## Introduction

Mathematical description of two-phase flow (immiscible fluids) in porous media requires appropriate governing equations for the conservation of fluids mass and momentum as well as other constitutive equations, e.g., capillary pressure (*P*_{c})–saturation (*S*_{w}) relationship.[1-3] Traditionally, an extended version of Darcy's law is used as the governing equation of motion for the fluid phases. The conservation of fluid mass is given by an equation for the conservation of phase saturation, i.e., the ratio of the volume of a fluid phase in a given domain to the pore volume in the domain. The relationship between *P*_{c} and *S*_{w} is described by various empirical models such as the Brooks–Corey[4] and van Genuchten[5] formulations. For these relationships, capillary pressure is generally calculated by an empirical relationship obtained under equilibrium condition between the individual fluid phase pressures as follows[1, 6, 7]

where *P*_{nw} and *P*_{w} are the average pressures of the nonwetting and wetting phases, respectively. In Eq. (1), *P*_{c} is defined to be a function of *S*_{w} in the porous medium. Experimentally, the *P*_{c}*–S*_{w} curves can be obtained by taking a porous medium initially saturated with a wetting fluid (e.g., water) and then letting it to gradually drain off by increasing the capillary pressure and displacing the wetting fluid (e.g., water) by a nonwetting fluid (e.g., oil). The resulting curve representing the corresponding values of wetting phase saturation and capillary pressure at equilibrium condition (i.e., *dS*_{w}/*dt* = 0) is known as the primary drainage (PD) *P*_{c}*–S*_{w} curve. In most cases, higher capillary pressure does not lead to any further displacement of the wetting phase due to phase separation and/or strong wetting phase attachment to the grain particles. The saturation at which this condition occurs is known as the irreducible wetting phase saturation, *S*_{wir}.[6, 8] Once the stage of irreducible wetting phase saturation is reached, the saturation of the wetting phase in the sample may be increased by displacing the nonwetting phase by decreasing capillary pressure. A plot of these experimental results provides a main imbibition (MI) *P*_{c}*–S*_{w} curve. The drainage and imbibition *P*_{c}*–S*_{w} curves generally do not coincide because the pore spaces of the porous medium imbibe and drain differently.[1, 6-9] Further, an imbibition experiment does not lead to full wetting phase saturation (*S*_{w} = 1) at zero capillary pressure because some of the nonwetting phase will be entrapped as isolated bubbles in the pores and thereby not displaced. This saturation value is referred to as residual nonwetting phase saturation (*S*_{wr}). The PD and MI curves start from full saturation with wetting fluid and from irreducible wetting phase saturation, respectively. The difference in the *P*_{c}*–S*_{w} curves is referred to as hysteresis. The hysteresis is a result of a number interdependent pore scale mechanisms during drainage and imbibition (e.g., dynamic contact angle) and has been the subject of many studies in the literature.[9-12] Further, the hysteresis is not just a matter of the PD and MI curves as the *P*_{c}*–S*_{w} relationships may follow infinite number of scanning curves depending on when the drainage or imbibition process is reversed. However, these two curves can be considered as the limiting curves, and as such, they are useful in determining the extent of hysteresis *P*_{c}*–S*_{w} relationships. The present study will limit itself to these curves and will not consider primary, secondary, or any other scanning *P*_{c}*–S*_{w} curves.

The hysteresis in equilibrium and dynamic *P*_{c}*–S*_{w} relationships and, in particular, the ways to measure/quantify the hysteresis are matters of great research interests in the literature.[13-23] There have been many discussions on the validity of Darcy's law and Eq. (1) for two-phase flow processes in the last two decades.[7, 20, 24] It has been argued that at shorter time duration when the two-phase flow may not necessarily be at equilibrium, the traditional approaches as described above cannot be applied. In fact, the early experiments of Topp et al.[14], Smiles et al.[15], Vachaud et al.[16] and Stauffer[17] have indicated that *P*_{c}*–S*_{w} curves at equilibrium and dynamic conditions are different. Further, it has been shown that at short time durations the *P*_{c}*–S*_{w} relationships depend on the rate of change of fluid saturation (*dS*_{w}/*dt*). The dependence of the *P*_{c}*–S*_{w} curves on the rate of change of saturation (*dS*_{w}/*dt*) is known as the dynamic effect in capillary pressure relationship.[7, 25] The dynamic effect has been discussed significantly in the last two decades and it has been shown to be of importance, as discussed below.

Hassanizadeh and Gray[24] presented the following relationship, which included the dynamic effect indicated by a dynamic coefficient in the conventional *P*_{c}–*S*_{w} relationship

