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-throat-size characteristics and the snap-o process occurring in pore throats strongly in¯uence the manifestation of pore-body-pore-throat-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 proper-ties of the ¯uids are altered [8]. This entrapped or re-sidual 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 compre-hensive assessment and analysis of the contamination source, and for a realistic simulation of remediation.
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 hys-teretic 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 atpc0. Fig. 1(b) is a plot of the residual
NAPL saturation snr at pc0 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±funi-cular 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].
Thesnr
±snfrelation 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
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*
Corresponding author.
E-mail addresses: markus_hilpert@unc.edu (M. Hilpert), jmcbride@tbcnet.com (J.F. McBride), casey_miller@unc.edu (C.T. Miller).
¯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 re-sidual NWP saturation,snr
max, obtained by MI top c0, is greater for consolidated porous media than for un-consolidated porous media [44]. Porous media with uniform particle size, whether unconsolidated or con-solidated, exhibit lower values of snr
max, presumably be-cause greater connectivity among pores decreases the probability of entrapment [37].snr
maxis 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 thesnr
±snf relation can be made with the same retention-cell apparatus used to measure the pc
±sw relation, but this approach is very time con-suming: the equilibration time for each data pair can range from hours to weeks. Rate-controlled porosim-etry, 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]. Al-though good agreement with experimental data has been obtained, this method has not received much attention
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
hc capillary pressure head, cm H2O
m van Genuchten parameter inpc
±sw relation
n van Genuchten parameter inpc
±sw relation
p dimensionless critical capillary pressure for
snap-o
pc capillary pressure
pp critical capillary pressure for piston
displacement
pr critical capillary pressure for retraction
ps critical capillary pressure for snap-o
rb pore-body radius, sphere radius
rt pore-throat radius, cylinder radius
s saturation
Z coordination number
a van Genuchten parameter inpc
±swrelation
b van Genuchten parameter insnr
±snf relation
c interfacial tension
k estimate for the ratio of pore body to pore
throat radius
m van Genuchten parameter insnr
±snf relation
q density
x van Genuchten parameter insnr
±snf relation
Subscripts and superscripts
max maximum value of the variable
n NWP
nf funicular NWP
nr residual NWP
w WP
wr irreducible WP (more exact: WP atpc
PD;max)
50 value of the variable at sw0:5
Fig. 1. (a) Idealized hystereticpc
in the groundwater hydrology community. In practice, thesnr
±snfrelation is rarely measured. Rather, functional forms for thesnr
±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
snr
maxand the irreducible WP saturation,s wr.
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
porous-medium columns to examine thesnr
±snf andpc±sw rela-tions 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 col-umn takes advantage of the correspondence between (1)
the ¯uid saturation distribution with elevation in a po-rous medium with respect to constant-elevation,
con-stant-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 po-rous medium in the long vertical column. Various ver-tical 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 nonde-structively 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 po-tential of 45 kV was used to produce a broad spectrum of photon energies. This X-ray spectrum was condi-tioned 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% sa-marium chloride) contained in a vial with 1-cm cross-section. The X-ray beam was collimated to a 5-mm by 5-mm cross-section by using lead collimators. A liquid-nitrogen-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
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 de-viation 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 loca-tion 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 ofpc
±sw relations and blob-size distributions of radiation-polymerized residual styrene. They used 5-cm-long retention cells to measure thepc
±sw relations. Although a snr±snf relation and SD curve are not part of their data set, the measured blob-volume distributions allow an evaluation of the pore-network model simulations. Table 1 summarizes the properties of the glass-bead and sand multiphase sys-tems (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 con-nected 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 me-dium 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 addi-tion 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 po-rous 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 equi-librium.
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 dy-namic models also account for the viscous pressure drop in the ¯uids. For experiments with equilibrium ¯uid distributions, such as those reported in this work, quasi-static 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 ap-proaches exist:
1. By assuming a model of non-intersecting capillary tubes for the porous medium, the pore-size distri-bution 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) 0:1150:0121 0:1150:0121 0:240:11
NWP Styrene Dyed PCE Dyed PCE
WP Water Water Water
qn g=cm3 0:9050:002 1:6130:002 1:6130:002
qw g=cm3 0:9980:001 0:9980:002 0:9980:002
c(dyn/cm) 33.3 36:230:21 36:230:21
connectivity of the pore space [59] and ignores infor-mation 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]. Be-cause 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 corre-lation structure [54±56].
2. The actual three-dimensional pore geometry, which may be obtained by serial-section technique or com-puted tomography, can be used to obtain the
pore-size distributions and the pore connectivity
[5,27,28,53].
3. The unknown pore-body and pore-throat size distri-butions 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 Z6. We use a
similar approach in this work.
3.2. Network structure
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 retention-cell 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¯uencepc
±sw curves [10±12,50].
