Research Article

Min Li*, Hemiao Yu, and Hongpu Du

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

https://doi.org/10.1515/phys-2020-0203received August 11, 2020; accepted October 23, 2020

Abstract: Moisture variation in porous media dependsmainly on the pore characteristics. This article used themicro-computed tomography (micro-CT) (a non-destruc-tive imaging technique to generate a three-dimensionalvirtual model) and the Brooks–Corey model to deduce themoisture migration in sand. Relationship between capil-lary rise height and time (h–t)was achieved by numericalsimulation in the capillary suction process, where theparameters fractal dimension, porosity, and air–waterinterfacial area were obtained by the micro-CT scanning.Meanwhile, experiments of capillary rise in sand columnwere 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 theaperture size and phase morphology simultaneously, andparticle size has obvious effect on capillarity, and thewetting front lowers with the increase in grain size andthe decrease in rising rate. Parameters air entry pressureand pore-size distribution index obtained by micro-CTscanning technology and empirical formula are accurate.Method of combing micro-CT images and Brooks–Coreymodel can predict well the capillary suction of porousmedia. It is also proved that the capillary suction isdecided by the aperture size and phase morphologysimultaneously.

Keywords: micro-CT, fractal dimension, Brooks–Coreymodel, porous media, capillary suction

1 Introduction

A porous medium contains many pores (voids) filledtypically with a fluid (liquid or gas), and the skeletalportion of the material is often called matrix. Many nat-ural substances such as rocks and soil are porous media.Engineering properties of the materials (e.g., perme-ability, tensile strength, and electrical conductivity)depend on their constituents, porosity, and pore struc-ture mainly. Because of non-homogeneity and randomdistribution, fluid flow in these media attracts muchattention and brings up an extensive study [1,2] such asrainfall filtration, subgrade, slope stability, and energystorage in aquifer [3,4].

Capillary action is the ability of a liquid flowing innarrow spaces without the assistance of and in opposi-tion to external forces like gravity [5]. Capillary rise canbe seen as the adhesive forces between walls of capillarytube and fluid [6]. They prompt the edges of the fluidupwards, while the surface tension (cohesive forces) con-stantly pulls molecules from the surface inward andholds the molecules together. The aforementioned twoforces fight against with each other, and then the fluidwill stop rising when the net force caused by the weightof the fluid column is greater than that of cohesive forces.The capillary rise is based mainly on measurements ofcapillary pressure difference or velocity of liquid penetra-tion. Bartell and Whitney [7] were the authors whoadvanced this method in research. Washburn [8] intro-duced the velocity of penetration based on the assumptionof n-cylindrical capillary pipes. The Washburn equationwas used frequently to model the capillary rise in porousmedia. However, capillary process also pointed out thatonly the rise curve agreed initially with the Washburnequation [9]. Xue et al. [10] indicated that porous mate-rials behave more like a bundle of capillaries and thecapillary rise was no longer considered as an approxi-mate analytical tool. Mualem [11] and Siebold [12] alsoverified that the Washburn equation could not provide afine-tuned and precise information on surface modification

* Corresponding author: Min Li, School of Civil and TransportationEngineering, Hebei University of Technology, Tianjin 300401, China;Hebei Research Center of Civil Engineering Technology, Tianjin300401, China, e-mail: limin0409@hebut.edu.cnHemiao Yu: School of Civil and Transportation Engineering, HebeiUniversity of Technology, Tianjin 300401, ChinaHongpu Du: Project Management Department, Henan InvestmentGroup Co., Ltd, Zhengzhou, 450008, China

Open Physics 2020; 18: 906–915

Open Access. © 2020 Min Li et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 InternationalLicense.

or on different kinds of particles. The observed under-prediction of front position was due to the neglect ofdynamic saturation gradients. Lockington and Parlange[13] put forward an analytical formula for the position offront, speed of propagation, and cumulative uptake.However, most of these models were based upon theassumption of symmetrical capillary tube and regulargeometrical cross section and cannot describe the capil-lary permeability in reality.

