capillary pressure-saturation relations in porous media

Journal of Hydrology

ELSEVIER Journal of Hydrology 178 (1996) 33-53

Capillary pressure-saturation relations in porous media including the effect of wettability

Wil l i am A. Moseleya; Vi jay K . D h i r b'* aRadian Corporation, 300 North Sepulveda Boulevard, El Segundo, CA 90245, USA

bMechanical, Aerospace, and Nuclear Engineering Department, University of California, Los Angeles, CA 90024, USA

Received 6 August 1993; revision accepted 2 June 1995

Abstract

In this work a functional relationship between capillary pressure and wetting phase saturation has been developed. Both the imbibition and drainage of a wetting phase liquid into a porous medium consisting of clean unconsolidated silica particles have been considered. In all, two bead packings, with average particle diameters of 120 #m and 360 #In, and three liquids--water, ethanol, and refrigerant-113--were studied. It was found that the Leverett function worked very well in describing the relation between capillary pressure and saturation for each liquid using both packings; however, it did not provide adequate correlation for all three liquids. The functional relationship presented here considers the dependence of capillary pressure upon the contact angle of the fluid-solid interface, and also conveys how this dependence varies with saturation. The inclusion of the contact angle dependence provides extension of the correlation to any wetting phase liquid of uniform properties.

1. Introduction.

Numerous investigations f rom a broad range of technical fields have been under- taken to describe and model accurately the phenomena of immiscible, multiphase flow through a porous medium. An important part of these investigations is the determination of the relationships between permeability, saturation and capillary pressure. In this work, emphasis is placed upon the relationship between saturation and capillary pressure.

In the field of heat and mass transfer, Dhir and Cat ton (1977) have considered saturat ion-capil lary pressure ( S - P c ) relationships in the study of dryout heat fluxes

* Corresponding author.

0022-1694/96/$15.00 © 1996 - Elsevier Science B.V. All rights reserved SSDI 0022-1694(95)02823-4

34 W.A. Moseley, V.K. Dhir / Journal of Hydrology 178 (1996) 33-53

Nomenclature

Dp mean particle diameter (m) g gravitational acceleration (m s -2) h height above free liquid level (m) J Leverett function, defined in Eq. (8) K permeability (m 2) M modified I.everett function P~ bubbling pressure (Pa) Pc capillary pressure (Pa) Ph hydrostatic pressure (Pa) r average pore radiu s (m) r b average pore radius corresponding to the bubbling pressure (m) R radii of curvature (m) R s radius of spherical particle S saturation Se effective saturation Sr residual saturation

Greek letters 7 interfacial tension (N m -1) Ap pressure difference (Pa) Ap density difference (kg m -3) E porosity 0 angular position of fluid-solid interface (deg) A pore size distribution index p density (kg m -a) ~sa static advancing contact angle (deg) ~bsr static receding contact angle (deg)

Subscripts c capillary e effective h hydrostatic 1 liquid nw non-wetting phase v vapor phase w wetting phase

within heated particulate beds. Bau and Torrance (1982) have also considered the effect o f capillarity th rough the inyestigation o f boiling in low-permeabil i ty porous materials. Historically, a knowledge o f the S - P c relationship has been an impor tan t aspect in the field o f pet roleum recovery, which o f t en requires the determinat ion o f the distr ibution o f fluids within a virgin reservoir (Corey, 1977). More recently, the s tudy o f this relationship has received at tent ion in its applicat ion to the mathemat ica l model ing o f the t ranspor t o f non-aqueous phase con taminants in unsa tura ted soils (Abriola and Pinder, 1988).

P robab ly the mos t wel l ,known correlat ion specifying the behav ior o f capillarity in porous solids is tha t presented by Leverett (1941). Leverett developed a semi- empirical relation correlat ing capillary pressure and saturat ion da ta for clean

W.A. Moseley, V.K. Dhir / Journal o f Hydrology 178 (1996) 33-53 35

unconsolidated sands of various permeability and porosity. The saturation data used in this development were obtained under both imbibition and drainage conditions, using water as the wetting liquid. Leverett plotted the saturation data against a 'dimensionless capillary pressure function'. This function was determined by physical parameters including densities of the wetting phase (water) and the non- wetting phase (air), interfacial tension, gravitational acceleration, and sand porosity and permeability. In doing this, Leverett found that the saturation data fell satis- factorily along one of two curves, depending on whether saturations were obtained under imbibition or drainage conditions. These curves (also known as J-curves) have been correlated as a function of saturation by Lipinski (1984) and by Udell (1985).

