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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 (Pc)–saturation (Sw) 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 Pc and Sw is described by various empirical models such as the Brooks–Corey and van Genuchten 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 Pnw and Pw are the average pressures of the nonwetting and wetting phases, respectively. In Eq. (1), Pc is defined to be a function of Sw in the porous medium. Experimentally, the Pc–Sw 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., dSw/dt = 0) is known as the primary drainage (PD) Pc–Sw 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, Swir.[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) Pc–Sw curve. The drainage and imbibition Pc–Sw 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 (Sw = 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 (Swr). The PD and MI curves start from full saturation with wetting fluid and from irreducible wetting phase saturation, respectively. The difference in the Pc–Sw 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 Pc–Sw 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 Pc–Sw relationships. The present study will limit itself to these curves and will not consider primary, secondary, or any other scanning Pc–Sw curves.
The hysteresis in equilibrium and dynamic Pc–Sw 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., Smiles et al., Vachaud et al. and Stauffer have indicated that Pc–Sw curves at equilibrium and dynamic conditions are different. Further, it has been shown that at short time durations the Pc–Sw relationships depend on the rate of change of fluid saturation (dSw/dt). The dependence of the Pc–Sw curves on the rate of change of saturation (dSw/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 presented the following relationship, which included the dynamic effect indicated by a dynamic coefficient in the conventional Pc–Sw relationship
where, is the dynamic capillary pressure defined as and is the capillary pressure at equilibrium conditions (dSw/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. Hassanizadeh et al. interpreted the imbibition experiments (displacement of oil by water) of Kalaydjian and reported that the average value of the dynamic coefficient in Kalaydjian's experiments was 2 × 106 kg/m.s. Hassanizadeh et al. 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 m2; porosity = 0.35–0.4). In this study, τ values were reported to be in the range 5 × 104−6 × 104 Pa s for PD, 1.3 × 105−1.9 × 105 Pa s for MD, and 3.1 × 105−7.7 × 105 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. Another study which is relevant to the current article was conducted by Sakaki et al. 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 (τB) of fluids defined originally by Barenblatt et al.[45-47] Sakaki et al. carried out a comparison of τB for PD and MI which showed differences in τB for the two cases suggesting hysteretic τB 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. used two sets of hypothetical τ values of 0, 105, 107 Pa s, and 0, 107 Pa s to simulate imbibition in homogeneous porous domains with permeabilities of 9.4 × 10−10 m2 and 1 × 10−12 m2, respectively.
In order to have a better understanding of the hysteretic behavior of τ–Sw relationships, it seems we need more experiments, which can investigate the extent of this hysteresis. Given that both τ and dSw/dt change with Sw and time, a direct comparison of τB for PD and MI as done by Sakaki et al. 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. and Sakaki et al. were done for homogeneous domain of a single permeability. Further, the results of Sakaki et al. 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–Sw 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–Sw curves. Many studies which reported τ values for drainage suggest that τ increases as Sw 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 Sw decreases, see, e.g., Camps-Roach et al. and Bottero et al.
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 (Pc–Sw) relationships were used to obtain the dynamic coefficient (τ) as a function of water saturation (Sw) 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–Sw curves are then compared for PD and MI to determine the hysteresis in the homogeneous and heterogeneous sand samples. We determine τ–Sw relationships at three different heights within both the homogeneous and heterogeneous domains and, use the local τ–Sw data to calculate the effective τ–Sw 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–Sw 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–Sw curves for imbibition for the same domains as used by Das and Mirzaei.[41, 42] The τ–Sw 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–Sw 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–Sw 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 (τB) 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 and the behavior of the dynamic coefficient (τ). As such, we do not discuss the work related to the Barenblatt model in this article.
Materials and Methods
Detailed descriptions of the materials and methods for drainage experiments were described by Das and Mirzaei.[41, 42] The experiments that were designed to obtain an understanding of the hysteresis in τ–Sw relationships simply included additional experiments for dynamic and quasi-static imbibition of two-phase flow. In consistent with our previous experiments[41, 42], we performed the flow experiments (Figure 1) in a cylindrical cell packed with the chosen porous sample. In each cycle of experiment, silicone oil was injected in the sample through a hydrophobic filter and water drained out of the cell though a hydrophilic filter. Three mini-time domain reflectometer (TDR) probes were installed at three different heights in the sample to determine water content during the experiments. Pressure transducers equipped with either hydrophilic or hydrophobic filter mounted on the wall of the cell at the same height of mini-TDR probes were used to selectively monitor water and oil pressure, respectively. As the descriptions of the methods and materials have been given earlier by Das and Mirzaei[41, 42], we provide only brief discussions on materials and methods in this article for its completeness.
