Curvilinear Local Grid Refinement

Abstract

Abstract A local orthogonal curvilinear coordinate system is used for Local Grid Refinement (LGR) of rectangular wellblocks in a finite difference grid. The refinement is hybrid since the curvilinear well-subgrids are approximately radial close to wells, while approximately rectilinear near wellblock boundaries. A composite of a local stretching and a conformal transformation is used to map individual wellblocks onto the unit disk on the complex plane. The pre-image of a radial grid on the complex plane becomes an orthogonal curvilinear grid on a k-isotropic real plane, centered at the well. The method allows us to avoid use of a numerical correlation for effective wellblock radius, and the associated limitations of the conventional Peaceman well model no longer apply. Radial gridblock centers on the complex plane can be distributed logarithmically. This allows for an optimum spacing of gridblocks, and is a major advantage over Cartesian near-well LGR. For the well-subgrids, curvilinear gridblock pore volumes are obtained by a numerically inexpensive integration on the complex plane. The details of the numerical methods that make pore volume determination rapid and viable for realistic field-scale simulations, will be presented elsewhere. Inter-block transmissibilities, half-boundary transmissibilities and well indices are obtained from an analytic potential for radial flow on the unit disk. This scheme is both more accurate and less cumbersome than those based on curvilinear grid geometry. Furthermore, the determination of gridblock centers on the real plane becomes unnecessary. The hybrid nature of the well-subgrids reduces grid discretisation errors near wellblock boundaries. The method easily accommodates k-anisotropy and off-centered wells. For multi-phase problems, when near-well gridblocks become small the fully implicit solution technique is applied to the well-subgrids, while an implicit pressure, explicit mass (IMPEM) technique is applied to the parent coarse grid. Owing to orthogonality, adding curvilinear LGR capability does not require extensive reprogramming of a five-point (on the plane) finite difference simulator. This is important from a practical standpoint. The present curvilinear LGR formulation has been integrated into a general purpose full-field finite difference simulator (with existing Cartesian LGR capability) without any reprogramming of the simulator kernel. The method applies to well regions comprising several coarse gridblocks in the vicinity of wells, provided these well regions do not have areally varying k-anisotropy. Extensions to polygon well regions are also possible, making the method applicable to cornerpoint or general flexible grids. This enhancement is only outlined in this paper. Several applications have been chosen to demonstrate the potential of curvilinear LGR. Comparisons are also made between curvilinear and Cartesian near-well LGR. P. 29