Typical solutions of the differential equations of statistical theories of flow through porous media

Abstract

The description of the miscible displacement process in porous media (such as occurs, for example, during the pollution of ground‐water resources with wastes) by methods of statistical mechanics has met with considerable success. The method usually employed is that of considering the motion of each particle through the medium as a random‐walk process. Recently, however, it has been suggested that autocorrelation should also be taken into account and a differential equation allowing for this effect to take place has been arrived at. The purpose of the present paper is to give solutions of the autocorrelation equation for some typical cases and to compare the result with that obtained for the random walk model without autocorrelation. These cases refer to the passage of a thin slug of ‘tagged’ fluid through a porous medium and to the intrusion of a sharp concentration front into a medium. It is shown that autocorrelation has the effect that a concentration‐discontinuity proceeds at a finite speed into the medium which cuts off the infinite ‘tail’ of the solution without autocorrelation. Graphs to illustrate the situation are given.