Water Resources Research

Influence of wettability variations on dynamic effects in capillary pressure

Authors


Abstract

[1] Traditional continuum-based multiphase simulators incorporate a capillary pressure-saturation relationship that assumes instantaneous attainment of equilibrium following a disturbance. This assumption may not be appropriate for systems where the capillary pressure is a function of the rate of change of saturation, a phenomenon referred to as dynamic capillary pressure. Previous studies have investigated the impact of soil and fluid properties on dynamic effects in capillary pressure; however, the impact of wettability on this phenomenon has not been investigated to date. In this study, two-phase multistep outflow (MSO) experiments conducted in chemically treated sands with different equilibrium contact angles were used to investigate the influence of wettability variations on dynamic effects in capillary pressure during displacement of water by tetrachloroethene (PCE). Data from the MSO experiments were modeled with a multiphase flow simulator that includes dynamic effects and were also analyzed through comparisons with theoretical model predictions for interface movement in a single capillary tube. Results showed that a faster approach to equilibrium, characterized by smaller fitted damping coefficients, occurred in sands with larger equilibrium contact angles. Damping coefficients for sands with an operational contact angle greater than 80° were found to be an order of magnitude smaller than those with an operational contact angle less than 65°. These results suggest that it may be possible to neglect dynamic effects in capillary pressure in systems that approach intermediate-wet conditions but that these effects will be increasingly important in more water-wet systems.

1. Introduction

[2] Continuum-based multiphase flow simulators used to predict the migration of nonaqueous phase liquids (NAPLs) in porous media commonly incorporate a constitutive expression relating capillary pressure and saturation (Pc-S). These relationships have been shown to be a function of the rate of change in saturation [e.g., Topp et al., 1967; Smiles et al., 1971; Vachaud et al., 1972; Stauffer, 1978; Hassanizadeh et al., 2002; Berentsen et al., 2006; Sakaki et al., 2010], which creates a nonuniqueness in Pc-S relationships that is not related to hysteresis (flow reversals). This dependence on the flow conditions has been attributed to dynamic effects in capillary pressure [Kalaydjian, 1987; Hassanizadeh and Gray, 1990]. A number of expressions that are similar in mathematical form have been proposed to model the nonuniqueness associated with dynamic effects, including dynamic capillary pressure [Stauffer, 1978; Barenblatt and Gil'man, 1987; Kalaydjian, 1987; Hassanizadeh and Gray, 1990]. These expressions incorporate a finite relaxation time required for a system to return to an equilibrium state following a disturbance. The relationship employed in this study is expressed as [Hassanizadeh and Gray, 1993]:

equation image

where Pcd is the dynamic capillary pressure, Pcs is the static capillary pressure, Sw is the water saturation, t is time, and τ is a material coefficient, often referred to as the damping coefficient [Hassanizadeh and Gray, 1993]. In this study the static capillary pressure (Pcs) is defined as the difference between the nonaqueous and aqueous phase pressures (PnPw) when there is no change in fluid saturation. Dynamic capillary pressure (Pcd) is the difference between the nonaqueous and aqueous phase pressures measured at any time. The static and dynamic capillary pressures are only equivalent at equilibrium (i.e., when ∂Sw/∂t = 0). Static capillary pressure, however, can provide a reasonable approximation of dynamic capillary pressure for systems that reestablish equilibrium quickly (i.e., τ ≈ 0). The assumption of τ = 0, and an instantaneous return to equilibrium, is implicit in common multiphase flow simulators that employ constitutive relationships based on equilibrium capillary pressure measurements. This assumption may be inappropriate for systems where high rates of saturation change are expected, such as for water infiltration into dry soil [Hassanizadeh et al., 2002], a rapid release of dense nonaqueous phase liquid into a water saturated soil, or CO2 sequestration in deep subsurface systems.

[3] Dynamic effects in capillary pressure have been investigated in multiple laboratory [e.g., Stauffer, 1978; Hassanizadeh et al., 2002; Manthey et al., 2004; O'Carroll et al., 2005b; Berentsen et al., 2006; Bottero et al., 2006; Sakaki et al., 2010] and modeling [e.g., Dahle et al., 2005; Manthey et al., 2005; Das et al., 2007; Mirzaei and Das, 2007; Juanes, 2008] studies; however, the underlying mechanisms and controlling parameters remain unclear. These investigations suggest that the magnitude of the dynamic effects can be related to fluid properties (viscosity, density, and interfacial tension), porous medium properties (intrinsic permeability, porosity, and pore-size distribution), water saturation, and the size of the system under consideration. Some of these studies have proposed functional relationships for the damping coefficient, based on experimental [Stauffer, 1978] or modeling [Das et al., 2007] work. Many of the modeling investigations have considered macroscopic contributions to dynamic effects, including viscous fingering [Das et al., 2007], macroscopic heterogeneities [Manthey et al., 2005], and boundary pressures [Manthey et al., 2005]. Other studies, however, have explored the contributions of smaller-scale phenomena, including fine-scale heterogeneities [Mirzaei and Das, 2007], forces acting at liquid-liquid and liquid-solid interfaces [Hassanizadeh and Gray, 1993], and dynamic contact angles [Friedman, 1999; Wildenschild et al., 2001; Hassanizadeh et al., 2002; Manthey et al., 2008]. To date, however, the impact of wettability variations on dynamic effects in capillary pressure has not been explored. While the theory presented by Hassanizadeh and Gray [1993] implicitly incorporates wettability by including forces acting at liquid-liquid and liquid-solid interfaces, these investigators did not explicitly examine the influence of wettability.

[4] Wettability, the “tendency of one fluid to spread on or adhere to a solid surface in the presence of another immiscible fluid” [Craig, 1971], is typically represented by the contact angle θ, the angle made by a fluid-fluid interface in contact with a solid surface [Hiemenz and Rajagopalan, 1997]. In a NAPL-water-solid system, the solid surface is referred to as strongly water-wet for θ approaching 0° and strongly NAPL-wet for θ approaching 180°. Solid surfaces in systems where 60° < θ < 130° are typically referred to as intermediate wet [Morrow, 1976]. Contact angles here refer to equilibrium contact angles measured through the water phase. Although it is often assumed that θ = 0° in many porous media systems, where silica is the dominant material [Anderson, 1986], this assumption may be inappropriate for materials that do not exhibit strongly water-wet behavior such as calcite, dolomite, coal, and talc [e.g., Anderson, 1986; Gooddy et al., 2002], materials coated with natural organic material [e.g., Abriola et al., 2005; Ryder and Demond, 2008], or surfaces that have been exposed to surfactants or NAPLs [e.g., Treiber et al., 1972; Barranco and Dawson, 1999; Lord et al., 2000; Harrold et al., 2001; Dwarakanath et al., 2002; Lord et al., 2005; Hsu and Demond, 2007; Ryder and Demond, 2008]. Laboratory experiments and theoretical studies have shown Pc-S relationships to be a function of wettability; lower entry pressures and lower capillary pressures have been observed at intermediate saturations during drainage and imbibition for both uniformly and fractionally wet media that are characterized by larger contact angles [e.g., Morrow, 1976; Anderson, 1987a; Bradford and Leij, 1995; Bradford et al., 1997; Ustohal et al., 1998; O'Carroll et al., 2005a; Hwang et al., 2006; Gladkikh and Bryant, 2007]. These studies, however, have typically quantified the Pc-S relationship under equilibrium conditions.

