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OF IMMISCIBLE FLUIDS BASED ON
PRESSURE CELL DATA
J.H. DANE and M. JALBERT
Department of Agronomy and Soils
Auburn University
Auburn, Alabama 36849-5412
tel: 1-334-844-3974
fax: 1-334-844-3945
email:jdane@acesag.auburn.edu
ABSTRACT
The capillary pressure head - volumetric fluid
content relation, defined at a physical point, is of widespread interest for
the prediction of multiphase fluid flow.
The use of a pressure cell, a common procedure to determine this relation,
yields inaccurate results for certain combinations of multiphase fluids and
porous media (Dane et al., 1992). A
computational scheme was recently developed by Liu and Dane (1995a,b) to
overcome this inaccuracy. In this paper we will examine examples of
differences between capillary pressure head - volumetric fluid content
relations calculated in the traditional manner and by the method of Liu and
Dane (1995a). We will also examine
their effect on the prediction of the relative permeability relations.
Keywords: Capillary pressure - saturation, Brooks -
Corey relation, van Genuchten relation, dense aqueous phase liquids, porous
media, wetting fluids, non-wetting fluids.
INTRODUCTION
Accurate determination of the capillary pressure (Pc) - saturation (S) relation, or the capillary pressure
head (hc) - volumetric
fluid content (θ) relation, is important for modeling multiphase
fluid flow. The relation between S and θ is S
= θ/φ, where φ is the porosity of the porous medium. Pressure cells are often used to determine
this relation (Lin et al., 1982; Lenhard and Parker, 1987; Turrin, 1988;
Kuepper et al., 1989; Demond and Roberts, 1991). This procedure, however, can give inaccurate results for
multiphase fluids with small interfacial tensions (Dane et al., 1992). An improved experimental method to determine
this relation, using a gamma radiation technique, was developed by Dane et al.
(1992). Their method determines the
relation for a physical point. A
computational procedure to determine the relation for multiphase fluids, using
commonly obtained pressure cell data, was presented by Liu and Dane
(1995a,b). The advantage of their
computational scheme is that previously determined data can still be used to
obtain relations applicable to physical points instead of to the sample as a
whole. As relative permeability
relations are often determined from parameters associated with the capillary
pressure - saturation relations, the accuracy of these relations should have a
direct bearing on the relative permeability values.
THEORY
We assume that the θw(hc)
relation for the wetting fluid at a physical point can be approximated by the Brooks - Corey
expression [1964]:
for
hc ³ hd
[1]
for
hc < hd
where λ is the pore size distribution index, θw (L3L-3) is the
volumetric wetting fluid content, θws is the
saturated volumetric wetting fluid content, θwr is the irreducible volumetric wetting fluid content,
and hd (L) is the
displacement capillary pressure head.
The capillary pressure head hc
is defined by:
[2]
where ρw (ML-3) is the wetting fluid density, g (LT-2) is the gravitational
field strength, and Pn and
Pw (ML-1T-2)
are the wetting and nonwetting fluid pressure at the same physical point,
respectively. In Equation [1], the Brooks - Corey parameters are θwr, θws, hd,
and λ. Once these parameters are
known, the θw(hc) relation is
completely determined by Equation [1].
Figure 1. Diagram representing a pressure cell
with height zc, while zn and zw are the elevations at which the nonwetting fluid
pressure (n)
and the wetting fluid pressure (w),
respectively, are known.
In many instances, pressure cell data are used to
determine the Brooks - Corey parameters. In reality, however, the data obtained with a pressure cell
(Figure 1) are not applicable to a physical point, i.e., the data refer to an average volumetric
wetting fluid content for the pressure cell (height zc) as a whole, defined as:
[3]
while a corresponding capillary pressure
head value is calculated from:
[4]
In Equation [4], w [ML-1T-2] is the wetting fluid
pressure at elevation zw
[L], and n [ML-1T-2]
is the nonwetting fluid pressure at elevation zn [L] (Figure 1).
Traditionally, the measured relation w(c)
has been assumed to be applicable to a physical point. Although this assumption is most likely
accurate enough for soil water - air systems in porous media with a gradual
change in pore size, it can be highly inaccurate if organic liquids with low
surface tensions are involved in relatively coarse porous media (Dane et al.,
1992). In other words, the pressure
cell should not be treated as a physical or macroscopic point.
To overcome the inaccuracy in determining the
capillary pressure head - volumetric fluid content relation with a pressure
cell, Liu and Dane [1995a] developed relations between the volumetric fluid
content and capillary pressure head values obtained with a pressure cell and
those that are applicable to a physical point.
In simplified form, these relations can all be expressed as:
[5]
They outlined how Equation [5] can be used
in a curve fitting procedure, using pressure cell data, to obtain the
Brooks-Corey parameters as they apply to a physical point.
Generally speaking, pressure cell
retention curves are more or less "S"-shaped.
For that reason, the most commonly used curve fitting procedure at the
moment is probably the one proposed by van Genuchten (1980). It fits the following equation directly
through the data points as measured with, e.g., pressure cells:
[6]
where α, m,
and n are curve fitting
parameters. The corresponding relative
permeability for the wetting phase, krw,
associated with the pore size distribution model of Mualem (1976) is:
[7]
whereas the associated expression for the
relative permeability of the non-wetting fluid, krnw, is
(Oostrom and Lenhard, 1998):
[8]
where
[9]
is referred to as the effective saturation.
