Published in Petroleum Transactions, AIME, Volume 216, 1959, pages 290–296. Introduction The fundamental equations that are used to describe two-phase fluid flow in porous media are Darcy's law for each phase and an equation of continuity for each component. The special case of one-dimensional, incompressible, two-phase flow has received much attention in the petroleum engineering literature. The basic paper on the subject is that of Buckley and Leverett. Buckley and Leverett showed that one could eliminate the pressure and obtain a single partial differential equation for the saturation: They solved this equation for the case when gravity and capillary forces are negligible. A frequently encountered property of the solution of the basic partial differential equation is that, as time progresses, the saturation becomes a multiple-valued function of the distance coordinate, x. Buckley and Leverett interpreted the formation of multiple values as an indication that the saturation-distance curve had become discontinuous. They developed a method of determining the position of the discontinuity from a material balance. We solve the Buckley-Leverett partial differential equation using the method of characteristics and the concept of shocks. Our treatment is analogous to treatments used in the theory of supersonic compressible flow. Although the numerical results obtained are the same as those obtained by previous methods, we believe that this approach is more logical and has broader applicability. It can, for example, be generalized to treat the case when the fluids are compressible. It is not clear how one would go about doing this using the method of Buckley and Leverett. To illustrate the power of the method of characteristics with shocks, we solve a problem in gravity segregation which has not been treated previously in the literature. A special case of the general method developed here was used by Welge to solve a particular fluid displacement problem. Fundamental Equations We consider two-phase incompressible flow along the vertical direction x in a porous medium. We assume that there is no flow transverse to x. We measure x positive in the upward direction. The fundamental equations are obtained from Darcy's law and the mass conservation law.

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A theoretical model of the positive column of a rare gas glow discharge is used to calculate metastable atom densities in the pressure x radius range from 0.01 to 1.0 Torr cm. Two body quenching is shown to be an important loss process for metastables in neon and argon discharges above 0.1 Torr cm. Measurements of metastable atom flux from an anode slit are reported for the title gases and shown to represent the metastable density in the anode fall region of the discharges.

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