Modeling Nonwetting-Phase Relative Permeability Accounting
for a Discontinuous Nonwetting Phase
Ulrich Fischer,* Olivier Dury, Hannes Fliihler, and Martinus Th. van Genuchten
ABSTRACT
A model for the wetting- and nonwetting-phase constitutive relationships is presented. The nonwetting-phase relative permeability inthe model is a function of the degree of continuous nonwetting-phasesaturation. Different formulations of the continuous nonwettingphase model were evaluated by comparison of calculated and measured air permeabilities as well as discontinuous air saturations. A
Brooks and Corey-Burdine type formulation of the nonwetting-phaserelative permeability was more accurate than the corresponding van
Genuchten-Mualem equation. Estimated discontinuous air saturations were higher for drying than for wetting, thus reflecting hysteresis
in the water-retention and relative air permeability functions. The
continuous nonwetting-phase model provided a much better prediction of relative air permeabilities than a formulation that neglects thepresence of a discontinuous nonwetting phase. The emergence point
model for the nonwetting-phase relative permeability provided a good
approximation of the continuous nonwetting-phase model.

T

WO-PHASE AS WELL AS MULTIPHASE FLOW has long

been of interest in petroleum production. Becauseof an increase in the number of spills of nonaqueousphase liquids (NAPLs) into the subsurface during thelast decades, multiphase flow has also become an important topic in soil physics and hydrology. Besides theinfiltration of NAPLs into soil, such remedial procedures as soil vapor extraction, air sparging, and soil
flushing also deal with multiphase flow.
Two-phase flow models not only differ in the calculation of permeability but also vary in the definitions ofresidual and effective saturations. The relative permeability of a porous medium to a given phase in a multiphase system is generally considered to be a functionof the degree of saturation of that phase. Since at highwetting-phase saturations, the nonwetting phase may bediscontinuous with a considerable fraction of nonwetting phase being trapped (e.g., Stonestrom and Rubin,
1989a), the nonwetting-phase permeability should bea function of the continuously distributed nonwettingphase saturation, rather than the total nonwetting-phasesaturation. Lenhard and Parker (1987) described thenonwetting-phase relative permeability as a function of
the continuous nonwetting-phase saturation and of thecorresponding apparent wetting-phase saturation, thelatter being the sum of the wetting-phase saturationand those amounts of nonwetting phase that do notcontribute to nonwetting-phase flow.
U. Fischer, Dep. of Civil Engineering and Operations Research,
Princeton Univ., Princeton, NJ 08544; O. Dury, and H. Fliihler, Inst,of Terrestrial Ecology, Swiss Federal Inst, of Technology ETH, Grabenstr. 3/lla, CH 8952 Schlieren, Switzerland; and M.Th. van Genuchten, U.S. Salinity Lab., USDA-ARS, 450 W. Big Springs Road, Riverside, CA 92507-4617. This work was conducted at the Inst, ofTerrestrial Ecology, Swiss Federal Inst, of Technology, Schlieren,
Switzerland. Received 31 Oct. 1996. *Corresponding author (fischer@karst.princeton.edu).

In practice it is difficult to separate the total nonwetting-phase content into continuous nonwetting phase(which should be connected to at least two open boundaries of the domain under consideration), a locally accessible nonwetting phase (connected to only one open
boundary), and a trapped nonwetting phase (connected
to none of the open boundaries). Stonestrom and Rubin(1989a,b) determined the locally accessible air content
and the relative air permeabilities of two soil materials
as a function of water saturation. For zero air permeability, the difference between total air and locally accessible air accounted completely for trapped air. For air
permeabilities greater than zero, the nontrapped fraction of air could not be separated into continuous andlocally accessible air. We will use the term discontinuous
nonwetting phase to refer to the sum of trapped andlocally accessible nonwetting phases.
Stonestrom and Rubin (1989b) investigated the dependence of air permeability on water saturation andintroduced the term emergence point for the water saturation where air flow first becomes detectable duringdrying, and the term extinction point for the water saturation where air flow first becomes unmeasurable (zero)
during wetting. Several previous experimental studies
on multiphase flow have shown that flow of the nonwetting phase could be detected only as long as the wettingphase saturations did not exceed about 70 to 80% (e.g.,Odeh, 1959; Corey, 1957; Morgan and Gordon, 1970;
Stonestrom and Rubin, 1989b). Despite these data sets
showing the important effect of a discontinuous nonwetting phase, to our knowledge only Fischer et al. (1996)
presented a comparison of nonwetting-phase permeability data and calculations obtained with parametricmodels that accounted for the presence of a discontinuous nonwetting phase. These same researchers presented an empirical model of the nonwetting-phase relative permeability that is based on the knowledge of theemergence or extinction point.
The objective of this study was to develop a modelof the wetting- and nonwetting-phase constitutive relationships accounting for a discontinuous nonwetting
phase, and to evaluate the capability of this model todescribe nonwetting-phase relative permeabilities anddiscontinuous nonwetting-phase saturations. Different
formulations of the continuous nonwetting-phase model
were evaluated by comparison of calculated and measured air permeabilities as well as discontinuous air saturations. We also compared the proposed model to othermodels describing nonwetting-phase relative permeability, and studied how well these models approximate thecontinuous nonwetting-phase concept.
Abbreviations: BCB, Brooks and Corey-Burdine; CNP, Continuous
Nonwetting Phase; EP, Emergence Point; TNP, Total Nonwetting
Phase; VGM, van Genuchten-Mualem.