Brooks and Corey (1964), using the pore distribution
model of Burdine (1953), derived for the relative permeability of the wetting
phase:
for
hc £hd
[10]
for
hc ³hd
and for the relative permeability of the nonwetting
phase:
for
hc ³hd [11]
In this paper, we compare the retention curves
obtained by fitting pressure cell data with Equation 1, 5 and 6, and their
corresponding relative permeability curves as calculated from Equation 7 - 11,
for different combinations of wetting and non-wetting fluids. Measurements obtained at physical points
will be used as reference data.
MATERIALS AND METHODS
Capillary pressure head - volumetric fluid content
data were obtained at physical points for trichloroethylene (TCE, density 1.456
g/cm3) and air, and for TCE and water (density 1.0 g/cm3),
according to the method of Dane et al. (1992).
For the TCE - air combination, a long column (inside diameter = 8.0 cm),
filled with sand, was first saturated with TCE. It was then step wise drained by reducing the outflow level,
connected to the bottom of the column, in increments of 2 to 3 mm. Air was allowed to enter through the top of
the column to displace the TCE. Each
time static equilibrium had been reached, gamma radiation measurements were
taken to determine the volumetric TCE content values at a number of
elevations. The gamma radiation passed
through a 6-mm collimator on the side of the sources as well as on the side of
the detector. The volume of sand over
which the volumetric fluid contents were measured was thus determined by the
diameters of the column and the collimator. We assumed this volume to be large
enough to represent a physical or macroscopic point. Corresponding hc
values were determined from the known fluid pressure distributions in the
column at each static equilibrium situation.
Consequently, we were able to determine a retention curve for each
elevation where gamma radiation measurements were made. A similar procedure was followed for the TCE
- water combination. TCE was now displaced by water entering from the top of
the column under a constant head, and subsequently by reducing the outflow
level again. This process was then
reversed to obtain the retention data during drainage of water. The curves
selected for the purpose of this paper were fitted with Equation 1 (Figure 2
and 3), and the obtained curve fitting
parameters are presented in Table 1.
These data were then used to construct the pressure cell (zc = 6.0 cm, zn = 0.0 cm, zw = 6.0 cm) data (Figure 2
and 3), which were fitted with Equation 1, 5, and 6. The obtained curve fitting parameters were then used to determine
and compare relative permeability functions.
Figure 2. Capillary pressure - saturation data
for air displacing TCE as determined at a physical point by gamma radiation,
the fitted Brooks-Corey curve and the curve obtained by integrating the
physical point data over the pressure cell domain (zc = 6 cm, zn
= 6 cm and zw = 3 cm).
Figure 3. Capillary pressure - saturation data
for TCE displacing water as determined at a physical point by gamma radiation,
the fitted Brooks-Corey curve and the curve obtained by integrating the
physical point data over the pressure cell domain (zc = 6 cm, zn
= 0 and zw= 6 cm).
Table 1. Parameter values obtained with Equations 1,
5, and 6 for a porous medium containing TCE and air, and for one containing TCE
and water. The last column contains values for the Root Mean Square Error on
volumetric content.
RESULTS AND DISCUSSION
The curve fitting parameters associated with Equations
1, 5, and 6 are presented in Table 1.
It is obvious that the Brooks-Corey parameter values (Equation 1)
related to the data obtained at a
physical point and those obtained with the pressure cell data (Equation 5) are
very much the same. The small
differences are attributed to the differences in the equations to be fitted and
the error minimization procedure of the parameter values. The Brooks-Corey (Equation 1) curve fitting
parameters determined directly on the pressure cell data are generally quite
different from those determined for a physical point. Lastly, the pressure cell data fitted with Equation 6 produced
the van Genuchten curve fitting parameters (Table 1). These values are not directly comparable with the Brooks-Corey
parameters, but the results will be used to compare their effect on the
calculated relative permeability functions.
Values for 1/α are often assumed to be equivalent to the
displacement capillary pressure head value as defined by Brooks and Corey.
Values of the obtained parameters not only affect the
shape of the retention curves, but also the relative permeability functions as
shown in Figure 4 for TCE (wetting fluid) in a
TCE - air system and in Figure 5 for TCE (non-wetting fluid) in a TCE -
water system. As was to be expected, in
both cases the relative permeability functions based on the parameters
determined with Equations 1 and 5 yielded similar results. The functions based on direct fitting
through the pressure cell data were quite different from those based on a
physical point. The functions also show
the importance of having knowledge of the displacement capillary pressure head
value, since no displacement, and therefore no flow of the displacing fluid can
occur until this value has been exceeded.
Figure 4. TCE (wetting fluid) relative
permeability in the case of air displacing TCE. Equations 1, 5 and 6 were used to fit the pressure cell data.
Figure 5. TCE (non-wetting fluid) relative permeability in the case of TCE displacing water. Equations 1, 5 and 6 were used to fit the pressure cell data.
CONCLUSIONS
The information presented shows that for the fluids
and porous media used in this paper large errors will occur in estimating curve
fitting parameters if either the Brooks-Corey or the van Genuchten equations
are directly applied to pressure cell data.
This not only affects the retention data, but also the relative
permeability functions. In order to use
pressure cell data for determining relations valid at a physical point, Liu and
Dane (1995) proposed a computational procedure, assuming a Brooks-Corey
relation for retention at a physical point.
An advantage of the Brooks - Corey expression is that it has a clearly
defined displacement capillary pressure head value. This is of particular importance
with regard to non-aqueous phase liquid (NAPL) spills. The NAPL will not enter
a saturated porous medium, or a layer with different hydraulic properties,
unless the displacement pressure head value is exceeded.
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