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 S4 and P3. [681.1.9.4] The primary variables are S1,S2,S3 and P1. [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].

[681.1.9.11] Dirichlet boundary conditions for the pressure P1 of the percolating water phase are imposed at the lower boundary (x=0⁢m), where pressure is determined by the water reservoir. [681.1.9.12] Dirichlet boundary conditions for the atmospheric pressure P3 of the percolating air phase are chosen at the upper boundary (x=0.72⁢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 S1⁢x,0=0.997, S2⁢x,0=0.001, S3⁢x,0=0.001, S4⁢x,0=0.001 for all x∈0⁢cm,72,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 αdr=4.28×10-4⁢Pa-1, αim=8.56×10-4⁢Pa-1, ndr=5.52, nim=5.52, mdr=0.82, mim=0.82, Sw⁢i=0.17, Sn⁢r=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 R11≈μW/k, R33≈μO/k, where k=33.7-12⁢m2 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 R22,R44 ≫R11,R33. [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].