[5] Under nonequilibrium (i.e., moving interface) conditions, the contact between the fluids and the solid can be characterized by a dynamic contact angle θd, which is not necessarily equal to the equilibrium contact angle [e.g., Weisbrod et al., 2009]. The dynamic contact angle is associated with the dependence of the contact angle on the velocity of the contact line [Hoffman, 1975; Dussan, 1979; de Gennes, 1985], the line that defines the intersection of the two fluid phases and the solid. This concept differs from what is commonly referred to as contact angle hysteresis, the dependence of the contact angle on the direction of the contact line movement. The dynamic contact angle has been shown to decrease with increasing velocity during drainage and increase with increasing velocity during imbibition [e.g., Dussan, 1979]. The fundamental disagreement between conventional hydrodynamic theory, which assumes a no-slip condition at the fluid-solid boundary, and the existence of a moving contact line has inspired substantial research relating to the dynamics of wetting processes, and to dynamic contact angles in particular (see recent reviews by Blake [2006] and Ralston et al. [2008]). The dynamic contact angle has been related to the contact line velocity and the static (i.e., equilibrium) contact angle through various functional relationships [e.g., Blake and Haynes, 1969; Hoffman, 1975; Voinov, 1976; Cox, 1986; Zhou and Sheng, 1990; Brochard-Wyart and de Gennes, 1992]. The approaches used to generate these relationships can be generally divided into two categories: hydrodynamic and molecular kinetic. Hydrodynamic approaches emphasize energy dissipation due to viscous effects in a wedge of liquid near the contact line, and molecular-kinetic approaches emphasize a friction force due to molecular-scale adsorption of the advancing fluid and desorption of the receding fluid at the contact line [Brochard-Wyart and de Gennes, 1992; Blake, 2006; Ralston et al., 2008].

[6] Friedman [1999] hypothesized that dynamic contact angles could account for dynamic effects in capillary pressure. That work suggested that decreased contact angles produced during dynamic drainage would yield higher water saturations, at a given capillary pressure, when compared to those produced during equilibrium drainage. Friedman [1999] suggested that the dynamic effects produced by dynamic contact angles were likely to be more significant for NAPL-gas systems than water-air systems because of the lower interfacial tension. The work of Wildenschild et al. [2001] and Hassanizadeh et al. [2002] supports dynamic contact angles as a plausible mechanism contributing to dynamic effects in capillary pressure but suggests that it is likely to be more significant in NAPL-water systems than in water-gas systems, where equilibrium contact angles are expected to be larger. In a recent theoretical study by Blake and De Coninck [2004], the combined effect of wettability and dynamic contact angle on contact line displacement was investigated. Using a molecular-kinetic approach, they showed that the contact line will move more slowly during drainage or imbibition in systems with a smaller equilibrium contact angle, all other conditions being the same. On the basis of these results, it is hypothesized that the wettability of porous media systems will influence dynamic effects in capillary pressure.

[7] Multistep outflow (MSO) experiments have been proposed as a rapid method to obtain porous media constitutive relationships. In a MSO experiment, a packed column is subjected to successive increases in pressure at the column inlet and the cumulative outflow is measured. A numerical solution of Richards' equation for air-water systems, or multiphase flow equations for any two-fluid system, is then used to match observed outflow response. This approach has been used to rapidly estimate Pc-S and kr-S constitutive relationship parameters in air-water and NAPL-water systems [Kool et al., 1985; Parker et al., 1985; Eching and Hopmans, 1993; Eching et al., 1994; van Dam et al., 1994; Liu et al., 1998; Chen et al., 1999; Hwang and Powers, 2003]. The estimation of these parameters is based on the best fit to cumulative outflow data. The approach to steady state, however, is frequently modeled to be more rapid than observed [e.g., Chen et al., 1999; Schultz et al., 1999; Hwang and Powers, 2003]. Recently, MSO experiments have also been used to estimate the damping coefficient related to dynamic effects in capillary pressure [O'Carroll et al., 2005b]. In that study, incorporation of equation 1 into the governing differential equations resulted in a better fit to observed outflow, particularly for the nonequilibrium portion of the outflow curve, than alternative model fits obtained assuming validity of equilibrium capillary pressure and relative permeability relations (τ = 0). Equilibrium model fits were nearly identical when the capillary pressure-saturation and relative permeability-saturation relations were decoupled, introducing four additional relative permeability-fitting parameters. These modeling results suggest that dynamic capillary effects were particularly important to capturing the MSO data.

[8] In this paper the influence of wettability variations on dynamic effects in capillary pressure is explored through the analysis of the results of MSO experiments, conducted on treated sands characterized by different equilibrium contact angles. MSO data are modeled with a numerical multiphase flow simulator that includes dynamic effects [O'Carroll et al., 2005b]. Differences between experiments are evaluated in the context of a theoretical relationship between dynamic effects and wettability based on interface movement in a single capillary tube.

2. Conceptual Model Based on a Single Capillary Tube Model

[9] Although a single capillary tube does not adequately represent the complexity of a real porous medium, relationships defined at the pore scale can provide insight into processes that manifest as macroscale behavior, including capillary pressure hysteresis [Morrow, 1976], forces acting at fluid interfaces [Hassanizadeh and Gray, 1993], and fluid-fluid interfacial area [Cary, 1994]. Dahle et al. [2005] used a model derived for a single capillary tube to provide insight into dynamic effects in capillary pressure for a constant equilibrium contact angle. They used the Washburn equation [Washburn, 1921] to develop an analogous expression to equation 1, thereby elucidating possible relationships between the damping coefficient (τ) and properties of the medium and fluids. This study follows the methodology of Dahle et al. [2005] but employs a modified form of the Washburn equation [e.g., Blake and De Coninck, 2004] to incorporate dynamic contact angles and their relationship to wettability.