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 dis-placement. Invasion occurs if
pcPpp 2c
rt
cosh 1
where rt is the pore-throat radius, c the interfacial
tension between the ¯uids, and h the contact angle
measured in the WP.
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
pore-body ®lling were investigated. The ®rst criterion sim-ply assumes that a pore body is invaded by WP if
pc6p r
2c rb
cosh 2
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:
pc6pr 2c Znwrb
cosh 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), respec-tively.
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
pc6ps
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
p1:558. But p has also been considered a ®tting
parameter, becausepsof a pore throat in the network model does generally not equalps of the pore throat
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 de-scribed 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 hZiand
the standard deviation r Z. The cumulative
distribu-tion 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
F r ÿ 1 ahcnÿm; 5
wherehc pc= gq
wis the capillary pressure head, and
pc2ccosh=r. All pressures in this work are expressed as heights of a corresponding water column. The dis-tribution of pore-throat radii can be estimated by using the van Genuchten parametersaPD;mPD, andnPDfor the PD curve [4]. Likewise, the distribution of pore-body radii can be estimated by using the van Genuchten parametersaMI;mMI andnMIfor the MI curve.mandn, 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) p1:558, the measured value for a cylindrical tube; and (b)
p3:3, 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 pore-network model by minimizing the deviation between simulated and measured PD and MI curves. We de®ned the objective function
where wPD and wMI are weighting functions, and DswPD and Dsw
MI are the dierences between measured and
simulation saturations. The geometric network param-eters were subject to optimization, but not the retraction rule, because simulation results are not continuous with
respect to a change in the rule. The snap-o parameter
pwas 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 fusing the constrained
opti-mization algorithm IFFCO [7], which solves the mini-mization 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 be-havior were chosen. The quality of the calibrations de-pended 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 onp, we used
the calibration runs with p as an optimization
param-eter 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,
wPDwMI1, 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 sat-uration deviations where the pc sw curve is ¯at were weighted less than where thepc swcurve is steep, i.e., in the ranges of irreducible water saturation and maximum NAPL residual saturation. The shortest distance be-tween points on the measured PD curve and the simu-lated PD curve weights the error at these points:
wPD pc sin/PD
w is a rescaled slope of
the measured PD curve. Note that for the calculation of the slope angle/, thepc axis was rescaled to 1. For the MI curve, the weight was de®ned in an analogous way:
wMI pc sin/MI
Both measurements and simulations provide discrete data, and because measured and simulated capillary pressures are generally not identical, simulated satura-tions for the measuredpcvalues were obtained by linear interpolation. Because the saturations obtained by using the X-ray instrument display random scatter, the
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 thepc
±sw curves. The network generation was not successful for arbi-trary combinations of a0
PD;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
a0
PD;m0PDnPD0 ;n0PD;a0MI=a0PD;m0MIn0MI andn0MI. The van Genuchten parameters of the PD and MI curves, shown in Table 2, were used as starting points for the optim-ization algorithm. a0
PD was searched in the interval
[0.7aPD; 1:3aPD;m0PDn0PD in 0:1mPDnPD;2:0mPDnPD;
Wardlaw and Taylor [56] quanti®ed snr
±snf relations from hystereticpc
±sw relations measured by using mer-cury 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
snr
and the funicular NWP saturation by
snfSD p
SDare the WP saturations of the PD and SD curves, respectively. Further, they assume that the
residual NWP saturation obtained by PD up topc and
back topc0 can be estimated by
±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 in-terpretation of the WTM. The basic physical under-pinning of the WTM is that for any capillary pressurepc,
the funicular NWP during SD up to pc occupies the
same pore space as the funicular NWP during PD up to the samepc. 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
residual NWP saturation, snr
SD pc. This residual NWP lies in that portion of the pore space that was invaded during PD for capillary pressures ranging frompcto the
maximum valuepc
PD;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
had undergone PD to pc
PD;max and MI to p
c0. At pc 1, funicular NWP displaced some WP, but residual NWP
did not reconnect with the funicular NWP. At pc
2, funi-cular NWP displaced more WP, and some residual
NWP reconnected with the funicular NWP. At pc
3, funicular NWP displaced more WP, and more residual
NWP reconnected with the funicular NWP. Oncepc
PD;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) 0.0260 0.0228 0.0585
mPD 0.5604 0.01298 0.01969
nPD 19.23 639.1 478.3
aMI(1/cm) 0.0476 0.0490 0.1199
mMI 0.2945 0.07138 0.01360
nMI 17.1 61.0 248.4
RMSLOFIT insw 0.014 0.021 0.019
Fig. 4. NWP distribution during SD up to three dierent capillary pressurespc
1<p c 2<p
c
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 conceptualiza-tion further requires that once residual NWP is recon-nected with funicular NWP during SD, no regions of the pore space that are inaccessible during PD are invaded by funicular NWP. Otherwise,sw
PD pc ÿswSD pccould 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 yieldsnr pc and the MI curve that yieldssnr
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
imbib-ition to zeropc. This can be accomplished by an inter-face between the funicular phases that advances reversibly when switching from drainage to imbibition, and vice versa. Consequently, one can write
snr pc snr
maxis 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 thatsnr pcfor the snr
±snf relation is re-lated to the horizontal distance between the PD and SD curves atpc0 andpc.
