Spontaneous Imbibition of Water and Determination of Effective Contact Angles in the Eagle Ford Shale Formation Using Neutron Imaging

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Journal of Earth Science, Vol. 28, No. 5, p. 874–887, October 2017 ISSN 1674-487X Printed in China https://doi.org/10.1007/s12583-017-0801-1

Victoria H. DiStefano *1, 2, Michael C. Cheshire1, Joanna McFarlane3, Lindsay M. Kolbus4, 5, Richard E. Hale6, Edmund Perfect7, Hassina Z. Bilheux5, Louis J. Santodonato8, Daniel S. Hussey9, David L. Jacobson9, Jacob M. LaManna9, Philip R. Bingham10, Vitaliy Starchenko1, Lawrence M. Anovitz1 1. Physical Sciences Directorate, Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge TN 37830-6110, USA 2. Bredesen Center, University of Tennessee, Knoxville TN 37996-3394, USA 3. Energy & Environmental Sciences Directorate, Energy & Transportation Sciences Division, Oak Ridge National Laboratory, Oak Ridge TN 37830-6181, USA 4. STEM Educator, Skateland, Indianapolis IN 46254, USA 5. Neutron Sciences Directorate, Chemical and Engineering Materials Division, Oak Ridge National Laboratory, Oak Ridge TN 37830-6475, USA 6. Nuclear Science & Engineering Directorate, Reactor & Nuclear Systems Division, Oak Ridge National Laboratory, Oak Ridge TN 37830-6165, USA 7. Department of Earth and Planetary Science, University of Tennessee, Knoxville TN 37996-1410, USA 8. Neutron Sciences Directorate, Instrument and Source Division, Oak Ridge National Laboratory, Oak Ridge TN 37830-6430, USA 9. Physical Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg MD 20899, USA 10. Energy & Environmental Sciences Directorate, Electrical & Electronics Systems Research Division, Oak Ridge National Laboratory, Oak Ridge TN 37830- 6075, USA Victoria H. DiStefano: http://orcid.org/0000-0001-6023-9716

ABSTRACT: Understanding of fundamental processes and prediction of optimal parameters during the horizontal drilling and hydraulic fracturing process results in economically effective improvement of oil and natural gas extraction. Although modern analytical and computational models can capture fracture growth, there is a lack of experimental data on spontaneous imbibition and wettability in oil and gas reservoirs for the validation of further model development. In this work, we used neutron imaging to measure the spontaneous imbibition of water into fractures of Eagle Ford shale with known geometries and fracture orientations. An analytical solution for a set of nonlinear second-order differential equations was applied to the measured imbibition data to determine effective contact angles. The analytical solution fit the measured imbibition data reasonably well and determined effective contact angles that were slightly higher than static contact angles due to effects of in-situ changes in velocity, surface roughness, and heterogeneity of mineral surfaces on the fracture surface. Additionally, small fracture widths may have retarded imbibition and affected model fits, which suggests that average fracture widths are not satisfactory for modeling imbibition in natural systems. KEY WORDS: spontaneous imbibition, effective contact angle, neutron imaging, Eagle Ford shale, rock fractures.

0 INTRODUCTION optimize recovery, models have been developed to simulate The combination of horizontal drilling and hydraulic fracture growth and fluid movement in oil and gas reservoirs fracturing, or fracking, has greatly increased the productivity under subsurface conditions. However, these models must em- of oil and natural gas wells, especially in tight gas shales. To ploy a multitude of assumptions about poorly understood rock properties that are highly dependent on micro-scale fluid-rock *Corresponding author: [email protected] interactions. A quantitative understanding of these interactions, © China University of Geosciences and Springer-Verlag GmbH including spontaneous imbibition and wettability, is a key to de- Germany 2017 veloping better models and improving hydraulic fracturing. For instance, the performance of hydraulic fracturing fluids, both Manuscript received June 15, 2017. water and gas based, can be enhanced by understanding the beha- Manuscript accepted July 30, 2017. vior of the 3-D anisotropic rock-fluid interactions through charac-

