ABSTRACT
Numerical dispersion and grid orientation problems 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 diffusion 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 system of first order partial differential equations. By solving for u = −(k/µ)Vp as one term, we minimize the difficulties occurring in standard methods caused by differentiation or differencing of p and multiplying by rough coefficients k/µ.
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.