Whereas the ®rst assumption that residual NWP re-connects 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 consti-tutes a data point on thesnr
±snf relation.
Figs. 6(a) and (e) show the observed pc
±sw and
snr
±snf relations obtained from data in Fig. 5 and data for the elevations of the air±water and air±PCE con-stant-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 constant-head reservoirs. The SD curve rejoins the PD curve in the vicinity of sw0:50, an indication that residual PCE has reconnected with funicular PCE. Behavior in hysteretic pcÿsw 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 re-gion with negativesnr values. Negative snr values result
because over some range sw
PD p
Figs. 6(b) and (f) present the observed pc
±sw and
snr
±snf relations in the long-column of C-109 sand. The SD rejoins the PD curve in the vicinity ofsw0:50. The C-109 data is less smooth than the GB1b data, probably due to the lower sphericity of the sand grains. Once again, thesnr
±snfrelation estimated from the PD and SD curve by using the WTM displays a negativesnr.
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 pro-cedure. 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 observedsnr
±snf data, we tested the ad-equacy of Land's [19] curvilinear model, which was also used by Kaluarachchi and Parker [14]:
^ saturation. The range of behavior in the observedsnr
±snf
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Fig. 6. GB1b system: (a) observedpc
±swrelations (PD, MI, SD); (c) observed hystereticpc±swrelations; (e) observed and estimatedsnr±snfrelation; (g) ®ttedsnr
±snfrelations. C-109 system: (b) observedpc±swrelations (PD, MI, SD); (d) observed hystereticpc±swrelations; (f) observed and estimated
snr
relations indicated that a more ¯exible function class is needed to describe the experimental data more accu-rately. Upon review of [52], which discusses the various curvilnear and sigmoid functions used to describepc
±sw relations, we adapted a function of the van Genuchten type to describe thesnr
±snf relation:
For m<1 a curvilinear function is obtained, and for
m>1 a sigmoid function is obtained. The values ofswr andsnr
maxwere determined from subsets of thepc±swdata showing constancy at high and lowpc, respectively. The values were used in the Land equation (13) without any further attempt to reduce the sum of squared error by additional adjustment inswrandsnrmax. The same value of
snr
maxwas used as a ®xed value in the ®t ofb,m, andxto 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 weresnr
max0:1752,
b0:2477, m1:1849, and x31:0065. The root of
mean standard error (RMSE) in snr of the van
Ge-nuchten function ®t was 0.0064, and the RMSE insnrof the Land equation ®t was 0.0085. For comparison, the standard deviation of snr
max was 0.0027. The van
Ge-nuchten 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 ex-perimental error in the GB1b data. For the ®t of the C-109 data, the parameters were snrmax0:1641;b 0:6752;m1:3978; and x15:4031. Again, the van Genuchten function yielded a better ®t than the Land
equation. The RMSE insnrof 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
maxwas 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 combina-tions of multiphase ¯uids and porous media, the length of the retention cell can cause an averaging of a non-uniform vertical saturation, distribution and yield a dierent result than the local saturation measured in the long column with X-ray attenuation [24]. For the sty-rene±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 Genuchtenpc
±swrelation to PD and MI data for the GB1a, GB1b and C-109 systems. These parameter
val-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 Ge-nuchten 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 de-termines the imbibition displacement patterns [21]. Some forward simulations were performed to under-stand better the impact of the retraction rule andp 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;h0°, 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, p1 and p10; and an
in-termediate case, p1:558, 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 ap-proximated from the van Genuchten parameterization
of the PD and MI curves: kaMI=aPD. For the GB1a
system, k1:78. In addition, a median capillary
pressure at sw0:50 is de®ned for the simulated PD
Fig. 7. pc
curve,pc
PD;50, and for the simulated MI curve,pMIc ;50. The imbibition displacement patterns produced by the pore-network model are discussed for the following cases:
1. pk,e.g.p1, LM and JS retraction rule:
Snap-o 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)), because ps equals pp. Thus, pcPD;50=p
c
MI;50 is approxi-matelypfor 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 smallswand high at largesw.
2. p<k, e.g. p1:558, LM and JS retraction rule:
NWP displacement is still dominated by snap-o.
pc PD;50=p
c
MI;50for the simulation is again approximately
p (see Figs. 8(c) and (d)). Because of the variability
of the pore-body sizes, one would not expect retrac-tion to be so negligible for this case; i.e., one would expect a greater value of pc
PD;50=p c
MI;50. Two observa-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 ex-ists, whereas retraction only takes place at the dis-placement front. Second, even though individual pore throats have a smaller volume than the pore
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)p1, LM retraction rule; (b)p1, JS retraction rule; (c)p1:558, LM retraction rule; (d)p1:558, JS
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 out-weighs the larger volume displaced from pore bodies during NWP retraction.
