Seepage flow around twin circular tunnels in anisotropic ground revealed by an analytical solution

Abstract

A new analytical solution is proposed for steady seepage flow around twin circular tunnels in fully saturated anisotropic ground. The solution is an exact one that fully satisfies all the boundary conditions and precisely considers the different permeabilities along two directions and the interactions between twin tunnels. The solution provides a fast approach for the estimation of the seepage field and a useful tool for design optimization. The solution is successfully addressed using problem equivalence and the Schwartz alternating method combined with a mapping function. Using a coordinate transformation of the governing equation, the anisotropic problem of circular tunnels is first equivalent to that of isotropic elliptical tunnels, and the length of the ellipse along the anisotropic axis depends on the anisotropic permeability ratio. The Schwartz alternating method is then employed to address the solution of equivalent elliptical twin tunnel problems, where a mapping function, with which an elliptical tunnel in the half-plane can be mapped into an annulus in the image plane, is introduced to solve the single tunnel problems in each iterative step. The iterative procedure is quite simple and efficient in its calculations and achieves good convergence, and the analytical solution agrees very well with the numerical results to reflect its high precision in the entire ground. Finally, parametric studies are performed to investigate the influences of the anisotropic permeability and tunnel spacing on the seepage field. This is the first study to provide the exact analytical solution of the seepage field of twin tunnel problems in anisotropic ground, and the procedure can be extended to multiple tunnel problems.