Development and Application of Variational Methods for Simulation of Miscible Displacement in Porous Media
Authors: Settari A. , Price H.S. , Dupont T. 
Published: December 2007

Abstract
Many reservoir engineering problems involve solving fluid flow equations whose solutions are characterized by sharp fronts and low dispersion levels. This is particularly important in tracking small slugs that are characteristic of chemical floods, polymer floods, first- and multiple-contact hydrocarbon miscible polymer floods, first- and multiple-contact hydrocarbon miscible displacements, and most thermal processes. The use of finite-difference approximations to solve these problems when low dispersion levels and small slugs need to be modeled accurately may be prohibitively expensive. This paper shows that the use of high-order variational approximations is a very effective means for economically solving these problems. This paper presents some numerical results that demonstrate that high-order variational methods can be used to solve two-dimensional reservoir engineering problems where finite-difference approximations would require 104 problems where finite-difference approximations would require 104 to 105 blocks. The variational solutions are shown to be essentially insensitive to grid orientation for unfavorable mobility ratios up to M = 100.

Introduction
The equations describing miscible displacement in a porous medium (convection-diffusion equations) are among the more difficult to solve by numerical means. The character of the concentration equation ranges from parabolic to almost hyperbolic depending on the ratio of convection to diffusion (Peclet number). Consequently, the finite-difference techniques developed for solving the convection diffusion problem can be divided into two categories: those solving the problem as parabolic and those treating the problem as hyperbolic. The parabolic techniques are unsatisfactory when the diffusion becomes small compared with the convection. The methods using central difference approximations for the convection terms oscillate. Price et al. have shown that these oscillations can be eliminated only by using small spatial increments. Methods using upstream difference approximations do not oscillate, but they introduce large truncation errors that have the character of a large diffusion term. Lantz has shown that for many practical problems, reducing the magnitude of numerical dispersion problems, reducing the magnitude of numerical dispersion sufficiently so that it does not mask the physical dispersion will force an impractically fine grid. Several improvements have been suggested, such as transfer of overshoots truncation-error cancellation, and two-point upstream approximations; but none of these is quite satisfactory in the general case. The hyperbolic methods (method of characteristics, point tracking, etc.) also pose many practical problems. These include the complex treatment required for sources and sinks, the need to redistribute points continually when modeling converging and diverging flow, the problem of maintaining a material balance, problems created by complex geometries, and the practical limitation problems created by complex geometries, and the practical limitation of the time-step size. Moreover, these schemes cannot be shown to converge, thereby making the choice of grid size and point distribution fairly arbitrary. Finally, many nonlinearities, such as reactions and adsorption, need to be treated point-by-point, requiring large amounts of computer time and storage. Because the major difficulty in solving the miscible displacement problem is the determination of an accurate approximation to a very sharp concentration front, one of the most promising alternatives to the schemes mentioned above is the use of promising alternatives to the schemes mentioned above is the use of high-order variational approximations, such as those proposed by Ciarlet et al. These methods (which include Galerkin and finite-element methods) are potentially far more accurate for a given amount of computation than the standard finite-difference techniques and, therefore, more able to solve problems that would otherwise be impractical.
SPEJ
P. 228