Prediction of capillary suction in porous media based on micro-CT technology and B–C model

Min Li; Hemiao Yu; Hongpu Du

doi:10.1515/phys-2020-0203

Article Open Access

Prediction of capillary suction in porous media based on micro-CT technology and B–C model

Min Li , Hemiao Yu and Hongpu Du

Published/Copyright: December 5, 2020

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From the journal Open Physics Volume 18 Issue 1

Abstract

Moisture variation in porous media depends mainly on the pore characteristics. This article used the micro-computed tomography (micro-CT) (a non-destructive imaging technique to generate a three-dimensional virtual model) and the Brooks–Corey model to deduce the moisture migration in sand. Relationship between capillary rise height and time (h–t) was achieved by numerical simulation in the capillary suction process, where the parameters fractal dimension, porosity, and air–water interfacial area were obtained by the micro-CT scanning. Meanwhile, experiments of capillary rise in sand column were performed using four different sizes washed sand, and the capillary heights at different times were recorded. Results show that the capillary suction is decided by the aperture size and phase morphology simultaneously, and particle size has obvious effect on capillarity, and the wetting front lowers with the increase in grain size and the decrease in rising rate. Parameters air entry pressure and pore-size distribution index obtained by micro-CT scanning technology and empirical formula are accurate. Method of combing micro-CT images and Brooks–Corey model can predict well the capillary suction of porous media. It is also proved that the capillary suction is decided by the aperture size and phase morphology simultaneously.

Keywords: micro-CT; fractal dimension; Brooks–Corey model; porous media; capillary suction

1 Introduction

A porous medium contains many pores (voids) filled typically with a fluid (liquid or gas), and the skeletal portion of the material is often called matrix. Many natural substances such as rocks and soil are porous media. Engineering properties of the materials (e.g., permeability, tensile strength, and electrical conductivity) depend on their constituents, porosity, and pore structure mainly. Because of non-homogeneity and random distribution, fluid flow in these media attracts much attention and brings up an extensive study [1,2] such as rainfall filtration, subgrade, slope stability, and energy storage in aquifer [3,4].

Capillary action is the ability of a liquid flowing in narrow spaces without the assistance of and in opposition to external forces like gravity [5]. Capillary rise can be seen as the adhesive forces between walls of capillary tube and fluid [6]. They prompt the edges of the fluid upwards, while the surface tension (cohesive forces) constantly pulls molecules from the surface inward and holds the molecules together. The aforementioned two forces fight against with each other, and then the fluid will stop rising when the net force caused by the weight of the fluid column is greater than that of cohesive forces. The capillary rise is based mainly on measurements of capillary pressure difference or velocity of liquid penetration. Bartell and Whitney [7] were the authors who advanced this method in research. Washburn [8] introduced the velocity of penetration based on the assumption of n-cylindrical capillary pipes. The Washburn equation was used frequently to model the capillary rise in porous media. However, capillary process also pointed out that only the rise curve agreed initially with the Washburn equation [9]. Xue et al. [10] indicated that porous materials behave more like a bundle of capillaries and the capillary rise was no longer considered as an approximate analytical tool. Mualem [11] and Siebold [12] also verified that the Washburn equation could not provide a fine-tuned and precise information on surface modification or on different kinds of particles. The observed under-prediction of front position was due to the neglect of dynamic saturation gradients. Lockington and Parlange [13] put forward an analytical formula for the position of front, speed of propagation, and cumulative uptake. However, most of these models were based upon the assumption of symmetrical capillary tube and regular geometrical cross section and cannot describe the capillary permeability in reality.

Richards model describing the permeable process of unsaturated porous media was derived from the basis of potential function and applied widely in the field of agriculture and rainfall [14]. The key to the mathematical model depended on the relationship between hydraulic conductivity and water content. However, such parameters were obtained only by data fitting and deducing basic property. However, the maneuverability of experiment and complexity of deduction restricted the application.

