Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol24.Issue2.2000:
Advances in Water Resources 24 (2001) 157±177
www.elsevier.com/locate/advwatres
Investigation of the residual±funicular nonwetting-phase-saturation
relation
Markus Hilpert *, John F. McBride, Cass T. Miller
Center for Advanced Study of the Environment, Department of Environmental Sciences and Engineering, University of North Carolina, CB 7400,
104 Rosenau Hall, Chapel Hill, NC 27599-7400, USA
Received 1 November 1999; received in revised form 13 June 2000; accepted 13 June 2000
Abstract
The constitutive relation that describes the amount of nonwetting ¯uid phase entrapment is critical to the modeling of multiphase
¯ow in porous media, but it has received insucient attention in the literature. We studied this relation using both experimental and
modeling approaches: we used a nondestructive, X-ray monitored, long-column experiment that yielded a rich data set for two
dierent porous media; we also used a quasi-static network model to simulate the experimental data and examine mechanisms
aecting the relation. The experimental work yielded a signi®cant data set for residual nonwetting phase (NWP) saturation as a
function of maximum funicular nonwetting phase saturation. We suggest a functional form that represents the observed data sets
accurately. Network model calibration to experimental data yields acceptable model-data agreement and a clear understanding of
constraints that should be satis®ed when using such models to avoid physically unrealistic behavior. We found that the pore-throatsize characteristics and the snap-o process occurring in pore throats strongly in¯uence the manifestation of pore-body-size
characteristics during imbibition and nonwetting ¯uid phase entrapment. We examined an estimation method proposed by Wardlaw
and Taylor for the residual±funicular relation. We observed that the method yields an unrealistic relation for each porous media in
the long-column experiments, and we used network modeling to understand the criteria that ensure a realistic estimate. Ó 2000
Elsevier Science Ltd. All rights reserved.
Keywords: Capillarity; Hysteresis; NAPL; Network model; Residual; Snap-o; X-ray attenuation
1. Introduction
In groundwater systems, non-aqueous-phase liquids
(NAPLs) are often the nonwetting phase (NWP) and
therefore subject to capillary trapping [5]. Entrapped
NAPL cannot be displaced by the surrounding
groundwater ¯ow unless the physico-chemical properties of the ¯uids are altered [8]. This entrapped or residual NAPL is a source of groundwater contamination
because the NAPL dissolves into the surrounding
groundwater and is dispersed downstream by natural
groundwater ¯ow [30]. Accurate modeling of how the
residual NAPL forms is a prerequisite for a comprehensive assessment and analysis of the contamination
source, and for a realistic simulation of remediation.
*
Corresponding author.
E-mail addresses: markus_hilpert@unc.edu (M. Hilpert),
jmcbride@tbcnet.com (J.F. McBride), casey_miller@unc.edu (C.T.
Miller).
Entrapment of NAPL as a disconnected residual
phase in water-wet porous media occurs during water
imbibition after an initial water displacement (drainage)
by funicular NAPL. Fig. 1(a) shows an idealized hysteretic capillary pressure±saturation pc ±sw relation, with
the main imbibition (MI) curve and two imbibition
scanning curves originating at dierent capillary
pressures (pc ) on the primary drainage (PD) curve and
terminating at pc 0. Fig. 1(b) is a plot of the residual
NAPL saturation snr at pc 0 observed in Fig. 1(a)
versus the initial funicular NAPL saturation snf at the
start of water imbibition. In this paper, the functional
form between snr and snf is termed the ``residual±funicular NWP saturation relation'' or the snr ±snf relation;
the functional form has also been termed the ``residual±
initial NWP relation'' in the petroleum engineering
literature [37,44].
The snr ±snf relation is needed to predict residual NWP
saturations correctly in simulations of multiphase ¯ow
[51]. The snr ±snf relation is also an important submodel
in hysteretic pc ±sw relations [17,36] used in multiphase
0309-1708/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 0 9 - 1 7 0 8 ( 0 0 ) 0 0 0 3 5 - X
158
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
Nomenclature
Abbreviations
Hg
mercury
MI
main imbibition
NAPL non-aqueous phase liquid
NWP
nonwetting phase
PD
primary drainage
RMSE root of mean squared error
SD
secondary drainage
WP
wetting phase
WTM
Wardlaw±Taylor method
Variables
capillary pressure head, cm H2 O
hc
m
van Genuchten parameter in pc ±sw relation
n
van Genuchten parameter in pc ±sw relation
dimensionless critical capillary pressure for
p
snap-o
capillary pressure
pc
critical capillary pressure for piston
pp
displacement
critical capillary pressure for retraction
pr
ps
rb
rt
s
Z
a
b
c
k
m
q
x
critical capillary pressure for snap-o
pore-body radius, sphere radius
pore-throat radius, cylinder radius
saturation
coordination number
van Genuchten parameter in pc ±sw relation
van Genuchten parameter in snr ±snf relation
interfacial tension
estimate for the ratio of pore body to pore
throat radius
van Genuchten parameter in snr ±snf relation
density
van Genuchten parameter in snr ±snf relation
Subscripts and superscripts
max
maximum value of the variable
n
NWP
nf
funicular NWP
nr
residual NWP
w
WP
c
)
wr
irreducible WP (more exact: WP at pPD;max
w
50
value of the variable at s 0:5
Fig. 1. (a) Idealized hysteretic pc ±sw relations (PD, SD, MI) with scanning curves from the PD curve to zero capillary pressue; (b) plot of snr ±snf
relation obtained from MI curve and two imbibition scanning curves in Fig. 1(a).
¯ow simulations. The particle-size distribution and the
extent of the porous medium's consolidation in¯uence
the slope of the snr ±snf relation [44], the maximum rec
sidual NWP saturation, snr
max , obtained by MI to p 0,
is greater for consolidated porous media than for unconsolidated porous media [44]. Porous media with
uniform particle size, whether unconsolidated or consolidated, exhibit lower values of snr
max , presumably because greater connectivity among pores decreases the
probability of entrapment [37]. snr
max is also a function of
the ¯uid±solid properties, especially wettability; ¯uid±
¯uid properties, including viscosity ratio, interfacial
tension, and density dierence; and displacement rates
[44].
The measurement of the snr ±snf relation can be made
with the same retention-cell apparatus used to measure
the pc ±sw relation, but this approach is very time consuming: the equilibration time for each data pair can
range from hours to weeks. Rate-controlled porosimetry, in which the pressure ¯uctuations during slow-rate
mercury injection are monitored and analyzed, has also
been used to predict the snr ±snf relation [47,58]. Although good agreement with experimental data has been
obtained, this method has not received much attention
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
in the groundwater hydrology community. In practice,
the snr ±snf relation is rarely measured. Rather, functional
forms for the snr ±snf are assumed, such as various linear
[16,20,26,42,46] or curvilinear forms [14,17,19].
In this work, we evaluate various methods to estimate
the snr ±snf relation from a limited set of experimental
data, such as the PD, MI, or secondary drainage (SD)
curve. Speci®cally, we investigate
1. the use of pore-network modeling to estimate the
snr ±snf relation from the PD and MI data;
2. the applicability of a method by Wardlaw and Taylor
[56] that estimates the snr ±snf relation only from the
PD and SD curve; and
3. the adequacy of the curvilinear form in the Land
equation [19], which requires measured values of only
wr
snr
max and the irreducible WP saturation, s .
We compare the estimated snr ±snf relations to those
obtained experimentally by (1) generating funicular and
then residual NAPL distributions in the same long
vertical porous-medium column and (2) measuring each
vertical NAPL distribution nondestructively with an
X-ray attenuation instrument.
2. Experimental methods
A series of NAPL±water displacement experiments
was performed in 1-m long, 2.5-cm-diameter porousmedium columns to examine the snr ±snf and pc ±sw relations in a glass-bead porous medium and in a coarse
silica-sand porous medium. Tetrachloroethylene (PCE)
dyed red with Oil RedO at a concentration of 0.267 g/l
was used as the NAPL. The bottom and the top of the
columns were connected to constant head PCE and
water reservoirs, respectively. Using a long vertical column takes advantage of the correspondence between (1)
159
the ¯uid saturation distribution with elevation in a porous medium with respect to constant-elevation, constant-head ¯uid reservoirs and (2) a pc ±sw relation
measured in a retention cell by using step-changes in the
elevation of the constant-head ¯uid reservoirs [40,41].
This correspondence requires homogeneity of the porous medium in the long vertical column. Various vertical equilibrium saturation pro®les, corresponding to
water drainage and imbibition curves, were obtained by
successive changes of the reservoir elevations. The PCE
fraction at locations along the vertical length of the
stationary glass-bead column was determined nondestructively by measuring the attenuation of X-rays
through the column cross-section at each location. Fig. 2
shows the experimental setup. Schiegg [40,41] made
similar measurements for an air±water system to obtain
hysteretic pc ±sw relations for a coarse sand, but used a
gamma-ray source (cesium-137) and a sand column with
a 15-cm cross-section.
The X-ray attenuation instrument depicted in Fig. 2
was designed and operated in a manner similar to the
instrument described by Oak and co-authors [33,34]. A
tungsten-target X-ray tube operating at a voltage potential of 45 kV was used to produce a broad spectrum
of photon energies. This X-ray spectrum was conditioned to yield an energy spectrum with two narrow
energy bands by using a 3-mm Al plate and an aqueous
salt solution (22 wt% cesium chloride and 11 wt% samarium chloride) contained in a vial with 1-cm crosssection. The X-ray beam was collimated to a 5-mm by
5-mm cross-section by using lead collimators. A liquidnitrogen-cooled germanium detector was used with a
photon-energy resolution of 0.8 keV. To determine one
single unknown phase volume fraction (PCE volume
fraction or the solid-phase volume fraction to determine
porosity), photon count rates were integrated within the
Fig. 2. Schematic of the experiment setup showing X-ray source, germanium detection, and porous-medium column with constant-head PCE delivery reservoir and constant-head water collection reservoir to establish equilibrium water drainage pro®le (apparatuses not to scale).
160
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
lower (30±36 keV) of the two narrow energy bands. A
similar use of this X-ray instrument can be found in
[9,35]. The measurement of one unknown phase fraction
at each location required approximately 15 s (real time)
of X-ray photon counting. The observed standard deviation of three repeat measurements at each location
was 0.00057 for porosity and 0.0017 for PCE volume
fraction. The standard error of the mean at each location was thus 0.00039 for porosity and 0.00099 for PCE
volume fraction.
The glass beads for these experiments came from the
same original stock of glass beads used by Mayer and
Miller [29] in their measurement of pc ±sw relations and
blob-size distributions of radiation-polymerized residual
styrene. They used 5-cm-long retention cells to measure
the pc ±sw relations. Although a snr ±snf relation and SD
curve are not part of their data set, the measured blobvolume distributions allow an evaluation of the porenetwork model simulations. Table 1 summarizes the
properties of the glass-bead and sand multiphase systems (labeled as GB1b and C-109, respectively) in the
long column and of the glass-bead multiphase system in
the 5-cm-retention-cell (labeled as GB1a).
The long columns were chromatography columns
(ACE Glass, Vineland, NJ). An individual column was
packed with the water-inlet plunger at the bottom. The
plunger had a porous glass frit and two sets of O-ring
seals. A 1-cm layer of more ®nely grained material was
added ®rst to serve as a capillary barrier to PCE entry.
Then, the column was ®lled with the desired porous
medium by using a funnel out®tted with screens and a
tube extension to deliver the material at a constant rate.
The delivery tube was kept approximately 5 cm above
the porous medium as its level rose in the column. Then,
three screens were placed (80-mesh, 360-mesh, 24-mesh)
on top of the porous medium and TEFLON-seal
plunger was inserted to con®ne the porous medium. The
TEFLON-seal plunger served as the PCE outlet/inlet.
