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4 Simulation Setup

[681.1.9.1] In this section we discuss how the experiment is represented mathematically. [681.1.9.2] The four mass balance equations (1) are solved numerically. [681.1.9.3] First, equations (6) and (7) are used to eliminate S_{4} and P_{3}. [681.1.9.4] The primary variables are S_{1},S_{2},S_{3} and P_{1}. [681.1.9.5] The mass balances are discretized in space by cell centered finite volumes with upwind fluxes. [681.1.9.6] They are discretized in time with a first order implicit fully coupled scheme. [681.1.9.7] The corresponding system of nonlinear equations is solved with the Newton-Raphson method. [681.1.9.8] The whole scheme is implemented in Matlab. [681.1.9.9] The simulation is run with a resolution of one cell per centimeter, i.e. with N=72 collocation points. [681.1.9.10] Details of the algorithm are given elsewhere [3].

Table 1: List of parameters with units and their numerical values used for simulating the experiment. Note that \epsilon _{{\rm M}} is a mathematical regularization parameter, i.e. the limit \epsilon _{{\rm M}}\to 0 is implicit and it has been tested that the numberical results do not change in this limit.
Parameters Units Values
\phi 0.36
\epsilon _{{\rm M}} 0.01
\mathbb{W} \mathbb{O} \mathbb{W} \mathbb{O}
\varrho _{{\mathbb{W}}} \varrho _{{\mathbb{O}}} \textrm{kg}\,\textrm{m}^{{-3}} 1000 1.2
S_{{\mathbb{W}\,\rm dr}} S_{{\mathbb{O}\,\rm im}} 0.13 0.21
{\eta _{{2}}} {\eta _{{4}}} 6 4
\alpha \beta 0.42 1.6
P_{{\rm a}} P_{{\rm b}} Pa 2700 3
\gamma \delta 2.4 2.9
P^{*}_{2} P^{*}_{4} Pa 11000 3000
R_{{11}} R_{{33}} \textrm{kg}\,\textrm{m}^{{-3}}\,\sec^{{-1}} 3.83\times 10^{{6}} 6.99\times 10^{{4}}
R_{{22}} R_{{44}} \textrm{kg}\,\textrm{m}^{{-3}}\,\sec^{{-1}} 10^{{16}} 10^{{16}}
R_{{ij}}, i\not=j \textrm{kg}\,\textrm{m}^{{-3}}\,\sec^{{-1}} 0

[681.1.9.11] Dirichlet boundary conditions for the pressure P_{1} of the percolating water phase are imposed at the lower boundary (x=0\,\textrm{m}), where pressure is determined by the water reservoir. [681.1.9.12] Dirichlet boundary conditions for the atmospheric pressure P_{3} of the percolating air phase are chosen at the upper boundary (x=0.72\,\textrm{m}) of the column. [681.1.9.13] All the other boundaries are impermeable so that the flux across them must vanish.

[681.1.9.14] The initial conditions are S_{1}(x,0)=0.997, S_{2}(x,0)=0.001, S_{3}(x,0)=0.001, S_{4}(x,0)=0.001 for all x\in[0\,\textrm{cm},72,\textrm{cm}] [681.1.9.15] Initial conditions for the pressures are not required because of the implicit formulation. [681.1.9.16] Before the protocol for the pressure is started, the system is given one day under hydrostatic water pressure conditions to equilibrate.

[681.1.10.1] The parameters for the simulation are given in Table 1. [681.1.10.2] They were obtained by fitting the primary drainage curve of the capillary pressure saturation relationship obtained in the residual decoupling approximation [9] to the primary drainage curve of van Genuchten parametrization that [15] obtained by a fit to data of the first drainage process in the experiment. [681.1.10.3] The van Genuchten parameters in [15] are \alpha^{{{\mathrm{dr}}}}=4.28\times 10^{{-4}}\,\textrm{Pa}^{{-1}}, \alpha^{{{\mathrm{im}}}}=8.56\times 10^{{-4}}\,\textrm{Pa}^{{-1}}, n^{{{\mathrm{dr}}}}=5.52, n^{{{\mathrm{im}}}}=5.52, m^{{{\mathrm{dr}}}}=0.82, m^{{{\mathrm{im}}}}=0.82, S_{{wi}}=0.17, S_{{nr}}=0.25. [681.1.10.4] The resulting capillary pressure curves are compared in Figure 2. [681.1.10.5] The viscous resistance coefficients were obtained through R_{{11}}\approx\mu _{{\mathbb{W}}}/k, R_{{33}}\approx\mu _{{\mathbb{O}}}/k, where k=33.7^{{-12}}\textrm{m}^{2} was again taken from [15]. [681.1.10.6] The viscous resistance coefficients for the non-percolating phases are assumed to be much larger than those for the percolating phases R_{{22}},R_{{44}} \gg R_{{11}},R_{{33}}. [681.1.10.7] For the time-scale of the experiment the results do not depend on the numerical values of the resistance coefficients given in Table 1 [3].

Figure 2: Illustration of the capillary pressure-saturation relationship in the residual decoupling approximation of our theory for the parameters given in Table 1 (solid lines) for primary and secondary drainage and imbibition respectively. Also shown are the primary and secondary drainage and the secondary imbibition curve of the hysteresis model used in [15] (dashed curves).