In an investigation similar to Leverett's, Brooks and Corey (1966) concluded from comparisons with a number of experimental data that the relationship between capillary pressure and drainage saturations could be described reasonably well using the following empirical equation:

S~ = 1 (Pc <~ Pb)

(Pb) (1) Se= :'

In this equation the variable Se is referred to as the effective saturation, which is a linear transformation of the saturation (S) and is given by

S - Sr Se - - 1 - S/ (2)

where Sr is the wetting phase residual saturation. The variables Pc and Pb represent the capillary pressure and the bubbling pressure, respectively, and the exponent )~ is the pore-size distribution index. As acknowledged by Brooks and Corey the variables Po, Sr, and )~ are scaling parameters which must be experimentally determined for each porous medium as well as for each wetting phase liquid. The residual saturation has been defined in a number of ways. White et al. (1970) described the residual saturation as the saturation of the porous medium at which the wetting phase becomes nearly immobile. However, a more applicable definition, and the one that will be used in this work, is the saturation at which the gradient dS/dPc becomes zero (excluding the.region near S = 1, which also has a zero gradient). The bubbling pressure was defined by White et al. as the capillary pressure at the inflection point on the S-Pc curve, at which the value of dS/dPc is a maximum. The bubbling pressure corresponds to the highest saturation at which the non-wetting phase is interconnected (continuous). The exponent ),, is a characteristic value of the porous medium which is related to the pore-size distribution and thus has been designated by Brooks and Corey as the pore-size distribution index.

Another empirical expression relating saturation and capillary pressure was proposed by Van Genuchten (1980) and is given by the following general equation:

1 " Se = [.l + ~h)nl (3)


where h is the capillary pressure head and % n, and m are experimentally determined parameters. Van Genuchten has described how they may be obtained from capillary pressure-saturation data.

Laliberte (1969) suggested an empirical relationship for determining drainage saturations as a function of Pc having the form

S e = 0.5(1 - erf~), S e <~ 0.5 (4a)

Se = 0.5(1 + erf~), Se ~>0.5 (4b)

where ot

- - - ~ ( 4 c )

r

in which a, ~, and ( are empirical constants of the porous material, r is the mean pore radius, and rb is the pore radius corresponding to the bubbling pressure.

In many applications the Brooks-Corey and Van Genuchten relationships have proven useful because of the significant amount of work which has been performed in obtaining values for the empirical scaling parameters. For example, studies conducted for the US Department of Agriculture (Maidment, 1993) and the US Environmental Protection Agency (Carsel and Parrish, 1988) have provided a basis for estimation of these parameters from basic information on soil texture. In addition, Mualem (1976), Van Genuchten (1980), and Parker and Lenhard (1987) have used S - P c relationships of the forms presented above in deriving analytical expressions for hydraulic conductivities.

The disadvantage of these developments, with the exception of Leverett's, is that to predict the relationship between saturation and capillary pressure, it is necessary first to know the values of the scaling parameters. However, these values can only be determined after saturation data have already been obtained.

The objective of this investigation is to provide a relationship between capillary pressure and saturation that is based on known physical properties and which takes into account fluid-solid systems having different wettabilities. This will provide a means of predicting the S - P c curve without using experimentally determined scaling parameters. It is .proposed here that this may be accomplished by incorporating a contact angle dependence term into the Leverett function.

The inclusion of a contact angle dependence for the analysis of capillarity in a porous solid appears to have first been introduced by Rose and Bruce (!949). More recently, Demond and Roberts (1991) have correlated capillary pressure with saturation by incorporating a theoretical relationship for an ideal packing developed by Melrose (1965) which accounts for non-zero contact angles. Morrow (1976) has also demonstrated the importance of contact angle as a scaling parameter in S - P c relationships.

This work presents a functional relationship between capillary pressure and wetting phase saturation. It has been developed for both the imbibition and drainage of a wetting phase liquid into a porous medium consisting of clean unconsolidated silica particles. This medium was chosen to provide a foundation for further extension to

W.A. Moseley, V.K. Dhir / Journal of Hydrology 178 (1996) 33-53 37

typical uniform vadose zone soil types. The functional relationship presented here considers the dependence of capillary pressure upon the contact angle of the fluid- solid interface, and also includes how this dependence varies with saturation. The inclusion of the contact angle dependence provides extension of the correlation to any wetting phase liquid of uniform properties.

2. Theoretical background

When two immiscible fluids coexist within the voids of a medium, a discontinuity in pressure exists at the interface between the two fluids. This pressure difference, termed the capillary pressure, is defined as

Pc = P,w - Pw (5)

where Pw and Pnw are respectively the pressures in the wetting and non-wetting phases. In general, the wetting phase is determined by the contact angle of the fluid-solid interface. If the fluid has a contact angle less than 90 °, then it will preferentially wet the solid surface (Fetter, 1993). In considering a liquid-gas system, the liquid will in most cases be the wetting phase.

The interracial tension that exists at the boundary between the two immiscible fluid phases results in a curvature of the interracial surface. The dependence of capillary pressure on this curvature is defined by Laplace's equation

Pc = 7 + (6)

where 7 is the interfacial tension between the two phases, and R l and R 2 are the principal radii of curvature of the surface.

Under static equilibrium, the capillary and gravitational potentials can be equated. The hydrostatic pressure (Ph) can be written as

Ph = Apgh (7)

where Ap is the difference in density between the two phases, and h is the hydrostatic head (height above the free liquid surface).