DiStefano, V. H., Cheshire, M. C., McFarlane, J., 2017. Spontaneous Imbibition of Water and Determination of Effective Contact Angles in the Eagle Ford Shale Formation Using Neutron Imaging. Journal of Earth Science, 28(5): 874–887. https://doi.org/10.1007/s12583-017-0801-1. http://en.earth-science.net Spontaneous Imbibition of Water and Determination of Effective Contact Angles in the Eagle Ford Shale Formation 875 terization and dynamic studies. with the fluid rather than with air. This preference also influ- This study uses neutron imaging, a non-destructive, rapid- ences many aspects of reservoir performance, particularly in ly developing capability, to verify and modify critical model- enhanced oil recovery techniques such as hydraulic fracturing. ing parameters for fluid flow in subsurface environments. For instance, making the assumption that a reservoir is water- Spontaneous imbibition of water into fractures in the Eagle wet, when it is not, can lead to irreversible reservoir damage Ford Shale Formation, a vitally important shale gas reservoir, and less than optimum recovery (Abdallah et al., 2007). Reser- was imaged to quantitatively measure in-situ imbibition rate. A voir rock formations are complex structures, and the wettabili- model of capillary uptake was then fit to the measured imbibi- ty of each differs. They typically contain multiple mineral tion rate to determine wettability through effective, in-situ types, each of which may wet differently. This makes estima- contact angles. These imbibition rates and contact angles are of tion of their overall wettability difficult. highly relevant to subsurface hydraulic fracturing models. Wettability is important because it is one of the primary variables controlling spontaneous imbibition (i.e., capillary uptake). 0.1 Spontaneous Imbibition In the simplest case, fluid in a narrow, smooth, cylindrical, capil- Hydraulic fracturing involves injecting high-pressure fluids lary with diameter, D, the Washburn-Lucas Equation, provides a into shale reservoirs to create fracture networks, liberating oil measure of the height of the wetting front due to capillary forces and gas reserves. Some of the injected fluid is never recovered. as a function of time (Washburn, 1921; Lucas, 1918) This missing fluid is termed leak-off. If not controlled properly,  σDcosθ  leak-off can exceed 70% of the injected volume, potentially h 2 =  t (1)  4η  decreasing well productivity by blocking oil and/or gas egress,   causing formation damage, and/or contaminating ground water where h is the height of the capillary, t is time, σ is surface ten- (Cheng, 2012; Penny et al., 1984). This loss thus presents a po- sion, and η is viscosity. Wettability is measured by the contact tential major barrier to oil and gas recovery. The processes by angle, θ, discussed in detail below. While this equation is insuf- which this loss occurs are, however, poorly understood. ficient to describe capillary uptake in real-world, or even two- One possible mechanism for the escape of fluid into a re- dimensional planar fractures in real rocks (see below), the impor- servoir is spontaneous imbibition into initially dry porous me- tance of wetting angle on the process of capillary uptake is clear. dia and fractures (Cheng, 2012). Spontaneous imbibition oc- Measuring the static contact angle of a liquid on a surface curs when a wetting fluid displaces a nonwetting fluid, such as is the most common method to measure the wettability of re- air, under the influence of capillary suction (Gao and Hu, servoir rocks. The liquid is placed on a uniform, flat, rock sur- 2016). This has been shown to strongly affect the production of face, and the angle between the tangent to the edge of the drop oil and gas by water blockage of oil and gas escape pathways and the solid substrate, the static contact angle relative to air, is (Cheng, 2012; Pordel Shahri et al., 2012; Li, 2007). The rate of measured (Fig. 1). Different liquids can exhibit different con- imbibition, however, is strongly dependent on the multiscale tact angles on the same surface and a single liquid can exhibit properties of the rock matrix. It depends on mineralogy of the different contact angles on different materials, or may change source rock, total organic carbon, distribution of pore throat as a function of other surface properties such as roughness, sizes and fractures, and wettability. Experimental analysis of ionic strength, and mineralogy (Chen et al., 2015; Hamraoui et spontaneous imbibition into porous media has been done on a al., 2000; Wenzel, 1936). Additionally, in systems with fluid number of rocks including shales, but these experiments usual- flow, it has been demonstrated that dissipation of frictional and ly determine spontaneous imbibition into porous media, not viscous forces with an advancing fluid front results in contact fractures such as those examined here (Javaheri et al., 2017). angles that change with time (Hamraoui and Nylander, 2002; While a number of models have been developed in the litera- Hamraoui et al., 2000; Joos et al., 1990). Such time-dependent ture to predict the rate of imbibition (Cheng et al., 2015; Cai et values are referred to as dynamic contact angles, and these can al., 2014, 2010a; Standnes, 2010; Xiao et al., 2006; Benavente differ significantly from measured static contact angles. As et al., 2002; Dreyer et al., 1994; Handy, 1960; Brittin, 1946), fluids in real reservoirs are likely to both be in long-term con- experimental data are needed to verify these models. tact with reservoir materials and, in many cases, flowing, these dynamic contact angles can alter effective contact angles, or 0.2 Wettability: Static, Effective, and Dynamic Contact the contact angle measured in in-situ conditions. Determination Angles of effective contact angles is critical to modeling fluid flow in The wettability of an oil and gas reservoir rock controls subsurface oil and gas reservoirs. Despite its importance, how- imbibition and must be considered to optimize oil and gas re- ever, measurement of effective contact angles on real surfaces covery. Wettability describes the preference of a solid to be in in reservoir-like conditions, rather than flat surfaces under contact with one fluid rather than another. With multiple phas- laboratory conditions, remains difficult. Actual reservoir rocks es present in the reservoir, understanding wettability becomes exhibit micro-heterogeneities in orientation, surface roughness, very important (Abdallah et al., 2007). In a system with two and mineralogy that are not always present in carefully- fluids, air and water, rocks can be classified as water-wet, air- prepared laboratory samples. Thus, characterization of the wet, or intermediate in nature (Fig. 1), depending on the mine- wettability as it applies to reservoir processes requires more ralogy, and this greatly affects the movement of fluid through complex, in-situ measurements. the rock formation. Where air is present, wettability describes Numerous studies have measured static contact angles on the extent of preference for a given surface to be in contact a host of minerals and realistic rock materials, most recently in

876 Victoria H. DiStefano, and et al.

Figure 1. Static contact angle (θ) of a water drop for water-wet, intermediate, and air-wet surfaces.

carbon dioxide (CO2)-brine systems for carbon sequestration ture systems will be investigated in future works. and storage reservoirs (Tokunaga and Wan, 2013). Wan et al.

(2014) measured CO2-brine contact angles on muscovite, a 1 METHODS AND MATERIALS common aluminosilicate mineral, noting the reproducibility of 1.1 Sample Preparation contact angle measurements on these surfaces are difficult Shale samples were obtained from an outcrop of the because clean and pristine mineral surfaces do not exist in-situ. Eagle Ford shale Formation (purchased from Kocurek Indus-

Yang et al. (2008) and Broseta et al. (2012) measured CO2- tries). The Eagle Ford Shale Formation is one of the most brine contact angles on carbonate rocks. However, very few actively drilled plays for oil and gas recovery in the United studies have investigated effective contact angles for reservoir States. As of April 2017, the Eagle Ford shale Play was pro- rocks in-situ. Andrew et al. (2014) measured effective contact ducing 1 177 312 bbl/day of oil and 5 852 211 Mcf/day of angles in limestone-carbon dioxide-brine systems using X-ray gas (U.S. Energy Information Administration (EIA), April micro-computed tomography (CT) to measure the effective 2017). The Eagle Ford was deposited in the Late Cretaceous contact angle of carbon dioxide as a residual phase on the li- in a marine continental shelf environment and is rich in hy- mestone surface. While this was a novel approach to in-situ drocarbons (Ergene, 2014). It underlays much of Southeast contact angle measurements, no fluid flow occurred. Texas into Mexico and outcrops in an arc from north of Aus- tin, through San Antonio and then west toward Kinney Coun- 0.3 Experimental Design ty (Anovitz et al., 2015). In 2011, the USGS estimated that To measure wettability of real rock materials under dynamic the Eagle Ford contained 853 million barrels of oil, 51 conditions, we have used the neutron imaging facilities at the 926 billion cubic feet of natural gas, and 2 043 million bar- National Institute of Standards and Technology (NIST) and the rels of natural gas liquids (Dubiel et al., 2012). While this Oak Ridge National Laboratory (ORNL) to measure fracture formation contains extensive oil and gas reserves, it was not imbibition rates of water, in samples from the Eagle Ford Shale considered economic for recovery until the recent coupling of Formation, Texas. Imbibition was monitored by neutron radio- hydraulic fracturing and horizontal drilling. graphy, which has been shown to be a highly-accurate method to Two samples from the Eagle Ford Formation were pre- quantitatively determine the rate of spontaneous imbibition of pared for analysis from paired shale blocks; each block was hydrogen rich fluids, such as water, into fractured media in real 12.7 mm×12.7 mm×152.4 mm. Prior to assembly, the fracture time because of the large neutron cross-section of hydrogen surface on each block was polished with a 180-grit lapping (Cheng et al., 2015; Perfect et al., 2014; Kang et al., 2013; plate until almost no light passed through the fracture when the Hassanein et al., 2006; Middleton et al., 2005). This means that blocks were held together. The blocks were then clamped to- water readily attenuates neutrons through incoherent scattering, gether and the seam taped with Kapton® tape (DuPont, Wil- allowing dynamic imaging of water movement. Attenuation can mington, DE) to create a nearly planar synthetic fracture with be modeled using the Lambert-Beer Law (Swinehart, 1962) an opening of about 50 μm. Kapton® tape is ideal to fasten the shale blocks together because the tape is made of a material I − σ T = e N cts (2) with a low neutron cross section and is only 25 µm thick, thus I 0 it minimally attenuates neutrons. This makes it almost completely invisible in neutron images. In the first sample, Eagle where I is the measured intensity and I0 the incident intensity, Ford 19 (EF 19) the synthetic fracture was aligned perpendicu- T is the transmission, N is the atom density, σc is the total neu- lar to bedding, while the fracture in the second, Eagle Ford 20 tron cross section, and ts is the thickness of the sample. The overall goal of this study was to establish a funda- (EF 20), was aligned parallel to bedding. The two samples, mental understanding of imbibition and effective contact an- with synthetic fractures, are pictured in Fig. 2. Before imbibi- gles in gas shales; information that can be used to develop tion, samples were allowed to equilibrate with ambient condi- robust poroelastic models of rock behavior that can be em- tions. The mineral composition of this shale formation was ployed for prediction and enhancement of hydrocarbon recov- measured with X-ray diffraction (XRD) and quantified via ery through development of more efficient fracturing metho- Rietveld refinement (Rietveld, 1969) (Table 1). dologies. Since the complex fracture geometries of natural systems complicates imbibition of fluids into a system, syn- 1.2 Static Contact Angle Measurement thetic fractures of known geometries were used. Complex frac- Static contact angles were measured on both sides of the

