Stability of Displacement Fronts in Porous Media—Growth of Large Elliptical Fingers

Abstract

Abstract A theoretical analysis is presented for growth of a single, large elliptical finger in a porous medium. The objective is to obtain an understanding of how large fingers develop once a displacement front has become unstable. The results provide information beyond that available from linear analysis, which has been used in most previous work and applies only to infinitesimal perturbations. Separate analyses are given for finger growth during instability of (1) the front accompanying displacement of one immiscible fluid by another, (2) the combustion front in a dry, forward combustion process, (3) a condensation front, and (4) the combustion front in dry, reverse combustion as employed in the linking step of underground coal gasification. In Cases 1 and 2, the fastest-growing finger is found so that growth rate and finger size can be estimated. In contrast, the linear analyses for these cases predict no fastest-growing disturbance. Thus, important additional information is provided by the elliptical finger model. In Case 3, the condensation front, a fastest-growing disturbance is found as well. Results show that effects on stability of the mobility ratio and of the volume decrease accompanying condensation depend only on the shape of the finger (ratio of major and minor axes) while heat transfer effects also depend on the absolute size of the finger. Since the stabilizing effect of heat transfer is least for large fingers, such fingers are favored. For Case 4, we found that the channel in underground coal gasification always advances at the maximum speed permitted by the reaction kinetics. Actual results confirm the basic validity of those obtained by Gunn and Krantz using linear analysis, but the elliptical finger model provides a clearer understanding of the physical phenomena controlling finger size and speed.