Porous media and fluid properties
The laboratory experiments were performed using Leighton Buzzard DA 14/25 sand as coarse-grained and Leighton Buzzard DA 30 as fine-grained sand (WBB Minerals, Cheshire, UK). They provide good uniformity of particles (uniformity coefficient (d60/d10) of 1.21 and 1.32 for fine- and coarse-grained particles), high sphericity, high chemical purity, and very low organic matter content. The nonwetting phase liquid selected for this study was silicone oil (viscosity: 200 cSt) with a negligible solubility in water, a low volatility at room temperature, and a low health hazard. The aqueous phase in our experiments was distilled water. Relevant properties of the porous samples and fluids are listed in Table 1.
Table 1. Fluids and Porous Media Properties which are Relevant in this Study
The medium properties are determined experimentally and the fluid properties are either standard values or taken from literature.[41, 42, 50-52]
The flow experiments were carried out in a cylindrical cell of 10 cm internal diameter (ID) and 12 cm height filled with a porous sample (homogeneous or heterogeneous) made by sand particles. The sample was first purged of air by stirring sand in distilled water reservoir until no air bubble came out of mixture. Subsequently, the mixture was put in a vacuum unit to release any trapped air bubble for 24 h. The cell was carefully mounted on a plate equipped with an O-ring to prevent any air inflow and a de-aired and fully water saturated hydrophilic filter to facilitate water drainage and prevent oil flow at outflow. Subsequently, the column was filled with distilled water. Wet and de-aired sand poured into water whilst column was on top of a mechanical shaker to provide even and dense fully water saturated packing. Visual observation confirmed no gas bubbles in the sand pack following this procedure.
While preparing the homogeneous porous samples, either coarse-grained or fine-grained sand was deposited in the cell. To prepare the heterogeneous (layered) domain, a predefined mass of de-aired and wet coarse sand was put in the cell to cover 3 mm above the radius of influence of lowest mini-TDR probe. The radius of influence is the distance measured from outer probe rods, within which TDR reflection is not interfered with the reflection of other TDR probes and/or reflection of probe is not affected by the texture of sand out of radius of influence. It is defined by the manufacturer to be 3 mm for the type of the mini-TDR probes used. Subsequently, a known quantity of fine sand is added to the cell to make a fine sand layer of 39 mm thickness to cover the middle mini-TDR probe. After this, another layer of coarse sand is deposited on the top of the find sand.
An X-ray test to measure packing density along the cell was also conducted before any two-phase flow experiment to ensure homogeneity in packing within either the whole domain or a layer in a heterogeneous domain.
For homogeneous domain, the porosity of the sand pack was calculated from the total mass of each sand pack. For heterogonous domain, the porosities of individual coarse or fine sand layers are not measured separately and are defined to be the same as the corresponding homogeneous domains of coarse or fine sand domains. This is because the mass/volume ratios of the homogeneous coarse and fine sand samples were the same way as those of individual coarse and fine sand layers, respectively.
Before conducting two-phase flow experiment, single phase flow using water and closely following constant head permeability test was performed to measure intrinsic permeability of the sand pack. To perform this experiment, a hydrophilic filter was placed on top of the sand column and the inflow reservoir was filled with distilled water.
Quasi-static and dynamic two-phase flow experiments
In the designed cell (Figure 1), three mini-TDR probes are installed at three different heights in the domain to measure water content during the experiment. Pressure transducers equipped with either hydrophilic or hydrophobic filters are mounted on the wall of the cell at the same height as mini-TDR probes which are used to monitor water and oil pressure, respectively. [41, 42] Locally measured water saturation and, water and oil pressures are used to calculate local saturation and capillary pressure, respectively, which are then used to construct either dynamic or quasi-static Pc–Sw curves. These curves are subsequently used to calculate dynamic coefficient.
