f ]
r
w
S
w
]t þ =
· r
w
q
w
¼ 1b
q
nw
¼ ¹
k
nw
m
nw
· [
=P
nw
þ r
nw
g
] 1c
f ]
r
nw
S
nw
]t þ =
· r
nw
q
nw
¼ 1d
In eqns 1a, 1b, 1c and 1d, the subscripts w and nw denote the wetting and non-wetting fluids, respectively; P
i
i ¼ w,nw denotes pressure M L
¹ 1
T
¹ 2
; S
i
i ¼ w,nw is the degree of fluid saturation relative to the porosity f
volumetric fluid content v
i
¼ fS
i
; q
i
is the flux density
vector LT; g is the gravitational acceleration vector; m
i
and r
i
denote dynamic viscosity M L
¹ 1
T
¹ 1
and density M L
¹ 3
, respectively; k
i
¼ k
ri
k is the effective permeabil-
ity tensor L
2
, where k is the intrinsic permeability L
2
and k
ri
¼ k
ri
S
i
is the relative permeability. Assuming that the porous media is nondeformable
implies that S
w
þ S
nw
¼ 1 or v
w
þ v
nw
¼ f
2 Assuming one-dimensional vertical flow and that the wet-
ting fluid is incompressible, substitution of eqn 1a into eqn 1b yields
]v
w
]t ¼
] ]z
k
w
m
w
]P
w
]z þ
r
w
g 3
When replacing fluid pressures P
w
and P
nw
and capillary pressure P
c
¼ P
nw
¹ P
w
with pressure head, defined by h
i
¼ P
i
r
H
2
O
g i ¼ w, nw or c, and defining the hydraulic conductivity of fluid i K
i
, LT by K
i
¼ k
i
r
H
2
O
g m
i
i ¼ w ,
nw 4
transient flow of the wetting fluid is described by C
w
]h
nw
]t ¹
]h
w
]t ¼
] ]z
K
w
]h
w
]z þ
1 5
where the capillary capacity C
w
¼ dv
w
dh
c
is negative
27
. If the non-wetting fluid is compressible e.g. air is the non-
wetting fluid, substitution of eqn 1c into eqn 1d yields ]
r
nw
v
nw
]t ¼
] ]z
r
nw
k
nw
m
nw
]P
nw
]z þ
r
nw
g 6
For air as the non-wetting fluid, it is furthermore assumed that its density is a linear function of the pressure head
28
, or
r
nw
¼ r
o ,
nw
þ r
o ,
nw
h
o
h
nw
7 where r
o,nw
and h
o
are the reference density and pressure head at atmospheric pressure, respectively. The ratio
r
o,nw
h
o
is defined as the compressibility l l ¼ 1.24 3 10
¹ 6
gcm
4
. Subsequently, using relationships eqns 4 and 7, and writing water potential in terms of pressure head
27
yields the flow equation for the non-wetting fluid: f ¹ v
w
ÿ l ¹ r
nw
C
w
]h
nw
]t þ
r
nw
C
w
]h
w
]t ¼
] ]z
r
nw
K
nw
]h
nw
]z þ
r
nw
r
H
2
O
ð 8Þ
Eqns 5 and 8 can be solved simultaneously for the unknown pressure heads h
w
and h
nw
, and thus for h
c
as well. The upper boundary condition at the top of the soil core
z
top
to be used in the inverse modeling of two-fluid flow is described by a zero wetting fluid flux and by a sequence of
imposed non-wetting fluid pressures, defined by P
nw
T
j
, or q
w
z
top
, t
¼ 9a
h
nw
z
top
, t
¼ h
nw
T
j
¼ P
nw
T
j
r
H
2
O
g j ¼ 1
, 2
, …, M
9b where M is the total number of pressure steps used during
time period T
j
in the multi-step experiment. At the lower boundary of the core z
bottom
, the flux of the non-wetting fluid is zero since the entry value of the ceramic plate for
the non-wetting fluid is greater than any of the imposed pressures. The wetting fluid pressure at z
bottom
is determined by the height of the wetting fluid in the burette, which is a
function of time as the wetting fluid drains into the burette Fig. 1, or
q
nw
z
bottom ,
t
¼ 10a
h
w
z
bottom
, t
¼ r
w
r
H
2
O
h
wjoutflow
t 10b
Finally, the initial condition for the multi-step method is given by the static pressure of each fluid at time t
1
: h
w
z ,
t
1
¼ h
w
z
bottom
, t
1
¹ r
w
z r
H
2
O
11a
h
nw
z ,
t
1
¼ h
nw
z
top
, t
1
þ r
nw
r
H
2
O
z
top
¹ z
11b where z is assumed positive upwards and z ¼ 0 at the
bottom of the ceramic plate. h
w
z
bottom
,t
1
and h
nw
z
top
,t
1
denote the boundary pressure head values at the start of the transient outflow experiment.
