for estimating constitutive relationship parameters of multi- fluid flow systems, we also consider the applicability of
several different capillary pressure–saturation and perme- ability relationships to this approach.
2 MATERIALS AND METHODS 2.1 Experimental apparatus
Multi-step transient outflow experiments were modified from Eching and Hopmans
9,10
to include measurement of the non-wetting fluid pressure Fig. 1, and were conducted
in a constant-temperature 208C laboratory
22
. Soils used in the transient outflow experiments were air-dried, sieved to
less than 2 mm in diameter, and uniformly packed in a 6.0 cm inside-diameter and 7.6 cm high brass soil core
that was placed in a standard Tempe pressure cell. Two soils, a Columbia fine sandy loam 63.2 sand, 27.5 silt
and 9.3 clay and a Lincoln sand 88.6 sand, 9.4 silt and 2.0 clay were used in the multi-step experiments.
These experiments were conducted using air or oil as the non-wetting nw fluid, displacing either water, or oil as the
wetting w fluid, assuming one-dimensional vertical flow in the soil core. Soltrol 130 and Soltrol 220 were used as the oil
phase for the Columbia and Lincoln soils, respectively. The density, viscosity, and interfacial tension values for the
experimental fluids are listed in Table 1. A 0.74 cm thick porous ceramic plate at the base of the soil core allowed
drainage of the wetting fluid, but was impermeable to the non-wetting fluid.
Applied air pressures expressed in cm water height for the air–water system were 60, 80, 120, 200, 400, and
700 cm above atmospheric pressure for the Columbia soil. Selected equivalent air pressure steps for air–water of the
Lincoln soil were 40, 60, 80, 100, 150, 200 and 400 cm above atmospheric pressure. These steps were smaller
than those for Columbia soil, because of Lincoln soil’s coarser texture. For both soil types, applied air pressure
increments were chosen such that drainage volumes were approximately equal for each pressure step. The pressure
steps for the other fluid pairs were reduced, to account for differences in interfacial tension values between the tested
fluid pair and air–water, thereby creating approximately equal drainage volumes for each wetting fluid during each
pressure step. A pressure step increment required from 5 to 36 h to reach equilibrium or near-equilibrium conditions,
depending on the pressure of the wetting fluid and soil type. After the drainage rate reduced to near zero, the
non-wetting fluid pressure was incrementally increased for the next drainage step. Initially the soil was fully saturated
with the wetting fluid, and then subsequently drained to a saturation corresponding to slightly in excess of the non-
wetting fluid entry pressure, thereby ensuring continuity of the non-wetting fluid throughout the soil core at the onset
of wetting fluid displacement
24
. Both cumulative outflow from the base of the soil core and pressures of both the
wetting and non-wetting fluid at the center of soil core were measured continuously using pressure transducers
T
1
, T
2
and T
3
in Fig. 1 coupled to a data logger. Cumula- tive drainage of the wetting fluid h
w|outflow
as a function of time was measured by the wetting fluid pressure in the bur-
ette. The two tensiometers enabled calculation of the capil- lary pressure P
c
as a function of time from the wetting fluid P
w
and the non-wetting fluid P
nw
pressures in the center of the core. Additional details about the experimental
methods are presented by Liu et al.
22
.
2.2 Governing equations
Assuming that the porous medium is nondeformable constant porosity and that cross-product permeability
terms associated with the viscous drag tensor
25
can be neglected, the general form of the two-fluid flow equations
without source–sink terms is described by the two-fluid, volume-averaged momentum and continuity equations
26
q
w
¼ ¹
k
w
m
w
· [
=P
w
þ r
w
g
] 1a
Table 1. Physical properties of fluids at 208C
Interfacial tension Nm Air–oil
Oil–water Air–water
0.0681
a
Soltrol 130 Columbia 0.0239
a
0.0259
a
Soltrol 220 Lincoln 0.0259
a
0.0364
23
Viscosity Nsm
2
3 10
¹ 3
Oil Air
Water 0.0181
1.00 Soltrol 130
1.44 Soltrol 220
3.92 Density kgm
3
1.20 998
Soltrol 130 762
Soltrol 220 803
a
Independently measured.
Pressure–saturation and permeability functions 481
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