The curvature at the interface between the two phases is dependent upon several factors, including pore size and geometry, wettability of the medium, saturations of the respective phases, and the manner in which saturation is obtained. Using Eqs. (6) and (7), Leverett developed a semi-empirical relationship between the mean interfacial curvature and saturation, defined as

J(S) Apgh ( K ) 1/2 = - - (8)

This dimensionless parameter, referred to here as the 'dimensionless capillary pressure function' or 'Leverett function', was supported by data obtained for a number of clean unconsolidated sands, in which water was used as the wetting fluid. In reference to Eq. (8), the term Apgh/7 represents the mean interfacial


curvature (from Eqs, (6) and (7)) and (K/e) 1/2 can be shown, from the Poiseuille and Darcy equations, to be equal to the 'average pore radius' of the medium. Rose and Bruce have modified Eq. (8) by including a dependence upon wettability as

J(S) ' Apgh ( g ) 1/2 - 7 cos ~b (9)

where ~ is the contact angle. The presence of the cos ~b term in the denominator of Eq. (9) can be derived through analogy with a capillary tube, and is often referred to as the capillary tube model. For a capillary tube of radius r c, combining Eqs. (6) and (7) gives

Apgh = 27 cos_______~ (10) r

where h is the liquid rise height in the tube. It should be noted, however, that this analogy is only partially correct, as the capillary tube analysis does not adequately represent the behavior of a porous medium. Thus, the dependence of capillary pressure on contact angle is different than that suggested by Eq. (9).

Realizing the limitations of applying the capillary tube model to a porous medium, Melrose derived a theoretical relationship predicting capillary rise for an ideal packing consisting of perfectly spherical particles of identical size and known packing formation. Melrose's development relates capillary pressure to a normalized interface curvature (N) given by

R s ( 1 1 ) (11)

which is a ratio between the spherical particle radius and the mean interface curvature. Combining this with Eq. (6), the capillary pressure can be determined from

Pc - 27N (12) Rs

The normalized interface curvature based on an ideal packing is a function of the interface position within the ,pore space and the contact angle of the fluid-solid interface. The general form of the equation developed by Melrose specifying the normalized interface curvature is given by

cos(O+ ~b) (13) N(O, ~b) = (cos ~.])'1 " COS 0

where 0 is the angle specifying the position of the fluid-solid interface relative to the center of the spherical particle representative of the pore opening (see Melrose (1965)). The angle ~7 specifies the type of array defining the pore opening 07 is 45 ° for a square array and 30 ° for a triangular array).

The limiting conditions for interface stability under imbibition and drainage cases can be applied to Eq. (13) to derive the maximum normalized interface curvature. Once the maximum curvature has been derived, Eq. (12) can be used to determine the capillary pressure corresponding to these limiting conditions. An important aspect of

W.A. Moseley, V.K. Dhir / Journal of Hydrology 178 (1996) 33-53

Table 1 Properties of glass beads used in height-saturation experiments

39

Packing 1 Packing 2

Composition sio 2 SiO 2 Structure Clean, unconsolidated Clean, unconsolidated Mesh (pm) 100-140 320-400 Average grain diameter (#m) 120 360 Permeability (x 10 -12 m 2) 9.24 4- 0.22 112 + 3.2 Approximate porosity a 0.41 4- 0.02 0.41 + 0.02

a See Moseley (1993) for exact data.

Eq. (13) is that it demonstrates that the effect of contact angle on capillary pressure is not constant, as suggested by the capillary tube model, but varies with saturation.

3. Experimental investigation

Two sets of experiments were conducted to develop capillary pressure-saturat ion profiles for both the imbibition of a liquid into a dry porous medium and the gravity drainage of a liquid f rom an initially saturated porous medium. For both sets of experiments, two porous media were used with three liquids. Thus, six experiments were conducted for each of these sets. Two sizes of glass beads were used to represent uniform and homogeneous porous media. Properties of the glass beads (from here on referred to as Packing 1 and Packing 2) are given in Table 1. The three liquids used in these experiments were water, ethanol, and refrigerant-113. Properties of these liquids are presented in Table 2.

The physical parameters used to correlate the saturat ion-capil lary pressure data include air and liquid densities, interfacial tension, porosity, permeability, and contact angle. Values for porosity were determined through saturation measurements as described below. Permeability data were experimentally determined by passing air at varying flow rates through the test packing and measuring the associated pressure drop. Contact angles for the three liquids were also experimentally determined as described below.

Table 2 Phase properties used in height-saturation experiments at 300 K and 1 atm

Water Ethanol R- 113

Molecular formula H20 C2H 6 FCCIECC1F 2 Interfacial tension (N m -1) 0.0717 0.022 0.017 Liquid density (kg m -3) 997 799 1569 Vapor density (kg m -3) 0.596 1.08 3.09

Air density as 1.1614 kg m -3.


