## 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 (*P*_{c}-*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 *P*_{c}-*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]:

where *P*_{c}^{d} is the dynamic capillary pressure, *P*_{c}^{s} is the static capillary pressure, *S*_{w} 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 (*P*_{c}^{s}) is defined as the difference between the nonaqueous and aqueous phase pressures (*P*_{n} − *P*_{w}) when there is no change in fluid saturation. Dynamic capillary pressure (*P*_{c}^{d}) 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 ∂*S*_{w}/∂*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 CO_{2} 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 *P*_{c}-*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 *P*_{c}-*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 θ_{d}, 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 *P*_{c}-*S* and *k*_{r}-*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.