X-ray computed tomography (CT) was a medical imaging procedure during the seventies of last century. It utilizes computer-processed X-rays to produce tomographic images of body’s specific areas. Digital geometry processing can generate a three-dimensional (3D) image inside the object from a large series of two-dimensional (2D) images taken from a single axis of rotation. CT can produce a volume of manipulated data through a process known as “windowing” and then demonstrate various body structures based on their ability to block the X-ray beam. CT scanner allows the volume of data to be reformatted in volumetric (3D) representations and is used gradually in other fields such as non-destructive materials testing [15]. In the field of geotechnical engineering, CT was mainly used to study the properties of density, porosity, and crack width [16,17]. Okabe and Blunt [18] obtained the internal structure of porous media through the 3D reconstruction and then studied its permeability according to the multipoint statistical theory. Seol and Kneafsey [19] used X-ray CT to image and quantify the effect of a heterogeneous sand grain-size distribution on the formation and dissociation of methane hydrate. Bera [20] analyzed the pore structure distribution features of sandstone with the aid of 3D microstructure. As for the mechanisms leading to capillary collapse in the loose unsaturated sand, Bruchon et al. [21] had research on it applying X-ray CT. The microstructure was analyzed to assess the volume of water filling the pores and deformation of the granular skeleton using Volumetric Digital Image Correlation tools. According to the trinarization of micro-CT images for partially saturated sand, Higo et al. [22] discussed the form of the existing pore water at different water-retention states. Zhou et al. [23] had analyzed a new CT method to study pollutant migration in unsaturated sand by scanning the specimens. The results show that the computerized tomography is a feasible method for studying the characteristics of KI solution in unsaturated sands. Generally, CT technology can reflect the internal structure of porous media accurately.

CT technology and Brooks–Corey model are introduced to predict the capillary suction in porous media like sand. Parameters mean grain size, porosity, and fractal dimension related to the hydraulic conductivity are obtained by micro-CT scan and the 3D reconstruction. In addition, from the statistical quantization of the air–water interfacial area by CT, the difference between the larger size and smaller is analyzed at the same saturation. Therefore, capillary suction model can be derived gradually by the expression of water retention in the B–C model. In order to verify the model scientifically, experiments of capillary rise were performed synchronously using four different interval sizes of screening washed sand, and capillary rise heights at different time were observed.

The use of micro-scale simulation method could be better to analyze the properties of porous media and the state of internal fluid. Aldakheel [24] studied the micro-scale model of concrete failure in pore elastic–plastic medium and found that regarding the concrete solid skeleton at the microscale, only the hydrated cement paste undergoes elastic–plastic–fracture deformations. Saenger et al. [25] carried out micro-scale finite-difference simulation of wave propagation, including pore scale propagation of elastic wave in digital rock sample and dynamic elastic characteristics of viscous saturated rock. Gao et al. [26] carried out 3D micro-scale flow simulation in porous media of saturated soil. When using MRT-LBM for flow simulation, stronger steady-state viscous flow solutions could be generated under a wider range of relaxation parameters (or viscosity settings).

2 Capillary phenomenon in porous media

From the view of microstructure, the capillary rise starts in the collection of juxtaposed particles and presents menisci as the soil corner contacts the wetting liquid (Figure 1).

Figure 1

Capillary rise in corners.

The driving capillary pressure gradient (γ/rh ^r) is balanced by both the forces of gravity and the viscous friction [27]:

(1) γ r h r ∼ ρ g + η h r r 2 .

For short time, gravity can be neglected ( ∂ h r / ∂ r ) r = r L = 0 . According to the Lucas–Washburn behavior, equation can be written in the scaling form:

(2) r L ∼ 1 8 η γ ρ 2 g 2 t 1 3 .

And the dynamics of the capillary rise can be deduced as follows:

(3) h ( t ) ∼ γ 2 t η ρ g 1 3 .

Thus, the capillary pressure is balanced by the inertial forces, the viscous forces, and the hydrostatic pressure.

(4) 2 σ cos θ R = d ( ρ h h ′ ) d t + 8 μ h R 2 h ′ + ρ g h ,

where σ is the surface tension (N/m), R is the inner tube radius (m), ρ is the fluid density (kg/m³), g is gravity (N/kg), and μ is the fluid viscosity (Pa s).

The capillary rise relies on the diameter of particles, viscosity, and dominant force. The whole process can be divided into four stages.

2.1 Purely inertial time stage

For the very first moments after the contact of the porous media with the liquid, the viscous and the gravity term can be neglected, the capillary rise purely dominated by inertial forces, and the capillary rise with constant velocity can be expressed as follows [28]:

(5) 2 σ cos θ R = ρ d ( h h ′ ) d t = ρ h ′ 2 + ρ h h ′ ′

and

(6) h = t 2 σ cos θ ρ R .