Published in Soil Sci. Soc. Am. J. 61:1348-1354 (1997).
1348

1349

FISCHER ET AL.: MODELING NONWETTING-PHASE RELATIVE PERMEABILITY

THEORY
Parametric Model for Two-Phase Flow Accounting
for a Discontinuous Nonwetting Phase
In our parametric model for two-phase flow, we considered
only single branches of the wetting and drying retention andpermeability functions. The relationship between the positive
value of the matric^potential head, h, and the effective wettingphase saturation, 5W, is described by the model of van Genuchten (1980):

SwaterSwater.a

Sw = [1 + (a ft)"]~m[1]where a, n, and m are empirical parameters, and 5W is defined
as

ow

c

_

C

_

J

w

0.0

^W^

[2]

C
'-'W.rin which 5W is the wetting-phase saturation, S WI is the residual
wetting-phase saturation, and 5WjS is the maximum wettingphase saturation. The relationship between matric potential
head and effective apparent wetting-phase saturation, 5 wa , ismodeled either with a van Genuchten type equation:
Sw,a = [1 + (a. h)"*]-™

[3]

or according to the model of Brooks and Corey (1964):*Jw.a —

for h >

[4a]for h < h,[4b]where aa, na, ma, ha, and Xa are empirical parameters, and 5wa
is defined as—r— 11 _ JC
—
w.a
n,c

J

9w.a — >Jw,r
S
i

[5]

O

in which 5 nc is the effective continuous nonwetting-phase saturation, and 5w,a is the apparent wetting-phase saturation. As
shown in Fig. 1 for an air-water system, the difference between
5wa and Sw is the discontinuous nonwetting-phase saturation,
§ •

S

=S

—S

F61

Wetting-phase relative permeability, kws(Sv), is modeled asa function of 5W since it was concluded that hysteresis in wetting-phase permeability is insignificant (e.g., Demond andRoberts, 1987,1993) and is described as a combination of thevan Genuchten retentivity model and the conductivity modelof Mualem (1976) to give, for fixed m = 1 - I/n (van Genuchten, 1980),[7]MS.)= (Sw)' [1 - (1 - SI"")"']
in which exponent a is a fitting parameter accounting forthe effects of tortuosity and connectivity with respect to thewetting phase. Mualem (1976) found that a value of 0.5 provided a good average of the relative water permeability formany soils.
Lenhard and Parker (1987) presented a model for the relative air permeability that was derived by application of thevan Genuchten-Mualem (VGM) approach (Mualem, 1976;
van Genuchten, 1980) to the air phase and accounts for discontinuous air. Their equation can be written as a general modelfor the non wetting phase relative permeability, knf(Snx), giving
for fixed m, = 1 — l/«a:
12ma

[8]

Saturation [-]Fig. 1. Domains of water saturation, £„„„.„ discontinuous air saturation, Sair,d, and continuous air saturation, Sai,f. The two air domains
are separated by the function of apparent water saturation, $„,„,,.
The curves were obtained for the data measured for the quartz

sand mixture and pure water during drying and using the Brooks
and Corey-Burdine formulation of the Continuous Nonwetting