Spontaneous Imbibition of Water and Determination of Effective Contact Angles in the Eagle Ford Shale Formation 877 fracture surfaces of samples EF 19 and EF 20. These mea- contact angles. No effort was made to evaluate whether the surements were made after the imbibition experiments, and the measured contact angles were time-dependent. fracture surfaces were assumed not to be modified from the condition under which imbibition measurements had been 1.3 Non-Destructive Fracture Characterization conducted. Sufficient time had elapsed for the samples to dry Characterization of the fracture width for EF 19 was done out and to equilibrate with ambient air. In order to perform the using X-ray CT data produced at the High-Resolution X-ray measurements, the samples were first mounted on a lab bench Computed Tomography Facility of the University of Texas at and leveled. A series of 10 μL droplets of deionized water (DI) Austin. The sample was imaged with the fracture plane per- were pipetted along the length of each sample and photo- pendicular to the CT slice plane at 140 keV in a North Star graphed with the camera centered on the top surface of the Imaging, Inc. X-ray scanner. A complete scan was obtained by rock surface as shown in Fig. 3. The resultant photographs taking 3 600 projections from -10º to 450º. The projections were magnified, and the contact angles were measured using were then reconstructed to provide a stack of 2D images, each ImageJ (Schneider et al., 2012). The pipetted droplets appeared image representing a slice through the sample. The resultant to be stable over the few minutes necessary for each measure- image had a voxel edge length of 9.49 μm. ment, allowing images of about five droplets to be captured. Characterization of fracture width for EF 20 was done us- No significant differences were observed between droplets ing X-ray CT scans done at the Oak Ridge National Laboratory. place on either side of the engineered fracture or systematically The sample was imaged at 140 keV in a Zeiss Xradia 520 Ver- along the length of the fracture. However, visible surface fea- sa X-ray system. A high-energy filter was used to prevent tures translated to a noticeable variability in the measured beam hardening. The sample was imaged with the fracture plane perpendicular to the CT slice plane. Three scans of the Table 1 Mineral composition of the Eagle Ford Formation fracture were obtained at the bottom, middle, and top, each about 9 mm tall and approximately 5 cm apart, using 3 200, Quartz Calcite Smectite Kaolinite Pyrite 1 600, and 1 600 projections, respectively, from 0º to 360º. (%) (%) (%) (%) (%) Final images had a voxel edge length of 5.35 μm. Eagle Ford 22 63 14 1 <1* Fracture roughness was characterized using a Keyence *Detectable but difficult to quantify. VR-3100 non-contact surface profilometer. This instrument provides 3D measurements with 0.1 μm vertical resolution using three, double-telecentric lenses and multi-triangulation to provide a 3D scan of a sample surface. Using this instrument, the surface roughness was measured in accordance with ISO 25178 (International Organization for Standardization, 1997). For both the EF 19 and 20, fracture roughness was calculated from six different areas along both sides of the fracture surface. For EF 19, each area was approximately 43.09 mm2 and, for EF 20, each was approximately 42.91 mm2. Thus, approximately 6.7% of the total fracture surface of each sample was analyzed. Based on visual observation, these areas were representative of the fracture surface. Two different surface roughness parameters where determined, the arithmetic mean height (Sa) and the root mean squared height (Sq). Sa is the average area Figure 2. Eagle Ford shale samples with the fracture oriented perpendicular above and below the mean plane while Sq is the average mean to bedding, EF 19 (a) and parallel to bedding, EF 20 (b). Left-hand figure square of the 3D area above and below the mean plane. Sq is shows the samples upright and right-hand figure shows the samples lying often higher than Sa. Several related parameters were calculated, down so that the bottom of the fracture, where contact occurred, is visible. including the maximum peak height (Sp), and the maximum Samples are 25.4 mm×12.7 mm×152.4 mm. valley height (Sv), and the maximum height (Sz=Sp+Sv)

(International Organization for Standardization, 1997).

1.4 Spontaneous Imbibition Measured with Neutron Imaging Spontaneous imbibition was measured for both samples at the BT-2 neutron imaging facility at the National Institute of Standards and Technology (NIST) Center for Neutron Re- search (NCNR) and the CG-1D neutron imaging facility at the High Flux Isotope Reactor (HFIR) at the Oak Ridge National Laboratory (ORNL). As was shown by Cheng et al. (2015), Figure 3. DI water droplets stable on fracture surface of EF 19 used in neutron imaging is an effective method for imaging water static contact angle measurements. The thickness of the sample is 12.5 mm. movement in empty fractures as neutrons are strongly atte- Kapton® tape is visible in on the outside of the sample, which was used to nuated by hydrogen. The two fractured samples were oriented hold the sample together for the imbibition measurements. in the neutron beam with the fracture plane parallel to the path

