Effects of fractional wettability on capillary pressure–saturation–relative permeability relations of two-fluid systems

Advances in Water Resources 29 (2006) 212–226 www.elsevier.com/locate/advwatres

Effects of fractional wettability on capillary pressure–saturation–relative permeability relations of two-fluid systems Sang Il Hwang a, Kwang Pyo Lee b, Dong Soo Lee b, Susan E. Powers

c,*

a

Korea Environment Institute, 613-2 BulGwang-Dong, EunPyung-Gu, Seoul 122-706, Republic of Korea Graduate School of Environmental Studies, Seoul National University, Seoul 151-742, Republic of Korea Department of Civil and Environmental Engineering, Clarkson University, 8 Clarkson Ave., Potsdam, NY 13799-5710, USA b

c

Received 8 February 2005; accepted 11 March 2005 Available online 15 July 2005

Abstract Capillary pressure (Pc)–saturation (S)–relative permeability (kr) relationships must be quantified to accurately predict non-aqueous phase liquid (NAPL) distribution in the subsurface. Several experimental techniques are presented here for two-fluid Pc–S–kr relationships for various saturation paths to better define the effect of fractional wettability on these relationships. During the primary drainage path of the Pc–S curves, the air–water system showed no distinct trend as a function of the fraction of sand treated by organosilane (S) to render it non-water wetting. In a NAPL–water system, however, a consistent decrease of capillary pressure with increase of the fraction of non-water wetting sands was observed. The much lower contact angle for air–water (a–w) system may result in the observed insensitivity of the a–w Pc–S curves to fractional wettability, at least for the PD pathway. For the main imbibition path of NAPL–water system, capillary pressure decreased as the fraction of the S component increased, requiring forced imbibition (negative capillary pressures) for a certain range of saturations. Systems with an increasing percentage of the S component also exhibited a higher water kr and lower NAPL or air kr at a given saturation for the primary drainage and main imbibition paths in both air–water and NAPL–water systems. The increase of water kr with increase of the fraction of the S component can be explained by the ability of water to occupy larger and highly conductive pores in such a system. Experimental kr–S data for the primary drainage path of NAPL–water system presented here were used to test the Bradford et al. [Bradford SA, Abriola LM, Leij FJ. Wettability effects on two- and three-fluid relative permeabilities. J Contam Hydrol 1997;28:171–91] model and the modified Mualem model for estimating the kr–S curves from measured Pc–S data as a function of fractional wettability. Both models predicted significantly less variation in the kr–S curves than measured indicating that they did not adequately represent the system under investigation.
2005 Elsevier Ltd. All rights reserved. Keywords: NAPL; Capillary pressure; Relative permeability; Fractional wettability

1. Introduction The process by which non-aqueous phase liquids (NAPLs) infiltrate into and redistribute within the sub*

Corresponding author. Tel.: +1 315 268 6542; fax: +1 315 268 7985. E-mail address: [email protected] (S.E. Powers). 0309-1708/$ - see front matter
2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2005.03.020

surface depends on capillary pressure–saturation–relative permeability (Pc–S–kr) relationships. The Pc–S–kr relations are affected by interfacial (or surface) tension and wettability. These interfacial properties can vary considerably due to trace and primary constituents in the NAPL and surface properties of the porous media comprising the subsurface. Various experimental attempts have been made to quantify trends between

S.I. Hwang et al. / Advances in Water Resources 29 (2006) 212–226

interfacial properties and Pc–S–kr relationships. The results are conflicting, however, reducing any potential to generalize the overall effects. Several experimental studies have been devoted to quantifying the effect of interfacial tension on the Pc– S–kr relation. It is clear that decreasing interfacial tension lowers the capillary pressure [20,23,40,41], although relative permeabilities are not affected except for extremely low values of the interfacial tension [2,19,23]. The wettability of a solid refers to the tendency of one fluid to spread on or adhere to the solid surface in the presence of other immiscible fluids [18]. It is important to note that the term wettability is used for the wetting preference of the solid and does not necessarily refer to the fluid that is in contact with the solid at any given time [4]. Generally, the wettability of porous systems can be divided into two basic classes: uniform and non-uniform. A porous system with uniform wettability shows consistent wettability condition (e.g., strongly waterwet, strongly oil-wet, or intermediate-wet) throughout the system considered. Non-uniform or fractional wettability is defined as the condition when different regions of the system have different wetting preferences [4]. Fractional wettability is often encountered in natural porous media as a result of the spatial variation in mineral composition or roughness of the solid surface, or the presence of residual NAPL films, adsorbed surfactant groups, or microorganisms on the solid surface [51]. A few experimental investigations have been conducted to elucidate the effect of fractional wettability on the Pc–S–kr relation of two-fluid systems [10,51]. Significant differences in the conclusions from these studies, however, prevent us from drawing generalities describing the nature of the influence of fractional wettability. These differences could be due to the experimental fluid pairs, fluid paths, and/or measurement techniques. It is

213

thus very important at this stage to clearly elucidate the effect of fractional wettability on the constitutive relationships. Models that estimate the Pc–S–kr relations in porous systems with fractional wettability or with mixed wettability (systems where small pores are water-wet and larger pores are intermediate- to oilwet) have been developed [8,9,11,37,51]. Some have resulted in useful predictions on the effect of wettability variation on the Pc–S–kr relation. Others, however, need to be tested against appropriate experimental data in order to be used more confidently for numerical simulation or estimation.

