NELSON, R. WILLIAM, GENERAL ELECTRIC CO., RICHLAND, WASH.
Introduction
Methods for determining permeability presented by W. D. Krugers in a paper entitled "Determining Areal Permeability Distribution by Calculation" have considerable merit. They can be expected to contribute significantly to analysis of flow in heterogeneous systems. Some rather subtle yet important conditions must be met, however, to assure that the computed permeabilities are correct. The conditions which insure a unique permeability distribution may have been overlooked in the method as proposed and presented in the subject paper. The subtlety of the special requirements may permit good agreement between computed and measured pressures in the reservoir; but the computed permeability may still be in error. The general theory and discussion to the requirements for a unique determination of the permeability distribution are presented elsewhere in more detail. Accordingly, only major features will be presented here in the notation of Kruger.
GENERAL DESCRIPTION OF REQUIREMENTS
A brief mention of the route to be followed in showing the special requirements may be helpful. The equation to be solved is a quasilinear, first-order, partial differential equation in the unknown permeability. This equation can be solved through an extension of Lagrange's method by reduction to a system of subsidiary ordinary differential equations. Through consideration of the identity of one of the Lagrange subsidiary differential equations with the stream function, a special interrelationship can be shown. The interrelationship has special significance with respect to the boundary condition. If the boundary function satisfies part of the subsidiary differential equation, no unique solution for the permeability distribution exists. In the physical problem this requirement for uniqueness indicates that the permeability measurements used as a boundary condition can not be along a stream tube. The boundary condition must be along a line, or series of lines, which completely traverse the region and pass through every stream tube.
DERIVATION OF SPECIAL REQUIREMENTS ON THE BOUNDARY CONDITION
If the velocity components in the directions of increasing x and y are vx and vy, the equation for the streamlines is
dx/vx = dy/vy ..............................(1)
Eq. 1 of Kruger's paper can be expanded to give:
.............................(2)
khEq. 2, when is the dependent variable, is a quasilinear first-order partial differential equation. The analytical method of solution is described in several texts with Goursat's treatment as translated by Hedrick and Dunkel, being very readable and complete. Eq. 2 is reducible to the system of subsidiary differential equations.
..............................(3)
Through assuming the form of the solution to be
......(4)
where F is an arbitrary function of two independent integrals of the system of Eq. 3.
SPEJ
P. 223^
Invariant measures for the flow of a first order partial differential equation