Richards model describing the permeable process ofunsaturated porous media was derived from the basisof potential function and applied widely in the field ofagriculture and rainfall [14]. The key to the mathema-tical model depended on the relationship betweenhydraulic conductivity and water content. However,such parameters were obtained only by data fittingand deducing basic property. However, the maneuver-ability of experiment and complexity of deductionrestricted the application.

X-ray computed tomography (CT) was a medical ima-ging procedure during the seventies of last century.It utilizes computer-processed X-rays to produce tomo-graphic images of body’s specific areas. Digital geometryprocessing can generate a three-dimensional (3D) imageinside the object from a large series of two-dimensional(2D) images taken from a single axis of rotation. CT canproduce a volume of manipulated data through a processknown as “windowing” and then demonstrate variousbody structures based on their ability to block the X-raybeam. CT scanner allows the volume of data to be refor-matted in volumetric (3D) representations and is usedgradually in other fields such as non-destructive mate-rials 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 throughthe 3D reconstruction and then studied its permeabilityaccording to the multipoint statistical theory. Seol andKneafsey [19] used X-ray CT to image and quantify theeffect of a heterogeneous sand grain-size distribution onthe formation and dissociation of methane hydrate. Bera[20] analyzed the pore structure distribution features ofsandstone with the aid of 3D microstructure. As for themechanisms leading to capillary collapse in the looseunsaturated sand, Bruchon et al. [21] had research on itapplying X-ray CT. The microstructure was analyzed toassess the volume of water filling the pores and deforma-tion of the granular skeleton using Volumetric DigitalImage Correlation tools. According to the trinarizationof micro-CT images for partially saturated sand, Higoet al. [22] discussed the form of the existing pore water

at different water-retention states. Zhou et al. [23] hadanalyzed a new CT method to study pollutant migrationin unsaturated sand by scanning the specimens. Theresults show that the computerized tomography is a fea-sible method for studying the characteristics of KI solu-tion in unsaturated sands. Generally, CT technology canreflect the internal structure of porous media accurately.

CT technology and Brooks–Corey model are intro-duced to predict the capillary suction in porous medialike sand. Parameters mean grain size, porosity, andfractal dimension related to the hydraulic conductivityare 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 thelarger size and smaller is analyzed at the same saturation.Therefore, capillary suction model can be derived gradu-ally by the expression of water retention in the B–Cmodel. In order to verify the model scientifically, experi-ments of capillary rise were performed synchronouslyusing four different interval sizes of screening washedsand, and capillary rise heights at different time wereobserved.

The use of micro-scale simulation method could bebetter to analyze the properties of porous media and thestate of internal fluid. Aldakheel [24] studied the micro-scale model of concrete failure in pore elastic–plasticmedium and found that regarding the concrete solidskeleton at the microscale, only the hydrated cementpaste undergoes elastic–plastic–fracture deformations.Saenger et al. [25] carried out micro-scale finite-differ-ence simulation of wave propagation, including porescale propagation of elastic wave in digital rock sampleand dynamic elastic characteristics of viscous saturatedrock. Gao et al. [26] carried out 3D micro-scale flow simu-lation in porous media of saturated soil. When usingMRT-LBM for flow simulation, stronger steady-stateviscous flow solutions could be generated under a widerrange of relaxation parameters (or viscosity settings).

2 Capillary phenomenon in porousmedia

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

The driving capillary pressure gradient (γ/rhr) isbalanced by both the forces of gravity and the viscousfriction [27]:

Prediction of capillary suction in porous media based on micro-CT technology and B–C model 907

∼ +

γrh

ρg η hr

.r

r2 (1)

For short time, gravity can be neglected((∂ /∂ ) = )

=

h r 0r rr L . According to the Lucas–Washburnbehavior, equation can be written in the scaling form:

∼r ηγ

ρ g t18

.L 2 2

13 (2)

And the dynamics of the capillary rise can bededuced as follows:

( ) ∼h t γ t

ηρg.

