Changes in residual air saturation after thorough drainage processes in an air–water fine sandy medium

Journal of Hydrology 519 (2014) 271–283

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Changes in residual air saturation after thorough drainage processes in an air–water fine sandy medium Yan Li a,⇑, Pengfei Wu a, Zhen Xia b, Qingshu Yang a, Giancarlo Flores c, Haoyu Jiang a, Masashi Kamon d, Baozheng Yu e a Guangdong Provincial Key Laboratory of Marine Resources and Coastal Engineering, Key Laboratory for Aquatic Product Safety of Ministry of Education, School of Marine Sciences, Sun Yat-sen University, 135 Xin’gang RD.W., Guangzhou 510275, China b Guangzhou Marine Geological Survey, 188 Guanghai RD., Guangzhou 510760, China c Graduate School of Engineering, Kyoto University, Yoshida-Honmachi, Kyoto 606-8501, Japan d National College of Technology, 355 Chokushicho, Takamatsu-shi, Kagawa 761-8058, Japan e School of Resources and Environment, Agriculture and Animal Husbandry College of Tibet University, 8 College Road, Nyingchi 86000, China

a r t i c l e

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Article history: Received 8 March 2014 Received in revised form 26 May 2014 Accepted 13 July 2014 Available online 21 July 2014 This manuscript was handled by Peter K. Kitanidis, Editor-in-Chief, with the assistance of Nunzio Romano, Associate Editor Keywords: Initial saturation Residual saturation Drainage–imbibition cycle Saturation–capillary pressure (S–p) relation

s u m m a r y In a previous study we investigated the unstable and stable residual air saturations in an air–water twophase system in a sand medium during a series of consecutive drainage–imbibition cycles with gradually increasing initial air saturations. In a reciprocal study reported here we extended the previous investigation by determining residual air saturations in consecutive imbibition processes starting from four gradually decreasing levels of initial air saturation (and thus increasing water saturation). Three parallel column tests with 9–12 consecutive drainage–imbibition cycles were performed, in which the first three imbibition processes started from the highest initial air saturation that could be obtained with our experimental system. The results show that all the residual air saturations resulting from the imbibition processes were almost constant after thorough drainage processes (even those following imbibition processes starting from low initial air saturations), and thus independent of the initial air saturation. The results also indicate that once the residual air in interconnected pores at the end of an imbibition process was present in the form of connected, pore network-scale air globules, the residual air remained in this state in subsequent imbibition processes, even if they started from low initial air saturations. It may be deduced that the presence of thin water films on the walls surrounding large pores and large volumes of air in their central parts during an imbibition process resulted in residual air being in the form of connected, pore-network scale air globules in interconnected pores. In contrast, thick water films and small volumes of air in the central parts of the pores resulted in residual air in the form of single porescale air globules in interconnected pores. Thus, stronger dynamic flow conditions (e.g., higher capillary numbers) may be required to remobilize connected, pore network-scale air globules than single porescale air globules in a forced imbibition process. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction During drainage and imbibition processes of two-phase porous media there are two kinds of residual saturations: the residual wetting fluid (WF) and nonwetting fluid (NWF) saturations that occur at the end of the drainage and imbibition processes, respectively. The former refers to the WF saturation at an arbitrarily high capillary pressure that may be reached in a drainage process and the latter to the NWF saturation, the volume that cannot be displaced by the WF and is trapped in pore networks of a porous ⇑ Corresponding author. Tel.: +86 20 39332201; fax: +86 20 39332159. E-mail address: [email protected] (Y. Li). http://dx.doi.org/10.1016/j.jhydrol.2014.07.019 0022-1694/Ó 2014 Elsevier B.V. All rights reserved.

medium at zero capillary pressure/the end of the imbibition process (Corey, 1994; Bear, 1972; Guarnaccia et al., 1997). The two residual saturations are key parameters for characterizing and modeling the migration of immiscible fluids in a porous medium. In a previous study (Li et al., 2013), we investigated the saturation–capillary pressure (S–p) relationship, including residual air saturations, under gradually increasing initial air saturations in a sandy medium during three series of consecutive drainage–imbibition cycles, in which the starting points of the drainage processes were the end points of the preceding imbibition processes, and vice versa. The results showed that the residual air saturation changed suddenly from an unstable to a stable and constant state when the initial air saturation exceeded a threshold level (0.49 in our

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experimental system). We also found that the Land model (Land, 1968) needs modification to describe phenomena under randomly starting imbibition path and dynamic flow conditions, and the entry pressure of a sandy medium is both medium-specific and independent of the starting points and paths of drainage and imbibition processes. Furthermore it was deduced that the air in the unstable and stable residual air saturation states may be mainly composed of individual air globules trapped in the center of the pores and connected air globules in interconnected pores, respectively. X-ray computed tomography (XCT) is a powerful tool for detecting micro-scale pore characteristics that has been applied in investigations of pore structure and tortuosity (Provis et al., 2012), twophase relative permeability (Schembre and Kovscek, 2003), and estimation of the average water film thickness in pore structures, employing the Beer–Lambert law (Jung et al., 2012). Tippkötter et al. (2009) used microfocus CT to visualize both soil matrices and soil water in natural soil aggregates, and were able to resolve films and estimate film thicknesses (at 3 and 10.6 lm, respectively) for two soils. Zhou et al. (2010) investigated pore structure and trapped gas bubbles in Berea sandstones using a microfocused X-ray CT scanner, and discovered that trapped bubbles have a pore-network scale size and are distributed in several pores at the end of an imbibition process. However, Peng et al. (2012) reported that, due to the resolution limitations, either a non-representative view of the sample or inaccurate results could be produced from XCT image processing. Low-resolution XCT can capture the large-pore porosity, but overestimates the pore size and pore connectivity. High-resolution XCT provides more accurate descriptions of the pore shape, porosity, and pore size. Brusseau et al. (2006) also suggested that the interfacial area associated with films and surface roughness cannot be accurately represented using MCT, possibly due to inadequate resolution. Wildenschild and Sheppard (2013) reported that open questions still remain, especially with respect to the thickness of water films, although water films in pores are clearly detectable. Thus, XCT technologies are available for characterizing pore structures and flow types in pore networks, but high and appropriate resolution XCT is required for accurate measurements of the thickness of water films in pore structures. The capillary number (Ca) — an important hydrodynamic parameter for describing distributions and volumes of trapped nonwetting fluid phases in porous media — represents the strength of viscous forces relative to surface tension acting across an interface between two immiscible liquids. Numerous studies have revealed that dimensionless parameters such as the capillary number and viscosity ratio can influence displacement flow patterns and consequently capillary pressure and relative permeability functions (Aggelopoulos and Tsakiroglou, 2008). Along the center line of a capillary in a pressure-driven flow of a different fluid, Lac and Sherwood (2009) found that if gravity effects are negligible, the motion of a drop is determined by three independent parameters: the size of the undeformed drop relative to the radius of the capillary, the viscosity ratio between the drop phase and the wetting phase, and the capillary number. Dawson et al. (2013) reported that the bubble movement in a low velocity wetting carrier fluid in a porous medium is governed primarily by viscous forces, surface tension forces, buoyancy forces and pressure drag resulting from the bubble bypassing the wetting fluid flow. The strength of the viscous forces on the capillary scale determines the level of broadening of the bubble upon dynamic encounter of a channel expansion, as well as the bubble’s ability to squeeze into a contraction of the tube, with small values of Ca facilitating entrapment of bubbles in expanded sections of the tube. If the pore is not too saturated a blob will also move when the capillary pressure difference between its ends is less than the