2.3 Constitutive relationships—parametric models
Parameter estimation requires the functional description of the capillary pressure–saturation, h
c
S
w
, and permeability functions, k
i
S
w
, in eqns 5 and 8. The choice of a sui- table parametric model involves collection of available can-
didate models, followed by a selection of the appropriate models using techniques to be described later. The consti-
tutive relationships considered below are listed in Table 2. 482
J. Chen et al.
Perhaps one of the most commonly used models of capil- lary pressure–saturation function is the van Genuchten
29
equation VG S
ew
¼ [
1 þ a
vg
h
c n
]
¹ m
12 where S
ew
denotes the effective saturation of the wetting fluid, S
ew
¼ ð v
w
¹ v
wr
Þ v
ws
¹ v
wr
, where v
ws
and v
wr
are the saturated and residual wetting fluid saturation, respectively;
a
vg
and n are fitting parameters, that are inversely propor- tional to the non-wetting fluid entry pressure value and the
width of pore-size distribution, respectively. In subsequent analysis, we assume that m ¼ 1 ¹ 1n, and that the effective
saturation of the non-wetting fluid S
en
is derived from S
en
¼ 1 ¹ S
ew
. The Brooks and Corey BC model
30
is based on empiri- cal observations that logS
ew
is linearly related to logh
c
, yielding a power form of S
ew
h
c
S
ew
¼ h
e
= h
c l
, h
c
. h
e
13a S
ew
¼ 1
, h
c
h
e
13b where h
e
and l are characteristic soil parameters with l characterizing the pore-size distribution and h
e
equal to the non-wetting fluid entry pressure. The VG model is asymp-
totically equal to the BC model in the dry range when h
c
becomes large, as then S
ew
[1ah
c
]
mn
, such that 1a ¼ h
e
and mn ¼ l. The lognormal distribution LN model
31,32
is based on the assumption that soils are represented by a lognormal
pore-radius distribution yielding a two-parameter model for the capillary pressure–saturation function
S
ew
¼ F
n
lnh
m
¹ lnh
c
j 14
where F
n
x ¼ ð
1=
2p p
Þ
x ¹ `
exp ¹
y
2
2
dy is the normal distribution function that can also be calculated from
F
n
x ¼
1=2erfc x=
2
p . In eqn 14, the parameters h
m
and j are physically based and are related to the geometric
mean and variance of the pore-size distribution. The Brutsaert BR capillary pressure–saturation func-
tion
33
is of the form S
ew
¼ b
b þ h
g c
15 where b and g are fitting parameters characterizing the
porous medium. While the capillary pressure–saturation function can be
considered a static soil property, the permeability function is a hydrodynamic property describing the ability of the soil to
conduct a fluid. Capillary pressure–permeability prediction models were developed from conceptual models of flow in
