A mathematical model for a steady-state seepage flow of groundwater under a reinforced concrete dam

Abstract

A numerical technique is applied to solve a biharmonic equation problem of a steady-state seepage flow of water through a thin layer of soil, under an impermeable dam. Due to the existence of a point boundary singularity and particular boundary conditions in the problem of the present study, the biharmonic equation is selected to be the governing equation. The potential function of the problem is approximated by the leading terms of the local asymptotic solution expansion. These terms are also used to weight the governing biharmonic equation in the Galerkin sense. Implementing Green’s theorem twice, the discretized equations are reduced to boundary integrals. The system of linear equations is completed by weakly enforcing the Dirichlet boundary conditions by means of Lagrange multipliers. The values of the latter are calculated together with the singular coefficients. The numerical results obtained with the method, indicate high accuracy and very fast convergence with the number of singular functions and the number of Lagrange multipliers. The volume of water per unit time q flowing through the soil in the horizontal direction, under the dam, which is directly proportional to the directional derivative of the potential function, is approximated by the derivatives of the leading terms of the local asymptotic solution expansion. Values of q and the corresponding discharge velocity, obtained with the method, compare favourably with the values given by FEM and soil Mechanics theory. The favourable numerical behaviour of this numerical technique, which in the bibliography is known as singular function boundary integral method (SFBIM), can be exploited in numerous ways. Combining the SFBIM with standard schemes, such as the FEM, appears to be an interesting extension of the method for future work, aiming at enforcing the strengths of the SFBIM in problems with boundary singularities.