hydrodynamic pressure produced by the viscous fluid flowing in the pore space beside it (Dullien, 1991). In addition, capillary and drag forces promote entrapment of nonwetting fluid blobs in porous media, while buoyant and push forces promote their mobilization (Corapcioglu et al., 2009). Thus, they are mobilized if the sum of push and buoyant forces exceeds the sum of capillary and drag forces. The flow pushing a finite bubble in motion in a tube of square cross-section is reportedly 30–100 times greater than the flows bypassing the bubble in the corner regions, if effects of gravity are neglected (Wong et al., 1995). However, all of the constant volume-flux flow in a tube has to by-pass a large trapped bubble, thereby generating significant drag forces. When a bubble has a stress-free interface, all the fluid bypassing it contributes to a pressure drag, which builds up a pressure difference between its tail and tip. Moreover, for a large droplet confined in a porous medium the contribution to viscous drag of shear forces at the interface is negligible compared to the pressure drag (Dangla et al., 2011). It should also be noted that body forces such as gravity and inertia are negligible compared with surface forces, due to the small size of capillaries (Wong et al., 1995). Thus, in more detail, the net effect of the pressure drag difference resulting from bypassing of the water phase between the two ends of a blob and the hydrodynamic pressure exerted by the carrier fluid flowing in the pore space alongside it will determine whether it is mobilized or trapped during an imbibition process of an air–water two-phase porous system. An important trapping mechanism generally in displacements of non-wetting fluids during imbibition processes is snap-off ahead of the displacement front (Chatzis et al., 1983; Chatzis and Morrow, 1984; Kamath et al., 2001; Lenormand and Zarcone, 1984; Mohanty et al., 1980). Smoothly constricted pores with small throats, high pressure gradients in the liquid phase, and lamella or lens movements that establish gradients in the axial profile of interfacial curvatures can all promote snap-off (Kovscek and Radke, 2003). The competition between snap-off events and frontal displacements determines the displacement pattern and value of residual saturation (Hughes and Blunt, 2000). Dynamic effects resulting from flow through wetting films can also affect this competition (Blunt and Scher, 1995; Constantinides and Payatakes, 2000; Mogensen and Stenby, 1998; Hughes and Blunt, 2000, 2001). During strong preferential wetting, in quasi-static displacements or displacements at very low capillary numbers (of the order of 107), movement of the wetting film also reportedly plays major roles in both displacement and trapping of the nonwetting phase (Dullien, 1991). All the cited studies offer pore size-scale understanding of the mechanisms involved in entrapment of the nonwetting fluid in an imbibition process. However, as we discussed in detail in a previous paper (Li et al., 2013), the soil-moisture dynamics of two- or three-phase porous media are highly complex, because they are affected by numerous interacting factors. Thus, despite intensive investigation there are still major uncertainties regarding key parameters involved and their interactive relationships, notably the residual saturations of given fluids in such media in given initial conditions are flow-path and system-history dependent, and they vary with position and time in the medium (Hilfer, 2006). Clearly, these uncertainties need to be resolved to improve our understanding and modeling of flows of fluids and contaminants in porous media. In the study presented here, as in our previous investigation (Li et al., 2013), we have examined the residual saturations and related phenomena of two phases (air and water) in a column of an undisturbed fine sandy medium. The primary focus has been on the effects of the initial saturation, which is considered as one of the key parameters affecting residual air and water saturations in a porous medium (Land, 1968).

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The differences from the previous study (Li et al., 2013) were that we studied the residual air saturations resulting from consecutive imbibition processes starting from four gradually decreasing levels of initial air saturation (and thus increasing water saturation), using the same experimental apparatus and medium. The results corroborate and extend the findings of our previous study in which the initial air saturation was gradually increased (Li et al., 2013). 2. Materials and methods Capillary pressure, Pc, was calculated from the measurements using the following formula:

Pc ¼ Pa  Pw

ð1Þ

a

where P is the pore air pressure (assumed to equal atmospheric pressure as the pores in the sandy medium were considered to be connected and open to the air) and Pw is the pore water pressure. The capillary pressure head, hc (cm), is defined on an equivalent water height basis by: c

h ¼ Pc =qw g

ð2Þ 3

where qw is the water density (g cm ), and g is the scalar magnitude of gravitational acceleration (m s2). To facilitate comparison of the data obtained in this and our previous study (Li et al., 2013), we used the same materials, experimental apparatus (Fig. 1), and methods for time domain reflectometer (TDR) and tensiometer calibration (before and after the tests), as described in detail in the cited paper. The water saturation and pore water pressure were measured with Trime-IT TDR probes (IMKO, Micromodultechnik GmbH, Germany) and SWT5 tensiometers (Delta-T Devices Ltd.), respectively, and recorded with a DT 80 data logger (Thermo Fisher Scientific Australia Pty Ltd.). The tensiometer calibration results were the same as in the previous investigation (Li et al., 2013), while the TDR was calibrated with more sampling points. The relation between the output signals of the TDR probe and the water saturation in the air–water system before the column tests is shown in Fig. 2, where R is the output signal, and R0 is the signal when the sand was fully saturated with water. A blank test (>2000-h) was also performed to estimate TDR and Tensiometer measurement errors in the range 15–30 °C. During the test, the largest water saturation and capillary pressure measurement errors were ±5.09% and ±6.95%, respectively. However,

Fig. 2. Relation between TDR signals and water saturation.

to minimize effects of temperature changes on the results, the laboratory temperature was kept within the range 20–25 °C during the column tests. At the end of each column test, the sand around the TDR probe was also sampled. The differences in saturation measurements by the TDR and the gravimetric methods were less than 1%, and the pressure drifts of the T5 tensiometers after the column tests were less than 0.8 cm water column equivalents. Three parallel column tests, designated Tests 1, 2 and 3, were conducted in the study reported here. The bulk density of the samples used in the tests was as consistent as possible, and to minimize variation in porosity the void ratio was minimized by tapping the column when introducing the sand. The sand samples in column Tests 1–3 had compacted-state void ratios of 0.739, 0.733 and 0.742, respectively, and dry bulk densities of 1.518, 1.523 and 1.515 g cm3, respectively. It should be noted that the sand samples contained a few clay particles, which could have influenced the stability of their pore structure due to shrinkage when drying and swelling when wetting. Changes in clay contents of mixtures can also affect the slope of the dielectric constant-water content relationship curve (Ponizovsky et al., 1999). Thus, before each test the sand sample was washed and dried to remove clay in order to minimize these effects.

Fig. 1. Apparatus and boundary conditions used for online measurement of saturation and capillary pressure: (a) drainage process and (b) imbibition process.

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3. Consecutive drainage–imbibition cycles In a system composed of air and water in a porous medium, the Primary Drainage Process (PDP) refers to the displacement of water by air from a thoroughly water-saturated condition until no more water can be displaced, even under an arbitrarily high pressure. The water saturation still remaining in the porous medium is defined as the residual water saturation. The subsequent displacement of air by water imbibition to a zero air–water capillary pressure (with no other disturbance of the medium) is called the Main Imbibition Process (MIP). The air saturation in the medium at zero capillary pressure is defined as the residual air saturation (Guarnaccia et al., 1997). From the residual air saturation of the MIP, a following displacement of the water in the medium by air to an arbitrarily high pressure is called the Ultimate Drying Process (UDP) (Poulovassilis, 1970). Only one pore fluid is mobile in the residual states at the ends of the PDP, MIP and UDP from a macroscopic view. The PDP and UDP are thus considered to be thorough drainage processes and the MIP a thorough imbibition process. The S–p curves corresponding to the PDP, MIP and UDP are named the PDC, MIC and UDC, respectively. As in the previous investigation (Li et al., 2013) we explored factors influencing the residual water saturation during drainage–imbibition cycles in three column tests, using the routine illustrated in Fig. 3, but with decreasing rather than increasing residual air saturations. The sandy medium was fully saturated with water before the tests. The first three cycles included three thorough drainage and three imbibition processes, to obtain a series of one PDC, two UDCs and three MICs, with corresponding residual water and air saturations. In the drainage processes of the following cycles, the medium was drained to successively lower scheduled air saturations (with two or three repetitions at each level), from which new imbibition processes were started to obtain new residual air saturations. This procedure was continued until the scheduled water saturation in the drainage process was too low to be obtained manually. Thus, the end point of each imbibition was the starting point of the following drainage process, and vice versa, and a series of residual air saturations were obtained at the ends of the imbibition processes, starting from gradually

Fig. 3. Test routine for obtaining residual air saturation values under gradually decreasing initial air saturations in successive drainage–imbibition cycles.

