Discrete Fracture Method for Petroleum Reservoir Simulation Using an Element-based Finite Volume Method

Any porous media flow is inherently complex to model due to the impossibility of giving the real flow geometry to the simulation model. If heterogeneous media is involved, as is the case of naturally fractured petroleum reservoirs, the difficulty increases even more, since large fractures can be seen as discontinuities, having as background the porous matrix, in which many smaller sized fractures are present. The porous matrix can be treated by a stochastic procedure and are, normally, large deposits of oil, while the large fractures are better solved through a deterministic treatment. A network of connected large fractures linked to the porous matrix may be the most important flow path for oil production. In the other hand, depending on the physical properties, capillarity and permeability, the fractures and porous matrix combination may lead to a undesirable secondary oil recovery, leaving considerable amount of oil in the porous matrix. Therefore, the prediction of this combined flow (fractures +porous matrix) is of utmost importance for the oil industry. There are several approaches to solve this combined flow, all of them based, of course, on a idealized fracture configuration, which gets more and more realistic as the characterization methods evolves, due to the specialization of well-logging, 4D seismic and other methods. The final goal would be to solve the local flow for any single fracture, irrespective its size, nowadays an impossible task due to the lack of characterization methods and computer capacity. However, as computational power and characterization techniques evolve, methods able to solve the details of the flow should be devised. This paper follows this route and presents a DFM (Discrete Fracture Method) in the framework of an Element-based Finite Volume Method (EbFVM) using unstructured grids. The EbFVM is per se a multi-point flux approximation, avoiding the usual two-point approach, which is conceptually wrong, since the errors do not vanish as the grid is refined. The EbFVM also avoids the need of more complex MPFA algorithms for having correct flux evaluation. Additionally, the EbFVM framework allows the use of truly directional upwind and higher order schemes with no extra efforts. The 2D oil-water flow using the DFM method with superposition for connecting the fractures and the porous matrix is solved. The fractures are assumed to be 1D considering its real thickness. Several aspects of the model are investigated, as the capillarity effects, especially in the situation in which imbibition of the porous matrix occurs and the anisotropy of the coefficients of the linear system resulting from the superposition of the equations. Since IMPES method is used, comments on the time step adaption is also given.