An Analysis Of Buckley-Leverett Theory

Abstract

Abstract The Buckley-Leverett theory is discussed in light of the recently proposed flow equations involving "generalized relative permeabilities". In particular it is argued that because capillary pressure was ignored in the Buckley-Leverett analysis, internal consistency requires that the parameters be corrected for the removal of interfacial tension. The results indicate a tendency toward removing the inflection point all the fractional flow curve, which is the mathematical source of an unphysical saturation profile. Introduction Any attempt to describe the flow of multiple phases through a porous medium must address the complex problem of incorporating interfacial phenomena from the pore scale into a larger-scale description of the dynamics. In particular, for water-oil displacements, motions of the fluid-fluid interfaces, the formation of oil ganglia and the wetting characteristics of the medium introduces "capillarity" into the dynamics. The importance of these processes in oil recovery mechanisms has been the subject of a great deal of research (e.g. Melrose and Brandner, 1974; Stegemeier, 1977; Mohanty et al., 1980; Takamura and Chow, 1983). However, difficulties in incorporating this information into larger-scale descriptions of the flow dynamics have been a stumbling block in attempts to describe the dynamics of multiphase flow. Furthermore, pore size distribution and pore geometry have been shown to have a large effect on the trapping of oil ganglia (cf. Chatzis et al. 1983, Wardlaw, 1982) and thus residual oil saturations. The description of multiphase flow which has been generally accepted is based on an analogy with single-phase flow. Proposed by Muskat (1946, 1953) over thirty years ago, it assumes that multiphase flow can be adequately described by flow equations of the same form as Darcy's equation. Permeability is replaced by parameters called relative permeabilities which are taken as functions of fluid saturation. Thus all the dynamic interfacial phenomena which occur at the pore scale during multiphase flow are to be handled by this single modified term in each of the flow equations. A more rigorous understanding of the dynamics of multiphase flow, in terms of pore scale phenomena, could not be considered at that stage since Darcy's equation itself was only understood as an empirical relationship. A number of researchers subsequently speculated that Darcy's equation was illustrating the average behaviour of the well understood flow dynamics at the pore scale (i.e. the Navier-Stokes equation) and the effects of the complex boundary between the fluid and the solid medium (e.g. Hubbard, 1956; Hall, 1956; Scheidegger. 1974). The main obstacle in these analyses to a clear formulation of the problem, was how to relate the averaged effect of the differential equations at the pore scale to the Darcy description which expressed the problem as a differential equation of the averaged quantities. A breakthrough came when Whittaker (1969) and Slattery (1969) proved what has come to be known as the Whittaker-Slattery averaging theorem. This mathematical result expresses the volume average of the spatial derivative of a microscopic quantity in terms of the derivative of the volume averaged quantity plus a surface integral over the interfaces between the fluid and solid.