Application of the Boundary Integral Method to Immiscible Displacement Problems

Abstract

Summary. This paper presents an application of the boundary integral method (BIM) to fluid displacement problems to demonstrate its usefulness in reservoir simulation. A method for solving two-dimensional (2D), piston-like displacement for incompressible fluids with good accuracy has piston-like displacement for incompressible fluids with good accuracy has been developed. Several typical example problems with repeated five-spot patterns were solved for various mobility ratios. The solutions were patterns were solved for various mobility ratios. The solutions were compared with the analytical solutions to demonstrate accuracy. Singularity programming was found to be a major advantage in handling flow in the vicinity of wells. The BIM was found to be an excellent way to solve immiscible displacement problems. Unlike analytic methods, it can accommodate complex boundary shapes and does not suffer from numerical dispersion at the front. Introduction The influence of pattern geometry and mobility ratio on reservoir displacement performance has been a great concern of reservoir engineers in developing improved recovery processes. Many efforts have already addressed the quantitative estimation of the displacement performance by analytic methods and numerical finitedifference methods. However, none of these methods are fully satisfactory in both accuracy and adaptability. Analytic methods have an advantage of providing exact solutions; however, the application is usually limited to some simple geometries or unit-mobility-ratio displacement. On the other hand, finite-difference methods have great flexibility but are inherently inaccurate because of numerical dispersion. Other numerical methods, like the boundary integral method (BIM) and the finite-element method, have been successfully introduced to various engineering fields as powerful alternatives to the finite-difference method. In this paper, application of the BIM to fluid displacement problems is presented to show its usefulness in reservoir simulation. The method has the following advantages.Accurate solutions can be obtained because the solution does not have numerical dispersion, which is inherent to the finite-difference method. In addition, the solution does not suffer the effects of grid orientation.The method has few restrictions on field geometries. Any combination of Dirichlet- and Neumann-type boundary conditions can be handled.Because the method uses only a boundary grid, the domain of calculation can be reduced by one dimension [2D problems to one dimension; three-dimensional (3D) problems to two dimensions], so that both time and storage area of a computer can be saved.Flow potential and velocity at any point in the domain can be solved by the method. Thus, movement of fluid can be tracked as the solution of a simple trajectory problem. The solutions for typical 2D, piston-like displacement example problems with repeated five-spot patterns are presented for various problems with repeated five-spot patterns are presented for various nonunit mobility ratios. The solutions are compared with the analytic solutions to demonstrate their accuracy. The application of the methods would be similar, although more complex, for 3D problems. This paper has not considered the extension of the method problems. This paper has not considered the extension of the method to multiple-region problems, such as areally heterogeneous reservoirs, or to miscible displacement problems. Problem Definition Problem Definition We consider fluid displacement problems simplified with the following assumptions:2D flow.incompressible fluids,immiscible piston-like displacement of two fluids,homogeneous fluid and reservoir properties over the entire reservoir, andnegligible capillary and gravity effects. With these assumptions, the governing flow equation is expressed as Darcy's law and the Laplace equation:andAs typical problems, idealized reservoir models of linear, radial, and repeated five-spot flooding patterns are taken for example solutions. Solution Method for Unit-Mobility-Ratio Problems Mobility ratio is defined as the ratio of the mobility of the displacing fluid to the mobility of the original fluid. In unit-mobility-ratio problems, the reservoir and fluid properties are uniform over the problems, the reservoir and fluid properties are uniform over the entire reservoir. If the injection and production rates are held constant, the potential distribution and the fluid velocity over the fid stay constant regardless of time. The potential distribution and the fluid velocity over the field can be solved by the BIM. After the fluid velocity is calculated, tracking the front is a simple trajectory problem. problem. BIM. To solve problems with numerical methods, we need discretized values that describe the problem. In the finite-difference method, a 2D grid is used for solving 2D problems. In the BIM, however, only the discretization along the boundary is necessary. The boundary of the reservoir is divided into n sections by n "boundary nodes," as shown in Fig. 1. The location of the interface between the zones is defined by n "interface nodes," which include two end nodes on the boundary. In a discretized system, boundary conditions can be defined by three parameters, phi, phi +, and phi, at each node. phi + and phi are the normal derivatives of the potential outward to the boundary elements on both sides of the node. They are not necessarily equal because the direction of the boundary elements on the two sides may be different. Suppose boundary conditions are given in the form of either the Dirichlet type (phi is specified) or the Neumann type (phi, is specified); there are four possible types of boundary conditions at each node. as shown in Fig. 2. In Types 1. 3, and 4, only one of three parameters is unknown. SPERE P. 1069