878 Victoria H. DiStefano, and et al.

of the incoming neutrons. This provided a flattened 2D image where IS is the measured intensity of the sample, IDF is the of the 3D phenomena of fracture imbibition. To take the mea- intensity of a dark field image, the image obtained with the surements, the image acquisition was first initiated and an alu- shutter closed to measure background radiation effects, and IR minum pan of water was then slowly raised using a remotely- is the intensity of a reference image. This latter was an image controlled vertical stage until the water barely touched the of the rock/fracture system taken before imbibition, and it al- bottom of the fracture and water spontaneously imbibed into lows any contributions of the rock to the overall image to be the sample. To prevent inducing hydrostatic pressure, the water removed. The resulting stack of images constitutes a time re- level was monitored using both light and neutron cameras as it solved sequence of water imbibition into the fracture, with a slowly approached the sample. As soon as contact was made, frame rate of 10 images per second. The imbibition is visually the elevation of the aluminum pan was stopped. Due to the distinguishable as a dark front gradually progressing upward stark contrast between empty fracture and water, the spontane- (Fig. 4). Approximately 2 000 frames were taken during each ous imbibition of water was easily visualized in the neutron experiment, with a run time of 3 to 4 min. images (Fig. 4). Images were taken every 0.1 s so that the uptake rate could be quantified. The resultant images had a pixel 2 RESULTS edge length of 55 μm. EF 19 was then soaked in DI water, 2.1 Static Contact Angles dried overnight at 105 °C, and the water uptake was repeated. For Eagle Ford 19, seventy-one contact angle measure- For each sample, all the images in the time sequence were ments were taken, which varied between 21º and 44º with an normalized according to Eq. (3) to form the transmission image, average of 35º±5º. For Eagle Ford 20, sixty-four contact angle

Ti. The transmission image is the image were each pixel cor- measurements were taken, with a range of 19º to 43º and an responds to the transmission, T, in Eq. (2). average of 31º±6º. The similar range and average static contact angles of the two samples is expected since they have the same I − I T = S DF (3) composition. These static contact angles indicate a water-wet i I − I R DF surface.

Figure 4. Time series of neutron images that show water imbibition into EF 19 with fitted plots. The black line on the fitted plots shows the three-point running average of the data, which was only used for visualization, not for fitting the data. Plots also show parameters for fitting.

Spontaneous Imbibition of Water and Determination of Effective Contact Angles in the Eagle Ford Shale Formation 879

2.2 Fracture Description The measured surface roughness parameters are listed in The width of the fractures in EF 19 and EF 20 were meas- Table 2. The average Sq for EF 19 was 5.76 μm, while that for ured using the reconstructed X-ray CT images of the sample. EF 20 was much higher at 18.3 μm. The maximum height val- Each 2D image is a horizontal cross section of the fracture as ue obtained for EF 20 is also about twice as large as that for EF can be seen in Fig. 5. For EF 19, eleven equidistant slices, 19. Thus, the surface of EF 20 is much rougher than EF 19. evenly spaced from the bottom to the top of the sample, were analyzed. The pixel width of the fracture in each image was 2.3 Quantitatively Determining the Height of the Wetting measured in the front, the center, and the back of the fracture Front from Neutron Images and converted to micrometers (Fig. 5). The average width was To quantitatively calculate the height of the wetting front 33±8 μm, ranging from 19 to 48 μm. The median width was as a function of time for each sample, a straight line was first calculated as 29 μm. Figure 6 shows a histogram of the deter- drawn along the imbibition path in the neutron images from the mined fracture widths for EF 19. The width of the fracture in EF 20 was calculated from Table 2 Surface roughness for EF 19 & 20 the three X-ray CT scans of the bottom, middle, and top of the Sa (µm) Sq (µm) Sz (µm) Sp (µm) Sv (µm) sample. Figure 5 shows the area along the sample where the scans were taken. Due to the increased resolution of the CT EF 19 than for EF 19, many more width measurements could be made. Area 1 3.6 4.8 58.9 14.1 44.9 The width of the fracture was measured in all of the recon- Area 2 5.4 6.6 64.2 37.9 26.3 structed 2D images (about 1 700 images for each scan) at the Area 3 7.4 8.9 60.3 27.7 32.5 front, center, and back of the fracture in each scan (Fig. 5). The Area 4 3.2 4.1 37.1 14.6 22.5 bottom of the sample had the smallest width, with an average Area 5 4.4 5.6 56.4 29.6 26.8 of 41±24 μm, a minimum below the resolution of the CT (be- Area 6 3.5 4.6 54.5 16.7 37.9 low 5.35 μm), and a maximum of 123 μm. At the middle and Average 4.6 5.8 55.2 23.4 31.8 top of the sample the fracture was wider, with average widths EF 20 of 75±25 and 62±19 μm, respectively. Figure 7 show a histo- Area 1 15.2 18.4 113 35.9 77.1 gram of the determined widths for each CT scan, the bottom, Area 2 13.7 16.0 99.2 36.9 62.3 top, and the middle. The average width from all the measured scans was 59±27 μm, ranging from less than 5.35 to 145 μm, Area 3 14.6 17.4 98.8 35.6 63.2 and the median width was the same as the average width Area 4 14.2 17.0 101 42.0 58.8 (59 μm). Figure 8 shows a vertical cross section of the front, Area 5 17.8 22.3 127 53.0 74.2 center, and back of the top of the sample, which is only a small Area 6 15.8 18.9 124 40.7 82.8 sub-volume of the sample. Average 15.2 18.3 110 40.7 69.7

Figure 5. Location of CT scans in EF 20 (left), with example of reconstructed cross-sections and locations of fracture width measurements in EF 20 and EF 19 (right).

880 Victoria H. DiStefano, and et al. well of water at the bottom of the sample to the highest point water, and values of 1 indicated complete transmission of the of uptake. For each frame, corresponding to a time point, a plot beam—the absence of water. As can be seen in Fig. 4, the was then made of the normalized intensity of each pixel along height of the imbibition front was not a sharp boundary, but the imbibition line. For the transmission images, values of 0 more closely approximated an error function, and the T values denoted complete attenuation of the beam, representative of along the uptake line at each time were, therefore, fitted as

m − n  x − c  m + n T = erf   + (4) 2  w  2

where m, n, c, and w correspond to the shape of the error function, with m being the maximum, n the minimum, c the center, and w the width. The value for x is the pixel distance along the imbibition lineand T corresponds to the transmission at pixel x. Figure 4 shows how these parameters were fitted to the transmission graphs of EF 19 and how they affect the shape of the error function. The parameters in Eq. (4) are used to determine the height Figure 6. Histogram of the fracture widths determined from EF 19. The of the wetting front. The minimum is the T value corresponding distribution is Gaussian. to the part of the fracture filled with water, which typically has a

Eagle Ford 20 histogram of fracture widths 3 000

2 500

2 000

1 500

Counts 1 000

500

0-7 7-14 14-21 21-28 28-35 35-42 42-49 56-63 63-70 77-84 84-91 91-98 49-56 70-77 98-105 105-112112-119119-126126-133133-140140-147 Fracture width (μm)

Bottom Middle Top Total

Figure 7. Histogram of the fracture widths determined from EF 20 for the bottom, middle, and top of the sample.

Figure 8. Vertical cross sections of the top of the fracture in EF 20. Images are segmented to just show the fracture.