3.1. Apparatus

In the conducted experiments, the glass beads were packed in a vertical Plexiglas column having an inner diameter of 9.53 cm, as is illustrated in Fig. 1. A fine mesh screen was placed between the two PVC flanges in the lower portion of the test section. This allowed the wetting liquid to enter the bead-filled column freely. A large cylindrical polyethylene tank having a 53.34 cm inner diameter was filled with the test liquid and was connected to the bot tom of the column.

A gamma densitometer was used to measure local saturations, and is illustrated in Fig. 2. It consisted of a 150 mCi cesium-137 source and a NaI detector, both encased in lead. The diameter of the columnator for the source and for the detector was 3.18 mm. The source and the detector were mounted on a base plate which could be moved along the column axis in the vertical direction. Gamma rays are absorbed by the detector after passing through the test section. A multichannel analyzer was used to count and display the signal magnitude ( 'attenuation count') of the absorbed gamma rays.

3.2. Procedure

For the imbibition experiments, the saturation profile was achieved by starting with an initially dry packed column and then allowing liquid from the reservoir to be imbibed into the packing column. These experiments were conducted for all six l iquid-packing combinations (two packing sizes x three liquids). In all cases the packing was filled into the empty test column and an eccentric motor was used to

plexiglass column 9.5 era. I.D.

1.5 m .glass be, ads

serP.~n

free liquid level

vent holG -.... reservoir 53.3 cm. I.D.

base plate

Fig. 1. Experimental apparatus for imbibition and drainage experiments.


Base Plate

columnator

! Test section

lead casing

Fig. 2. Gamma densitometer apparatus for saturation measurement.

vibrate the system, thus creating a more uniform packing structure as well as minimizing pore space. The reservoir valve was then opened, allowing the liquid to enter the column below the screen and be imbibed upward into the packed column. The free liquid level within the reservoir was set to be approximately 8-10 cm above the screen, and the height of this level was used as the zero reference at which the capillary pressure is defined to be zero (at this level h = 0). To minimize evaporation losses from both the reservoir tank and the test column, each was capped, leaving only small vent holes to maintain atmospheric pressure. For Packing 1 (K = 9.24 × 10 -12 m2), equilibrium was reached within 14 days for all three liquids. For Packing 2 (K = 112 x 10 -12 m 2) equilibrium was reached within 8 days for all three liquids. In all cases, saturation measurements were made after an additional 7 days to insure that equilibrium was obtained.

For the drainage experiments, the saturation profile was achieved by starting with a completely saturated packed column and then allowing the system to come to capillary equilibrium through gravity drainage of the liquid into a reservoir having a constant liquid level. As in the imbibition experiments, drainage experiments were conducted with all six liquid-packing combinations. In all cases, the reservoir tank was raised so that the test column was initially filled with liquid. The packing was then slowly and steadily poured, without interruption, into the liquid-filled column. During this time, an eccentric motor was used to vibrate the system, to achieve a uniform porosity and minimize pore space. By pouring the packing into the liquid- filled column, it was assured that no air bubbles would become trapped in the column, which would result in a non-uniform porosity. After achieving a saturated packed column, the reservoir tank was lowered so that the liquid level was approximately


8-10 cm above the screen. As described in the imbibition experiments, both the column and the reservoir were capped, leaving only small vent holes, thus minimizing evaporation losses. For the experiments conducted with the lower-permeability packing (Packing 1), capillary equilibrium was achieved after about 10-12 days, at which time the saturation profile was determined using the gamma densitometer. The same process was followed for the higher-permeability packing (Packing 2), for which capillary equilibrium was reached within approximately 2-3 days for all three liquids. Saturation measurements were taken again after a further period of 7 days for the higher-permeability packing and of 10 days for the lower-permeability packing to insure that equilibrium had been achieved.

3.2.1. Saturation and porosity measurement Saturations at incremental heights along the column were determined using a

gamma densitometer. In this approach, a narrow circular beam of gamma rays is passed through the diameter of the column and the attenuated beam is absorbed by a detector on the opposite side of the column. Saturations are then calculated from the attenuation counts. Attenuation counts are taken at representative locations along the height of the column to determine the saturation profile. Ryan and Dhir (1992) have described this method in detail.

3.2.2. Contact angle measurement The experimental determination of contact angles is complicated by two factors as

detailed by Adam (1968). First, the angle depends on the quality of the liquid, which if contaminated will act to diminish the contact angle. Second, the contact angle is not definite and may have any Value between two extremes, depending on whether the liquid is tending to advance over a dry surface or recede from a previously wetted one. Advancing angles are often much larger, sometimes as much as 40 ° . This difference between advancing and receding angles is referred to as hysteresis in contact angle. In this investigation, both the wetted surface and the dry surface contact angles were experimentally determined.

The primary experimental methods that have been previously used in determining contact angles include capillary tube rise measurements, the plate method, the cylin- der method, Taggart's bubble method, drop thickness correlations, and microscopic observation (see Adam (1968)). A variation of microscopic observation has been used here. As contact angles are influenced by gravity, as suggested by Adamson (1967), contact angle measurements were made using spherical glass particles to represent better the physical contact angles actually existing within the glass bead packing.