decreasing initial air saturations. In total, there were 12, 9 and 9 drainage–imbibition cycles in Tests 1, 2 and 3, respectively. The drainage–imbibition cycles were generated by scheduled water level changes in the column. The water level in the water tank was maintained at a fixed height of 147.3 cm above the bottom of the column (16.3 cm above the surface of the sand sample) throughout each column test. During each drainage process, the valve connected to the water tank was closed to prevent the water in the tank from flowing into the column, and the water in the sand flowed out of the column to the measuring flask at a flux rate controlled by a rotary pump. The water level moved downward, passing through the TDR and T5 probes without any stops, resulting in a smooth pressure reduction at the bottom boundary of the column. The position to which the water level fell depended on the value of the scheduled water saturation at the measurement location, since a lower water saturation required a lower water level from the bottom of the column. During this test routine, the final water level was above the bottom of the column in all except the first three drainage processes, in which a suction pump was used to draw as much pore water out of the column as possible to obtain more thorough drainage, after which the entire sand column was unsaturated. The boundary conditions of both the drainage and imbibition processes were designed to reflect those of groundwater movement in the aeration zone of ground with a fluctuating water table. In drainage processes, the upper boundary was open to air, while the lower boundary was open to both air and water (Fig. 1a). Before the drainage started, the pressure head difference between the two boundaries was the height of the spillway (h0). During the subsequent imbibition processes, the valve to the water tank was opened, and the water in the tank was infiltrated into the sand from the bottom of the column at a controlled low flux. The pressure head at the bottom condition was fixed at 143.0 cm throughout each imbibition process. The water level in the column increased smoothly, passing through the TDR and T5 probes until it reached the spillway, and then remained stable at the spillway, resulting in a smooth decrease of pressure head difference initially, followed by a stable pressure head difference of 16.3 cm (the water level difference between the spillway and the water tank) between the top and bottom boundaries. When the air in the sand approached its residual status, water was infiltrated into the column for a further time (more than 80 h), while excess water in the column flowed out from the top spillway, to obtain the stable residual air saturation. In imbibition processes, the upper boundary was open to both air and water, while the lower boundary was open to water only (Fig. 1b). Before starting, the pressure head difference between the upper and lower boundaries was the difference between the height of the water level in the water tank and the final water level in the previous drainage process (hw–ht), while the pressure head at the lower boundary was fixed constantly at 143.0 cm. The velocities from the start to the end of the imbibition processes were adjusted with a peristaltic pump operating at constant speed. The pore water flowing out of the sand was collected in a graduated cylinder to estimate the average water flow speed in the sand medium. In Tests 1, 2 and 3 the Darcy velocities were maintained at less than 4.07  105 m s1, 1.48  105 m s1 and 1.38  105 m s1, respectively. The velocities approached zero at the end of each drainage process with the reduction in water saturation. The Darcy velocities in the imbibition processes were slightly higher than those in the drainage processes to obtain stable residual air saturations as quickly as possible, and the velocity at the beginning was slightly higher than at the end of each imbibition process due to the reduction in the pressure head difference between the two ends of the column.

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In the drainage–imbibition cycles the water may have more easily penetrated the relatively large spaces between the porous medium and the column wall than the pores inside the medium. However, the 5.0 cm long needles of the TDR probes and the 0.5 cm long ceramic cup at the end of the 3.0 cm long shaft of the T5 tensiometers were completely buried in the sand medium and thus isolated from the boundary zones between the wall and the medium. Moreover, the upward water flow from the water tank displaced as much air from pores as possible under gravity in the imbibition processes, resulting in stable saturation of the medium. Consequently, effects of the boundary conditions on the TDR and T5 values were assumed to be negligible.

4. Results and discussion 4.1. Dynamic nonequilibrium effects All influential parameters except for the water/air saturation and capillary pressure were considered to be constant during the column tests, although the temperature varied in the range 20– 25 °C. Therefore, the saturations and capillary pressure were used to characterize the monitored processes in the sandy medium. In each drainage–imbibition cycle, the air saturation at the point of reversal from drying to wetting (Fig. 3) was defined as the initial air saturation (IAS) of the cycle, and used to explore the effects of initial air saturation on the measured residual air saturations during the tests. The IASs were decreased gradually to vary the drainage–imbibition cycle in the three tests, and there were two or three repetitions of cycles with each IAS (which was set as accurately as possible by using the rotary pump). Four IASs were applied in total in 12, 9 and 9 successive cycles in Tests 1, 2, and 3, respectively. The reductions and following increases in water saturation (and vice versa for capillary pressure) in the sandy medium recorded during the series of drainage and imbibition processes in the three tests are shown in Fig. 4. The changes in capillary pressure in the first three drainage processes varied slightly among the three tests due to variations in the suction pump’s performance. Notably, in the 6th drainage in Test 1 the velocity of water flowing out of the column was adjusted more slowly than usual, resulting in blips in the water saturation and capillary pressure changes. A reduction and subsequent increase (to the right in Fig. 4) in water saturation (correspondingly, an increase and subsequent decrease in capillary pressure) form a drainage–imbibition cycle. As shown in Fig. 4, the reduction in water saturation was followed by a slight increase at the ends of all drainage processes except in the first three drainage–imbibition cycles, when a suction pump was applied, resulting in increasing pressure followed by a slight reduction. As we also noted in our previous study (Li et al., 2013), these phenomena clearly reveal dynamic nonequilibrium effects (DNE), as reported by Diamantopoulos and Durner (2012), of the gradually decreasing initial air saturations. The effects are attributable at microscale to the time required for curvature of the air–water interfaces in pores to respond to changes in capillary pressure (Barenblatt, 1971; Sakaki et al., 2010), and at macroscale to some water remaining in the surface grooves, edges, or wedges of the emptying pores when air displaces the water in a porous medium, and some of this water being conducted slowly toward the continuous water body by film flow. Both of these processes reduce the capillary pressure and increase the water saturation under stopped-flow conditions in dynamic drainage processes (Poulovassilis, 1974; O’Carroll et al., 2005). At the end of each imbibition process, the residual air was presumably present in the form of small air blobs surrounded by the continuous water phase. Thus, the size of the air–water interfaces fell from pore cross sectionscale at the end of drainage processes to air blob-scale, becoming

Fig. 4. Variation of water saturation and capillary pressure with time during the column tests.

increasingly responsive to the high water pressure. Further, a continuous water phase increasingly occupied pores of all sizes, from their inner walls to their centers. Therefore, the DNE effects were strong in the drainage processes but inconspicuous in the imbibition processes. 4.2. Residual air saturations under decreasing levels of initial air saturation The conditions during the first three drainage–imbibition cycles illustrated in Fig. 4 were set to provide PDP + MIP, followed by two UDP + MIP cycles, with corresponding residual water and air saturations (Swr and Sgr) in the thorough drainage and imbibition processes. In total, the drainage–imbibition cycles generated a series of residual air saturations under the following gradually decreasing levels of IAS: 0.85–0.39 in Test 1, 0.85–0.38 in Test 2, and 0.88–0.42 in Test 3. This resulted in ranges of residual air saturations of 0.25– 0.27, 0.22–0.25 and 0.30–0.33, respectively, and ranges of residual air saturations of the MIPs (thorough imbibition processes in the tests) of 0.25–0.26, 0.22–024, and 0.30–0.31, respectively. Although the porosity and bulk density of the sand samples in the three tests were as consistent as possible, the series of residual air saturations in Test 3 were slightly larger than those of the other two tests, presumably due to differences in the diameters of pores around the TDR probes in the three tests. At the end of each drainage process, the invading air phase preferentially occupied the large pores in the sand medium. In the following imbibition process, the water was preferentially transported to the throats of pores in the sandy medium in the form of films in spaces between air and the solid surfaces, and gradually occupied increasingly large pores. Thus, the smaller the pores around the TDR probes

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were, the more water remained around them, explaining the smaller residual air saturations in Tests 1 and 2. S–p curves with similar forms and values were obtained from all three tests. Therefore, representative S–p curves (illustrating drainage from the fully water-saturated medium to the reversal point, followed by imbibition to the starting point of the next cycle), obtained for six of the nine drainage–imbibition cycles in Test 2 are shown in Fig. 5. The initial air saturations at the transition points from drying to wetting were 0.85, 0.85, 0.86, 0.78, 0.56 and 0.38, respectively, resulting in residual air saturations of 0.22, 0.23, 0.24, 0.24, 0.24, and 0.24, respectively. Slight reductions of the residual air saturations occurred in the first three imbibition processes. These may be attributable to dissolution of air into the pore water of the medium (Lenhard et al., 1991) in these imbibition processes, when the water level moved up and the sandy medium was gradually immersed in water. As the successive imbibition processes progressed further the air approached saturation in the pore water, thus air solvation in the residual air saturations became negligible in the last imbibition processes. The largest water saturation measurement error in the 15–30 °C temperature range was ±5.09% during the >2000 h blank test. However, the temperature during the three laboratory tests was kept in the range 20–25 °C, and the differences between the TDR and gravimetric saturation measurements were less than 1% (lower than variations in the blank test). Thus, all the residual air saturations in both the MIPs and following imbibition processes starting from low initial air saturations were clearly stable and almost identical. The residual air saturations were independent of the initial air saturations in the imbibition processes of all three column tests. The residual air saturations measured in the column tests reported here differ from our previous findings (Li et al., 2013). As shown in Fig. 6, in the cited study the residual air saturation changed suddenly from an unstable to a stable state during a series of consecutive imbibition processes starting from gradually increasing initial air saturations. It also gradually fell to zero with

