## 1. Introduction

[2] Two-phase flow processes are usually modeled by the mass balance equation and Darcy's equation for each phase. When combined, they result in the following equation [*Bear*, 1972; *Helmig*, 1997]:

where denotes porosity, *S* is the saturation, *k _{r}* is the relative permeability,

**k**is the intrinsic permeability, denotes the dynamic viscosity,

**g**is the gravity acceleration, indicates the density, and is the pressure gradient. The subscripts

*w*and

*n*denote wetting and nonwetting phases, respectively. All quantities appearing in these two equations are macroscale quantities. Assuming that porosity, relative permeability, and fluids properties are known, and considering that

*S*+

_{w}*S*= 1, the unknowns in this set of equations are:

_{n}*P*,

_{n}*P*, and

_{w}*S*. Obviously, we are short of one equation. Commonly this deficit is eliminated by an equation which states that the pressure difference between the two phases is a function of wetting phase saturation:

_{w}This is also known as “the capillary-pressure saturation relationship” because in the literature, the fluids pressure difference is commonly assumed to be the macroscale “capillary pressure” *P _{c}*. Thus, one commonly writes:

This expression is a source of some of the misunderstandings associated with two-phase flow theory. In fact, the macroscale pressure difference *P _{n}* –

*P*is not the same as capillary pressure

_{w}*P*. This holds even at the pore scale where the capillary pressure originally is defined. The capillary pressure is an intrinsic property of the porous medium and the two fluids. For a meniscus, it is defined, independently of the fluid pressures, by the Young-Laplace equation:

_{c}where represents the fluid–fluid interfacial tension, *r*_{1} and *r*_{2} denote the principal radii of curvature of the meniscus, and *R _{m}* is the mean radius of curvature. Thus, through

*r*

_{1}and

*r*

_{2}the capillary pressure depends on the pore dimension and through the interfacial tension it depends on the interfacial properties of the fluids and the porous solid. Equation (4) is assumed to be valid whether the interface is moving or not. The relationship between

*p*–

_{n}*p*and

_{w}*p*in a pore, however, depends on flow conditions. This relationship can be derived from the force balance in the direction normal to the interface. In this respect,

_{c}*Hassanizadeh and Gray*[1993] derived the following force balance equation in the direction N normal to an interface:

where *p _{n}* and

*p*are microscale pressures of the two fluids on two sides of the meniscus, and are their corresponding viscous stress tensors, and is the interfacial viscous stress tensor. It is evident that in such a case

_{w}*p*–

_{n}*p*is not equal to . Even if interfacial viscous forces (last term in equation (5)) are negligible or not present at all,

_{w}*p*–

_{n}*p*may still be different form

_{w}*p*because of dissipative forces within the fluids surrounding the meniscus. The significance of the effect of these forces has been shown, for instance, by

_{c}*Sheng and Zhou*[1992], who have studied the motion of a meniscus in a tube during piston displacement of a wetting phase by a nonwetting phase or vice versa. They found:

where *p _{n}* and

*p*are pressures on the two sides of the interface, is viscosity,

_{w}*q*is fluid velocity through the tube, and B and A are coefficients that control the velocity-dependent “capillary pressure.” Thus, the relationship

*p*–

_{n}*p*=

_{w}*p*is valid at the meniscus under static conditions only. Under dynamic conditions,

_{c}*p*–

_{n}*p*depends on the flow velocity.

_{w}[3] Even if the pore scale nonequilibrium term in equation (6) is negligible, so that we have *p _{n}* –

*p*=

_{w}*p*at all menisci within a representative elementary volume (REV) under flow conditions, it does not mean that the same can be said at macroscale. Obviously, since fluid pressures are spatially variable within each flowing phase, macroscale (or average) pressure values will be different from pressure values at the interface. This has been shown by

_{c}*Dahle et al.*[2005] through a calculation of average pressures in a bundle of tubes model. In fact, on the basis of thermodynamic considerations,

*Hassanizadeh and Gray*[1990] and

*Kalaydjian*[1992] derived an equation which relates the difference in the macroscale fluids pressure,

*P*–

_{n}*P*, to the macroscale capillary pressure

_{w}*P*and the rate change of saturation. A linear approximation is given as

_{c}where is a dynamic capillarity coefficient, also called a damping coefficient. In this formulation, *P _{c}* is an intrinsic property of the porous medium-fluids system, whereas fluids pressure difference

*P*–

_{n}*P*is dependent on flow dynamics (and thus initial and boundary conditions). In the literature, the terms “static capillary pressure” (referring to

_{w}*P*) and “dynamic capillary pressure” (referring to,

_{c}*P*–

_{n}*P*), have been used. This is potentially confusing terminology; in fact, there is only one capillary pressure, as explained above. So, we propose to refer to

_{w}*P*–

_{n}*P*as nonequilibrium

_{w}*fluids pressure difference*or simply as

*pressure difference*.

*P*(

_{c}*S*) denotes the capillary pressure which is measured experimentally as a function of saturation in equilibrium experiments. Obviously, the pressure difference between the wetting and nonwetting phase is equal to capillary pressure only under equilibrium conditions. In fact, equation (7) suggests that when the equilibrium is disturbed, the saturation will change to re-establish the equilibrium condition, and the coefficient controls this process.

_{w}[4] Most nonequilibrium experiments reported in the literature deal with unsaturated flow [*Topp et al.*, 1967; *Smiles et al.*, 1971; *Vachaud et al.*, 1972; *Stauffer*, 1978; *Wildenschild et al.*, 2001; *Chen*, 2006; *Sakaki et al.*, 2010; *Camps-Roach et al.*, 2010]. A comprehensive review of nonequilibrium laboratory experiments was given by *Hassanizadeh et al.* [2002]. A review of computational models is found in the work of *Joekar-Niasar and Hassanizadeh* [2011]. Only a few nonequilibrium experiments have involved two phases, such as oil and water [*Kalaydjian*, 1992; *Hassanizadeh et al.*, 2004; *O'Carroll et al.*, 2005]. In this work, we performed a series of nonequilibrium drainage experiments, in order to investigate the nonequilibrium capillarity effect in two-phase flow involving water and PCE. Nonequilibrium primary drainage experiments were carried out in a homogeneous sand column with a range of (large) injection pressure values. Experimental results are presented and compared with equilibrium experimental data. First, local-scale values of the damping coefficient were calculated on the basis of locally measured values of pressures and saturations. Next, column-scale values of were calculated on the basis of average pressures and saturation values. The results are compared to determine scale dependence of the dynamic coefficient.