[10] Capillary flow of two immiscible fluids in a single cylindrical pore can be described by the Washburn equation [Washburn, 1921]:

equation image

where ΔP is an externally applied pressure, r is the radius of the capillary tube, γ12 is the interfacial tension between fluids 1 and 2, θ is the contact angle (measured through fluid 1), μ1 and μ2 are the viscosities of fluids 1 and 2, respectively, and l1 and l2 are the lengths of the tube occupied by fluids 1 and 2, respectively, as shown in Figure 1. Fluid 1 is taken to be the wetting fluid and fluid 2 the nonwetting fluid. Equation 2 has been expressed in terms of the viscosities of both fluids [Adamson, 1982] and can be simplified by defining a length-averaged viscosity as

equation image

[Dahle et al., 2005], where L = l1 + l2 is the total length of the capillary tube, to yield

equation image

where the subscript on l1 has been removed (i.e., l = l1). Equation 4 is typically applied using a constant contact angle, taken to be the static contact angle [Washburn, 1921; Blake and De Coninck, 2004; Dahle et al., 2005; Ralston et al., 2008]. However, equation 4 can also be applied using the dynamic contact angle

equation image

[Martic et al., 2003; Blake and De Coninck, 2004; Lee and Lee, 2007]. Several relationships have been proposed for θd = f(dl/dt,θ), based on hydrodynamic and molecular-kinetic approaches [e.g., Blake and Haynes, 1969; Hoffman, 1975; Voinov, 1976; Cox, 1986; Zhou and Sheng, 1990; Brochard-Wyart and de Gennes, 1992].

Figure 1.

Schematic representation of a capillary tube containing a wetting fluid 1 and a nonwetting fluid 2, where the dotted line represents the fluid-fluid interface at equilibrium and the solid line represents the fluid-fluid interface in motion during drainage.

[11] A general form of the relationship used in the hydrodynamic approach, which emphasizes energy dissipation due to viscous effects in a wedge of liquid near the contact line, is

equation image

[Blake, 2006], where Ca is the capillary number (Ca = 1/γ12), U is the velocity of the contact line (U = dl/dt), LM is a macroscopic length scale, Lm is a microscopic length scale that defines a slip length or cutoff length [Ralston et al., 2008], and θm is a microscopic contact angle typically taken to be equal to the static contact angle.

[12] In the molecular-kinetic approach, the movement of the contact line is assumed to depend on the displacement of fluid molecules between adsorption sites within the three-phase (fluid-fluid-solid) region at the contact line [Brochard-Wyart and de Gennes, 1992]. The characteristic frequency and distance between these displacements are used to define a coefficient of contact line friction, which then relates the dynamic contact angle and the contact line velocity

equation image

[Blake, 2006], where ζ is a coefficient of contact line friction. Blake and Haynes [1969] successfully applied the molecular-kinetic approach to describe the dynamic contact angles for a benzene-water interface in a glass capillary treated with trimethylchlorosilane. The value of ζ is given by

equation image

[Blake and De Coninck, 2004], where Δgs* is the specific activation energy of wetting, λ is the average distance between displacements, kB is the Boltzmann constant, and T is temperature. β = μ1υ1/λ3 for gas-liquid systems, and β = μ1μ2υ1υ2/3 for liquid-liquid systems, where υ is the molecular volume and h is Planck's constant (see Blake and Haynes [1969] for further discussion of molecular-kinetic parameters). In the molecular-kinetic approach, the microscopic contact angle changes with velocity and is equal to the dynamic contact angle. Note that U = dl/dt < 0 for drainage here, as l refers to the length of tube occupied by the wetting fluid and θd < θ for drainage.

[13] Blake [2006] also summarized work by Brochard-Wyart and de Gennes [1992] who proposed the following expression that combines both hydrodynamic and molecular-kinetic approaches:

equation image

Equations 6, 7, and 9 have all been used to describe dynamic contact angles; however, a single preferred approach has not been recognized due, in part, to the ability of each model to provide adequate fits to experimental data [Blake, 2006; Ralston et al., 2008]. Equations 6, 7 and 9 have typically been applied to gas-liquid systems [e.g., Martic et al., 2003; Lunati and Or, 2009], and little information is available concerning the values of key parameters (e.g., LM, Lm, and ζ) for application to liquid-liquid systems. However, equation 7 offers an advantage for use in this study because the value of ζ has been linked to wettability for use in gas-liquid systems [Blake and De Coninck, 2004] and may offer the opportunity for similar links in liquid-liquid systems. Ranabothu et al. [2005] found that a molecular-kinetic model was able to provide better fits to experimental dynamic contact angle data when compared to the hydrodynamic model for simple fluids (including water). However, for polydimethylsiloxane oils, their results suggested that neither the molecular-kinetic nor hydrodynamic model adequately represented observed dynamic contact angle behavior for the range of contact velocities tested. Thus, although the molecular-kinetic approach was selected for the present study, the range of fluids to which it can be applied is still a subject of investigation. A combination of equations 5 and 7 gives a modified Washburn equation

equation image

[e.g., Martic et al., 2003; Blake and De Coninck, 2004] and rearrangement gives

equation image

Similar expressions could be derived using other approaches to describe the dynamic contact angle, but modifications would be required to accommodate two liquids [e.g., Cox, 1986]. Although expressions to estimate ζ in liquid-liquid systems have not been established [Blake and De Coninck, 2004], expressions derived for gas-liquid systems suggest that ζ will be greater for systems characterized by smaller equilibrium contact angles [Blake and De Coninck, 2004]. That is, systems with a smaller equilibrium contact angles will experience greater resistance to movement of the contact line.

[14] Following the approach of Dahle et al. [2005], equation 11 can be considered as a pore-scale analogue to the continuum-scale rate of change in saturation, as expressed by manipulating equation 1

equation image

Here −Pcd is analogous to ΔP, Pcs is analogous to 2γ12cosθ/r, and τ is analogous to (2ζ/r + 8μL/r2) in equation 11. Note that, for negligible contact line friction (ζ ≈ 0), τ is analogous to 8μL/r2, as presented by Dahle et al. [2005]. For nonnegligible contact line friction (ζ > 0), the analogy suggests that τ will be a function of not only porous media (r), fluid (μ), and scale (L) properties but also the strength of interactions between the fluids and the solid (ζ).