sufficient imbibition time when the initial air saturation of the imbibition process was 60.49 in that study. Entry pressure (also known as bubbling pressure) refers to the capillary pressure beyond which the water in the medium can be driven out by air in a drainage process (Bear, 1972). In the three tests reported here, the average entry pressures were 20.33 cm, 21.24 cm, and 20.07 cm, respectively. All the imbibition processes after the first three started from the reversal points in which the sand was partially drained. The last ISAs were 0.40, 0.39 and 0.39 in Test 1, 0.38 and 0.38 in Test 2, and 0.42 and 0.42 in Test 3 (Table 1). The corresponding capillary pressures were similar to (or less than) the corresponding entry pressures: 13.75, 13.75 and 14.16 cm in Test 1, 22.08 and 22.7 cm in Test 2 and 20.83 and 20.64 cm in Test 3. Thus, the results confirm that once stable residual air saturation was achieved in a thorough imbibition process starting from high initial air saturation the residual air saturations in all of the following imbibition processes remained constant. This applied even for subsequent imbibition processes that started from low initial air saturations or capillary pressures that were lower than the entry pressure. These findings are consistent with results of our previous tests with initial air saturations exceeding 0.49 (Li et al., 2013) and the assumption of Poulovassilis (1970). Thus, we conclude that the residual air saturation may change suddenly from an unstable to a stable state with changes in initial air saturation during a series of consecutive imbibition processes starting from gradually increasing initial air saturations. Moreover, once a stable residual air saturation has been reached after an imbibition starting from high initial air saturation, all the subsequent residual air saturations may remain constant (even in imbibition processes starting from low initial air saturations). The movements of the two fluids in our sandy medium can be considered in greater (pore-scale) detail using the descriptions provided by Dullien et al. (1986) and Melrose (1987). Pores in real porous media usually have irregular cross-sections and rough

Fig. 5. Six representative S–p curves obtained from Test 2.

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Fig. 6. Difference in the changes of water saturation and capillary pressure between the two test routines. Test 1 refers to a series of cycles with gradually decreasing initial air saturations after thorough drainage processes, Test A refers to a series of cycles with gradually increasing air saturations (Li et al., 2013).

Table 1 Initial air saturations and residual air saturations measured in the column tests and estimated with the Land model. Cycle

1 2 3 4 5 6 7 8 9 10 11 12

Test 1

Level 1

Level 2

Level 3

Level 4

Test 2

Test 3

IAS

RAS

RAS*

Error

IAS

RAS

RAS*

Error

IAS

RAS

RAS*

Error

0.85 0.85 0.85 0.73 0.72 0.73 0.54 0.54 0.55 0.40 0.39 0.39

0.25 0.25 0.26 0.26 0.26 0.27 0.26 0.26 0.27 0.27 0.27 0.27

0.24 0.24 0.25 0.24 0.24 0.24 0.22 0.22 0.22 0.19 0.19 0.19

4.00 4.00 3.85 7.69 7.69 11.11 15.38 15.38 18.52 29.63 29.63 29.63

0.85 0.85 0.86 0.78 0.79

0.22 0.23 0.24 0.24 0.24

0.21 0.22 0.23 0.23 0.23

4.55 4.35 4.17 4.17 4.17

0.87 0.88 0.88 0.76 0.75

0.30 0.30 0.31 0.30 0.30

0.28 0.29 0.29 0.28 0.28

6.67 3.33 6.45 6.67 6.67

0.56 0.56

0.24 0.24

0.20 0.20

16.67 16.67

0.57 0.58

0.31 0.31

0.26 0.26

16.13 16.13

0.38 0.38

0.24 0.25

0.17 0.18

29.17 28.00

0.42 0.42

0.32 0.33

0.22 0.23

31.25 30.30

IAS is initial air saturation, RAS is the residual air saturation measured in the column tests. RAS* is the residual air saturation estimated with the Land model, and Error is the percentage error (%) between RAS and RAS* compared to RAS.

surfaces. In the air–water two-phase natural sandy medium of our three tests, the sand was initially saturated with water before the primary drainage process. Thus, the grooves, edges, and wedges on the pore surfaces were presumably covered with water films in all the following drainage–imbibition cycles. Water in soils and aggregates can be classified as hygroscopic water (also known as adsorption water), capillary water (viscous water), and free water (Brady and Weil, 1974). A hygroscopic water layer consists of an extremely well arranged monomolecular layer on negatively charged mineral surfaces and adsorption layers (film water), bound with varying strength by combinations of van der Waals, electrostatic and so-called structural forces (Mitchell, 1992; Derjaguin et al., 1987). Capillary water is essentially immobile and held by capillary forces in pendular rings around regions

of grain-to-grain contacts. Two layers (inner and outer) of capillary water can be distinguished. The inner layer is slightly diffusive, controlled solely by colloidal forces and acts as a transitional zone between the film water and outer capillary water, while the latter has been defined as loosely held water controlled by surface tensional and colloidal forces (Lyon and Buckman, 1937). The air– water interface formed by capillary menisci between the particles and air in a partially saturated soil mass is important for soil mechanics because it exerts a tensile pull (Fredlund and Rahardjo, 1993). Free water, also called gravitational water, is attracted to the soil solids so loosely that it may respond to the pull of gravity and move downwards in the soil (Lyon and Buckman, 1937). It should be noted that the hygroscopic water and film water, as defined above, differ from the macroscopic water film

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considered here: the water between pores’ inner surfaces and the air–water interface beyond them, which can move under the pressure gradient and is composed of the free water and some (but not all) of the capillary water and film water. At the end of each thorough drainage process the water films were mainly composed of parts of the film water and inner layer of capillary water, which were difficult to remove from the pore structure, even when using the suction pump (less than 100.0 cm water column) (Fig. 4). At the beginning of each PDP the water was the only phase present, and flow of the air phase only started after the breakthrough capillary pressure (bubbling pressure) was reached. From that point onward, both air and water phases were continuous. The invading air phase preferentially occupied the central portion of relatively large pores, bypassing and cutting off the path of the water phase. The displacement of water by air progressed through the air–water interfaces penetrating successively narrower pore throats from adjacent pore bodies in a piston-like manner, with occasional trapping of the water phase. When air had penetrated the central part of a pore, some water was always left in the surface grooves, edges, and wedges on the pore surface, which formed a continuous network throughout the porous medium. Thus, increasing portions of the water phase became disconnected until finally all water was discontinuous, except for the thick films of water in the surface capillaries. If the capillary pressure had been increased sufficiently, for sufficient time, the excess water could have escaped from the sand medium via the network of surface grooves, edges, and wedges. The UDP differed from the PDP only in the beginning, when disconnected, trapped air blobs surrounded by the continuous water phase would have been present in the sandy medium. When the sand was drained from a fully saturated to a residual state at the end of a thorough drainage process (one PDP and two UDPs) in Tests 1, 2 and 3, the largest pore spaces were left for air, most air invaded the connected pores, and the water phase was trapped in grooves, edges, and wedges on the pore surfaces. The solubility of a gas in a liquid increases with the partial pressure of the gas above the liquid, while decreases with the temperature of the liquid. The air pressure in the pore networks was considered to be connected with outside and maintained 1.0 atm. in the three tests. Thus the solubility of air in pore water of the sand medium was mainly affected by the temperature of the pore water. Since the solubility of air at laboratory temperature (in the range 20–25 °C) is in the range 18.68–17.09 ml L1 (Battino, 1981, 1982), variations in air solubility would have had negligible effects on air or water saturation (less than 0.02). The effects of air solubility on the residual air saturations were minor during the 2000 h tests. As stated by Dullien (1991), when a bubble moves in a tube of a porous medium under constant-flux flow with negligible inertial forces, the bubble configuration is primarily governed by the capillary number. The capillary number is widely defined as Ca = lwqw/Acaw (Aggelopoulos and Tsakiroglou, 2008), where qw is the flow rate of injected water (m3 s1), A is the total crosssectional area (m2), lw is the water dynamic viscosity (1.00  103 Pa s at 20 °C, and 0.89  103 Pa s at 25 °C (Kestin et al., 1978), and caw is the air/water interfacial tension (72.75  103 N s1 at 20 °C, and 71.99  103 N s1 at 25 °C (Vargaftik et al., 1983)). In the three tests, the capillary numbers of the imbibition processes at 20 °C were kept at less than 5.59  107, 2.03  107, and 1.90  107, respectively, and at 25 °C, less than 5.03  107, 1.83  107, and 1.71  107, respectively. All these values suggest that, during the three tests (in the range 20–25 °C), the surface tension force was substantially stronger than the viscous forces in all the imbibition processes in the three tests. Furthermore, when the displacing phase is strongly wetting, the capillary number is low (less than the order of

