Capillary Imbibition in Layered Sandstone

: Spontaneous capillary imbibition in rocks is fundamental to numerous geomorphological processes and has caused numerous engineering problems. Sedimentary rocks are widely distributed across the Earth’s surface and usually bear layer structures that make the pore structure anisotropic. Understanding the effects of the anisotropic pore structure on capillary imbibition in sedimentary rocks is crucially important but remains inadequate, especially on larger scales than a single tube. In this study, the capillary imbibition process in sandstone was monitored by measuring the water absorption mass, height of the water absorption front, NMR (nuclear magnetic resonance) T 2 spectra, and stratiﬁed moisture distribution. The results demonstrate that (1) the layer structure had a signiﬁcant effect on the capillary imbibition process by altering water absorption rate and water redistribution mode, as the time of the water front reaching the top of Sample A1 lagged behind Sample A2 by 500 min; (2) vapor diffusion and condensation occurred ahead of the water-absorption front, which was more obvious in samples with well-developed beddings; (3) in sandstone samples with bedding planes perpendicular to the height (P er samples), internal water migration lagged behind superﬁcial water migration and was longer in sandstones with well-developed beddings, such as the case of Sample A2, for which the time lag was as large as 280min. Based on a combination of observations of the sandstone structure at pore scale and layer scale with results calculated from the Lucas–Washburn equation, we propose the concept of the representative pore-structure element (RPE). Based on analysis on water migration in RPEs, we suggest that the effects of the layer structure on capillary imbibition in sandstone are embedded in the different water migration modes in P ar (samples with bedding planes parallel to the height) and P er samples. The water migration mode in P ar samples can be simpliﬁed as primary upward intra-layer migration followed by intra-layer horizontal migration, while that in P er samples is primary intra-layer horizontal migration followed by intra-layer upward migration.

Water 2023, 15 There has been plentiful research on capillary water migration in rocks, mainly focusing on the effects of liquid properties (salt content, viscosity, pH, et al.) [27][28][29], pore structure (size, shape, connectivity) [30][31][32][33][34][35], and environmental conditions (temperature, humidity) [36][37][38] on the capillary imbibition process.The influence of pore structure on capillary imbibition is particularly important and has received much research attention.The Young-Laplace equation explains the source of capillary suction; i.e., the surface tension causes the liquid surface to bend, thus creating a pressure difference that allows the liquid to rise [39].Bell and Cameron [40] found the proportional relationship between water migration height and square root of time.Based on the Young-Laplace and Hagen-Poiseuille equations, Lucas [41] and Washburn [42] proposed the Lucas-Washburn (LW) equation, which describes the capillary water migration process while considering the influence of the capillary radius.The LW equation assumes that the section of a capillary tube is uniform and circular, which is different from the actual situation, generating disagreements between the predicted results and the experimental ones.Therefore, many scholars have sought to improve the LW equation by considering the geometrical features of capillary pores [43][44][45][46][47][48].These features include changes in the cross-sectional area [49][50][51], branching and intersection of pores [52], and tortuosity [53,54].Some scholars introduced the fractal theory to describe the tortuousness of capillaries [55,56].As an alternative, Lundblad and Bergman (1997) [57] proposed that the complex effects of geometrical features of capillaries can be equalized by employing an effective radius, so that the capillary imbibition equation stays as simple as the LW law.
These studies deepen the understanding of capillary imbibition process; however, these classic models provide primarily one-dimensional (1D) solutions, which ignores the effect of capillary interactions in pore networks.Recently, more attention has been paid to water migration in two-or three-dimensional (2D or 3D) capillary networks [58].Among them, one of the most fundamental studies was on the capillary competition phenomenon of a Y-shaped bifurcation [59].Despite these attempts, capillary imbibition in rocks with anisotropic pore network can still not be predicted perfectly.Anisotropy is an inherent attribute of the pore structure in layered rocks and is characterized by the significant difference in pore radius and connectivity along the bedding and perpendicular to the bedding, as well as between adjacent layers [60,61].Therefore, capillary imbibition in layered rocks should be studied at a scale larger than one single tube or one Y-shaped bifurcation.Still, research on the capillary imbibition in layered rocks at larger scales is inadequate, and the way that the layer structure modifies capillary water migration remains uncertain.
In addition, the capillary imbibition in rock pores should be related to water vapor adsorption and condensation as well, whose effects are commonly neglected.The movement of water vapor is mainly affected by the relative humidity of the environment (under constant-temperature conditions) [62,63].At low relative humidities, water molecules are bound to the particle surface in the form of adsorption film [64,65].Along with increases in relative humidity, water-vapor adsorption approaches its maximum value gradually; afterwards, a free-water layer appears.Capillary condensation occurs via liquid bridges when the thickness of adsorption water film reaches a critical value.In our experiment, the vapor adsorption and condensation during water migration was verified by analyzing the bound water content through the T 2 spectrum.
Apart from theoretical investigations on capillary imbibition in porous rocks, experimental studies promote understanding of this phenomenon as well.Among them, weighing and visual observation methods were initially adopted to clarify the migration process and moisture distribution in porous rocks, as they are intuitive and convenient.However, absorption water mass or rising height of the absorption front cannot accurately reflect the water migration process inside samples, and especially cannot measure the migration process of water vapor.Therefore, some technologies such as electrical resistivity tomography (ERT) [66], nuclear magnetic resonance imaging (MRI) [67], X-ray computerized tomography CT [68], and infrared thermography (IRT) are employed to monitor the capillary imbibition process.Some of the above methods, for instance, MRI [69][70][71] and X-ray CT, measure variation of moisture distribution along the imbibition direction directly, while others adopt parameters such as resistivity (ERT) and infrared rays (IRT), which are sensitive to moisture content, to demonstrate moisture movement indirectly [72][73][74].Nuclear magnetic resonance technology has great advantages in such experiments, since it not only quickly obtains the water distribution in rock without damaging it, but also accurately characterizes the rock's pore structure.In this study, two types of sandstone samples with different layer orientations were used in capillary imbibition experiments.During the experiments, the water migration process was carefully observed by measuring water absorption mass, height of the absorption front, lightness of the sample tops, nuclear magnetic resonance T 2 spectra, and the stratified moisture distribution.The typical pore structure of layered sandstone was obtained through multi-scale observation of the pore structure with MRI and cast thin sections.Based on the results, the concept of the representative pore-structure element (RPE) is proposed.Combined with some basic theories of capillary water movement, effects of layer structure on the moisture migration process were interpreted at the sample scale and pore structure scale.