Phase (CNP_BCB) model for calculating S^,f (cf. Fig. 2).
where exponent b accounts for tortuosity and connectivity
with respect to the nonwetting phase. The corresponding equation obtained by applying the Brooks and Corey-Burdine(BCB) approach (Burdine, 1953; Brooks and Corey, 1964) is

The Continuous Nonwetting Phase (GNP) models of the nonwetting-phase relative permeability represented by combination of either Eq. [3], [5], and [8], or Eq. [4], [5], and [9],are referred to as the CNPJVGM model and the CNP_BCB
model, respectively. In predictions, we used a value of 0.5
for b in the CNP_VGM model and a value of 2.0 in the
CNP_BCB model.
Alternative Models for the Nonwetting-Phase
Relative Permeability
As already pointed out, the amount of continuous nonwetting phase (or apparent wetting phase) is difficult to measure.
Therefore, the nonwetting-phase relative permeability is frequently described as a function of the total nonwetting-phasesaturation. The assumption that any nonzero nonwettingphase saturation, 5n = 1 — Sw, allows advective flow of thenonwetting phase leads to the following concept presented
originally by Parker et al. (1987) for the modeling of relative
air permeability:= (So)* [1 ~ (1 ~[10]

in which the effective nonwetting-phase saturation, 5n, is given
as
_

JC

[11]

w.r

This model is referred to as the Total Nonwetting-Phase(TNP) model.
Recently, Fischer et al. (1996) presented an equation for
calculating relative air permeability that can be written in a
generalized form as a model of the nonwetting-phase relative
permeability based on knowledge of the emergence or extinction point of the nonwetting-phase flow, 5wi. (and therefore
further referred to as the Emergence Point [EP] model):1350

SOIL SCI. SOC. AM. J., VOL. 61, SEPTEMBER-OCTOBER 1997

Table 1. Summary of gas permeability data. The values of A(5«,e) were calculated from the fitted water retention curves for which thevan Genuchten parameters a and in were obtained. The values of h,(Svf) were calculated according to Eq. [14].
m
Soil materialcth(S,, ,)
Wetting fluid
Process
Sw
Referencet
No.
S.,,*,<«„„)s..
— cm ————
1
2

3
4
5

6
7
8

quartz sand mixture
quartz sand mixture
quartz sand mixture
quartz sand mixture
quartz sand mixture
quartz sand mixture
Oakley sandOakley sand

A

A
A
A
A
A
B
B

water
2% butanol
6% butanol
water
2% butanol
6% butanol
water
water

dryingdryingdryingwetting
wetting
wetting
dryingwetting

0.79
0.78
0.77
0.78
0.81
0.80
0.88
0.87

0.04
0.04
0.04
0.04
0.04
0.04
0.29
0.29

0.53
0.57
0.59
0.69

30
17

0.73

13
13
9

0.77
0.77
0.80

7
37
7

19
13
8

6
-

cm '
0.031
0.054
0.051

0.81
0.72

0.081
0.068
0.096
0.023
0.068

0.78
0.79
0.77
0.82
0.60

0.81

tA: Dnry (1997, unpublished data); B: Stonestrom and Rubin (1989b).
Kn,r(3 n,c)&n,r(S*n,c)

I/m 2m= (S*nc)6 [1 - (1 — 5*c) ]''for 5W r < 5W < Sw,e=0

for 5W > "w,e

[12a]
[12b]

where S*c is an estimate of the effective continuous nonwetting-phase saturation, 5n?c, and is defined as
c

J

w,e

_ o

^w

[13]

For predictions, we again used a value of 0.5 for b in the TNP
and the EP models.

EXPERIMENTAL DATA
The different constitutive relationships were tested using
water-retention and air permeability data obtained for twoquartz sands. Stonestrom and Rubin (1989a,b) (numerical values and additional experimental details are given in Stonestrom [1987]) investigated Oakley sand, and Dury (1997, unpublished data) used an artificial quartz sand mixture andimplemented the experimental procedure presented by
Fischer et al. (1996). The constitutive relationships in bothstudies were determined for wetting as well as drying. In addition to pure water, the constitutive relationships of the quartz
sand mixture were determined using aqueous solutions containing 2 or 6% butanol by weight as the wetting fluid. Relative
air permeabilities, /cait „ were obtained by normalizing absolute
air permeabilities with the permeability value determined at
residual water saturation. The relative air permeability data
modeled in this study are summarized in Table 1. More details
about the experimental procedures are given in the references
cited above.
The emergence and extinction points for air flow were determined as the mean values of the lowest water saturation
at which air flow could not be detected and the highest water
saturation at which air flow was detectable. Parameter optimization was done using least square procedures. The relative
permeability models were not fitted to log(fcairr) since thisprocedure generally resulted in a poor description of highpermeability data and in most cases the improvement in thedescription of low permeability data (compared with fitting^air.r) was limited to one single permeability value. Fitted parameter values are eiven in Tables 1 and 2.