[15] On the basis of the above discussion, a damping coefficient for a single capillary tube is defined as

equation image

Equation 13 is a new expression presented here to link the previously presented equation 11 for capillary tubes (developed through previously presented equations 25, 7, 8, and 10) to equation 1 used to model macroscopic porous media. A relationship between ζ and θ is required to investigate the effect of wettability variations on τtube using equation 13. Expressions for Δgs* in liquid-liquid systems are not available in the literature [Blake and De Coninck, 2004]. The work of adhesion [WA = γ12(1 + cosθ)] has been successfully used to approximate Δgs* in gas-liquid systems [Blake and De Coninck, 2002] and is assumed here to be a reasonable first-order estimate for liquid-liquid systems. Blake and De Coninck [2002] do not report a theoretical basis for equating Δgs* and WA but were able to closely match experimental results using this assumption, which provides empirical support for the use of this assumption here. Substituting WA for Δgs* in equation 8 yields

equation image

It is important to emphasize that equations 11 and 12 are only analogous, and the units of τ and τtube are not the same. To facilitate the comparison of theoretical τtube values with those calculated from the experiments presented in section 4, dimensionless values of τtube/τtube(θ = 83°) were calculated using equations 13 and 14 and are presented in Figure 2 for L/r = 1. Air-water values were calculated using γ12 = 72 mN/m, υ1 = 3.0 × 10−29 m3, λ = 0.36 nm [Blake and De Coninck, 2002], T = 298 K, and μ1 = μ. PCE-water values were calculated using γ12 = 47.5 mN/m [Demond and Lindner, 1993], υ1 = 3.0 × 10−29 m3, υ2 = 1.7 × 10−29 m3, λ = 0.55 nm, T = 298 K, μ1 = μ, and μ2 = 0.0009 kg/m·s [Pankow and Cherry, 1996]. The values of υ1 and υ2 were approximated based on density and molecular weight, and the λ value of 0.55 nm used for the PCE-water calculations is the estimated size of a PCE molecule based on λ3υ2 [Blake and De Coninck, 2004]. A reference value of τtube(θ = 83°) was chosen to correspond to operative contact angles in the experiments conducted during this work, as discussed in section 3.2. It is implicitly assumed that τtube/τtube(θ = 83°) will behave in a similar manner as τ/τ(θ = 83°). The curves in Figure 2 show that larger damping coefficients are expected for smaller equilibrium contact angles. According to equation 1, larger damping coefficients will result in greater dynamic effects in capillary pressure. This analysis suggests that τ values in water-wet systems (approaching θ = 0°) could differ by an order of magnitude when compared to intermediate-wet systems (approaching θ = 90°).

Figure 2.

Dimensionless damping coefficient for a single capillary tube as a function of the equilibrium contact angle for air-water and PCE-water systems.

[16] It is important to note that the magnitude of the τtube/τtube(θ = 83°) value in Figure 2 is sensitive to the choice of λ, the value of which is not clear for liquid-liquid systems [Blake and De Coninck, 2004]. For example, using a λ value based on water instead of PCE changes the maximum τtube/τtube(θ = 83°) value for PCE-water from approximately 22.1 to 3.7. Similarly, τtube/τtube(θ = 83°) is also sensitive to L/r. For example, increasing L/r by an order of magnitude lowered the maximum τtube/τtube(θ = 83°) value for PCE-water from 22.1 to 21.8 and for air-water from 5.2 to 2.1.

3. Materials and Methods

3.1. Fluids and Porous Media

[17] All MSO and equilibrium experiments were conducted using laboratory grade (99%) tetrachloroethene (PCE) (Aldrich Chemical, Milwaukee, WI) and Milli-Q water, with properties listed in Table 1. The columns for all MSO and equilibrium experiments were packed with a mixture of F35-F50-F70-F110 Ottawa sand (US Silica, Ottawa, IL), with a mean grain size of 0.026 cm, and a uniformity index of 2.79 [O'Carroll et al., 2005a]. The same sand was used by O'Carroll et al. [2005b] to investigate dynamic effects in capillary pressure in water-wet sand.

Table 1. Fluid Properties
PropertyWateraPCEb
Density (kg/m3)9991630
Viscosity (N·s/m2)1.12 × 10−39.0 × 10−4

[18] A batch of the sand was treated with a 5% (by volume) solution of octadecyltrichlorosilane (OTS) (ICN Biomedicals Inc, Aurora, OH) in ethanol [Anderson et al., 1991]. Two additional batches of the sand were treated by immersing the sand in a 5% (by volume) solution of Rhodorsil Siliconate 51T (RDS) (Rhodia Silicones, Rock Hill, SC) in Milli-Q water and lowering the pH below 8 [Fleury et al., 1999]. The two batches of RDS-treated sand were prepared separately (referred to here as batches A and B), which may have resulted in the development of different surface properties, as discussed in section 4.1.

[19] The wetting conditions produced by the OTS and RDS treatments were evaluated by O'Carroll et al. [2005a] by measuring advancing and receding contact angles on smooth quartz slides (Fisher Scientific, Pittsburg, PA) treated using the same procedures used for the sands. The contact angles were measured using the axisymmetric drop shape analysis technique [Cheng et al., 1990; Lord et al., 1997] as described by O'Carroll et al. [2005a]. The measured values of the water receding contact angles are listed in Table 2. These measured contact angles suggest that the OTS treatment produced an organic-wet surface on the slides, and the RDS treatment produced an intermediate-wet surface on the slides according to the contact angle classification of Morrow [1976].

Table 2. Contact Angles Measured Through the Aqueous Phasea
TreatmentReceding Contact Angle on Glass SlidebOperative Contact Angle During Drainage Based on Pc-S Scalingc
  • a

    Standard deviation in parentheses.

  • b

    O'Carroll et al. [2005a].

  • c

    Interfacial tension was equivalent in all systems (41 mN/m).

  • d

    Calculated by scaling water-wet Pc-S fit to match EQ-RDS-A data.

  • e

    Calculated by scaling water-wet Pc-S fit to match EQ-RDS-B2 data [O'Carroll et al., 2005a].

Rhodorsil Siliconate 51T (RDS)66.4° (4.4°)64.4° (1.7°)d, 82.3° (1.4°)e
Octadecyltrichlorosilane (OTS)137.6° (18.0°)83.4° (0.7°)

3.2. Equilibrium Drainage Experiments

[20] Four equilibrium drainage experiments were conducted: one using RDS-treated sand from batch A, two using RDS-treated sand from batch B, and one using OTS-treated sand, referred to here as EQ-RDS-A, EQ-RDS-B1, EQ-RDS-B2, and EQ-OTS, respectively. Results from EQ-RDS-B2 were previously reported by O'Carroll et al. [2005a]. The replicate experiments EQ-RDS-B1 and EQ-RDS-B2 were performed on subsamples of the same treated sand but represent different packings. Primary drainage capillary pressure-saturation data were measured using a pressure cell system based on the design of Salehzadeh and Demond [1999]. Dry sand from the OTS or RDS batches were packed into a column (length = 1.27 cm and ID = 2.54 cm), flushed with carbon dioxide, and flushed with 200 pore volumes of deaired Milli-Q water to begin each experiment at 100% water saturation [O'Carroll et al., 2005a]. The sands in each of the equilibrium drainage experiments were subsamples of the sands used in the MSO experiments, to facilitate comparison between the results obtained using the small pressure cell system and those from the larger MSO columns.