The column was saturated from below with de-gassed
water until all the trapped gas visible along the walls of
the column disappeared. The column was then inverted,
a water constant-head reservoir connected to the top of
the column, and a PCE constant-head reservoir connected to the bottom. Using X-ray attenuation, we
measured the solid fraction at 160 vertical locations in a
long water-saturated column. These measurements
yielded the porosity, which we then used to assess medium homogeneity. Then, various equilibrium ¯uid
distribution were established, and the resulting PCE and
water saturations were quanti®ed by X-ray attenuation
at the same 160 locations used for the porosity
measurement. We raised the PCE reservoir in order to
establish a PD pro®le and then lowered it in four steps,
waiting 24 h for water-imbibition equilibrium. In addition to 160 scanning paths, each de®ned by the four data
points from this four-step transition, the experiments
also yielded a well-de®ned snr ±snf relation for each porous medium. By using the same reservoir elevations
from the PD pro®le, we next obtained a SD curve by
drainage to irreducible water saturation, imbibition to
maximum residual PCE saturation, and drainage equilibrium.
3. Pore-network-model formulation
3.1. Overview
Pore-network models are simpli®ed representations
of natural porous media, such as spherical pore bodies
that are connected by cylindrical pore throats. But there
are also models where the elements have rectangular or
triangular cross-sections and converging±diverging
characters. See Celia et al. [2] for an overview. There are
two approaches for simulating multiphase ¯ow in pore
networks: quasi-static models [1,12,25,38] update ¯uid
distribution by using a stability analysis for the menisci
based on the external capillary pressure, whereas dynamic models also account for the viscous pressure drop
in the ¯uids. For experiments with equilibrium ¯uid
distributions, such as those reported in this work, quasistatic pore-network modeling is adequate [2].
Perhaps the most challenging task when using a
quasi-static pore-network model as a predictive tool is
the calibration of its geometry. The following approaches exist:
1. By assuming a model of non-intersecting capillary
tubes for the porous medium, the pore-size distribution can be estimated from PD data [4,31]. This
method is inexpensive but does not account for the
Table 1
Properties of the three experimental multiphase systems
Parameter
GB1a
GB1b
C-109
Grain diameter (mm)
NWP
WP
qn g=cm3
qw g=cm3
c (dyn/cm)
Porosity
0:115 0:0121
Styrene
Water
0:905 0:002
0:998 0:001
33.3
0:372 0:001
0:115 0:0121
Dyed PCE
Water
1:613 0:002
0:998 0:002
36:23 0:21
0:356 0:002
0:24 0:11
Dyed PCE
Water
1:613 0:002
0:998 0:002
36:23 0:21
0:346 0:002
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
connectivity of the pore space [59] and ignores information from MI data [13]. PD and MI data permit
estimates of the statistical distribution of pore-throat
sizes and pore-body sizes, respectively [13,56]. Because of NWP entrapment, the MI data only yields
a partial distribution. Imbibition scanning paths in
the zone of NWP entrapment are needed to complete
the pore-body size distribution and analysis of correlation structure [54±56].
2. The actual three-dimensional pore geometry, which
may be obtained by serial-section technique or computed tomography, can be used to obtain the poresize distributions and the pore connectivity
[5,27,28,53].
3. The unknown pore-body and pore-throat size distributions can be determined by matching simulated
PD, MI, or SD curves with the measured ones, as
demonstrated by Fischer and Celia [6] for a network
model with coordination number Z 6. We use a
similar approach in this work.
161
Fig. 3. Schematic of implemented pore-level events: (a) piston displacement; (b) ®lm ¯ow; (c) retraction; (d) snap-o. Shaded area represents NWP.
2. Displacement of isolated WP by ®lm ¯ow: The walls of
the pore network were assumed to be covered by a
WP ®lm. A ®lm-¯ow mechanism for displacement
was implemented by using Eq. (1) as the criterion,
the same as for piston displacement.
3. Retraction during imbibition: Two criteria for porebody ®lling were investigated. The ®rst criterion simply assumes that a pore body is invaded by WP if
2c
cos h
rb
3.2. Network structure
pc 6 pr
The network structure was the same as that used by
Lowry and Miller [25] with the addition of an adjustable
coordination-number distribution and a more general
parameterization for imbibition displacement events.
The pore bodies were represented by spheres and the
pore throats by cylinders. The pore-body locations were
random in space. The three-dimensional network was
periodic in all directions except the top and bottom,
where the connections were cut and connected to a WP
reservoir and a NWP reservoir to simulate a retentioncell experiment. In order not to introduce further
parameters, we did not account for spatial correlations
among pore bodies and pore throats, although these
correlations in¯uence pc ±sw curves [10±12,50].
where rb is the pore-body radius (Lowry and Miller
[25]). The second criterion accounts for the eect of
the local ¯uid distribution on the curvature of the
meniscus:
3.3. Pore-level events
The same pore-level events as described in Lowry and
Miller [25] were implemented in the pore-network
model. In addition, a more general parameterization of
retraction and snap-o during imbibition was used.
Fig. 3 shows the following four displacement events:
1. Piston displacement during drainage: During drainage,
the menisci are positioned at the entrance of the pore
throats emanating from the pore body. The invasion
of the pore throats with NWP is called piston displacement. Invasion occurs if
pc P pp
2c
cos h
rt
1
where rt is the pore-throat radius, c the interfacial
tension between the ¯uids, and h the contact angle
measured in the WP.
pc 6 pr
2c
cos h
Znw rb
2
3
where Znw is the number of NWP-®lled pore throats
connected to the NWP-®lled pore body (Jerauld and
Salter [12]). There are also displacement rules that
account for the irregular structure of real pore spaces
by using a randomization of the displacement criteria
[1]. In the following, we will also use the terms LM
and JS retraction rule for Eqs. (2) and (3), respectively.
4. Snap-o during imbibition: Snap-o occurs if the
pressure of the WP ®lm covering the solid surface
or the pressure of the WP in the corners of the throats
is smaller than the pressure of the WP reservoir [39].
Then, the WP ¯ows into the throat and pinches o
the NWP. A simple criterion for snap-o in pore
throats is
pc 6 ps
pp
;
p
4
where p is a dimensionless number. p has been
shown, experimentally and theoretically, to be a
function of pore geometry and contact angle
[22,23,48,49]. For an air±water system in a water-wet
cylindrical tube, Li and Wardlaw [23] measured
p 1:558. But p has also been considered a ®tting
parameter, because ps of a pore throat in the network
model does generally not equal ps of the pore throat
162
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
in the corresponding natural pore space because of
the dierent curvature of the solid phase [12,25].
The ¯uid con®guration in the network responded to a
change in pc following the quasi-static approach as described in Lowry and Miller [25].
3.4. Calibration of network model
For a simulation, one has to specify the sizes of the
pore bodies and pore throats and the coordination
number Z. The coordination-number distribution was
Gaussian and characterized by the mean value hZi and
the standard deviation r Z. The cumulative distribution of Z was cut o for values larger than 0.95 and
values smaller than 0.05. The standard deviation of the
coordination number, r Z, had minimum in¯uence on
simulated pc ±sw curves and was set equal to 1. The
minimum value of Z was two, i.e., no dead-end pores
were allowed. We used the van Genuchten relation [52]
as the cumulative pore-size distribution function
n ÿm
F r ÿ 1 ahc
c
;
5
c
where h p = gqw is the capillary pressure head, and
pc 2c cos h=r. All pressures in this work are expressed
as heights of a corresponding water column. The distribution of pore-throat radii can be estimated by using
the van Genuchten parameters aPD ; mPD , and nPD for the
PD curve [4]. Likewise, the distribution of pore-body
radii can be estimated by using the van Genuchten
parameters aMI ; mMI and nMI for the MI curve. m and n,
when used as independent parameters, provide more
¯exibility in matching observed data.
We investigated various parameterizations for the
displacement rules during imbibition. For the snap-o
parameter, we considered in particular (a) p 1:558,
the measured value for a cylindrical tube; and (b)
p 3:3, the value used by Jerauld and Salter [12] for
unconsolidated porous media. But we also used other
values in order to improve the quality of the calibration.
Either Eq. (2) or Eq. (3) was implemented for retraction.
We determined the unknown parameters of the porenetwork model by minimizing the deviation between
simulated and measured PD and MI curves. We de®ned
the objective function
f a0PD ; m0PD ; n0PD ; a0MI ; m0MI ; n0MI ; hZi; p
Z pc
PD;max
1
2
wPD DswPD dpc
c
pPD;max 0
Z pc
PD;max
1
c
wMI DswMI 2 dpc ;
pMI;max 0
respect to a change in the rule. The snap-o parameter
p was not always used as an optimization parameter (so
it is thus listed in parentheses), because values of the
optimal p could yield nonphysical behavior of the
simulated scanning curves. The ®tting parameters were
determined by minimizing f using the constrained optimization algorithm IFFCO [7], which solves the minimization problem at dierent scales to avoid local
minima solutions. These local minima are inherent to be
problem but are also caused by the noisiness of the
random network geometry. Calibration was performed
for various parameterizations of the displacement rule.
Then, the rules yielding the most realistic physical behavior were chosen. The quality of the calibrations depended considerably on the choice of the IFFCO
parameters, such as the minimum stepsize and the error
tolerance. Because the ability to match experimental
data turned out to depend considerably on p , we used
the calibration runs with p as an optimization parameter to choose appropriate values for the IFFCO
parameters. The IFFCO parameter values were changed
until satisfying optimization results were achieved; the
parameters were then used for the calibration runs using
dierent ®xed p . We are aware that this procedure is
subjective, but it seems appropriate, given the problem
of an objective function with the possibility of multiple
minima.
The eect of two dierent weighting functions on the
calibration results was investigated. In one case,
wPD wMI 1, which is the standard approach for de®ning an objective function. In the other case, the
weighting function was de®ned in such a way that saturation deviations where the pc sw curve is ¯at were
weighted less than where the pc sw curve is steep, i.e., in
the ranges of irreducible water saturation and maximum
NAPL residual saturation. The shortest distance between points on the measured PD curve and the simulated PD curve weights the error at these points:
wPD pc sin /PD R pc
c
pPD;max
PD;max
0
c
=dsw is a rescaled slope of
where tan /PD d pc =pPD;max
the measured PD curve. Note that for the calculation of
the slope angle /, the pc axis was rescaled to 1. For the
MI curve, the weight was de®ned in an analogous way:
wMI pc sin /MI R pc
c
pMI;max
MI;max
0
6
where wPD and wMI are weighting functions, and DswPD
and DswMI are the dierences between measured and
simulation saturations. The geometric network parameters were subject to optimization, but not the retraction
rule, because simulation results are not continuous with
7
sin /PD dpc0
sin /MI dpc0
:
8
Both measurements and simulations provide discrete
data, and because measured and simulated capillary
pressures are generally not identical, simulated saturations for the measured pc values were obtained by linear
interpolation. Because the saturations obtained by using
the X-ray instrument display random scatter, the
weights wPD and wMI were determined from the ®tted
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
van Genuchten PD and MI relations, respectively, and
not from the discrete data constituting the measured PD
and MI curves. Otherwise, we would have seen signi®cant slopes even in the ¯at portion of the pc ±sw curves.
The network generation was not successful for arbitrary combinations of a0PD ; m0PD ; n0PD ; a0MI ; m0MI and n0MI .
