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Foamy oil flow in porous media II: Nonlinear Relaxation Time Model of Nucleation

D.D. Joseph, A. Kamp & R. Bai

Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455 (USA)



In a previous communication, here called Part I, we presented a model of the flow of foamy oil in porous media in situations in which the bubbles do not coalesce to produce the percolation of free gas so that the gas moves with the oil as it evolves. A central role in that theory is an equation of state, called the solubility isotherm, which describes an equilibrium between the fraction of dispersed gas and the pressure below the bubble point pressure. A rate equation governing the return to equilibrium was postulated and it requires a value for the relaxation constant multiplying . The theory developed in Part I was applied to experimental data and good agreements were achieved except for sharp transients at early times such as those that occur for sudden drops of pressure at the open end of the closed sand pack. In the present theory we introduce two rates and two relaxation times to describe the dynamics of relaxation of the system to an equilibrium state; one rate for and the other for the pressure as was suggested already in Part I. However, we found that constant relaxation times could not be found to fit all the available data. We interpret this in terms of bubbles nucleating more slowly at initial drawdowns, and more rapidly as gas and vapor is released when the pressure is held below the bubble point. This feature has been more or less successfully addressed by the introduction of two relaxation functions of the gas fraction which allows us to describe the low rates of evolution of when is zero or close to it. The relaxation functions were fit to one rate of depletion in a depletion experiment in which oil is pulled out of a closed sand pack at a constant rate. With this selection of the relaxation function established, the set of governing PDEs is fixed and may be used to predict the results of other experiments. The prediction of the pressure profiles for other greatly different rates of depletion is satisfactory. Moreover, the experimental results processed in Part I, are improved by the new theory with the same fitting.