where, is the dynamic capillary pressure defined as and is the capillary pressure at equilibrium conditions (*dS*_{w}/*dt* = 0). The coefficient is called the dynamic coefficient which is a measure of the rate of change in saturation and hence the speed to equilibrium condition. In the last decade, a number of studies have discussed the significance of dynamic capillary pressure and in different circumstances.[7, 19-21, 23, 25-43] For example, hysteretic dynamic effect in the capillary pressure relationship was discussed in a theoretical study by Beliaev and Hassanizadeh[19]. Hassanizadeh et al.[20] interpreted the imbibition experiments (displacement of oil by water) of Kalaydjian[44] and reported that the average value of the dynamic coefficient in Kalaydjian's experiments was 2 × 10^{6} kg/m.s. Hassanizadeh et al.[21] carried out a laboratory study to determine values for PD, main drainage (MD), and MI for tetrachloroethylene (PCE) and water flow in a homogeneous porous medium (permeability =10^{−11} m^{2}; porosity = 0.35–0.4). In this study, *τ* values were reported to be in the range 5 × 10^{4}−6 × 10^{4} Pa s for PD, 1.3 × 10^{5}−1.9 × 10^{5} Pa s for MD, and 3.1 × 10^{5}−7.7 × 10^{5} Pa s for MI, which imply that the dynamic coefficient was found to be higher for imbibition than that for drainage by Hassanizadeh et al.[21] Another study which is relevant to the current article was conducted by Sakaki et al.[23] Sakaki et al. conducted PD, MD, and MI experiments and measured *τ* values for these three cases for a single porous domain of saturated hydraulic conductivity 0.016 cm/s. They calculated a characteristic redistribution time (*τ*_{B}) of fluids defined originally by Barenblatt et al.[45-47] Sakaki et al.[23] carried out a comparison of *τ*_{B} for PD and MI which showed differences in *τ*_{B} for the two cases suggesting hysteretic *τ*_{B} and hence, hysteretic *τ*. We are not aware of any other paper, which has conducted experiments for measuring dynamic coefficient for imbibition and/or hysteretic behavior of the dynamic coefficient. There is, however, some studies which attempted to simulate dynamic imbibition of porous medium using hypothetical values of *τ*. For example, Manthey et al.[48] used two sets of hypothetical *τ* values of 0, 10^{5}, 10^{7} Pa s, and 0, 10^{7} Pa s to simulate imbibition in homogeneous porous domains with permeabilities of 9.4 × 10^{−10} m^{2} and 1 × 10^{−12} m^{2}, respectively.

In order to have a better understanding of the hysteretic behavior of *τ–S*_{w} relationships, it seems we need more experiments, which can investigate the extent of this hysteresis. Given that both *τ* and *dS*_{w}/*dt* change with *S*_{w} and time, a direct comparison of *τ*_{B} for PD and MI as done by Sakaki et al.[23] may not reflect the true extent of hysteretic behavior of *τ*. A number of recent studies[32, 41, 42] on experimental determination of values for drainage have indicated that depends on the medium properties (e.g., particle size, permeability, and heterogeneity) and size of the domain. However, it is not clear how do for imbibition and the hysteretic behavior of depend on the medium properties (e.g., particle size, permeability, heterogeneity). Note that the experiments of Hassanizadeh et al.[21] and Sakaki et al.[23] were done for homogeneous domain of a single permeability. Further, the results of Sakaki et al.[23] were based on the measurements at the middle height of a 10 cm column and it is not clear to us how do these local measurements may determine the effective *τ–S*_{w} relationships at the scale of the experimental domain (core scale). This issue may become particularly important if the domain is heterogeneous in nature. Another important issue that should be revisited in the context of both drainage and imbibition is the trend/shape of *τ–S*_{w} curves. Many studies which reported *τ* values for drainage suggest that *τ* increases as *S*_{w} decreases.[7, 20, 23, 26, 27, 41, 42] However, some recent studies on the measurements of *τ* for drainage have found different trends where *τ* values decrease, fluctuate, or remain almost constant as *S*_{w} decreases, see, e.g., Camps-Roach et al.[32] and Bottero et al.[49]

This article aims to address the above points by using carefully controlled laboratory experiments of two-phase flow in homogeneous and layered porous media. In this work, dynamic and quasi-static capillary pressure–saturation (*P*_{c}–*S*_{w}) relationships were used to obtain the dynamic coefficient (*τ*) as a function of water saturation (*S*_{w}) for PD and MI. Homogeneous porous samples composed of either fine-grained (low permeability) or coarse-grained (high permeability) sand have been used for the experiments. Furthermore, the same coarse and fine sand particles were used to create a heterogeneous (layered) domain composed of a fine sand layer sandwiched between two coarse sand layers. The coarse and fine sand layers in the heterogeneous domain are created so that the material properties (particle-size distribution, porosity, and intrinsic permeability) are the same as in the homogeneous coarse and fine sand domains. The *τ–S*_{w} curves are then compared for PD and MI to determine the hysteresis in the homogeneous and heterogeneous sand samples. We determine *τ–S*_{w} relationships at three different heights within both the homogeneous and heterogeneous domains and, use the local *τ–S*_{w} data to calculate the effective *τ–S*_{w} curve for these domains. Please note that the homogeneous and heterogeneous domains used for the purpose of this article are the same as presented earlier by Das and Mirzaei[41, 42] who reported *τ–S*_{w} curves for drainage for homogeneous and layered domains. Also, as discussed by Das and Mirzaei[41, 42], the layered domain in our work should be treated as a “weakly layered” domain as the contrast in permeability between the layers is not significant.

For the purpose of this article, additional experiments have been conducted to obtain *τ–S*_{w} curves for imbibition for the same domains as used by Das and Mirzaei.[41, 42] The *τ–S*_{w} curves for imbibition are then compared with the curves for drainage reported earlier by Das and Mirzaei[41, 42] to determine the significance of hysteresis in *τ–S*_{w} curves. In effect, the results presented in this article not only provide continuity of the results presented by us earlier but also allow easy comparison of *τ–S*_{w} curves for drainage and imbibition for the same conditions.

Please also note that we are aware that the dynamic capillary pressure effect is often interpreted according to a characteristic redistribution time (*τ*_{B}) using a model by Barenblatt et al.[45-47] However, this article is concerned with the dynamic effect as presented by the dynamic capillary pressure relationship (Eq. (2)) proposed by Hassanizadeh and Gray[24] and the behavior of the dynamic coefficient (*τ*). As such, we do not discuss the work related to the Barenblatt model in this article.