Spontaneous Imbibition of Water and Determination of Effective Contact Angles in the Eagle Ford Shale Formation 881 value around 0.7 to 0.9. This value would be 0 if all neutrons visually observed height of the wetting front. Therefore, this were attenuated, however, the small volume of water in the was taken to be the height of the wetting front for each frame fracture prevents complete attenuation of the neutron beam. for the purposes of further analyzing our results. The maximum corresponds to the T value where the fracture is Because of the large number of image frames that needed completely empty of water, and is always approximately 1. to be analyzed, an automated program was written to fit the Additionally, as the error function is the integral of the Gaus- error function to each frame using a least squares fit (Fig. 4). sian distribution, the parameters of the error function corres- Figure 9 shows the height of the wetting front as a function of pond to those of the Gaussian. The center, c, corresponds to the time for EF 19 and EF 20. The visible outliers are not signifi- mean, and the width, w, corresponds to 2σ , where σ is the cant, but are where the least squares algorithm failed to fit the standard deviation. The center is the point along the uptake data. However, after about 150 s in EF 19 (frame 1 500) and path when the fracture is about 50 percent full. In general, we 140 s in EF 20 (frame 1 400), the fracture is mostly filled and found that the center plus one standard deviation, the point the error function approximation no longer accurately fits the when the facture is about 32 percent full, corresponded to the data. These data were, therefore, excluded from Fig. 9.

Figure 9. The height of the wetting front as a function of time for EF 19, uptake 1 and uptake 2, and EF 20. The visible outliers are not significant, but are where the least squares algorithm failed to fit the data.

2.4 Determining the Effective Contact Angle through ity, φ is the porosity, A is the cross-sectional area, Sw is the

Modeling the Wetting Front fractional water content, and ηw is the viscosity of water. The To estimate the effective contact angle from the measured expression is often abbreviated to the Handy Equation as uptakes rates, a model of capillary uptake appropriate to the (Handy, 1960) geometry of the experiment is needed. In this system, fluids H = at 0.5 (6) can be imbibed not only into the main fracture, but into micro- fractures as well as the porous media. The classic Washburn- where H is the height of the fluid front (Qw/A) and a is the con- Lucas Equation (Eq. (1)) models the rise of water in a single stant commonly referred to as sorptivity, which is defined as straight capillary tube (Washburn, 1921; Lucas, 1918). Numer- 2P k ϕS ous modifications of this model have been proposed to better a = c w w (7) η model porous media and fracture imbibition (Standnes, 2010; w Benavente et al., 2002). Additionally, Handy (1960) proposed the following equation to describe the imbibition of a fluid into These expressions assume that capillary forces are much porous media. This suggests that the volume of water imbibed, greater than the gravitational force, that there is no pressure gradient ahead of the rising fluid front, and that imbibition Qw, can be expressed as the time (t) dependent function occurs in a piston-like manner. This model, and the Washburn- 2 2  2P k ϕA S  Lucas model, indicates that the height of the fluid column Q =  c w w t (5) w  η   w  should increase as a linear function of the square root of time. That is, that capillary uptake is, essentially, a diffusive process. where Pc is the capillary pressure, kw is the effective permeabil- This has been demonstrated in experimental studies measuring

882 Victoria H. DiStefano, and et al. the imbibition of fluids into a porous media matrix, but only at These values are calculated from parameters in the experiment, early times where gravity is negligible (Cheng et al., 2015; Cai et including fracture characteristics, width and depth, and fluid al., 2014). As shown in Fig. 9, however, the uptake rate exhi- characteristics, density, surface tension, and viscosity (Table 3). bited by our samples is not proportional to the square root of the Equations and values for h0 and t0 are found in Tables 4 and 5. imbibition time because gravity, which is not accounted for in The analytical solution is given by Xiao et al. (2006) as the Washburn-Lucas, Handy, or similar models, is not taken into n1 account. Additional models, usually for spontaneous imbibition * ≈ + * + * h a0,0 a0,1 exp(r2t ) ak ,0 exp(r1t ) (9) into porous media, take into account the fractal nature of pores k=1 which leads to a relationship where imbibition is proportional to This analytical solution requires several additional defini- a time exponent of 0.5 multiplied by the fractal dimension for tions to interpret. Firstly, n1 is the number of iterations. As itera- tortuosity (Cai and Yu, 2011; Cai et al., 2010b). However, the tions increase, the calculated function converges. For this system, models discussed above are for uptake in porous media or nar- n1 was kept at 31, higher than the n1 values reported in Xiao et al. row, cylindrical capillaries, not planar fractures with rough sur- (2006). Additionally, am,n is solved for according to faces. In order to take these variables into account, therefore, we = have adopted the model of Xiao et al. (2006), who developed a am,n m n generalized theoretical model and analytical solution for capil- {}+ 2 − + + + + ak ,l am−k ,n−1 (kr1 lr2 ) (1 c2 ) (kr1 lr2 )[c2 (mr1 nr2 ) c3 ] lary flow between parallel plates, based on a set of nonlinear k==00l + + 2 + + + + second-order differential equations (Dreyer et al., 1994; Brittin, (a0,0 c1 )(mr1 nr2 ) (c3a0,0 c4 )(mr1 nr2 ) c5 1946). This analytical solution was applied and then fitted to the (10) imbibition data measured here. The analytical solution is given as normalized height, h*, where (k,l)≠(m,n) and (k,l)≠(0,0), (0,1), (1,0). c1 through c6 as as a function of normalized time, t*. They are normalized ac- well as the Bond number (Bo) and Ohnesorge number (Oh) are cording to parameters used often in the analytical solution. The calcula- tion for each of these parameters and the values for these expe- h t h* = , t * = (8) riments are listed in Tables 4 and 5. Additionally, c6 is depen- h0 t0 dent on the effective contact angle (θe) and is solved for after fitting the data. a0,0, a1,0, and a0,1 must be solved for using where h0 is characteristic height and t0 is characteristic time.

Table 3 Parameters of the analytical solution

Sample Fracture width, 2B Fracture depth, 2w Aspect ratio, γ Fluid Density, ρ Surface tension, σ Viscosity, η (μm) (mm) (B/W) (kg/m3) (N/m) (Pa·s) Eagle Ford 19 19–48 12.7 1.5 to 3.7E-3 Water 998.2 0.072 0.001

Eagle Ford 20 0–145 12.7 0.42 to 11E-3 Water 998.2 0.072 0.001

Table 4 Calculated parameters for Eagle Ford 19-uptake 1 & 2

Width c1 c2 c3 c4 Bo Oh c5 Characteristic time, Characteristic height,

(μm) t0 (s) h0 (μm)

0.555 / r 0.958 1 0.295 r [ρg(2B)2]/σ η / 2Bρσ Bo/(144Oh2)12η 2B 33 Ave. 10.8 0.958 1 0.015 1 1.51E-4 0.020 4 2.52E-3 9.27E-5 33.4 29 Median 11.7 0.958 1 0.014 0 1.10E-4 0.022 1 1.57E-3 6.76E-5 28.5 48 Max 9.08 0.958 1 0.018 0 3.07E-4 0.017 1 7.27E-3 1.88E-4 47.5 19 Min 14.4 0.958 1 0.011 4 4.90E-5 0.027 1 4.65E-4 3.00E-5 19

*All parameters without units are dimensionless.

