Foamy oil flow in porous media

Certain heavy oils which foam under severe depressurization give rise to increased recovery factor and an increased rate of production under solution gas drive. These oils not only stabilize foam, but also stabilize dispersion of gas bubbles at lower volume ratios. The way this phenomenon is related to the chemistry of the oil and its viscosity is presently not understood. We present here a mathematical model of reservoir flow of foamy oil which depends only on the velocity through Darcy's law, the pressure and the dispersed gas fraction. The theory governs only in situations in which the bubbles do not coalesce to produce the percolation of free gas. In this theory the bubbles move with the oil as they evolve. The main empirical content of the theory enters through the derivation of solubility isotherms which can be obtained from PVT data; modeling of nucleation, coalescence, bubble drag laws and transfer functions are avoided. The local pressure difference and dispersed gas fraction are in equilibrium on the solubility isotherm. In a pressure drawdown the time taken for the system to return to equilibrium is described by a rate law characterized by an empirical relaxation time (rate constant). The resulting systems of equations can be reduced to a coupled pair of nonlinear PDE's for the dispersed gas fraction and pressure difference, which can further be reduced in the equilibrium case to a second order evolution equation for the pressure difference. This system of equations can also be derived from usual theory of two-phase flow in a porous media based on relative permeability under the assumption that the bubbles and oil move in lock step. We propose a reformulation of the conventional theory in which the concept relative permeability of the porous media is replaced with the more familiar concept of an effective phase viscosity. The equations of our relaxation theory are solved numerically, and the mixture viscosity function and relaxation time are selected to match the sandpack experiments of Maini and Sarma [1994].