The glass spheres used for the contact angle measurement were approximately 2 cm in diameter and of the same silicon dioxide composition as the glass beads used in the packed column. The spheres, which were well cleaned before liquid was introduced onto the solid, were set on top of a flat glass plate, and a syringe was used to introduce the test liquid between the sphere and the plate as shown in Fig. 3. Advancing contact angles were achieved by injecting the test liquid at the bottom of the initially dry sphere, and receding angles were obtained by drawing liquid away from the bottom of the initially wetted sphere. Photographs taken of the static meniscus using macro lens

W . A . M o s e l e y , V . K . D h i r I J o u r n a l o f H y d r o l o g y 178 ( 1 9 9 6 ) 3 3 - 5 3 43

Glass Plate

Fig. 3. Contact angle measurement.

Liquid

photography allowed the contact angle to be determined by measuring the angle between the tangent to the sphere at the point of contact with the liquid and the line extending in the direction at which the liquid contacts the sphere (see Fig. 3). These experiments were also conducted with 1.5 mm glass spheres, and a low- powered microscope with photographic capability was used for contact angle determination. There was no appreciable difference in results when the 1.5 mm spheres were used. However, in using the smaller sphere size, a greater measurement error was introduced.

0.5

0 .4

0.3

O.

II 0.2

0. I

l i ~ i i i i i

_ A 0

..................................... ~ " " " ' ! .................. i ....................................................... T ...................

I i I I

A Packing #2

- - - - - smoo th fit " d - - i~ ! !

...................................................................... ¢-----~--.~ . . . . . ~..!..~-.-.-i--.~_ ............................. ~ ................................... t .......... : ......

" ? ' 0 ~ ............................................................................................................ i ................................... ~..-..---~.~..4 ............... 4 ................. ~ ............ i o i x i i

............................................................................................................... i ........................................................ ~ : ^ ~ . ~ . . x . . - . : ............ i . . . . . . . . . . . . . .

ion A_ . ........................................................................ ................ .......................................................................................... L . . . x { . + ................

i o i -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T . . . . . . . . . . . . . . .

IO I ~ "

I I I I I I I I I I I ~ t

0 0.1 0.2 0.3 0 .4 0.5 0 .6 0 .7 0 .8 0 .9 1

Saturation

Fig. 4. J-Curves for imbibition experiments using ethanol.


4. Results and discussion

The imbibition saturation data for ethanol with both packings are presented in Fig. 4. This figure plots the saturation data against the 'dimensionless capillary pressure function' (Eq. (8); also referred to as the 'Leverett function' or 'J-curves') for both sizes of glass beads. Inspection of these results demonstrates that the Leverett function works very well in describing the relationship between capillary pressure ( A p g h )

and saturation for a specific liquid. Similar results were obtained for water and R- 113. Yet when the dimensionless data for all three liquids are compared as demonstrated by Fig. 5, a significant difference is readily noted.

The drainage saturation data obtained for R-113 with both packings have been plotted against the dimensionless capillary pressure function as shown in Fig. 8, below. This figure illustrates that for the drainage experiment, as was also the case for imbibition, the Leverett function works very well in describing the relationship between capillary pressure and saturation for a specific liquid. Similar results were obtained for water and ethanol. However, as was the case with the imbibition data, when the J-curves for all three liquids are compared as demonstrated by Fig. 9 below, a significant difference is again noted.

Scheidegger (1974) suggested that it 'is impossible to correlate capillary pressure curves that have been obtained with different fluids. Others, including Purcell (1949), Rose and Bruce (1949), and Brown (1951), have proposed a capillary tube model

0.5

0.4

0.3 ,-~

lm 0.2

0.1

. . . . . . . . i ' i '

............... ~ ............. " !i .................................................................................................................. ETHANOL

................... ~ ............. ::....::~,,,-- .................................................................................. R-l 13

-- .~ ........... ,.~ .................................... ~ .................. i ........................................... ....... :'""'"'7 ..................... ~ ........................................ i ~ ~ ! - :: i

i " -i . . . . i . . . . ! . . . . . . . i ~ ~ ............. i ...................

...................................................................................................................................... ~ ..................... J ............... :'""~"~ ......... "~"i ....................

.................... T ..................... i ...................... i ...................... i ..................... i ..................... ..................... T ..................... ) ........................... i ..................... i ...................... i ...................... i ..................... i ..................... i ..................... i ..................... ~ .................................. :..~:i. 2.,

.7 ,, ...................................................................................................................................................................................................... .................

E I I I i I I I i I I

0 0.2 0.4 0.6 0.8 Saturation

Fig. 5. Imbibition J-curves for all three liquids.


which includes a contact angle dependence with the Leverett function as described by Eq. (9), to provide a general correlation for various wetting phase liquids. However, as described previously, Melrose demonstrated that even for an ideal soil packing, the capillary tube model is not valid. The relationship presented by Melrose (Eq. (13)) provides a means of correcting predictions given by the capillary tube model for ideal packing structures. This correction is achieved by incorporating a constant factor into the capillary tube model which accounts for the actual maximum interface curvature for imbibition and drainage cases. As these maximum curvatures correspond to the maximum capillary rise, they are referred to here as the limiting conditions for imbibition and drainage cases.

