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3 Initial Conditions, Boundary Conditions and Model Parameters

[513.4.1] Consider a cylindrical column containing a homogeneous, isotropic and incompressible porous medium. [513.4.2] The column is closed at both ends and filled with two immiscible fluids. [513.4.3] Assume that the surface and interfacial tensions are such that the capillary fringe is much thicker than the column diameter so that a onedimensional description is appropriate.

[513.5.1] The experimental situation considered here is that of raising a closed column as described in [26, p. 223]. [513.5.2] It is illustrated in Figure 1. [513.5.3] Initially, at instant , both fluids are at rest. [513.5.4] The column is oriented horizontally (), i.e., perpendicular to the direction of gravity. [513.5.5] The displacement processes are initiated by rotating the column into a vertical position. [513.5.6] The time protocol for rotating the column may generally be written as

 (14)

where is the instant of most rapid rotation and is the inverse rate of the rotation.

[513.6.1] The constitutive equations yield the following system of coupled nonlinear partial differential equations

 (15a) (15b) (15c) (15d) (15e) (15f) (15g) (15h) (15i) (15j)

[page 514, §0]    where , and the quantities , are defined in eq. (7). [514.0.1] This system of equations is reduced to a system of only equations by inserting eq. (15j) into eqs. (15) and (15) to eliminate . [514.0.2] The remaining unknowns are and .

[514.1.1] The system (15) has to be solved subject to initial and boundary data. [514.1.2] No flow boundary conditions at both ends require

 (16a) (16b)

[514.1.3] The fluids are incompressible. [514.1.4] Hence the reference pressure can be fixed to zero at the left boundary

 (17)

[514.1.5] The saturations remain free at the boundaries of the column.

[514.2.1] Initially the fluids are at rest and their velocities vanish. [514.2.2] The initial conditions are ()

 (18a) (18b) (18c)

[514.2.3] In the present study the initial saturations will be taken as constants, i.e. independent of .

[514.3.1] The model parameters are chosen largely identical to the parameters in [26]. [514.3.2] They describe experimental data obtained at the Versuchseinrichtung zur Grundwasser- und Altlastensanierung (VEGAS) at the Universität Stuttgart [40]. [514.3.3] The model parameters are , , , , , , , , , , , , , and . [514.3.4] In [26] only stationary and quasistationary solutions were considered, and the viscous resistance coefficients remained unspecified. [514.3.5] In order to [page 515, §0]    find realistic values remember that [25, 26]. [515.0.1] Realistic values for the viscosity and permeability are kgms and m. [515.0.2] Based on these orders of magnitude the viscous resistance coefficients are specified as kg ms, and kg ms.

[515.1.1] The column is filled with water having total saturation and oil with saturation . [515.1.2] Two different initial conditions will be investigated that differ in relative abundance of the nonpercolating phase. [515.1.3] The saturations for initial condition A and B are in obvious notation given as

 (19a) (19b) (19c) (19d)

[515.2.1] These phase distributions can be prepared experimentally by an imbibition process for A or by a drainage process for inital condition B. [515.2.2] The values for the nonpercolating saturations were chosen from the nonpercolating saturations predicted within the residual decoupling approximation. [515.2.3] They can be read off from Figure 5 in [26]. [515.2.4] The values for the percolating phases follow from the requirement that the total water saturation is . [515.2.5] The time scales for raising the column are chosen as corresponding to roughly 3 hours.