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1 Introduction

[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[43]. [507.1.2] Despite the fact that Wyckoff and Botset (1936) strongly emphasized the viriation of hydraulically disconnected fluid regions [30], 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 [43], 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” [43]. [508.2.5] Rather surprisingly, the authors of [43] 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 [23].

[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 [26]. [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.