3. p>k, e.g.,p10, LM retraction rule: The
retrac-tion criterion, Eq. (2), is ful®lled earlier than the snap-o criterion, Eq. (4). Thus,pc
PD;50=p c
MI;50 approx-imately equalsk(see Fig. 8(e)). This kind of behavior was also observed by Lowry and Miller [25], who usedp3:3 in their simulations.
4. p1;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). Thus, pc
PD;50=pcMI;50 is approximately equal to p, as in Figs. 8(b), (d) and (f). Such behavior was also ob-served in thepc
±sw curves simulated by Jerauld and Salter [12]. Under the JS retraction rule, imbibition would be governed bykonly forp>Zk.
Table 3 summarizes the simulation results. It shows how the value ofpc
PD;50=p c
MI;50is aected byp for
snap-o dominated imbibition. Speci®cally, pc
PD;50=p c MI;50 is
shown to be governed byk only for the LM retraction
rule andp10:0. Because of the dominant eect ofp
in the pore-network model as formulated, one should expect a greater lack-of-model ®t error ifpis ®xed at a
value less thanpc
PD;50=pcMI;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].
Table 3 Thepc
PD;50=pcMI;50ratio produced by the forward simulations shown in Fig. 8 usingk1:78, various values ofp, and the LM and JS retraction rules
Retraction rule LM LM LM JS JS JS
p 1.0 1.558 10.0 1.0 1.558 10.0
pc PD;50=p
c
MI;50 1.013 1.444 1.542 1.003 1.555 8.000
Table 4
Calibration results for the GB1a system
p 1.558 Opt 1.558 1.558 1.650 1.700 1.7500 1.800 3.300 Opt
Retraction rule JS JS LM LM LM LM LM LM LM LM
Weighting Yes Yes Yes No Yes Yes Yes Yes Yes Yes
aPD(1/cm) 0.0263 0.0270 0.0271 0.0258 0.0270 0.0263 0.0268 0.0263 0.0252 0.0262
mPD 0.207 0.0902 0.130 0.280 0.070 0.192 0.100 0.179 0.173 0.184
nPD 29.2 46.1 40.6 28.7 69.22 28.5 50.9 29.7 19.5 25.6
aMI(1/cm) 0.0517 0.0530 0.0555 0.0559 0.0560 0.0541 0.0553 0.0545 0.0500 0.0547
mMI 0.170 0.180 0.223 0.252 0.110 0.222 0.230 0.209 0.642 0.198
nMI 24.3 23.7 32.9 26.4 48.3 22.9 22.8 24.3 6.6 25.6
Z 12.4 12.9 17.3 14.9 15.5 14.4 15.0 14.1 11.3 12.7
p ± 1.72 ± ± ± ± ± ± ± 1.99
pc
PD;50(cm H2O) 39.8 39.6 39.3 39.7 39.5 39.9 39.7 39.9 40.6 39.8
pc
MI;50(cm H2O) 25.6 23.1 25.7 25.9 25.1 25.2 24.3 24.3 23.2 23.3
f 0.0028 0.0009 0.0033 0.0080 0.0022 0.0022 0.0013 0.0012 0.0026 0.0007
Intersection No No No No No No No Yes Yes Yes
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 size-independent results were obtained. Table 4 summarizes the calibration runs on the GB1a system.
For all simulations, the eect of weighting the ob-jective function, given by Eq. (6), was examined. As expected, the weighted optimization yielded the best ®t topc
±sw data in the range of maximum NAPL residual,
whereas the unweighted optimization yielded a better match in the horizontal portions of thepc±swcurves, as
shown in Fig. 9 for p1:558 and the LM retraction
rule.
Figs. 10(a) and (b) show that p1:558 determines
pc PD;50=p
c
MI;50 for the calibration. For p3:3 and the
LM retraction rule, k approached pc
PD;50=p c
MI;50 for the calibration (see Fig. 10(c)). The van Genuchten
pa-rameter aMI for the MI curve (see Table 2) was
0:0476 cmÿ1. The weighted calibration with the LM rule yielded aMI0:0555 cmÿ1 for p1:558 and aMI 0:0500 cmÿ1 for p3:3. Even though p controlled
pc
PD;50=pcMI;50 in the p1:558 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 presum-ably are constraints on the calibrated pore-body size distributions.
We gained further insight by looking at the imbib-ition scanning curves. Forp3:3 and the LM
retrac-tion rule, imbibiretrac-tion scanning curves intersected the MI curve (see Fig. 10(c)). The results of the forward simu-lation of the scanning curves using p1:558 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
NAPL during PD up to pcpc
PD;max, at the earliest if
pcpc
PD;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 withp3:3 because of the ®ndings in the preliminary
investigations.
