Abstract
This paper describes the application of the fourth-order operator compact implicit (OCI) method to multiphase flow through porous media. A conservative form of the method is developed and used to solve a one-dimensional, two-phase, immiscible waterflood problem.The results of numerical experiments demonstrate the ability to track sharp fronts with relatively few mesh points. Comparisons with conventional finite-difference methods and variational methods are given.
Introduction
In recent years, new work in the numerical solution of boundary-value problems for differential equations has been in the direction of high-order methods - that is, methods whose order of accuracy is higher than the first- and second-order methods generally in use. These high-order methods have been developed for both finite-difference, finite-element, and collocation techniques.High-order finite-difference techniques are formed by expanding the computational molecule of the problem. For example, while a second problem. For example, while a second order-accurate formula for a second derivative involves three points, a fourth-order-accurate formula involves five points. This growth in the number of points in the approximation leads to an increase in points in the approximation leads to an increase in the bandwidth of matrices and the need to impose special difference formulas near the boundary.Variational methods achieve higher-order accuracy by using a higher-degree piecewise polynomial subspace for its approximation. The problem of special treatment near the boundary does not exist, but the increase in the degree of the polynomial also leads to an increase in the work for a given number of mesh points. However, this increase is offset by the need for fewer mesh points to achieve a given accuracy. Thus, on model test problems it has been shown that the total work is less for high-order methods.An alternative to the approach of increasing bandwidth to increase order of accuracy is the operator compact implicit (OCI) method. In this method, an approximation to the spatial part of the differential operator is sought using not only the value of the function at adjacent points but also the value of the operator at these points. By this technique, an implicit relationship is defined between the differential operator and the function on the most compact set of points possible. For example, an implicit fourth-order relationship for a second derivative may be derived using only three points.The OCI method has been applied successfully to two-point boundary-value problems, first- and second-order wave equations, diffusion-convection equations, and viscous flow problems. In this paper, a form of the OCI method problems. In this paper, a form of the OCI method applicable to equations in conservation form is presented and the method is applied to the equations presented and the method is applied to the equations of flow through porous media.The specific example considered is a one-dimensional, two-phase waterflood problem studied by Spivak et al. After presenting the form of the two-phase flow equations to be used, the OCI method is developed. In the implementation of the method it is necessary to define transmissibilities at half mesh points. Therefore, we next examine the problem of interpolation. Finally, numerical results problem of interpolation. Finally, numerical results are presented on the performance of the method for the one-dimensional waterflood problem.
Development of the Differential Equations
The physical problem that the OCI method is applied to in this study is a one-dimensional, compressible, two-phase, immiscible waterflood problem.
SPEJ
P. 120
Inverses of SBP-SAT Finite Difference Operators Approximating the First and Second Derivative
