The variational inequality formulation for unconfined seepage through three-dimensional dense fracture networks

Abstract

The space block search technology is used to determine a connected three-dimensional fracture network in polygonal shapes, i.e., seepage paths. After triangulation on these polygons, a finite element mesh for 3D fracture network seepage is obtained. Through introduction of the generalized Darcy’s law, conservative equations for both fracture surface and fracture interactions are established. Combined with the boundary condition of Signorini’s type, a partial differential equation (PDE) formulation is presented for the whole domain concerned. To solve this problem efficiently, an equivalent variational inequality (VI) formulation is given. With the penalized Heaviside function, a finite element procedure for unconfined seepage problem in 3D fracture network is developed. Through an example in a homogeneous rectangular dam, validity of the algorithm is verified. The analysis of an unconfined seepage problem in a complex fracture network shows that the proposed algorithm is very applicable to complex three-dimensional problems, and is effective in describing some interesting phenomenon usually encountered in practice, such as “preferential flow”.