2. Background 2.1. Experimental investigations Table 1 summarizes experimental conditions considered by various authors for their measurements of the Pc–S–kr relationships in two-fluid systems as affected by fractional wettability. In all experiments reviewed, porous media with fractional wettability have been obtained by mixing different portions of two or three pure components (i.e., quartz sands (Q), organosilane-treated sands (S), or Teflon grains (T)). The Q component is strongly water-wet, while the S and T components are non-water wetting. 2.1.1. Capillary pressure–saturation relations Little consistency exists among published experimental results describing the effect of fractional wettability on the primary drainage (PD) path of the Pc–S relation for air–water systems (Table 1). Bradford and Leij [10] found that the PD curves for pure Q, QS 3:1, and QS 1:1 media were similar, as were the curves for pure S

Table 1 Experimental conditions performed for the Pc–S–kr measurements of two-fluid systems by various authors Authors

Pc–S–kr

Fractional wettability conditionsa

Fluid pairsb

Fluid pathsc

Measurement techniques

Bradford and Leij [10]

Pc–S

‘‘Brooks method’’ [13]

Pc–S

a–w, a–o, and o–w (oil: Soltrol 220) a–w

PD and MI

Bauters et al. [6]

PD and MI

Using experimental chamber

Ustohal et al. [51]

Pc–S

Pure Q, pure S, QS 3:1, QS 1:1, and QS 1:3 Pure Q and four QS mixtures (3.1%, 5.0%, 5.7%, and 9.0% S component) Pure Q, pure S, pure T, QS 1:1, QT 1:1, QS 1:2, QT 1:2, and QST 1:1:1

a–w

MI and MD

Pure Q, pure S, QS 3:1, QS 1:1, and QS 1:3

o–w (oil: kerosene)

PD

Pc–S: by adjusting the boundary conditions (upper: no flux for water, lower: changing water pressure) kr–S: steady-state, pressure equilibrium approach [17] Adjusting the boundary conditions (upper: no flux for water, lower: decreasing water pressure)

kr–S Fatt and Klikoff [25]

a b c

Pc–S

Q: quartz sand, S: organosilane-treated sand, T: Teflon component. a–w: air–water, a–o: air–oil, o–w: oil–water systems. PD: primary drainage, MD: main drainage, MI: main imbibition.

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and QS 1:3 media (herein, QS 1:3 stands for a mixture of the Q and S components in a mass ratio 1:3). Also, the more hydrophobic pure S and QS 1:3 media had a lower capillary pressure at a given water saturation than systems with higher mass fractions of quartz. On the other hand, Bauters et al. [6] conducted the Pc–S measurements with pure Q medium and four QS mixtures (3.1%, 5.0%, 5.7%, and 9.0% organosilane-treated sands). Although Bauters et al. [7] noted that ‘‘once water repellent soils are fully wet, the hydrophobicity disappears and thus, the drainage curves should be the same,’’ their experimental Pc–S data in the PD path were not the same. The PD curves for four QS mixtures were all within their 95% confidence intervals, whereas the pure Q medium had significantly lower air entry capillary pressure head than the QS mixtures. The opposite trend observed by Bauters et al. [6] versus Bradford and Leij [10] suggests that experimental measurements are difficult to accurately complete, even for the most basic PD pathway. For the main drainage (MD) path in air–water systems (Table 1), Ustohal et al. [51] showed that a pure S medium had a lower capillary pressure than pure Q medium, whereas there was no distinct difference among various QS mixtures. With Teflon however, samples with increasing mass fractions of T had lower capillary pressures. Ustohal et al. [51] found a similar dependence of main imbibition (MI) Pc–S curves in air–water systems on fractional wetting—more hydrophobic media had a lower capillary pressure, with forced imbibition required for the most hydrophobic systems. Furthermore, the range of saturations with negative capillary pressures increased with an increase of the fraction of the S or T components. Bauters et al. [6] also found that capillary pressures were negative for the entire saturation range, even for two QS mixtures with a small fraction of the S component. In contrast, Bradford and Leij [10] found that MI capillary pressure curves for all media were always greater than zero, indicating that spontaneous imbibition occurred. These inconsistencies add to the overall conclusion that the experimental results can by strongly influenced by the experimental techniques. Oil–water systems. For the PD path in oil–water systems (Table 1), results of existing experimental investigations showed that more hydrophobic media generally had a lower capillary pressure at a given water saturation [10,25,50]. Bradford and Leij [10] found that the PD curves for all systems with at least 50% quartz were similar, showing the same trend as their PD curves for air–water systems. For the MI path, Bradford and Leij [10] found that capillary pressures decreased with an increase in the S fraction for a particular saturation, although pure Q and QS 3:1 media had similar curves. Also, the range of saturations with negative capillary pressures increased with the mediumÕs hydrophobicity. Several experiments

reported that negative capillary pressures were required for imbibition for media with more than half of the fraction of the hydrophobic component [3,26,48]. 2.1.2. Relative permeability–saturation relations The effect of fractional wettability on the kr–S relations has been studied in relatively few studies in comparison with the numerous experimental studies for the Pc–S relations. To our knowledge, there have been no experimental investigations for the PD path of the kr–S relations for air–water systems as a function of fractional wettability. Ustohal et al. [51] conducted experiments only for the MD and MI paths in air–water systems. For a particular saturation, they determined that kr values for both the MD and MI paths in a pure T medium were distinctly higher than in a pure Q medium. For a pure S medium, the kr values were between those for pure T and Q media in each MD and MI path. They also found that the hysteresis between the MD and MI paths was significant for pure T medium, whereas there was minimal hysteresis for the pure Q medium. For various mixtures, the hysteresis effects were intermediate of that for the respective pure components. Very few experiments have been conducted for the effect of fractional wettability on the kr–S relations of oil– water systems. Fatt and Klikoff [25] and Singhal et al. [49] found that the changes of the ratio of relative permeability of water to oil are similar to those in uniform wettability systems, with this ratio greater for a pure S medium than for a pure Q medium. Relative permeability curves for the mixtures were between these two extremes. 2.2. Methods to estimate Pc–S–kr relationships Models that estimate the Pc–S relations in porous systems with fractional wettability or with mixed wettability have been developed and successfully verified using experimental data [10,37]. A few researchers have also developed models that estimate the kr–S relations in porous systems with fractional wettability [9,51] or with mixed wettability [1,8,32,35,37,46]. Among the models listed, all models except for the model of Bradford et al. [9] have been successfully tested or verified against experimental data. Bradford et al. [9] developed a model that estimates two- and three-fluid kr–S relations for media with nonzero contact angles and/or fractional wettability by using measured Pc–S data and the mass fraction of organosilane-treated sands. This model still needs to be tested using appropriate experimental data. 2.3. Aims of this research Based on the conflicting nature of the results described above, one of the objectives of this study was

Steady-state method [21,45] PD MI a–w o–w Pure Q, pure S, QS 3:1, QS 1:1, and QS 1:3

c

kr–S D

Q: quartz sand, S: organosilane-treated sand. a–w: air–water, o–w: oil–water systems. PD: primary drainage, MI: main imbibition.