Fig. 11 presents thesnr
±snf relation predicted for the GB1a system by the pore-network model using the dif-ferent values ofpand the two retraction rules: ansnr
±snf relation was not measured by Mayer and Miller [29].
Fig. 10. Weighted optimization for the GB1a system (a)p1:558, LM retraction rule; (b)p1:558, JS retraction rule; (c)p3:3, LM retraction
The predictedsnr
±snfrelation was relatively insensitive to the choice ofp and the retraction rule, as long as the
calibration yielded a good ®t topc
±sw data in the range of maximum NAPL residual saturation so that the pla-teau in thesnr
±snfrelation 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 whenp is optimized.
Fig. 12 presents the blob-volume distribution esti-mated 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 re-traction rule in all further calibrations and simulations because of the unreasonably steep scanning curves ob-tained 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 experimental pc
±sw data could not be matched up to arbitrary accuracy; the error caused by the lack-of-®t of the pore-network model was larger than the random measurement error, as was the straight ®t of the van Genuchten function reported in Table 2. The measurement error was less than 0.01, as
quanti®ed by the standard error in the mean water saturation observed among three replications. If the lack-of-®t error is zero, the fvalue resulting from ran-dom measurement error should not be greater than
0:012 0:012, equal to 0.0002. The lack-of-®t error can be seen in the failure of the pore-network model to match pc
±sw data when the slope of the data changes
from a vertical to a horizontal direction. The lack-of-®t is attributable to conceptual limitations of the pore-network model used, as well as the function used to describe the pore-size distribution. For example, the pore-network model produces a large jump in water saturation (in the range of 0.1±0.2) at the NAPL-entry pressure during PD if one uses large networks. De-creasing the pressure steps did not change this behavior, which is consistent with percolation theory: in the case of large networks, there is a sharply de®ned threshold radius for which a NAPL cluster, spanning the entire network, is generated [43]. The large saturation jump at the NAPL-entry pressure (or the angular shoulder of the PD curve) would become more gradual if spatial cor-relations among pore bodies and pore throats were ac-counted for [50] or if a network model with alternative geometry was used for the pore bodies and pore throats.
(a) (b)
Fig. 12. Observed and simulated cumulative blob-volume distribution for the GB1a system: (a) LM retraction rule; (b) JS retraction rule.
(a) (b)
Fig. 11. Simulatedsnr
For the weighted optimization using the LM retrac-tion rule, the calibraretrac-tion could be improved further (as indicated by thefvalue) by increasing the ®xed value of
p in small increments, as long as the imbibition
scan-ning curves do not intersect the MI curve. In these
cal-ibration runs, the initial ®xed p was 1.65 and the
increments were 0.05. The calibration withp1:8 was the ®rst calibration in this series to yield imbibition scanning curves that interested the MI curve. Ifp was included in the weighted objective function as an opti-mization parameter, the smallest value offwas obtained
at p1:99. Fig. 10(d) shows that a good agreement
was achieved between simulated and experimental PD and MI curves though once again the imbibition scan-ning curves intersected the MI curve.
Although considering p an adjustable parameter is
inconsistent with the pore-network geometry, there is good reason to do so: a model cyclindrical throat does not generally possess the same ps as a throat in a real
porous medium. A calibratedp that deviates from the
pappropriate to the pore-network model indicates that
the imbibition displacement patterns in natural porous media are dierent than in the pore-network model. Our
inability to match snr
max and p c PD;50=p
c
MI;50 of the GB1a system withp1:558 is evidence that a cylindrical tube
geometry does not represent well the pore throats of the
GB1a porous medium. An adjustable p assimilates the
lack of correspondence between the real porous medium and the pore-network model geometry. The criterion of non-intersecting imbibition scanning curves places an upper bound onp.
In all the remaining calibrations we (1) used the LM retraction rule to obtain a reasonable curvature for the scanning curves; (2) used the weighted objective function because snr
max can be matched more accurately; and (3)
increased p until the imbibition scanning curves
inter-sected with the MI curve since the prediction of the
snr
±snf relation relies also on a good match of
pc
Having established the LM retraction rule's
robust-ness, and a maximum value forprequiring the
imbib-ition scanning curves not to intersect the MI curve, the pore-network model was calibrated to the PD and MI curves in the GB1b and C-109 systems. Tables 5 and 6 show the calibration results for the GB1b and C-109 systems, respectively. Next, the snr
±snf relation was de-termined by simulating imbibition scanning curves originating from the PD curve. The snr
±snf relation was also derived from the calibrated PD and simulated SD curve by using the WTM.