For example, a very narrow pore-body size distribution
cannot be combined with a very wide pore-throat size
distribution because pore-body radii need to be larger
than the radii of all adjacent pore throats. Hence,
the bounding hyperbox may not be de®ned arbitrarily
large. The parameter space could be investigated more
wisely by optimizing for the parameter combinations
a0PD ; m0PD n0PD ; n0PD ; a0MI =a0PD ; m0MI n0MI and n0MI . The van
Genuchten parameters of the PD and MI curves, shown
in Table 2, were used as starting points for the optimization algorithm. a0PD was searched in the interval
[0.7aPD ; 1:3aPD ;m0PD n0PD in 0:1mPD nPD ;2:0mPD nPD ;
n0PD in 0:2nPD ;5:0nPD ; a0MI in 1:5a0PD ;2:5a0PD ;m0MI n0MI
in 0:1mMI nMI ;2:0mMI nMI ; n0MI in 0:2nMI ;5:0nMI ; Z
in [5,20], and p in [1.5,2.5].
4. The Wardlaw±Taylor method
4.1. Principles
Wardlaw and Taylor [56] quanti®ed snr ±snf relations
from hysteretic pc ±sw relations measured by using mercury porosimetry and proposed a method to estimate a
snr ±snf relation from PD and SD curves. The Wardlaw±
Taylor method (WTM) assumes that during SD, the
residual NWP saturation is given by
c
w
c
w
c
snr
SD p sPD p ÿ sSD p ;
9
and the funicular NWP saturation by
c
w
c
snf
SD p 1 ÿ sPD p ;
10
where swPD and swSD are the WP saturations of the PD and
SD curves, respectively. Further, they assume that the
163
residual NWP saturation obtained by PD up to pc and
back to pc 0 can be estimated by
w
c
w
c
snr pc snr
max ÿ sPD p ÿ sSD p :
11
The snr ±snf relation shown in Fig. 1(b) was constructed
using the PD and SD curves shown in Fig. 1(a) and
based upon the assumptions of the WTM.
4.2. Interpretation
Wardlaw and Taylor [56] did not provide a detailed
explanation of their method. We present here our interpretation of the WTM. The basic physical underpinning of the WTM is that for any capillary pressure pc ,
the funicular NWP during SD up to pc occupies the
same pore space as the funicular NWP during PD up to
the same pc . And with SD, additional NWP exists in the
form of residual NWP trapped during the prior MI. The
amount of residual NWP at pc is called the cumulative
c
residual NWP saturation, snr
SD p . This residual NWP
lies in that portion of the pore space that was invaded
during PD for capillary pressures ranging from pc to the
c
maximum value pPD;max
.
Fig. 4 illustrates this concept. The NWP distribution
in a porous medium is shown at three dierent capillary
pressures during SD. Prior to SD, the porous medium
c
and MI to pc 0. At p1c ,
had undergone PD to pPD;max
funicular NWP displaced some WP, but residual NWP
did not reconnect with the funicular NWP. At p2c , funicular NWP displaced more WP, and some residual
NWP reconnected with the funicular NWP. At p3c ,
funicular NWP displaced more WP, and more residual
c
NWP reconnected with the funicular NWP. Once pPD;max
was achieved during SD (not shown), all of the residual
NWP reconnected with funicular NWP.
The conceptualization presented above requires all
residual NWP (resulting from prior PD and MI) to
become reconnected during SD. This constraint requires
a ®lm-¯ow mechanism that permits drainage of a pore
Table 2
Initial estimates of van Genuchten parameters obtained by using
RETC computer code [52]
Parameter
GB1a
GB1b
C-109
aPD (1/cm)
mPD
nPD
aMI (1/cm)
mMI
nMI
swmin
snr
max
RMSE in sw
RMSLOFIT in sw
0.0260
0.5604
19.23
0.0476
0.2945
17.1
0.001
0.209
0.011
0.014
0.0228
0.01298
639.1
0.0490
0.07138
61.0
0.055
0.195
0.005
0.021
0.0585
0.01969
478.3
0.1199
0.01360
248.4
0.100
0.160
0.008
0.019
Fig. 4. NWP distribution during SD up to three dierent capillary
pressures p1c < p2c < p3c . Correspondence to NWP distribution obtained
by PD up to pc and imbibition to pc 0.
164
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
throat separating a pore body with residual NWP from
a pore body with funicular NWP. This process usually
happens on a very slow time scale. The conceptualization further requires that once residual NWP is reconnected with funicular NWP during SD, no regions of the
pore space that are inaccessible during PD are invaded
by funicular NWP. Otherwise, swPD pc ÿ swSD pc could
increase with pc , which would yield negative residual
NWP saturations in Eq. (11) when used to estimate the
snr ±snf relation.
The WTM also links the PD and SD curves through
Eq. (11) to imbibition scanning curves that yield snr pc
and the MI curve that yields snr
max . For Eq. (11) to hold,
the residual NWP that becomes connected during SD up
to an arbitrary pc must equal the residual NWP that is
generated by PD up to that pc and subsequent imbibition to zero pc . This can be accomplished by an interface between the funicular phases that advances
reversibly when switching from drainage to imbibition,
and vice versa. Consequently, one can write
nr
c
snr pc snr
max ÿ sSD p ;
12
where snr
max is the maximum residual saturation caused by
PD to the maximum capillary pressure and successive
MI. Eq. (11) follows by substituting Eq. (9) into Eq. (12),
and we can see that snr pc for the snr ±snf relation is related to the horizontal distance between the PD and SD
curves at pc 0 and pc .
Whereas the ®rst assumption that residual NWP reconnects with funicular NWP during SD is reasonable,
the second assumption that the ¯ow behavior of the
displacement front between the funicular NWP and
funicular WP is reversible is open to discussion.
5. Experimental results
Fig. 5 shows the PCE volume fraction pro®les at
water drainage and imbibition equilibrium along with
the porosity pro®le in the long column of glass beads
(GB1b system). The greater porosity in the top section
of the column ®lled with the same glass beads implies
that this section has a lower NAPL entry capillary
pressure; thus, water and PCE pressure-heads were
controlled to ensure that no PCE entered this section.
The funicular PCE-content pro®le (open circles) at
water drainage equilibrium takes the shape of an inverted
air-water drainage pro®le because PCE, more dense
than water, was introduced from the bottom of the
column. In the PCE-content pro®le (solid circles) at
water imbibition equilibrium, a residual PCE zone is
observed between )300 and )600 mm. Within this zone,
a data pair consisting of the funicular PCE fraction and
the residual PCE fraction at the same elevation constitutes a data point on the snr ±snf relation.
Fig. 5. Porosity and PCE-content pro®les, measured by using X-ray
attenuation instrument, in 90-cm vertical column of glass beads (GB1b
system).
Figs. 6(a) and (e) show the observed pc ±sw and
s ±snf relations obtained from data in Fig. 5 and data
for the elevations of the air±water and air±PCE constant-head reservoirs. The SD curve was obtained from
a pro®le measured at a later time after a prescribed set
of movements of the air±PCE and air±water constanthead reservoirs. The SD curve rejoins the PD curve in
the vicinity of sw 0:50, an indication that residual
PCE has reconnected with funicular PCE. Behavior in
hysteretic pc ÿ sw data similar to that in Figs. 6(a) and
(c) has been reported elsewhere [32,40,41,45]. The
snr ±snf relation estimated from the PD and SD curve by
using the WTM shows a physically nonreasonable region with negative snr values. Negative snr values result
because over some range swPD pc ÿ swSD pc > snr
max in
Eq. (11).
Figs. 6(b) and (f) present the observed pc ±sw and
nr nf
s ±s relations in the long-column of C-109 sand. The
SD rejoins the PD curve in the vicinity of sw 0:50. The
C-109 data is less smooth than the GB1b data, probably
due to the lower sphericity of the sand grains. Once
again, the snr ±snf relation estimated from the PD and SD
curve by using the WTM displays a negative snr .
Figs. 6(c) and (d) present some of the 160 imbibition
scanning paths obtained for the GB1b system and for
the C-109 system by using a four-step imbibition procedure. Although individual imbibition scanning paths
displayed dierent curvature, none were observed to
intersect with each other or to cross the MI curve.
By using the observed snr ±snf data, we tested the adequacy of Land's [19] curvilinear model, which was also
used by Kaluarachchi and Parker [14]:
nr
^snn
^snf
;
1 1=^snr
snf
max ÿ 1^
13
where ^s s= 1 ÿ swr , and swr is the irreducible WP
saturation. The range of behavior in the observed snr ±snf
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
(a)
(c)
165
(b)
(d)
(e)
(f)
(g)
(h)
Fig. 6. GB1b system: (a) observed pc ±sw relations (PD, MI, SD); (c) observed hysteretic pc ±sw relations; (e) observed and estimated snr ±snf relation;
(g) ®tted snr ±snf relations. C-109 system: (b) observed pc ±sw relations (PD, MI, SD); (d) observed hysteretic pc ±sw relations; (f) observed and estimated
snr ±snf relation; (h) ®tted snr ±snf relations.
166
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
relations indicated that a more ¯exible function class is
needed to describe the experimental data more accurately. Upon review of [52], which discusses the various
curvilnear and sigmoid functions used to describe pc ±sw
relations, we adapted a function of the van Genuchten
type to describe the snr ±snf relation:
nr
snr
m ÿx
max ÿ s
1 bsnf :
nr
smax
14
For m < 1 a curvilinear function is obtained, and for
m > 1 a sigmoid function is obtained. The values of swr
c w
and snr
max were determined from subsets of the p ±s data
c
showing constancy at high and low p , respectively. The
values were used in the Land equation (13) without any
further attempt to reduce the sum of squared error by
additional adjustment in swr and snr
max . The same value of
was
used
as
a
®xed
value
in
the
®t of b, m, and x to
snr
max
the van Genuchten function. Figs. 6(g) and (h) show the
results of the ®t to the GB1b and the C-109 data. For the
®t of the GB1b data, the parameters were snr
max 0:1752,
b 0:2477, m 1:1849, and x 31:0065. The root of
mean standard error (RMSE) in snr of the van Genuchten function ®t was 0.0064, and the RMSE in snr of
the Land equation ®t was 0.0085. For comparison, the
standard deviation of snr
max was 0.0027. The van Genuchten function yielded a slightly better ®t of the GB1b
data than the Land equation, but the lack-of-model ®t
error for both models was greater than the random experimental error in the GB1b data. For the ®t of the
C-109 data, the parameters were snr
max 0:1641; b
0:6752; m 1:3978; and x 15:4031. Again, the van
Genuchten function yielded a better ®t than the Land
equation. The RMSE in snr of the van Genuchten ®t was
0.0080, and the RMSE in snr of the Land equation ®t
was 0.0108. For comparison, the standard deviation of
snr
max was 0.0078. Thus for the van Genuchten function,
there was less lack-of-model ®t error than for the Land
equation. In the ®t of the van Genuchten function to the
GB1b data and the C-109 data, m > 1:0 indicates that a
sigmoid function was necessary to obtain the best ®t.
Fig. 7 presents the PD and MI curves measured in the
5-cm long retention cell, the PD curve has a rounded
shoulder for the initial portion of NAPL entry. In
contrast, the PD curve in Fig. 6(a) has an angular
shoulder at NAPL entry. Depending on the combinations of multiphase ¯uids and porous media, the length
of the retention cell can cause an averaging of a nonuniform vertical saturation, distribution and yield a
dierent result than the local saturation measured in the
long column with X-ray attenuation [24]. For the styrene±water and PCE±water systems, the smoothing of
the PD curve could not be explained by this artifact of
cell-volume averaging.
