This paper describes a hybrid finite volume method, designed to simulate multi-phase flow in a field-scale naturally fractured reservoir. Lee et al. (WRR 37:443–455, 2001) developed a hierarchical approach in which the permeability contribution from short fractures is derived in an analytical expression and that from medium fractures is numerically solved using a boundary element method. The long fractures are modeled explicitly as major fluid conduits. Reservoirs with well-developed naturally fractures include many complex fracture networks that cannot be easily modeled by simple long fracture formulation and/or homogenized single continuity model. We thus propose a numerically efficient hybrid method in which small and medium fractures are modeled by effective permeability and large fractures are modeled by discrete fracture networks. A simple, systematic way is devised to calculate transport parameters between fracture networks and discretized, homogenized media. An efficient numerical algorithm is also devised to solve the dual system of fracture network and finite volume grid. Black oil formulation is implemented in the simulator to demonstrate practical applications of this hybrid finite volume method.
Many reservoirs are highly fractured due to the complex tectonic movement and sedimentation process the formation has experienced. The permeability of a fracture is usually much larger than that of the rock matrix; as a result, the fluid will flow mostly through the fracture network, if the fractures are connected. This implies that the fracture connectivities and their distribution will determine fluid transport in a naturally fractured reservoir (Long et al.1). Due to statistically complex distribution of geological heterogeneity and multiple length and time scales in natural porous media, three approaches (Smith and Schwartz2) are commonly used in describing fluid flow and solute transport in naturally fractured formations:
discrete fracture models,
continuum models using effective properties for discrete grids, and
hybrid models that combine discrete large features and equivalent continuum.
Currently, most reservoir simulators employ dual continuum formulations (i.e., dual porosity/permeability) for naturally fractured reservoirs, in which matrix blocks are divided by very regular fracture patterns (Kazemi et al.3, Van Golf-Racht4). Part of the primary input into these simulation models is the permeability of the fracture system assigned to the individual grid-blocks. This value can only be reasonably calculated if the fracture systems are regular and well connected. However, field characterization studies have shown that fracture systems are very irregular, often disconnected and occur in swarms (Laubach5 Lorenz and Hill6, and Narr et al.7).
Most naturally fractured reservoirs include fractures of multiple length-scales. The effective grid-block permeability calculated by the boundary element method becomes expensive as the number of fractures increases. The calculated effective properties for grid-blocks also underestimates the properties for long fractures whose length scale is much larger than the grid-block size.