Figs. 13(a), (c) and (e) present observed and simu-lated hysteretic pc
±sw relations in the GB1b system,
along with observed and simulatedsnr
±snf relations. The calibration yielded p2:1, which is in the vicinity of
the observedpc PD;50=p
c
MI;502:0. The calibrated PD and MI curves match the experimental ones very well. The observed imbibition scanning curves show more curva-ture than the simulated ones causing an underprediction of residual saturations in thesnr
±snfrelation. The WTM, which had failed for the experimental data, also failed for the network modeling data.
Figs. 13(b), (d) and (f) present observed and simu-lated hysteretic pc±sw relations in the C-109 system, along with observed and simulatedsnr±snf relations. The
calibration yielded p1:85 which equals
approxi-mately the observed pc
PD;50=p c
MI;501:84. The match between simulated and experimental PD and MI curves is only acceptable. The same holds true for snr
max. The observed imbibition scanning curves show more curva-ture than the simulated ones. Again, the WTM failed both for the experimental and simulated data.
The calibration for the GB1b system yielded a larger
value for p and a better match of the observed
pc
PD;50=pMIc ;50 than the calibrations for the GB1a and C-109 systems. This result stems from the critical
in-Table 5
Optimization parameters for the ®nal simulations of the GB1b multiphase system
p 2.0 2.1 2.15 Opt
aPD(1/cm) 0.0232 0.02279 0.02304 0.0230
mPD 0.0637 0.0780 0.0606 0.0489
nPD 75.3 75.2 78.0 75.8
aMI(1/cm) 0.0531 0.0570 0.0521 0.0535
mMI 0.185 0.0850 0.127 0.145
nMI 38.9 58.5 48.3 37.6
f 0.0017 0.0009 0.0009 0.0004
Intersection No No Yes Yes
Table 6
Optimization parameters for the ®nal simulations of the C-109 multiphase system
p 1.75 1.85 1.9 Opt
aPD(1/cm) 0.0625 0.0619 0.0623 0.0620
mPD 0.0295 0.0329 0.0266 0.0225
nPD 105.5 99.1 104.2 91.5
aMI(1/cm) 0.135 0.136 0.135 0.130
mMI 0.102 0.101 0.0849 0.130
nMI 47.6 43.9 51.4 34.5
f 0.0028 0.0023 0.0019 0.0012
teraction betweenpandkin controlling the occurrence of intersecting imbibition scanning curves. The
cali-brated k value for the GB1b system was larger than
those for the other systems. This largerk value permit-ted a larger range of possible values of p and possible
reduction in the fvalue without producing intersecting scanning curves.
6.4. Analysis of the Wardlaw±Taylor method
The WTM using the calibrated PD curve and lated SD curve yielded a good prediction of the simu-latedsnr
±snf relation in the pore-network modeling that
usedp1:558, both for the LM and the JS retraction rule. The WTM failed in the pore-network modeling that used p3:3 and the LM retraction rule. To
un-derstand these results, simulations were performed using a small pore network (200 nodes) with a small coordi-nation number (Z 6), which allowed us to visualize the distribution of funicular and residual NWP. The pore-body and pore-throat size distributions were determined by using Eq. (5) from the measured PD and MI curves of the GB1a system. The simulations were run with the LM retraction rule and p1:3; p1:558 was not
used because the pore-level characteristics produced by smallp values become more visible with decreasingp.
(a) (b)
(c) (d)
(e) (f)
Fig. 13. GB1b system: (a) observed and calibratedpc
±swrelations; (c) simulated hystereticpc±swrelations; (e) observed and simulatedsnr±snfrelation. C-109 system: (b) observed and calibratedpc
The calibrated PD curve was independent ofpbecause
snap-o does not occur during PD.
One constraint of the WTM is that snr obtained for
any imbibition scanning curve is less thansnr
maxobtained from the MI curve, i.e., all imbibition scanning curves are bounded by the MI curve. The phenomenon of im-bibition scanning curves intersecting with the MI curve was investigated because such behavior implies thatsnris not necessarily a monotonic function of the initial snf. The investigation was performed by comparing the re-sults of similar sequences inpcforp3:3 andp1:3.
PD up to a certainpc was simulated to obtain the ¯uid
distribution shown in Fig. 14(c) for p1:3 and
Fig. 15(c) for p3:3. Then, the simulation was
con-tinued with imbibition to pc0 to obtain the residual saturation state shown in Figs. 14(d) and 15(d) for
p1:3 andp3:3, respectively. Figs. 14(a) and 15(a)
show the ¯uid distribution upon PD to pc
PD;max and subsequent MI forp1:3 and p3:3, respectively.
