An experimental study on the rate and mechanism of capillary rise in sandstone
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The Lucas-Washburn equation is a fundamental expression used to describe capillary rise in geologic media on the basis of pore radius, liquid viscosity, surface tension, contact angle, and time. It is known that a radius value significantly smaller than the main pore radius must be used in the equation in order for the predictions to fit the experimentally measured values. To evaluate this gap between theoretical predictions and experimental data, we conducted several capillary rise experiments using Berea sandstone. First, to investigate conditions in which pores of any size are available for capillary rise, an experiment was conducted using a dried core. Next, to adjust the size distribution of pore water before the capillary rise, gas pressure was applied to a water-saturated core and water was expelled from pores of r > 10 μm; then, capillary rise was initiated. Under this condition, capillary rise occurred only in the pores of r > 10 μm. The same experiment was conducted for r = 3, 1, and 0.36 μm. When narrower pores were made available for capillary rise, the overall rate of rise decreased and approached the rate observed when the sample was dry initially. This observation suggests that the capillary rise in narrow pores plays a significant role in the overall rate. Based on these results, we propose a conceptual capillary rise model that considers differing radii in branched pores and provide an example of a quantitative description of capillary rise.
KeywordsCapillary rise Capillary pressure Lucas-Washburn equation Water expulsion method Sandstone
Geologic media commonly have pores of various sizes and shapes. When water comes in contact with a geologic medium, it is drawn into the pores by capillary force. At equilibrium, the height of capillary rise in a vertical pipe with a radius of r at 20 °C is estimated to be 15, 1.5, and 0.15 m for r = 1, 10, and 100 μm, respectively. An understanding of the mechanism and rate of capillary rise is important when considering the transport of water in geologic media near the ground surface (i.e., imbibition and drying), associated processes such as rock weathering and soil formation, water availability to plants, and contaminant migration.
This is a fundamental equation for understanding capillary rise in pores. However, it has been found that the x-t relationship predicted by the L-W equation with simple assumptions (e.g., a constant, realistic r and a constant θ) does not agree well with measured, experimental data (Dullien et al. 1977; Hammecker et al. 1993). Methods to improve the L-W equation involving change in contact angle (Einset 1996; Siebold et al. 2000; Heshmati and Piri 2014), non-uniformity of pore radius (Dullien et al. 1977; Erickson et al. 2002), tortuosity of pores (Benavente et al. 2002; Cai et al. 2014), non-circularity of the cross section of pores (Benavente et al. 2002; Cai et al. 2014), and inertial effect and pore wall roughness especially at the earliest stage of capillary rise (Szekely et al. 1971) have been discussed. A geologic medium usually has a complex pore structure, and any of the variables τ, r, and θ can change depending on the position in the medium and the elapsed time of capillary rise. Therefore, it is often not easy to determine which factor most affects the overall rate of capillary rise.
For this study, several different experiments were performed using sandstone, each designed to evaluate the effects of pore size on the rate of capillary rise. We compare experimental results with theoretical predictions based on the L-W equation, discuss the mechanism of capillary rise, and propose a conceptual model that can account for the experimental results.
Experiment 1: measurement of capillary rise using a dried core
Experiment 2: measurement of capillary rise after controlling for the size distribution of pore water
where r is the radius of pore from which water is expelled (Yokoyama and Takeuchi 2009; Nishiyama et al. 2012; Nishiyama and Yokoyama 2014). Equation 7 shows that the minimum radius of empty pores after water expulsion decreases as ΔP gas increases (Fig. 4b). Expelled water was wiped away with tissue, and pores larger than a given radius emptied. After this water expulsion treatment, the sample was weighed, and the capillary rise experiment was initiated (Fig. 3b). The water expulsion treatment was conducted for four pore radii using the same sample: the values of ΔP gas applied were 146, 485, 1441, and 4000 hPa, corresponding to radii of 10, 3, 1, and 0.36 μm, respectively. For comparison, the capillary rise experiment was also carried out with the fully dried sample. In experiment 2, the position of the wet front could not be seen because the sample was sealed with resin. Therefore, the sample was removed from the apparatus intermittently and weighed to determine the amount of water absorbed. The temperature and relative humidity were 17.6–22.0 °C and 33.1–44.2 %, respectively (humidity was not measured during the experiments for r = 1 and 3 μm).