Pc–S C

a

Pc–S B

b

Pure Q, pure S, and QS 1:1 Diameter: 3.0 cm Length: 12.0 cm

Pure Q and QS 1:1 Diameter: 5.5 cm Length: 3.0 cm

Diameter: 5.5 cm Length: 3.0 cm

d50: 0.27 mm (0.11 6 d 6 0.60 mm) qb: 1.74 g cm
3 d50: 0.27 mm (0.11 6 d 6 0.60 mm) qb: 1.74 g cm
3 d50: 0.44 mm (0.05 6 d 6 1.0 mm) qb: 1.86 g cm
3 d50: 0.12 mm (0.05 6 d 6 0.3 mm) qb: 1.64 g cm
3 Pc–S kr–S A

Porous medium

The multi-step outflow technique has been widely used by many researchers for air–water and/or oil–water systems [16,24,30,39,52,55], though not for porous media with fractional wettability. A multi-step outflow technique was used here to simultaneously estimate parameters for the a–w primary drainage path of Pc–S and kr–S constitutive relationships in porous media with fractional wettability (Table 2). The fundamental assumption of the multi-step outflow technique is that the Pc–S–kr relationships can be described by parametric functions for which the unknown parameters can be estimated by minimizing the

Pc–S–kr

3.2. Experiment A: Multi-step outflow method

Experiments

Porous media with fractional wettability were obtained by mixing different proportions (0%, 25%, 50%, and 100%) of the Q and S components (Table 2). The organosilane-coated sand was obtained by shaking quartz sands in a solution of octadecyltrichlorosilane (5%) in ethanol for 5 h and then air-dried [10]. Deionized degassed water was used as the aqueous phase in all experiments and Soltrol 220, an oil composed of a mixture of C13–C17 hydrocarbons, was used for oil–water systems. Interfacial tension (r) for these fluid pairs included: raw = 72 mN/m; rao = 24 mN/m; rwo = 26 mN/m [10].

Table 2 Experimental matrix for Pc–S–kr measurements and verification in this study

3.1. Materials

Column designs

Fractional wettability conditionsa

Table 2 summarizes the experimental conditions for each of the various experiments used in this study to provide and verify Pc–S–kr results as a function of fractional wettability.

Pure Q, pure S, QS 3:1, and QS 1:1

3. Materials and methods

Diameter: 2.5 cm Length: 26.0 cm

PD MI

‘‘Brooks method’’ [10,13,38]

215

o–w

Measurement techniques

Hanging column method PD a–w

PD

Fluid pathsc Fluid pairsb

a–w

to verify the effects of fractional wettability on various saturation paths of the Pc–S–kr relations in air–water and oil–water systems. A range of experimental techniques was employed to separate experimental artifact from significant mechanisms. These techniques included: (1) multi-step outflow experiments to estimate the Pc–S– kr relations of the PD path for air–water systems [30]; (2) hanging column experiments to confirm a–w Pc–S data from multi-step outflow experiments; (3) Brooks method for Pc–S measurements with MD and MI curves; and, (4) steady-state MD and MI o–w kr–S experiments. The model developed by Bradford et al. [9] to estimate two- and three-fluid kr–S relations for media with fractional wettability from measured Pc–S data has not been tested against experimental data. Thus, another objective of this study was to evaluate the suitability of this model to predict kr–S data for a fractionally wet system based on measured Pc–S data.

Multi-step outflow method [30]

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difference between the predicted and observed flow-controlled variables such as flow rate, fluid content, or capillary pressure head. Soils used were packed in a stainless steel column with 5.5 cm diameter and 3.0 cm length. Nylon membrane filter at the base of the soil column allowed drainage of water, but was impermeable to air. A micro-tensiometer, connected to a pressure transducer, was inserted into the middle of the soil column. Four QS mixtures (i.e., pure Q, pure S, QS 3:1, and QS 1:1 media) were used in this experiment. Initially the sand was fully saturated with water, and then subsequently drained to a capillary pressure slightly greater than the air entry pressure, thereby ensuring continuity of air along the length of the soil column. Subsequent stepwise increases of air pressure to initiate outflow of water were applied only after each fluid was in hydraulic equilibrium. As air pressure increments were applied to the top of the soil column, cumulative outflow of water was collected in a flask and its mass monitored as a function of time. Applied air pressure increments were chosen such that outflow volumes were approximately equal for each pressure step. After the outflow rate was reduced to near zero, the air pressure was incrementally increased for the next pressure step. Details of the method are provided by Hwang and Powers [30]. Data collected included continuous readings of cumulative outflow and capillary pressure head at the middle of the soil column in response to step changes in the applied air pressure. Table 3 summarizes experimentally determined saturated (hs), initial (hi), and final (hf) water contents, initial (hc,i) and final (hc,f) capillary pressure heads, and cumulative outflow (Qc) for four media. There were no trends in any of these parameters with fractional wetting. The TF-OPT model developed by Hopmans et al. [29] was used for numerical simulation and inverse modeling to estimate Pc–S–kr parameters based on observed data. Experiments were conducted only for monotonic PD path because the TF-OPT simulator does not include a routine for hysteresis associated with multiple drainage and imbibition pathways. Hwang and Powers [30] found that, among several commonly used Pc–S–kr functions, the Lognormal Distribution-Mualem (LDM) function [34] provided the best, unique parameter sets for both

