The Use of the Method of Characteristics in Determining Boundary Conditions for Problems in Reservoir Analysis

Abstract

Published in Petroleum Transactions, AIME, Volume 216, 1959, pages 284–289. Abstract Problems in analysis can usually be expressed in terms of a system of nonlinear partial differential equations. A method for setting physically reasonable boundary conditions for these systems by application of the theory of characteristics is discussed. A short description is given of the technique for finding characteristics of general first order and second order equations. Specific examples are quoted from equations occurring in theoretical studies. The principles associated with the use of characteristics to set physically realistic boundary conditions are developed, and the method is discussed in terms of boundary conditions for the noncapillary, incompressible, two-phase flow problem; for the capillary, incompressible, two-phase flow problem; for the detergent flooding problem; and for the compressible, noncapillary, two-phase flow problem. Introduction In most problems in reservoir analysis the physical situation is described by a system of nonlinear partial differential equations. The general solution to the set of equations usually comprises a wide range of functions and a particular solution is normally selected by applying suitable boundary conditions to the problem. In some cases it is clear from the physical situation what form the boundary conditions should take, but it is usually desirable to have an independent mathematical check of the suitability of the chosen conditions. From the mathematical standpoint, a system of partial differential equations with boundary conditions is said to be properly set if the combination determines a unique solution which is continuously dependent on the equation and the boundary conditions. Such a solution is sometimes termed "reasonable". In investigating the properties of the mathematical representation, it is useful to determine the structure of the system by the application of the theory of characteristics. This theory determines in the space of the independent variables, lines or surfaces which play a special role in the system's behavior. The characteristics determine the possibility and nature of the propagation of discontinuities in the derivatives of the dependent variables.