inequalities into the (7.15) and using the triangle inequality we have

We easily deduce,using (7.19)

which completes the proof.   □.

From [7], we state two lemmas for the properties of the dispersion-diffusion tensor, which will be used to prove error estimates for the concentration.

Lemma 7.1 [7] (Uniform positive definiteness of D(u)) Let D(u) defined as in Equation 2.6, where ϕd_m(x) ≥ 0, d_l(x) ≥ 0 and d_t(x) ≥ 0 are non-negative functions of x ∈ Ω. Then

If, in addition, φd_m(x) ≥ d_m,* > 0 uniformly in the domain Ω, then D(u) is uniformly positive definite in Ω:

Lemma 7.2 [7] (Uniform Lipschitz continuity of D(u)) Let D(u) defined as in Equation 2.6, where d_m(x) ≥ 0, d_l(x) ≥ 0 and d_t(x) ≥ 0 are non-negative functions of x ∈ Ω, and the dispersivities d_l and d_t is uniformly bounded, i.e., d_l(x) ≤ d_l^* and d_t(x) ≤ d_t^*. Then

where is a fixed number (d = 2 or 3 is the dimension of domain Ω).

Theorem 7.2 (Error estimate for concentration) Let (u, p, c) be the solution to (2.1)-(2.5), and assume p ∈ L²(0, T; H^l(T_h)), u ∈ (L²(0, T; H^l-1(T_h)))^d and c ∈ L²(0, T; H^k(T_h)). We further assume that p, ∇p, c and ∇c are essentially bounded. If the constant M for the cut-off operator is sufficiently large, then there exists a constant C independent of h and Δt such that

Proof The first equation of (5.1) is

It can be written as

Subtracting the DG scheme equation from the weak formulation, we have for any w ∈ D_k-1(T_h)

that is

Choosing w = τ₁ⁿ⁺¹, we obtain

Let us first consider the left side of the error equation (7.25). The first term can be bounded as

The second term of Equation 7.25 is

The second term of B(·,·;·) can be estimated using the boundedness of u_M and :

Thus

Let us bound the right side of the error equation (7.25).

Using Taylor series expansion, we have

The fourth term in the right side of the error equation (7.25) is

Terms T₁ through T₃ can be bounded by using Cauchy-Schwartz inequality and approximation results,

and

Terms T₄ and T₅ can be estimated using inverse inequalities,

and

Using Cauchy-Schwartz inequality and the trace inequality, we have

Noting that [[cⁿ⁺¹]] = 0, if the constant M for the cut-off operator is sufficiently large, we write the last two terms in the right side of the error equation (7.25) as

Noting that point-wise if the constant M for the cut-off operator is sufficiently large, we can bound term S₁ as

Term S₂ can be bounded in a similar way as that for S₁

Term S₃ can be bounded using the penalty term and continuity of dispersion-diffusion tensor

Combining all the terms in (7.25), we have

Suppose that m is an integer, 0 ≤ m ≤ N - 1. Multiplying by 2Δt, summing from n = 0 to n = m, we obtain

The theorem follow from (7.3), the discrete Gronwall's lemma and the triangle inequality.    □

Theorem 7.3 (Error estimate for flow in coupled system) Let (u, p, c) be the solution to (2.1)-(2.5), and assume p ∈ L²(0, T; H^l(T_h)), u ∈ (L²(0, T; H^l-1(T_h)))^d and c ∈ L²(0, T; H^k(T_h)). We further assume that p, ∇p, c and ∇c are essentially bounded. If the constant M for the cut-off operator is sufficiently large, then there exists a constant C independent of h and Δt such that

Proof Taking L^∞ norm with time in (7.3), we have

Substituting (7.24) into the above inequality, we obtain (7.39).    □

The authors declare that they have no competing interests.

YL participated in the design and theoretical analysis of the study, drafted the manuscript. MF conceived the study, and participated in its design and coordination. YX participated in the design and the revision of the study. All authors read and approved the final manuscript.

The work was supported by National Natural Science Foundation of China (Grant Nos.11101069, Grant Nos.11126105)and the Youth Research Foundation of Sichuan University (no. 2009SCU11113).

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