213 (3)

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

=

( ′)

+ ′ +

σ θR

ρhht

μhR

h ρgh2 cos dd

8 ,2(4)

where σ is the surface tension (N/m), R is the inner tuberadius (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 bedivided into four stages.

2.1 Purely inertial time stage

For the very first moments after the contact of the porousmedia with the liquid, the viscous and the gravity termcan be neglected, the capillary rise purely dominatedby inertial forces, and the capillary rise with constantvelocity can be expressed as follows [28]:

=

( )

= ′ +′

′

′

σ θR

ρd hht

ρh ρhh2 cosd

2 (5)

and

=h t σ θρR

2 cos . (6)

The purely inertial flow period shows a rise with con-stant velocity and linear behavior at the beginning ofcapillary suction.

2.2 Visco-inertial time stage

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

( ′) + ′ =

thh ahh bd

d. (7)

While

= =a μR ρ

b σ θRρ

8 ; 2 cos .2 (8)

Then

= − ( − )

−h ba

ta

2 1 1 e .at2 (9)

2.3 Purely viscous time stage

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

=h σR θμ

tcos2

.2 (10)

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. Analyticsolutions are given by Washburn in explicit form:

( ) = + −

− −h t cd

W1 e .tc1 d2

(11)

While

= =c σR θμ

d σgRμ

cos4

,8

,2

(12)

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

x

y

D/2

Quadratic corner y = x2/D

x

y

Figure 1: Capillary rise in corners.

908 Min Li et al.

3 Capillary suction modeling

Ignoring the influence of temperature, one-dimensionalunsaturated capillary water rise model can be expressedby the Richards equation [32]:

∂

∂

=

∂

∂

( )

∂

∂

+

∂

∂

θt z

K h hz

Kz

, (13)

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

Soil–water characteristic curve indicates the relation-ship between water content and matric suction. Accordingto the B–C model, suction is used as a water coefficientfunction or a hydraulic conductivity coefficient function.The relationship of water content vs suction and perme-ability vs suction can be expressed, respectively, as follows:

< − = ∣ ∣

−hα

S αh1 , ,ne (14)

≥ − =hα

S1 , 1,e (15)

=

/ + +K K S ,n ms e

2 2 (16)

=

−

−

S θ θθ θ

,er

s r(17)

where Ks is the saturated permeability coefficient (mm/s), θris the residual water content (%), θs is the saturated watercontent (%), α is the reciprocal of air entry value, n is thepore size distribution index, m is the connected coefficient(always equal to 1) [33], and Se is the effective saturation.

Considering the characteristics of soil fractal dimen-sion, porosity, and permeability, the relationship betweensaturated hydraulic conductivity and water retentionparameters in the B–C model was established by Rawlsand Brakensiek [34]. Parameters reciprocal of air entryvalue and pore distribution index can be obtained bymeasuring the fractal dimension, porosity, and saturatedpermeability coefficient.

= ×K ϕ

lR4.41 10 ,s

7c

22 (18)

= = ( − )l D n1.86 1.86 2 ,5.34 5.34 (19)

=Rh

0.148 ,α

(20)

where l is a parameter related to fractal dimension, c is theconstant, often value for 4/3, R is the radius of the largestopen pore (mm), and ha is the air entry pressure head (mm).

Therefore, capillary suction model in this article isbuilt on the following assumptions: (i) the height of

sand column is 500mm. (ii) The bottom is immersedcompletely in water and presents saturated state, butthe upper shows dry condition. (iii) The influence of en-vironment temperature and surface evaporation is ne-glected. During the computational process, the boundaryspace calculation step length and the gridding numberare 0.01 mm and 50,000, respectively, and that of solvingtime and time step is 36,000 s and 0.01 s. Using the finitedifference method to solute discretely the aforementionedequation, it can be expressed as equations (22) and (23):

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

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

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, por-osity, and water distribution patterns. This scanner usesan X-ray source with adjustable voltage and a range offilters for versatile adaptation to different object densi-ties. A sensitive 1.3 megapixel X-ray camera allows scan-ning of the whole sample volume in several minutes.Variable magnification (6–30 µm pixel size) is combinedwith object positioning for easy selection of the objectpart to be scanned. The scanner can run from any desktopor portable computer, requiring just one USB (or serial)port and a FireWire (IEEE1394) input. The full range ofSkyscan software is supplied, including fast volumetricreconstruction and software for 2D/3D quantitative ana-lysis and for realistic 3D visualization.