Comparing Figs. 14(a)±(d) from the p1:3
simu-lation, one observes that the form and number of NWP ganglia obtained by the imbibition scanning curve are such that a smaller volume of residual NWP was created than by MI, implying that thesnr
±snfrelation would be a
monotonic function of snf. Comparing Figs. 15(a)±(d)
from thep3:3 simulation, one observes that the form and number of NWP ganglia obtained by the imbibition scanning curve are such that a greater volume of
resid-Fig. 14. The state of the pore-network p1:3within Scanning Loop I: (a) PD toapc
PD;maxand MI top
c0; (b) PD topc
PD;max, MI top
c0, then SD up topc. Within Scanning Loop II: (c) PD up to the samepcas in Loop I; (d) PD up to the samepcas in Loop I, and imbibition topc0. WTM works because total NWP in (b) minus total NWP in (c), then subtracted from residual NWP in (a) is equal to residual NWP in (d).
ual NWP was created than by MI, implying that the
snr
±snf relation would not be a monotonic function of
snf.
The dierent behavior can be explained by consider-ing the reversibility of the displacement front between funicular phases during drainage and imbibition. Re-versibility of the displacement front is expected if im-bibition is dominated by snap-o in pore throats, the same pore throats that govern the displacement front during PD. Reversibility of the displacement front is not expected if retraction from pore-body radii dominates imbibition, because the pore bodies display a dierent spatial structure than the pore throats. Thus, imbibition in thep1:3 simulation was dominated by snap-o in
pore throats, and imbibition in the p3:3 simulation
was dominated by retraction in pore bodies. If the dis-placement front moves approximately reversibly (ideally
for p1 and thus no retraction), one can expect the
®nal residual NWP saturation, snr, to be a monotonic
function of the initial funicular NWP saturation,snf.
Another constraint for the WTM is that
sw
PD pc ÿswSD pc is equal to snrmax at pc0, and the quantity decreases with increasing pc. To investigate conditions that violate this constraint, the ¯uid distri-bution for PD up to a certainpcforp1:3 (Fig. 14(c)) and for p3:3 (Fig. 15(c)) was compared to the ¯u-id distribution for SD up to the same pc for p1:3
(Fig. 14(b)) andp3:3 (Fig. 15(b)).
Comparing the individual ®gures in Fig. 14 for
p1:3, one observes that:
1. the total NWP volume is greater in (b) than in (c); 2. the dierence between the total NWP volume in (b)
and (c) is the residual NWP volume (black); and 3. the sum of the residual NWP volume in (b) and (d)
yields the residual NWP volume in (a). Thus, in the p1:3 simulation, (a)
)[(b))(c)](d),
which conforms to the aassumptions of the WTM. Comparing the individual ®gures in Fig. 15 forp3:3, one observes that (a))[(b))(c)] is not equal to (d). In
addition, (a))[(b))(c)] yields a negative value. This
result is obtained because the residual NWP in (a) at the start of SD has made more pore space accessible to funicular NWP as residual NWP reconnected with funicular NWP. This is best observed in the central re-gion in the pore network devoid of NWP as shown in Fig. 15(c). The residual NWP ganglion in that location in Fig. 15(a), once reconnected in Fig. 15(b), permits adjacent pores to ®ll with NWP. In this case, the exact NWP front advance during drainage is not repeated during SD.
For the case ofp1:3 and front movement during
PD, MI and SD controlled by pore throats, it appears that a NWP ganglion connects to funicular NWP during SD at that throat from which it was disconnected during MI, this throat is the largest of the pore throats that surround the NWP ganglion. Such a connection at the
largest pore throat makes no additional pore space ac-cessible to funicular NWP during SD. For the case of
p3:3, and front movement controlled by pore bodies
during MI, it appears that reconnection to a NWP ganglion during SD can lead to more pore space being made accessible to the advancing NWP front. In sum-mary, the above investigation suggests that the WTM succeeds for porous media systems where snap-o dominates the imbibition process: in this case, the front between funicular NWP and funicular water moves re-versibly upon changing from PD to imbibition and from imbibition to SD.
Ifp were to be increased incrementally between 1.3
and 3.3, one would expect the eventual outright failure
of the WTM at some value of p as indicated by a
cal-culated value of snr<0. As observed in the calibration and forward simulations of the GB1b and C-109 sys-tems, failure of the WTM is not necessarily brought about by the occurrence of intersecting imbibition scanning curves. Our pore-network-model investigation
of the WTM and its success in estimating the snr
±snf relation of a sandstone from Wardlaw and Taylor's [56] experiments suggest that snap-o dominated during Hg withdrawal in those experiments. But spatial correla-tions among pore throats and pore bodies in consoli-dated porous media such as sandstones could also favor the reversible displacement front between funicular phases required for the WTM's success.
