An Analytical Solution for the Steady Seepage of Localized Line Leakage in Tunnels

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This paper proposes an analytical solution for the seepage field when a localized line leakage occurs in a tunnel by accurately considering the boundary conditions at the leakage site, which overcomes the problem of current methods, such as the equivalent method or methods improving on the existing analytical solution for fully drained tunnels, being unable to give an accurate analytical solution. First, the semi-infinite seepage region is converted into a rectangular seepage region using two conformal transformations. Subsequently, in order to accurately consider the boundary conditions at the leakage site, the rectangular seepage region with a discontinuous boundary is divided into three subregions with continuous boundaries, and the water head solution for each subregion is given by using the separated variable method. Finally, the principle of orthogonality of trigonometric functions is specially adopted to construct a non-homogeneous system of equations to solve the unknowns in the analytical solution, and through the inverse transformation of the conformal transformation, an analytical solution for the steady-state seepage field when localized line leakage occurs in a tunnel is obtained. The solution proposed is verified by its satisfactory agreement with the numerical simulation results and existing experimental results, and is much more accurate than the existing analytical solution. In addition, the proposed analytical solution is much less computationally demanding compared to numerical simulations. Finally, the capability of the proposed analytical solution is demonstrated by a parametric analysis of the tunnel burial depth, leakage location, and leakage width, and some meaningful conclusions are drawn.