The paper presents a class of control-volume discretization methods for general quadrilateral grids in three space dimensions. The methods allow full permeability tensor and arbitrary orientation of the grid. The paper discusses handling of inactive cells and flow barriers. Numerical examples are used to compare the new methods with conventional comer-point geometry methods.
The development of increasingly complex geological models and the appearance of advanced well technology are two major challenges for reservoir simulation technology. Enhanced flexibility in the design of computational grids is absolutely necessary to meet these challenges; a grid must be allowed to adapt to layer boundaries, faults, well trajectories, and other geometric constraints. To achieve this flexibility without losing accuracy in the numerical solution, the model equations must be discretized with a method that allows for the irregular or distorted cells created by the geometrical modelling.
Commercially available simulators predominantly are based on a structured grid with a corner-point formulation. This formulation may introduce significant errors if the cell shapes deviate too much from an ideal orthogonal situation or if the flow direction is unfavorable compared with that of the grid. The structured grids do, however, have computational advantages over triangle-based, unstructured grids, which could fit the geometry better. At present, available engineering software for grid construction and upscaling also strongly favors structured gridding techniques.
In the corner-point formulation, the rock permeability is assumed to be defined with values along the local cell axes. This assumption is not fulfilled if the grid is aligned with a curved well trajectory instead of a layer boundary. The assumption may also introduce large errors in nonorthogonal cells when the permeability anisotropy is strong. Increased flexibility in cell orientation and cell shape hence requires proper handling of anisotropy. In general, the permeability should be represented by a tensor K with principal axes independent of local grid axes.
Proper handling of nonorthogonal cells and permeability anisotropy will generally require a flux approximation involving more than two grid cells, as applied in the corner-point formulation. For a three-dimensional (3D) structured grid, this will lead to cell molecules involving more than the standard seven cells. To reduce the computational overhead, the more accurate discretization method should be used only locally in areas of special importance, for instance, in the vicinity of wells.