The Mathematical Model of Nonequilibrium Effects in Water-Oil Displacement

Abstract

Summary Forced oil-water displacement and spontaneous countercurrent imbibition are the crucial mechanisms of secondary oil recovery. Classical mathematical models of both these unsteady flows are based on the fundamental assumption of local phase equilibrium. Thus, the water and oil flows are assumed to be locally distributed over their flow paths similarly to steady flows. This assumption allows one to further assume that the relative phase permeabilities and the capillary pressure are universal functions of the local water saturation, which can be obtained from steady-state flow experiments. The last assumption leads to a mathematical model consisting of a closed system of equations for fluid flow properties (velocity, pressure) and water saturation. This model is currently used as a basis for numerical predictions of water-oil displacement. However, at the water front in the water-oil displacement, as well as in capillary imbibition, the characteristic times of both processes are, in general, comparable with the times of redistribution of flow paths between oil and water. Therefore, the nonequilibrium effects should be taken into account. We present here a refined and extended mathematical model for the nonequilibrium two-phase (e.g., water-oil) flows. The basic problem formulation, as well as the more specific equations, are given, and the results of comparison with an experiment are presented and discussed. Introduction The problem of simultaneous flow of immiscible fluids in porous media, and, in particular, the problem of water-oil displacement, both forced and spontaneous, are both fundamental to the modern simulations of transport in porous media. These problems are also important in engineering applications, especially in the mathematical simulation of the development of oil deposits. The classical model of simultaneous flow of immiscible fluids in porous media was constructed in late 30s and early 40s by the distinguished American scientists and engineers M. Muskat and M.C. Leverett and their associates.1–3 Their model was based on the assumption of local equilibrium, according to which the relative phase permeabilities and the capillary pressure can be expressed through the universal functions of local saturation. The Muskat-Leverett theory was in the past of fundamental importance for the engineering practice of the development of oil deposits, and it remains so. Moreover, this theory leads to new mathematical problems involving specific instructive partial differential equations. It is interesting to note that some of these equations were independently introduced later as simplified model equations of gas dynamics. Gradually, it was recognized that the classical Muskat-Leverett model is not quite adequate, especially for many practically important flows. In particular, it seems to be inadequate for the capillary countercurrent imbibition of a porous block initially filled with oil, one of the basic processes involved in oil recovery, and for the even more important problem of flow near the water-oil displacement front. The usual argument in favor of the local equilibrium is based on the assumption that a representative sampling volume of the water-oil saturated porous medium has the size not too much exceeding the size of the porous channels. In fact, it happens that it is not always the case and that the nonequilibrium effects are of importance. A model, which took into account the nonequilibrium effects, was proposed and developed by the first author and his colleagues4–8 ; see also Ref. 9. This model was gradually corrected, modified, and confirmed by laboratory and numerical experiments. In turn, this model leads to nontraditional mathematical problems. In this paper, the physical model of the nonequilibrium effects in a simultaneous flow of two immiscible fluids in porous media is presented as we see it now. We also relate the new asymptotic time scaling of oil recovery by countercurrent imbibition in water-wet rock (Eq. 25) to experimental data. We discuss some peculiar properties of the solutions to the capillary imbibition problem clearly demonstrating nonequilibrium effects.

Upscaling of fractured oil reservoirs using homogenization including non-equilibrium capillary pressure and relative permeability

“…The effective saturation η is always larger than the water saturation S w or equal to it when steady state is attained. The concept of effective saturation was first introduced by Barenblatt and his colleagues [6,7,9,10]. It is equivalent to an alternative formulation introduced by Hassanizadeh and Gray [20] (see also [21]).…”

“…However, following Barenblatt [6], Barenblatt and Gilman [7], and Barenblatt et al [10], we assume that the effective saturation is the same for all three functions. Thus, instead of the relationships in Eq.…”

“…Due to non-equilibrium effects, the relationship between η and S w must be a time-dependent function. We use the hypothesis [6,9,10] that there is a relationship between the local effective water saturation η and the actual water saturation S w and its rate of change ∂ S w /∂t. Dimensional analysis suggests that such a relationship must include a characteristic redistribution time.…”

“…Hassanizadeh and Gray [19,20] do not deal with relative permeabilities, but their theory is based on fundamental non-equilibrium thermodynamic principles, whereas the theory of Barenblatt et al [9,10] is based on insight that initially larger pores will be filled with water before the water withdraws into the smaller pores. If we would have a linear saturation dependence of the capillary pressure, e.g.,…”

“…Reservoir interpretation that does not recognize the potential for reduced recovery because of non-equilibrium effects may lead to an overestimation of the recovery. The concept of the nonequilibrium effect was first introduced by Barenblatt and his colleagues [6,7,9,10]. It is equivalent to an alternative formulation introduced by Hassanizadeh and Gray [19,20] and Pavone [33].…”