The quasi-static two-phase flow experiments during drainage were conducted as follows. The outflow valve of the cell was opened to provide very low flow rate during experiment and it was leveled with the top of the sand to overcome hydrostatic head pressure gradient to minimize gravity effect. The initial pressure of silicone oil was zero within sand column. Then, the pressure of silicone oil on top boundary was gradually increased by increasing air pressure at the top of an oil reservoir, which was the Mariotte bottle. This resulted in infiltration of silicone oil through hydrophobic filter at the top of cell. Injected oil also displaced water out of sand. The experiment was carried out until steady-state flow condition was reached; that was, a constant rate water flows from the outflow valve. Water saturation was calculated considering initial water content and outflow water volume and compared with the mini-TDR readings. Silicone oil and water reservoir and pressure transducers pressure facilitated Pc measurement for comparison. Calculated Sw and Pc from the reservoirs and recorded sensors provide one point of the equilibrium Pc–Sw curve. The experiment was continued until a new steady-state was reached for higher fluids pressures. A second point for Pc–Sw curve was thus obtained. This procedure was repeated until displacing oil front reached bottom hydrophilic filter; that was, measured water content at lower TDR probe reached irreducible water content. At this point, the sample was assumed to have reached its irreducible wetting phase saturation (Swir).
The procedures adopted to determine the dynamic Pc–Sw curves for drainage are as follows. Silicone oil pressure at the top of domain was increased to a high pressure at once and maintained constant by constant air pressure on Mariotte bottle provided through a manual pressure regulator. Table 2 shows the applied boundary condition for different dynamic flow experiments. The drainage displacement was continued until the saturation at the lower mini-TDR probe reached its Siw. Mini-TDR measured water content and pressure transducer recorded oil and water pressure provide data to calculate three local dynamic Pc–Sw curves for each boundary pressure. Dynamic experiment was repeated for four different pressures.
Table 2. Summary of our Experimental Results on Dynamic Coefficient, τ, (Pa s) and Water Saturation (Sw) for Fluids and Porous Materials in Table 1
Type of Domain
Type of Data
Level of Measurement
Range of Water Saturation (–)
Range of Dynamic Coefficient (Pa s)
Arithmetic Average of Dynamic Coefficient ± Standard Deviation
Primary drainage (PD)
Coarse grained sand
Local measurement (experimental)
84855.8 ± 148861.2
74125.8 ± 135903.7
77542.4 ± 143887.2
Effective data (calculated)
78784.6 ± 142150.3
Fine grained sand
Local measurement (experimental)
125444.7 ± 164697.5
118259.6 ± 159699.6
116802.3 ± 167783.0
Effective data (calculated)
120097.0 ± 163537.0
Heterogeneous (layered) domain
Local measurement (experimental)
Upper level (coarse sand)
106092.6 ± 16298.07
Middle level (fine sand)
174900.3 ± 213720.3
Bottom level (coarse sand)
111687.1 ± 174497.3
Effective data (calculated)
132005.8 ± 184421.8
Main imbibition (MI)
Local measurement (experimental)
275278.9 ± 58420.3
287362.6 ± 47410.7
277250.0 ± 56412.2
Effective data (calculated)
279966.6 ± 53829.5
Local measurement (experimental)
317262.7 ± 57553.6
311362.0 ± 57892.3
316754.7 ± 56711.9
Effective data (calculated)
315095.0 ± 57130.5
Heterogeneous (layered) domain
Local measurement (experimental)
Upper level (coarse sand)
276636.1 ± 41075.5
Middle level (fine sand)
326589.3 ± 47467.6
Bottom level (coarse sand)
257082.3 ± 41257.4
Effective data (calculated)
286910.0 ± 43457.4
Following the drainage experiment, imbibition experiments are performed in the same domain, which is at its irreducible water saturation. For this, the direction of fluid displacement is reversed. Water is injected from bottom of the domain through the hydrophilic filter and oil is ejected through the hydrophobic filter at the top of the cell. Imbibition experiment ends when water front reached hydrophobic filter at the top boundary. In this condition, the domain has reached its residual saturation (Swr). The same approach was adopted for obtaining both the quasi-static and dynamic Pc–Sw curves.
