Simulation With a Full-Tensor Coefficient Velocity Field Recovered From a Diagonal-Tensor Solution
- M.G. Edwards (Stanford U.)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 2000
- Document Type
- Journal Paper
- 387 - 393
- 2000. Society of Petroleum Engineers
- 5.3.2 Multiphase Flow, 1.2.3 Rock properties, 4.3.4 Scale, 5.5 Reservoir Simulation
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Commercial simulators generally assume that the pressure equation always has a diagonal-tensor permeability field. However, cases that involve the use of non-Cartesian grids, crossbedding, or upscaling (or any combination thereof) will give rise to a full-tensor coefficient velocity field. Discretization of a full tensor increases the scheme stencil, simulation cost, and complexity of implementation. The focus of this paper is on the development of fully implicit finite-volume schemes that can allow full-tensor fluxes while retaining standard-matrix inversion.
A major assumption in today's commercial simulators is that the pressure equation always has a diagonal tensor. The design and efficiency of such codes is intrinsically linked to the diagonal-tensor assumption. However, this assumption is only true if the computational grid is aligned with the principal axes of the tensor.
In general, a full-tensor pressure equation arises in reservoir simulation whenever (a) the medium is anisotropic and nonaligned with the local frame of reference, or crossbedded, (b) fine-scale crossflow upscaling is performed, and (c) when non-K-orthogonal structured and unstructured grids are employed. Consequently, in general, all diagonal-tensor simulators will suffer from O(1) errors in flux1-4 when applied to cases involving these major effects. For example, while these simulators appear to allow for nonorthogonal grids through the definition of corner-point geometry, only the diagonal-tensor permeability-geometry contribution to the flux is included, thus leading to an O(1) error in flux (even for Laplaces equation) on a nonorthogonal grid.4
The introduction of a full tensor typically increases the support of the standard scheme on a logically rectangular grid from 5 to 9 nodes in two-dimensions (2D) and from 7 to 19 or 27 nodes in three-dimensions (3D) and, therefore, represents a potentially huge increase in computational cost, as the pressure field is recalculated at every time step of the simulation.
Earlier work on nine-point schemes mainly focused on improving the diagonal-tensor approximation, e.g.,5.6 the most popular application being studies of grid orientation effects on Cartesian grids. Full-tensor equations and approximations have been the subject of more recent developments.1-4,7-18
The focus of this paper is on the development of fully implicit finite-volume schemes that can allow full-tensor fluxes while retaining standard-matrix or reduced Jacobian-matrix inversion. The schemes are developed within a structured block-centered fully implicit formulation. Extension to unstructured grids is presented in Ref. 16. The methods presented here enable full-tensor solutions to be computed while using reduced operator matrices and generally reduce computation time compared to full-matrix inversion. In addition, full-tensor operators can be included with minimal code changes. Properties of the finite-volume schemes are presented together with analysis of the operator splitting and discussion of the iteration strategy. The schemes are applied to a number of problems involving strong crossflow due either to the local or global orientation of the grid relative to the problem, and benefits of the method are clearly demonstrated.
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