error-free numerically generated and inherently errorcontaining experimental multi-step outflow data for a quartz sand system. The LDM function is based on the assumption that soils are represented by a lognormal pore-size distribution, yielding a two-parameter model for the Pc–S function: S
w ¼ F n ½ln ðhm =hc Þ=rc  ð1aÞ hw
hr S
w ¼ ð1bÞ hs
hr where S
w is the effective water saturation, hc is the capillary pressure head, hm is the hc at S
w ¼ 0.5, rc is the standard deviation of the lognormal pore-size distribution, Fn is the cumulative normal distribution function, hw is the volumetric water content, and hr is the residual water content. The associated kr–S function is
 2 l 
ð2Þ k rw ¼ S
w F n F
1 n ð S w Þ þ rc where l is the exponent parameter in the pore tortuosity factor [33] and F
1 n is the inverse function of cumulative normal distribution. The unknown parameters defining the Pc–S–kr function are hm, rc, hs, hr, and l. In our study, hs was obtained experimentally from Qc, and hi was fixed to measured values (Table 3). The l value was optimized rather than set at 0.5, and intrinsic permeability (k) was set at its independently measured value (5.0 · 10
7 ± 5.0 · 10
8 cm2) to minimize errors and define unique parameter sets [30]. Consequently, the LDM function was defined by four fitting parameters (hr, hm, rc, and l). The final parameter set was taken as that having the minimum optimization error, where error was defined by the normalized standard deviation (NSD) [16], which was calculated as (ri/bi) · 100%, where ri is the standard deviation of parameter bi as estimated from the parameter covariance matrix. 3.3. Experiment B: Hanging column method Equilibrium Pc–S measurements were conducted using the hanging column method (Experiment B) to confirm a–w Pc–S results from the multi-step outflow experiments (Experiment A). This allowed an evaluation of potential flow-rate dependence of measured Pc–S–kr

Table 3 Experimentally determined water contents, capillary pressure heads, and cumulative outflow data for four fractional wettability media in Experiment A (multi-step outflow experiments) 3


3

Saturated water content, hs (cm cm ) Initial capillary pressure head, hc,i (cm water) Initial water content, hi (cm3 cm
3) Final capillary pressure head, hc,f (cm water) Final water content, hf (cm3 cm
3) Cumulative outflow, Qc (ml)

Pure Q

QS 3:1

QS 1:1

Pure S

0.33 23.7 0.28 55.7 0.03 20.73

0.33 17.8 0.27 64.0 0.02 21.51

0.32 15.5 0.29 57.2 0.02 21.12

0.34 17.8 0.30 54.8 0.01 23.46

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data [27,55]. The same column was used for the hanging column method as used for Experiment A. Hanging column experiments were completed for an air–water system with pure Q and QS 1:1 media (Table 2). 3.4. Experiment C: Brooks method The ‘‘Brooks method,’’ which has been used in Lenhard and Parker [38] and Bradford and Leij [10] was employed to measure the Pc–S curves of both PD and MI paths of oil–water systems for pure Q, pure S, and QS 1:1 media (Table 2). Although this method is much slower than the multi-step outflow method and does not allow simultaneous evaluation of both Pc–S and kr–S relationships, it does provide a mechanism to assess hysteresis between drainage and imbibition curves. The Brooks method requires the equilibrium measurement of fluid mass displaced following a change in the capillary pressure [13]. The soil column (3.0 cm diameter and 12.0 cm length) was outfitted two hydrophilic and two hydrophobic ring tensiometers for independent control of the saturation and monitoring of the pressure for both water and oil phases. These tensiometers were connected to their respective pressure transducers, burettes, and vacuumpressure regulators. The regulators were connected to burettes that were filled with water or oil phases. Three-way valves below the burettes were opened for 2 h at each measurement point to allow fluid flow into or from the porous medium until a new saturation level was established. The valves were then closed to achieve equilibrium fluid distribution in the medium. Fluid pressures in the porous medium were monitored with pressure transducers. Saturation values of each fluid were obtained from the burette reading. Detail of experimental setup and procedure is provided by Lenhard and Parker [38] and Bradford and Leij [10]. 3.5. Experiment D: Steady-state kr–S method Relative permeability was assessed in columns containing a known and constant saturation of fluids [21]. This steady-state method provides opportunities to measure both the PD and MI paths of the kr–S curves of air–water and oil–water systems for a variety of fractional wet media (Table 2). We adopted the experimental apparatus and setup of Demond and Roberts [21] and used an additional closed loop system for flowing water and oil phases into or from the soil column in order to maintain the material balance [45]. The 2.5 cm diameter by 26.0 cm length soil column was horizontally oriented. The column was much longer than used for steady-state kr experiments in petroleum engineering to allow for sufficiently high and measurable pressure drops in media with a relatively high hydraulic

217

conductivity. The small diameter (2.5 cm) minimized variation in saturations with increasing depth across the column diameter. Muqeem [45], who measured steadystate oil–water kr–S curves in very fine sand, adopted a column size (length 40.7 cm and diameter 2.8 cm) similar to ours. For oil–water systems, the chemical composition of the water and oil phases were equilibrated and pumped from the oil–water separator by two programmable digital pumps (Model MCP-Z Standard drive; Ismatec Co, CH-8152, Glattbrugg-Zurich, Switzerland). The water started to flow from the bottom of the windowed separator cell and passed successively through the water pump and column before returning to the separator cell. Therefore, any change in the volume of water inside the column was accompanied by an equal change in the volume of water in the separator cell. Oil and humidified air were also retained within closed-loop systems. The pressure drop inside the column was measured using four tensiometers, a pair for each phase. Two pairs were located on opposite sides of the column, with members of a pair 20 cm apart and 3 cm from either end of the core. Porous ceramic cups for hydrophobic tensiometers were made by treatment with organosilane compound. The tensiometers were inserted directly into the sand via O-ring seals in stainless steel taps screwed into the soil column. The stainless steel tap was connected to pressure transducer and then CR-10X data logger (Campbell Scientific, Inc., Logan, UT 84321-1784, USA). The whole experimental setup was placed in a constant temperature room with an average deviation of 1 C. The packed column initially fully water saturated. The pressure drop was recorded at the four different flow rates to obtain a value for intrinsic permeability. The air or oil phase was introduced at a flow rate just high enough to enable the bleeding of the hydrophobic tensiometers and transducers. The flow rate of water was simultaneously reduced by the same amount to maintain a constant overall flow rate. The pressure drop in both phases was stabilized as a result of keeping the fluid flow rates constant. These steps were repeated until the flow rate of water was equal to zero. The outflow was monitored until the outflow of each phase equaled the inflow and the pressure readings stabilized. The excess produced of either phase determined the change of saturation inside the column. To measure relative permeabilities for the MI path, the direction of the step changes in flow rate was reversed. Demond and Roberts [21] provide details of experimental procedure. 3.6. Testing the Bradford et al. [9] model Bradford et al. [9] developed a kr–S model to estimate kr values from measured Pc–S data, using the pore-size distribution and saturation independent (or dependent) contact angle estimates. They modified the original

