Variational Simulation With Numerical Decoupling And Local Mesh Refinement
Authors: Chase Curtis A. 
Published: December 2007

Abstract
The Variational Chemical Flood Simulator VCHFLDI solves from two to six coupled nonlinear parabolic partial differential equations in two space partial differential equations in two space dimensions and time. The program was written principly to solve chemical flood problems. It may be used, however, for any system of up to six parabolic partial differential equations because of the ease with partial differential equations because of the ease with which problem description can be changed.
Variable order function spaces were considered for local accuracy but were rejected in favor of Co piecewise bilinear function spaces. Local refinement on a rectangular mesh provides resolution where needed.
The set of equations are decoupled numerically and solved sequentially for each dependent variable. This approach achieves considerable savings in linear algebra work over a fully simulataneous solution; however, it appears that this savings is negated by the tremendous increase in coefficient calculation work as a result of the numerical decoupling.
The partially explicit nature of the sequential approach introduces stability time-step limitations which coupled with the excessive coefficient calculations are unacceptable for practical problem solving. Incorporation of a second order stiffly stable time approximation significantly reduced the stability limitations and cut computer execution times by a factor of four to five.
Quality of computer results with VCHFLDI is good as demonstrated by the sharpness of concentration fronts when compared with a finite difference program using an equivalent spatial discretization. program using an equivalent spatial discretization

Introduction
There are certain reservoir processes the simulation of which requires accurate characterization of front location and shape. Primary among these are chemical flood processes, in which the interfacial tension and hence mobilization of oil is extremely sensitive to injected surfactant concentration. Because of the high cost of injected chemicals, economics dictate as small a surfactant slug as possible yet one which still maintains concentrations above critical values needed for oil mobilization. To simulate such a process, a numerical technique must accurately track the front and model the true physical dispersion without being overwhelmed by physical dispersion without being overwhelmed by numerical dispersion. To obtain sufficient accuracy with conventional finite difference techniques often requires a fine mesh with attendent high computer costs. In some cases mesh size is so fine that the resulting problem cannot be solved within practical time or monetary constraints on existing practical time or monetary constraints on existing computers.
In the past few years, theoretical and numerical work have suggested that Variational/Galerkin methods might provide sufficient accuracy to solve difficult front tracking problems at a cost less than conventional finite difference methods. The purpose of the project discussed in this paper was to develop, using Variational/Galerkin methods, a numerical simulator capable of modeling actual chemical flood processes. The conservation equations describing such processes. The conservation equations describing such processes are three or more in number and are highly processes are three or more in number and are highly nonlinear and strongly coupled. Except for the application of Galerkin methods to simulate conventional black oil processes, little has been reported in the literature on attempts to solve more than three coupled conservation equations. In the following, we describe a general approach which employs Galerkin methods to solve m coupled equations. The Variational Chemical Flood Programs, VCHFLD, and an improved version VCHFLDI, solve from two to six conservation equations.
We examined two different approaches as a means of obtaining resolution locally in the vicinity of fronts. The first employed a fixed mesh and increased the order of the function space from linear to cubic in the vicinity of fronts.