RESULTS AND DISCUSSION
Since no data of continuous nonwetting-phase saturations and corresponding nonwetting-phase permeabilities were available, both GNP models could only befitted to the permeability data selected for this study
and no predictions could be done. In this study, twocriteria were used to judge the quality of the fits, namely
the description of the relative permeability data andhow well the calculated emergence or extinction of continuous air, 5airiC, coincided with the experimentally observed emergence or extinction of air flow. Three parameters (aa, ma, and b, or /za, \a, and b) could be
optimized in both models. It was found that fitting all
three parameters resulted in an only slightly better description of relative air permeabilities than fitting only
two parameters. On the other hand, the second criterion
was fulfilled much better when l/cta or /za were fixed in
Eq. [3] and [4], respectively, as the value of the matricpotential head at the experimentally determined emergence or extinction point. Therefore this latter option
was used in all calculations presented here.
First, we compared both formulations of the GNP
model to each other and used this model to estimate
discontinuous air and apparent water saturations. Afterward, we compared the GNP model to the TNP and EP
models and investigated how well the GNP model could
be approximated by TNP and EP models.

Evaluation of CNP Models — Estimationof the Discontinuous Air Saturation
Figure 2 compares relative air permeabilities measured for the quartz sand mixture and for pure water
during main drainage with model calculations obtainedwith both CNP models. The CNP BCB model describes

Table 2. Parameter values obtained by fitting the Total Nonwetting Phase (TNP), the Emergence Point (EP), and the Continuous
Nonwetting Phase (CNP) models to the data listed in Table 1.
Data no.
Model
TNP
EP
CNP_VGM

OptionParameter

1

2

3

4

b fittedb fitteda, = 1/A(5»,«)

b

1.8

1.8

b
m*

0.14

0.42

1.3
0.26

6.3
3.7
4.1

8.7
11
7.4
9.8

0.85
0.02
4.6
6.9
3.5
6.2

b

CNP BCBh. = A(X,,)

X,
6

2.1

5.8
5.1
4.4
4.2

6

7

8

0.41

0.38

-0.18

-0.06
5.1
4.7
3.5

-0.09
0.15

0.12
0.20
3.4
19
2.4
19

5

5.0
4.3
3.6
3.4

3.4

5.6
6.0
4.3
5.6

1351

FISCHER ET AL.: MODELING NONWETTING-PHASE RELATIVE PERMEABILITY

.ff

°

a.
< 0.01 -50

Experimental-----

CNP_BCB Model....................

CNP_VGM Model

75

Matric Potential Head h [cm]Fig. 2. Comparison of relative air permeabilities determined for thequartz sand mixture and pure water during drying with model
calculations using the Brooks and Corey-Burdine (BCB) and van
Genuchten-Mualem (VGM) formulations of the Continuous Nonwetting Phase (CNP) model. The parameters h, and l/aa were
defined as h(Sv) (Table 1) and X,, m,, and b were fitted (Table 2).

the experimental data slightly better than the
CNPJVGM model. The corresponding saturations of
discontinuous air were calculated with Eq. [6] and areshown in Fig. 3 as a function of water saturation. The
total air saturation, S^,, and the experimentally determined emergence point are also shown. For water saturations above the emergence point, the discontinuous
air saturation should be equal to the total air saturation
since no air flow could be detected, and thus no continuous air existed. According to the estimate obtained with
the CNP_VGM model, large amounts of continuous air
existed already at water saturations above the emergence point. In contrast, the emergence of continuous
air as predicted with the CNP_BCB model agrees withthe emergence of air permeability. Similar findings as
those shown in Fig. 2 and 3 were also obtained for allother investigated data sets. Hence, we conclude that the
CNP_BCB model gives the more accurate descriptionof discontinuous air saturation. The curve of apparent
water saturation according to the CNP_BCB model isshown in Fig. 1 and compared with the fitted waterretention curve. This plot shows that small amounts of
discontinuous air, as indicated by the difference between

