Conditions Under Which The Capillary Term May Be Neglected

Abstract

Abstract The effect the capillary term in the fractional flow equation has on saturation profits is important, because these profiles determine the ultimate economic oil recovery. Buckley-Levereu's original solution to the non-capillary, two-phase flow problem became multiple-valued in saturation. As it is physically unrealistic for the saturation to have more than one value at a given position, Buckley and Leverett resolved this difficulty by introducing a saturation discontinuity or shock. This paper demonstrates that the Buckley-Leverett solution is, in reality, the steady-state solution to a secondorder, non-linear, parabolic, partial differential equation. It also demonstrates that the steady-state (non-capillary, Buckley-Leverett) solution is an acceptable approximation to the transient solution, provided the capillary number is sufficiently small. Finally, it is shown that the discrepancy between the transient and steady-state solutions increaseswith decreasing mobility ratio for any given value of the capillary number. Introduction In 1942, Buckley and Leverett'" presented the first approach to predicting the linear displacement of one fluid by another. Their original solution to the non-capillary, two-phase flow problem became multiple-valued in saturation.As it is physically unrealistic for the saturation to have more than one value at a given position, Buckley and Leverett resolved this difficulty by introducing asaturation discontinuity or shock. They evaluated the strength and position of the shock from material-balance considerations. The Use of the Buckley-Leverett approach did not immediately become widespread because of the lack of a theoretical justification for the introduction of a saturation discontinuity. Consequently, it was not until 1951 that two papers(2,3) were published that made use of Buckley~Leverett theory. Holmgren and Morse(2) used the BUCkley-Leverett frontal-drive method, with simplifications added by Pirson(4), to compute the average water saturation at water breakthrough. The computed average water saturation at breakthrough was found to be slightly larger than the experimentally measured value. The authors attributed this difference to "dispersion of the flood front caused by capillary forces neglected in the simplified calculation". In the second paper, by Terwilliger et al.(3), saturation profiles were calculated which correlated- closely with those obtained experimentally by displacing water vertically downward with gas. These authors found that, for some rates of flow, all points of saturation in the lower range moved down the column at the same rate. Consequently, the shape of this portion of the saturation distribution curve became constant with time. They called this portion of the saturation distribution the "stabilized" zone. In addition, these authors demonstrated that the saturation at the upper end of the stabilized zone could be defined by laying a tangent to the non-capillary fractional flow curve from the Sw corresponding to the initial displacing fluid saturation and Fw equal to zero. In 1952, Welge(5) described a simplified method for obtaining the average saturation in an oil reservoir at water breakthrough.. He showed that the tangent construction suggested by Terwilliger et a1.(3) was equivalent to introducing a saturation discontinuity, as suggested by Buckley and Leverett(1).