Calculation of dynamic coefficient
Equation (2) shows that if Pc,dyn−Pc,equ and dSw/dt are known at a Sw value, τ can be determined. From the experimental data for Pc,dyn, Pc,equ, Sw, and dSw/dt at different Sw, the local τ–Sw relationships are determined at different heights in the porous sample using Eq. (2). Furthermore, the average τ and Sw values are calculated for the whole domain by using the following equations [41, 42]
where is the volume weighted water saturation, which is also defined as the average/effective water saturation for the whole domain. is the saturation weighted dynamic coefficient (τ), which is the average/effective τ and presented as a function of average/effective saturation. is the dynamic coefficient and is the measured saturation at measurement volume Vi at corresponding measurement height i, and i=1, 2, 3, …, n is the number of measurements heights in which the τ– curves are calculated, i.e., the number of τ– curves, in which n = 3.
Results and Discussion
Transient saturation (Sw–t) profiles
In this section, we present the Sw–t curves obtained in this study. Although the trend of these curves are well understood, it is important to discuss them in the present context as they help to understand when the flow system reaches equilibrium (∂Sw/∂t = 0). These curves are also needed to calculate ∂Sw/∂t and the dynamic coefficient using Eq. (1). Figure 2 shows some typical Sw–t curves at three measurement heights during PD and MI in coarse sand domain. The drainage Sw–t curves in the domain have been discussed by Das and Mirzaei and as such, these curves are only included in this article to demonstrate the differences between the Sw–t curves for drainage and imbibition. As expected, the water saturation in the domain decreases during the drainage and increases during the imbibition cycle. Figures 2a, c use the time elapsed from the start of the experiment while Figure 2b uses the time from the moment the first change of saturation is recorded. Figure 2a demonstrates that the water saturation in the domain (i.e., top, middle, and bottom of the domain) at the end of an imbibition cycle for a pressure of 8 kPa is almost the same everywhere. However, the superimposed Sw–t curves in Figure 2b for the same boundary condition show that although the saturation is almost the same throughout the domain, saturation profiles at different measurement heights may vary. Consequently, contrary to the drainage displacement, the Sw–t curves do not completely overlap during imbibition. Figure 2c displays the Sw–t curves for imbibition displacement in the same domain for different boundary conditions of 8, 9, 10, and 11 kPa.
Similar to Figure 2a, Figure 3a shows typical drainage and imbibition Sw–t curves in fine sand domain for 8 kPa boundary pressure. At first glance, the Sw–t curves at different measurement heights seem to follow a consistent trend. However, superimposed Sw–t curves in Figure 3b show that although the curves follow a consistent trend and end up at almost similar saturation value, toward the end of an experiment, the shape of the Sw–t curve at the upper measurement point varies from the other curves slightly. A close comparison between Figures 2b and 3b shows that contrary to the overall behavior of all Sw–t curves in coarse sand domain, the curves in fine sand domain overlie for a large range of saturation. The Sw–t curves at different measurement heights in fine sand for 9, 10, and 11 kPa boundary pressure also follow similar trends. Again, for a large range of saturation, the curves overlap. Figure 3c shows the Sw–t curves at three different measurement heights for various boundary conditions. The figure clearly shows that saturation distribution along the homogeneous fine sand domain in imbibition is fairly uniform, which is consistent with the observations made for homogeneous coarse sand. The fact that the Sw–t curves for both drainage and imbibition at different heights within the coarse or fine sand domains in this work are not significantly different is not completely surprising given that the domain size is small and the distances between two measurement points are small.
As in the homogeneous domains, imbibition experiments were carried out in the layered domain (Figure 4) for oil pressures of 8, 9, 10, and 11 kPa. Figure 4a displays the drainage and imbibition transient saturation profiles in the domain for 8 kPa boundary pressure in real time scale. However, the superimposed curves in Figure 4b make the comparison of Sw–t curves easier. The figure shows that the shapes of Sw–t curves at different measurement heights are similar at first, which follow a consistent increasing trend. As the experiments progress and water saturations increase, the Sw–t curves at different measurement heights show slightly different behavior. This is obviously due to the variations in the properties of the sands in different layers, which affect the fluid flow behavior. The same trends in Sw–t curves for 9, 10, and 11 kPa boundary pressure are found. Sw–t curves for imbibition for all pressures are shown in Figure 4c.
