Finite Difference Simulation of Geologically Complex Reservoirs With Tensor Permeabilities
Authors: Lee S.H. , Durlofsky L.J. , Lough M.F. , Chen W.H. 
Published: December 2007

Abstract
The grid block permeabilities used in reservoir simulation are commonly determined via 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 using a method analogous to that of previous researchers (Edwards and Rogers; Aavatsmark et al.), 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 accurately capture the effects of the underlying geology. Finally, the significant errors which can be incurred through inaccurate representation of the full permeability tensor are demonstrated for several cases.

Introduction
Recent advances in reservoir characterization permit the construction of realistic, highly detailed, heterogeneous reservoir descriptions. Such models typically contain far too many grid blocks to simulate directly and therefore require some type of upscaling before they can be used for reservoir simulation. The most important of the upscaled rock properties, for purposes of flow simulation, is the absolute permeability. Accurate procedures for the scale up of permeability generate full tensor permeabilities on the coarse scale, even in cases where the underlying fine scale permeability description is isotropic. Therefore, simulation models generated through scale up of complex reservoir descriptions will in general be characterized by full tensor permeabilities.
For many models, however, the off-diagonal components of the effective (or equivalent grid block) permeability tensors can be expected to be small relative to the diagonal components and can generally be ignored. This will typically be the case, for example, when the fine scale permeability is correlated along the coordinate directions (e.g., strictly layered systems) or is correlated nearly isotropically. However, for other types of systems the cross terms of the permeability tensor can be expected to be quite significant. These include formations containing complex crossbedding, dipping layers not aligned with the coordinate system, or extensive fracturing. In these cases the upscaled simulation model will contain full tensor permeabilities with significant off-diagonal terms, which must be accommodated by the simulator if reservoir performance is to be predicted accurately.
Our intent here is to develop an approach for the modeling of complex geological systems using a general purpose reservoir simulator. Toward that end, we first develop and implement a formulation for full tensor permeability models, applicable for curvilinear grids, into a general purpose finite difference reservoir simulator. Second, we apply this formulation to the simulation of flow through a variety of upscaled geologic descriptions. P. 253^