capillary tubes combined with pore-size distribution knowl- edge derived from the capillary pressure–saturation rela-
tionship. Typical representations of this type of model include Burdine
4
and Mualem
34
Burdine B
: k
rw
¼ S
2 ew
Z
S
e
dS
e
h
2 c
Z
1
dS
e
h
2 c
2 6
6 6
4 3
7 7
7 5
,
k
rn
¼ 1 ¹ S
ew 2
Z
1 S
e
dS
e
h
2 c
Z
1
dS
e
h
2 c
2 6
6 6
4 3
7 7
7 5
ð 16Þ
Mualem M
: k
rw
¼ S
h ew
Z
S
e
dS
e
h
c
Z
1
dS
e
h
c
2 6
6 6
4 3
7 7
7 5
2
,
k
rn
¼ 1 ¹ S
ew h
Z
1 S
e
dS
e
h
c
Z
1
dS
e
h
c
2 6
6 6
4 3
7 7
7 5
2
ð 17Þ
Combining the van Genuchten capillary pressure–satura- tion eqn 12 with the Mualem VGM model yields
permeability functions as defined by
35
k
rw
¼ k
w
k ¼
S
h ew
1 ¹ 1 ¹ S
1 m
ew m
2
18a
k
rnw
¼ k
nw
k ¼
1 ¹ S
ew h
1 ¹ S
1 m
ew 2m
: 18b
Similarly, substituting eqn 13 into eqn 17 and integrat- ing yields the BCM formulation
34
k
rw
¼ S
h þ 2 þ 2=l ew
19a k
rn
¼ 1 ¹ S
ew h
1 ¹ S
1 þ
1 l
ew
2 6
4 3
7 5
2
19b Similarly, the BCB model is derived by substituting eqn
13 into eqn 16 and integrating
30
k
rw
¼ S
3 þ
2 l
ew
20a k
rn
¼ 1 ¹ S
ew 2
1 ¹ S
1 þ
2 l
ew
2 6
4 3
7 5
20b Comparison of eqns 18 and 19, indicates that the BCB
model differs from the BCM model only by the tortuosity parameter h.
The VGB model is derived by substitution of eqn 12 Pressure–saturation and permeability functions
483
into eqn 16 and integrating to obtain
36
k
rw
¼ S
2 ew
1 ¹ 1 ¹ S
1 m
ew m
21a k
rn
¼ 1 ¹ S
ew 2
1 ¹ S
1 m
ew m
21b Similarly, the LNM model is derived by substitution of eqn
14 into eqn 17 and integrating to obtain
32
k
rw
¼ S
h ew
F
n
F
¹ 1
n
S
ew
þ j
2
22a k
rn
¼ 1 ¹ S
ew h
1 ¹ F
n
F
¹ 1
n
S
ew
þ j
2
22b where F
n ¹
1
S
ew
is the inverse of F
n
S
ew
. No closed-form Burdine or Mualem permeability models
are available for the BR capillary pressure–saturation model eqn 15. However, through simplifying eqn 15,
Brutsaert
33
obtained a closed form Burdine model BRB k
rw
¼ S
2 ew
[ 1 ¹
1 ¹ S
ew 1 ¹
2 g
] 23a
k
rn
¼ 1 ¹ S
ew 3 ¹
2 g
23b Finally, combining the Gardner
37
equation k
rw
¼ e
¹ a
g
h
c
24a with eqn 17 and using h ¼ 0, Russo
21
derived a capillary pressure–saturation relation GD
S
ew
¼ 1 þ
1 2
a
g
h
c
e
¹
1 2
a
g
h
c
24b after which also the non-wetting permeabilitity relationship
can also be derived GDM
k
rn
¼ 1 ¹ e
¹
1 2
a
g
h
c
1 A
2
24c In subsequent analyses, for convenience in modeling and
due to lack of information to the contrary, we set the h value equal to 0.5 for both the wetting and non-wetting
permeability expressions
34
, despite
possible physical
reasoning suggesting that h values for wetting and non- wetting fluids may differ
38
.
2.4 Numerical solution of governing equations and optimization method