Experiment 3: measurement of the height profile of water saturation
Results and discussion
Height profile of water saturation
Time variation in capillary rise height in initially dry sandstone
The solution is equal to those of Hamraoui and Nylander (2002) and Fries and Dreyer (2008) if τ = 1. The most dominant pore radius in the rock sample is approximately 10 μm (Fig. 2); therefore, we initially assume r = 10 μm. As for θ, the assumption of cosθ = 1 (θ = 0°) has been used previously to analyze capillary rise in sandstone (Dullien et al. 1977; Hammecker and Jeannette 1994) and granitic rocks (Mosquera et al. 2000). In addition, Heshmati and Piri (2014) reported that if capillary numbers (μv/γ) are <~0.001, θ becomes ~10° (cosθ = 0.98) for the case of capillary rise in a glass tube. The capillary number of our sample was calculated to be <0.001 for x > 1 mm based on the measured value of v. The assumption of cosθ = 1, therefore, seems to be reasonable. However, the contact angle of quartz, the predominant mineral in Berea sandstone, has been reported to range between 0° and 54° (Jaňczuk et al. 1986). Therefore, we also considered the case of cosθ = 0.59 (θ = 54°) as an extreme case. Figure 6b shows the variation of water height with time, calculated by Eq. 11 with cosθ = 1 and 0.59 (γ = 0.0727 N/m at 20 °C) and plotted with the measured data. Calculated water heights were significantly higher than the measured values, both in the case of cosθ = 1 and 0.59.
Pore radius and cosθ (θ = 0°, 24°, 40°, 54°) values at which the measured and calculated values match best
Capillary rise after controlling for the size distribution of pore water
A conceptual model for capillary rise
where r and l are the radius and the length of a component of the unit pore, respectively (Dullien et al. 1977). The summations represent overall components of the unit pore. In Eq. 12, the factor of 1/3 originally included by Dullien et al. (1977) is excluded because the factor corresponds to the effect of tortuosity, which is already taken into account in this study (Eq. 4 or Eq. 11). If r wide = 10 μm, r 0 = 30 μm, and l 0 = l 1 = 200 μm are assumed on the basis of grain size (Fig. 1) and pore size distribution (Fig. 2), r eff is calculated to be 1.4 μm, which means that the rate of capillary rise in the rock is equivalent to that in a pore with a 1.4 μm radius, even though the pore radii of the rock are 10 and 30 μm. Therefore, this effect may partially explain the slow capillary rise indicated by our results. However, this effect is unlikely to be the sole factor; according to Dullien’s model (Dullien et al. 1977), the height of capillary rise is predicted to be similar between experiments started with dry samples and with samples subjected to water expulsion treatment prior to the experiment, as shown in Fig. 8a, b. This prediction is inconsistent with our results (Fig. 7f).
Inertia has also been considered to explain deviations between the measured rate of capillary rise and predictions made using the L-W equation (e.g., Bosanquet 1923; Quéré 1997). Theoretical studies and experiments conducted using a capillary pipe demonstrated that the height of capillary rise dominated by inertial force is proportional to time (Quéré 1997; Siebold et al. 2000), whereas the L-W equation shows that the height of capillary rise governed by viscous force is proportional to the square root of time. Because the height of capillary rise in our sample did not increase linearly with time (Fig. 6), the effect of inertia is likely to be negligible, at least for the time span considered in this study.
A pore radius value significantly smaller than dominant pore radius of the rock had to be used to reproduce the experimental results using the L-W equation, assuming cosθ = 1 and a uniform pore radius.
In the experiment that was initiated with a dry sample, water saturation in the sample was relatively constant below approximately 70 % of the height of the wet front and decreased rapidly with increasing height toward the wet front.
As the radius of the pores available for capillary rise decreased, the rate of capillary rise decreased and approached that of the initially dry sample.
The effect of the nonuniformity of pore radii in a single flow path can explain the result of (1) at least partly but is unlikely to account for the result of (3). These results can be better explained by considering the capillary rise in branched pores with different pore radii.
We thank two anonymous reviewers for their helpful comments.
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