A new variational inequality formulation for unconfined seepage flow through fracture networks

Abstract

Darcy’s law only applying to the flow domain is extended to the entire fracture network domain including the dry domain. The partial differential equation (PDE) formulation for unconfined seepage flow problems for discrete fracture network is established, in which a boundary condition of Signorini’s type is prescribed over the potential seepage surfaces. In order to reduce the difficulty in selecting trial functions, a new variational inequality formulation is presented and mathematically proved to be equivalent to the PDE formulation. The numerical procedure based on the VI formulation is proposed and the corresponding algorithm has been developed. Since a continuous penalized Heaviside function is introduced to replace a jump function in finite element analysis, oscillation of numerical integration for facture elements cut by the free surface is eliminated and stability of numerical solution is assured. The numerical results from two typical examples demonstrate, on the one hand the effectiveness and robustness of the proposed method, and on the other hand the capability of predicting main seepage pathways in fractured rocks and flow rates out of the drainage system, which is very important for performance assessments and design optimization of complex drainage system.