Analytical solutions of seepage theory problems. Inverse method, variational theorems, optimization and estimates (a review)

The results of analytical studies of the problems arising in connection with the prediction of ground water flow in civil engineering, hydrogeology and irrigation engineering are reviewed. Numerical techniques have become of ever greater significance in solving practical problems of seepage theory since the introduction of powerful computers in the sixties. However, even so analytical methods have proved to be necessary not only to develop and test the numerical algorithms but also to gain a deeper understanding of the underlying physics, as well as for the parametric analysis of complex flow patterns and the optimization and estimation of the properties of seepage fields, including in situations characterized by a high degree of uncertainty with respect to the porous medium parameters, the mechanisms of interaction between the fluid and the matrix, the boundary conditions and even the flow domain boundary itself. The review covers studies of ground water dynamics directly related to the problems of flow in domains with incompletely specified boundaries related to the author's interests. Mathematically, these problems reduce to boundary value problems for partial differential equations of elliptic type in domains with unknown boundaries found using specified boundary conditions. These are either deduced from the physical model of the process (the depression surface being an example) or determined by structural considerations (such as the underground shape of a dam or embankment).