7. Discussion and conclusions
7.1. Experimental aspects
1. Our technique of using a long column for measuringpc
±swrelations is a dierent method than that used by Dane et al. [3]. They also used a long column but measured ¯uid saturations at a few locations in the vertical center of the column after each of many pres-sure increments. Thus, the Dane et al. approach in analogous to existing retention-cell procedures except that a radiation attenuation instrument is used to measure a local saturation in the retention cell, instead of using a gravimetric or volumetric mass balance to calculate an average saturation in the retention cell. The Dane et al. approach and the retention-cell technique would require a separate scanning loop for each discrete point on thesnr
±snfrelation, a drainage scanning path to get snf, and an imbibition scanning path to get snr. Moreover, each such scanning loop would not be in-dependent from others, making the measurement prone to cumulative errors. In contrast, each column section for which a ¯uid saturation can be measured in our long-vertical-column technique yields a discrete point on thesnr
2. The procedure using the long column is limited to ¯uid phases and porous media that yield a wide range of equilibrium ¯uid saturations along the vertical length of the column (i.e., great density dierence between the two ¯uid phases, large mean pore size, and narrow pore-size distribution).
3. In the glass-bead porous medium, the PD curve measured in the retention cell displayed a dierent cur-vature in the zone of NAPL entry than the PD curve measured in the long-column setup. We hypothesize that the dierent behavior is caused by dierent initial and boundary conditions in the retention cell and the long-column setup. Stainless steel is less strongly wet by water than glass is, and as a consequence, styrene in the stainless-steel retention cell probably occupied the pore space along the sidewalls as well as the cross-sectional boundary during PD. In contrast, the water-wet glass walls in the long column prevented early invasion of the pore space along the sidewalls during PD.
The observations of Wardlaw and Taylor [56] support this hypothesis. They measured Hg injection and with-drawal curves on two similar cylindrical limestone sam-ples; Hg injection corresponds to drainage, and Hg withdrawal corresponds to imbibition. One sample was coated with an epoxy ®lm except for one end, and the other sample was untreated, so that Hg could invade anywhere. According to the authors, the coated sample best approximates an in®nite porous medium. The coated sample yielded an injection curve with an angular shoulder at the Hg-entry pressure, whereas the untreated sample yielded an injection curve with a rounded shoul-der. The withdrawal curves were similar for both samples. 4. The measured imbibition scanning curves for the GB1b and C-109 systems did not intersect with the measured MI curve (or each other). Following our network modeling results, this indicates that snap-o dominated imbibition. We point out that major as-sumptions for this conclusion are a ®lm-¯ow mechanism for the WP and the absence of spatial correlations among pore bodies and pore throats.
5. The WTM used with the long-column data may have failed because of the dierent macroscopic initial and boundary conditions during PD and SD ± the same underlying reason that dierent PD curves were ob-tained by Wardlaw and Taylor [56] during Hg injection in coated and uncoated samples of limestone. Recall that in the pore-network modeling, NAPL invasion only occurred from the cross-sectional boundary during PD and during SD. In the long-column experiments, funi-cular NAPL during PD invaded initially only along the cross-section of the column because no NAPL occupied the pore space along the column wall. During SD, the residual NAPL along the column wall allowed invasion along the column walls as well as the cross-section, once reconnected with the invading funicular NAPL (because of the greater porosity along the wall boundary, perhaps
at lower capillary pressure than the NAPL entry pressure for the porous media). More surface area for the NAPL-water displacement front increased the prob-ability of encountering a displacement front during SD that was dierent, however slightly, than during PD.
Thus, in addition to the requirement that snap-o of NWP dominates imbibition in a multiphase porous media system, success of the WTM requires that PD and SD curves be measured in an apparatus that yields the same initial and boundary conditions for NWP invasion during PD and SD. This might be a retention-cell but not the long vertical column of this work.
6. For both porous media systems investigated, the
Land equation yields good predictions of the snr
±snf
relation and only requires knowledge of snr
max and s wr.
The accuracy achieved appears satisfactory for the purpose of closing models for hystereticpc
±sw relations, although details of thesnr
±snfrelation, such as a sigmoid shape, cannot be described. We suggest the use of a van Genuchten type function, if the data are available for appropriate ®tting.
7.2. Network-modeling aspects
1. Snap-o dominated imbibition causes a reversible front between the funicular phases during imbibition and drainage, thereby ensuring non-intersecting im-bibition scanning curves. This displacement can then be described as a bond-percolation process [21]. 2. The WTM succeeded for snap-o dominated
imbib-ition but not for the network models calibrated to the experimental porous media systems of this work. We believe that those results might change if spatial correlations between pore bodies and pore throats are taken into account.
3. The problem of simulated imbibition scanning curves intersecting with the MI curve has not been acknowl-edged explicitly in the literature. In many cases, pub-lished work [25] only looked at PD and MI curves. In the work that did explore scanning curves, slight in-tersection of the imbibition curves with the MI curve has been tolerated [15,38], but they have not investi-gated the degree to which intersection depends on how the network-model parameters are selected. We recommend the simulation of scanning curves as a standard test for pore network models. We believe that spatial correlations between pore bodies and pore throats diminish the likelihood of intersecting imbibition curves.