[21] An operative contact angle was calculated using Leverett scaling for each of the equilibrium drainage experiments. The operative contact angle is defined as the contact angle required to scale the van Genuchten Pc-S curve, fit to water-wet (i.e., θ = 0°) sand data, to measured drainage data for the treated sands. van Genuchten model parameters and operative contact angles were fitted by minimizing the square difference between measured and fit water saturation at a given capillary pressure. These operative contact angles represent equilibrium receding contact angles that are characteristic of each porous medium and include the effects of both the OTS or RDS treatment and the porous medium surface. In this work, all drainage curves were fit with van Genuchten [1980]Pc-S expressions [O'Carroll et al., 2005b].

3.3. Multistep Outflow (MSO) Experiments

[22] Three MSO experiments were conducted: one using RDS-treated sand from batch A, one using RDS-treated sand from batch B, and one using OTS-treated sand, referred to here as MSO-RDS-A, MSO-RDS-B, and MSO-OTS, respectively. The experiments were conducted using the approach described by O'Carroll et al. [2005b]. Dry sand from the OTS or RDS batches was packed into a custom-designed aluminum column (length = 9.62 cm and ID = 5.07 cm), flushed with carbon dioxide, and then flushed with a minimum of 30 pore volumes of deaired Milli-Q water to completely saturate the column. PCE flowed from a constant pressure reservoir into the bottom of the column through a PCE-wet PTFE membrane (0.45 μm pore size; Pall Corporation, Ann Arbor, MI), displacing water from the top of the column through a water-wet nylon membrane (0.45 μm pore size; Pall Corporation, Ann Arbor, MI) and into a reservoir with an overflow weir.

[23] Fluid flow was induced by imposing a fixed air pressure above the PCE in the constant pressure reservoir. Water and PCE pressures were measured in the top and bottom column end plates, respectively, using pressure transducers (MicroSwitch, Freeport, IL). The PCE pressure was increased in a series of steps (between 1.0 and 8.6 cm H2O) during each experiment; 9 steps were used in MSO-RDS-A, 4 steps in MSO-RDS-B, and 10 steps in MSO-OTS (Table 3). Steady state was not necessarily achieved prior to the initiation of subsequent steps. Each experiment was stopped when additional pressure increases did not result in significant additional water outflow, at which point it was assumed that residual water saturation had been achieved.

Table 3. Pressure Steps Employed During MSO Experiments
Step No.Time (h)PCE Pressure at Bottom of Column (cm H2O)Change in PCE Reservoir Pressure (cm H2O)Column-Averaged Effective Water Saturation at Start of Pressure Step
MSO-RDS-A
1041.82.61.00
20.144.93.00.98
30.547.32.50.96
412.449.21.90.61
516.152.13.00.49
638.855.13.00.21
745.361.96.80.14
861.567.96.00.02
966.275.98.00.00
MSO-RDS-B
1046.83.91.00
20.0348.41.60.98
322.353.44.90.03
433.959.05.60.00
MSO-OTS
1034.81.41.00
20.735.71.00.95
34.136.91.20.87
416.840.03.00.80
525.641.81.80.34
627.444.72.90.24
740.347.52.80.10
841.652.04.60.04
942.556.84.80.01
1043.165.48.60.00

3.4. Numerical Model

[24] A 1-D, fully implicit, point-centered, fully coupled finite difference multiphase flow simulator was used to analyze the MSO experimental results [O'Carroll et al., 2005b]. The development of the model is discussed by O'Carroll et al. [2005b], and essential elements are presented here for clarity. The governing equation for the aqueous phase is

equation image

where ϕ is porosity, ρw is water density, λw = kw/μw is water mobility, μw is the water viscosity, kw = kkrw is the water permeability, k is the intrinsic permeability, krw is the water relative permeability, g is gravity, and z is the spatial dimension. In the governing equation for the NAPL phase, the NAPL phase pressure is expanded in terms of the aqueous phase and dynamic capillary pressures (Pn = Pcd + Pw). Defining Pcd using equation 1 gives an expression in terms of Pcs and Pw

equation image

where ρn is NAPL density, Sn is NAPL saturation, λn = kn/μn is NAPL mobility, μn is the NAPL viscosity, kn = kkrn is the NAPL permeability, and krn is the NAPL relative permeability. In this expansion, dynamic effects in the NAPL compressibility term are ignored. Static capillary-saturation relations were represented by a van Genuchten [1980]Pc-S function (VG)

equation image

where α, m, and n are the van Genuchten model parameters, Se = (SwSr)/(1 − Sr) is the effective water saturation, and Sr is the residual water saturation. Relative permeability was expressed by the Burdine [1953] relationship

equation image
equation image

where m = 1 − 2/n. The relative permeability-saturation expressions were not varied as a function of system wettability during this study. The appropriateness of this approach was confirmed by observing that use of wettability-modified relatively permeability functions for the MSO-OTS experiment had a very minor impact on the magnitude of the damping coefficient estimated through inverse modeling (further discussed in section 4.4).

[25] The damping coefficient (τ) has sometimes been treated as a constant for the modeling of dynamic effects [e.g., Hassanizadeh et al., 2002]. However, it is widely believed that τ = τ(Sw) [Hassanizadeh et al., 2002; Dahle et al., 2005; Manthey et al., 2005; O'Carroll et al., 2005b; Berentsen et al., 2006; Mirzaei and Das, 2007], although the form of this function remains under investigation [Berentsen et al., 2006]. In this study the following functional form was assumed [O'Carroll et al., 2005b]:

equation image

where the constant A is a fitting parameter. The use of equation 20 was previously found to significantly improve the ability of the numerical simulator to fit outflow data from MSO experiments in water-wet media compared to the use of a constant τ value [O'Carroll et al., 2005b]. Although experimental studies support the use a saturation dependent τ [e.g., Manthey et al., 2004; Sakaki et al., 2010], the assumption of a linear form needs further study.

[26] The MSO experiments were simulated using a column domain discretized with 145 nodes in a variable grid spacing. A maximum grid spacing of 10−3 m was located at the center of the domain, and a minimum grid spacing of 8 × 10−6 m was located near the column boundaries. A dynamic time step adjustment algorithm was employed using time steps between 10−12 and 10 s.

