The fluid flow processes in porous media are determined by the interplay of various forces (e.g., pressure, gravitational, and viscous forces) and factors such as temperature, medium permeability, and heterogeneity.[1-8] In the case of single-phase flow in porous media, the flow behavior is generally described by the Darcy law[9, 10]
where pressure gradient ∇P is the driving force, q is the Darcy flux defined as the flow rate per unit cross-sectional area, μ is the fluid viscosity, and K is the intrinsic permeability of the porous medium. In the case of saturated two-phase flow in porous media, Eq. (1) can be extended based on various assumptions including that the driving force for the fluid phases is determined by the gradient in phase pressures for that particular phase and other forces such as gravity. The extended form of Darcy's law for two-phase flow can be expressed in the following form
where ∇z is the upward unit vector, ργ is the fluid density, and Krγ is the relative permeability. When the pore space is occupied by two immiscible fluid phases, an interface exists between the two phases. The pressures in the two phases near the interface will not be the same, and a pressure difference will exist across the interface. This pressure difference is referred to as capillary pressure (Pc). In the context of two-phase flow in porous medium, Pc is the pressure required to drive a fluid through the pores and displace the pore-wetting fluid. For example, as the pore size becomes smaller, higher capillary pressures are generally required to displace fluids. Therefore, in the description of two-phase flow in porous media, capillary forces play a central role. The main theoretical tool currently used to quantify the capillary pressure function is an empirical relationship obtained under equilibrium conditions between the average pressures of the individual phase in the form of
where Pnw and Pw are the average pressures of the nonwetting and wetting phases, respectively, and S is the wetting phase saturation. As discussed by various authors (e.g., Das et al.), Eq. (3) is defined to account for any parameter that influences the equilibrium distribution of the fluid phases in the porous domain. However, it has been shown that the fluid phases do not necessarily flow under steady condition (∂S/∂t = 0), particularly at smaller time periods when the time derivative of saturation (∂S/∂t) may not be ignored.[4, 6, 7] This fact has been supported by a number of experimental and modeling studies that show that the capillary pressure relationships depend on the flow conditions, that is, whether it is at steady or unsteady state.[7, 11, 12] Consequently, there are many authors who suggest that the conventional steady-state capillary pressure relationship (Eq. (3)), which defines that Pc is a function of fluid equilibrium saturation alone, must be modified to account for the two-phase flow behavior under dynamic conditions.[5, 7, 13, 14] To take into account, the dynamic capillary pressure effects on two-phase flow in porous media, Hassanizadeh and Gray proposed a relationship, which indicates that the conventional Pc–S relationships (Eq. (3)) can be generalized to include a capillary damping or dynamic coefficient τ as follows
where Pc,dyn is the dynamic capillary pressure ( − ), Pc,equ is the capillary pressure at equilibrium conditions ( − ), which is calculated from Eq. (4), and ∂S/∂t is the time derivative of saturation, all measured at the same fluid saturation value (S). As evident, Eq. (4) has the general form of a straight line, and it should pass through the origin in this form. The slope of this linear relationship is the capillary damping coefficient or the dynamic coefficient (τ). If τ is small, the equivalence between Pc,dyn and Pc,equ is established quickly. On the other hand, the necessary time period to reach the equilibrium is high for larger τ values. Thus, the dynamic coefficient (τ) behaves as a capillary damping coefficient and indicates the dynamics of the two-phase flow system. In the last decade or so, there have been significant interests in determining the range of validity of Eq. (4) as well as obtaining τ values under different situations.[4-7, 11-23] For example, τ has been shown to depend on the medium and fluid properties,[4, 18, 20] degree of heterogeneity,[4, 21] wettability and contact angles in porous medium, temperature,[6, 22] and scales. The dynamic capillary pressure relationship has been studied further using thermodynamically constrained averaging theory where the effects of interfacial area between fluid phases have been discussed. The significance of the dynamic capillary pressure effect is also evidenced by the fact that a number of recent studies have focussed on developing numerical simulators[25-27] and artificial neural network tools for two-phase flow in porous media based on the dynamic capillary pressure relationship (Eq. (4)).
Despite the importance and interests in the topic of this article there seems to be a lack of experimental studies on the measurements of dynamic effects in heterogeneous porous sample. It is known that there are many types of heterogeneity, for example, fractures, microscale heterogeneities, layers, and so forth. Although there is clear indication that τ depends on the porous medium heterogeneity, most of these studies in this area are theoretical in nature involving numerical simulations[5, 21] as far as the authors are aware of and there seems to be a lack of study which reports how to measure, and provides experimental evidence of, these dependencies. The present article aims to eliminate this gap by reporting an experimental study of the effects of heterogeneity on the dynamic effect in capillary pressure relationship for two-phase flow behavior. For the purpose of the article, a weakly layered porous domain is chosen as a model of heterogeneous porous sample where the contrast in permeability of various layers is not significant. These types of heterogeneity are easy to prepare in the laboratory and mimic real heterogeneity in the subsurface, for example, lamina, alternating layers of fine and coarse sand, and so forth.