CVD-MPFA full pressure support, coupled unstructured discrete fracture–matrix Darcy-flux approximations

Two novel control-volume methods are presented for flow in fractured porous media, involving coupling the control-volume distributed multi-point flux approximation (CVD-MPFA (c.f. Edwards et al.)) constructed with full pressure support (FPS), to two types of discrete fracture-matrix approximation for flow simulation on unstructured grids; (i) involving hybrid grids and (ii) a lower dimensional fracture model. Flow is governed by Darcy's law together with mass conservation both in the rock matrix and in fractures, where large discontinuous permeability tensors can occur. Finite-volume FPS schemes are more robust than the earlier CVD-MPFA triangular pressure support (TPS) schemes for problems involving strongly anisotropic homogeneous and heterogeneous full-tensor permeability fields. We use a cell-centred hybrid-grid method, where fractures are represented by lower-dimensional interfaces between matrix grid cells in the physical mesh, and expanded to equi-dimensional cells in the computational domain. We present a simple procedure to form a consistent hybrid-grid locally for a dual-cell. We also propose a novel hybrid-grid for intersecting fractures, for the FPS method, which improves the condition number of the global linear system and permits larger time steps for tracer transport. The tracer flow transport equation is coupled with the pressure equation and the results provide flow parameter assessment of the fracture models. Transport results obtained via TPS and FPS hybrid-grid formulations are compared with corresponding results of fine-scale explicit equi-dimensional formulations. The results show that the hybrid-grid FPS method applies to general full-tensor fields and provides improved robust approximations compared to the hybrid-grid TPS method for fractured domains, for both weakly anisotropic permeability fields and in particular for very strong anisotropic full-tensor permeability fields where the TPS scheme exhibits spurious oscillations. The hybrid-grid FPS formulation is extended to compressible flow and the results demonstrate the method is also robust for transient flow. Furthermore, FPS is coupled with a lower-dimensional fracture model, where fractures are strictly lower-dimensional in the physical mesh. Comparisons of the hybrid-grid FPS method and the FPS lower-dimensional fracture model are presented for several cases of isotropic and strongly anisotropic fractured media which illustrate the benefits of the respective methods