Darcian flow under/through a leaky cutoff wall: Terzaghi–Anderson's seepage problem revisited

An analytical solution is obtained for 2-D steady Darcian flow under and through a cutoff wall partially obstructing a homogeneous isotropic foundation of a dam. The wall is leaky; that is, flow across it depends on the ratio of hydraulic conductivity of the wall and the wall thickness that results in the third-type (Robin) boundary condition along the wall, as compared with the Terzaghi problem for an impermeable wall. The Laplace equation for the hydraulic head is meshlessly solved in a non-standard flow tube. A Fredholm equation of the second kind is obtained for the intensity of leakage across the wall. The equation is tackled numerically, by adjusted successive iterations. Flow characteristics (total Darcian discharge and its components through the wall and the window between the wall top and horizontal bedrock, stream function, head distribution, and Darcian velocity along the wall and tailwater bed) are obtained for various conductivity ratios, head drops across the structure, thicknesses of the foundation, and the degree of its blockage by the wall. Comparisons with the Terzaghi limit of an impermeable wall show that for common wall materials and thicknesses, the leakage may constitute tens of percent of the discharge under the dam. The through-flow hydraulic gradients on a vertical wall face (Robin's boundary condition) as well as the exit gradients along a horizontal tailwater boundary (Dirichlet's boundary condition) acting for decades have deleterious impacts on dam stability because of potential heaving, piping, and mechanical–chemical suffusion.