[27] Best fits to observed data were obtained by varying three parameters (Sr, α, and n) for the τ = 0 simulations and four parameters (Sr, α, n, and A) for the τ ≠ 0 simulations to minimize the root-mean-square error based on the difference between measured and predicted values of the cumulative water outflow (RMSEQ). The Pc-S parameters derived from fitting the MSO data were then used to generate Pc-S curves for comparison to Pc-S data obtained from independent equilibrium measurements. Sand and membrane intrinsic permeability, measured independently according to the method of O'Carroll et al. [2005b], were inputs for the model and are listed in Table 4. It is important to note that the τ values obtained in the study, calculated using equation 20 and fitted A values, represent small-scale (i.e., local) damping coefficients [O'Carroll et al., 2005b; Berentsen et al., 2006]. This is in contrast to other studies that have reported macroscopic τ values based on averaging pressures and saturations over length scales of 3 to 100 cm [Manthey et al., 2005; Das et al., 2007; Mirzaei and Das, 2007].

Table 4. Numerical Model Parameters
ParameterMSO-RDS-AMSO-RDS-BMSO-OTS
Input parameters
Sand permeability at Sw =1 (m2)1.08 × 10−111.23 × 10−111.11 × 10−11
Nylon membrane permeability at Sw =1 (m2)3.48 × 10−156.48 × 10−156.62 × 10−15
Teflon membrane permeability at Sw =Sr (m2)7.12 × 10−142.77 × 10−144.70 × 10−14
Porosity0.3110.3190.301
Fitting parameters and RMSE (τ = 0)
α (cm H2O)−16.35 × 10−21.38 × 10−12.25 × 10−1
n7.178.04.42
Sr0.240.00.14
RMSEQ1.502.081.05
RMSES1.72 × 10−13.56 × 10−11.69 × 10−1
Fitting parameters and RMSE (τ ≠ 0)
α (cm H2O)−16.69 × 10−21.50 × 10−12.28 × 10−1
n6.618.04.74
Sr0.240.00.17
A (kg/m·s)3.43 × 1072.76 × 1063.74 × 106
RMSEQ0.571.200.70
Reduction in RMSEQ compared to τ = 0 fit (%)62%42%33%
RMSES1.51 × 10−13.34 × 10−11.60 × 10−1

4. Results and Discussion

4.1. Equilibrium Drainage Experiments

[28] Capillary pressure/saturation data from the four equilibrium drainage experiments are shown in Figure 3, along with data from an equivalent experiment previously conducted in the same size fraction of water-wet sand [O'Carroll et al., 2005b], referred to here as EQ-WW. Experiments with the RDS- and OTS-treated sands exhibited decreased capillary pressures at all effective saturation values, compared to those in the water-wet sand. This decrease in capillary pressure, at a given water saturation, is consistent with other experiments conducted on non-water-wet material [e.g., Morrow, 1976; Anderson, 1987a; Bradford and Leij, 1995; Bradford et al., 1997; Ustohal et al., 1998; O'Carroll et al., 2005a; Hwang et al., 2006] and supports the expectation that less energy, characterized by the area under the Pc-S curve [Leverett, 1941], is required for drainage in such materials [Donaldson and Alam, 2008]. Capillary pressure/saturation data for the two RDS treatments are distinctly different. Sands from these two experiments were from different batches leading to these observed differences. Because the same sand treatment procedure was used for both batches, it is unclear why each batch yielded differing wettability behavior. For the treatments used in this study, the OTS-treated sand had the largest operative contact angle (Table 2). This is qualitatively consistent with the contact angles measured on glass slides, where the OTS-treated slides also exhibited the largest contact angle. However, fitted contact angles exhibit two important discrepancies with those measured on glass slides. First, the operative contact angle for the OTS-treated sand falls within the range for an intermediate-wet system (i.e., 60° < θ < 130°) rather than the NAPL-wet system (i.e., θ > 130°) that was suggested by the glass slide measurement. This discrepancy was previously noted by O'Carroll et al. [2005a], who stated that the operative contact angle of θ ≈ 90° was consistent with the negligible capillary pressure behavior observed during the infiltration of PCE into OTS-treated sand [O'Carroll et al., 2004]. In O'Carroll et al. [2004], use of an operative contact angle greater than 90° for OTS-treated sand in the numerical simulator led to results that were inconsistent with experimental sandbox behavior. Second, the two separate RDS treatments (A and B) did not produce similar operative contact angles; treatment B produced an operative contact angle similar to the OTS treatment. As discussed above, the cause of the discrepancy in operative contact angles between the two separate RDS treatments is unknown.

Figure 3.

Comparison of equilibrium drainage measurements (symbols) and van Genuchten curves (lines) derived from parameter optimization in the simulation of multistep outflow data, assuming no dynamic effects. Parameters derived from MSO data simulations incorporating dynamic effects, as described by equations 1 and 20, produced similar van Genuchten curves, which are not shown for clarity.

4.2. Multistep Outflow (MSO) Experiments

[29] The outflow response from the three MSO experiments is shown in Figure 4 as cumulative outflow (Q) versus time. Although the results from MSO-RDS-B appear to have resulted from a single pressure step, four pressure steps were used. The majority of the water outflow occurred during the second step, with the next step being initiated 22.3 h into the experiment (Table 3). Each of the experiments exhibits an initially rapid increase in outflow, followed by a slower approach to zero outflow (i.e., steady cumulative outflow), subsequent to each step change in the PCE boundary pressure. Previous studies of water-wet media have shown that the slow approach to zero outflow is indicative of dynamic effects in capillary pressure if the relative permeability to each phase is not small (i.e., at intermediate fluid saturations) [O'Carroll et al., 2005b].

Figure 4.

Data from the multistep outflow experiments (symbols) compared to best fit numerical simulations (lines) assuming (a) no dynamic effects and (b) dynamic effects described by equations 1 and 20. The measured data in Figures 4a and 4b are the same, and only selected data points (approximately every 60th point) have been plotted for clarity.

