Numerical Three-Phase Model of the Two-Dimensional Steamflood Process
Authors: Shutler N.D. 
Published: December 2007

Abstract
This paper describes a numerical mathematical model that is a significant extension of a previously published one-dimensional model of the steamflood published one-dimensional model of the steamflood process. process. The model describes the simultaneous flow of the three phases - oil, water and gas - in two dimensions. Interphase mass transfer between water and gas phases is allowed, but the oil is assumed nonvolatile and the hydrocarbon gas insoluble in the liquid phases. The model allows two-dimensional heat convection within the reservoir and two-dimensional heat conduction in a vertical cross-section spanning the oil sand and adjacent strata. Example calculations are presented which, on comparison with experimental results, tend to validate the model. Steam overriding due to gravity effects is shown to significantly reduce oil recovery efficiency in a thick system while jailing to do so in a thinner system. A study of the effect of capillary pressure indicates that failure to scale capillary forces in laboratory models of thick sands may lead to optimistic recovery predictions, while properly scaled capillary forces may be sufficiently low as to play no important role in oil recovery. Calculations made with and without vertical permeability show that failure to account for vertical fluid flow can lead to predictions of pessimistic oil recovery efficiency. pessimistic oil recovery efficiency

Introduction
Mathematical tools of varying complexity have been used in studying the steamflood process. A "simplified" class of mathematical models has served primarily as aids in engineering design. A more comprehensive class of models has improved understanding of the nature of the process. The model described in this report is of the latter class, but it is more comprehensive than any previously published model. published model. All previously available calculations of the steamflood process are confined to one space dimension in their treatments of fluid flow. Thus all previous models necessary ignore all effects of gravity reservoir heterogeneity, and nonuniform initial fluid-phase distributions on fluid flow in a second dimension. This model, an extension of a previously published model accounts for heat and previously published model accounts for heat and fluid transfer in two space dimensions and, hence, can evaluate these effects on simultaneous horizontal and vertical flow. While the model can describe the areal performance of a steamflood (in which case the heat transfer is described in three dimensions), this aspect will not be considered in this paper. Rather, this paper will describe the model in its application to a vertical cross-section through the reservoir and will consider some preliminary investigations to demonstrate the importance of being able to simultaneously account for horizontal and vertical fluid flow. Mathematical details are given in appendices.

MATHEMATICAL DESCRIPTION OF STEAMFLOODING
Darcy's law provides expressions for the velocities of the three phases (oil, water and gas), which, when combined with oil, water and gas mass balances give the partial differential equations governing Now of the three phases within a reservoir sand:

OIL PHASE
..(1)

WATER PHASE
..(2)
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