Table 2 lists the parameters obtained by ®tting the
van Genuchten pc ±sw relation to PD and MI data for the
GB1a, GB1b and C-109 systems. These parameter val-
Fig. 7. pc ±sw relation (PD, MI) of glass beads measured in 5-cm retention cell for a styrene±water system (GB1a system). Data from
Mayer and Miller [29].
ues were used as the initial estimates in the calibration of
the pore-network model. Table 2 also lists the RMSE,
which is an estimate of the true random error in the sw
data, and lists the root of mean standard lack-of-model
®t error (RMSLOFIT) as an estimate of the error caused
by the inability of the model (in this case the van Genuchten relation) to ®t the shape delineated by the data.
For each case, the lack-of-model ®t error was greater
than the random experimental error. The statistics were
calculated according to Whitmore from the ®t of the PD
data [57].
6. Pore-network modeling results
6.1. Preliminary investigations on imbibition displacement
rules
The competition between snap-o and retraction determines the imbibition displacement patterns [21].
Some forward simulations were performed to understand better the impact of the retraction rule and p on
the shape of the simulated pc ±sw curves. The pore-body
and pore-throat size distributions were, as described in
Section 3.4, obtained from the measured PD and MI
curves for the GB1a medium shown in Fig. 7. Following
Lowry and Miller [25], we chose Z 9; h 0°, and a
network size of 8000 nodes. Each of the two retraction
rules was examined in combination with three values of
p : two limiting cases, p 1 and p 10; and an intermediate case, p 1:558, the measured value for a
cylindrical tube [22].
The results of the forward simulations are shown in
Fig. 8. To aid in the interpretation of these results, k, the
mean ratio of pore body to pore throat radius, is approximated from the van Genuchten parameterization
of the PD and MI curves: k aMI =aPD . For the GB1a
system, k 1:78. In addition, a median capillary
pressure at sw 0:50 is de®ned for the simulated PD
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
167
Fig. 8. Forward simulations in a pore network with ®xed geometry in order to understand the impact of the imbibition displacement rules on
simulated PD and MI curves. (a) p 1, LM retraction rule; (b) p 1, JS retraction rule; (c) p 1:558, LM retraction rule; (d) p 1:558, JS
retraction rule; (e) p 10, LM retraction rule; (f) p 10, JS retraction rule.
c
c
curve, pPD;50
, and for the simulated MI curve, pMI;50
. The
imbibition displacement patterns produced by the porenetwork model are discussed for the following cases:
1. p k, e.g. p 1, LM and JS retraction rule: Snapo dominates imbibition whereas retraction does not
have a signi®cant impact on the MI curve. The MI
curve follows the PD curve (see Figs. 8(a) and (b)),
c
c
=pMI;50
is approxibecause ps equals pp . Thus, pPD;50
mately p for the simulation. Snap-o in a pore throat
does not necessary lead to NWP entrapment, which
only occurs if the pore throat turns out to be the last
link to funicular NWP. The probability of this being
the case is low at small sw and high at large sw .
2. p < k, e.g. p 1:558, LM and JS retraction rule:
NWP displacement is still dominated by snap-o.
c
c
=pMI;50
for the simulation is again approximately
pPD;50
p (see Figs. 8(c) and (d)). Because of the variability
of the pore-body sizes, one would not expect retraction to be so negligible for this case; i.e., one would
c
c
=pMI;50
. Two observaexpect a greater value of pPD;50
tions can explain the results. First, in the quasi-static
pore-network model formulation, snap-o is possible
everywhere in the domain where funicular NWP exists, whereas retraction only takes place at the displacement front. Second, even though individual
pore throats have a smaller volume than the pore
168
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
Table 3
c
c
The pPD;50
=pMI;50
ratio produced by the forward simulations shown in Fig. 8 using k 1:78, various values of p , and the LM and JS retraction rules
Retraction rule
p
c
c
pPD;50
=pMI;50
LM
LM
LM
JS
JS
JS
1.0
1.013
1.558
1.444
10.0
1.542
1.0
1.003
1.558
1.555
10.0
8.000
Table 4
Calibration results for the GB1a system
p
Retraction rule
Weighting
1.558
JS
Yes
Opt
JS
Yes
1.558
LM
Yes
1.558
LM
No
1.650
LM
Yes
1.700
LM
Yes
1.7500
LM
Yes
1.800
LM
Yes
3.300
LM
Yes
Opt
LM
Yes
aPD (1/cm)
mPD
nPD
aMI (1/cm)
mMI
nMI
Z
p
c
pPD;50
(cm H2 O)
c
pMI;50
(cm H2 O)
f
Intersection
0.0263
0.207
29.2
0.0517
0.170
24.3
12.4
±
39.8
25.6
0.0028
No
0.0270
0.0902
46.1
0.0530
0.180
23.7
12.9
1.72
39.6
23.1
0.0009
No
0.0271
0.130
40.6
0.0555
0.223
32.9
17.3
±
39.3
25.7
0.0033
No
0.0258
0.280
28.7
0.0559
0.252
26.4
14.9
±
39.7
25.9
0.0080
No
0.0270
0.070
69.22
0.0560
0.110
48.3
15.5
±
39.5
25.1
0.0022
No
0.0263
0.192
28.5
0.0541
0.222
22.9
14.4
±
39.9
25.2
0.0022
No
0.0268
0.100
50.9
0.0553
0.230
22.8
15.0
±
39.7
24.3
0.0013
No
0.0263
0.179
29.7
0.0545
0.209
24.3
14.1
±
39.9
24.3
0.0012
Yes
0.0252
0.173
19.5
0.0500
0.642
6.6
11.3
±
40.6
23.2
0.0026
Yes
0.0262
0.184
25.6
0.0547
0.198
25.6
12.7
1.99
39.8
23.3
0.0007
Yes
bodies they are connected to, there are many more
pore throats than pore bodies. Consequently, the
number of pore throats in which snap-o occurs outweighs the larger volume displaced from pore bodies
during NWP retraction.
3. p > k, e.g., p 10, LM retraction rule: The retraction criterion, Eq. (2), is ful®lled earlier than the
c
c
=pMI;50
approxsnap-o criterion, Eq. (4). Thus, pPD;50
imately equals k (see Fig. 8(e)). This kind of behavior
was also observed by Lowry and Miller [25], who
used p 3:3 in their simulations.
4. p 1; 1:558, and 10, JS retraction rule: During MI,
the number of NWP-®lled pore throats is greatest
at low sw and least at high sw . Because of this, the
snap-o criterion, given by Eq. (4), is usually ful®lled
earlier than the retraction criterion given by Eq. (3).
c
c
=pMI;50
is approximately equal to p , as
Thus, pPD;50
in Figs. 8(b), (d) and (f). Such behavior was also observed in the pc ±sw curves simulated by Jerauld and
Salter [12]. Under the JS retraction rule, imbibition
would be governed by k only for p > Zk.
Table 3 summarizes the simulation results. It shows
c
c
=pMI;50
is aected by p for snaphow the value of pPD;50
c
c
=pMI;50
is
o dominated imbibition. Speci®cally, pPD;50
shown to be governed by k only for the LM retraction
rule and p 10:0. Because of the dominant eect of p
in the pore-network model as formulated, one should
expect a greater lack-of-model ®t error if p is ®xed at a
c
c
=pMI;50
.
value less than pPD;50
6.2. Calibration
For an appropriate choice of p and the retraction
rule, the pore-network geometry was calibrated to the
PD and MI data measured by Mayer and Miller [29] for
styrene-water in glass beads, labeled as GB1a. Then, the
calibrated pore-network model was used to simulate
imbibition scanning curves originating from the PD
curve. For the ®nal selection of the best imbibition
displacement rule, the imbibition scanning curves were
examined for reasonable shape (curvature) and absence
of intersection. In addition, the simulated residual
NAPL blob-volume distribution obtained by MI was
compared to that measured by Mayer and Miller [29].
Fig. 9. Comparison between the weighted and unweighted optimization for the GB1a system for the LM retraction rule and p 1:558.
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
169
Fig. 10. Weighted optimization for the GB1a system (a) p 1:558, LM retraction rule; (b) p 1:558, JS retraction rule; (c) p 3:3, LM retraction
rule; (d) p optimization parameter, LM retraction rule.
The pore network had 6000 pore bodies, the contact
angle was 0°, and the pressure increment was chosen
such that 400 steps were used for simulating the PD or
MI curves. Because of the computational expense, our
networks were slightly smaller than those used by Lowry
and Miller [25], with 8000 pore bodies for which sizeindependent results were obtained. Table 4 summarizes
the calibration runs on the GB1a system.
For all simulations, the eect of weighting the objective function, given by Eq. (6), was examined. As
expected, the weighted optimization yielded the best ®t
to pc ±sw data in the range of maximum NAPL residual,
whereas the unweighted optimization yielded a better
match in the horizontal portions of the pc ±sw curves, as
shown in Fig. 9 for p 1:558 and the LM retraction
rule.
Figs. 10(a) and (b) show that p 1:558 determines
c
c
for the calibration. For p 3:3 and the
pPD;50 =pMI;50
c
c
=pMI;50
for the
LM retraction rule, k approached pPD;50
calibration (see Fig. 10(c)). The van Genuchten parameter aMI for the MI curve (see Table 2) was
0:0476 cmÿ1 . The weighted calibration with the LM rule
yielded aMI 0:0555 cmÿ1 for p 1:558 and aMI
0:0500 cmÿ1 for p 3:3. Even though p controlled
c
c
pPD;50
=pMI;50
in the p 1:558 simulation, the mean
pore-body size, as indicated by aMI , did not increase
markedly. The value of snr
max , the shape of the PD and
MI curves, and the starting values for IFFCO presumably are constraints on the calibrated pore-body size
distributions.
We gained further insight by looking at the imbibition scanning curves. For p 3:3 and the LM retraction rule, imbibition scanning curves intersected the MI
curve (see Fig. 10(c)). The results of the forward simulation of the scanning curves using p 1:558 and the
two retraction rules are shown in Figs. 10(a) and (b).
For both retraction rules, non-intersecting scanning
curves were obtained. The JS retraction rule yielded
imbibition scanning curves with little or no curvature,
because snap-o dominated. If only snap-o took place,
WP would invade a throat, which was invaded by
c
, at the earliest if
NAPL during PD up to pc pPD;max
c
c
p pPD;max =p . The small curvature results from the few
retraction events taking place. The LM rule yielded
scanning curves with more curvature, because retraction
played a more important role. We did not use the JS rule
with p 3:3 because of the ®ndings in the preliminary
investigations.
Fig. 11 presents the snr ±snf relation predicted for the
GB1a system by the pore-network model using the different values of p and the two retraction rules: an snr ±snf
relation was not measured by Mayer and Miller [29].
170
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
(a)
(b)
Fig. 11. Simulated snr ±snf relation for the GB1a system from the weighted optimization results presented in Table 4: (a) LM retraction rule; (b) JS
retraction rule.
(a)
(b)
Fig. 12. Observed and simulated cumulative blob-volume distribution for the GB1a system: (a) LM retraction rule; (b) JS retraction rule.
The predicted snr ±snf relation was relatively insensitive to
the choice of p and the retraction rule, as long as the
calibration yielded a good ®t to pc ±sw data in the range
of maximum NAPL residual saturation so that the plateau in the snr ±snf relation was matched. This constraint
required the use of the weighted optimization scheme.
The dash-dot line shows the eect of the scanning curves
intersecting with the MI curve when p is optimized.
Fig. 12 presents the blob-volume distribution estimated by the pore-network model from the model ®t to
the observed PD and MI data. The simulations match
the experimental results better quantitatively, if one uses
the JS retraction rule. Neverthless, we used the LM retraction rule in all further calibrations and simulations
because of the unreasonably steep scanning curves obtained with the JS retraction rule. The mismatch of the
blob-volume distribution may be attributed to a lack of
reality in the pore-network model.
