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 controlvolume 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.

A multipoint flux approximation with diamond stencil finite volume scheme for the two‐dimensional simulation of fluid flows in naturally fractured reservoirs using a hybrid‐grid method

“…Our convergence ratios and errors magnitude are very close to those presented by Ahmed et al, 20 as shown in Figure 7, despite the errors magnitude, for = 1, of both CVD-MPFA versions (TPS and FPS) being slightly lower than those of the MPFA-D. Otherwise, for the other values of (for all values of a) the errors magnitude obtained by the MPFA-D are basically the same and never bigger than those of the CVD-MPFA (Figure 7).…”

“…In the convergence test, the proposed formulation performed very well when compared to the CVD-MPFA, method that has been recently adapted in the Hybrid-Grid context, 20 being capable to return similar error magnitudes and equivalent convergence rates.…”

“…In this section, the results of the application of the MPFA-D in the context of HyG for the simulation of one-and two-phase flows in naturally fractured reservoirs are presented. The first example is a convergence test, comparing the results of the MPFA-D/HyG with some results of Ahmed et al 20 The second and the third examples are one-phase flow problems. In the second one, we compare the MPFA-D with the MPFA-O (multipoint flux approximation with "O" stencil), [4][5][6] both in HyG context.…”

“…The errors magnitudes of all tested cases are close to each other. The results of the MPFA-D were compared with those presented, for the same test, by Ahmed et al 20 These authors obtained their results using the CVD-MPFA (control-volume distributed multipoint flux aproximation), also in a hybrid-grid context, in two different versions, the triangle-pressure support and the full-pressure support.…”

“…17 In the LDFMs, 8,16,18 the flow equations for the fractures are discretized separately from the matrix, in a (n − 1)-dimensional subdomain. On the other hand, in the HyGs, 12,19,20 the fractures are expanded to n-dimensional objects in the computational domain, and the equations for fractures and the rock matrix can be discretized together, taking advantage from full pressure support (FPS) MPFA methods to solve the pressure equation. 20 Therefore, the HyG seems to present a good compromise between accuracy and computational costs, due its capability to properly represent complex geometries and anisotropy within fractures, without the computational overhead associated to the explicit discretization of fractures of the equidimensional approach.…”