This paper was prepared for presentation at the 1999 SPE Reservoir Simulation Symposium held in Houston, Texas, 14-17 February.

We describe a hierarchical approach for modeling fluid flow in a naturally fractured reservoir with multiple length-scale fractures. Based on fracture length (lf) relative to the finite-difference grid size (lg), fractures are classified as belonging to one of three groups:short disconnected fractures (lf << lg),medium-length fractures (lf ~ lg), andlong fractures (lf >> lg). Effective grid-block permeabilities, associated with the short and medium length fractures, are used as input to a finite-difference simulator. We also present a separate transport equation for flow through long fractures to capture effects of large-scale high permeability pathways. Our new approach provides an improved means to include realistic and explicit fracture descriptions. Previously, Lough et al. (1997, 1998) developed a numerical method to compute the effective permeability of simulation grid-blocks with realistic fracture characterizations. Although this method handles generalized fracture geometries, it is numerically inefficient for the case where many small fractures exist. The method can also underestimate the flow contribution from long fractures. Assuming a linear potential gradient along the short fractures, we derive an analytical solution for the permeability contribution from short fractures. The solution becomes more accurate as fractures become randomly distributed and asymptotically small in length. The permeabilities from this analytical solution are used as the effective matrix permeability in computing a combined effective grid-block permeability that includes medium-length fractures. The method of Lough et al. is used for this computation. This hierarchical approach takes into account coupled flow between the rock matrix, short fractures and medium-length fractures. Long fractures are modeled explicitly in the reservoir simulator, using a transport equation that describes flow between long fractures and surrounding simulation grid-blocks. Simulation results from our new hierarchical approach are compared with those from the conventional dual porosity/permeability model. Numerical efficiency and accuracy are also examined.

The gridblock permeabilities used in reservoir simulation are commonly determined through the upscaling of a fine scale geostatistical reservoir description. Though it is well established that permeabilities computed in this manner are, in general, full tensor quantities, most finite difference reservoir simulators still treat permeability as a diagonal tensor. In this paper, we implement a capability to handle full tensor permeabilities in a general purpose finite difference simulator and apply this capability to the modeling of several complex geological systems. We formulate a flux continuous approach for the pressure equation by use of a method analogous to that of previous researchers (Edwards and Rogers 1 ; Aavatsmark et al. 2 ), consider methods for upwinding in multiphase flow problems, and additionally discuss some relevant implementation and reservoir characterization issues. The accuracy of the finite difference formulation, assessed through comparisons to an accurate finite element approach, is shown to be generally good, particularly for immiscible displacements in heterogeneous systems. The formulation is then applied to the simulation of upscaled descriptions of several geologically complex reservoirs involving crossbedding and extensive fracturing. The method performs quite well for these systems and is shown to capture the effects of the underlying geology accurately. Finally, the significant errors that can be incurred through inaccurate representation of the full permeability tensor are demonstrated for several cases.