[page 2333, §1]

[2333.1.1] The preceding theoretical considerations are confirmed by numerical solutions of eq. (23) with a simple jump-type hysteresis, as formulated in eq. (29). [2333.1.2] The initial conditions are monotone and of the form

(41) |

where is given in Table 1. [2333.1.3] Note that all initial conditions are monotone and do not have an overshoot. [2333.1.4] Two cases are, labelled A and B, will be illustrated in the figures, namely

(A) | |||

(B) |

[2333.1.5] Case A is chosen to illustrate travelling wave solutions with . [2333.1.6] The second initial condition illustrates a general overshoot solution with two wave speeds . [2333.1.7] Moreover it illustrates one possibility for the limit and in which a very thin saturated layer (that may arise from sprinkling water on top of the column) initiates an overshoot profile. [2333.1.8] This could provide a partial answer to the question of how saturation overshoot profiles can be initiated.

[2333.2.1] The numerical solution of eq. (29) was performed using the open source toolkit for computational fluid mechanics OpenFOAM [82]. [2333.2.2] This requires to develop an appropriate solver routine. The solver routine employed here was derived from the solver [page 2334, §0] scalarTransportFoam of the toolkit. [2334.0.1] Two different discretizations of eq. (29) have been implemented. One is based on a direct discretization of using the fvc::div-operators, the other on using fvc::grad-operators. [2334.0.2] For the discretization of the time derivatives an implicit Euler scheme was chosen, for the discretization of an explicit least square scheme, and for the discretization of the second order term an implicit Gauss linear corrected scheme has been selected. [2334.0.3] The second order term had to be regularized by replacing the function with . [2334.0.4] This avoids oscillations at the imbibition front. [2334.0.5] The discretized system was solved with an incomplete Cholesky conjugate gradient solver in OpenFOAM . [2334.0.6] The numerical solutions were found to differ only in the numerical diffusion of the algorithms. [2334.0.7] The divergence formulation seems to be numerically more stable and accurate.

[2334.1.1] Equation (38) can have one solution, several solutions or no solution. [2334.1.2] This can be seen numerically, but also analytically. [2334.1.3] Depending on the values of the parameters the velocity of the imbibition front may be larger or smaller than that of the drainage front. [2334.1.4] With the parameters from Table 1 the overshoot saturation for a travelling wave as computed from (38) is found to be and its velocity is .

[2334.2.1] The travelling wave solution expected from the graphical construction in Figure 1a) and b) for the initial condition A with and width is displayed in Figure 1c) at the initial time and after dimensionless time corresponding to s. [2334.2.2] It confirms a travelling wave of the form (36) whose velocity agrees perfectly with that predicted from eq. (38). [2334.2.3] We have also checked that the solution does not change with grid refinement.

[2334.3.1] For different initial conditions the numerical solutions also agree with the theoretical considerations. [2334.3.2] Saturation overshoot is found also when initially a thin saturated layer with steplike or linear profile is present. [2334.3.3] These solutions, however, do not form travelling waves. [2334.3.4] Instead they have for their imbibition and drainage front as predicted analytically. [2334.3.5] For the overshoot region grows linearly with time at the rate . [2334.3.6] Such an overshoot solution is generated by the second initial condition B and compared with the travelling wave solution in Figure 2. [2334.3.7] The non-travelling overshoot for initial condition B with and is shown as the solid profiles at times corresponding to s. [2334.3.8] The profile quickly approaches an overshoot solution with plateau value close to the saturation of the Welge tangent construction. [2334.3.9] This is higher than the plateau value . [2334.3.10] The difference will be difficult to observe experimentally because of the large uncertainties in the measurements [54, 55]. [2334.3.11] The velocities of the numerical imbibition and drainage fronts in the case of initial conditions B with and (solid curves in Fig. 2) again agree perfectly with and from the theoretical analysis.

[2334.3.12] Equation (29) with initial and boundary conditions for and an initial condition for is conjectured to be a well defined semigroup of bounded operators on on a finite interval of time. [2334.3.13] The conjecture is supported by the fact that each of the equations (23) individually defines such a semigroup, and because multiplication by or are projection operators.

[2334.3.14] Equation (29) with initial and boundary conditions for and an initial condition for is conjectured to be a well defined semigroup of bounded operators on on a finite interval of time. [2334.3.15] The conjecture is supported by the fact that each of the equations (23) individually defines such a semigroup, and because multiplication by or are projection operators.

[2334.4.1] Other values of the initial saturation have also been investigated. [2334.4.2] A rich phenomenology of possibilities indicates that the existence or nonexistence of overshoot solutions depends very sensitively on the parameters of the problem and the initial conditions.

[2334.5.1] Note also the asymmetric shape of the overshoot solutions (travelling or not). [2334.5.2] This asymmetry resembles the asymmetry seen in experiment [54, 55]. [2334.5.3] While the leading imbibition front is steep, the trailing drainage front is more smeared out. [2334.5.4] This results from the capillary flux term whose values at are smaller than at .