Sample Preparation
Four sandstone samples were selected for the test, as shown in Figure 1a [75].Cylindrical samples with a diameter of 50 mm were drilled from two sandstone blocks both perpendicular and parallel to the bedding plane.After that, the cylindrical cores were cut into standard samples with a height of 100 mm.The top and bottom surface of the samples were polished to be smooth.Samples A1 and A2 were taken from the same coarse-grained sandstone, which had well-developed bedding with alternating coarse-grained strata and fine-grained strata.The mineral composition of Sample A1 and A2, derived from XRD experimental results, are shown in Table 1.After the samples were saturated in a vacuum chamber, we could obtain their porosity, and the pore size distribution curves (Figure 1b) could be calculated from the NMR T 2 distributions (Figure 1b); the principle is introduced in Section 2.2.The porosity of Sample A1 was 16.97%; while the porosity of Sample A2 was 18.60%.Samples B1 and B2 were taken from the same fine-grained sandstone, which had obvious bedding as well but with no great differences in grain size between the adjacent strata.Mineral composition of Sample B1 and B2 derived from XRD experimental results are shown in Table 2.The porosity of Sample B1 was 14.93%, and the porosity of Sample B2 was 14.59%.Sample A1 and B1 were sandstones with bedding planes parallel to the axis of the cylindrical core (hereinafter marked by P ar samples), while Sample A2 and B2 were sandstones with bedding planes perpendicular to the axis of the cylindrical core (hereinafter marked by P er samples).Pore size distribution curves of the four samples are shown in Figure 1b (the principle is introduced in Section 2.2).

Background Theory of Nuclear Magnetic Resonance (NMR)
(1) Measurement of pore size distribution Generally, we study the moisture changes of rock and soil by measuring their relaxation time, T2, which can reflect the water content of samples [76,77].In capillary tubes of different radii, the T2 of water is different.Therefore, the pore size distribution of rock can be calculated according to T2 spectra [78], which is related to the average pore size as follows: where s F is a geometric factor ( s F = 3 for spherical pores and 2 for columnar pores; hence, s F = 2 in this study); 2  is the surface relativity of pore walls and is strongly influenced by paramagnetic ions or electrons on the solid surface; and c r is the average radius.According to Equation (1), there is a one-to-one correspondence between T2 and pore size.
(2) Measurement of stratified moisture distribution Like the basic phenomena of T2 spectra, stratified moisture content can be reflected by the signal value.On the basis of the static magnetic field B0, a gradient magnetic field Gy is superimposed along the axis of the cylindrical core of the sample.Assume that the magnetic field strength of cross section A, which is along the sample's axis of cylindrical core, is BA.When a pulse of frequency rBA is applied (where r is a positive integer), only the hydrogen protons at cross section A will undergo magnetic resonance.Therefore, the stratified moisture distribution of the sample can be obtained from the signal intensity at different height sections of the sample.

Background Theory of Nuclear Magnetic Resonance (NMR)
(1) Measurement of pore size distribution Generally, we study the moisture changes of rock and soil by measuring their relaxation time, T 2 , which can reflect the water content of samples [76,77].In capillary tubes of different radii, the T 2 of water is different.Therefore, the pore size distribution of rock can be calculated according to T 2 spectra [78], which is related to the average pore size as follows: 1 where F s is a geometric factor (F s = 3 for spherical pores and 2 for columnar pores; hence, F s = 2 in this study); ρ 2 is the surface relativity of pore walls and is strongly influenced by paramagnetic ions or electrons on the solid surface; and r c is the average radius.According to Equation (1), there is a one-to-one correspondence between T 2 and pore size.
(2) Measurement of stratified moisture distribution Like the basic phenomena of T 2 spectra, stratified moisture content can be reflected by the signal value.On the basis of the static magnetic field B 0 , a gradient magnetic field G y is superimposed along the axis of the cylindrical core of the sample.Assume that the magnetic field strength of cross section A, which is along the sample's axis of cylindrical core, is B A .When a pulse of frequency rB A is applied (where r is a positive integer), only the hydrogen protons at cross section A will undergo magnetic resonance.Therefore, the stratified moisture distribution of the sample can be obtained from the signal intensity at different height sections of the sample.(3) Nuclear magnetic resonance imaging (MRI) An NMR imaging sequence was used to obtain the layered structure of a sample [79], in which, the brighter a pixel was, the higher the moisture content was interpreted to be.On the basis of the static magnetic field B 0 , the phase-encoding gradient G p was applied in the y-direction.Therefore, protons along this direction had different phase values.In the process of collecting the sample signal, a gradient magnetic G r was applied along the x-direction, and the precession frequencies of protons at different positions along this direction were different.So, the frequency and phase values could be used to determine the unique position of the proton.Then, through Fourier transformation of the position information, an image illustrating the layer structure was obtained.

