Relaxation methods applied to engineering problems. VII. Problems relating to the percolation of fluids through porous materials

Abstract

The accepted theory of percolation of fluids through porous materials (which is based on Darcy’s law of resistance) indicates that the velocities can be calculated from a velocity-potential which, in two-dimensional motion, is plane-harmonic within the fluid field. The associated stream function, and the fluid pressure, are also plane-harmonic, so in cases where all boundaries are known their determination is an ordinary problem in plane-potential theory. But in cases where a free surface exists (as in the percolation of water through earth dams), its shape is not known a priori, consequently orthodox methods cannot be applied. Here the relaxation method developed in earlier papers is shown to be applicable without special assumptions, and to yield results of more than sufficient accuracy. Three typical examples are treated, the third involving ‘refraction’ of the lines of flow and pressure at the junction of two materials of different porosity.