Reservoir simulation is accomplished by solving a set of simultaneous partial differential equations, usually using finite difference methods. The PDE"s are well known and widely documented. However, their boundary conditions, which are an integral part of the solution, have received little attention. Most simulators use boundary conditions that insure that there is no flow of each phase across the boundaries. Saturation equations use no-flow pressure boundary conditions for the phase whose conservation they represent. Pressure equations, use no-flow pressure boundary conditions that insure no bulk flow. However, the pressures of each phase are related by capillary pressure and phase densities. They cannot be specified independently. To do so at the boundary is erroneous.
This error can be avoided by choosing boundary conditions that curtail flow through the boundaries with specifications on both pressures and saturations. For example, along the bottom of the reservoir, pressures can be specified so that there is no flow of the most dense, mobile phase. The saturations of the lighter phases are set at their interstitial or residual values or less. This paper demonstrates that although such new boundary conditions can significantly affect reservoir simulation results, they are difficult to incorporate in finite difference reservoir simulators.
Although the partial differential equations which constitute reservoir simulators, and their finite difference approximations, have been widely discussed in the literature1–5.their accompanying boundary conditions have not. This lack of attention is probably the result of the simplicity with which impermeable reservoir boundaries are treated. Peaceman4 suggests the modeling of a rectangular volume larger than the reservoir and that cells exterior to the reservoir should have zero permeability and porosity. In this manner the explicit handling of boundary conditions is avoided. Perhaps the most popular way in insuring no flow across reservoir boundaries is to simply excluding the terms in the finite difference equation that correspond to the flow coming through the side of the cells coincident with the boundary. Equivalently, fictitious grid points are sometimes located in a mirror image position across the boundary and their flow potentials (P + ρz) are set equal to their interior image cells. However, the pressures and flow potentials of the three phases are related to one another through capillary pressure, pc, and through the phase densities, ρ. They cannot be specified independently. If cells contains more than one mobile phase, and those phases have different densities, then only the flow of one phase can be set equal to zero with pressure or flow potential specifications. If the pressures are such that oil does not flow, then water and gas will. If water won't, oil and gas will, etc.
If the specification of pressures is insufficient to insure that there is no flow through the boundaries, then the other dependent variables, i.e. the saturations, must also be specified. The saturations of two out of the three phases must be sufficiently low that these phases are immobile.
Hence, it is hypothesized that along the bottom of the reservoir the gradient in the flow potential of the densest mobile phase is zero.
