Physical Theory for Capillary Flow Phenomena

From the assumption that the microscopic behavior of the liquid in an unsaturated porous medium is controlled by the physical laws of surface tension and viscous flow, differential equations governing the macroscopic flow in such a medium are deduced. No special pore-shape assumptions are required, but one topological approximation is needed; i.e., that neither isolated drops nor isolated bubbles occur. Several nonessential simplifying assumptions are used; i.e., that the macroscopic properties of the medium, the character of the liquid, and the pressure of the gas are independent of position, time, and direction. The macroscopic equations are obtained in a fully reduced form, permitting comparison between two media— or between two flow systems—that differ only by scaling factors. A novel feature of this calculation is its prediction that the liquid-transmission and liquid-capacity properties of an unsaturated medium will exhibit hysteresis in their dependences upon the liquid-gas pressure differential, p. The properties of the medium depend upon the pressure history but are invariant to monotonic time-scale distortions of that history. Such time-invariant functionals have been termed by the authors ``hysteresis functions,'' symbolized by the subscript, H, e.g. FH(p). Although methods for measuring and describing the characteristics of specific ``hysteresis functions'' have not yet been developed, the general validity of this analysis can be studied experimentally by testing predictions that are contained in the reduced variables.