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

Abstract

Two novel control-volume methods are presented for flow in fractured media, and involve coupling the control-volume distributed multi-point flux approximation (CVD-MPFA) constructed with full pressure support (FPS), to two types of discrete fracture–matrix approximation for 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 matrix and the fractures, where large discontinuities in permeability tensors can occur. Finite-volume FPS schemes are more robust than the earlier CVD-MPFA triangular pressure support (TPS) schemes for problems involving highly anisotropic homogeneous and heterogeneous full-tensor permeability fields. We use a cell-centred hybrid-grid method, where fractures are modelled by lower-dimensional interfaces between matrix cells in the physical mesh but 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 reduces the condition number of the global linear system and leads to larger time steps for tracer transport. The transport equation for tracer flow is coupled with the pressure equation and provides flow parameter assessment of the fracture models. Transport results obtained via TPS and FPS hybrid-grid formulations are compared with the 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 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, we present FPS coupled with a lower-dimensional fracture model, where fractures are strictly lower-dimensional in the physical mesh as well as in the computational domain. We present a comparison of the hybrid-grid FPS method and the lower-dimensional fracture model for several cases of isotropic and anisotropic fractured media which illustrate the benefits of the respective methods.