Experimental Design 2.3.1. Capillary Imbibition Test
The four dried samples were placed on a permeable stone in a petri dish.The water surface in the petri dish only just diffused through the permeable stone, and its height was kept constant during the tests (Figure 1a).The following parameters were tested under different water absorption times: (1) Water absorption mass and height rising of the water absorption front The water absorption mass of a sample equals the mass of the sample after water absorption minus that at dry state.The mass of samples was measured with a balance at regular intervals.The height of the water-absorption front on the surface of the samples was measured at the same intervals.The interval time between measurements was 10-30 min in the early stage and 300 min in the later period.Four fixed measuring lines were arranged on the side of the sample, and the height of water migration between the water-absorption front and the ground was measured along the four measuring lines.The average value was taken as the height increase of the water absorption front at a certain time.
(2) Lightness of the sample's top surface To observe the water redistribution process on the top surface of samples, the lightness of the top surface was measured during the water migration process.The lightness of a surface is determined by its ability to reflect light.The good light-reflection ability of water causes the reflection of the upper surface of sandstone to vary according to water conditions.Therefore, by measuring the lightness of the upper surface, the moisture distribution process at that location can be determined.The lightness value was measured by colorimeter (3nh NR145), and the lightness value of the upper surface was obtained by taking the average value of the three values.

Observations of the Layer Structures and Pore Structures of Samples
The layer structure of the sandstone was observed by NMR imaging after the samples were completely saturated with distilled water via the vacuum method.The equipment was the same as above, and the test parameters were set in accordance with Jia et al. [79].The pore structure of sandstone was observed by optical microscopy using cast thin sections.

Water-Absorption Mass Changes with Time
The water migration processes in the four samples were similar (Figure 2), showing a three-stage trend of (1) rapid increasing, (2) slow increasing, and (3) stabilization.The stages were divided according to the change in the water-absorption rate.When the water-absorption rate is much higher than 0.01, this stage is the stage of rapid growth.However, the layer orientation had a significant effect on the capillary imbibition process.The layer orientation mainly affected the duration time, water absorption mass, and the rate of water-absorption mass in each stage.In sandstone with well-developed bedding (Samples A1 and A2), the layer orientation caused significant differences in water absorption mass and its rate of change at each stage.In P ar sample (Sample A1), shown in Figure 2a, the water mass mainly increased during the rapid increase stage, while for the P er sample (Sample A2), shown in Figure 2b, the water absorption mass increased mainly in the slow increasing stage.Counting the water-absorption mass and its proportion at each stage, shown in Table 3, in the P ar sample (Sample A1), water-absorption mass increase during the rapid increase stage accounted for 81.37% of the total water-absorption mass, while that absorbed in the slow increasing stage accounted for 4.81%.The parameters in Table 3 are defined in the note below the table.In the P er sample (Sample A2), water-absorption mass increase during the rapid increase stage accounted for 41.35% of the total water-absorption mass in sandstone, while that absorbed in the slow increasing stage accounted for 44.03%.In addition, in each stage, the water increase rate (the increase in the water absorption mass per unit time) is significantly greater in P ar sample than that in P er sample (Table 3).In sandstone with poorly developed bedding (Samples B1 and B2), the layer orientation caused significant differences in the duration of the slow increasing stage (Figure 2c,d).In addition, the layer orientation still affected the water-absorption mass and its rate at each stage, but the influence is not as pronounced in these sandstones with poorly developed bedding.In the P ar sample (Sample B1), the slow increasing stage lasted for 800 min, while in the P er sample (Sample B2), the slow increasing stage lasted for 1640 min.In the P ar sample (Sample B1), the water-absorption mass increase during the rapid increasing stage accounted for 46.42% of the total water-absorption mass, while that in the slow increasing stage accounted for 38.39%.In the P er sample (Sample B2), the water-absorption mass increase during the rapid increasing stage accounted for 36.21% of the total water absorption mass, while that in the slow increasing stage accounted for 50.57% (Table 3).In each stage, the average rate of change in water absorption was greater in the P ar sample than in the P er sample, but the difference is smaller compared to the well-developed sandstone.
In addition, the total water-absorption mass of the samples was influenced by porosity.The water-absorption mass of the four samples were ranked in descending order as A2, A1, B1, and B2, which is the same order as for porosity.