Dependence of the rate change of saturation (dSw/dt) on time (t)
The time derivative of saturation (dSw/dt) is the gradient at a point on a Sw–t curve. As explained before,[41, 42] one needs the values of dSw/dt at different saturation and time to determine dynamic coefficient values. The importance of the dSw/dt−t curves and their implications are also discussed earlier[20, 41, 42] and are not discussed in detail in this article. In general, the rate of change of saturation (dSw/dt) changes with saturation and time and this determines when a two-phase flow equilibrium is reached (dSw/dt = 0). This section reiterates this point and reports some typical dSw/dt−t curves for MI and compare them with the curves for PD with a view to demonstrate the hysteresis in these curves which would be related to the hysteresis in the τ–Sw relationship (discussed later). The values of dSw/dt are approximated as shown below
where , , and are the average wetting phase saturations at tn − 1, tn, and tn+1 calculated using Eq. (3) in different heights. It should be noted that the dSw/dt values are also related to the mobility ratios, which depend on the relative permeabilities of the fluid phases and the fluid viscosity ratios. In other words, if the mobility ratios change then dSw/dt and, hence, the dynamic coefficient may also change. In the present article, no attempt is made to address this point as it requires experiments with different fluid pairs.
Figures 5a, b show some typical imbibition and drainage dSw/dt−t curves in the coarse sand domain for 8 kPa boundary pressure. As expected, the drainage dSw/dt have negative values, while the values of dSw/dt during imbibition are positive. This is logical as Sw decreases with time in drainage and increases during imbibition. As the flow reaches equilibrium, dSw/dt values for both drainage and imbibition reach almost zero. These points are further illustrated in Figure 5b, which shows the superimposed imbibition and drainage dSw/dt−t curves. The magnitudes of dSw/dt vary for drainage and imbibition (Figure 5b) at different heights within the domain (Figure 5c). The differences in the dSw/dt values at the same time and saturation give rise to different dynamic effects, which is discussed later. Figure 5c shows the dSw/dt−t curves for different pressures of 8, 9, 10, and 11 kPa. Within the pressure range tried in this work, dSw/dt values do not seem to be significantly different. The observations related to the dSw/dt−t curves for coarse sand domain seem to be valid for fine sand (Figure 6) and the layered domain too (Figure 7).
Dynamic and quasi static capillary pressure (Pc)–saturation (Sw) curves
Figures 8a, b present the hysteresis in the dynamic drainage and imbibition Pc–Sw curves at 8 and 11 kPa pressures. Similar patterns are also observed for oil pressures of 9 and 10 kPa. Figure 8c shows the hysteresis in quasi-static Pc–Sw curves for drainage and imbibition. Figures 8a, b show the curves at the upper, middle, and lower measurement heights of coarse sand domain. As demonstrated in the figures, the drainage Pc–Sw curves lie above the imbibition Pc–Sw curves, demonstrating hysteresis in the drainage and imbibition curves. In general, the hysteresis in the Pc–Sw curves can arise from the fact that capillary pressure depends on the adhesive forces between the fluids and the minerals which make up the pore walls, the pressure difference between oil and water phase on the pore spaces and, hence the curvature of meniscus between oil and water in the pore space, etc. Although the porous medium and fluids are the same in drainage and imbibition, it is difficult to pinpoint exactly which of these factors is the least or most significant in causing the hysteresis in this study. However, it is clear that the difference of water and oil pressure and, consequently the saturation distribution in filling the pore spaces during drainage and imbibition result in different capillary pressure behavior resulting in different Pc–Sw curves during drainage and imbibition. Similar interpretation is made from Figure 8c, which demonstrates that the quasi static Pc–Sw curves are also hysteretic in nature.
The imbibition experiments following the drainage experiments in coarse sand domain has been conducted for fine sand (Figure 9), which also show hysteresis in Pc–Sw curves. Figure 10 shows some typical graphs to compare the dynamic and quasi static drainage and imbibition Pc–Sw curves at the three measurement heights in layered media. As in the case for coarse sand domain, hysteresis in imbibition and drainage Pc–Sw curves in heterogeneous domain is present in these figures.
The hystereses in Pc–Sw curves in Figures 8-10 mean that the average saturation in the domain is not the same for drainage and imbibition at the same capillary pressure, which is due to different pore filling and emptying rates. There have been a number of pore-scale studies in the literature, which may explain the hysteresis in terms of various pore scale physics.[9-12] However, in the context of this study, we measure averaged Pc–Sw curves at the core scale which are then used to determine the hysteretic dynamic coefficient (τ)–saturation (Sw) relationships. Difference in average Sw along with different dSw/dt at the same Pc or/and time level give rise to hysteretic dynamic effect in capillary pressure relationships as discuss in the next section.