107), and the viscosities are not very high, the trapped phase is present in the form of disconnected blobs or ganglia of various sizes and shapes, held in inter-granular spaces by capillary forces (Dullien, 1991). Therefore the trapped air phase at the end of imbibition processes in our test system may have been in the form of disconnected blobs or ganglia of various sizes and shapes held in the inter-granular spaces by capillary forces. This summary of the probable movements during imbibition is also consistent with findings by Zhao and Rezkallah (1993), Lowe and Rezkallah (1999) and Rezkallah (1996) that air commonly occurs as bubbles and ‘slugs’ in surface tension-dominated air–water two-phase porous systems. In the following MIPs starting from reversal points at which air occupied most of the connected pores, and the water phase was largely trapped in grooves, edges, and wedges on the pore surfaces, the water was preferentially transported to the pore throats inside the sandy medium in the thick films between the air and the solid surfaces. The water filled the throats and pores where break-off and snap-off events disconnected the air phase, resulting in the formation of new air–water interfaces in throats, piston-type water motions in the pores and eventually (in conjunction with the water film on the pore surfaces) construction of a continuous water phase as imbibition progressed. Accordingly, the air phase gradually lost its continuity, and broke into individual globules that were trapped in the larger pores. Viscous forces, surface tension forces, buoyancy forces and pressure drag resulting from a bubble bypassing the wetting fluid flow primarily govern its movement in a low velocity wetting carrier fluid in a porous medium (Dawson et al., 2013). In our test system, the capillary force can be neglected at the end of the imbibition processes, where a continuous water phase occupied the interconnected pores from their inner surfaces to their central parts, and the buoyant force can also be neglected due to the small diameters of the pores. The movement of bubbles in this wetting carrier fluid depended most strongly on the size of the bubbles, and the interaction between the pressure drag force (a combination of surface force and viscous force resulting in the shear force exerted by the pore surfaces on the fluid films surrounding the bubbles) and the push force exerted by the difference in fluid pressure between two ends of the bubbles. The viscous force, surface tension force, and the push force resulting from the pressure gradient between the two ends of a bubble were accordingly the main forces driving the constant volume-flux flow at the end of the imbibition processes of the three tests. Therefore, under static or quasi-static flow conditions, all the residual displaced phase and displacing phase in the pores are immobile. More precisely, when the capillary number falls towards the critical value for trapping a given bubble, the bubble will slow, temporarily stop moving, and the push forces acting on the quasiarrested bubble will be insufficient to force it through a constriction of a tube (or pore). Thus, it will be trapped and the process will lead to stable residual displaced phase saturation. However, when the flow rate of the displacing phase is increased and/or the interfacial tension between the two phases is decreased sufficiently, the pressure forces acting over the length of the trap will exceed the surface energy required for the trapped bubble to reenter the constricted tube/pore (Dawson et al., 2013). Thus, the blobs or ganglia will start flowing along with the displacing phase again. At this stage, the two-phase flow degenerates into one-phase flow in a pore network, which contains obstacles in the form of the blobs or ganglia of the disconnected displaced phase (Dullien, 1991). However, a long bubble in a polygonal tube behaves like a leaky piston. The fluid will preferentially bypass it through leaky corners because of its large drag. In such cases the movement of the liquid consists of plug flow, which pushes the bubble and occupies the main cross-sectional area of the capillary, and the corner flow

Y. Li et al. / Journal of Hydrology 519 (2014) 271–283

which bypasses the bubble in the leaky corners of the capillary. The plug flow will obey the bubble’s pressure–velocity relationship, while the corner flow will follow a linear pressure–velocity function. Furthermore, a small, quantum-like jump may occur whenever a meniscus at the advancing end of the blob is forced through a constriction at pore throats, as visually observed in columns of perspex spheres (Ng et al., 1978; Yadav and Mason, 1983; Chatzis and Morrow, 1984). All these considerations indicate that in a given air–water two-phase porous system the trapped air blobs at the end of an imbibition process may remobilize if the water flow rate generates sufficient push force to exceed the pressure drag force. In addition, the mode of air blob motion depends on the ratio of blob size to the pore throat radius. Two types of trapped gas bubbles can occur in a porous medium: pore-scale bubbles occupying single pores, and pore network-scale bubbles occupying multiple interconnected pores at the end of an MIP (Suekane et al., 2009; Zhou et al., 2010). In the three tests, at the start of the MICs the largest pore spaces were occupied by air and the minimal water phase was trapped in grooves, edges, and wedges on the pore surfaces. The water film in the walls surrounding large pores would have been much thinner than in the other partial drainage processes. The large amounts of continuous air phase and thin water films surrounding the pore surfaces resulted in the trapped air consisting mainly of pore network-scale bubbles, clogging the entire cross-sections of pores except small corner regions. The water flow then had to squeeze through greatly reduced cross-sections, and the bypassing of the water flow led to a large pressure drag. As stated by Ryan and Dhir (1993), remobilization of residual ganglia can also occur when the capillary number increases beyond a critical value. In our test system, since the capillary number was in the order of 107, the water pressure gradient transferred to the connected air globules was insufficient to exceed the pressure drag difference between the two ends of these air globules, thus the trapped air globules remained in the medium, resulting in stable residual air saturations. In the following partial drainage processes, although these connected pore network-scale bubbles became larger and occupied central portions of most large pores again, the connections between them still remained in the interconnected pores of the medium. In a given air–water two-phase medium system, the capillary number is primarily determined by the fluid velocity, and thus by the water pressure gradient (hw–h0) between the two ends of the column. Since the water pressure gradient between the two ends of each connected air globule (also hw–h0) was not large enough to produce a push force exceeding the pressure drag difference between the two ends of the connected air globules to drive them out of the sandy medium, these pore network-scale bubbles remained throughout the partial imbibition processes. This also resulted in constant residual air saturations in all the following imbibition processes. It should be noted that if the water pressure gradient between the two ends of the column had been increased these trapped pore network-scale air globules may have remobilized in a leaky piston or even quantum-like jump manner as they were forced through constrictions at pore throats. Regarding the residual air saturations following gradually increasing initial air saturations in successive drainage–imbibition cycles we previously examined (Li et al., 2013), since the sand was partially drained at the beginning of each series of tests only small fractions of the large pores were occupied by the air phase (Fig. 6). Minute pores, dead ends, and the grooves, edges, and wedges on the walls surrounding large pores were occupied by water or a water film, which still formed a continuous phase in the pores of the medium. Furthermore, the water films on the surfaces of these large pores were much thicker than the films after a thorough drainage processes. Then in the following partial imbibition

279

processes the trapped air probably consisted of single pore-scale air globules due to snap-off or choke-off releases to the air phase promoted by the small amounts of continuous air phase and thick water films on the pore surfaces. These single pore-scale air globules did not clog the pore cross-sections completely, i.e. water films still remained in the spaces between the air globules and pore walls, and in corner regions. The pressure drag caused by the bypassing of the water flow was limited. Furthermore, in our tests, the water pressure gradient transferred to the air globules was sufficient to exceed the pressure drag difference, and the trapped air globules could be driven out of the sandy medium. Thus, the two-phase flow degenerated into one-phase flow in network pores, coupling with obstacles in the form of the blobs or ganglia of the disconnected displaced phase, which resulted in unstable residual air saturations. More air remained in the pores when more pore water was drained from the sand medium, after which more air blobs were produced and trapped in the pores of the medium. As stated by Stark and Manga (2000), when bubbles occupy low fractions of pore spaces, most of them follow a limited number of high-flow pathways through the pore network. As the volume fraction of bubbles increases, interactions between individual bubbles become important, resulting in residence times of the trapped air blobs in the porous media gradually increasing. In an imbibition process, the water phase under the water pressure gradient converges with the water film, then surrounds and cuts off the air phase, resulting in air globules. The thickness of the water film plays a positive role in the snap-off events of the air phase. As mentioned above, if thin water films are initially present in the grooves, edges, and wedges on the walls surrounding large pores of a sandy medium, and there are large volumes of air in the pores’ central parts, an imbibition process will result in pore network-scale air globules in interconnected pores. In such conditions snap-offs of the air phase ahead of the displacement front will be weak. In contrast, if the imbibition process starts from thick water films in the grooves, edges, and wedges on the walls surrounding large pores, with small volumes of air in their central parts, single pore-scale air globules will form in interconnected pores due to the stronger snap-offs of the air phase ahead of the displacement front. Thus, a stronger dynamic flow (higher capillary number) may be required to remobilize connected, pore networkscale air globules than single pore-scale air globules from a sandy medium in a forced imbibition process. This conclusion is consistent with previous findings that trapped pore network-scale air globules strongly contribute to residual air saturation in given hydraulic conditions (Suekane et al., 2009), and that trapped gas bubbles remained stable with increases in flow rate up to a capillary number of 1.0  105 in nitrogen–water two phase Berea sandstone systems (Zhou et al., 2010). 4.3. Residual air saturations measured in tests and estimated with the Land model Following the tests the acquired residual air saturation data were compared to values obtained using the Land model (Land, 1968), which is generally expressed as:

1 1 1  ¼ 1 Sgr Smax S gr g

ð3Þ

where Smax is the initial air saturation from which an imbibition g process starts, Swr is the residual water saturation of the PDC, and (hence) Smax should be equal to or less than 1  Swr. Sgr* is the residg ual air saturation of the imbibition process starting from Smax , and g Sgr is the residual air saturation of the MIC, which was obtained from the average value of the residual air saturations in the first

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three drainage–imbibition cycles (0.25 in Test 1, 0.23 in Test 2, and 0.30 in Test 3). Initial and residual air saturations measured in the column tests and estimated with the Land model (Fig. 7) show that the residual air saturations estimated with the Land model decreased with reductions in the initial air saturation, while the measured residual air saturations remained almost constant in the series of consecutive imbibition processes. As mentioned above, the measurement errors of the three laboratory tests were less than that of the blank test (±5.09%). Since the initial air saturations exceeded 0.72 in Test 1, 0.56 in Test 2, and 0.76 in Test 3, the differences between values measured and estimated with the Land model exceeded the blank test errors. Furthermore, when the initial air saturations decreased below 0.38 (Table 1), the largest deviations between measured values and model estimates from the three tests increased up to 29.63%, 28.00%, and 30.30%, respectively. In Test 2, the high value of initial air saturation and small differences in the initial air saturation among the first five imbibition processes resulted in smaller deviations between the measured and estimated values. Thus, results obtained from the Land model clearly conflict with the residual air saturations measured in the column tests reported here, and findings from our previous study that unstable residual air saturations increased with increases in the initial air saturations of consecutive imbibition processes (Li et al., 2013). As also mentioned above, under dynamic flow conditions the pressure gradient in the water phase transfers directly to trapped air globules (Diamantopoulos and Durner, 2012). In an imbibition process, when the water pressure gradient acting on air globules exceeds the pressure drag difference between their two ends, and there is sufficient time for the pressure transfer, the globules will be released from the medium, thus the residual air saturation will decrease. While the residual air is present in the form of connected, pore network-scale air globules, and the water pressure gradient acting on them does not exceed the pressure drag difference between their ends, they will remain trapped in the pores, resulting in stable, constant residual air saturations. Macroscopically, the residual air saturation is related to the applied boundary condition, i.e. the water pressure gradient between the top and bottom of the column. Thus, the Land model may not be valid in some dynamic flow conditions. Under quasi-static or steady-state conditions, no water pressure gradient was applied to the blobs in the air–water two-phase medium system. As stated by Dullien (1991), owing to various effects of pore morphology (e.g., snap-off or choke-off), the displacing phase tends to surround and cut off portions of the phase originally present in the pore space. Once cut off and isolated, these so-called blobs or ganglia of the displaced phase normally become stationary because of the action of interfacial forces that are too great for the

viscous or gravitational forces present to overcome. Whether present in the form of single pore-scale globules or connected pore network-scale air globules, the residual air saturation remains and is stable in the medium. With increases in the air volume left in the pores at the end of a drainage process more air globules are trapped in the following imbibition process. As also partially demonstrated by Li et al. (2013), generally when a sample partially drains there may be a relationship between the initial water saturation and residual air saturation of an imbibition process under quasi-static or steady-state conditions (Lenhard et al., 1991). As we noted in our previous study (Li et al., 2013), the Land model (Land, 1968) was based on integrated results of several studies under hydrostatic flow conditions, and it may only be valid for imbibition processes starting from a point on the PDC (Lenhard et al., 1991). Our findings presented here further suggest that relations between initial saturation and residual saturation in porous media may not always be monotonous, but may shift in dynamic flow conditions. Of course, more research is required to fully elucidate these relations in imbibition processes, especially along random paths, or under dynamic nonequilibrium conditions. The microscopic liquid geometry determines macroscopic properties, such as saturation and phase permeability. However, the macroscopic properties in a porous medium depend on capillary pressure history rather than on the momentary value of the capillary pressure alone (Dullien, 1991). In our tests, the residual air saturations ranged from 0.25 to 0.27 in Test 1, 0.22 to 0.25 in Test 2, and 0.30 to 0.33 in Test 3 (Table 1). Thus, the differences in residual air saturation among the consecutive drainage–imbibition cycles were very small, indicating that the capillary pressure history and number of imbibition processes had minimal effects on the residual air saturation in the undisturbed sandy medium. This finding conflicts with the assumption that the residual saturation of a NWF is influenced by the number of drainage and imbibition cycles (Helmig, 1997). 4.4. Parameters of the van Genuchten model under consecutive drainage–imbibition cycles A widely accepted S–p relationship model, the van Genuchten (VG) model (Van Genuchten, 1980) is expressed as follows: c

h ¼

1=n 1  1=m Swe  1

a

hc > 0

ð4Þ

Here, hc is defined in Eq. (2), a (1/cm) and n are fitted parameters, m = 1  1/n, Sw is the water saturation, and Swe is the effective water saturation, which is defined as:

Swe ¼

Sw  Swr 1  Swr  Sgr

Swr 6 Sw 6 1  Sgr

Fig. 7. Initial air saturations and corresponding residual air saturations.

ð5Þ

281

0.969 0.987 0.306 0.323 0.122 0.122 9.821 2.844 0.037 0.038 0.877 0.940 0.239 0.246 0.141 0.141 9.388 3.665 0.034 0.034

e

c

d

a

b

Swr is the residual water saturation measured in the column tests. Sar is the measured residual air saturation. R is the correlation coefficient. D is the drainage process. I is the imbibition process.

0.938 0.980 0.261 0.272 0.147 0.147 7.209 2.664 D I 10

0.035 0.034

0.990 0.994 0.298 0.319 0.122 0.122 9.002 3.159 0.038 0.053 0.933 0.990 0.236 0.240 0.141 0.141 8.851 3.269 0.037 0.045 0.941 0.986 0.260 0.268 0.147 0.147 8.848 2.587 D I 7

0.034 0.048

0.932 0.979 0.309 0.308 0.122 0.122 8.770 4.194 0.038 0.059 0.950 0.990 0.244 0.238 0.141 0.141 9.264 3.661 0.036 0.056 0.956 0.973 0.258 0.260 0.147 0.147 8.606 4.826 D I 4

0.035 0.055

0 0.295 0.126 0.126 10.762 6.770 0.036 0.092 0.985 0.979 0 0.219 0.146 0.146 8.900 3.886 0.040 0.067 0.952 0.977 0 0.246 0.139 0.147 10.287 6.577 D Ie 1

d

0.036 0.069

Sar Swr n a (cm1)

Test 3

R Sar Swr a (cm1)

Test 2

Rc Sarb Swra n Test 1

a (cm1)

where Pn, Pw, and Pc are nonwetting fluid pressure, wetting fluid pressure and capillary pressure under static flow conditions, respectively. s, a non-equilibrium capillarity coefficient, is a material property but may still be related to saturation. Eq. (6) has been widely used to relate differences in macroscale pressures between nonwetting and wetting fluids to the macroscale capillary pressure and the rate change of saturation under dynamic flow conditions. However, it has not been widely used for numerical simulations of the movement of immiscible fluids in porous media under dynamic flow conditions, since it is relatively difficult to obtain accurate estimates of the dynamic capillarity coefficient (s). The correlation coefficients obtained from fitting the VG model to the measured S–p relation data from Tests 1, 2 and 3 exceed 0.94, 0.88 and 0.93, respectively. Based on the difference in the S–p relationship between static and dynamic conditions, as reported by Hassanizadeh et al. (2002), significant DNE effects occur in the range between the entry pressure and the pressure at which the saturation does not change any more, regardless of the magnitude of the pressure drop between the two ends of the column, i.e. about 10 cm difference in the 20–70 cm capillary pressure range, according to Eq. (1) (Topp et al., 1967; Smiles et al., 1971; Vachaud et al., 1972). Furthermore, DNE effects have only been empirically examined over rather limited ranges of length and time scales, usually in sandy materials, and their influence on water flows at greater (field) scale is still unclear (Diamantopoulos and Durner, 2012). All these considerations suggest that the VG model, generally used in static or quasi-static flow conditions, can satisfactorily model immiscible fluids in large-scale domains of porous media until a more accurate constitutive equation is developed to describe the relationship between saturation and capillary pressure under dynamic fluid conditions.

n

ð6Þ

Cycle

@Sw @t

Table 2 Parameters of S–p relations fitted with the Van Genuchten model.

