[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.
[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
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![]() |
(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]
[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.