[page 681, §1]

[681.1.1.1] A quantitative prediction of fluid saturation profiles during
immiscible displacement remains a fundamental open problem in the
physics of porous media.
[681.1.1.2] Despite its well known limitations regarding
hysteresis and trapping phenomena, the traditional approach [2, 23, 20, 26]
has remained the most popular mathematical model for more than 70 years in
applications such as reservoir engineering [13] or
groundwater hydrology [1].

[681.1.2.1] Many authors have emphasized the importance of hysteresis between drainage and imbibition [25, 7, 21, 18]. [681.1.2.2] Ad hoc extensions of the existing two phase flow model [21, 24, 19, 14] yield reasonable agreement with laboratory measurements. [681.1.2.3] However, the physical foundations for such hysteresis models are not clear.

[681.1.3.1] A recent macroscopic theory of capillarity in porous media [8] proposes to take into account the different hydrodynamic properties of percolating (=connected) and non-percolating (=not connected) fluid parts. [681.1.3.2] This provides the physical basis for hysteresis in our approach. [681.1.3.3] In the residual decoupling limit, the traditional constitutive relations between relative permeability and saturation as well as between capillary pressure and saturation are recovered including hysteresis [8, 9]. [681.1.3.4] Approximate analytical results for a quasi-static displacement have been calculated in [10]. In [11] and [5], numerical solutions were calculated for experiments with a closed homogeneous column in gravity. [681.1.3.5] Further, analytic and quasi analytic solutions for the theory were developed in [4, 6, 12]. [681.1.3.6] However, an explicit comparison with experimental data is pending.

[681.1.4.1] The objective of this paper is to close this gap. [681.1.4.2] We show that the theory is able to model a laboratory experiment [15] with a porous column that shows hysteretic behavior.

[681.1.5.1] The manuscript is structured as followed. [681.1.5.2] First, we illustrate the experimental setup [15] that has been used to illustrate hysteretic phenomena in fluid distributions in porous media flow. [681.1.5.3] Section 3 briefly presents the theory of percolating and non-percolating phases and its mathmatical formulation for incompressible immiscible two-phase flow. [681.1.5.4] In Section 4 we discuss the numerical representation of the experiment and its implementation. The results are presented and discussed in Section 5. [681.1.5.5] The paper closes with concluding remarks.