In the calibration runs, the experimenta
www.elsevier.com/locate/advwatres
Investigation of the residual±funicular nonwetting-phase-saturation
relation
Markus Hilpert *, John F. McBride, Cass T. Miller
Center for Advanced Study of the Environment, Department of Environmental Sciences and Engineering, University of North Carolina, CB 7400,
104 Rosenau Hall, Chapel Hill, NC 27599-7400, USA
Received 1 November 1999; received in revised form 13 June 2000; accepted 13 June 2000
Abstract
The constitutive relation that describes the amount of nonwetting ¯uid phase entrapment is critical to the modeling of multiphase
¯ow in porous media, but it has received insucient attention in the literature. We studied this relation using both experimental and
modeling approaches: we used a nondestructive, X-ray monitored, long-column experiment that yielded a rich data set for two
dierent porous media; we also used a quasi-static network model to simulate the experimental data and examine mechanisms
aecting the relation. The experimental work yielded a signi®cant data set for residual nonwetting phase (NWP) saturation as a
function of maximum funicular nonwetting phase saturation. We suggest a functional form that represents the observed data sets
accurately. Network model calibration to experimental data yields acceptable model-data agreement and a clear understanding of
constraints that should be satis®ed when using such models to avoid physically unrealistic behavior. We found that the pore-throatsize characteristics and the snap-o process occurring in pore throats strongly in¯uence the manifestation of pore-body-size
characteristics during imbibition and nonwetting ¯uid phase entrapment. We examined an estimation method proposed by Wardlaw
and Taylor for the residual±funicular relation. We observed that the method yields an unrealistic relation for each porous media in
the long-column experiments, and we used network modeling to understand the criteria that ensure a realistic estimate. Ó 2000
Elsevier Science Ltd. All rights reserved.
Keywords: Capillarity; Hysteresis; NAPL; Network model; Residual; Snap-o; X-ray attenuation
1. Introduction
In groundwater systems, non-aqueous-phase liquids
(NAPLs) are often the nonwetting phase (NWP) and
therefore subject to capillary trapping [5]. Entrapped
NAPL cannot be displaced by the surrounding
groundwater ¯ow unless the physico-chemical properties of the ¯uids are altered [8]. This entrapped or residual NAPL is a source of groundwater contamination
because the NAPL dissolves into the surrounding
groundwater and is dispersed downstream by natural
groundwater ¯ow [30]. Accurate modeling of how the
residual NAPL forms is a prerequisite for a comprehensive assessment and analysis of the contamination
source, and for a realistic simulation of remediation.
*
Corresponding author.
E-mail addresses: markus_hilpert@unc.edu (M. Hilpert),
jmcbride@tbcnet.com (J.F. McBride), casey_miller@unc.edu (C.T.
Miller).
Entrapment of NAPL as a disconnected residual
phase in water-wet porous media occurs during water
imbibition after an initial water displacement (drainage)
by funicular NAPL. Fig. 1(a) shows an idealized hysteretic capillary pressure±saturation pc ±sw relation, with
the main imbibition (MI) curve and two imbibition
scanning curves originating at dierent capillary
pressures (pc ) on the primary drainage (PD) curve and
terminating at pc 0. Fig. 1(b) is a plot of the residual
NAPL saturation snr at pc 0 observed in Fig. 1(a)
versus the initial funicular NAPL saturation snf at the
start of water imbibition. In this paper, the functional
form between snr and snf is termed the ``residual±funicular NWP saturation relation'' or the snr ±snf relation;
the functional form has also been termed the ``residual±
initial NWP relation'' in the petroleum engineering
literature [37,44].
The snr ±snf relation is needed to predict residual NWP
saturations correctly in simulations of multiphase ¯ow
[51]. The snr ±snf relation is also an important submodel
in hysteretic pc ±sw relations [17,36] used in multiphase
0309-1708/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 0 9 - 1 7 0 8 ( 0 0 ) 0 0 0 3 5 - X
158
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
Nomenclature
Abbreviations
Hg
mercury
MI
main imbibition
NAPL non-aqueous phase liquid
NWP
nonwetting phase
PD
primary drainage
RMSE root of mean squared error
SD
secondary drainage
WP
wetting phase
WTM
Wardlaw±Taylor method
Variables
capillary pressure head, cm H2 O
hc
m
van Genuchten parameter in pc ±sw relation
n
van Genuchten parameter in pc ±sw relation
dimensionless critical capillary pressure for
p
snap-o
capillary pressure
pc
critical capillary pressure for piston
pp
displacement
critical capillary pressure for retraction
pr
ps
rb
rt
s
Z
a
b
c
k
m
q
x
critical capillary pressure for snap-o
pore-body radius, sphere radius
pore-throat radius, cylinder radius
saturation
coordination number
van Genuchten parameter in pc ±sw relation
van Genuchten parameter in snr ±snf relation
interfacial tension
estimate for the ratio of pore body to pore
throat radius
van Genuchten parameter in snr ±snf relation
density
van Genuchten parameter in snr ±snf relation
Subscripts and superscripts
max
maximum value of the variable
n
NWP
nf
funicular NWP
nr
residual NWP
w
WP
c
)
wr
irreducible WP (more exact: WP at pPD;max
w
50
value of the variable at s 0:5
Fig. 1. (a) Idealized hysteretic pc ±sw relations (PD, SD, MI) with scanning curves from the PD curve to zero capillary pressue; (b) plot of snr ±snf
relation obtained from MI curve and two imbibition scanning curves in Fig. 1(a).
¯ow simulations. The particle-size distribution and the
extent of the porous medium's consolidation in¯uence
the slope of the snr ±snf relation [44], the maximum rec
sidual NWP saturation, snr
max , obtained by MI to p 0,
is greater for consolidated porous media than for unconsolidated porous media [44]. Porous media with
uniform particle size, whether unconsolidated or consolidated, exhibit lower values of snr
max , presumably because greater connectivity among pores decreases the
probability of entrapment [37]. snr
max is also a function of
the ¯uid±solid properties, especially wettability; ¯uid±
¯uid properties, including viscosity ratio, interfacial
tension, and density dierence; and displacement rates
[44].
The measurement of the snr ±snf relation can be made
with the same retention-cell apparatus used to measure
the pc ±sw relation, but this approach is very time consuming: the equilibration time for each data pair can
range from hours to weeks. Rate-controlled porosimetry, in which the pressure ¯uctuations during slow-rate
mercury injection are monitored and analyzed, has also
been used to predict the snr ±snf relation [47,58]. Although good agreement with experimental data has been
obtained, this method has not received much attention
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
in the groundwater hydrology community. In practice,
the snr ±snf relation is rarely measured. Rather, functional
forms for the snr ±snf are assumed, such as various linear
[16,20,26,42,46] or curvilinear forms [14,17,19].
In this work, we evaluate various methods to estimate
the snr ±snf relation from a limited set of experimental
data, such as the PD, MI, or secondary drainage (SD)
curve. Speci®cally, we investigate
1. the use of pore-network modeling to estimate the
snr ±snf relation from the PD and MI data;
2. the applicability of a method by Wardlaw and Taylor
[56] that estimates the snr ±snf relation only from the
PD and SD curve; and
3. the adequacy of the curvilinear form in the Land
equation [19], which requires measured values of only
wr
snr
max and the irreducible WP saturation, s .
We compare the estimated snr ±snf relations to those
obtained experimentally by (1) generating funicular and
then residual NAPL distributions in the same long
vertical porous-medium column and (2) measuring each
vertical NAPL distribution nondestructively with an
X-ray attenuation instrument.
2. Experimental methods
A series of NAPL±water displacement experiments
was performed in 1-m long, 2.5-cm-diameter porousmedium columns to examine the snr ±snf and pc ±sw relations in a glass-bead porous medium and in a coarse
silica-sand porous medium. Tetrachloroethylene (PCE)
dyed red with Oil RedO at a concentration of 0.267 g/l
was used as the NAPL. The bottom and the top of the
columns were connected to constant head PCE and
water reservoirs, respectively. Using a long vertical column takes advantage of the correspondence between (1)
159
the ¯uid saturation distribution with elevation in a porous medium with respect to constant-elevation, constant-head ¯uid reservoirs and (2) a pc ±sw relation
measured in a retention cell by using step-changes in the
elevation of the constant-head ¯uid reservoirs [40,41].
This correspondence requires homogeneity of the porous medium in the long vertical column. Various vertical equilibrium saturation pro®les, corresponding to
water drainage and imbibition curves, were obtained by
successive changes of the reservoir elevations. The PCE
fraction at locations along the vertical length of the
stationary glass-bead column was determined nondestructively by measuring the attenuation of X-rays
through the column cross-section at each location. Fig. 2
shows the experimental setup. Schiegg [40,41] made
similar measurements for an air±water system to obtain
hysteretic pc ±sw relations for a coarse sand, but used a
gamma-ray source (cesium-137) and a sand column with
a 15-cm cross-section.
The X-ray attenuation instrument depicted in Fig. 2
was designed and operated in a manner similar to the
instrument described by Oak and co-authors [33,34]. A
tungsten-target X-ray tube operating at a voltage potential of 45 kV was used to produce a broad spectrum
of photon energies. This X-ray spectrum was conditioned to yield an energy spectrum with two narrow
energy bands by using a 3-mm Al plate and an aqueous
salt solution (22 wt% cesium chloride and 11 wt% samarium chloride) contained in a vial with 1-cm crosssection. The X-ray beam was collimated to a 5-mm by
5-mm cross-section by using lead collimators. A liquidnitrogen-cooled germanium detector was used with a
photon-energy resolution of 0.8 keV. To determine one
single unknown phase volume fraction (PCE volume
fraction or the solid-phase volume fraction to determine
porosity), photon count rates were integrated within the
Fig. 2. Schematic of the experiment setup showing X-ray source, germanium detection, and porous-medium column with constant-head PCE delivery reservoir and constant-head water collection reservoir to establish equilibrium water drainage pro®le (apparatuses not to scale).
160
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
lower (30±36 keV) of the two narrow energy bands. A
similar use of this X-ray instrument can be found in
[9,35]. The measurement of one unknown phase fraction
at each location required approximately 15 s (real time)
of X-ray photon counting. The observed standard deviation of three repeat measurements at each location
was 0.00057 for porosity and 0.0017 for PCE volume
fraction. The standard error of the mean at each location was thus 0.00039 for porosity and 0.00099 for PCE
volume fraction.
The glass beads for these experiments came from the
same original stock of glass beads used by Mayer and
Miller [29] in their measurement of pc ±sw relations and
blob-size distributions of radiation-polymerized residual
styrene. They used 5-cm-long retention cells to measure
the pc ±sw relations. Although a snr ±snf relation and SD
curve are not part of their data set, the measured blobvolume distributions allow an evaluation of the porenetwork model simulations. Table 1 summarizes the
properties of the glass-bead and sand multiphase systems (labeled as GB1b and C-109, respectively) in the
long column and of the glass-bead multiphase system in
the 5-cm-retention-cell (labeled as GB1a).
The long columns were chromatography columns
(ACE Glass, Vineland, NJ). An individual column was
packed with the water-inlet plunger at the bottom. The
plunger had a porous glass frit and two sets of O-ring
seals. A 1-cm layer of more ®nely grained material was
added ®rst to serve as a capillary barrier to PCE entry.
Then, the column was ®lled with the desired porous
medium by using a funnel out®tted with screens and a
tube extension to deliver the material at a constant rate.
The delivery tube was kept approximately 5 cm above
the porous medium as its level rose in the column. Then,
three screens were placed (80-mesh, 360-mesh, 24-mesh)
on top of the porous medium and TEFLON-seal
plunger was inserted to con®ne the porous medium. The
TEFLON-seal plunger served as the PCE outlet/inlet.