[30] Direct comparisons of the approach rate to zero outflow among the MSO experiments is not easily accomplished using Figure 4 due to the different magnitudes of the pressure steps, the different magnitudes of the outflow response produced, and the different water saturation conditions when the pressure steps were initiated. To facilitate a comparison of the experimental data, the outflow response from one pressure step in each experiment was chosen. The outflow responses selected were those that achieved near-steady cumulative outflow (∂Q/∂t ≈ 0) before the initiation of the next pressure step and took place at effective water saturations of Se ≈ 0.4 to Se ≈ 0.8. On the basis of these criteria, the selected curves represent reasonably similar conditions. While the achievement of ∂Q/∂t ≈ 0 is not required for the determination of Pc-S parameters by inverse modeling, its attainment facilitates the direct comparison among MSO experiments by allowing the normalization of the outflow as described below. The effective water saturations, and those listed in Table 3, are column-averaged values based on the cumulative outflow during the experiment. The selected outflow responses (Figure 5) have been shifted such that t = 0 corresponds to the initiation of the pressure step (referred to here as relative time), and normalized to the cumulative change in outflow, such that the initial normalized outflow is zero, and the final normalized outflow is 1, or

equation image

where QN is the normalized cumulative outflow, Q is the cumulative outflow at any time, Qi is the cumulative outflow at the initiation of the pressure step, and Qf is the cumulative outflow immediately prior to the initiation of the next pressure step. In addition to the MSO experiments conducted in this study, selected outflow responses from water-wet experiments (referred to here as MSO-WW-A and MSO-WW-B) [O'Carroll et al., 2005b] are also plotted in Figure 5 for comparison. The outflow step selected from MSO-WW-A (step 4) resulted from a PCE pressure change of 6.84 cm of water and produced a decrease in effective water saturation from 0.48 to 0.14. For MSO-WW-B, the selected step (step 6) resulted from a PCE pressure change of 4.53 cm of water and produced a decrease in effective water saturation from 0.73 to 0.52.

Figure 5.

Selected, scaled outflow steps from the multistep outflow experiments in RDS- and OTS-treated sand, compared to outflow steps from previously conducted multistep outflow experiments in water-wet sand [O'Carroll et al., 2005b].

[31] The scaled outflow curves in Figure 5 reveal that zero outflow was approached faster in the experiments with the OTS-treated sand than with the water-wet sand. This faster approach is consistent with a decrease in dynamic capillary pressure effects associated with a larger equilibrium contact angle, as discussed in section 2. However, it could also be attributed to an increase in relative permeability, as larger contact angles allow the drainage of water through larger pores. The outflow response in the RDS-treated sand varied considerably between the experiments conducted using material from batches A and B. The response in MSO-RDS-A was similar to the water-wet experiments, but the response in MSO-RDS-B was similar to the experiments with OTS-treated sand. The similar behavior of the MSO-RDS-B and MSO-OTS experiments is consistent with their similar equilibrium drainage Pc-S curves (Figure 3) and the correspondingly similar operative contact angles of 82.3° and 83.4°, respectively. The faster response in MSO-RDS-B and MSO-OTS is consistent with the theory presented in section 2, where decreased dynamic effects are expected for systems characterized by larger equilibrium contact angles.

4.3. Numerical Simulations

[32] The MSO experimental data were first fit using the model described in section 3.4, using the van Genuchten-Burdine Pc-S-kr relationships, and τ = 0 (i.e., no dynamic capillary pressure effects). This is representative of the conventional approach for modeling multiphase flow, where Pcs is achieved instantaneously following a disturbance to a system at equilibrium [Hassanizadeh and Gray, 1993]. The best fit simulation results for the τ = 0 case are plotted in Figure 4a, and the best fit values of the parameters are listed in Table 4 along with the RMSEQ values. The simulated outflow curves for each of the experiments fails to capture the approach rate to zero outflow and the steady cumulative outflow value. Simulated curves tend to exhibit a more rapid approach to steady cumulative outflow and achieve a lower cumulative outflow plateau as the system reaches equilibrium. Similar differences have been observed for simulations of MSO outflow data during the drainage of water-wet sands [Chen et al., 1999; Schultz et al., 1999; Hwang and Powers, 2003; O'Carroll et al., 2005b]. It has been previously reported that this slower approach to zero outflow in a PCE/water/quartz sand system could not be matched by a simulator that neglects dynamic effects [O'Carroll et al., 2005b]. The lower steady cumulative value obtained in the simulations is a consequence of the root-mean-square error minimization procedure, where minimization of the objective function causes the model to undershoot the plateau while overshooting its rate of approach. The differences between the observed and simulated values is more apparent in data from MSO-RDS-A than from MSO-RDS-B or MSO-OTS, but the differences are generally less than those reported for MSO experiments in water-wet sands [O'Carroll et al., 2005b].

[33] Fit values of the Pc-S function parameters using τ = 0 were used to generate Pc-S curves using equation 17 for each of the three sands and are compared to data from the equilibrium drainage experiments in Figure 3. Also included, for comparison, is the Pc-S curve for the water-wet sand generated using parameters fit using inverse modeling to data from MSO-WW-A [O'Carroll et al., 2005b]. The root-mean-square errors for the Pc-S curves were calculated based on the difference between the measured and VG model-generated values of Se (RMSES) and are listed in Table 4. The RMSES was based on differences in Se rather than Pc to reduce the importance of data at high and low effective water saturations and increase the importance of data in the intermediate effective water saturation range [O'Carroll et al., 2005b]. The VG Pc-S functions based on fits to the MSO data using τ = 0 were generally consistent with the static Pc-S measurements, despite the inability of the multiphase flow simulator to fit the outflow data using τ = 0. Note that, in the equilibrium capillary pressure/saturation experiments, large decreases in water saturation were observed following a small increase in capillary pressure, when the operative contact angle was large. The VG Pc-S curves generated from the MSO approach, however, do not conform to these abrupt changes in water saturation following small increases in capillary pressure. Nevertheless, the saturation levels at higher capillary pressures are consistent with the static measurements. Thus, this analysis indicates that Pc-S functions fit using MSO data without considering dynamic effects in capillary pressure can yield Pc-S curves that are in good agreement (i.e., similar capillary pressure at 50% effective water saturation) with independent Pc-S experiments for a broad range of wettability conditions.

[34] MSO experimental data were next fit using the same approach as described above (i.e., fitting VG parameters α and n as well as residual water saturation) and assuming τ ≠ 0 (equation 20) to investigate whether dynamic effects could improve the fit. The best fit simulation results are plotted in Figure 4b, and the best fit values of the parameters are listed in Table 4. The inclusion of τ as a linear function of Se significantly improved the fits to the MSO data. The RMSEQ values for MSO-RDS-A, MSO-RDS-B, and MSO-OTS decreased by 62%, 42%, and 33%, respectively. In addition, the simulation results provided much better fits to the approach to zero outflow and the magnitude of the steady cumulative outflow (Figure 4b). Previous work demonstrated that a modeling approach that decoupled the capillary pressure and relative permeability constitutive relationships, adding three additional fitting parameters, produced nearly identical agreement between the numerical model and experimental MSO data [O'Carroll et al., 2005b]. Thus, this alternative modeling approach was not considered herein.

[35] Only relatively small differences exist between the fit values of the Pc-S function parameters for τ = 0 and τ ≠ 0 (Table 4). The resulting Pc-S curves are similar to those shown in Figure 3 and, thus, a separate set of curves is not shown. These results demonstrate that the static Pc-S data can also be well matched using the fit parameter values based on τ ≠ 0 simulations, as indicated by the similar RMSES values calculated for the τ = 0 and τ ≠ 0 parameter fits (Table 4).

