[page 507, §1]
[507.1.1] Almost all accepted and applied theories of multiphase flow in porous media are based on generalized Darcy laws and the concurrent concept of relative permeabilities. [507.1.2] Despite the fact that Wyckoff and Botset (1936) strongly emphasized the viriation of hydraulically disconnected fluid regions , almost all subsequent applications of the relative permeability concept treat the residual nonwetting (or irreducible wetting) saturations as material constants [5, 11, 16, 20, 42, 39, 32].
[page 508, §1] [508.2.1] Modern theories of multiphase flow in porous media often resort to microscopic models (e.g., network models) [6, 8, 9, 13, 14, 17, 18, 21, 22, 31, 38]. [508.2.2] An important motivation is the need to derive or estimate macroscopic relative permeabilities from pore scale parameters. [508.2.3] It was emphasized in , however, that the “possibility of determining the overall dynamical behavior of nonhomogeneous fluids from a study of microscopic detail” is remote [43, p. 326]. [508.2.4] One should instead consider saturation, velocities, and pressure gradient “to derive therefrom the overall or macroscopic behavior of the system” . [508.2.5] Rather surprisingly, the authors of  emphasize the important difference between “continouos moving” fluids and “stationary or locked” fluids in their introduction, but later cease to distinguish between them in the main body of the paper. [508.2.6] Experimentally, the volume fraction of stationary, locked, trapped, or nonpercolating fluid phases varies strongly with time and position [1, 3, 41, 43]. [508.2.7] Modeling such variations of trapped or nonpercolating fluid phases explicitly is the main objctive of this paper.
[508.3.1] Dispersed droplets, bubbles, or ganglia of one fluid phase obstruct the motion of the other fluid phase. [508.3.2] Extensive experimental and theoretical studies of the simple phenomenon exist [4, 19, 30, 33, 34, 35, 36, 37]. [508.3.3] It is, therefore, surprising that the concept of hydraulic percolation has been neglected in the modeling of two phase flow until 10 years ago .
[508.4.1] Given that the basic concept of hydraulic percolation for macroscpic capillarity has been discussed extensively in [26, 25, 24] our objective in this paper is to find approximate numerical solutions of the mathematical model. [508.4.2] Let us, therefore begin the discussion by formulating a set of mathematical equations for the hydraulic percolation approach in Sect. 2. [508.4.3] One also needs to specify initial and boundary conditions representing a realistic experiment. [508.4.4] Raising a closed column from a horizontal to a vertical orientation causes simultaneous imbibition and drainage processes inside the medium as emphasized already in . [508.4.5] In this paper, we report approximate numerical results for the full time evolution of such simultaneous imbibition and drainage processes. [508.4.6] As expected, the resulting equilibrium saturations depend strongly on the initial conditions. [508.4.7] Moreover, they differ significantly from the equilibrium profiles of the traditional theory.