The column was saturated from below with de-gassed
water until all the trapped gas visible along the walls of
the column disappeared. The column was then inverted,
a water constant-head reservoir connected to the top of
the column, and a PCE constant-head reservoir connected to the bottom. Using X-ray attenuation, we
measured the solid fraction at 160 vertical locations in a
long water-saturated column. These measurements
yielded the porosity, which we then used to assess medium homogeneity. Then, various equilibrium ¯uid
distribution were established, and the resulting PCE and
water saturations were quanti®ed by X-ray attenuation
at the same 160 locations used for the porosity
measurement. We raised the PCE reservoir in order to
establish a PD pro®le and then lowered it in four steps,
waiting 24 h for water-imbibition equilibrium. In addition to 160 scanning paths, each de®ned by the four data
points from this four-step transition, the experiments
also yielded a well-de®ned snr ±snf relation for each porous medium. By using the same reservoir elevations
from the PD pro®le, we next obtained a SD curve by
drainage to irreducible water saturation, imbibition to
maximum residual PCE saturation, and drainage equilibrium.
3. Pore-network-model formulation
3.1. Overview
Pore-network models are simpli®ed representations
of natural porous media, such as spherical pore bodies
that are connected by cylindrical pore throats. But there
are also models where the elements have rectangular or
triangular cross-sections and converging±diverging
characters. See Celia et al. [2] for an overview. There are
two approaches for simulating multiphase ¯ow in pore
networks: quasi-static models [1,12,25,38] update ¯uid
distribution by using a stability analysis for the menisci
based on the external capillary pressure, whereas dynamic models also account for the viscous pressure drop
in the ¯uids. For experiments with equilibrium ¯uid
distributions, such as those reported in this work, quasistatic pore-network modeling is adequate [2].
Perhaps the most challenging task when using a
quasi-static pore-network model as a predictive tool is
the calibration of its geometry. The following approaches exist:
1. By assuming a model of non-intersecting capillary
tubes for the porous medium, the pore-size distribution can be estimated from PD data [4,31]. This
method is inexpensive but does not account for the
Table 1
Properties of the three experimental multiphase systems
Parameter
GB1a
GB1b
C-109
Grain diameter (mm)
NWP
WP
qn g=cm3
qw g=cm3
c (dyn/cm)
Porosity
0:115 0:0121
Styrene
Water
0:905 0:002
0:998 0:001
33.3
0:372 0:001
0:115 0:0121
Dyed PCE
Water
1:613 0:002
0:998 0:002
36:23 0:21
0:356 0:002
0:24 0:11
Dyed PCE
Water
1:613 0:002
0:998 0:002
36:23 0:21
0:346 0:002
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
connectivity of the pore space [59] and ignores information from MI data [13]. PD and MI data permit
estimates of the statistical distribution of pore-throat
sizes and pore-body sizes, respectively [13,56]. Because of NWP entrapment, the MI data only yields
a partial distribution. Imbibition scanning paths in
the zone of NWP entrapment are needed to complete
the pore-body size distribution and analysis of correlation structure [54±56].
2. The actual three-dimensional pore geometry, which
may be obtained by serial-section technique or computed tomography, can be used to obtain the poresize distributions and the pore connectivity
[5,27,28,53].
3. The unknown pore-body and pore-throat size distributions can be determined by matching simulated
PD, MI, or SD curves with the measured ones, as
demonstrated by Fischer and Celia [6] for a network
model with coordination number Z 6. We use a
similar approach in this work.
161
Fig. 3. Schematic of implemented pore-level events: (a) piston displacement; (b) ®lm ¯ow; (c) retraction; (d) snap-o. Shaded area represents NWP.
2. Displacement of isolated WP by ®lm ¯ow: The walls of
the pore network were assumed to be covered by a
WP ®lm. A ®lm-¯ow mechanism for displacement
was implemented by using Eq. (1) as the criterion,
the same as for piston displacement.
3. Retraction during imbibition: Two criteria for porebody ®lling were investigated. The ®rst criterion simply assumes that a pore body is invaded by WP if
2c
cos h
rb
3.2. Network structure
pc 6 pr
The network structure was the same as that used by
Lowry and Miller [25] with the addition of an adjustable
coordination-number distribution and a more general
parameterization for imbibition displacement events.
The pore bodies were represented by spheres and the
pore throats by cylinders. The pore-body locations were
random in space. The three-dimensional network was
periodic in all directions except the top and bottom,
where the connections were cut and connected to a WP
reservoir and a NWP reservoir to simulate a retentioncell experiment. In order not to introduce further
parameters, we did not account for spatial correlations
among pore bodies and pore throats, although these
correlations in¯uence pc ±sw curves [10±12,50].
where rb is the pore-body radius (Lowry and Miller
[25]). The second criterion accounts for the eect of
the local ¯uid distribution on the curvature of the
meniscus:
3.3. Pore-level events
The same pore-level events as described in Lowry and
Miller [25] were implemented in the pore-network
model. In addition, a more general parameterization of
retraction and snap-o during imbibition was used.
Fig. 3 shows the following four displacement events:
1. Piston displacement during drainage: During drainage,
the menisci are positioned at the entrance of the pore
throats emanating from the pore body. The invasion
of the pore throats with NWP is called piston displacement. Invasion occurs if
pc P pp
2c
cos h
rt
1
where rt is the pore-throat radius, c the interfacial
tension between the ¯uids, and h the contact angle
measured in the WP.
pc 6 pr
2c
cos h
Znw rb
2
3
where Znw is the number of NWP-®lled pore throats
connected to the NWP-®lled pore body (Jerauld and
Salter [12]). There are also displacement rules that
account for the irregular structure of real pore spaces
by using a randomization of the displacement criteria
[1]. In the following, we will also use the terms LM
and JS retraction rule for Eqs. (2) and (3), respectively.
4. Snap-o during imbibition: Snap-o occurs if the
pressure of the WP ®lm covering the solid surface
or the pressure of the WP in the corners of the throats
is smaller than the pressure of the WP reservoir [39].
Then, the WP ¯ows into the throat and pinches o
the NWP. A simple criterion for snap-o in pore
throats is
pc 6 ps
pp
;
p
4
where p is a dimensionless number. p has been
shown, experimentally and theoretically, to be a
function of pore geometry and contact angle
[22,23,48,49]. For an air±water system in a water-wet
cylindrical tube, Li and Wardlaw [23] measured
p 1:558. But p has also been considered a ®tting
parameter, because ps of a pore throat in the network
model does generally not equal ps of the pore throat
162
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
in the corresponding natural pore space because of
the dierent curvature of the solid phase [12,25].
The ¯uid con®guration in the network responded to a
change in pc following the quasi-static approach as described in Lowry and Miller [25].
3.4. Calibration of network model
For a simulation, one has to specify the sizes of the
pore bodies and pore throats and the coordination
number Z. The coordination-number distribution was
Gaussian and characterized by the mean value hZi and
the standard deviation r Z. The cumulative distribution of Z was cut o for values larger than 0.95 and
values smaller than 0.05. The standard deviation of the
coordination number, r Z, had minimum in¯uence on
simulated pc ±sw curves and was set equal to 1. The
minimum value of Z was two, i.e., no dead-end pores
were allowed. We used the van Genuchten relation [52]
as the cumulative pore-size distribution function
n ÿm
F r ÿ 1 ahc
c
;
5
c
where h p = gqw is the capillary pressure head, and
pc 2c cos h=r. All pressures in this work are expressed
as heights of a corresponding water column. The distribution of pore-throat radii can be estimated by using
the van Genuchten parameters aPD ; mPD , and nPD for the
PD curve [4]. Likewise, the distribution of pore-body
radii can be estimated by using the van Genuchten
parameters aMI ; mMI and nMI for the MI curve. m and n,
when used as independent parameters, provide more
¯exibility in matching observed data.
We investigated various parameterizations for the
displacement rules during imbibition. For the snap-o
parameter, we considered in particular (a) p 1:558,
the measured value for a cylindrical tube; and (b)
p 3:3, the value used by Jerauld and Salter [12] for
unconsolidated porous media. But we also used other
values in order to improve the quality of the calibration.
Either Eq. (2) or Eq. (3) was implemented for retraction.
We determined the unknown parameters of the porenetwork model by minimizing the deviation between
simulated and measured PD and MI curves. We de®ned
the objective function
f a0PD ; m0PD ; n0PD ; a0MI ; m0MI ; n0MI ; hZi; p
Z pc
PD;max
1
2
wPD DswPD dpc
c
pPD;max 0
Z pc
PD;max
1
c
wMI DswMI 2 dpc ;
pMI;max 0
respect to a change in the rule. The snap-o parameter
p was not always used as an optimization parameter (so
it is thus listed in parentheses), because values of the
optimal p could yield nonphysical behavior of the
simulated scanning curves. The ®tting parameters were
determined by minimizing f using the constrained optimization algorithm IFFCO [7], which solves the minimization problem at dierent scales to avoid local
minima solutions. These local minima are inherent to be
problem but are also caused by the noisiness of the
random network geometry. Calibration was performed
for various parameterizations of the displacement rule.
Then, the rules yielding the most realistic physical behavior were chosen. The quality of the calibrations depended considerably on the choice of the IFFCO
parameters, such as the minimum stepsize and the error
tolerance. Because the ability to match experimental
data turned out to depend considerably on p , we used
the calibration runs with p as an optimization parameter to choose appropriate values for the IFFCO
parameters. The IFFCO parameter values were changed
until satisfying optimization results were achieved; the
parameters were then used for the calibration runs using
dierent ®xed p . We are aware that this procedure is
subjective, but it seems appropriate, given the problem
of an objective function with the possibility of multiple
minima.
The eect of two dierent weighting functions on the
calibration results was investigated. In one case,
wPD wMI 1, which is the standard approach for de®ning an objective function. In the other case, the
weighting function was de®ned in such a way that saturation deviations where the pc sw curve is ¯at were
weighted less than where the pc sw curve is steep, i.e., in
the ranges of irreducible water saturation and maximum
NAPL residual saturation. The shortest distance between points on the measured PD curve and the simulated PD curve weights the error at these points:
wPD pc sin /PD R pc
c
pPD;max
PD;max
0
c
=dsw is a rescaled slope of
where tan /PD d pc =pPD;max
the measured PD curve. Note that for the calculation of
the slope angle /, the pc axis was rescaled to 1. For the
MI curve, the weight was de®ned in an analogous way:
wMI pc sin /MI R pc
c
pMI;max
MI;max
0
6
where wPD and wMI are weighting functions, and DswPD
and DswMI are the dierences between measured and
simulation saturations. The geometric network parameters were subject to optimization, but not the retraction
rule, because simulation results are not continuous with
7
sin /PD dpc0
sin /MI dpc0
:
8
Both measurements and simulations provide discrete
data, and because measured and simulated capillary
pressures are generally not identical, simulated saturations for the measured pc values were obtained by linear
interpolation. Because the saturations obtained by using
the X-ray instrument display random scatter, the
weights wPD and wMI were determined from the ®tted
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
van Genuchten PD and MI relations, respectively, and
not from the discrete data constituting the measured PD
and MI curves. Otherwise, we would have seen signi®cant slopes even in the ¯at portion of the pc ±sw curves.
The network generation was not successful for arbitrary combinations of a0PD ; m0PD ; n0PD ; a0MI ; m0MI and n0MI .