4.4. Effect of Contact Angle on the Damping Coefficient (τ)

[36] The RMSE-optimized A values differed significantly among experiments (Table 4). MSO-RDS-A, characterized by the lowest operative contact angle of these experiments, had a RMSE-optimized value of A = 3.43 × 107 kg/m·s. This lies between the values of A = 5.64 × 107 kg/m·s and A = 1.99 × 107 kg/m·s reported for water-wet material [O'Carroll et al., 2005b]. The MSO-RDS-B and MSO-OTS experiments, with higher operative contact angles, had RMSE-optimized values of A = 2.76 × 106 kg/m·s and A = 3.74 × 106 kg/m·s, respectively.

[37] It is important to examine whether discrepancies between observed and simulated cumulative water outflow can be attributed in part to the selected relative permeability functions, which determine the rate of fluid flow and, therefore, the transient drainage of a two-fluid system. The selection of wetting and nonwetting fluids used to assign these relative permeability functions becomes increasingly ambiguous as a system approaches an operative contact angle of 90°, as was the case for the OTS and RDS-B experiments. It is possible that the faster approach to steady state observed in these experiments, and the correspondingly smaller τ values, were due in part to the increased permeability of the treated sand to NAPL, associated with the increased operative contact angle. To test the sensitivity of the A values to the relative permeability expressions used in the multiphase flow simulator, the MSO-OTS experiment was also simulated with the wettability of the fluids reversed. That is, water permeability was calculated using the nonwetting phase relative permeability expression (equation 19), and the NAPL permeability was calculated using the wetting phase relative permeability expression (equation 18). This is expected to bound the dynamic behavior, based upon theoretical [e.g., Bradford et al., 1997] and experimental [e.g., Anderson, 1987b] work that has shown that water relative permeability tends to increase and NAPL relative permeability to decrease as wettability changes from water-wet to NAPL-wet. Results of simulations employing this relative permeability reversal revealed that the new RMSE-optimized value of A = 1.83 × 106 kg/m·s was not significantly different, based on a 95% confidence limit, from the former value of A = 3.74 × 106 kg/m·s. This result demonstrates that the choice of wetting fluid did not affect the magnitude of τ determined using the simulations.

[38] The τ values, based on equation 20 and the A values in Table 4, for MSO-RDS-A (operative contact angle of 64.4°), and those of the two previously reported water-wet sand experiments, are an order of magnitude greater than those for MSO-RDS-B and MSO-OTS (operative contact angles of 82.3° and 83.4°, respectively). This trend is consistent with the values presented in Figure 2 for a PCE-water system, where τtube/τtube(θ = 83°) increases as the system becomes more water wetting. Here it is expected that τtube/τtube(θ = 83°) behaves in a similar manner as τ/τ(θ = 83°). While the observed increase in τ with decreasing equilibrium contact angle is consistent with Figure 2, the small difference between the τ values for MSO-RDS-A and the two previously water-wet sands is not. For example, based on the operative contact angles and trends presented in Figure 2, τ values for MSO-RDS-A should be approximately 3 times larger than the τ values for MSO-RDS-B and MSO-OTS [θ = 64.4°, τtube/τtube(θ = 83°) = 3.0] and approximately 4 times less than those in the water-wet system [θ = 34.4°, τtube/τtube(θ = 83°) = 11.9]. This suggests that while the capillary tube model provides qualitative insight, it is insufficient to provide a quantitative analogy to actual porous media in this context. Overall, however, the increase in experimentally determined τ values with decreasing equilibrium contact angle is consistent with the conceptual model presented in section 2 and the expectation of increased dynamic capillary pressure effects with decreased equilibrium contact angle.

5. Summary and Conclusions

[39] A series of MSO experiments conducted in 1-D columns was used to explore the influence of wettability on dynamic effects in capillary pressure. These effects were qualitatively compared to a theoretical model based on the consideration of dynamic contact angles during interface movement in a single capillary tube. The dynamic response in the cumulative column outflow was found to be faster in experiments in an OTS-treated sand, characterized by a higher operative contact angle, than in experiments conducted in water-wet or RDS-treated sands. As has been reported for water-wet sands [O'Carroll et al., 2005a], a numerical model using traditional constitutive relationships, which neglect dynamic effects (τ = 0), failed to adequately fit the dynamic outflow data. However, the discrepancy was less pronounced in experiments using RDS and OTS sand with higher operative contact angles. Despite the poorer representation of outflow data when assuming τ = 0, Pc-S function parameters estimated through inverse modeling were consistent with equilibrium experiments, regardless of wettability. Including dynamic effects in the numerical simulations, using a damping coefficient that is a linear function of effective water saturation (τ = −ASe + A), greatly improved the numerical fit to the measured MSO outflow data. These fits yielded damping coefficient values for material with equilibrium contact angles greater than 80° that were an order of magnitude less than those for material with equilibrium contact angles less than 65°.

[40] The MSO experiment results reported here show that an increased equilibrium contact angle produced a faster approach to steady state during drainage, a reduced discrepancy between observed and simulated outflow values using τ = 0 and smaller τ values as a function of saturation. These previously unreported effects are consistent with the model presented here based on interface movement in a single capillary tube, which predicts increased damping coefficients with decreasing equilibrium contact angle based on dynamic contact angles. While the consideration of dynamic contact angles alone does not explain all observations related to dynamic capillary pressure effects, this model does help to explain observations of this study related to wettability. These results suggest that dynamic effects in capillary pressure are likely not only a function of material, fluid, and scale properties but also depend on the interaction of the fluids and solid and that this interaction may be largely responsible for local-scale dynamic capillary pressure effects. Furthermore, the theoretical and experimental results presented here suggest that it may be possible to neglect dynamic effects in systems that approach intermediate-wetting conditions but that they will be increasingly important for more water-wet systems. The understanding of dynamic effects in capillary pressure and the implications of including those effects in numerical simulators would benefit from additional experiments in materials of differing wettability, theoretical τ expressions that incorporate wettability at the REV-scale (e.g., application of the molecular-kinetic approach at the macroscopic scale and developing a functional relationship between τ(Sw) and operative contact angle), and field-scale simulations that include dynamic effects.

Acknowledgments

[41] This research was supported by grant DE-FG07-96ER14702, Environmental Management Science Program, Office of Science and Technology, Office of Environmental Management, U.S. Department of Energy (DOE), and by the Natural Sciences and Engineering Research Council (NSERC) of Canada through a Strategic Grant (O'Carroll/Gerhard) and a Postdoctoral Fellowship (Mumford). Any opinions, findings, conclusions, or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the DOE or NSERC.

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