In order to understand the hysteretic behavior of the dynamic coefficient, the τ–Sw data for homogeneous and layered domains for PD and MI are presented in Figures 11 and 12. We also present a table (Table 2) which summarizes the important data ranges and statistics of our experimental results for the dynamic coefficient (τ). The table would be invaluable in following the discussions in this section.
Figure 11a displays the drainage and imbibition τ–Sw curves at three different heights in the coarse sand domain. The effective τ–Sw curves are also presented in the figure, which are determined using Eqs. (3) and (4) as mentioned earlier. The hystereses in the local and the effective τ–Sw curves are found, which imply that the energy required to achieve flow equilibrium (dSw/dt = 0) at the same water saturation is different for drainage and imbibition. As discussed earlier, the dSw/dt values vary slightly for drainage and imbibition (Figures 5-7) at the same saturation and time which cause the hysteresis in the τ–Sw curves. Analyzing the pore scale processes is beyond the scope of this article. However, it can be safely stated that these differences in dSw/dt values and, hence, the τ–Sw curves, are due to the difference in the pore-scale flow processes during drainage and imbibition, e.g., difference in pore filling and emptying rates, and mechanisms (e.g., percolation-type piston drive in drainage, snap-off, and cluster growth in imbibition), wettability, etc. What we observe in the τ–Sw curves is the lumped affect of all possible pore scale processes. In consistent with the conclusion of Hassanizadeh et al., the imbibition τ values in Figure 11a are found to be bigger than the drainage τ values at the same Sw, meaning that the speed to flow equilibrium during imbibition is slower than the drainage at that Sw. The imbibition τ–Sw curves for coarse sand show that the dynamic coefficient varies nonlinearly with the water saturation similar to the drainage τ–Sw curves.[41, 42] But, as the water saturation increases from irreducible water saturation, the imbibition τ–Sw curves break away from that for drainage and they remain almost constant. Similar trends are observed for fine sand domain.
Table 2 shows that our imbibition τ values for both coarse- and fine-grained sands are consistent with the results of Hassanizadeh et al. who reported τ values in the range 3.1 × 105−7.7 × 105 Pa s for MI. In the imbibition experiments of Sakaki et al., the dynamic coefficient at the middle of their domain was found to be relatively constant in the Sw range of 0.3–0.8 with the τ values ranging between 1.0 × 106 and 6.0 × 106 Pa s. Our local and effective τ values for imbibition are smaller than those obtained by Sakaki et al. However, the slopes of our imbibition τ–Sw curves for both coarse and fine sands are small in the range 0.5 < Sw < 0.8 which imply that we also observe almost horizontal τ–Sw curves in consistent with the results of Sakaki et al. This practically means that the energy required for the two-phase flow system to reach equilibrium is almost the same in the middle saturation points during imbibition.
In Figure 11b, we demonstrate the local and effective imbibition τ–Sw curves for the coarse and fine sand domains. In effect, the figure shows if the effective τ–Sw curves for homogeneous porous media at the core scale are any different from the local τ–Sw relationships while also demonstrating any effect of the particle size and permeability on the imbibition τ–Sw curves. The results show that there are not significant differences among the local and effective τ values at different saturations for most practical purposes. This observation is somewhat expected as the domain size is not very big (12 cm long) and it is homogeneous in its properties (e.g., particle size and shape). Further, the sample volumes over which each set of pressure transducers and TDRs measure the τ–Sw curves are small which are also equal. Similar qualitative conclusion has been made by others[32, 41] for drainage in homogeneous domains who have found that there are no significant differences among τ–Sw curves at different levels of a core scale domain. Whether or not one obtains significantly different τ values at different points in a larger homogeneous domain or for different boundary conditions is not certain and it should be explored separately. Bottero et al. have calculated effective τ for PD in a bigger homogeneous domain of 21 cm; however, it is not clear to us if τ values at different points within the domain are different and, if so how significantly.
Figure 11b suggests that our τ values obtained for imbibition are higher in fine sand than the values at the corresponding heights in coarse sand at a given saturation. As such, the average τ values in Table 2 are also higher for fine sand than those for coarse sand. These values are smaller than those of Kalaydjian but similar in magnitude to those obtained by Hassanizadeh et al. and Sakaki et al.
