We present a method for handling non-linear partial differential equations having solutions that develop steep fronts. Traditional methods for solving such problems (the finite difference and finite element methods) implicitly assume a simple form for the solution over a limited domain. Instead of using a simple basis set and many small spatial elements, we use a complex basis set and no spatial discretization. The method produces accurate results because we are able to include in the basis set a non-rational function that can represent a step.

We have chosen to use the Buckley-Leverett equations to model the quarter 5-spot problem of oil reservoir modeling in two dimensions. The problem is a coupled system of partial differential equations – the flow equation which is nearly hyperbolic, and the pressure equation which is elliptic. In this paper we discuss the solution of the flow equation; the solution of the elliptic equation is discussed elsewhere.

The test problem is two dimensional and restricted to the unit square. Extensions to three dimensions and more general geometries will be discussed.

Keywords:
discretization, partial differential equation, dimension, nugget, equation, reservoir modeling, spe 13506, contour, differential equation, initial condition
Subjects:
Reservoir Simulation
Copyright 1985, Society of Petroleum Engineers
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