For all packings other than the ideal packing described by Melrose, irregularities in pore size will be present. As a result of these irregularities in pore size, pendular saturations may exist in what is predominantly a region of funicular saturation and vice versa. In these instances, the interface curvature, and therefore saturation, may be different from that suggested by the limiting conditions for imbibition or drainage. Thus, for real packing structures, the presence of pendular rings in predominantly funicular regions coupled with varying pendular saturations requires a correction factor which varies with saturation. This point is illustrated further in the following discussions.

4.1. Imbibition experiments

Through inspection of Fig. 5, it can be noted that the deviation between the data for the three liquids is not a constant factor over the entire saturation range as is suggested by the presence of the cos ~b term in Eq. (9). Rather, the difference in the data increases significantly at low saturation. Thus, in addition to the contact angle dependence, a dependence upon saturation must also be included in the dimensionless capillary pressure function to reduce the data to a common curve.

Results of the contact angle measurements are given in Table 3 for both static advancing (non-wetted) and static receding (previously wetted) surfaces. In comparing these two extremes, the effect of hysteresis is most evident. It is difficult to determine exactly which contact angle, either advancing or receding, is present for the imbibition of liquid into the packed column. However, it is plausible to assume that for the imbibition of liquid in a previously unwetted packing, an advancing contact angle would exist at the fluid-solid interface. Thus, in developing a relationship

Table 3 Measured contact angles for water, ethanol and R-113 with glass sphere

Liquid Contact angle (deg)

Receding Advancing

Water 29 ± 3 64 4- 5 Ethanol 22 4- 3 47 4- 4 R-113 174-2 25±3


between sa tura t ion and capi l lary pressure for the imbibi t ion o f liquid, static advanc ing contac t angles (~bsa) were used.

The sa tura t ion da ta for the three liquids have been correlated th rough a modif ica t ion o f the Leveret t funct ion (J, defined in Eq. (8)), as follows: Region 1: for 0.2 ~< S <~ 1

Ml =- (c°sOJ ~)sa ") (14a)

= 0.404(1 - 8) 0.23

Region 2: for 0 ~< S < 0.2

M2 -- C°s°'q~sa 0.383 / (cos 1"23 Osa) + 0.383 (14b)

= [(1 - 8) - 0.8] TM "1- 0.383

where M1 and M2 are the modif ied Leveret t funct ion for the range of sa tura t ion specified. This funct ion has been plot ted against sa tura t ion in Fig. 6. Here it can be noted tha t the corre la t ion provides a very accurate relat ionship between capil lary pressure and sa tura t ion for var ious liquids having measurab le contac t angles. I t is noted here tha t da ta have been correlated in two distinct regions: Region 1 for sa tura t ion between 20 and 100% and Region 2 for sa tura t ion between 0 and 20%.

M

0.6

0.5

0.4

0.3

0.2

0.1

0 -- 0

Region 1 0.2<S<1.0

M 1 = J/COS0" 40

i i !

- - -- -Water

m . -Ethanol

. . . . . R-I13

11. Empirical

Region 2 0 < S < 0 . 2

M 2 = { [(J/COS 0" 40)-0.383]/COS 1.230 } +0.383

0.2 0.4 0.6 0.8 Saturation

Fig. 6. Modified capillary pressure function for imbibition saturations.


From Eq. (14a), it is apparent that in Region 1 the capillary pressure is weakly dependant upon contact angle as suggested by the cos °'4 q~sa term in the denominator. In comparison, Region 2 shows a much stronger dependence upon contact angle as demonstrated by COS 1"63 q~sa in the denominator of Eq. (14b). This point is illustrated further by noting that as the saturation of the wetting phase changes from high to low, the interstitial configuration of the wetting phase changes from funicular rings to pendular rings as shown in Fig. 7. In the pendular region the contact angle exists, having a definite value, whereas in the funicular region the solid surface is almost completely wetted, such that the effect of contact angle is less significant.

The correlated data have been empirically fitted as described in Eq. (14) to obtain a functional form of capillary pressure vs. saturation. In the limit as the contact angle

(a)

surface is ~ ~ " ~ / completely / ~ ~. wetted

/-- wetting / hose

definite ongle

Fig. 7. (a) Funicular saturation region. (b) Pendular saturation region.


Table 4 Properties of tetrahydronaphthalene at 300 K and 1 atm

Molecular formula C12H22 Interracial tension (N m -l) 0.0472 Liquid density (kg m -a) 990 Receding contact angle a (deg) 16 + 2 Advancing contact angle a (deg) 23 4- 3

a Contact angles were measured as described previously.

a p p r o a c h e s ze ro , Eq . (14) b e c o m e s

l im M = J fo r 0 ~ < S ~ < l ~,a --,0

Thus , as dPsa a p p r o a c h e s zero , the c o r r e l a t i o n is i den t i ca l to the L e v e r e t t func t ion .

