An approximation of incompressible miscible displacement in porous media by mixed finite element method and characteristics-mixed finite element method

Abstract

MSC: 65N30 65N12Keywords: Incompressible miscible displacement Characteristics-mixed finite element method Mixed finite element method Post-processing step a b s t r a c t An approximation scheme is defined for incompressible miscible displacement in porous media. This scheme is constructed by using two methods. Standard mixed finite element is used for the Darcy velocity equation. A characteristics-mixed finite element method is presented for the concentration equation. Characteristic approximation is applied to handle the convection part of the concentration equation, and a lowest-order mixed finite element spatial approximation is adopted to deal with the diffusion part. Thus, the scalar unknown concentration and the diffusive flux can be approximated simultaneously. In order to derive the optimal L 2 -norm error estimates, a post-processing step is included in the approximation to the scalar unknown concentration. This scheme conserves mass globally; in fact, on the discrete level, fluid is transported along the approximate characteristics. Numerical experiments are presented finally to validate the theoretical analysis.

Unconditional Convergence and Optimal Error Estimates of a Galerkin-Mixed FEM for Incompressible Miscible Flow in Porous Media

“…Optimal error estimates were obtained under a similar time-step condition τ = o(h). There are many other numerical methods in literature for solving the equations of incompressible miscible flow, such as [39,42] with an ELLAM in two-dimensional space, [40] with an MMOC-MFEM approximation for the 2D problem, [13,36] with a characteristic-mixed method in two and three dimensional spaces, respectively, and [28,29] with a collocationmixed method and a characteristic-collocation method, respectively. In all those works, error estimates were established under certain time-step conditions.…”

A MCC finite element approximation of incompressible miscible displacement in porous media

“…Example 5.3. Finally we apply our methods to a real two-dimensional miscible displacement problem of water-oil system in porous media [21][22][23][24]. we simulate problem (2.1) on a horizontal reservoir of one unit thickness, for one-quarter of (a) Contour plot at 10 steps.…”

Optimal error analysis of Crank–Nicolson lowest‐order Galerkin‐mixed finite element method for incompressible miscible flow in porous media

“…Theoretical analysis for the existence of weak solutions of the system was studied by Feng [3], while the existence of semi-classical/classical solutions is unknown so far. Numerical solutions of the system (1.1)-(1.5) attracted much attention due to its wide applications in many engineering areas, see [4][5][6][7][8][9][10][11][12][13][14] and references therein. Two review articles on numerical methods used in these areas were presented by Ewing and Wang [15] and Scovazzi et al [16].…”