American Institute of Mining, Metallurgical and Petroleum Engineers Inc.
A fundamental flow equation can be derived for each of the oil, water, and gas phases by combining the law of conservation of mass, a law of force, and thermodynamic relationships that describe the pressure-volume-temperature behavior of the fluids. The law of conservation of mass states that the sum of mass flow into a cell equals the change of mass within a cell. The law of force describing flow through porousmedia is Darcy's law, which assumes that flow is within the laminar flow regime described by the law. The thermodynamic relationships used to describe the fluid behavior must (because of the lack of accurate analytical expressions) be those found experimentally in a "PVT" study. The first part of this report shows how these laws can be combined to give equations describing fluid flow through porous media.
The next step in the development is the cabining of these equations with auxiliary equations so that the number of dependent variables is reduced. The result is an equation that can be assumed to be in terms of oil pressure only, so that it may be solved implicitly pressure only, so that it may be solved implicitly for pressures at the new time level. Separate from this pressure equation are the equations that are solved explicitly for saturations, using the newly computed pressures. Hence, there are three equations developed in final, finite difference form in this paper. 1) a pressure equation, 2) an oil saturation equation, pressure equation, 2) an oil saturation equation, and 3) a water saturation equation.
The logic used in the computation of all pertinent terms is given in the final section pertinent terms is given in the final section of this report.
The general aspects of numerical fluid flow simulation has been given elsewhere (1). The present paper gives the derivation of the fluid present paper gives the derivation of the fluid flow simulation equations. The numerical solution of these equations is given in (2).
Fig. 1 shows a differential element of porous media. Pounds of oil (w o) are flowing porous media. Pounds of oil (w o) are flowing into or out of two faces, with a well (w op)in the center of the elements.
Weight rates of flow, rather than mass rates, are used here to simplify the final algebra; of course, this necessitates the assumption of constant acceleration of gravity. The rate of depletion (ROD) of oil from this element can then be defined as:
where ROD is the rate of oil removal from the element in lb/day at reservoir conditions.