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Quasi K-orthogonal grid generation is presented, to improve grid quality and method stability with respect to flux approximation in the presence of strongly anisotropic full-tensor permeability fields.K-orthogonal grid generation is only possible for low anisotropy ratios. Quasi K-orthogonal grid generation involves satisfying the K-orthogonal condition approximately, resulting in grids that place less demand on an approximation with respect to stability conditions, and therefore improve grid quality with respect to flux approximation in the presence of anisotropic permeability fields. The method employed enables Delaunay grid generation principles to be employed in a locally transformed system according to local permeability tensor variation. The resulting method has great flexibility for handling complex geometries and can handle jumps in permeability tensor principal axes orientation and jumps in coefficients and details will be presented. Results are presented that demonstrate the benefit of a quasi K-orthogonal grid. Highly challenging cases involving strong full-tensor permeability fields where control-volume distributed multi-point flux approximation (CVD-MPFA) schemes exceed their stability limits and yield solutions with spurious oscillations when using conventional grids, are solved using the new grid generation method. CVD-MPFA schemes are still required as the grids are only approximately K-orthogonal in such cases, however the schemes retain a discrete maximum principle on the new quasi-K-orthogonal grids and yield well resolved solutions that are free of spurious oscillations. While the two-point flux approximation (TPFA) requires strict K-orthogonality, results using both CVD-MPFA and TPFA will be presented. New Quasi K-orthogonal grid generation methods are presented that satisfy the K-orthogonal condition approximately, resulting in practical grids that restore a discrete maximum principle (stability) for the CVD-MPFA schemes when applied to cases involving general full-tensor permeability fields. Results are presented for a variety of test cases that confirm the validity of the grids.
Three-dimensional unstructured grid generation for reservoirs with geological layers, faults, pinchouts, fractures and wells is presented. Grids are generated for example cases, and pressure fields and flow fields computed by the cell-centered and vertex-centered control-volume distributed multi-point flux approximation (CVD-MPFA) schemes are compared and contrasted together with the methods. Grid generation for reservoir simulation, must honour classical key geological features and multilateral wells. The geological features are classified into two groups; 1) involving layers, faults, pinchouts and fractures, and 2) involving well distributions. In the former, control-volume boundary aligned grids (BAGs) are required, while in the latter, control-point well aligned grids (WAGs) are required. In reservoir simulation a choice of grid type and consequent control-volume type is made, i.e. either primal or dual-cells are selected as control-volumes. The control-point is defined as the centroid of the control-volume for any grid type. Three-dimensional unstructured grid generation methods are proposed that automate control-volume boundary alignment to geological features and control point alignment to wells, yielding essentially perpendicular bisector (PEBI) meshes either with respect to primal or dual-cells depending on grid type. Both primal and dual-cell boundary aligned grid generators use primal-cells (tetrahedra, pyramids, prisms and hexahedra) as grid elements. Dual-cell feature aligned grids are derived from underlying primal-meshes, such that features are recovered, with control-volume faces aligned with interior feature boundaries. The grids generated enable a comparative performance study of cell- vertex versus cell-centered CVD-MPFA finite-volume formulations using equivalent degrees of freedom. The benefits of both types of approximation are presented in terms of flow resolution relative to the respective degrees of freedom employed. Stability limits of the methods are also explored. For a given mesh the cell-vertex method uses approximately a fifth of the unknowns used by a cell-centered method and proves to be the most beneficial with respect to accuracy and efficiency, which is verified by flow computation. Novel techniques for generating three-dimensional unstructured hybrid essentially PEBI-grids, honouring geological features are presented. Geological boundary aligned grid generation is performed for primal and dual-cell grid types. Flow results show that vertex-centered CVD-MPFA methods outperform cell-centered CVD-MPFA methods.