Table 5 Calculated parameters for Eagle Ford 20

Width c1 c2 c3 c4 Bo Oh c5 Characteristic Characteristic

(μm) time, t0 (s) height, h0 (μm) 0.555 / r 0.958 1 0.295 r [ρg(2B)2]/σ η / 2Bρσ Bo/(144Oh2)12η 2B 59 Ave./median 8.15 0.958 1 0.020 1 4.71E-4 0.015 4 1.39E-2 2.89E-4 58.9 145 Max 5.20 0.958 1 0.031 5 2.84E-3 0.009 81 2.05E-1 1.74E-3 144.5 >5.35 Min 27.0 0.958 1 0.006 05 3.89E-6 0.051 0 1.04E-5 2.38E-6 5.35

*All parameters without units are dimensionless.

Spontaneous Imbibition of Water and Determination of Effective Contact Angles in the Eagle Ford Shale Formation 883

Eqs. (11)–(13). widths (Tables 4 and 5), the range of which provides an estimate of the uncertainties in the calculated values. The RMSE a = −c / c (11) 0,0 6 5 of each fit is reported in Table 6, however, these values are not comparable between widths since all height values are in h*, a1,0 is solved for using Eq. (12). By recursively applying Eq. reported in Tables 4 and 5. To compare the RMSE values be- (10) for ak,0, Eq. (12) becomes a polynomial with variable, a1,0. tween widths, the RMSE was multiplied by the fracture width The real root with the smallest magnitude is taken to be a1,0. to negate the effects of the normalization (giving units of dis- n1 + − = tance, and reported as RMSE_c in Table 6). a0,0r2 ak ,0 (r2 kr1 ) 0 (12) k=1 3 DISCUSSION a is solved for using Eq. (13). 1,0 3.1 Eagle Ford 19

n1 + + = For EF 19, with the fracture perpendicular to bedding, the a0,0 a0,1 ak ,0 0 (13) k=1 initial spontaneous imbibition experiment, uptake 1, yielded effective contact angles that ranged from 72.3º to 80.9º. The Finally, r1 and r2 are the roots of the quadratic Eq. (14) and analytical solution fit the data best at a median width of 29 μm |r1|<|r2|. (RMSE_c=4.86 mm) and an effective contact angle of 77.7º (Fig. + 2 + + + = 10). This contact angle is much higher than the static equilibrium (a0,0 a0,1 )r (c3a0,0 c4 )r c5 0 (14) contact angle measured on the same sample, 35º (±5º), indicating This analytical solution was applied to the data and the best fit a water-wet system that is more intermediate in nature, i.e., a of the data was used to determine effective contact angles. decrease in the wettability of water compared to air (Fig. 1). This The best fit of the model toeach data set was determined difference is most likely due to changes in the velocity of the by minimizing the root mean square error (RMSE), or the flow, surface roughness, and possibly changes in the mineralogy square root of the mean of the squares of the residuals, between or surface chemistry of the fracture, as discussed previously. the model prediction and the data, Eq. (15) Overall, the analytical solution fit the data reasonably well, with RMSE_c ranging from 4.86 to 6.27 mm. There is, however, a n (hˆ − h )2 region (h*≈1 750 to 2 500, Fig. 10) where the model over- i=1 i i (15) n predicts the height of the wetting front. That is, the model predicts a faster uptake rate than was observed. There may be sever- ˆ where hi is the height predicted by the model at time ti, hi is al reasons for this overprediction. Hamraoui et al. (2000) noted a the experimental height at the same time, ti, and n is the total similar region where flow was retarded and they attributed this number of data points. RMSE is a good measure of the accura- difference to a dynamic contact angle, which changed as a func- cy of a model in predicting a response, which is the goal for tion of time. They added a correction term to their model, a time- this analytical solution. Due to the range of fracture widths dependent contact angle which becomes smaller as the liquid measured on each sample, several different contact angles rises higher into the capillary. This, therefore, affected the early could be obtained, we, therefore, determined values appropri- time data more than the later, although, it is, at best, totally ate to the average, median, maximum, and minimum fracture

Table 6 Best fit effective contact angles for EF 19 & EF 20

Width (μm) Effective contact angle, θe (º) c6 RMSE RMSE_c (mm) 2 EF 19-uptake 1 (γ–cosθe)/72Oh 33 76.2 -7.52 159 5.32 29 77.7 -6.00 170 4.86 48 72.3 -14.2 124 5.90 19 80.9 -2.97 330 6.27 EF 19-uptake 2 33 60.6 -13.9 594 19.8 29 No fit - - - 48 65 -12.0 222 10.6 19 No fit - - - EF 20 59 70.7 -19.2 167 9.84 145 No fit - - - >5.35 No fit - - -

*c6 and RMSE are dimensionless. RMSE not comparable.

884 Victoria H. DiStefano, and et al.

Eagle Ford 19: fracture width=29μ m 3 500 diffusion into the matrix from slowing down the wetting front. 3 000 This implies that spontaneous imbibition can occur more quickly 2 500 in fractures or other porous materials that have already been 2 000 hydrated, whether by previous hydraulic fracture operations of

* h 1 500 other processes. However, replicated experiments are needed to 1 000 confirm this finding. Eagle Ford 19 (perpendicular) experimental data The analytical solution shown above was also applied to 500 θe=77.7º 0 the second uptake. The fit of this data set was poorer than for 0 24681012 14 uptake 1, failing to fit the data for the minimum and median t* (×105 ) fracture widths (Table 6). The best fit to the data was at the Figure 10. The best fit of the analytical solution to the EF 19 data for up- maximum fracture width of 48 μm. This yielded an effective take 1 at the median fracture width of 29 μm. The effective contact angle contact angle of 65º (Fig. 11), which is slightly closer to the

