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# 3 Saturation Overshoot

## 3.1 Experimental Observations

[2328.2.1] Infiltration experiments [59] on constant flux imbibition into a very dry porous medium report existence of non-monotone travelling wave profiles for the saturation [55, 62]

 (20)

as a function of time and position along the column. [2328.2.2] Here denotes the similarity variable

 (21)

and the parameter is the constant wave velocity. [2328.2.3] From here on all quantitites are dimensionless. [2328.2.4] The relation

 (22a)

defines the dimensional depth coordinate increasing along the orientation of gravity. [2328.2.5] The length is the system size (length of column). [2328.2.6] The dimensional time is

 (22b)

where denotes the total (i.e. wetting plus nonwetting) spatially constant flux through the column in m/s (see eq.  (13)). [2328.2.7] The dimensional similarity variable reads

 (22c)

[2328.2.8] Experimental observations show fluctuating profiles with an overshoot region [59, 73, 54, 55, 74, 75]. [2328.2.9] It can be viewed as a travelling wave profile consisting of an imbibition front followed by a drainage front.

## 3.2 Mathematical Formulation in d=1

[2328.3.1] The problem is to determine the height of the overshoot region (=tip) and its velocity given an initial profile , the outlet saturation , and the saturation at the inlet as data of the problem. [2328.3.2] Constant and constant are assumed for the boundary conditions at the left boundary. [2328.3.3] Note, that this differs from the experiment, where the flux of the wetting phase and the pressure of the nonwetting phase are controlled.

[2328.4.1] The leading (imbibition) front is a solution of the nondimensionalized fractional flow equation (obtained from eq.  (17) for )

 (23a)

[page 2329, §0]    while

 (23b)

must be fulfilled at the trailing (drainage) front. [2329.0.1] Here is the porosity and the variables and have been nondimensionalized using the system size and the total flux . The latter is assumed to be constant. [2329.0.2] The functions are the fractional flow functions for primary imbibition and secondary drainage. [2329.0.3] They are given as

 (24a) (24b)

with and the dimensionless gravity number [76, 37]

 (25)

defined in terms of total flux , wetting viscosity , density , acceleration of gravity and absolute permeability of the medium. [2329.0.4] The functions with are the relative permeabilities. [2329.0.5] The capillary flux functions for drainage and imbibition are defined as

 (26)

with and a minus sign was introduced to make them positive. [2329.0.6] The functions are capillary pressure saturation relations for drainage and imbibition. [2329.0.7] The dimensionless number

 (27)

is the macroscopic capillary number [76, 37] with representing a typical (mean) capillary pressure at (see eq.  (34)). [2329.0.8] For one has and eqs. (23) reduce to two nondimensionalized Buckley-Leverett equations

 (28a)

for the leading (imbibition) front and

 (28b)

for the trailing (drainage) front.

[page 2330, §1]

## 3.3 Hysteresis

[2330.1.1] Conventional hysteresis models for the traditional theory require to store the process history for each location inside the sample [77, 78, 79]. [2330.1.2] Usually this means to store the pressure and saturation history (i.e. the reversal points, where the process switches between drainage and imbibition). [2330.1.3] A simple jump-type hysteresis model can be formulated locally in time based on eq.  (23) as

 (29)

[2330.1.4] Here denotes the left sided limit

 (30)

is the Heaviside step function (see eq.  (35)), and the parameter functions with require a pair of capillary pressure and two pairs of relative permeability functions as input. [2330.1.5] The relative permeability functions employed for computations are of van Genuchten form [80, 81]

 (31a) (31b)

with and

 (32)

the effective saturation. [2330.1.6] The resulting fractional flow functions with parameters from Table 1 are shown in Figure 1b. [2330.1.7] The capillary pressure functions used in the computations are

 (33)

with and the typical pressure in (27) is defined as

 (34)

[2330.1.8] 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. [2330.1.9] 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.