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Burdine model [14] to incorporate wettability variations. In their model, wetting and less wetting (non-wetting or intermediate) fluid pore classes are used to calculate kr for water or oil. The wettability of the porous medium is used to determine the contributions of the pore classes to kr. Water relative permeability, krw, and oil relative permeability, kro, are defined, respectively, as ow

ow 2

k rw ðS
w Þ ¼ ðS
w Þ 

½1
f ð/sow Þ

R S
ow w 0

2

RðxÞ dx þ f ð/sow Þ R1 2 RðxÞ dx 0

R1

ow

1
S
w

2

RðxÞ dx ð3aÞ

ow ow 2 k ro ðS
w Þ ¼ ð1
S
w Þ R 1
S
ow R1 2 2 f ð/sow Þ 0 w RðxÞ dx þ ½1
f ð/sow Þ S
ow RðxÞ dx w  R1 2 RðxÞ dx 0

ð3bÞ where the superscripts indicate the fluid phase involved in the system (ow—oil–water system), the subscripts deow notes the fluid phases, RðS
w Þ is the pore-radius distribution, and f(/sow) is an empirical weighting function which depends upon the macroscopic contact angle ow (/sow) of the medium. RðS
w Þ can be estimated from the Pc–S data for the PD path according to [36] ow

RðS
w Þ ¼

2row cosð/sow Þ ow P ow ðS
w Þ

ð4Þ

where r is the interfacial tension, Pow is the capillary pressure, and f(/sow) is calculated by [42]: f ð/sow Þ ¼ 0.5½1
cosð/sow Þ

ð5Þ

Both Eqs. (4) and (5) need values of /sow, which can be estimated by scaling method which uses observed oil– water and air–oil Pc–S data and interfacial tensions (assuming that /sao = 0), using the following relation [10]: cosð/sow Þ ¼

ow rao P ow ðS
w Þ ao row P ao ðS
o Þ

ð6Þ

ow ao at points where S
w ¼ S
o . For systems with fractional wettability (referred to herein as having saturationdependent contact angles), the values of /sow can be calculated at each saturation according to Eq. (6).

4. Results and discussion 4.1. Capillary pressure–saturation relations 4.1.1. Air–water systems In the multi-step outflow experiments (Experiment A), the inverse parameter estimation procedure was applied to data for the PD path of air–water systems with

Table 4 Final optimized parameters with the degree of goodness-of-fit expressed by RMS and their uncertainties expressed by the NSD in Experiment A (multi-step outflow experiments) Parameters
1

hm (cm ) r (–) hr (–) l (–) RMS

Pure Q a

31.9 (0.4) 0.32 (0.9) 0.00 (–b) 2.27 (4.1) 0.570

QS 3:1

QS 1:1

Pure S

29.2 (1.1) 0.51 (1.6) 0.00 (–) 1.92 (7.3) 0.849

32.4 (2.6) 0.48 (3.1) 0.00 (–) 1.56 (22.6) 1.351

30.0 (1.1) 0.39 (1.8) 0.00 (–) 0.80 (18.8) 1.081

a

The number within parenthesis indicates the normalized standard deviation (NSD, %), calculated from (ri/bi) · 100%, where ri is the standard deviation of parameter bi as estimated from the parameter covariance matrix. b Optimized parameter was close to zero, thereby providing a meaningless NSD value.

four QS media to estimate parameters for the LDM function. Final optimized parameter sets for four QS media are listed in Table 4. Uncertainty, which is expressed as NSDs for the optimized parameters, was generally within just a few percent of the parameter values, except for l. Hwang and Powers [30] also found significant error associated with fitting a value to this parameter, but determined that using l as a fitting parameter provided a better overall fit than setting it at 0.5. Fig. 1 illustrates the typical comparison of measured with optimized cumulative outflow and capillary pressure head for the QS 3:1 medium. The RMS value for this experiment (0.85; Table 4), was much smaller than those reported by Chen et al. [16] (1.56–5.94) for multi-step outflow experiments for air and water in two natural soils with uniform wettability, indicating that the estimated Pc–S–kr parameters are reasonable for characterizing soil hydraulic properties of the fractional wettability systems investigated in our study. Fig. 2 presents optimized Pc–S curves for the PD path of air–water systems corresponding to the parameters listed in Table 4 and Eq. (1). These curves suggest that there is little difference among the various QS mixtures. Pc–S curves generated from the multi-step outflow experiments and inverse modeling all show the same capillary pressure at Sw  0.5. Small differences in the slope of the curves, which leads to slight spread in the curves at low and high Sw, shows no consistent trend with the percentage of non-water wetting media in the sample. To verify results from the multi-step experiments, we conducted equilibrium Pc–S measurements (Experiment B—hanging column) for pure Q and QS 1:1 media (circular symbols in Fig. 2). These data also show virtually no differences among the samples and confirm the results of the multi-step outflow experiment provide accurate Pc–S curves. From experiments A and B, we can conclude that the Pc–S curves of the PD path for air–water systems are not significantly affected by fractional wettability condition. This result is different from those from Bradford

S.I. Hwang et al. / Advances in Water Resources 29 (2006) 212–226

15

10

5

Observed Optimized

0 0

20

40

60

80

100

120

140

80 70 60 50 40 30 20 Observed Optimized

10 0 0

20

40

b

60

80

100

120

140

Time, hr

Capillary pressure head, cm water

Fig. 1. Comparison of measured with optimized (a) cumulative outflow and (b) capillary pressure head during the primary drainage path of air–water system for the QS 3:1 medium (Experiment A).

pure Q (Exp. A) QS 3:1 (Exp. A) QS 1:1 (Exp. A) pure S (Exp. A) pure Q (Exp. B) QS 1:1 (Exp. B)

