Finite Difference Simulation of Geologically Complex Reservoirs With Tensor Permeabilities

Abstract

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.

Symmetric Positive Definite General Tensor Discretization Operators on Unstructured and Flow Based Grids

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S ummaryThe derivation of algebraic flux continuity conditions for full tensor discretization operators has lead to efficient and robust locally conservative flux continuous finite volume methods for determining the discrete velocity field in subsurface reservoirs e .g [1][2][3][4][5][6][7][8] .

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Limitations to upscaling of groundwater flow models dominated by surface water interaction

“…An important aspect of coarsening is that K c becomes often anisotropic, even if the original hydraulic permeability K is isotropic. Recently, methods became available that cope with the off-diagonal components in a tensor [Aavatsmark et al, 1996;Lee et al, 1998;Anderman et al, 2002].…”

A Black-Oil Reservoir Simulator for Hexahedral Multiblock Grids

“…Flux-continuons methods, designed to handle full tensor permeability and/or non-orthogonal grids, have been presented previously for two dimensional [8,10] and three dimensional systems [1,111 . Though there are differences between these various scheures, they all provide continuity of flux across cell boundaries and reduce to the usual weighted harmonic mean transmissibilities in the appropriate limit (orthogonal grid, diagonal permeability tensors) .…”

“…Multi-point discretization scheme is used for the flux formulation in order to treat grid non-orthogonalities and anisotropic permeability fields correctly . A general finite-volume discretization scheme for nonorthogonal hexahedron grids with tensor permeability leads to wide stencils (9-pt in 2D [8,10] and 27-pt in 3D [1,11]) . We employ the wide stencil derived by the finite-volume method of Lee et al .…”