Variation in the Height of the Absorption Front
The orientation of the bedding plane has a significant effect on the rising rate and morphology of the water-absorption front, and this effect varies with the degree of layer development (Figure 3).The rising rate in the water-absorption front was higher in the P ar sample than in the P er sample.This phenomenon is more obvious in well-developed bedding sandstone.The rising of the water-absorption front in Sample A1 can be divided into two stages: (1) rapid rising and (2) stabilization.The front reached the top of the sample within 105 min.The rise in Sample A2 can be divided into three stages: (1) rapid rising, (2) slow rising, and (3) stabilization.It took 500 min for the front to reach the top of the sample.Samples B1 and B2 also featured these three stages, and it took 700 min and 1380 min, respectively, for the fronts to reach the tops of the samples.
The influence of layer orientation on the morphology of the water absorption fronts was mainly evident in sandstone with well-developed bedding.The height of the water-absorption front varied significantly within the different layers of the Par sample, reflecting different absorption rates.Meanwhile, the front height increased uniformly layer by layer in the Per sample.In sandstone with poorly developed bedding, there was no obvious difference in the morphologies of the water absorption fronts, demonstrating that the water absorption rates were relatively uniform.

Water Distribution at the Top of Samples According to Lightness
The lightness of the top surface of the samples was negatively correlated with the water content of that surface.The change in lightness of the top surface may reflect the moisture migration process along the radial direction of the specimen.The change in lightness of the top surface corresponds to the phases of the height increase of the water absorption front and can be divided into three stages: (1) Water migrates along the axis of cylindrical core.In this stage, water mainly migrates upward along the dominant seepage channels in the axis of cylindrical core.The absorption front gradually rises to the top surface, but its lightness is largely unchanged.As shown in Figure 4a The influence of layer orientation on the morphology of the water absorption fronts was mainly evident in sandstone with well-developed bedding.The height of the waterabsorption front varied significantly within the different layers of the P ar sample, reflecting different absorption rates.Meanwhile, the front height increased uniformly layer by layer in the P er sample.In sandstone with poorly developed bedding, there was no obvious difference in the morphologies of the water absorption fronts, demonstrating that the water absorption rates were relatively uniform.

Water Distribution at the Top of Samples According to Lightness
The lightness of the top surface of the samples was negatively correlated with the water content of that surface.The change in lightness of the top surface may reflect the moisture migration process along the radial direction of the specimen.The change in lightness of the top surface corresponds to the phases of the height increase of the water absorption front and can be divided into three stages: (1) Water migrates along the axis of cylindrical core.In this stage, water mainly migrates upward along the dominant seepage channels in the axis of cylindrical core.The absorption front gradually rises to the top surface, but its lightness is largely unchanged.As shown in Figure 4a,b, red star A is the top surface of the sample in this state.(2) Water migrates along the radial direction.In this stage, water mainly migrates along the radial seepage channels.The height of the water absorption front no longer changes, but the lightness of the top surface decreases significantly.As shown in Figure 4a,b, red star B is the top surface of the sample in this state.(3) Water redistributes.Water migration in this stage mainly occurs between pores of different sizes.The lightness and water absorption mass of the samples are also stable (Figure 4).As shown in Figure 4a, red star C is the top surface of the sample in this state.
of the water absorption front no longer changes, but the lightness of the top surface decreases significantly.As shown in Figure 4a,b, red star B is the top surface of the sample in this state.(3) Water redistributes.Water migration in this stage mainly occurs between pores of different sizes.The lightness and water absorption mass of the samples are also stable (Figure 4).As shown in Figure 4a, red star C is the top surface of the sample in this state.It should be noted that during the stage of water migration along the height, there was still minor radial migration.In addition, the initial difference in lightness between Specimens A and B was caused by a difference in the grain thickness of the sandstone samples.

Variation in T2 Spectra During Water Adsorption
The T2 spectra of the samples at different moments reflect the capillary imbibition process (Figure 5).To study the pore water transport process more clearly, pore water was classified into three types-bound water, capillary water, and bulk water-according to the T2 spectra.The T2 of bound water in sandstone is generally < 3 ms, whereas those of capillary water range between 3 and 33 ms, and values of > 33 ms are associated with bulk water or free water [75].The bound water content was the first to reach stabilization in all four samples during the capillary imbibition process, followed The variation in lightness in different types of layered sandstones indicates that (1) the layer orientation has a significant effect on the capillary imbibition process in the axis of cylindrical core; (2) the layer orientation also has a significant influence on the capillary imbibition process in the radial direction.The main features of this phenomenon are (1) in sandstone with well-developed bedding, radial water migration in the P ar sample (Sample A1) lasted for 130 min after the water absorption front reached the top, while in the P er sample (Sample A2), it lasted 240 min; (2) in sandstone with poorly developed bedding, radial water migration in the P ar sample (Sample B1) lasted 440 min after the water absorption front reached the top, while in the P er sample (Sample B2), it lasted 810 min.
It should be noted that during the stage of water migration along the height, there was still minor radial migration.In addition, the initial difference in lightness between Specimens A and B was caused by a difference in the grain thickness of the sandstone samples.