80 60

Capillary pressure head, cm water

Capillary pressure head, cm water

a

4.1.2. Oil–water systems For the PD path of oil–water systems, results of the Brooks method experiments (Experiment C) showed that the capillary pressure head at particular water saturation decreases as the fraction of the S component increased (Fig. 3a). Other experiments conducted by Fatt and Klikoff [25], Talash and Crawford [50], and Bradford and Leij [10] showed a similar trend with ours. Bradford and LeijÕs [10] result showing no difference of the Pc–S curves between pure Q and QS 1:1 media was the only a difference from ours. If we can assume that the physical dimensions of the pores remain constant for different QS mixtures, the only variable potentially responsible for changing the Pc–S curves is the fractional wettability condition. At a particular capillary pressure head, oil would move into a smaller pore which the surface of the S component exists at its pore opening, resulting in a lower water saturation at a given capillary pressure head. The dependence of the Pc–S PD curve for an oil– water system on the fractional wettability is different than the independence observed for the air–water fluid pair. The contrast between these systems is partly due to their effective contact angles. The large difference in

40

80

40 20 0 -20 -40 0.0

a 20 0 0.0

0.2

0.4 0.6 0.8 Effective water saturation

1.0

Fig. 2. Comparison of optimized Pc–S curves (Experiment A) with measured Pc–S data (Experiment B) during the primary drainage path of air–water system.

and Leij [10] and Bauters et al. [6] who showed some, but inconsistent, dependence on the fractional wettability. We believe that our result can be explained by the insensitivity of primary drainage Pc–S curves to changes in the contact angle [43], due to the initial condition where all grains are wetted with water.

b

pure Q QS 1:1 pure S

60

Capillary pressure head, cm water

Cumulative outflow, ml

20

219

0.2 0.4 0.6 0.8 Effective water saturation

1.0

80 pure Q QS 1:1 pure S

60 40 20 0 -20 -40 0.0

0.2

0.4

0.6

0.8

1.0

Effective water saturation

Fig. 3. Measured Pc–S data for oil–water system during the (a) primary drainage and (b) main imbibition paths (Experiment C).

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4.2. Relative permeability curves Based on the consistency in Pc–S results between Experiment A (multi-step outflow) and B (hanging column), we can assume that the kr–S curves simultaneously estimated from Experiment A are also

adequate for the PD path of air–water system. From Experiment A, we found that estimated PD kr–S curves were clearly affected by fractional wettability (Fig. 4a). An increase in the fraction of the non-water wetting S component caused the kr for water to be greater at given ow water saturation. For example, at S
w  0.6, the relative permeability is twice as great for the non-water-wetting pure S sample, indicating significantly greater possible water flow through this sample than the water-wetting Q sample. This trend is confirmed by the steady-state relative permeability measurements (Experiment D) (Fig. 4b). As shown conceptually in Fig. 5, the increase of water kr with an increase of the fraction of the S component can be explained by the presence of these non-waterwetting S components. In a water-wet system, air moves into larger pores that drain under low capillary pressures, leaving water only in the smaller pores with lower permeability. In a system with both water and nonwater wetting grains, air moves into smaller pores near the S components. A greater fraction of water can move in larger pores with higher conductivity. Therefore, water kr increases as the fraction of the S component increases.

1.0 Water relative permeability

density between air and water facilitates the wetting of the solid by water [10]. Wei et al. [53] found that contact angles were much lower for air–water system than for oil–water system. For the PD path, the lower contact angle for the air–water system may result in an insensitivity of the Pc–S curves on the fractional wettability. The fractional wettability did not affect the Pc–S curves for the air–water system during the PD path (Fig. 2), thus, the pore-size distribution can be estimated from measured a–w Pc–S data of field samples without worry about variability in the wettability within the sample. On the other hand, great care must be taken in estimating a pore-size distribution from measured oil–water Pc–S data of field samples with unknown fractional wettability conditions due to the apparent greater importance of fractional wetting that also influences the shape of the Pc–S curve. Results of the MI path of oil–water systems (Fig. 3b) are consistent with the results presented by Bradford and Leij [10]. Capillary pressure head decreased at a given water saturation as the fraction of the S component increased in a manner similar to that observed for the PD path. The QS 1:1 and pure S media had negative capillary ow ow pressure heads for S
w P 0.5 and S
w P 0.1, respectively. Imbibing water beyond this saturation was only possible when oil was under suction relative to water. For the QS 1:1 medium, as water starts to imbibe into the medium at residual water saturation, water would first replace oil near the water-wetting quartz. Subsequently, the thickness of water films on the quartz grains would increase and water will spontaneously displace oil within the pore space near the quartz, entrapping oil. The entrapment depends on both the pore geometry and configuration of the Q and S components. ow This imbibition process continues until S
w  0.5. After this saturation, imbibing water would act as the non-wetting fluid and force is required for water to displace oil from the surface of and pore space near the S component. Two distinct regions (i.e., spontaneous and forced water imbibition regions), therefore, exist for the MI path. The range of saturations requiring forced water imbibition increased for pure S medium compared with that of the QS 1:1 medium (Fig. 3b). This indicates that although the same water pressure is applied for imbibing water into oil contaminated sites, the amount of water imbibed is a function of the hydrophobicity of the porous media. If the fraction of the S component increases, the amount of water imbibed will be smaller under the same capillary pressure.

0.8

pure Q QS 3:1 QS 1:1 pure S

0.6 0.4 0.2 0.0 0.0

a

0.2

0.4

0.6

0.8

1.0

Effective water saturation 1.0

Water relative permeability

220

0.8 0.6 0.4 0.2 0.0 0.0

b

pure Q QS 1:1 pure S

0.2

0.4 0.6 0.8 Effective water saturation

1.0

Fig. 4. (a) Optimized (Experiment A) and (b) measured kr–S curves (Experiment D) during the primary drainage path of air–water system.

S.I. Hwang et al. / Advances in Water Resources 29 (2006) 212–226

221

This finding is consistent with observations of others [22,54], even though the porous media and experimental methods used were different. The explanation for trends in the PD path of kr in an air–water system (Fig. 5) is also applicable to the oil– water system. The decrease of oil kr with an increased fraction of the S component can induce vertical spreading of the oil in porous systems with fractional wettability. 4.3. Testing the Bradford et al. model

Fig. 5. Schematic showing hypothetical distribution of air and water during the primary drainage path for water-wet and fractional-wet systems (adapted from [31]).

