Frontal Structure and Stability in Immiscible Displacement

Abstract

Abstract A theory of the stability or instability of plane saturation fronts in immiscible displacements is presented which is more realistic than earlier theories. For given two-phase relative permeability functions, capillary pressure function, viscosity ratio, saturation ahead of the front, and allowable front-propagation speed, the displacement-front saturation profile of permanent form is calculated from Rizhik et al.'s (1961) solution by quadrature of the nonlinear convective diffusion equations for two-phase flow with significant capillary effect. Stability of the saturation profile of permanent form with respect to small disturbances in pressure and saturation (as may be caused by minor heterogeneities in a reservoir) is found by solving the generalized eigenproblem to which linear stability theory and finite differencing lead. The results show that when the front is unstable there is a fastest growing wavelength of disturbance transverse to the flow. Shorter wavelengths are slowed or stabilized by capillary-pressure-gradient-driven flow; longer wavelengths grow more slowly because the transverse pressure gradients driving them are weaker. If the displacing fluid is viscous enough the front is stable, but the larger the scale of disturbance the more nearly it is marginally stable. An earlier theory that approximates the front as a discontinuity agrees roughly but differs in important details, among them the mildly stabilizing effect associated with the saturation profile in the frontal zone.