Variation in T 2 Spectra during Water Adsorption
The T 2 spectra of the samples at different moments reflect the capillary imbibition process (Figure 5).To study the pore water transport process more clearly, pore water was classified into three types-bound water, capillary water, and bulk water-according to the T 2 spectra.The T 2 of bound water in sandstone is generally <3 ms, whereas those of capillary water range between 3 and 33 ms, and values of >33 ms are associated with bulk water or free water [75].The bound water content was the first to reach stabilization in all four samples during the capillary imbibition process, followed by capillary water, and finally, bulk water (Figures 5 and 6).In addition, upon comparing the stabilization times of the bound water and water absorption mass and the time taken for the water absorption front to reach the top of the sample (Table 4), it can be found that the bound water content stabilized before the measured water absorption front reached the top.Given the fact that bound water may transport in the pores through vapor diffusion in addition to surface diffusion, the stabilization time of three types of water indicate that vapor diffusion did exist in the capillary imbibition process; i.e., vapor diffused and condensed to fill  Table 4. Stabilization time for bound water content (Figure 6), water-absorption mass (Figure 2), and height (Figure 3) of the four samples.The results in Figures 7 and 8 also show that the porosity and layer orientation of the samples had significant effects on the migration processes and final state of water distribution in the following manner: (1) Bound water content stabilization time is influenced by the porosity and layer orientation of the sample.The higher the porosity, the shorter the time required for the bound water to reach stability.The stabilization time in the P ar samples was significantly shorter than that in P er samples.In sandstone with well-developed bedding, it took 120 min for the bound water content to reach stability in the P ar sample (Sample A1) and 360 min in the P er sample (Sample A2).In sandstone with poorly developed bedding, it took 500 min for the bound water content to reach stability in the P ar sandstone (Sample B1) and 1020 min in the P er sample (Sample B2). ( 2) The time taken for the capillary water content to reach stability was mainly influenced by porosity.The higher the porosity of the sample, the longer it took for the capillary water content to reach stability.In sandstone with well-developed bedding (Samples A1 and A2), the stabilization time was relatively long due to the high porosity, being 3380 min in both.Sandstone with poorly developed bedding (Samples B1 and B2) had lower porosity, corresponding to shorter times required for their capillary water contents to stabilize, being 1020 min and 1860 min, respectively.

Sample
(3) The bulk water content stabilization time was unaffected by layer orientation and pore size.(4) The layer orientation did not affect the final content of each type of water.The results in Figures 7 and 8 also show that the porosity and layer orientation of the samples had significant effects on the migration processes and final state of water distribution in the following manner: (1) Bound water content stabilization time is influenced by the porosity and layer orientation of the sample.The higher the porosity, the shorter the time required for the bound water to reach stability.The stabilization time in the Par samples was significantly shorter than that in Per samples.In sandstone with well-developed bedding, it took 120 min for the bound water content to reach stability in the Par sample (Sample A1) and 360 min in the Per sample (Sample A2).In sandstone with poorly developed bedding, it took 500 min for the bound water content to reach stability in the Par sandstone (Sample B1) and 1020 min in the Per sample (Sample B2). ( 2) The time taken for the capillary water content to reach stability was mainly influenced by porosity.The higher the porosity of the sample, the longer it took for the capillary water content to reach stability.In sandstone with well-developed bedding (Samples A1 and A2), the stabilization time was relatively long due to the high porosity, being 3380 min in both.Sandstone with poorly developed bedding (Samples B1 and B2) had lower porosity, corresponding to shorter times required for their capillary water contents to stabilize, being 1020 min and 1860 min, respectively.(3) The bulk water content stabilization time was unaffected by layer orientation and pore size.(4) The layer orientation did not affect the final content of each type of water.
According to the pore water classification method mentioned above, the signal amplitude of different types of water can be calculated, and then its proportion in the total signal amplitude can be obtained.In sandstone with well-developed bedding, the contents of bound water, capillary water, and bulk water inside the Par sample (Sample A1) accounted for 30%, 46%, and 24%, respectively, of the total water; while in the Per sample (Sample A2), they accounted for 28%, 45%, and 27%.In sandstone with poorly developed bedding, the contents of bound water, capillary water, and bulk water inside the According to the pore water classification method mentioned above, the signal amplitude of different types of water can be calculated, and then its proportion in the total signal amplitude can be obtained.In sandstone with well-developed bedding, the contents of bound water, capillary water, and bulk water inside the P ar sample (Sample A1) accounted for 30%, 46%, and 24%, respectively, of the total water; while in the P er sample (Sample A2), they accounted for 28%, 45%, and 27%.In sandstone with poorly developed bedding, the contents of bound water, capillary water, and bulk water inside the P ar sample (Sample B1) accounted for 12%, 40%, and 48%, respectively, of the total water; while in the P er sample (Sample B2), they accounted for 12%, 40%, and 48%.This indicates that the proportions of different types of water within a rock were mainly related to its pore size distribution and had no direct correlation with layer orientation.

