Calculations on the flow of heterogeneous mixtures through porous media

Summary

Muskat and Meres¹) have formulated basic differential equations governing the motion of heterogeneous fluid mixtures through porous media. They obtained a solution of these equations by numerical integration for the problem of a column initially filled with liquid saturated with gas, which is closed at one end and kept at a constant low pressure at the other¹).

Buckley and Leverett²) obtained an analytical solution for the problem of a column initially filled with liquid which is flooded with a second immiscible liquid. They found a solution in which the saturation is a three-valued function of the coordinate along the column.

In our paper in the first place a discussion is given of Buckley and Leverett's solution. It appeared that the true solution which contains a discontinuity may be derived from the three-valued solution by a discussion of the integral relation which represents the total liquid recovery from the column. This discussion bears a formal resemblance to that occurring in the theory of van der Waal's equation of state.

The second problem treated in our paper is that of a vertical column initially filled with liquid saturated with gas under a high pressure, which is opened at its lower end. For this problem we succeeded in finding an analytical solution for low values of the pressure gradient. Here again the paradox of a three-valued solution occurred and led to a discontinuity in the saturation.

A general argument is given to the effect that three-valued solutions are unavoidable for a theoretical treatment based on Muskat's equations of problems which lead to discontinuities in the saturation.