Some cases of the steady two-dimensional percolation of water through ground

Abstract

In this paper we examine three cases of steady two-dimensional motion of water through uniform permeable ground and obtain exact mathematical solutions of the problems which are investigated. In the cases considered the ground soaked with water is bounded by impervious surfaces, boundaries of water-basins and free surfaces. (The term “free surface” is used to denote that surface in ground-water motions which engineers call the “water table”.) Further, with the exception of the free surfaces, the boundaries are restricted to be plane. These problems do not involve seepage surfaces, but the method can, in certain cases, be applied to problems in which plane seepage surfaces occur. As we are considering only two-dimensional problems the plane surfaces are represented by straight lines in the representative plane—as for example in figure 1. In this figure LBCM is an earth embankment. AB is the boundary of the head water-basin, BC is the impervious base of the embankment and CD is the boundary of the tail water-basin. AE is the free surface or water table, that is, the upper level of the water which is percolating through the embankment. Above AE the pores between the particles of soil are filled with air. ED is the seepage surface, that is, the surface of the embankment through which water is steadily leaking and falling into the tail water. We assume that in the region of flow the “inertia forces” are neglected and that Darcy’s law is satisfied, that is, that the frictional forces per unit volume between the water and the permeable ground are proportional to the first power of the “percolation velocity”. The percolation velocity is defined as the limiting value of the ratio of discharge to area as the area tends to zero, it being understood that the discharge is measured in a direction normal to the area and that “area” includes not only the clearance between particles of soil but also the soil itself. A general method of solution has been given by one of the authors in a previous paper (Davison 1936 a ). The present paper is a development of this work and contains, in § 2, a lemma which is useful in special cases. As examples of this, three problems are discussed below.