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

Abstract

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 throughflow 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.

Phreatic seepage flow through an earth dam with an impeding strip

“…Then, we re-assemble eqs. (11) and (4) to get the final solution in terms of Strack's potential (2).…”

“…A motivating example for this paper is the Oroville Dam (California) crisis of 2017. On the eve of this crisis, Yakimov and Kacimov [2] accentuated the necessity to study a potentially dangerous phenomenon of seepage across cutoff-walls in porous foundations of dams. In this type of Darcian flow, invisible but insidious hydraulic gradients across the barrier can cause its mechanical suffusion and deterioration of counter-seepage functions.…”

Analytical Method for Groundwater Seepage through and Beneath a Fully Penetrating Cut-off Wall Considering Effects of Wall Permeability and Thickness

“…In general, the permeability of the aquifer is much higher than that of the underlying aquitard. Thus, the head change within a small area near the cut-off wall in the aquifer can be overlooked [32]. Then, seepage flow near the cut-off wall in the aquitard can be modelled by the seepage model, as illustrated in Figure 1c.…”

“…The internal permeable boundary conditions make the seepage model in Figure 1c difficult to solve exactly. In previous studies, the cut-off wall was simplified to an impermeable sheet pile with no thickness [23,24], or the effect of the wall permeability or thickness on the seepage was considered separately [26,32]. Based on the assumption that cut-off walls are impermeable sheet piles with no thickness, Aravin and Numerov [23] listed fundamental analytical solutions for the seepage through foundations beneath dams.…”

“…Nevertheless, the leakage through the wall body cannot be counted. Yakimov and Kacimov [32] assumed that the cut-off wall was infinitely thin and derived exact solutions for the leakage through the wall body and seepage under the wall. The effect of the wall permeability on the seepage was studied.…”