Modified Method of Characteristics Combined with Finite Volume Element Methods for Incompressible Miscible Displacement Problems in Porous Media

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. 1. Introduction 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