both curves, also existed at relatively low water saturations.
The abruptness of air permeability emergence as reported by Stonestrom and Rubin (1989b) suggests a
sudden increase in continuous air saturation at water
saturations just below the emergence point. Probably,
during drying just prior to the emergence of nonwettingphase permeability, large amounts of discontinuous
nonwetting phase exist and continuity of this phase isimpeded by small amounts of wetting phase. Removalof these wetting-phase blockages will then result in an
abrupt emergence of the continuous nonwetting phase.
Similarly, during wetting at the extinction point, small
amounts of wetting phase separate large volumes ofnonwetting phase into discontinuous portions. Therefore, it is to be expected that the apparent wetting-phasesaturation curve is described better by the model of
Brooks and Corey than by van Genuchten's model since
the latter predicts a much more gradual increase indiscontinuous nonwetting phase with increasing matricpotential head.
Figure 4 shows discontinuous air saturations estimated with the CNP_BCB model from the air permeability data measured for the quartz sand mixture andfor both wetting and drying and involving pure water
as well as an aqueous solution containing 6% butanolas the wetting fluid. Notice the substantial difference
between wetting and drying. Apparently, much larger
amounts of discontinuous air existed during drying, andat much lower water saturations, than during wetting.
This finding agrees with the observed hysteresis in thecorresponding water retention and air permeability
functions (Dury, 1997, unpublished data), indicatingthat nonwetting-phase trapping is an important factor
contributing to hysteresis. Given the values of the emergence and extinction points of air flow as derived fromthe air permeability experiments (Table 1), in cases likethose shown in Fig. 4 it might be difficult to determine
the amount of discontinuous air at nonzero air permeabilities during wetting, while it should be possible todetermine this quantity during drying.

Salr
Sair.d ' CNP_BCB Model-----

Drying - Pure Water-------

Wetting - Pure Water....................

Drying - 6% Butanol------

Wetting - 6% Butanol

Sair,d - CNP_VGM Model
S water ,e
0.0

0.2

0.4

0.6

0.8

Wetting Phase Saturation S,, [-]Water Saturation [-]Fig. 3. Discontinuous air saturations, .S,lr.,(, calculated with Eq. [6]from the model curves shown in Fig. 2. The total air saturation,
Sai,, and the experimentally determined emergence point, .$„,„.,,,
are also shown.

Fig. 4. Discontinuous air saturations, 5,M, calculated from the relative
air permeability data measured for the quartz sand mixture during
wetting and drying involving pure water as well as an aqueous
solution containing 6% butanol as the wetting fluid. The functionsof S,i,,d were obtained with Eq. [6] using the Brooks and CoreyBurdine formulation of the Continuous Nonwetting Phase(CNP_BCB) model.

1352

SOIL SCI. SOC. AM. J., VOL. 61, SEPTEMBER-OCTOBER 1997

Experimental
TNP Model - b fixed

Sajri(j - Drying

TNP Model - b optimized
EP Model - b fixed< 0.01 -— — ——

Swater,e - Drying-------

SwatW]e - Wetting

EP Model - b optimized

Matric Potential Head h [cm]Water Saturation [-]Fig. 5. Discontinuous air saturations, 5,M, calculated from relative airpermeability data determined by Stonestrom and Rubin (1989b)
for Oakley sand during both wetting and drying. The functions of•SaM were obtained with Eq. [6] using the Brooks and CoreyBurdine formulation of the Continuous Nonwetting Phase(CNP_BCB) model. The total air saturation, S,in and the experimentally determined emergence and extinction points, Sw,,af, arealso shown.

Figure 5 shows discontinuous air saturations calculated using the CNP_BCB model for wetting and drying
air permeability data presented by Stonestrom and Rubin (1989b) for Oakley sand. Compared with the curves
shown in Fig. 3 and 4, the discontinuous air saturation
during drying approaches zero at much higher water
saturations, while during wetting again a steep increasein the discontinuous air saturation is apparent at water
saturations close to the extinction point. The calculated
emergence and extinction of continuous air did notagree as much with the air permeability emergence andextinction points, respectively, as was the case in Fig. 3
for the data obtained for the quartz sand mixture, but
notice that the experimental determination of emergence or extinction points strongly depends on the resolution in the nonwetting-phase permeability data. Still,
the curves presented in Fig. 5 again agree with the observed hysteresis in the corresponding water retention
and air permeability functions as reported by Stonestrom and Rubin (1989a,b).

