The Mathematical Model of Non-Equilibrium Effects in Water-Oil Displacement

Abstract

TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractForced oil-water displacement and spontaneous countercurrent imbibition are crucial mechanisms of secondary oil recovery. The classical mathematical models of these phenomena are based on the fundamental assumption that in both these unsteady flows a local phase equilibrium is reached in the vicinity of every point. Thus, the water and oil flows are locally redistributed over their flow paths similarly to steady flows. This assumption allowed the investigators 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 predictions of water-oil displacement with numerical simulations.However, at the water front in the water-oil displacement, as well as in capillary imbibition, the characteristic times of both processes are 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 experiments are presented and discussed.

Probability density function approach for modelling multi-phase flow with ganglia in porous media

“…It is interesting to note that (B 4) is similar to many other relaxation models, e.g. the ones proposed by Barenblatt et al (2003), Silin & Patzek (2004) and Schembre & Kovscek (2006), to describe non-equilibrium effects in relative permeability and macroscopic capillary pressure. However, here the macroscopic model parameters are rigorously derived from the microscopic physics by adopting a general statistical approach.…”

“…223 the previously proposed models (see Hassanizadeh, Celia & Dahle 2002;Barenblatt, Patzek & Silin 2003;Hilfer 2006b;Tyagi et al 2008, for example).…”

“…They argued that the time interval between successive appearances of traps can be significant; consequently, the average trapping process at macro-scale is governed by a kinetic equation. In appendix B, it is shown that for small τ ζ the above two-equation model reduces to the non-equilibrium model proposed by Barenblatt et al (2003), Silin & Patzek (2004) and Schembre & Kovscek (2006) and the well-known relative permeability-saturation hysteresis curves can naturally be obtained in this special case.…”

On the physics of unstable infiltration, seepage, and gravity drainage in partially saturated tuffs

“…The physical meaning of the infiltration threshold for gravity drainage is the same as that for the infiltration threshold for seepage to occur, and both values are approximately the same, 5 ml/min. The dependence of gravity drainage on the infiltration rate implies the need to simulate flow in fractured media using flow-rate dependent water-retention curves and models simulating non-equilibrium effects in the formation (Barenblatt et al, 2002).…”

Heavy and Thermal Oil Recovery Production Mechanisms

“…This is another possible cause for counter-current imbibition results to deviate from square-root-of-time behavior (cf. Barenblatt et al, 2002). Hence, before applying Eq.…”