Pn  Pw ¼ Pc  s

R

where Swr and Sgr are residual water saturation and residual air saturation, respectively. In the primary drainage process, Swr = 0. To test the validity of the VG model under dynamic flow conditions it was fitted to the S–p data measured in the three column tests reported here, using the modified Gauss–Newton method (Hartley, 1961) and parameters shown in Table 2. Among all the drainage processes, the parameter a ranged from 0.034 to 0.036 in Test 1, from 0.034 to 0.040 in Test 2, and from 0.036 to 0.038 in Test 3. This parameter is considered to be inversely related to the bubbling pressure (the capillary pressure at which the pore water can be displaced by air) in a drainage process (Van Genuchten, 1980). All the a values were almost identical, suggesting that differences in the bubbling pressure among all the drainage processes starting from different initial water saturations were minor. Moreover, as shown in Fig. 5, the bubbling pressure in all the drainage processes was consistently ca. 20.0 cm H2O. All of these findings suggest that the bubbling pressure was medium-specific, which further confirms that the entry pressure is related to the valid entry radius of a medium (Frette and Helland, 2010). Eq. (1) is widely applied to describe S–p relationships obtained experimentally under equilibrium conditions. However, it is not accurate for S–p relationships under dynamic flow conditions due to the dynamic effects. The S–p relationship depends on the flow dynamics, including both the capillary pressure history and the rate of change of saturation (Hassanizadeh et al., 2002). The dynamic effects are obvious in the middle section of an S–p curve for a drainage or imbibition process, but they are not important at the beginning and end due to the rate of change of saturation approaching zero. For non-equilibrium situations, the following equation for the difference in fluid pressures has been suggested (Hassanizadeh and Gray, 1990; Kalaydjian, 1992):

0.994 0.997

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In a comparison with an oil–water two-phase bead system, Culligan et al. (2006) found that drainage processes in a corresponding air–water system yielded higher entry pressures and lower residual water saturations, while imbibition processes yielded lower residual air saturations. However, similar S–p curves were obtained for both systems and the differences were apparently due to the higher interfacial tension between air and water in the air–water system than between oil and water in the oil–water system. Oddie et al. (2003) also observed qualitatively similar flow patterns of water–gas and oil–water–gas systems in a series of steady-state and dynamic analyses of multiphase (kerosene, tap water and nitrogen) flows in a transparent 11 m long, 15 cm diameter, inclinable pipe. Thus, the stable and constant residual air saturations we observed in our air–water two-phase fine sandy medium may also be applicable to the unsaturated hydraulic properties of porous media hosting dynamically changing oil–water mixtures. 5. Conclusions A series of column tests was conducted to extend our previous investigation and characterize relations between the initial water saturation and resulting residual air saturation, and S–p relation, in an air–water two-phase sandy medium. The main results obtained are as follows: (1) After a thorough drainage process, all the residual air saturations in the following imbibition processes under gradually decreasing initial air saturations were almost constant, and independent of the initial air saturations. (2) In imbibition processes, thin water films on the walls surrounding large pores and large volumes of air in their central parts may result in residual air in the form of connected, pore network-scale air globules in interconnected pores due to weak snap-off of the air phase ahead of the displacement front. In contrast, thick water films and small volumes of air in the central parts of the pores may result in residual air in the form of single pore-scale air globules in interconnected pores. (3) Stronger dynamic flow conditions (e.g., higher capillary numbers) may be required to remobilize connected, pore network-scale air globules than single pore-scale air globules in a forced imbibition process. (4) The bubbling pressure in a sandy medium is medium-specific, and independent of drainage process parameters under consecutive drainage–imbibition cycles. (5) The VG model generally applied under static or quasi-static flow conditions can satisfactorily model immiscible fluids in large-scale domains of porous media under dynamic fluid conditions.

Acknowledgements The authors gratefully acknowledge Dr. John Blackwell (Seesediting Ltd.) for his editing of this paper. This work was supported by the Chinese National Science Foundation (Grant No. 41072182), the Science-Technology Planning Project of Guangdong Province (Grant No. 2012A030700008), the Science and Technology Project of Daya Bay (Grant No. 20100103), Environmental Assessment of the Prospecting and Trial Mining of Gas Hydrates in South China Sea (Grant No. GZH201100311), and Special Foundation of Science and Technology Dissemination of Guangdong Provincial Marine and Fishery (Grant No. B201300A04).

References Aggelopoulos, C.A., Tsakiroglou, C.D., 2008. The effect of micro-heterogeneity and capillary number on capillary pressure and relative permeability curves of soils. Geoderma 148, 25–34. Barenblatt, G.I., 1971. Filtration of two nonmixing fluids in a homogeneous porous medium. Fluid Dyn. 6, 857–864. Battino, R. (Ed.), 1981. IUPAC Solubility Data Series. Oxygen and Ozone, vol. 7. Pergamon Press, Oxford, England. Battino, R. (Ed.), 1982. IUPAC Solubility Data Series. Nitrogen and Air, vol. 10. Pergamon Press, Oxford, England. Bear, J., 1972. Dynamics of Fluids in Porous Media. Dover Publications, INC., New York. Blunt, M.J., Scher, H., 1995. Pore-level modeling of wetting. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics. 52 (6), 6387–6403. Brady, N.C., Weil, R.R., 1974. The Nature and Properties of Soils. Prentice Hall, New Jersey. Brusseau, M.L., Peng, S., Schnaar, G., Costanza-Robinson, M.S., 2006. Relationships among air–water interfacial area, capillary pressure, and water saturation for a sandy porous medium. Water Resour. Res. 42 (3). http://dx.doi.org/10.1029/ 2005WR004058. Chatzis, I., Morrow, N.R., 1984. Correlation of capillary number relationships for sandstone. SPE 10114. SPE J. 24 (5), 555–562. Chatzis, I., Morrow, N.R., Lim, H.T., 1983. Magnitude and detailed structure of residual oil saturation. SPE 10681. SPE J. 23 (2), 311–326. Constantinides, G.N., Payatakes, A.C., 2000. Effects of precursor wetting films in immiscible displacement through porous media. Transp. Porous Media. 38, 291–317. Corapcioglu, M.Y., Yoon, S., Chowdhury, S., 2009. Pore-scale analysis of NAPL blob dissolution and mobilization in porous media. Transp. Porous Med. 79, 419–442. Corey, A.T., 1994. Mechanics of Immiscible Fluids in Porous Media. Water Resources Publications, Colorado. Culligan, K.A., Wildenschild, D., Christensen, B.S.B., Gray, W.G., Rivers, M.L., 2006. Pore-scale characteristics of multiphase flow in porous media: a comparison of air–water and oil–water experiments. Adv. Water Resour. 29, 227–238. Dangla, R., Lee, S., Baroud, C.N., 2011. Trapping microfluidic drops in wells of surface energy. Phys. Rev. Lett. 107, 124501. Dawson, G., Lee, S., Juel, A., 2013. The trapping and release of bubbles from a linear pore. J. Fluid Mech. 2013 (722), 437–460. http://dx.doi.org/10.1017/ jfm.2013.103, Cambridge University Press. Derjaguin, B.V., Churaev, N.V., Muller, V.M., 1987. Surface Forces. Consultants Bureau, New York. Diamantopoulos, E., Durner, W., 2012. Dynamic nonequilibrium of water flow in porous media: a review. Vadose Zone J. http://dx.doi.org/10.2136/ vzj2011.0197. Dullien, F.A.L., 1991. Porous Media: Fluid Transport and Pore Structure (2nd edition). Academic Press, Inc., San Diego, California, 92101. Dullien, F.A.L., Lai, F.S.Y., Macdonald, I.R., 1986. Hydraulic continuity of residual wetting phase in porous media. J. Colloid Interface Sci. 109, 201. Fredlund, D.G., Rahardjo, H., 1993. Soil Mechanics for Unsaturated Soils. Wiley, New York. Frette, O.I., Helland, J.O., 2010. A semi-analytical model for computation of capillary entry pressures and fluid configurations in uniformly-wet pore spaces from 2D rock images. Adv. Water Resour. 33, 846–866. Guarnaccia, J., Pinder, G., Fishman, M., 1997. NAPL: Simulator Documentation, EPA/600/R-97/102, Nationa l Risk Management Research Laboratory, USEPA. Hartley, H.O., 1961. The modified Gauss–Newton method for the fitting of non– linear regression functions by least squares. Technometrics 3, 269–280. Hassanizadeh, S.M., Gray, W.G., 1990. Mechanics and thermodynamics of multiphase flow in porous media. Adv. Water Resour. 13, 169–186. Hassanizadeh, S.M., Celia, M.A., Dahle, H.K., 2002. Dynamic effect in the capillary pressure–saturation relationship and its impacts on unsaturated flow. Vadose Zone J. 1, 38–57. Helmig, R., 1997. Multiphase Flow and Transport Process in the Subsurface: A Contribution to the Modeling of Hydrosystems. Springer. Hilfer, R., 2006. Capillary pressure, hysteresis and residual saturation in porous media. Phys. A 359, 119–128. Hughes, R.G., Blunt, M.J., 2000. Pore scale modeling of rate effects in imbibition. Transp. Porous Media. 40 (3), 295–322. Hughes, R.G., Blunt, M.J., 2001. Network modeling of multiphase flow in fractures. Adv. Water Resour. 24, 409–421. Jung, S.Y., Lim, S., Lee, S.J., 2012. Investigation of water seepage through porous media using X-ray imaging technique. J. Hydro. 452–453, 83–89. Kalaydjian, F., 1992. Dynamic capillary pressure curve for water/oil displacement in porous media, theory vs. experiments. In: Proc. SPE Conference, Washington DC. SPE, Brookfield, CT., SPE 24813, pp. 491–506. Kamath, J., Meyer, R.F., Nakagawa, F.M., 2001. Understanding waterflood residual oil saturation of four carbonate rock types. In: Proceedings of the SPE Annual Technical Conference and Exhibition, 30 September–3 October, New Orleans, Louisiana. Paper SPE 71505. Kestin, J., Sokolov, M., Wakehanm, W.A., 1978. Viscosity of liquid water in the range 8 C to 150 C. J. Phys. Chem. Ref. Data 7 (3), 941–948.

