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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]

 S⁢z,t=s⁢y (20)

as a function of time t0 and position 0z1 along the column. [2328.2.2] Here -<y< denotes the similarity variable

 y=z-c*⁢t (21)

and the parameter -<c*< is the constant wave velocity. [2328.2.3] From here on all quantitites z,t,y,S,c* are dimensionless. [2328.2.4] The relation

 z^=z⁢L (22a)

defines the dimensional depth coordinate 0z^L increasing along the orientation of gravity. [2328.2.5] The length L is the system size (length of column). [2328.2.6] The dimensional time is

 t^=t⁢LQ (22b)

where Q 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

 y^=y⁢L=z^-c*⁢Q⁢t^. (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 S* of the overshoot region (=tip) and its velocity c* given an initial profile Sz,0, the outlet saturation Sout, and the saturation Sin at the inlet as data of the problem. [2328.3.2] Constant Q and constant Sin 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 d=1)

 ϕ⁢∂⁡S∂⁡t+∂∂⁡z⁢fim⁢S-Dim⁢S⁢∂⁡S∂⁡z=0 (23)

[page 2329, §0]    while

 ϕ⁢∂⁡S∂⁡t+∂∂⁡z⁢fdr⁢S-Ddr⁢S⁢∂⁡S∂⁡z=0 (23)

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

 fi⁢S =μOμW⁢kWri⁢SkOri⁢S+kWri⁢SGrW⁢1-ϱOϱW1+μOμW⁢kWri⁢SkOri⁢S (24a) =1+kOri⁢SGrW⁢μWμO⁢1-ϱOϱW1+μWμO⁢kOri⁢SkWri⁢S (24b)

with iim,dr and the dimensionless gravity number [76, 37]

 GrW=μW⁢QϱW⁢g⁢k (25)

defined in terms of total flux Q, wetting viscosity μW, density ϱW, acceleration of gravity g and absolute permeability k of the medium. [2329.0.4] The functions kWriS,kOriS with iim,dr are the relative permeabilities. [2329.0.5] The capillary flux functions for drainage and imbibition are defined as

 Di⁢S=-kWri⁢SCaW⁢d⁢Pci⁢Sd⁢S1+μOμW⁢kWri⁢SkOri⁢S (26)

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

 CaW=μW⁢Q⁢LPb⁢k (27)

is the macroscopic capillary number [76, 37] with Pb representing a typical (mean) capillary pressure at S=0.5 (see eq.  (34)). [2329.0.8] For Ca= one has Di=0 and eqs. (23) reduce to two nondimensionalized Buckley-Leverett equations

 ϕ⁢∂⁡S∂⁡t+∂∂⁡z⁢fim⁢S=0 (28)

for the leading (imbibition) front and

 ϕ⁢∂⁡S∂⁡t+∂∂⁡z⁢fdr⁢S=0 (28)

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

 ϕ⁢∂⁡S∂⁡t+Ξ⁢S⁢∂∂⁡z⁢fim⁢S-Dim⁢S⁢∂⁡S∂⁡z+1-Ξ⁢S⁢∂∂⁡z⁢fdr⁢S-Ddr⁢S⁢∂⁡S∂⁡z=0. (29)

[2330.1.4] Here ΞS denotes the left sided limit

 Ξ⁢S=limε→0⁡Θ⁢∂⁡S∂⁡t⁢z,t-ε, (30)

Θx is the Heaviside step function (see eq.  (35)), and the parameter functions fiS,DiS with idr,im 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]

 kWri⁢Sei =KWei⁢Sei1/2⁢1-1-Sei1/αiαi2 (31a) kOri⁢Sei =KOei⁢1-Sei1/2⁢1-Sei1/βi2⁢βi (31b)

with iim,dr and

 Sei⁢S=S-SW⁢ii1-SO⁢ri-SW⁢ii (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

 Pci⁢Sei=Pbi⁢Sei-1/αi-11-αi (33)

with i=dr,im and the typical pressure Pb in (27) is defined as

 Pb=Pcim⁢0.5+Pcdr⁢0.52. (34)

[2330.1.8] Equation (29) with initial and boundary conditions for S and an initial condition for S/t is conjectured to be a well defined semigroup of bounded operators on L10,L on a finite interval 0,T 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 Ξ or 1-Ξ are projection operators.