For example, a very narrow pore-body size distribution
cannot be combined with a very wide pore-throat size
distribution because pore-body radii need to be larger
than the radii of all adjacent pore throats. Hence,
the bounding hyperbox may not be de®ned arbitrarily
large. The parameter space could be investigated more
wisely by optimizing for the parameter combinations
a0PD ; m0PD n0PD ; n0PD ; a0MI =a0PD ; m0MI n0MI and n0MI . The van
Genuchten parameters of the PD and MI curves, shown
in Table 2, were used as starting points for the optimization algorithm. a0PD was searched in the interval
[0.7aPD ; 1:3aPD ;m0PD n0PD in 0:1mPD nPD ;2:0mPD nPD ;
n0PD in 0:2nPD ;5:0nPD ; a0MI in 1:5a0PD ;2:5a0PD ;m0MI n0MI
in 0:1mMI nMI ;2:0mMI nMI ; n0MI in 0:2nMI ;5:0nMI ; Z
in [5,20], and p in [1.5,2.5].
4. The Wardlaw±Taylor method
4.1. Principles
Wardlaw and Taylor [56] quanti®ed snr ±snf relations
from hysteretic pc ±sw relations measured by using mercury porosimetry and proposed a method to estimate a
snr ±snf relation from PD and SD curves. The Wardlaw±
Taylor method (WTM) assumes that during SD, the
residual NWP saturation is given by
c
w
c
w
c
snr
SD p sPD p ÿ sSD p ;
9
and the funicular NWP saturation by
c
w
c
snf
SD p 1 ÿ sPD p ;
10
where swPD and swSD are the WP saturations of the PD and
SD curves, respectively. Further, they assume that the
163
residual NWP saturation obtained by PD up to pc and
back to pc 0 can be estimated by
w
c
w
c
snr pc snr
max ÿ sPD p ÿ sSD p :
11
The snr ±snf relation shown in Fig. 1(b) was constructed
using the PD and SD curves shown in Fig. 1(a) and
based upon the assumptions of the WTM.
4.2. Interpretation
Wardlaw and Taylor [56] did not provide a detailed
explanation of their method. We present here our interpretation of the WTM. The basic physical underpinning of the WTM is that for any capillary pressure pc ,
the funicular NWP during SD up to pc occupies the
same pore space as the funicular NWP during PD up to
the same pc . And with SD, additional NWP exists in the
form of residual NWP trapped during the prior MI. The
amount of residual NWP at pc is called the cumulative
c
residual NWP saturation, snr
SD p . This residual NWP
lies in that portion of the pore space that was invaded
during PD for capillary pressures ranging from pc to the
c
maximum value pPD;max
.
Fig. 4 illustrates this concept. The NWP distribution
in a porous medium is shown at three dierent capillary
pressures during SD. Prior to SD, the porous medium
c
and MI to pc 0. At p1c ,
had undergone PD to pPD;max
funicular NWP displaced some WP, but residual NWP
did not reconnect with the funicular NWP. At p2c , funicular NWP displaced more WP, and some residual
NWP reconnected with the funicular NWP. At p3c ,
funicular NWP displaced more WP, and more residual
c
NWP reconnected with the funicular NWP. Once pPD;max
was achieved during SD (not shown), all of the residual
NWP reconnected with funicular NWP.
The conceptualization presented above requires all
residual NWP (resulting from prior PD and MI) to
become reconnected during SD. This constraint requires
a ®lm-¯ow mechanism that permits drainage of a pore
Table 2
Initial estimates of van Genuchten parameters obtained by using
RETC computer code [52]
Parameter
GB1a
GB1b
C-109
aPD (1/cm)
mPD
nPD
aMI (1/cm)
mMI
nMI
swmin
snr
max
RMSE in sw
RMSLOFIT in sw
0.0260
0.5604
19.23
0.0476
0.2945
17.1
0.001
0.209
0.011
0.014
0.0228
0.01298
639.1
0.0490
0.07138
61.0
0.055
0.195
0.005
0.021
0.0585
0.01969
478.3
0.1199
0.01360
248.4
0.100
0.160
0.008
0.019
Fig. 4. NWP distribution during SD up to three dierent capillary
pressures p1c < p2c < p3c . Correspondence to NWP distribution obtained
by PD up to pc and imbibition to pc 0.
164
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
throat separating a pore body with residual NWP from
a pore body with funicular NWP. This process usually
happens on a very slow time scale. The conceptualization further requires that once residual NWP is reconnected with funicular NWP during SD, no regions of the
pore space that are inaccessible during PD are invaded
by funicular NWP. Otherwise, swPD pc ÿ swSD pc could
increase with pc , which would yield negative residual
NWP saturations in Eq. (11) when used to estimate the
snr ±snf relation.
The WTM also links the PD and SD curves through
Eq. (11) to imbibition scanning curves that yield snr pc
and the MI curve that yields snr
max . For Eq. (11) to hold,
the residual NWP that becomes connected during SD up
to an arbitrary pc must equal the residual NWP that is
generated by PD up to that pc and subsequent imbibition to zero pc . This can be accomplished by an interface between the funicular phases that advances
reversibly when switching from drainage to imbibition,
and vice versa. Consequently, one can write
nr
c
snr pc snr
max ÿ sSD p ;
12
where snr
max is the maximum residual saturation caused by
PD to the maximum capillary pressure and successive
MI. Eq. (11) follows by substituting Eq. (9) into Eq. (12),
and we can see that snr pc for the snr ±snf relation is related to the horizontal distance between the PD and SD
curves at pc 0 and pc .
Whereas the ®rst assumption that residual NWP reconnects with funicular NWP during SD is reasonable,
the second assumption that the ¯ow behavior of the
displacement front between the funicular NWP and
funicular WP is reversible is open to discussion.
5. Experimental results
Fig. 5 shows the PCE volume fraction pro®les at
water drainage and imbibition equilibrium along with
the porosity pro®le in the long column of glass beads
(GB1b system). The greater porosity in the top section
of the column ®lled with the same glass beads implies
that this section has a lower NAPL entry capillary
pressure; thus, water and PCE pressure-heads were
controlled to ensure that no PCE entered this section.
The funicular PCE-content pro®le (open circles) at
water drainage equilibrium takes the shape of an inverted
air-water drainage pro®le because PCE, more dense
than water, was introduced from the bottom of the
column. In the PCE-content pro®le (solid circles) at
water imbibition equilibrium, a residual PCE zone is
observed between )300 and )600 mm. Within this zone,
a data pair consisting of the funicular PCE fraction and
the residual PCE fraction at the same elevation constitutes a data point on the snr ±snf relation.
Fig. 5. Porosity and PCE-content pro®les, measured by using X-ray
attenuation instrument, in 90-cm vertical column of glass beads (GB1b
system).
Figs. 6(a) and (e) show the observed pc ±sw and
s ±snf relations obtained from data in Fig. 5 and data
for the elevations of the air±water and air±PCE constant-head reservoirs. The SD curve was obtained from
a pro®le measured at a later time after a prescribed set
of movements of the air±PCE and air±water constanthead reservoirs. The SD curve rejoins the PD curve in
the vicinity of sw 0:50, an indication that residual
PCE has reconnected with funicular PCE. Behavior in
hysteretic pc ÿ sw data similar to that in Figs. 6(a) and
(c) has been reported elsewhere [32,40,41,45]. The
snr ±snf relation estimated from the PD and SD curve by
using the WTM shows a physically nonreasonable region with negative snr values. Negative snr values result
because over some range swPD pc ÿ swSD pc > snr
max in
Eq. (11).
Figs. 6(b) and (f) present the observed pc ±sw and
nr nf
s ±s relations in the long-column of C-109 sand. The
SD rejoins the PD curve in the vicinity of sw 0:50. The
C-109 data is less smooth than the GB1b data, probably
due to the lower sphericity of the sand grains. Once
again, the snr ±snf relation estimated from the PD and SD
curve by using the WTM displays a negative snr .
Figs. 6(c) and (d) present some of the 160 imbibition
scanning paths obtained for the GB1b system and for
the C-109 system by using a four-step imbibition procedure. Although individual imbibition scanning paths
displayed dierent curvature, none were observed to
intersect with each other or to cross the MI curve.
By using the observed snr ±snf data, we tested the adequacy of Land's [19] curvilinear model, which was also
used by Kaluarachchi and Parker [14]:
nr
^snn
^snf
;
1 1=^snr
snf
max ÿ 1^
13
where ^s s= 1 ÿ swr , and swr is the irreducible WP
saturation. The range of behavior in the observed snr ±snf
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
(a)
(c)
165
(b)
(d)
(e)
(f)
(g)
(h)
Fig. 6. GB1b system: (a) observed pc ±sw relations (PD, MI, SD); (c) observed hysteretic pc ±sw relations; (e) observed and estimated snr ±snf relation;
(g) ®tted snr ±snf relations. C-109 system: (b) observed pc ±sw relations (PD, MI, SD); (d) observed hysteretic pc ±sw relations; (f) observed and estimated
snr ±snf relation; (h) ®tted snr ±snf relations.
166
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
relations indicated that a more ¯exible function class is
needed to describe the experimental data more accurately. Upon review of [52], which discusses the various
curvilnear and sigmoid functions used to describe pc ±sw
relations, we adapted a function of the van Genuchten
type to describe the snr ±snf relation:
nr
snr
m ÿx
max ÿ s
1 bsnf :
nr
smax
14
For m < 1 a curvilinear function is obtained, and for
m > 1 a sigmoid function is obtained. The values of swr
c w
and snr
max were determined from subsets of the p ±s data
c
showing constancy at high and low p , respectively. The
values were used in the Land equation (13) without any
further attempt to reduce the sum of squared error by
additional adjustment in swr and snr
max . The same value of
was
used
as
a
®xed
value
in
the
®t of b, m, and x to
snr
max
the van Genuchten function. Figs. 6(g) and (h) show the
results of the ®t to the GB1b and the C-109 data. For the
®t of the GB1b data, the parameters were snr
max 0:1752,
b 0:2477, m 1:1849, and x 31:0065. The root of
mean standard error (RMSE) in snr of the van Genuchten function ®t was 0.0064, and the RMSE in snr of
the Land equation ®t was 0.0085. For comparison, the
standard deviation of snr
max was 0.0027. The van Genuchten function yielded a slightly better ®t of the GB1b
data than the Land equation, but the lack-of-model ®t
error for both models was greater than the random experimental error in the GB1b data. For the ®t of the
C-109 data, the parameters were snr
max 0:1641; b
0:6752; m 1:3978; and x 15:4031. Again, the van
Genuchten function yielded a better ®t than the Land
equation. The RMSE in snr of the van Genuchten ®t was
0.0080, and the RMSE in snr of the Land equation ®t
was 0.0108. For comparison, the standard deviation of
snr
max was 0.0078. Thus for the van Genuchten function,
there was less lack-of-model ®t error than for the Land
equation. In the ®t of the van Genuchten function to the
GB1b data and the C-109 data, m > 1:0 indicates that a
sigmoid function was necessary to obtain the best ®t.
Fig. 7 presents the PD and MI curves measured in the
5-cm long retention cell, the PD curve has a rounded
shoulder for the initial portion of NAPL entry. In
contrast, the PD curve in Fig. 6(a) has an angular
shoulder at NAPL entry. Depending on the combinations of multiphase ¯uids and porous media, the length
of the retention cell can cause an averaging of a nonuniform vertical saturation, distribution and yield a
dierent result than the local saturation measured in the
long column with X-ray attenuation [24]. For the styrene±water and PCE±water systems, the smoothing of
the PD curve could not be explained by this artifact of
cell-volume averaging.