Variation in Stratified Moisture Distribution during Capillary Imbibition
The stratified moisture distribution curve obtained by NMR can intuitively reflect the water migration process along the axis of cylindrical core of the samples.Corresponding to changes in water-absorption height, the curve can be divided into two stages: (1) water migration and (2) redistribution.In the water migration stage, water primarily migrates in the axis of the cylindrical core, and the signal amplitude of the stratified moisture distribution curve also increases in this direction.In the redistribution stage, water migration in the main capillary channels is largely complete, and water mainly migrates into adjacent pores.
The stratified moisture distribution curve no longer changes in the axis of the cylindrical core while its overall signal amplitude increases (Figure 7).cates that the proportions of different types of water within a rock were mainly related to its pore size distribution and had no direct correlation with layer orientation.cates that the proportions of different types of water within a rock were mainly related to its pore size distribution and had no direct correlation with layer orientation.The stratified moisture distribution curves of samples with different layer directions show that the layer direction influences the water migration along the axis of the cylindrical core.In the P ar sample, the signal difference is relatively small along the axis of the cylindrical core; while in the P er sample, the difference is greater.In sandstone with well-developed bedding, the maximum-to-minimum signal amplitude ratio after water absorption stabilized was 0.54 in the P ar sample (Sample A1) and 0.20 in the P er sample (Sample A2).In sandstone with poorly developed bedding, the ratio was 0.82 in the P ar sample (Sample B1) and 0.63 in the P er sample (Sample B2).Here, it should be noted that since moisture dissipated more rapidly at the sample surface, the data from both ends of the samples were excluded when determining the maximum and minimum signal amplitudes.
Upon comparing the water-absorption height reflected by stratified moisture distribution curves with the directly measured height of the water-absorption front (Figure 8), we found that the internal and superficial capillary imbibition processes were basically synchronized in the P ar sample, while in the P er sample, internal water migration lagged behind superficial water migration.In sandstone with well-developed beddings, there was little difference between the directly measured height and NMR-derived height in the P ar sample (Sample A1), while in the P er sample (Sample A2), the NMR-derived height lagged behind the measured height.This phenomenon was not obvious in samples with poorly developed bedding (Figure 8).

Pore Structure of Layered Sandstone 4.1. Layer Structure According to MRI
Based on the above experimental results, the layer structure and porosity had important influences on the migration and distribution of water in the samples.To study the layer structures more closely, the samples were subjected to NMR imaging.Figure 9a,b show images from the MRI of two P ar samples: Samples A1 and B1, respectively.The coarse layers and fine layers of the samples can be clearly seen in the imaging results.

Pore Structure Observed in Cast Thin Sections
The sandstone samples exhibited a layer structure at the macroscopic scale, whic was reflected in different particle arrangements at the microscopic scale.Thus, the sam ples were cast into thin sections and then observed under the microscope (Figure 10).I the Par sample with well-developed bedding (Sample A1), the difference between th coarse-grained and fine-grained layers was more obvious.In the coarse-grained laye the quartz particles were mostly angular, less filled with fine grains, and the structur generally showed support from the particles, with large inter-granular voids and goo connectivity.There were obvious boundaries between the fine-grained an

Pore Structure Observed in Cast Thin Sections
The sandstone samples exhibited a layer structure at the macroscopic scale, which was reflected in different particle arrangements at the microscopic scale.Thus, the samples were cast into thin sections and then observed under the microscope (Figure 10).In the P ar sample with well-developed bedding (Sample A1), the difference between the coarse-grained and fine-grained layers was more obvious.In the coarse-grained layer, the quartz particles were mostly angular, less filled with fine grains, and the structure generally showed support from the particles, with large inter-granular voids and good connectivity.There were obvious boundaries between the fine-grained and coarse-grained layers.The mineral particles in the fine-grained layer were mainly sub-angular and the support particles formed a mosaic-like structure, resulting in a significant reduction in pore volume and pore connectivity.The result was processed to obtain a binarized image.Figure 10 shows that the radius of the capillary channels within the coarse-grained layer was larger than those in the fine-grained layer.According to the binarization results, the porosity of the coarse-grained layer was 20.48% and that of the fine-grained layer was 12.83%.For the P ar sample with poorly developed bedding (Sample B1), there was no significant difference between the coarse-grained and fine-grained layers.The quartz grains were similar in size and their shapes were mostly sub-angular.A binarized image was obtained after processing, which indicated a porosity of 14.44%.

Discussion
The major phenomena observed during the capillary imbibition experiments in this study are as follows: (1) the layer structure had a significant effect on the capillary imbibition process by altering water-absorption rate and water redistribution mode; (2) vapor diffused and condensed to fill partial micropores and adsorb on surface of pores ahead of the water absorption front, which was more obvious in samples with well-developed beddings; (3) in the Per sample, internal water migration lagged behind superficial water migration, which was longer in sandstones with well-developed beddings.The mechanisms behind the above phenomena were interpreted by addressing the effects of the pore/layer structure on the water migration process.

Basic Theories of Capillary Rise in a Tube and Calculation of the Equivalent Radius
When capillary water migrates upward along a tube with a radius of a, based on the law of conservation of momentum, the following equation of motion for the fluid in the tube is obtained [84]: According to the positive correlation between porosity and permeability [81,82], the pore connectivity of the coarse-grained layer is significantly higher than that of the finegrained layer.In addition, due to the directivity of mineral grains during sedimentation [83], pore connectivity is usually better along the bedding plane than perpendicular to the bedding plane.

Discussion
The major phenomena observed during the capillary imbibition experiments in this study are as follows: (1) the layer structure had a significant effect on the capillary imbibition process by altering water-absorption rate and water redistribution mode; (2) vapor diffused and condensed to fill partial micropores and adsorb on surface of pores ahead of the water absorption front, which was more obvious in samples with well-developed beddings; (3) in the P er sample, internal water migration lagged behind superficial water migration, which was longer in sandstones with well-developed beddings.The mechanisms behind the above phenomena were interpreted by addressing the effects of the pore/layer structure on the water migration process.

