Simulation of Miscible Displacement Using Mixed Methods and a Modified Method of Characteristics

Abstract

Members SPE-AIME Abstract Numerical dispersion and grid orientation problens with adverse mobility ratios are two of the problens with adverse mobility ratios are two of the major difficulties in the numerical simulation of enhanced recovery processes. An efficient method for modeling convection-dominated flows which greatly reduces numerical dispersion and grid orientation problems is presented and applied to miscible displacement in a porous medium. The base method utilizes characteristic flow directions to model convection and finite elements to treat the diffision and dispersion. The characteristic approach also minimizes certain overshoot difficulties which accompany many finite element methods for problems with sharp fluid interfaces. The truncation error caused by the characteristic time-stepping technique is small, so large stable time-steps can be taken as in fully-implicit methods without the corresponding loss in accuracy, A finite difference analogue can also be formulated, Since the computed fluid velocities help to determine the time-stepping procedure in the characteristic-based method and since accurate velocities are crucial in the method's ability to conserve mass, very accurate Darcy velocities are necessary. A mixed finite element method solves for the pressure and the Darcy velocity simultaneously, as a pressure and the Darcy velocity simultaneously, as a system of first order partial differential equations. By solving for u = -(k/o) p as one term, we minimize the difficulties occurring in standard methods caused by differentiation or differencing of p and multiplying by rough coefficients k/u. p and multiplying by rough coefficients k/u. Using a combination of characteristic-based time-stepping procedures and mixed methods for accurate velocities, a variety of problems with variable (or random) permeabilities, adverse mobility ratios, and tensor dispersion models are examined. A study of viscous fingering is presented. Computational results on a variety of two-dimensional problems show minimal grid-orientation effects, reduced numerical dispersion, minimal overshoot at the front, and very low mass balance errors. Introduction We shall discuss accurate time-stepping procedures for coupled systems of partial differential procedures for coupled systems of partial differential equations arising in reservoir simulation. We shall formulate our methods in the context of miscible displacement problems in porous media. Similar techniques are applicable for a wide variety of secondary and tertiary enhanced recovery procedures involving transport-dominated processes. Although we shall define method-of-characteristics-based finite difference time-stepping procedures in a coupled setting with finite element spatial methods, similar techniques can be formulated for finite difference spatial methods. In convection-diffusion equations, such as the concentration equation for miscible displacement problems, when convection dominates diffusion, problems, when convection dominates diffusion, standard finite difference or finite element procedures suffer from numerical dispersion, overshoot, and grid orientation difficulties, References 1 and 2 contain extensive bibliographies of work in this area. A different procedure, which concentrates on treating the convective terms efficiently and accurately, is presented. Our procedure, a modification of the method of characteristics which is so useful for hyperbolic equations, takes time steps in the direction of flow, along the characteristics of the velocity field of the total fluid. It then accounts for physical diffusion or dispersion in a more standard fashion. This amounts to a type of physical splitting of the spatial operator. The key to the ease of implementation of this technique is that we look backward in time, along an approximate flow path, instead of forward in time as in many methods of characteristics or moving mesh techniques, Thus the points at which the unknowns are determined need not change in time (though they may if desired) and stay in a regular grid pattern; no complex data structure is required to keep track of moving grid points. This makes implementation of these techniques significantly easier, in two and three space dimensions, than moving mesh methods. We extrapolate the velocity field in time from previous time levels to obtain an approximation to the velocity field at the advanced level along which to proceed in time.