Table 2 lists the parameters obtained by ®tting the
van Genuchten pc ±sw relation to PD and MI data for the
GB1a, GB1b and C-109 systems. These parameter val-
Fig. 7. pc ±sw relation (PD, MI) of glass beads measured in 5-cm retention cell for a styrene±water system (GB1a system). Data from
Mayer and Miller [29].
ues were used as the initial estimates in the calibration of
the pore-network model. Table 2 also lists the RMSE,
which is an estimate of the true random error in the sw
data, and lists the root of mean standard lack-of-model
®t error (RMSLOFIT) as an estimate of the error caused
by the inability of the model (in this case the van Genuchten relation) to ®t the shape delineated by the data.
For each case, the lack-of-model ®t error was greater
than the random experimental error. The statistics were
calculated according to Whitmore from the ®t of the PD
data [57].
6. Pore-network modeling results
6.1. Preliminary investigations on imbibition displacement
rules
The competition between snap-o and retraction determines the imbibition displacement patterns [21].
Some forward simulations were performed to understand better the impact of the retraction rule and p on
the shape of the simulated pc ±sw curves. The pore-body
and pore-throat size distributions were, as described in
Section 3.4, obtained from the measured PD and MI
curves for the GB1a medium shown in Fig. 7. Following
Lowry and Miller [25], we chose Z 9; h 0°, and a
network size of 8000 nodes. Each of the two retraction
rules was examined in combination with three values of
p : two limiting cases, p 1 and p 10; and an intermediate case, p 1:558, the measured value for a
cylindrical tube [22].
The results of the forward simulations are shown in
Fig. 8. To aid in the interpretation of these results, k, the
mean ratio of pore body to pore throat radius, is approximated from the van Genuchten parameterization
of the PD and MI curves: k aMI =aPD . For the GB1a
system, k 1:78. In addition, a median capillary
pressure at sw 0:50 is de®ned for the simulated PD
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
167
Fig. 8. Forward simulations in a pore network with ®xed geometry in order to understand the impact of the imbibition displacement rules on
simulated PD and MI curves. (a) p 1, LM retraction rule; (b) p 1, JS retraction rule; (c) p 1:558, LM retraction rule; (d) p 1:558, JS
retraction rule; (e) p 10, LM retraction rule; (f) p 10, JS retraction rule.
c
c
curve, pPD;50
, and for the simulated MI curve, pMI;50
. The
imbibition displacement patterns produced by the porenetwork model are discussed for the following cases:
1. p k, e.g. p 1, LM and JS retraction rule: Snapo dominates imbibition whereas retraction does not
have a signi®cant impact on the MI curve. The MI
curve follows the PD curve (see Figs. 8(a) and (b)),
c
c
=pMI;50
is approxibecause ps equals pp . Thus, pPD;50
mately p for the simulation. Snap-o in a pore throat
does not necessary lead to NWP entrapment, which
only occurs if the pore throat turns out to be the last
link to funicular NWP. The probability of this being
the case is low at small sw and high at large sw .
2. p < k, e.g. p 1:558, LM and JS retraction rule:
NWP displacement is still dominated by snap-o.
c
c
=pMI;50
for the simulation is again approximately
pPD;50
p (see Figs. 8(c) and (d)). Because of the variability
of the pore-body sizes, one would not expect retraction to be so negligible for this case; i.e., one would
c
c
=pMI;50
. Two observaexpect a greater value of pPD;50
tions can explain the results. First, in the quasi-static
pore-network model formulation, snap-o is possible
everywhere in the domain where funicular NWP exists, whereas retraction only takes place at the displacement front. Second, even though individual
pore throats have a smaller volume than the pore
168
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
Table 3
c
c
The pPD;50
=pMI;50
ratio produced by the forward simulations shown in Fig. 8 using k 1:78, various values of p , and the LM and JS retraction rules
Retraction rule
p
c
c
pPD;50
=pMI;50
LM
LM
LM
JS
JS
JS
1.0
1.013
1.558
1.444
10.0
1.542
1.0
1.003
1.558
1.555
10.0
8.000
Table 4
Calibration results for the GB1a system
p
Retraction rule
Weighting
1.558
JS
Yes
Opt
JS
Yes
1.558
LM
Yes
1.558
LM
No
1.650
LM
Yes
1.700
LM
Yes
1.7500
LM
Yes
1.800
LM
Yes
3.300
LM
Yes
Opt
LM
Yes
aPD (1/cm)
mPD
nPD
aMI (1/cm)
mMI
nMI
Z
p
c
pPD;50
(cm H2 O)
c
pMI;50
(cm H2 O)
f
Intersection
0.0263
0.207
29.2
0.0517
0.170
24.3
12.4
±
39.8
25.6
0.0028
No
0.0270
0.0902
46.1
0.0530
0.180
23.7
12.9
1.72
39.6
23.1
0.0009
No
0.0271
0.130
40.6
0.0555
0.223
32.9
17.3
±
39.3
25.7
0.0033
No
0.0258
0.280
28.7
0.0559
0.252
26.4
14.9
±
39.7
25.9
0.0080
No
0.0270
0.070
69.22
0.0560
0.110
48.3
15.5
±
39.5
25.1
0.0022
No
0.0263
0.192
28.5
0.0541
0.222
22.9
14.4
±
39.9
25.2
0.0022
No
0.0268
0.100
50.9
0.0553
0.230
22.8
15.0
±
39.7
24.3
0.0013
No
0.0263
0.179
29.7
0.0545
0.209
24.3
14.1
±
39.9
24.3
0.0012
Yes
0.0252
0.173
19.5
0.0500
0.642
6.6
11.3
±
40.6
23.2
0.0026
Yes
0.0262
0.184
25.6
0.0547
0.198
25.6
12.7
1.99
39.8
23.3
0.0007
Yes
bodies they are connected to, there are many more
pore throats than pore bodies. Consequently, the
number of pore throats in which snap-o occurs outweighs the larger volume displaced from pore bodies
during NWP retraction.
3. p > k, e.g., p 10, LM retraction rule: The retraction criterion, Eq. (2), is ful®lled earlier than the
c
c
=pMI;50
approxsnap-o criterion, Eq. (4). Thus, pPD;50
imately equals k (see Fig. 8(e)). This kind of behavior
was also observed by Lowry and Miller [25], who
used p 3:3 in their simulations.
4. p 1; 1:558, and 10, JS retraction rule: During MI,
the number of NWP-®lled pore throats is greatest
at low sw and least at high sw . Because of this, the
snap-o criterion, given by Eq. (4), is usually ful®lled
earlier than the retraction criterion given by Eq. (3).
c
c
=pMI;50
is approximately equal to p , as
Thus, pPD;50
in Figs. 8(b), (d) and (f). Such behavior was also observed in the pc ±sw curves simulated by Jerauld and
Salter [12]. Under the JS retraction rule, imbibition
would be governed by k only for p > Zk.
Table 3 summarizes the simulation results. It shows
c
c
=pMI;50
is aected by p for snaphow the value of pPD;50
c
c
=pMI;50
is
o dominated imbibition. Speci®cally, pPD;50
shown to be governed by k only for the LM retraction
rule and p 10:0. Because of the dominant eect of p
in the pore-network model as formulated, one should
expect a greater lack-of-model ®t error if p is ®xed at a
c
c
=pMI;50
.
value less than pPD;50
6.2. Calibration
For an appropriate choice of p and the retraction
rule, the pore-network geometry was calibrated to the
PD and MI data measured by Mayer and Miller [29] for
styrene-water in glass beads, labeled as GB1a. Then, the
calibrated pore-network model was used to simulate
imbibition scanning curves originating from the PD
curve. For the ®nal selection of the best imbibition
displacement rule, the imbibition scanning curves were
examined for reasonable shape (curvature) and absence
of intersection. In addition, the simulated residual
NAPL blob-volume distribution obtained by MI was
compared to that measured by Mayer and Miller [29].
Fig. 9. Comparison between the weighted and unweighted optimization for the GB1a system for the LM retraction rule and p 1:558.
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
169
Fig. 10. Weighted optimization for the GB1a system (a) p 1:558, LM retraction rule; (b) p 1:558, JS retraction rule; (c) p 3:3, LM retraction
rule; (d) p optimization parameter, LM retraction rule.
The pore network had 6000 pore bodies, the contact
angle was 0°, and the pressure increment was chosen
such that 400 steps were used for simulating the PD or
MI curves. Because of the computational expense, our
networks were slightly smaller than those used by Lowry
and Miller [25], with 8000 pore bodies for which sizeindependent results were obtained. Table 4 summarizes
the calibration runs on the GB1a system.
For all simulations, the eect of weighting the objective function, given by Eq. (6), was examined. As
expected, the weighted optimization yielded the best ®t
to pc ±sw data in the range of maximum NAPL residual,
whereas the unweighted optimization yielded a better
match in the horizontal portions of the pc ±sw curves, as
shown in Fig. 9 for p 1:558 and the LM retraction
rule.
Figs. 10(a) and (b) show that p 1:558 determines
c
c
for the calibration. For p 3:3 and the
pPD;50 =pMI;50
c
c
=pMI;50
for the
LM retraction rule, k approached pPD;50
calibration (see Fig. 10(c)). The van Genuchten parameter aMI for the MI curve (see Table 2) was
0:0476 cmÿ1 . The weighted calibration with the LM rule
yielded aMI 0:0555 cmÿ1 for p 1:558 and aMI
0:0500 cmÿ1 for p 3:3. Even though p controlled
c
c
pPD;50
=pMI;50
in the p 1:558 simulation, the mean
pore-body size, as indicated by aMI , did not increase
markedly. The value of snr
max , the shape of the PD and
MI curves, and the starting values for IFFCO presumably are constraints on the calibrated pore-body size
distributions.
We gained further insight by looking at the imbibition scanning curves. For p 3:3 and the LM retraction rule, imbibition scanning curves intersected the MI
curve (see Fig. 10(c)). The results of the forward simulation of the scanning curves using p 1:558 and the
two retraction rules are shown in Figs. 10(a) and (b).
For both retraction rules, non-intersecting scanning
curves were obtained. The JS retraction rule yielded
imbibition scanning curves with little or no curvature,
because snap-o dominated. If only snap-o took place,
WP would invade a throat, which was invaded by
c
, at the earliest if
NAPL during PD up to pc pPD;max
c
c
p pPD;max =p . The small curvature results from the few
retraction events taking place. The LM rule yielded
scanning curves with more curvature, because retraction
played a more important role. We did not use the JS rule
with p 3:3 because of the ®ndings in the preliminary
investigations.
Fig. 11 presents the snr ±snf relation predicted for the
GB1a system by the pore-network model using the different values of p and the two retraction rules: an snr ±snf
relation was not measured by Mayer and Miller [29].
170
M. Hilpert et al. / Advances in Water Resources 24 (2001) 157±177
(a)
(b)
Fig. 11. Simulated snr ±snf relation for the GB1a system from the weighted optimization results presented in Table 4: (a) LM retraction rule; (b) JS
retraction rule.
(a)
(b)
Fig. 12. Observed and simulated cumulative blob-volume distribution for the GB1a system: (a) LM retraction rule; (b) JS retraction rule.
The predicted snr ±snf relation was relatively insensitive to
the choice of p and the retraction rule, as long as the
calibration yielded a good ®t to pc ±sw data in the range
of maximum NAPL residual saturation so that the plateau in the snr ±snf relation was matched. This constraint
required the use of the weighted optimization scheme.
The dash-dot line shows the eect of the scanning curves
intersecting with the MI curve when p is optimized.
Fig. 12 presents the blob-volume distribution estimated by the pore-network model from the model ®t to
the observed PD and MI data. The simulations match
the experimental results better quantitatively, if one uses
the JS retraction rule. Neverthless, we used the LM retraction rule in all further calibrations and simulations
because of the unreasonably steep scanning curves obtained with the JS retraction rule. The mismatch of the
blob-volume distribution may be attributed to a lack of
reality in the pore-network model.
In the calibration runs, the experimenta