2 THEORETICAL CONSIDERATIONS
In this section, we first discuss the model used to solve the forward problem, i.e. the governing equations describing
unsaturated flow. Then we formulate the inverse problem, and finally introduce a number of criteria suitable for eval-
uating experiments designed for parameter estimation purposes.
2.1 Formulation of the forward problem
The general-purpose numerical simulator Tough2
33
is used to solve the forward problem. While Tough2 is able to
handle nonisothermal flow of multiple components in up to three phases, we discuss here only the terms involved
in unsaturated flow of water according to the Richards equation. The integral finite difference method employed
in Tough2 solves the following mass balance equation for an arbitrary subdomain V
n
bounded by the surface G
n
and an
inward normal vector n, where t is time and q is a local sink or source term:
d dt
Z
V
n
M dV ¼
Z
G
n
F ·n
dG þ Z
V
n
q dV
1 The accumulation term M represents mass per unit volume
M ¼ fr 2
where f is porosity and r is water density. The mass flux term is given by Darcy’s law
F ¼ ¹ k
abs
k
r
r m
=p ¹ rg 3
Here, k
abs
m
2
is the absolute permeability, k
r
is the rela- tive permeability, a dimensionless number between zero
and one as a function of saturation S, m Pa·s is the dynamic viscosity, p Pa is the soil water pressure, and g
m
2
s
¹ 1
is the gravitational acceleration vector. This form of the governing equation makes it explicit that the com-
monly used hydraulic conductivity m s
¹ 1
is not only a function of the porous material, but also of fluid density
and viscosity which are slightly affected by pressure and temperature changes. Estimating the absolute permeability
instead of hydraulic conductivity ensures that these depen- dencies are properly taken into account. While the effect is
minor for water flow under ambient conditions, it may become significant when inverting gas flow data for an
example, see Finsterle and Persoff
16
, or when performing predictions under strongly nonisothermal conditions such
as encountered during remediation of contaminated sites by means of steam flooding.
A crucial part of the conceptual model is the choice of the characteristic curves, expressing capillary pressure p
c
and relative permeability as a function of S. Many consistent
parametric models have been proposed in the literature for a comprehensive review, see Durner
8,9
. We restrict our discussion to the most widely used models. Brooks
and Corey BC introduced the following capillary pressure function:
2
p
c
¼ ¹ p
e
·S
¹
1 l
e
4 Here, p
e
and l are fitting parameters sometimes referred to as
air entry
pressure and
pore size
distribution index, respectively. The effective liquid saturation S
e
is defined as
S
e
¼ S
l
¹ S
lr
1 ¹ S
lr
5 where S
lr
is the residual liquid saturation. Introducing eqn 4 in the model suggested by Burdine
3
yields the following expression for relative permeability:
k
r
¼ S
2 þ 3l l
e
6 Applying Mualem’s model yields
39
k
r
¼ S
t þ 2 þ 2=l
e
7 where t is an additional fitting parameter which accounts
for pore connectivity and tortuosity effects. Note that with t¼
1, eqn 7 is identical to eqn 6. We will refer to eqns 5 and 6 as the BCB model, and eqn 5 in combination
with eqn 7 as the BCM model. Alternative expressions are given by van Genuchten VG:
49
p
c
¼ 1
a S
¹ 1=m
e
¹ 1
1=n
8 With m ¼ 1 ¹ 2n and applying Burdine’s model, the
relative permeability becomes k
r
¼ S
2 e
1 ¹ 1 ¹ S
1=m e
m
9 Assuming m ¼ 1 ¹ 1n and introducing eqn 8 into
Mualem’s model yields k
r
¼ S
t e
1 ¹ 1 ¹ S
1=m e
m 2
10 We will refer to eqns 8 and 9 as the VGB model, and
eqn 8 in combination with eqn 10 as the VGM model. For convenience and based on a weak analogy to the BC
model described by Morel–Seytoux et al.,
28
we will refer to parameter n as the pore size distribution index PSDI, 1a
as the air entry pressure AEP, and t as the tortuosity factor.
While the two models, BC and VG, exhibit only minor differences in the capillary pressure function for intermedi-
ate and low liquid saturations, the system behavior may differ significantly near full saturation.
8,9
Also note that the pore connectivity models by Burdine and Mualem
require a single value for the residual liquid saturation S
lr
. However, the physical meaning of this parameter is ambig-
uous.
30
In the water retention curve, it indicates the asymp- totic saturation at which the capillary pressure approaches
infinity. In the relative permeability function, it indicates the saturation at which the liquid phase becomes discontinuous,
i.e. where saturation cannot be further reduced by Inverse modeling of multistep outflow experiment
433
hydromechanical processes. It is obvious that the values for S
lr
do not have to coincide in the two different curves. It is important to realize that the parameters estimated by
inverse modeling are not intrinsic properties of the soil, but are model parameters strictly related to the specific formu-
lation as outlined in this section. Transferring parameters from one model to another may lead to conceptual and
thus numerical prediction inconsistencies. The same prob- lem arises when using ‘directly’ measured parameters
values in an analytical or numerical prediction model. On the other hand, the parameters estimated by inverse model-
ing can be considered optimal for the specific forward model used to simulate the experiment.
2.2 Formulation of the inverse problem
