The Meaning of the Triple Value in Noncapillary Buckley-Leverett Theory

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Published in Petroleum Transactions, AIME, Volume 216, 1959, pages 271–276. Abstract Physically absurd, triple-valued saturations appear in the straight-forward solution of the Buckley-Leverett equations for the displacement of oil by water or gas. From an engineering viewpoint, the triple value causes no difficulty. It is well known how it may be compensated in order to obtain physically meaningful, numerical results. From a scientific viewpoint, the question still arises: What did the triple value mean? This paper explains how and why the triple value arose in noncapillary Buckley-Leverett theory. The discussion should serve as a background for the understanding and use of the modern method of characteristics in displacement theory. Introduction Displacement theory was introduced to petroleum technologists in 1941 by Buckley and Leverett. Buckley and Leverett called Eq. 1 "a material balance equation". Actually it is derived from both a continuity equation, or material balance equation, and Darcy's law. Eq. 2 may be readily interpreted as saying that a point of constant saturation (on a saturation-vs-distance curve) moves with a constant velocity that is proportional to the total volumetric rate, inversely proportional to the porosity, and is otherwise a function of the saturation itself. In Fig. 1, the abscissas represent distances along a column of porous medium, which column is assumed to be uniform in cross-section perpendicular to the abscissal direction. The ordinates represent fractions of pore space occupied by the displacing phase, which may be either water or gas. It is assumed that the saturation is uniform over all planes perpendicular to the abscissal direction. The saturation is a function only of time and one space variable.

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