The purely inertial flow period shows a rise with constant velocity and linear behavior at the beginning of capillary suction.

2.2 Visco-inertial time stage

The capillary rise influenced by viscosity increases with time. A solution featuring the inertial and viscous term can be expressed in the following differential equation [29]:

(7) d d t ( h h ′ ) + a h h ′ = b .

While

(8) a = 8 μ R 2 ρ ; b = 2 σ cos θ R ρ .

Then

(9) h 2 = 2 b a t − 1 a ( 1 − e − a t ) .

2.3 Purely viscous time stage

When capillary rise affected by inertia vanishes, the flow will become as a pure viscous. Neglecting the influence of inertia and gravity, the intermediate flow period is given by Washburn [30]:

(10) h 2 = σ R cos θ 2 μ t .

2.4 Viscous and gravitational time stage

Gravity can no longer be neglected. Fries and Dreyer [31] showed that gravity had to be considered for h > 0.1. Analytic solutions are given by Washburn in explicit form:

(11) h ( t ) = c d 1 + W − e − 1 − d 2 t c .

While

(12) c = σ R cos θ 4 μ , d = σ g R 2 8 μ ,

where W(x) is the Lambert W function, and c and d are constants.

3 Capillary suction modeling

Ignoring the influence of temperature, one-dimensional unsaturated capillary water rise model can be expressed by the Richards equation [32]:

(13) ∂ θ ∂ t = ∂ ∂ z K ( h ) ∂ h ∂ z + ∂ K ∂ z ,

where θ is the volumetric water content (%), h is the matric suction (kPa), z is the vertical space coordinate, and K(h) is the hydraulic conductivity (mm/s).

Soil–water characteristic curve indicates the relationship between water content and matric suction. According to the B–C model, suction is used as a water coefficient function or a hydraulic conductivity coefficient function. The relationship of water content vs suction and permeability vs suction can be expressed, respectively, as follows:

(14) h < − 1 α , S e = α h − n ,

(15) h ≥ − 1 α , S e = 1 ,

(16) K = K s S e 2 / n + m + 2 ,

(17) S e = θ − θ r θ s − θ r ,

where K _s is the saturated permeability coefficient (mm/s), θ _r is the residual water content (%), θ _s is the saturated water content (%), α is the reciprocal of air entry value, n is the pore size distribution index, m is the connected coefficient (always equal to 1) [33], and S _e is the effective saturation.

Considering the characteristics of soil fractal dimension, porosity, and permeability, the relationship between saturated hydraulic conductivity and water retention parameters in the B–C model was established by Rawls and Brakensiek [34]. Parameters reciprocal of air entry value and pore distribution index can be obtained by measuring the fractal dimension, porosity, and saturated permeability coefficient.

(18) K s = 4.41 × 10 7 ϕ c l 2 R 2 ,

(19) l = 1.86 D 5.34 = 1.86 ( 2 − n ) 5.34 ,

(20) R = 0.148 h α ,

where l is a parameter related to fractal dimension, c is the constant, often value for 4/3, R is the radius of the largest open pore (mm), and h _a is the air entry pressure head (mm).

Therefore, capillary suction model in this article is built on the following assumptions: (i) the height of sand column is 500 mm. (ii) The bottom is immersed completely in water and presents saturated state, but the upper shows dry condition. (iii) The influence of environment temperature and surface evaporation is neglected. During the computational process, the boundary space calculation step length and the gridding number are 0.01 mm and 50,000, respectively, and that of solving time and time step is 36,000 s and 0.01 s. Using the finite difference method to solute discretely the aforementioned equation, it can be expressed as equations (22) and (23):

(22) S e ( z , t = 0 ) = 0.01 ,

(23) S e ( z , t > 0 ) = 1 ( z = 0 ) .

4 Tests

4.1 Micro-CT scanning

Skyscan 1174 compact micro-CT made in Belgium (Figure 2) is used to obtain the parameters fractal dimension, porosity, and water distribution patterns. This scanner uses an X-ray source with adjustable voltage and a range of filters for versatile adaptation to different object densities. A sensitive 1.3 megapixel X-ray camera allows scanning of the whole sample volume in several minutes. Variable magnification (6–30 µm pixel size) is combined with object positioning for easy selection of the object part to be scanned. The scanner can run from any desktop or portable computer, requiring just one USB (or serial) port and a FireWire (IEEE1394) input. The full range of Skyscan software is supplied, including fast volumetric reconstruction and software for 2D/3D quantitative analysis and for realistic 3D visualization.