U p o n e v a l u a t i n g the l imi t as the c o n t a c t ang le a p p r o a c h e s 7r/2 (90 °) u s ing

L ' H o s p i t a l ' s R u l e a n d a n y a r b i t r a r y v a l u e fo r J , Eq . (14) b e c o m e s

l im M = 0 fo r 0~<S~< 1 esa ---+ Ir/2

T h u s , as ~sa a p p r o a c h e s ~r/2, the re is no cap i l l a ry rise in the m e d i u m . T h i s is logica l ,

s ince at $sa = ~ ' /2 the re is n o ne t fo rce c o m p o n e n t in the ve r t i ca l d i rec t ion .

A s a m e a n s o f ve r i fy ing the a c c u r a c y o f the c o r r e l a t i o n de f ined by Eq . (14), an

0.8

0.6

II 0.4

0.2

i i - , I 1 i I I i I I

I "

............................... ................................................................................................................... o aokin # I-...-..-

', , i ! i i ~, i " i_ ................. i .......... °.l.....i .................... i ................... i .................... i .................................................. Packing #2 [ ........

i o~.i . i i i i i

................. i ............ .A~ .................. i ................... i .................... i ................................................... Smooth Fit l ........

.................. i .................. i ....................... o~z......~.~ ................ ,i .................... i .................... i ....................................... ~ .................... ~ ..................

Average residual saturation = 17.2% i , ' -8 " ....................................... i .................... i ................... ~ .................... J .................... . .................... , ......................................... - ............... 9 i ...................

! to

o!

! l

i I I I I I I I I I i ~i- I

0 0.2 0.4 0.6 0.8 1

Saturation

Fig. 8. J-Curves for drainage experiments using R-113.


experiment using another liquid, tetrahydronaphthalene (Tetramp), was conducted with Packing 2 (360 #m). The properties of this liquid are presented in Table 4. The saturation data obtained for Tetramp are in excellent agreement with the correlation defined by Eq. (14). It should be further noted that the contact angle for Tetramp is virtually the same as that obtained for R-113. Thus, it would be expected that the J-curve for Tetramp should match that for R-113, as the Leverett function does not contain a contact angle dependence. When comparing these two J-curves, this inference was supported.

4.2. Drainage experiments

Fig. 9, below, presents J-curves using best curve fits of the saturation data obtained for each liquid with both packings (e.g. Fig. 8). It can be noted from this figure that, in comparison with the J-curves obtained under imbibition (see Fig. 5), the difference between the drainage J-curves for the three liquids is not as pronounced.

It should be noted that in developing the correlation for the imbibition saturation profiles given by Eq. (14), static advancing contact angles were used, as the wetting liquid advanced into an initially dry medium. However, in the drainage experiments, the wetting liquid receded from an initially saturated medium. Thus, to correlate the drainage results illustrated in Fig. 9 onto a common curve, it seems appropriate to use static receding contact angles. Upon inspection of the data given in Table 3, it is seen

0.8

0 . 6

~ o.4 II

0.2

................ ] ............. .................... i ................... i ....................................... i ................... i .................. ................ i . . . . . . . . . . . . . . . ~i ..................... T ................... i .................... i ................... ] ..................

................ ~ ................ ~ ~ i - ~ i ~ ................. i ................... i .................... '

J ( W a t e r )

- - - - - J ( E t h a n o l )

. . . . . J ( R - 1 1 3 )

Water S r = 17.6% Ji

............. Ethanol S = 16.9% ............................................. ~ .............................................................................

r i R-113 S = 17.2% . . . . . . . . . . . . . . . . . . . . . . r . . ~ . . . . . . . . . . . . . . . . . . . . [ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i I t I i 1 I I

0.2 0.4 0.6 0.8 I

Saturation

Fig. 9. Drainage J-curves for all three liquids.

/

50 V.A. Moseley, V.K. Dhir / Journal of Hydrology 178 (1996) 33-53

that the values for the receding angles for the three liquids differ much less than those for the advancing angles. Considering this, the effect of contact angle is expected to be less apparent in the saturation data obtained under drainage conditions. Thus, the results presented in Fig. 9, which illustrate a significantly smaller deviation in the J-curves of the three liquids in comparison with the imbibition data, seems to be intuitively correct.

The drainage saturation data in Fig. 9 have been correlated using the Leverett function (,/), the static receding contact angle (~bsr), and the effective saturation (Se) as functional parameters, and is described with the following equations: Region 1: for 0 ~< M1 ~< 0.4

M 1 _= J and Se = 1 (15a)

Region 2: for 0.75 ~< Se < 1 J

M2~(COS°'74flPsr) (15b)

= 0.4+ 0~189(1 - --Se) 0"29

Region 3: for 0 ~< Se < 0.75

M3 - Cos°774 q~sr 0.526 / ( c o s 2"1 dPsr) + 0.526 (15c)

= 0.523[(1 -- Se) - 0.25] 2.3 + 0.526

where M l, M2 and M3 are the modified Leverett functions for the range of effective saturations specified. The effective saturation Se, is defined as

(1 - S) (16) Se = (1 - St)

,The residual saturation St, as defined previously, was determined for each liquid by averaging the residual saturations for both packings. In all cases, the residual saturations for both packings were very close and a dependence on the particle diameter could not be established. Fig. 10 presents the results obtained by applying Eq. (15) to the data given in Fig. 9.

