Interpretation of Well-cell Pressures on Hexagonal K-orthogonal Grids in Numerical Reservoir Simulation

Abstract

Peaceman's equivalent well-cell radius for 2D Cartesian grids has been generalized to 2D uniform hexagonal K-orthogonal grids in an anisotropic medium. An analytic expression for the equivalent well-cell radius for infinitely fine grids is derived. The derivation is performed by comparison of analytical and numerical solution for boundary value problems with one or two wells. The derivation for the anisotropic case is based on a transformation to an isotropic image space and follows Peaceman's derivation closely. Since the well-cell radius varies slowly with the grid fineness, the found formula can be considered representative for all grid sizes. Since 2D seven-point stencils are more rotationally invariant than five-point stencils, they are often preferred to reduce grid-orientation problems. The formula can be applied to calculate the correct difference between the bottomhole pressure and the numerical well-cell pressure for 2D hexagonal grids. Such a formula is necessary in case of pressure-controlled wells. It is also useful for rate-controlled injection wells with an upper pressure bound. The formula is easy to implement in a reservoir simulator.