Abstract
The problem of steady-state seepage flow underneath two intermediate filters located anywhere between two end cut-offs of a flat floor of a hydraulic structure on a permeable soil of a finite depth has not been previously solved. Conformal mapping has been applied in this paper to investigate the exact solution. Companion papers (M.I. Farouk, I.M. Smith, Journal of Hydraulic Division, ASCE (2000), submitted for publication; M.I. Farouk, I.M. Smith, Proceedings of the European Geophysical Society on Millennium Conference on Earth, Planetary and Solar Systems Sciences, Nice, France, April 2000) use numerical (finite element) techniques to study a wider range of practical problems involving anisotropy, arbitrary geometries and so on. These solutions are subject to idealization and discretization errors and it was of great value to have the exact results presented herein for comparison purposes. Equations for pore water pressures at key points and hydraulic exit gradient downstream of the hydraulic structure have been derived. Since these equations are not explicit, they are calculated numerically. Results are employed to create design charts in easy-to-use form. These charts indicate that potential heads along the entire floor and the hydraulic exit gradient downstream of the structure are considerably decreased by the provision of two filters even with very small length. Uplift pressures along the floor between the two filters are controlled by the location of both filters. Potential heads along the upstream floor are controlled by the upstream filter location and potential heads downstream, and the hydraulic exit gradient downstream by the downstream filter location. Potential heads downstream of the second filter and the hydraulic exit gradient are decreased with decreases in depth of the permeable strata. Pore water pressures upstream of the second filter are decreased with increases in depth of the upstream cut-off. The results clearly indicate that the use of more than one filter in these situations is worth practical consideration.