Figure 2

Operating principle of micro-CT (Type 1174).

4.2 Constant head permeability test

Permeability is measured using ASTM method D2434-68, during the test adjustable constant head reservoir and outlet reservoir are used to maintain a constant head and a loading piston is used to maintain a constant axial stress [35].

Constant head permeability test is carried out to measure the saturated permeability coefficient of four interval sizes washed sand (Figure 3). Sand size (0.15–0.22, 0.30–0.35, 0.40–0.45, and 0.50–0.60 mm) is controlled by sieving. The cylindrical soil sample is 75 mm in diameter and 260 mm in height.

Figure 3

Sand samples of different particle sizes under optical microscope: (a) 0.15–0.22 mm, (b) 0.30–0.35 mm, (c) 0.40–0.45 mm, and (d) 0.50–0.60 mm.

4.3 Capillary absorption test

A capillary rise test of porous media is designed in this article. Washed sand (0.15–0.22, 0.30–0.35, 0.40–0.45, and 0.50–0.60 mm) is chosen as the objective. Sand is washed completely and oven dried so as to eliminate the effect from clay in the sample during the preparation process. Sand sample is stratified, compacted into a grass tube with a length of 500 mm and an inner diameter of 25 mm (Figure 4). Sand materials are divided into three and compacted layer upon layer. The interlayer is roughed with a scraper knife. The bottom of the tube is fixed with filtration fabric and screen. Then, this installation is fixed on an iron stand and immersed partly in the storage water tank. Replenisher is used to maintain a constant water level and overflow to ensure a certain immersed depth of tube. The density of sample is controlled by backfill quality and volume. The parameters capillary rising height and time are recorded, and the wetting front is photographed at the same time.

Figure 4

Schematic diagram of capillary rise test.

5 Results

5.1 Determination of the parameters

Particle size has a significant influence on pore distribution. The quantity of macropore increases with the increase in grain diameter (Figure 5), while the fractal dimension decreases with the increase in particle diameter (Table 1).

Figure 5

Three-dimensional structure of four size washed sand.

Table 1

Fractal dimension and mean porosity of washed sand

Particles size (mm)	Parameters
Particles size (mm)	Fractal dimension	Mean porosity (%)
0.15–0.22	1.42	39.0
0.30–0.35	1.32	42.0
0.40–0.45	1.25	41.0
0.50–0.60	1.17	36.5

The saturated permeability coefficients of four samples are 0.10, 0.22, 0.40, and 1.33 mm/s, respectively. According to equations (18)–(20), B–C model parameters reciprocal of air entry value and pore distribution index are obtained. Saturated water content was deduced by micro-CT porosity scan (Figure 4) and residual water content value was taken to be equal to 0.02 (Table 2).

Table 2

B–C model parameters of four interval size washed sand

B–C parameters	Particles size (mm)
B–C parameters	0.15–0.22	0.30–0.35	0.40–0.45	0.50–0.60
Pore size distribution	0.580	0.680	0.750	0.830
Reciprocal of air entry value (mm⁻¹)	0.014	0.033	0.053	0.083
Saturated water content (m³/m³)	0.390	0.420	0.410	0.365
Residual water content (m³/m³)	0.020	0.020	0.020	0.020

Also, from the gray value determined, three-phase state is distinguished. The apparent distribution pattern could be seen directly through the 3D reconstruction image (Figure 6). Simultaneously, the relationship between the interfacial area of liquid–air and saturation is given in Figure 7, in which the white part represents the grain skeleton, the green part represents the water, and the rest represents air. It could be seen that there is no obvious difference at the lower saturation. At the range of intermediate moisture, the air–water interfacial area of washed sand (0.20–0.30 mm) is above the ones of larger size (0.50–0.60 mm). But the trend is contrary to the former from the cutoff point of 74% to nearly complete saturated.

Figure 6

Three-dimensional reconstruction of solid–liquid phases.

Figure 7

Relationship between air–water interfacial area and saturation.