To make a further analysis of the drainage and imbibition τ–Sw curves in fine- and coarse-grained porous domains, plots of the effective and local τ–Sw relationships measured at different domain heights are shown in Figures 11c, d. Figure 11c shows the drainage and imbibition τ–Sw curves at the middle measurement heights of both coarse and fine sands. On other hand, Figure 11d shows the effective drainage and imbibition τ–Sw relationship in the two sand domains. The figure shows that both the drainage and imbibition τ–Sw curves follow nonlinear functional dependence and there is clear hysteresis where the imbibition τ values are generally larger as compared to the τ values for drainage at the same saturation. It suggests that for the imposed boundary conditions, the two-flow system would require more energy to reach equilibrium during imbibition, as such, one should expect hysteresis in the τ–Sw curves.
As discussed earlier, the τ values depend on a number of other factors, e.g., permeability of the domain, fluid properties, domain size, temperature, etc. Therefore, the results of this study cannot be compared directly with values from other study, which may use different fluid, porous medium, or domain size. Nevertheless, comparisons of our results with those obtained by others reveal interesting facts. Hassanizadeh et al. used a domain of permeability 10−11 m2 while our fine-grained domain has a permeability of 3.1 × 10−10 m2. Hassanizadeh et al. used a fluid pair of PCE and water while we used silicone oil and water as fluids with properties as shown in Table 1. Although there are significant differences in media and fluids properties, it seems that our results are similar to those of Hassanizadeh et al. For example, both studies obtain an imbibition τ value of approximately 3 × 105 Pa s at 70% water saturation for the cases mentioned above. This implies that the speed to flow equilibrium during imbibition in these two systems at the conditions above would be almost the same despite difference in the properties of the flow system.
The observation in Figure 11 that the τ values during imbibition are higher for fine-grained sand (i.e., the domain with smaller particles and less permeability) than coarse-grained sand is consistent with previous studies which measured τ for drainage.[7, 32, 41] The uniformity coefficients, which indicate the uniformity of the particle shape, were 1.6 and 2.3 for the particles that Camps-Roach et al. have used. The particles that we have used have uniformity coefficients of 1.21 and 1.32 for fine and coarse particles, respectively, and these have better uniformity in shape than the particles of Camps-Roach et al. as indicated by smaller uniformity coefficient values, which are closer to unity. The average diameters of the particles that Camps-Roach et al. have used were 0.42 mm and 0.18 mm. The particles that we have used are bigger in size as compared to the particles that Camps-Roach et al. have used given that our particles have average diameters of 0.946 mm and 0.482 mm. Clearly, Camps-Roach et al., Das and Mirzaei[41, 42], and the current article imply that at a particular saturation the τ value is expected to be higher for fine sand domain (i.e., domain with smaller particles and less permeability) for both drainage and imbibition for a range of particles shape and sizes.
Although most results in this article are consistent with other results in the literature, the shapes of the τ–Sw curves for drainage and imbibition (and hence their interpretation) are not necessarily consistent with all studies.
Sakaki et al. report that the τ values from their MI experiments are slightly smaller than those in the PD. This is opposite to the observation of this article (Figure 11) as well as of Hassanizadeh et al.
The results for both PD and MI in our study suggest that the τ values are inversely proportional to the water saturations. Although the values of the dynamic coefficient may vary in different studies and similar trends in τ–Sw curves are found by most[7, 25-27, 34, 41, 42], some papers have been published recently where different trends of τ–Sw curves have been observed.[23, 32] During drainage, Camps-Roach et al. find that the τ values decrease as water saturation decreases. Sakaki et al. report that their τ values increase as water saturation decreases during drainage. However, the imbibition τ values of Sakaki et al. fluctuate, decrease, and then increase with decrease in water saturation. Bottero et al. reported τ values for PD for a saturation range of 0.55–0.85 and found that τ was almost constant in that saturation range. It is not exactly clear to us at this moment why there are distinctly different trends of τ–Sw curves. We are likely to revisit this question in another paper in the future. At this moment, it is suffice to say that the τ–Sw curves of this article are consistent with the majority of results found in the literature.