Basic Theories of Capillary Rise in a Tube and Calculation of the Equivalent Radius
When capillary water migrates upward along a tube with a radius of a, based on the law of conservation of momentum, the following equation of motion for the fluid in the tube is obtained [84]: ). ( The solution of this equation was given by Bosanquet [85] as where z is the height of capillary front from initial height, z 0 is the initial height, σ is the surface tension, a is the tube radius, η is the coefficient of the term representing the inertia of fluid in reservoir, ρ is the mass density, θ is the contact angle, and t represents the time. When the capillary tube is long enough and the flow approaches the Poiseuille flow, as was the case in this study, Equation (3) will reduce to the Lucas-Washburn formula [41]: Using Equation ( 4), assuming z 0 = 0, the curves of z vs. t with different tube radii can be obtained.Parameters used in the calculation are as follows: σ is the surface tension (at 25 • C, σ = 0.07197 N/m [86]; η is the coefficient of viscosity (at 25 • C, η = 0.899 × 10 3 Pa • s) [87]; ρ is the mass density of water (ρ = 1 × 10 3 kg/m 3 ); θ is the contact angle (between water and quartz, θ = 33 • ) [88].
By approximating the tested z vs. t curves with the calculated curves, an equivalent radius of each sample could be obtained (Figure 11).As we expected, the equivalent radius was larger along the bedding plane than perpendicular to the bedding plane, which is consistent with the results from direct observation with the cast thin sections in Figure 10.The equivalent radii in sandstones with well-developed bedding were much larger than those with poorly developed bedding.

Interpretating the Effects of the Bedding Plane on Moisture Migration with the Representative Pore-Structure Element
Based on the direct observation of pore structure of layered sandstone samples and calculation of the equivalent radii of samples, we propose the concept of the representative pore-structure element (RPE).An RPE consists of primary capillary channels (both parallel and perpendicular to the bedding plane) embedded in coarse-grained layers, secondary capillary channels embedded in fine-grained layers, and dead-end pores connected to the channels.The sizes of capillaries that develop in parallel to the bedding are larger than at other directions (but of the same magnitude as shown in Figure 11) in one layer (Figure 12).Accordingly, the layer structure of sandstone can be simplified as a combination of numerous RPEs.
the contact angle (between water and quartz,
By approximating the tested z vs. t curves with the calculated curves, an equivalent radius of each sample could be obtained (Figure 11).As we expected, the equivalent radius was larger along the bedding plane than perpendicular to the bedding plane, which is consistent with the results from direct observation with the cast thin sections in Figure 10.The equivalent radii in sandstones with well-developed bedding were much larger than those with poorly developed bedding.

Interpretating the Effects of the Bedding Plane on Moisture Migration with the Representative Pore-Structure Element
Based on the direct observation of pore structure of layered sandstone samples and calculation of the equivalent radii of samples, we propose the concept of the representative pore-structure element (RPE).An RPE consists of primary capillary channels (both parallel and perpendicular to the bedding plane) embedded in coarse-grained layers, secondary capillary channels embedded in fine-grained layers, and dead-end pores connected to the channels.The sizes of capillaries that develop in parallel to the bedding are larger than at other directions (but of the same magnitude as shown in Figure 11) in one layer (Figure 12).Accordingly, the layer structure of sandstone can be simplified as a combination of numerous RPEs.
By differentiating Equation (4) with t, the rising velocity of capillary water in a tube dz dt can be obtained:

Conclusions
The following conclusions were obtained.
(   . ( As indicated by Equation ( 5), dz dt correlates positively with the radius of the tube a.This correlation stipulates one of the water migration laws in the RPE that water migrates faster in primary capillary channels.Furthermore, when water migration front reaches the branching point, the capillary competition effect should be taken into account, which says water will flow in both branches when sizes of branches are close to that of the root tube but will flow primarily in the narrow branch when the thick branch is larger than the root tube [58].The water migration process in the RPEs of P ar and P er samples are deduced below, following the above migration laws.
In the RPEs of the P ar samples, water migrated primarily upwards in larger channels within coarse-grained layers, and upward migration in fine-grained layer was slower.Meanwhile, horizontal water movement in the form of both intra-layer and inter-layer migration was slower than upward migration.Intra-layer horizontal migration in coarse-grained layers was faster than in fine-grained layers as well.Inter-layer horizontal migration from coarse-grained layers to adjacent fine-grained layers may also have occurred.As the water-migration front reached the top of the samples, water migration continued by entering the dead-end pores until pore gas pressure balanced the driving pressure of water migration.The average water-migration rate in the P ar samples was mainly determined by the sizes of larger channels.Accordingly, the water-migration mode in P ar samples can be simplified as follows: primary upward intra-layer migration followed by intra-layer horizontal migration and then inter-layer horizontal migration.
In the RPEs of the P er samples, horizontal water migration should have been dominant, since it moved along larger channels.Upward water migration was much slower, alternating between intra-layer and inter-layer migration.Due to the slow upward migration rate, there was sufficient time for water entering the dead-end pores.The average watermigration rate in the P er samples was mainly determined by the sizes of smaller channels.Accordingly, the water migration mode in the P er samples can be simplified as follows: primary intra-layer horizontal migration followed by intra-layer upward migration and then inter-layer horizontal migration.
Due to horizontal water migration, pore water at the center of samples was sucked out towards the surface.This suction effect was more significant in P er samples, because horizontal capillaries are the preferential channels for water migration.This explains why internal water migration lagged behind surface water migration and the time lag was longer in sandstones with well-developed beddings.Although prior researchers proposed that capillary imbibition in rocks exhibits features of direction-dependence [84], this study explains in detail the origination of direction-dependence.
Except for capillary movement along pore channels, water-vapor diffusion is an important part of the water-migration process that cannot be ignored; this is consistent with previous studies [89,90].Ahead of the capillary water front, water vapor diffused to the particle surface, which is shown in blue lines in Figure 12.The driving force of vapor diffusion is the equilibrium tendency between a higher water vapor concentration and a lower one.The water diffusing on the pore walls favored capillary water migration; however, when the vapor condensed at the pore throat, the enclosed space hindered capillary water movement.

