Numerical analysis of finite element methods for miscible displacements in porous media

Abstract

Finite element methods are used to solve a coupled system of nonlinear partial differential equations, which models incompressible miscible displacement in porous media. Through a backward finite difference discretization in time, we define a sequentially implicit time-stepping algorithm that uncouples the system at each time-step. The Galerkin method is employed to approximate the pressure, and accurate velocity approximations are calculated via a post-processing technique involving the conservation of mass and Darcy's law. A stabilized finite element (SUPG) method is applied to the convection-diffusion equation delivering stable and accurate solutions. Error estimates with quasi-optimal rates of convergence are derived under suitable regularity hypotheses. Numerical results are presented confirming the predicted rates of convergence for the post-processing technique and illustrating the performance of the proposed methodology when applied to miscible displacements with adverse mobility ratios.

Reduced integration post‐processing techniques for Darcy's velocity recovery in bilinear quadrilaterals

“…It was shown earlier in [1] that u h may lead to non-physical oscillations which were usually attributed only to numerical dispersion. In order to recover the velocity approximation it was proposed in [10,11] an alternative to the usual finite element methods found in the literature. These new methodologies are based on global and local post-processing approaches with a standard finite element implementation (Lagrangian elements) and improved accuracy and regularity to approximate the velocity after computing an approximation to the pressure field, as we shall see in the next sub-sections and also illustrated in the numerical results section.…”

“…where is a positive parameter which depends on h and u P ∈ U l h,0 is the approximation of u. A complete analysis of this methodology can be found in [10]. Different choices for U l h can be made in Equation (10) [2].…”

“…Hence, this is the most economic option. However, it is not the most accurate approximation for the velocity field, as we have observed in [1,10,11]. Thus, if accuracy is recovered by using the global post-processing technique, we observe that the equation system (19) has to be solved with the coefficient matrix V g composed by two terms (22) and (23), which are usually evaluated at the four Gauss integration points.…”

“…To illustrate post-processing accuracy with the stabilized reduced integration strategy, we analyse a Poisson problem (4)-(5), considering k/ = 1, in a square domain = [0, L] × [0, L], where L = 1.854075. We consider a point source at x = y = 0.0 and a point sink at x = y = L. This problem has the analytical solution given in [10]. The influence of the lack of regularity of the exact solution on the rates of convergence of the finite element approximations and the efficiency of the post-processing were studied by Malta and Loula [10] and Coutinho et al [19], where the gain in accuracy of the post-processing in comparison with Galerkin and mixed methods was observed.…”

“…We consider a point source at x = y = 0.0 and a point sink at x = y = L. This problem has the analytical solution given in [10]. The influence of the lack of regularity of the exact solution on the rates of convergence of the finite element approximations and the efficiency of the post-processing were studied by Malta and Loula [10] and Coutinho et al [19], where the gain in accuracy of the post-processing in comparison with Galerkin and mixed methods was observed. Here, the finite element approximations were computed with an uniform and two distorted meshes (see Figure 2), all with 256 bilinear elements.…”