Figure 12a displays the imbibition and drainage τ–Sw curves at different measurement heights in the heterogeneous (layered) domain. As shown in the figure, imbibition τ–Sw curves lie higher than the drainage τ–Sw curves for the domain, which is consistent with the results obtained for homogeneous domain (Figure 11). Contrary to the drainage τ values, which increase as saturation decreases, the imbibition τ values decrease sharply initially as saturation increases from irreducible water saturation but as the imbibition proceeds and saturations increase further, the dynamic coefficients decline slowly and they become almost constant. In Figure 12a, the effective τ–Sw curves for both drainage and imbibition are also shown. For both drainage and imbibition, the effective curves fall between the curves for coarse sand (upper and lower measurement point) and fine sand (middle measurement point) for the heterogeneous domain used in this work. The results suggest that the effective τ–Sw curves are dominated by the τ values of the coarse sand as it occupies the maximum volume of the domain. The results also show that the curves are hysteretic in nature (similar to the local the τ–Sw curves).
In Figures 12b–d, we attempt to compare the τ values during imbibition at different measurement levels of the homogeneous and layered domains. As the properties of the coarse and fine sand in these experiments as well as the sample size, boundary condition, and fluid properties are the same, this comparison may reveal interesting information. In Figures 12b, d, we present the imbibition τ–Sw curves at the upper and lower measurement heights of the homogenous coarse sand and layered domain, which are occupied by coarse sand. As demonstrated in these figures, although the imbibition τ–Sw curves for the same sand type but different samples show similar trends, the imbibition τ–Sw curve of coarse sand in heterogeneous domain lie slightly higher than the imbibition τ–Sw curve of coarse sand in homogeneous domain. This means that the fluid displacement in the upper and lower layers (coarse sand) in heterogeneous domain needs more energy to reach equilibrium, which is logical considering the presence of fine sand layer at the middle of the heterogeneous domain. On the contrary, the imbibition τ–Sw curves at the middle measurement point of the homogeneous fine sand and heterogeneous domain show slightly different behavior (Figure 12c). When saturation is low at the beginning of the experiment, imbibition τ values in the fine sand layer at the middle of heterogeneous domain seem to be higher. However, as saturation increases, the two curves almost overlie and at some saturation values imbibition τ for fine sand layer in heterogeneous domain is less than the imbibition τ data at corresponding height in the homogeneous fine sand domain. Again, this is due to higher permeability of upper and lower coarse sand layers in the heterogeneous domain, which makes faster displacement than that in homogeneous fine sand domain filled with lower permeability sand. In general, however, we believe, there is not a significant difference between the τ values at the different heights in the homogeneous and heterogeneous domain in this study due to small size of the domain and weak permeability contrast between coarse and fine sand layers.
Most previous studies that attempted to relate the dynamic effects in capillary pressure relationships to process parameters (e.g., grain size, permeability, temperature, domain size, saturation, fluid properties, and boundary conditions) find that τ values depend on these parameters. As such it is expected that the τ–Sw curves for both drainage and imbibition cycles, and hence the hysteresis in the τ–Sw curves, will be affected by these parameters. However, it seemed there were not enough experiments to provide a detailed understanding of these effects. This article has attempted to address this point by carrying out a number of laboratory experiments. Dynamic and quasi-static capillary pressure–saturation (Pc–Sw) curves for PD and MI were determined in this work which were then used to obtain the dynamic coefficient (τ) as function of water saturation (Sw). Both homogeneous and heterogeneous (layered) domains were used in this work. The results in this article clearly show that the energy required for, or the speed to, flow equilibrium at the same water saturation varies depending on whether a drainage or imbibition takes place, which confirms that the dynamic effect in capillary pressure–saturation relationships for two-phase flow is hysteretic in nature. We believe further work is needed to relate the properties of the fluids and other important factors (e.g., boundary condition and domain size) to the hysteresis in the τ–Sw relationship. This is beyond the scope of this article. Nevertheless, it is envisaged that the purpose of this article, i.e., to investigate further the hysteresis in τ–Sw relationships in homogeneous and heterogeneous domains through laboratory experiments, has been achieved.
This study has been carried out in the framework of the EPSRC (UK) Project GR/S94315/01, “micro-heterogeneity and temperature effects on dynamic capillary pressure–saturation relationships for two-phase flow in porous media”. The EPSRC funding for this project is gratefully acknowledged. Emeritus Professor G. Sills (the University of Oxford, UK) is acknowledged for her help in completing the experiments in this work.