A stabilized mixed discontinuous Galerkin method for the incompressible miscible displacement problem
© Luo et al; licensee Springer. 2011
Received: 8 May 2011
Accepted: 25 November 2011
Published: 25 November 2011
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  presented a mixed finite element (MFE) method for the compressible miscible displacement problem. For the Darcy flow, 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. 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 , the method of  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 . In  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 , stable Crank-Nicolson discretization was given for incompressible miscible displacement problem.
The discontinuous Galerkin (DG) method was introduced by Reed and Hill , 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 .
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 . 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 . 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.
2 Governing equations
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 Discontinuous Galerkin method for the concentration
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).
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.
3.2 Discontinuous Galerkin schemes
here c11 > 0 is a constant independent of the meshsize.
4 A stabilized mixed DG method for the velocity and pressure
4.1 Elimination for the flux variable u
4.2 Stabilization of formulation (4.3)
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 , 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 , 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 . In this paper, we consider the following stabilized formulation which includes the methods of [2, 6] and .
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.
5 A mixed stabilized DG method for the incompressible miscible displacement problem
6 Stability and consistency
From , we can state the following results.
Theorem 6.1 (Stability) For δ = 1, problem (4.18) is stable for all θ ∈ (0,1).
and the stability in the norm (5.2) follows from .
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.
and since θ < 0 the result follows.
and since θ < 0 the result follows again from Lemma 6.3 and 6.1. □
provided that the constant M for the cut-off operator is sufficiently large.
7 Error estimates
hold for all t ∈ J with the constant C independent only on bounds for the coefficient α(c), but not on c itself.
The theorem follows from the triangle inequality.
which completes the proof. □.
From , we state two lemmas for the properties of the dispersion-diffusion tensor, which will be used to prove error estimates for the concentration.
whereis a fixed number (d = 2 or 3 is the dimension of domain Ω).