(θe) for this fit is 77.7º. static contact angle (also determined after previous wetting) than was obtained for the same sample in uptake 1. This val- empirical. In their study, they analyzed two separate uptake ue still indicates a surface with intermediate water-wet prop- experiments. In the first experiment, they calculated dynamic erties, but suggests that the surface is slightly more wetting contact angles that started at 82º then over time decreased to than the original. While this could simply reflect the in- the equilibrium contact angle of 0º. The second experiment creased velocity of the imbibition, the effective contact angle yielded similar results with the contact angle starting at a max- obtained is slightly closer to the measured static value. The imum of 33º then gradually decreasing to 0º. Hamraoui and reasons for the failure of the model at smaller fracture widths Nylander (2002) specifically identified frictional effects asso- is unknown. ciated with the moving liquid as responsible for these time- dependent changes. 3.2 Eagle Ford 20 The idealized geometry of the model system may also ac- As described above, a third uptake experiment was per- count for some of the failures of the model to perfectly fit the formed on EF 20, which consists of the same material as EF 19, data. The analytical solution models flow between two parallel but cut parallel to bedding. The best fit for the effective contact plates. It does not account for diffusion of fluid into the primary angle for this experiment, 70.7º, was obtained at the average porosity of the rock matrix. As water enters the matrix, it is re- and median fracture width of 59 μm (RMSE_c=9.84 mm) (Fig. moved from the vertical flow path, thus slowing the rise of the 12). This is, again, significantly larger than the static contact wetting front and causing the real uptake front to deviate from angle measured on the sample (19º–43º). However, during the the model prediction. Such diffusion would be expected to be initial time steps (h*≈0 to 1 100) the model fit the data very fastest during initial contact between the water and the matrix, poorly, predicting a much quicker uptake than was actually slowing as a function of the square root of time. However, the observed. There are several possible explanations for this result. same would be true at all distances along the fracture, and thus at It is possible that the observed differences could be due to all times, which is not what is observed here. While diffusion dynamic contact angles or diffusion into the matrix, described and dynamic changes in contact angle may affect the fit of the above. However, as the fracture in EF 20 was oriented parallel model, variations in facture width, discussed below, could also to bedding diffusion in this sample would be expected to be play a role. These phenomena are a subject for future study. lower than in EF 19. Alternatively, for EF 20 the problem A second imbibition experiment, uptake 2, was performed could be due to variations in fracture width not observed as after the initial imbibition experiment to determine if the results strongly in EF 19. As can be seen in Fig. 7, there is a bimodal could be replicated or if the initial hydration affected later uptake distribution of fracture widths in EF 20. In EF 19, by compari- rates. As can be seen in Fig. 9, the rate of uptake in experiment 2 son, this distribution is Gaussian (Fig. 6). This bimodal shape was much faster than that in uptake 1, filling the fracture in ap- largely reflects a dichotomy between widths from the bottom proximately one seventh the time. This suggests the first uptake caused significant alterations to the fracture surface, changing its Eagle Ford 19 uptake 2: fracture width=48μ m 2 500 wetting properties (Hamraoui et al., 2000). While the actual origin of this effect is unknown, several possibilities can be sug- 2 000 gested. This could be caused by water dissolving minerals during 1 500

* the initial imbibition and changing the surface roughness, forma- h tion of an alteration layer on the minerals themselves, or changes 1 000 in the hydration state of the electrical double layer (Mamontov et 500 Eagle Ford 19 (perpendicular) experimental data uptake 2 θe=65º al., 2009, 2008, 2007; Fischer and Gaupp, 2005; Murphy et al., 0 1989). Additionally, hydration of clays along the virgin fracture 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.4 4.5 5.0 5.5 t 4 and diffusion into the matrix could have increased the uptake * (×10 ) rate during the second imbibition since drying the sample at Figure 11. The best fit of the analytical solution to the EF 19 data for up- 105 ºC would not have removed all clay-bound water, and per- take 2 at the maximum fracture width of 48 μm. The effective contact angle haps not all of the water in the matrix pores. The increased mois- (θe) for this fit is 65º. ture in the system could have prevented water adsorption or

Spontaneous Imbibition of Water and Determination of Effective Contact Angles in the Eagle Ford Shale Formation 885

Eagle Ford 20: fracture width=59μ m where H is the equilibrium height, σ is surface tension, θ is ef- 1 600 e 1 400 fective contact angle, D is fracture diameter, ρ is density, and g is 1 200 the acceleration due to gravity. While water is expected to dif- 1 000 fuse more quickly into the matrix in EF 19 due to the orientation

h 800 of the fracture relative to bedding, thus slowing imbibition rela- 600 Eagle Ford 20 (parallel) experimental data tive to EF 20, the opposite is observed. This may imply that 400 θe=70.7º matrix uptake effects are minor relative to the other differences 200 between these samples (e.g., fracture width, roughness and mi- 0 0.0 0.5 1.0 1.5 2.02.5 3.0 3.5 4.0 4.5 5.0 cromineralogy). Additionally, while the Washburn-Lucas Equa- t 5 * (×10 ) tion (Eq. (1)) is not sufficient to quantitatively describe experi- Figure 12. The best fit of the analytical solution to the EF 20 uptake data at mental uptake in this system, qualitatively it can be used to see the average (& median) fracture width of 59 μm. The effective contact that the uptake rate should be increase with increasing fracture width. While the overall fracture width of EF 20 is, on average, angle (θe) for this fit is 70.7º. The poor fit at small heights is most likely due to dynamic contact angles or small, possibly closed, fracture widths at the larger than EF 19, the smaller pathways at the bottom of the bottom of the sample. sample discussed previously could have caused the decreased uptake rate in EF 20 during the initial times. 5 cm of the sample, which are very low (average of 41 μm), in some cases below the resolution of the X-ray CT, and those of 4 CONCLUSIONS the rest of the sample (average of 68.5 µm). These small This study has demonstrated the use of neutron imaging in widths may be retarding uptake at early times, causing the measuring the spontaneous imbibition of water into shale frac- model to fail to better fit the data. As can be seen in the Wash- tures with known, relatively simple, geometries. It has shown burn-Lucas Equation (Eq. (1)), the rate of uptake is proportion- that neutron imaging is an effective way to quantitatively al to the width of a capillary, suggesting that small widths can measure uptake height as a function of time during spontane- decrease the rate of uptake as demonstrated here. Not surpri- ous imbibition in shales in order to validate fluid flow models. singly, a model that assumes constant average fracture widths We have also showed how an available analytical solution to may not be satisfactory for modelling spontaneous imbibition the appropriate set of nonlinear second-order differential equa- in real fractures, even for a relatively simple experimental tions can be applied to the measured imbibition data to deter- geometry such as that employed here. mine an effective contact angle, and done so for two samples The best-fit effective contact angles determined from EF of the Eagle Ford shale, one in which the fracture is oriented 19 and 20, 77.7º and 70.7º, respectively, are very similar and perpendicular to bedding and the other with the fracture indicated a water-wet to intermediate system. The bulk mine- oriented parallel to bedding. Additionally, the imbibition into ralogy of these two samples are essentially identical but, as the perpendicular fracture was repeated after drying to deter- noted above, EF 19 was cut perpendicular to bedding, while EF mine whether the initial wetting caused changes to the matrix 20 was cut parallel to bedding. Thus, the micro-heterogeneity and, therefore, to the uptake rate. of the minerology is likely to be significantly greater in EF 19, While the quantitative utility of neutron imaging for these which is likely to affect the effective contact angles. Addition- experiments is clear, this work also suggests that the available ally, the surface roughness of EF 20 was greater than that of analytic models are not yet sufficient to fully describe the EF 19, which could cause increased wettability of water, thus process. While fitting the measured imbibition data yielded the observed lower effective contact angle. Wenzel (1936) usable effective contact angles, these were significantly higher described this effect, noting that the roughness of a hydrophilic than measured values. While we have suggested several possi- solid (water-wet, θ<90º) enhances its hydrophilicity, thus lo- ble explanations for this phenomenon, such as the effects of in- wering the contact angle. This effect has been attributed to the situ velocity changes, surface roughness, mineral heterogeneity, longer contact lines between the surface and the fluid, accentu- and surface alteration, the specific origins remain the subject of ating the surface properties (Gao and McCarthy, 2007). This is further investigation. Similarly, the proposed reasons for the also consistent with the measured differences in the static con- differences between the first and second imbibition into EF 19 tact angles (35º±5° for EF 19, and 31º±6º for EF 20). require additional evidential support. In addition, it is clear that Figure 9 shows a direct comparison between the first spon- the model employed does not, as yet, fully reproduce the ob- taneous imbibitions for EF 19 and 20. While EF 19 imbibed served rate of uptake as a function of time. Refinement of the quicker than EF 20 and reached a slightly higher equilibrium model, and a more detailed explanation of the effects of these height, the width of EF 19 was also much smaller than EF 20. variables awaits future work. The greater final height in EF 19 is expected, given its smaller fracture width originally described in a cylindrical capillary by COPYRIGHT James Jurin and known as Jurin’s Law (Jurin, 1717). This has This manuscript has been authored by UT-Battelle, LLC been described more quantitatively and is given as (c.f., under Contract No. DE-AC05-00OR22725 with the U.S. De- Rodríguez-Valverde and Tirado Miranda, 2010; Hardy, 1922) partment of Energy. The United States Government retains and the publisher, by accepting the article for publication, ac- 4σ cosθ H = e (16) knowledges that the United States Government retains a non- ρ D g exclusive, paid-up, irrevocable, world-wide license to publish

