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.
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".