Y. Li et al. / Journal of Hydrology 519 (2014) 271–283 Kovscek, A.R., Radke, C.J., 2003. Pressure-driven capillary snap-off of gas bubbles at low wetting-liquid content. Colloids and surfaces a: physicochem. Eng. Aspects 212, 99–108. Lac, E., Sherwood, J.D., 2009. Motion of a drop along the centreline of a capillary in a pressure–driven flow. J. Fluid Mech. 640, 27–54. http://dx.doi.org/10.1017/ S0022112009991212. Land, C.S., 1968. Calculation of imbibition relative permeability for two- and threephase flow from rock properties. Trans. Am. Inst. Min. Metall. Pet. Eng. 243, 149–156. Lenhard, R.J., Parker, J.C., Kaluarachchi, J.J., 1991. Comparing simulated and experimental hysteresis two–phase transient fluid flow phenomena. Water Resour. Res. 27 (8), 2113–2124. Lenormand, R., Zarcone, C., 1984. Role of roughness and edges during imbibition in square capillaries. In: Proceedings of the SPE Annual Technical Conference and Exhibition, Houston, Texas, 16–19 September. Paper SPE 13264. Li, Y., Flores, G., Xu, J., Yue, W.Z., Wang, Y.X., Luan, T.G., Gu, Q.B., 2013. Residual air saturation changes during consecutive drainage–imbibition cycles in an air– water fine sandy medium. J. Hydrol. 503, 77–88. http://dx.doi.org/10.1016/ j.jhydrol.2013.08.050. Lowe, D.C., Rezkallah, K.S., 1999. Flow regime identification in microgravity twophase flow using void fraction signals. Int. J. Multiphase Flow. 25, 433–457. Lyon, T.L., Buckman, H.O., 1937. The Nature and Properties of Soils. Macmillan, New York. Melrose, J.C., 1987. Characterization of petroleum reservoir rocks by capillary pressure techniques. In; Presented at IUPAC Symposium, Bad Solen, West Germany, April 26–29. Mitchell, J.K., 1992. Fundamentals of Soil Behaviour, 2nd ed. Wiley, New York. Mogensen, K., Stenby, E., 1998. A dynamic two-phase pore-scale model of imbibition. Transp. Porous Media. 32 (3), 299–327. Mohanty, K., Davis, H., Scriven, L., 1980. Physics of oil entrapment in waterwet rock. In: Proceedings of the 55th Annual Fall Technical Conference and Exhibition of the SPE of AIME, Dallas, Texas, 21–24 September. Paper SPE 9406. Ng, K.M., Davis, H.T., Scriven, L.E., 1978. Visualization of blob mechanics in flow through porous media. Chem. Eng. Sci. 33, 1009. O’Carroll, D.M., Phelan, T.J., Abriola, L.M., 2005. Exploring dynamic effects in capillary pressure in multistep outflow experiments. Water Resour. Res. 41, W11419. http://dx.doi.org/10.1029/2005WR004010. Oddie, G., Shi, H., Durlofsky, L.J., Aziz, K., Pfeffer, B., Holmes, J.A., 2003. Experimental study of two and three phase flows in large diameter inclined pipes. Int. J. Multiph. Flow. 29, 527–558. Peng, S., Hu, Q.H., Dultz, S., Zhang, M., 2012. Using X-ray computed tomography in pore structure characterization for a Berea sandstone: resolution effect. J. Hydrol. 472–473, 254–261. Ponizovsky, A.A., Chudinov, S.M., Pachepsky, Y.A., 1999. Performance of TDR calibration models as affected by soil texture. J. Hydrol. 218, 35–43. Poulovassilis, A., 1970. The effect of the entrapped air on the hysteresis curves of a porous body and on its hydraulic conductivity. Soil Sci. 100 (3), 154–162. Poulovassilis, A., 1974. Hysteresis in relationship between hydraulic conductivity and suction. Soil Sci. 117 (5), 250–256.

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Provis, J.L., Myers, R.J., White, C.E., Rose, V., van Deventer, J.S.J., 2012. X-ray microtomography shows pore structure and tortuosity in alkali-activated binders. Cem. Concr. Res. 42, 855–864. Rezkallah, K.S., 1996. Weber number based flow-pattern maps for liquid–gas flows at microgravity. Int. J. Multiph. Flow. 22, 1265–1270. Ryan, R.G., Dhir, V.K., 1993. The effect of soil-particle size on hydrocarbon entrapment near a dynamic water table. J. Soil Contam. 2 (1), 1–34. Sakaki, T., O’ Carroll, D.M., Illangasekare, T.H., 2010. Direct quantification of dynamic effects in capillary pressure for drainage–wetting cycles. Vadose Zone J. 9, 424–437. http://dx.doi.org/10.2136/vzj2009.0105. Schembre, J.M., Kovscek, A.R., 2003. A technique for measuring two-phase relative permeability in porous media via X-ray CT measurements. J. Petrol. Sci. Eng. 39, 159–174. Smiles, D.E., Vachaud, G., Vauclin, M., 1971. A test of the uniqueness of the soil moisture characteristic during transient, nonhysteretic flow of water in a rigid soil. Soil Sci. Soc. Am. Proc. 35, 534–539. http://dx.doi.org/10.2136/ sssaj1971.03615995003500040018x. Stark, J., Manga, M., 2000. The motion of long bubbles in a network of tubes. Transport Porous Med. 40, 201–218. Suekane, T., Thanh, N.H., Matsumoto, T., Matsuda, M., Kiyota, M., Ousaka, A., 2009. Direct measurement of trapped gas bubbles by capillarity on the pore scale. Energy Procedia 1, 3189–3196. Tippkötter, R., Eickhorst, T., Taubner, H., Gredner, B., Rademaker, G., 2009. Detection of soil water in macropores of undisturbed soil using microfocus X-ray tube computerized tomography (lCT). Soil Tillage Res. 105 (1), 12–20. Topp, G.C., Klute, A., Peters, D.B., 1967. Comparison of water content-pressure head data obtained by equilibrium, steady-state, and unsteady-state methods. Soil Sci. Soc. Am. Proc. 31, 312–314. http://dx.doi.org/10.2136/ sssaj1967.03615995003100030009x. Vachaud, G., Vauclin, M., Wakil, M., 1972. A study of the uniqueness of the soil moisture characteristic during desorption by vertical drainage. Soil Sci. Soc. Am. Proc. 36, 531–532. Van Genuchten, M. Th, 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil. Sci. Soc. Am. J. 44, 892–898. Vargaftik, N.B., Volkov, B.N., Voljak, L.D., 1983. International tables of the surface tension of water. J. Phys. Chem. Ref. Data 12 (3), 817–820. Wildenschild, D., Sheppard, A.P., 2013. X-ray imaging and analysis techniques for quantifying pore-scale structure and processes in subsurface porous medium systems. Adv. Water Resour. 51, 217–246. Wong, H., Radke, C.J., Morris, S., 1995. The motion of long bubbles in polygonal capillaries. Part 2. Drag, fluid pressure and fluid flow. J. Fluid Mech. 292, 95–110. Yadav, G.D., Mason, G., 1983. The onset of blob motion in a random sphere packing caused by flow of the surrounding liquid. Chem. Eng. Sci. 38 (9), 1461–1465. Zhao, L., Rezkallah, K.S., 1993. Gas–liquid flow patterns at microgravity conditions. Int. J. Multiphase Flow 19, 751–763. Zhou, N., Matsumoto, T., Hosokawa, T., Suekane, T., 2010. Pore-scale visualization of gas trapping in porous media by X-ray CT scanning. Flow Meas. Instrum. 21, 262–267.