For the PD and MI paths of the fractionally wet oil– water systems, the water kr tended to increase and the oil kr decreased at a given water saturation as the fraction of the S component increased (Fig. 6). These results are similar to those in uniform wettability systems [47]. Note that the water kr curve of the PD path was steeper for the pure S medium than in other media (Fig. 6a).

The Bradford et al. [9] model was tested against measured o–w kr–S data (Fig. 6a) in order to determine its effectiveness in predicting o–w kr–S curves from a–w Pc–S curves in fractionally wet systems. To do this, we measured the PD Pc–S curve (Experiment B) for air and water in a pure Q medium and steady-state relative permeability curves (Experiment D) for a range of QS media with the same grain size distribution. The Pc–S curve for the PD path of air–oil system in pure Q medium was estimated by scaling the a–w curve with a ratio of surface tensions [10]. Based on the results of Bradford and Leij [10] that showed that air–oil Pc–S curves were similar between pure Q and pure S media, it

0.8

1.0 pure Q QS 3:1 QS 1:1 pure S

Oil relative permeability

Water relative permeability

1.0

0.6 0.4 0.2 0.0 0.0

a

0.2 0.4 0.6 0.8 Effective water satuation

0.2

0.2 0.4 0.6 0.8 Effective water satuation

1.0

1.0 pure Q QS 3:1 QS 1:1 pure S

Oil relative permeability

Water relative permeability c

0.4

b

0.6 0.4 0.2 0.0 0.0

pure Q QS 3:1 QS 1:1 pure S

0.6

0.0 0.0

1.0

1.0 0.8

0.8

0.2 0.4 0.6 0.8 Effective water satuation

0.8 0.6 0.4 0.2 0.0 0.0

1.0 d

pure Q QS 3:1 QS 1:1 pure S

0.2 0.4 0.6 0.8 Effective water satuation

1.0

Fig. 6. Measured kr–S curves of the (a) water and (b) oil phases during the primary drainage path, and (c) water and (d) oil phases during the main imbibition path (Experiment D).

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was assumed that the air–oil Pc–S curve for pure Q medium estimated in our work can represent curves for all of the QS mixtures. This assumption suggests that the oil strongly wets both Q and S components in comparison with air [10]. The Pc–S curves for oil–water systems were calculated from the estimated Pc–S curve for air–oil system, using the following equations [11] row ao P ao ðS
o Þ
kD rao row  P ao ð0.5Þ ½0.1135  %QS þ 1.81 kD ¼ rao  15.17 ow

P ow ðS
w Þ ¼

ð7aÞ ð7bÞ

where row is the interfacial tension between oil and ao water, Pao(0.5) is the capillary pressure at S
o ¼ 0.5, and %QS is the percentage of the S component in the medium. Bradford and Leij [11] developed Eq. (7b) empirically based on measured Pc–S data for Soltrol– water systems in porous media with a range of fractions of organosilane-treated sands. The pore-size distribution (Eq. (4)) and saturationdependent contact angles in oil–water systems (Eq. (6)) were calculated from the estimated air–oil and oil–water Pc–S data. Relative permeabilities Eq. (3) for the PD path of oil–water systems were then calculated using these estimated pore-size distribution and saturationdependent contact angles.

Relative permeability

Contact angle results Fig. 7 presents the estimated contact angles as a function of saturation and fractional wettability for the PD path of oil–water systems. As expected, higher contact angles were observed in systems with a greater fraction of non-wetting sand grains, although none of the systems can be characterized by contact angles typically associated with non-water wetting systems (e.g., >150). For the primary drainage pathway, where all

Contact angle (degrees)

180 160 140 120 100

pure Q QS 3:1 QS 1:1 QS 1:3 pure S

80 60 40 20 0 0.0

the grains are initially wetted by water, even the pure S medium is characterized as an intermediate wetting (h  90) rather than an oil-wetting system. The basis of the Bradford et al. [9] model is the assumption that the effective contact angle that represents the bulk system behavior varies with saturation. Yet, Bradford and Leij [10], who estimated contact angles for the PD path of pure S medium (Eq. (6)) using a different data set than used here, found that the contact angle was independent of saturation for the PD path and remained constant around 65 over the entire saturation range. In contrast, our findings show that contact angle does depend on saturation—at least weakly—for the PD path, especially at very low and high effective water saturations. We believe that our results are more feasible based on the physical mechanism of contact of the two fluids within the fractionally wet porous medium. As oil starts to displace water during the PD path, oil readily enters pore spaces near the S solids at a low Pow (i.e., high contact angle). As oil moves from pore spaces near the S solids into pore bodies, an intermediate Pow (i.e., lower contact angle) is needed for oil to displace water. Then, as water saturation approaches its residual saturation, a relatively higher Pow, with a correspondingly small contact angle, is required for oil to displace most of water from pore bodies.

0.2 0.4 0.6 0.8 Effective water saturation

1.0

Fig. 7. Estimated saturation-dependent contact angles for the primary drainage path of oil–water system.

Fig. 8 shows comparison between measured kr–S curves and those estimated by Eqs. (3a) and (3b), which Bradford et al. [9] developed by modifying the Burdine model. Due to the relatively small differences in contact angles estimated for the various QS media (Fig. 7), the variation of water and oil kr curves over the range of fractionally wet systems varied much less than measured curves. In addition, the values of the estimated water kr curves were much less than measured. Overall, the relative permeability curves predicted by the Bradford et al. [9] model failed to adequately predict differences in relative permeability curves as a function of the fraction of non-wetting media and failed to adequately predict the measured values for any of the media used. One potential source for error in the Bradford et al. model is the use of the Burdine model as a basis for estimating relative permeability from pore-size distribution data. The Mualem pore-size distribution model [44] has been known to predict higher wetting phase kr values at a particular saturation than the Burdine model [9]. Thus, we modified the original Mualem model to incorporate fractional wetting variations to test it with the Bradford et al. model to assess if the predictions could be improved. The resulting modified Mualem model was defined as

