In this paper we present a three-dimensional flux-continuous finite-difference formulation designed for flow simulation of models with nonorthogonal hexahedron grids with general tensor permeability. Our development follows that of Aavatsmark et. al1 , but we do not operate in transformed space. The new 27-point discretization formula has been implemented in a finite difference reservoir simulator. This stencil has many desirable properties including collapsing into a consistent form in two-dimensions.

We demonstrate that there are many practical situations where neglecting the influence of the nonorthogonality and general tensors results in first-order errors in flow predictions. A rigorous implementation for this 27-pt difference operator as a control-volume finite-difference method determines the winded convection terms associated with multiphase computations. Results and issues associated with implementation of this operator in a conventional finite difference reservoir simulator are discussed.

As an alternative to directly solving the linear matrix associated with the 27-pt stencil of the flux-continuous operator, we examine iterative methods that split the matrix into a 7-pt stencil part and a remainder. The 7-pt stencil part is solved by a direct or iterative method with the remainder part updated from the previous time step or iteration. This split operator may permit retention of the linear solver for the standard 7-point formulation while retaining non-orthogonal tensor and grid information.

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