Comparison of CNP_BCB, TNP, and EP Models
We tested the CNP_BCB model and compared its
performance with the TNP and EP models. In Fig. 6
both models are compared with the data obtained forthe quartz sand mixture during drying and involving
pure water. The exponent b in both models was either
fixed to 0.5 or was optimized. Notice that the EP model
describes the data much better than the TNP model,
irrespective of whether or not b was fixed or optimized.
Even the optimized TNP model does not describe thedata correctly. Optimization of b only marginally improved the fit of the EP model to the data. The maindifference between the TNP and EP models is the inclusion of the discontinuous nonwetting phase in the EP
model. Hence, the results of Fig. 6 illustrate the needto include the effects of a discontinuous nonwetting
phase when developing nonwetting-phase relative permeability models. Similar results as those shown in Fig.

Fig. 6. Comparison of relative air permeabilities determined for thequartz sand mixture and pure water during drying with model
calculations obtained with the Total Nonwetting Phase (TNP) and
Emergence Point (EP) models for both a fixed and optimized
exponent b. Fitted values of b are given in Table 2.

6 were obtained for all other data sets investigated inthis study.
Predictions using the CNP_BCB, the TNP, and the
EP models were done for the air permeability data determined for the quartz sand mixture during drying using aqueous solutions containing either 2 or 6% butanol.
The matric potential head at the emergence point in
the GNP BCB model was calculated as
= ———

[14]

where /zs(5we) is the scaled matric potential head at the
emergence point of the aqueous solutions containing 2
or 6% butanol, cr2,6 is the corresponding interfacial tension, /z0(Sv/je) is the matric potential head at the emergence point determined for pure water, and (T0 is theinterfacial tension of pure water. In the CNP_BCB
model, we furthermore used the values of b and Xa as
fitted for pure water. Since the data measured for thequartz sand mixture did not indicate a significant dependence of the emergence point on wettability, for the EP
model we used the same values of the emergence point
as determined for pure water, took the value of b as
fitted for pure water, and used the value of m obtained
from fitting the retention data of the aqueous solutions
containing 2 or 6% butanol. In the TNP model, we usedthe value of b as fitted for pure water. Those predictions
were compared with calculations in which b was fixed
to 0.5 in the TNP and EP models and to 2.0 in the
CNP_BCB model while keeping the other parameters
as described above.
Figure 7 compares drying branch data obtained forthe aqueous solution containing 2% butanol with predictions obtained with the TNP, EP, and CNP_BCB
models using either fixed values of b or the values as
optimized for pure water. The TNP model describes thehigher relative permeability data somewhat better thanthe CNP_BCB and the EP models when a fixed valueof b was used in all three models (Fig. 7a) while thismodel fails to predict the low permeability data, which
are described quite well by the other two models. The
CNP_BCB model with fixed b slightly overestimates
the permeability data in the high range while the EP

FISCHER ET AL.: MODELING NONWETTING-PHASE RELATIVE PERMEABILITY

1353

r

7
0

'•S
0>

ExperimentalI....................

TNp Mode|-----

EP Model———-CNP_BCB Model I

r

0.1

|
I

0-

<! 0.01 .1- 0.01

CL

0.001
20

—I—
30

0.001

10

40

20

30

40

Matric Potential Head h [cm]Matric Potential Head h [cm]Fig. 7. Comparison of relative air permeabilities determined for the quartz sand mixture during wetting involving an aqueous solution containing
2% butanol as the wetting fluid with model predictions (set; text) obtained with the Brooks and Corey-Burdine formulation of the Continuous
Nonwetting Phase (CNP_BCB), Total Nonwetting Phase (TNP), and Emergence Point (EP) models. The exponent b was either (a) fixed
to 2.0 in the CNP_BCB model and to 0.5 in the TNP and EP models or (b) the values optimized for pure water were used.