886 Victoria H. DiStefano, and et al. or reproduce the published form of this manuscript, or allow Benavente, D., Lock, P., Angeles Garcia Del Cura, M., et al., 2002. others to do so, for United States Government purposes. The Predicting the Capillary Imbibition of Porous Rocks from Department of Energy will provide public access to these re- Microstructure. Transport in Porous Media, 49(1): 59–76 sults of federally sponsored research in accordance with the Brittin, W. E., 1946. Liquid Rise in a Capillary Tube. Journal of Applied DOE Public Access Plan (http://energy.gov/downloads/doe- Physics, 17(1): 37–44. https://doi.org/10.1063/1.1707633 public-access-plan). Broseta, D., Tonnet, N., Shah, V., 2012. Are Rocks still Water-Wet in the

Presence of Dense CO2 or H2S?. Geofluids, 12(4): 280–294. CONFLICT OF INTEREST https://doi.org/10.1111/j.1468-8123.2012.00369.x Certain trade names and company products are mentioned Cai, J. C., Perfect, E., Cheng, C. L., et al., 2014. Generalized Modeling of in the text or identified in an illustration in order to adequately Spontaneous Imbibition Based on Hagen-Poiseuille Flow in Tortuous specify the experimental procedure and equipment used. In no Capillaries with Variably Shaped Apertures. Langmuir, 30(18): 5142– case does such identification imply recommendation or en- 5151. https://doi.org/10.1021/la5007204 dorsement by the National Institute of Standards and Technol- Cai, J. C., Yu, B. M., 2011. A Discussion of the Effect of Tortuosity on the ogy and Oak Ridge National Laboratory, nor does it imply that Capillary Imbibition in Porous Media. Transport in Porous Media, the products are necessarily the best available for the purpose. 89(2): 251–263. https://doi.org/10.1007/s11242-011-9767-0 Cai, J. C., Yu, B. M., Mei, M. F., et al., 2010a. Capillary Rise in a Single ACKNOWLEDGMENTS Tortuous Capillary. Chinese Physics Letters, 27(5): 054701. This work was supported as part of the Center for Nanos- https://doi.org/10.1088/0256-307x/27/5/054701 cale Controls on Geologic CO₂ (NCGC), an Energy Frontier Cai, J. C., Yu, B. M., Zou, M. Q., et al., 2010b. Fractal Characterization of Research Center funded by the U.S. Department of Energy, Spontaneous Co-Current Imbibition in Porous Media. Energy & Fuels, Office of Science, Basic Energy Sciences (No. DE-AC02- 24(3): 1860–1867. https://doi.org/10.1021/ef901413p 05CH11231). Victoria H. DiStefano acknowledges a graduate Chen, C., Wan, J. M., Li, W. Z., et al., 2015. Water Contact Angles on fellowship through the Bredesen Center for Interdisciplinary Quartz Surfaces under Supercritical CO2 Sequestration Conditions: Research at the University of Tennessee. Vitaliy Starchenko Experimental and Molecular Dynamics Simulation Studies. was supported by the U.S. Department of Energy, Office of International Journal of Greenhouse Gas Control, 42: 655–665. Science, Office of Basic Energy Sciences, Chemical Sciences, https://doi.org/10.13039/501100001809 Geosciences, and Biosciences Division. Edmund Perfect’s Cheng, C. L., Perfect, E., Donnelly, B., et al., 2015. Rapid Imbibition of research was sponsored by the Army Research Laboratory (No. Water in Fractures within Unsaturated Sedimentary Rock. Advances in W911NF-16-1-0043). The views and conclusions contained in Water Resources, 77: 82–89. https://doi.org/10.13039/100006151 this document are those of the authors and should not be inter- Cheng, Y. M., 2012. Impact of Water Dynamics in Fractures on the preted as representing the official policies, either expressed or Performance of Hydraulically Fractured Wells in Gas-Shale Reservoirs. implied, of the Army Research Laboratory or the U.S. Gov- Journal of Canadian Petroleum Technology, 51(2): 143–151. ernment. The U.S. government is authorized to reproduce and https://doi.org/10.2118/127863-pa distribute reprints for government purposes notwithstanding Dreyer, M., Delgado, A., Path, H. J., 1994. Capillary Rise of Liquid any copyright notation herein. We acknowledge the support of between Parallel Plates under Microgravity. Journal of Colloid and the National Institute of Standards and Technology, U.S. De- Interface Science, 163(1): 158–168. partment of Commerce, in providing the neutron research facil- https://doi.org/10.1006/jcis.1994.1092 ities used in this work. A portion of this research used re- Dubiel, R. F., Pitman, J. K., Pearson, O. N., et al., 2012. 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