S.I. Hwang et al. / Advances in Water Resources 29 (2006) 212–226

1.0 Water relative permeability

Water relative permeability

1.0 0.8 0.6 0.4 0.2 0.0 0.0 a

0.2

0.4

0.6

0.8

1.0 a

Effective water saturation

pure Q (predicted) QS 3 : 1 QS 1 : 1 QS 1 : 3 pure S pure Q (measured) QS 3 : 1 QS 1 : 1 QS 1 : 3 pure S

0.8 0.6 0.4

Oil relative permeability

Oil relative permeability b

0.8 0.6 0.4 0.2 0.0 0.0

1.0

1.0

0.2 0.0 0.0

223

0.2 0.4 0.6 0.8 Effective water saturation

0.8 0.6 0.4

b

1.0

pure Q (predicted) QS 3 : 1 QS 1 : 1 QS 1 : 3 pure S pure Q (measured) QS 3 : 1 QS 1 : 1 QS 1 : 3 pure S

0.2 0.0 0.0

1.0

0.2 0.4 0.6 0.8 Effective water saturation

0.2 0.4 0.6 0.8 Effective water saturation

1.0

Fig. 8. Comparison of measured with estimated kr–S curves by the modified Burdine model.

Fig. 9. Comparison of measured with estimated kr–S curves by the modified Mualem model.

ow ow 0.5 k rw ðS
w Þ ¼ ðS
w Þ 8 92 R ow R <½1
f ð/sow Þ S
w RðxÞdx þ f ð/sow Þ 1
ow RðxÞdx= 0 1
S w  R1 : ; RðxÞdx

Micro-scale heterogeneous configuration of the Q and S components during packing of media [5,15,28] could have contributed to these discrepancies. For example, Caruana and Dawe [15] conducted immiscible displacement experiments for the PD and MI paths of oil–water systems with three different packing patterns (i.e., stripe, lens, and quadrant patterns of the S component within the Q component matrix). They found that even a single stripe pattern could lead to completely altered flow physics for immiscible flows, resulting in difference of the pore-scale fluid configuration and entrapment and, ultimately, variability in the kr–S relations. It is possible that some of the measured variability in our kr curves could be attributed to non-uniform distribution of Q and S grains within the packed sand samples even though we adopted a careful packing technique [56]. This undesired variable configuration of the Q and S components could result in variability in the measured kr curves that could not be reproduced by the Bradford et al. [9] model. More likely, the empirical correlation for kD (Eq. ow (7b)), which is used to determine P ow ðS
w Þ (Eq. (7a)), does not adequately represent the range of effective contact angles determined in our experiments. This empirical equation was developed with a limited data set. The generation of additional data for Pc–S–kr relationships

0

ð8aÞ ow ow k ro ðS
w Þ ¼ ð1
S
w Þ0.5 8 92 R ow R
ð8bÞ The resulting kr–S curves estimated from the modified Mualem model Eq. (8) were compared with measured data (Fig. 9). The modified Mualem model performed slightly better than the modified Burdine model (Fig. 8) for estimating water kr curves, but a little worse for estimating oil kr curves. The value for kD (Eq. (7b)) was used in both cases, therefore, there was no improvement in the predicted dependence of kr on the fraction of non-water wetting media in the sample. The poor performance of both of these models indicates that they did not adequately represent the system under investigation (oil–water–solid combination).

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in fractionally wet media could improve the predictive capacity of this empirical approach. The model will, however, still be limited by the constraint that the independent variable is the mass fraction of sand treated with organosilane to make it non-water wetting. While an improvement in this empirical model could be used to better assess the overall concept of using a saturation-dependent contact angle, it will not provide any practical application to natural systems.

5. Summary and conclusions Several experimental techniques were used to clarify the effect of fractional wettability conditions on the Pc–S–kr relations of two-fluid systems according to various saturation paths. During the PD path, the air–water system showed no distinct trends as a function of the fraction of the nonwater wetting S component, whereas oil–water system showed a consistent decrease of capillary pressure head with increase of the fraction of the S component. We believe that a much lower contact angle for air–water system may result in insensitivity of the a–w Pc–S curves to the fractional wettability. For the MI path of oil– water system, the capillary pressure decreased as the fraction of the S component increased, showing negative capillary pressures for a certain range of saturation, which is in agreement with results from Bradford and Leij [10]. Systems with an increasing percentage of the S component also exhibited a higher water kr and lower oil or air kr at a given saturation for the PD and MI paths in air–water and oil–water systems. The increase of water kr with an increase in the fraction of the S component can be explained by the ability of water to occupy larger and highly conductive pores in such a system. Experimental kr–S data for the PD path of oil–water system presented here were used to test the Bradford et al. [9] model and the modified Mualem model for estimating the kr–S curves from measured Pc–S data. Neither model resulted in accurate predictions, due to their inability to predict the dependence of kr on the fractional wettability. The error in these models is most likely from the empirical correlation used to estimate ow P ow ðS
w Þ based on a–o Pc–S data and the fraction of organosilane-treated sands. The empirical correlation is based on saturation-dependent contact angles in the fractionally wet systems. While it is likely that the use of such variable contact angles provides the best approximation of the physical phenomena that dictates multiphase flow in these complex systems, it appears that the empirical correlation developed by Bradford et al. [9] could benefit from the incorporation of more experimental data to improve its capability of representing the real systems.

This study has demonstrated the effect of fractional wettability on the constitutive relationships according to fluid pairs and saturation paths. Particularly, for air–water system, our findings can be used to better understand preferential flow paths or unstable flow phenomena as affected by fractional wettability. For NAPL–water system, combined effects of fractional wettability and other factors (temperature or interfacial tensions) can be further explored for identifying water and NAPL distributions in the subsurface. These results can be applied to some protective techniques such as NAPL spill containment as well as traditional remediation techniques (e.g., pump-and-treat or soil flushing). It is possible to separate NAPL that is retained by capillary forces in porous systems with high hydrophobic soil wettability. Therefore, the use of oil-wet capillary barriers [12] appears to become one of feasible options for hindering NAPL spreading.

Acknowledgements Financial support for this research through grants from the DOE Environmental Management Science Program (DE-FG07-99ER 15006) and the National Science Foundation (BES-9981494) is gratefully acknowledged. We are especially grateful to Dr. J.W. Hopmans of the University of California at Davis, for supplying us with his two-fluid flow model TF-OPT.

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