5.2 Simulated results

The saturation of granular finally reaches a constant situation along with time, and the closer it gets to water source, the quicker it gets wet (Figure 8). The variation gradient of water content vs time is getting slower successively with the increase in height and grain diameter. Particle size has obvious effect on capillary rise. The rising time of smaller and larger grain is shorter than that of intermediate one near water source. But, it is different when the rising increases with the increase in grain diameter. Taking the height of 200 mm, for example, the time rise of small grain (0.15–0.22 mm) need 2,931 s, but that of larger ones are not monitored until 35,971 s.

Figure 8

Relationship between moisture distribution and time: (a) h = 20 mm, (b) h = 50 mm, (c) h = 100 mm, (d) h = 150 mm, and (e) h = 200 mm.

In Figure 8a, it can be found that the capillary rates of three types of sand are not consistent with particle size except the minimum one in the virtual box. The position of 20 mm, which is at the early stage of capillary rise, shows that the phenomenon is caused by piecewise functions of (14–15). To some extent, the inertia force plays a significant role in large media channels at the initial phase. Also, according to statistical analysis of air–water interfacial area under different saturation conditions for two kinds of washing sand, it is found that there is cross in the larger saturation range. Joekar-Niasar et al. [36] indicated that the larger the air–water interfacial area, the smaller the capillary suction at the same saturation. The process of capillary rise depends on not only the particle size and pore characteristics but also the distribution phase of the fluid medium. Therefore, the experimental results explain the necessity of segmented function in the saturated-larger saturation range.

The location wetted by liquid at a time is defined as the height of wetting front. Particle size has obvious effect on wetting front (Figure 9), which lowers with the increase in grain size at the same time. The moisture gradient changes from steeply to gently along with time. The capillary height of the washed sand size of 0.15–0.22 mm is 320 mm after 10 h, but of 0.50–0.60 mm is only 130 mm. Moister of sand column near water source is stable latterly and presents a kind of overlap state. Such a region adsorbs little water and plays a role of water transfer in later stages.

Figure 9

Moisture distribution vs time of four samples: (a) 360 s, (b) 3,600 s, (c) 14,400 s, and (d) 36,000 s.

5.3 Comparison of the results of capillary absorption test and calculation with the parameters

The capillary rise results of calculation and testing are in good agreement (Figure 10). The change of height increases with time and later closes to stability gradually. The deviation degree of specimen composed of small grains is higher than that with large grains, and the tested value of small grains specimen is greater than the calculated one, which accounts for the cohesiveness force caused from tiny clay particle residual and the small passageway between grains. Capillary rise height increased with reducing passageway diameter. Generally, model deduced in this article can reflect actually the phenomenon of capillary rise for porous media.

Figure 10

Capillary rise results of calculation and testing.

6 Conclusion

Micro-CT technology is introduced to predict the capillary suction in porous media like sand. It indicates that parameters air entry pressure and pore size distribution index can be obtained acceptably by the micro-CT scanning, and the model can be well used to predict and analyze the dynamic capillary rise process of porous media together with theoretical deduction with experimental verification.

Capillary pressure is balanced by the inertial forces, the viscous forces, and the hydrostatic pressure, and capillary rise depends on the dominant force. Particle size has obvious effect on capillarity, and the wetting front lowers with the increase in grain size and the decrease in rising rate. Region near water source is wetted quickly and then saturated and stable latterly. This region presents a kind of overlap state and plays a role of water transfer. At the early capillary rise, the migration rate does not entirely correspond to the particle size. From the statistical analysis of air–water interfacial area obtained by tomography for two different sizes washing sand, the values were cross under the conditions of lager saturation. It is also proved that the capillary suction is decided by the aperture size and phase morphology simultaneously.

Generally, dynamic capillary process of porous media can be well predicted with micro-CT technology and B–C model. The model suggested has explicit meaning and convenient implementation. The achievements of this research can provide some reference value for quantizing the capillary rise along with different saturations in the future study.

Acknowledgments

This project was supported by the National Natural Science Foundation of China (51978235), the Natural Science Foundation of Hebei (E2018202274), Technology Innovation Strategy Foundation of Hebei Province (405211), and the Natural Science Foundation of Tianjin (17JCZDJC39200). The authors are grateful to the Hebei Research Center of Civil Engineering Technology for its support.

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Received: 2020-08-11

Revised: 2020-10-07

Accepted: 2020-10-23

Published Online: 2020-12-05

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2020-0203

Keywords for this article

micro-CT; fractal dimension; Brooks–Corey model; porous media; capillary suction

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