Upon inspection of Region 1 from Fig. 9, it is seen that for 0 <~ J ~< 0.4, there was no need to consider a contact angle dependence, as the data for all three liquids were undistinguishable. As this region is completely saturated, the effect of contact angle should indeed be non-existent. Although the form of Eq. (15) is slightly different from that given by Eq. (14) for the imbibition data, the trend in contact angle dependence is very similar; that is, at high saturations the contact angle is an insignificant parameter, whereas at lower saturations it becomes increasingly important.

In the limit as the contact angle approaches zero, Eq. (15) becomes

lim M = J for 0~<Se~<l ~ 0

Thus, as the contact angle approaches zero, the correlation is identical to the


M

0.8

0.6

0.4 ......

0.2

0 0

' ' .I ' ' ' ' ' ' W a t e r

................ - ................... i ......................................................................................................................... - E t h a n o l

............ i'""'";~':'"'~ ................... i ................... i ................... i .................... i ................... i .................. - - . . . . R - 1 1 3

R e g i o n 1 0<-M1 < 0 . 4 1" R ~ g i o n 2

M 1 -= J R e g i o n i3

...... Region2 0.75<Se<1 ....... i ........................................................... i ................. i .........................

M 2 - (J/COS0.74~r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i ............................................................ ] ................ ..R..~..o...n.....!. ...............

R e g i o n 3 0 -<Se<0 .75

M 3 --- { [ ( J / C O S 0 " 7 4 , r ) - 0 . 5 2 6 1 / C O S 2 " 1 , r } + 0 . 5 2 6 .................................... i ..................................

I , I , I , I , I t I i I , , I , ,

0 . 2 0 . 4 0 . 6 0 . 8

s e = (s-sp/(1-s r) Fig. 10. M o d i f i e d cap i l l a ry p re s su re f u n c t i o n fo r d r a i n a g e sa tu ra t ions .

Leverett function. In the limit as the contact angle approaches 7r/2 (90°), Eq. (15) becomes

lim M = 0 for O<~Se<~l 4,s~ ~ ~r/2

The physical interpretation of this situation is that for saturations obtained under drainage conditions, as the contact angle approaches 7r/2, the saturation will never be greater than the residual saturation. In other words, as the contact angle approaches re~2, Se will be zero everywhere.

The inherent strength of the functional relationships given by Eqs. (14) and (15) is that it should provide extension to the correlation between capillary pressure and saturation for any porous medium of uniform properties (assuming the properties remain unaltered upon the introduction of the wetting phase) for which the contact angle can be specified. Additional experimental verification for the extension of this correlation to porous material other than that used in this investigation will be valuable. The correlation developed here may not be fully applicable to all packing configurations, as pore size distribution and surface roughness parameters were not considered. However, these correlations demonstrate the importance of contact angle as a scaling parameter as well as relate how the effect of contact angle is not a constant, but rather a function of saturation.

Future emphasis should be placed on extending the application of the above


correlations to naturally occurring soil media. Although the contact angle is a physical parameter of the liquid-solid system, it is likely that this parameter must be related to some empirically derived wettability index, as it may be difficult to quantify the contact angle for many soil media. The incorporation of a pore size distribution index, similar to that of Brooks and Corey, would also prove useful to extend the application to non-uniform pore structures. These developments were beyond the scope of this work, in which the objective was to demonstrate how the effect of wettability varies with saturation, and how contact angle can be used as a physical scaling parameter in predicting the relationship between saturation and capillary pressure.

5. Summary

When each of the three liquids studied in this investigation was considered separately, it was found that Leverett's capillary pressure function (Eq. (8)) worked well in correlating both the imbibition and drainage saturation data obtained for the two different packing sizes investigated. However, a significant discrepancy was discovered when utilizing this same function to correlate the data obtained for all three liquids. It has been postulated here that this discrepancy is a result of the differences in contact angles of the three liquids.

In contrast to the suggestions given in previous investigations, it has been shown here that the effect of contact angle is not constant throughout the saturation profile. Rather, as the saturation regime changes from funicular rings to pendular rings, the effect of contact angle appears to increase significantly. The inclusion of a contact angle dependence as suggested by Eq. (14) for imbibition data and Eq. (15) for drainage data, provides a simple means of correlating the saturation data obtained with various liquids. Furthermore, this more generalized form should also provide extension to any porous medium of uniform properties for which the contact angle can be specified. These correlations are based on physical properties of the system rather than empirical scaling parameters which must be experimentally derived for each porous medium and each wetting phase liquid.

Acknowledgments

This work received support from the State of California under the Toxics Research Program.

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