Technology, October-December, 1978, Montreal 41-c>l In the second paper, by Terwilliger et al."', saturation profiles were calculated which correlated closely ...
CONDITIONS UNDER WHICH THE CAPILLARY TERM MAY BE NEGLECTED R.G. BENTSEN
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JCPT 78-04-01 JCPT-25 ADVANCES IN RESERVOIR ENGINEERING Conditions under which the capillary term may be neglected R.G. BENTSEN Department of Mineral Engineering The University of Alberta Edmonton, Alberta ABSTRACT The effect the capillary term in the fractional flow equa- tion has on saturation profiles is important, because these profiles determine the ultimate economic oil recovery. Buckley-Leverett's original solution to the non-capil- lary, 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 .vaturation discontinuity or shock. This paper demonstrates that the Buckley-Leverett solu- tion iv, in reality, the steady-state solution to a second- order. non-linear, parabolic, partial differential equation. It also demonstrates that the steady-state (non-capillary, Buckley-Leverett) solution is an accepta ble 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 increases with decreasing mobility ratio for any given value of the capillary number. Introduction In 1942, Buckley and Leverett"' presented the first ap- proach 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 satura- tion. 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. They evaluated the strength and position of the shock from materialbalance considerations. The use of the Buckley-Leverett approach did not im- mediately become widespread because of the lack of a theoretical justification for the introduction of a satura- tion di scontinuity. Consequently, it was not until 1951 that two papers"," were published that made use of Buckley-Leverett theory. Holmgren and Morse"' used the Buckley-Leverett frontal-drive method, with simplifica- tions added by Pirson"', 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". Technology, October-December, 1978, Montreal 41-c>l In the second paper, by Terwilliger et al."', saturation profiles were calculated which correlated closely with those obtained experimentally by displacing water verti- cally 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. Conse- quentiv, the shape of this portion of the saturation dis- tri@uti-on 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 I;iying a tangent to the non-capillary fractional flow curve from the S. corresponding to the initial dis- placing fluid saturation and F. equal to zero. In 1952, Welge"' described a simplified method for obtaining the average saturation in an oil reservoir at water break@through. He showed that the tangent construction suggested by Terwilliger et al."' was equivalent to introducing a saturation discontinuity, as suggested by Buckley an(i Leverett"'. In addition, Weige derived an equation thit relates the average displacing fluid satura- tion to the saturation existing at the outlet end of the system. With the advent of high-speed computers, it became possible to undertake numerical studies of o il displace- ment including the effects of capillary pressure. In 19'58, Douglas et al."' presented a novel Eulerian technique which involved using a transformed saturation variable. As a consequence of using this approach, the authors were oblige(i to use a small mobile water saturation ahead of the front to move the foot of the fl@ front. Fayers and Sheldon"' suggest that, for slow displacement rates, "this can lead to a large and possibly erroneous 'pile-up' of water at the outflow boundary". FayeTs and Sheldon"' have also obtained solutions to the one-diniensional displacement equation using a La- grangian approach. Using this approach, the elapsed time necessary tc, obtain a particular saturation profile cannot be determined, because the distance travelled by the zero saturation i@; governed by a separate equation. Moreover, at slow injection rates, the authors' use of the boundary condition, S. = constant, at the input boundary in the Lagrangian formulation is incorre