model underestimates those data. Using the values of bas optimized for pure water resulted in good predictions
with the EP and CNP_BCB models across the entire
range of permeability data (Fig. 7b). Again the optimized TNP model does not describe the data correctly.
This finding shows that b generally is a fitting parameterbut that for the same porous medium, the same process,and the same model, one single value may serve asa good estimate to describe nonwetting-phase relative
permeability in the presence of wetting fluids with different wettability. Similar results as those shown in Fig.
7 were also obtained for wetting as well as the casewhen water containing 6% butanol was used as thewetting fluid. Generally, the predictions obtained with
the EP model were as good as those calculated with the
CNP_BCB model, indicating that the EP model is agood approximation of the continuous nonwettingphase concept.
For a general predictive application of the GNP
model, experimental procedures for determining continuous and discontinuous nonwetting-phase contents haveto be developed. When an accurate description of relative permeability data is required for modeling purposes(e.g., Fischer et al., 1996), the GNP model can be usedby fitting all three parameters as discussed above. We
compared this option of the CNP_BCB model to calculations obtained with the EP model using a fitted valueof b. In all cases, the CNP_BCB model described thedata slightly better than the EP model.

CONCLUSIONS
The results presented here show that nonwettingphase relative permeability should be modeled as a function of the continuous nonwetting-phase saturation
rather than of the total nonwetting-phase saturation.
Large differences between these two approaches wereobtained in cases of low emergence or extinction pointsof the nonwetting-phase permeability. The calculated
contents of discontinuous nonwetting phase suggested

a more accurate description of the dependence of apparent wetting-phase saturation on matric potential headby the model of Brooks and Corey than that of van
Genuchten. Our calculations indicated that during drying, considerable amounts of discontinuous nonwetting
phase may exist at relatively low wetting-phase saturations, while during wetting, discontinuous nonwetting
phase may exist only at wetting-phase saturations closeto (as well as above) the emergence or extinction pointof the nonwetting-phase permeability. This result is consistent with the observed hysteresis in the wetting-phaseretention and nonwetting-phase permeability functions,indicating that nonwetting-phase trapping is a major
factor determining hysteresis. The experimental determination of discontinuous nonwetting-phase content
might be difficult for the wetting process while it should
be more easy for drying. There is a strong need todevelop such methods for a general predictive application of the GNP model.
The use of the TNP model for predictions may leadto drastic overestimation of the nonwetting-phase permeability, especially when the emergence or extinction
point of the nonwetting-phase permeability is low. The
EP model provided a good approximation of the GNP
model and can be used when the emergence or extinction point is known. Improved methods for rapidly estimating the emergence or extinction point would be helpful. Still, even when the emergence or extinction point
is known, methods for estimating the exponent b accounting for tortuosity and connectivity are missing. In
order to improve the predictive capability of parametricpermeability models, methods for estimating this parameter or new concepts accounting for tortuosity andconnectivity may have to be developed.
APPENDIXabhtortuosity exponent for the wetting phasetortuosity exponent for the nonwetting phasematric potential head [L]

1354
ha
A(Sw,e)hs(S^ e)^o(X,e)&ajr,r
fcn,r
&WiF
m
ma
n
«a
5air
5airc
Saw
5n
Sn
5nc
5*c
Snid
Sw
5W
S^a
5wa
5we
5wr
Sws
•Swater.a
Smtaf
a
aa

SOIL SCI. SOC. AM. J., VOL. 61, SEPTEMBER-OCTOBER 1997

empirical parameter in the CNP_BCB model [L]matric potential head at the emergence point [L]scaled matric potential head at the emergence point[L]matric potential head at the emergence point forpure water [L]relative air permeability
relative nonwetting-phase permeability
relative wetting-phase permeability
van Genuchten parameter for the wetting phasevan Genuchten parameter for the apparent wetting
phase
van Genuchten parameter for the wetting phasevan Genuchten parameter for the apparent wetting
phase
air saturation
continuous air saturation
discontinuous air saturation
nonwetting-phase saturation
effective nonwetting-phase saturation
effective continuous nonwetting-phase saturation
estimate of the effective continuous nonwettingphase saturation
discontinuous nonwetting-phase saturation
wetting-phase saturation
effective wetting-phase saturation
apparent wetting-phase saturation
effective apparent wetting-phase saturation
emergence/extinction point of nonwetting-phaseflow
residual wetting-phase saturation
maximum wetting-phase saturation
apparent water saturation
emergence point for water as the wetting fluid
van Genuchten parameter for the wetting phase[L-']
van Genuchten parameter for the apparent wetting
phase [L~'J

XaBrooks and Corey parameter for the apparent wetting phase [L"1]

(TO
cr2 6

interfacial tension of pure water [M L 2]
interfacial tension of aqueous solutins containing 2
or 6% butanol [M LT2]