This brief paper is entirely devoted to the mathematical theory of the partial differential equation for porous flow. The authors note that the permeability in the Darcy equation is a rapidly oscillating function in space and suggest that one should assume that it is a random field. They then define the problem as a random boundary value problem. In outline (this paper contains no detailed proofs), the authors show that a unique weak solution to this partial differential equation exists. They proceed to define a random total flux, then assume the permeability to be the product of a deterministic function and the exponential of a random function. Next, they calculate the total flux to within a defined remainder by using the solutions to three successively solved boundary value problems. Finally, they present formulations of the moments of the total flux.
This paper does not discuss numerical implementation. For this, the reader is referred to an as-yet-unpublished paper, referenced only by title and author, “to appear.” Similarly, the reader has to consult a thesis dissertation by one of the authors for the complete proofs of the lemmas and theorems cited.