Conclusions
The following conclusions were obtained.
(1) The layer structure had a significant effect on the capillary imbibition process by altering the water-absorption rate and water-redistribution mode.The water-migration rate along the axis of the cylindrical core was faster in the P ar samples, which is evidenced by the variation in water-absorption mass and water height increase.The change of lightness on the top of the samples with time indicated different moistureredistribution modes in P ar samples and P er samples.(2) Vapor diffused and condensed to fill partial micropores and adsorb on the surface of pores ahead of the water-absorption front.This was more obvious in samples with well-developed bedding.The content of bound water as measured by NMR signals achieved stability before the visible water-absorption front reached the samples' top; this was attributed to vapor diffusion and condensation above the capillary water front.

Figure 1 .
Figure 1.Test samples (a) and their pore size distributions and T2 spectra (b).

Figure 1 .
Figure 1.Test samples (a) and their pore size distributions and T 2 spectra (b).
Note.T = water-absorption time; M = water-absorption mass within each stage; MR = ratio of water-absorption mass in each stage to total water-absorption mass; R = the increase in the water-absorption mass per unit time = M/T.

Figure 4 .
Figure 4. Lightness vs. time reflecting the water reaching the top surfaces in (a) Sample A1; (b) Sample A2; (c) Sample B1 and (d) Sample B2.The variation in lightness in different types of layered sandstones indicates that (1) the layer orientation has a significant effect on the capillary imbibition process in the axis of cylindrical core; (2) the layer orientation also has a significant influence on the capillary imbibition process in the radial direction.The main features of this phenomenon are (1) in sandstone with well-developed bedding, radial water migration in the Par sample (Sample A1) lasted for 130 min after the water absorption front reached the top, while in the Per sample (Sample A2), it lasted 240 min; (2) in sandstone with poorly developed bedding, radial water migration in the Par sample (Sample B1) lasted 440 min after the water absorption front reached the top, while in the Per sample (Sample B2), it lasted 810 min.It should be noted that during the stage of water migration along the height, there was still minor radial migration.In addition, the initial difference in lightness between Specimens A and B was caused by a difference in the grain thickness of the sandstone samples.

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Figure 8 .
Figure 8.Comparison of NMR-derived and directly measured water-absorption front heights during the saturation process in (a) Sample A1; (b) Sample A2; (c) Sample B1 and (d) Sample B2.

Figure 8 .
Figure 8.Comparison of NMR-derived and directly measured water-absorption front heights during the saturation process in (a) Sample A1; (b) Sample A2; (c) Sample B1 and (d) Sample B2.Figure 8. Comparison of NMR-derived and directly measured water-absorption front heights during the saturation process in (a) Sample A1; (b) Sample A2; (c) Sample B1 and (d) Sample B2.

Figure 8 .
Figure 8.Comparison of NMR-derived and directly measured water-absorption front heights during the saturation process in (a) Sample A1; (b) Sample A2; (c) Sample B1 and (d) Sample B2.Figure 8. Comparison of NMR-derived and directly measured water-absorption front heights during the saturation process in (a) Sample A1; (b) Sample A2; (c) Sample B1 and (d) Sample B2.

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Water 2021 , 22 Figure 10 .
Figure 10.Cast thin sections and binary images of the pore structures of (a) Samples A1 and (b) B1.

Figure 10 .
Figure 10.Cast thin sections and binary images of the pore structures of (a) Samples A1 and (b) B1.

Figure 11 .
Figure 11.Comparison of the capillary imbibition curve reflected by the equivalent radius with the actual ones.

Figure 11 .
Figure 11.Comparison of the capillary imbibition curve reflected by the equivalent radius with the actual ones.Water 2021, 13, x FOR PEER REVIEW 18 of 22
) The layer structure had a significant effect on the capillary imbibition process by altering the water-absorption rate and water-redistribution mode.The water-migration rate along the axis of the cylindrical core was faster in the Par samples, which is evidenced by the variation in water-absorption mass and water height increase.The change of lightness on the top of the samples with time indicated different moisture-redistribution modes in Par samples and Per samples.(2) Vapor diffused and condensed to fill partial micropores and adsorb on the surface of pores ahead of the water-absorption front.This was more obvious in samples with well-developed bedding.The content of bound water as measured by NMR signals achieved stability before the visible water-absorption front reached the samples' top; this was attributed to vapor diffusion and condensation above the ca-

Figure 12 .
Figure 12.Capillary imbibition mechanisms in (a) P ar Sample; (b) P er Samples.By differentiating Equation (4) with t, the rising velocity of capillary water in a tube dz dt

Table 1 .
Mineral composition in Samples A1 and A2.

Table 2 .
Mineral composition in Samples B1 and B2.

Table 2 .
Mineral composition in Samples B1 and B2.

Table 3 .
Water-absorption time and mass of samples at each stage.

Table 3 .
Water-absorption time and mass of samples at each stage.