4.2 Constant head permeability test

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

Constant head permeability test is carried out tomeasure the saturated permeability coefficient of fourinterval sizes washed sand (Figure 3). Sand size (0.15–0.22,0.30–0.35, 0.40–0.45, and 0.50–0.60mm) is controlled bysieving. The cylindrical soil sample is 75mm in diameter and260mm in height.

Prediction of capillary suction in porous media based on micro-CT technology and B–C model 909

4.3 Capillary absorption test

A capillary rise test of porous media is designed in thisarticle. Washed sand (0.15–0.22, 0.30–0.35, 0.40–0.45,and 0.50–0.60mm) is chosen as the objective. Sand iswashed completely and oven dried so as to eliminatethe effect from clay in the sample during the preparationprocess. Sand sample is stratified, compacted into agrass tube with a length of 500mm and an inner diameterof 25 mm (Figure 4). Sand materials are divided intothree and compacted layer upon layer. The interlayer isroughed with a scraper knife. The bottom of the tube isfixed with filtration fabric and screen. Then, this installa-tion is fixed on an iron stand and immersed partly in the

storage water tank. Replenisher is used to maintain aconstant water level and overflow to ensure a certainimmersed depth of tube. The density of sample is con-trolled by backfill quality and volume. The parameterscapillary rising height and time are recorded, and thewetting front is photographed at the same time.

5 Results

5.1 Determination of the parameters

Particle size has a significant influence on pore distribu-tion. The quantity of macropore increases with theincrease in grain diameter (Figure 5), while the fractaldimension decreases with the increase in particlediameter (Table 1).

Figure 2: Operating principle of micro-CT (Type 1174).

Figure 3: Sand samples of different particle sizes under opticalmicroscope: (a) 0.15–0.22mm, (b) 0.30–0.35 mm,(c) 0.40–0.45mm, and (d) 0.50–0.60mm. Figure 4: Schematic diagram of capillary rise test.

910 Min Li et al.

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

Also, from the gray value determined, three-phasestate is distinguished. The apparent distribution patterncould be seen directly through the 3D reconstructionimage (Figure 6). Simultaneously, the relationship be-tween the interfacial area of liquid–air and saturation isgiven in Figure 7, in which the white part represents thegrain skeleton, the green part represents the water, andthe rest represents air. It could be seen that there is noobvious difference at the lower saturation. At the range ofintermediate moisture, the air–water interfacial area ofwashed sand (0.20–0.30mm) is above the ones of largersize (0.50–0.60mm). But the trend is contrary to theformer from the cutoff point of 74% to nearly completesaturated.

5.2 Simulated results

The saturation of granular finally reaches a constant si-tuation along with time, and the closer it gets to watersource, the quicker it gets wet (Figure 8). The variationgradient of water content vs time is getting slower suc-cessively with the increase in height and grain diameter.Particle size has obvious effect on capillary rise. The risingtime of smaller and larger grain is shorter than that ofintermediate one near water source. But, it is differentwhen the rising increases with the increase in grain dia-meter. Taking the height of 200mm, for example, the timerise of small grain (0.15–0.22mm) need 2,931 s, but that oflarger ones are not monitored until 35,971 s.

In Figure 8a, it can be found that the capillary rates ofthree types of sand are not consistent with particle sizeexcept the minimum one in the virtual box. The positionof 20mm, which is at the early stage of capillary rise,shows that the phenomenon is caused by piecewise func-tions of (14–15). To some extent, the inertia force plays asignificant role in large media channels at the initialphase. Also, according to statistical analysis of air–waterinterfacial area under different saturation conditions fortwo kinds of washing sand, it is found that there is crossin 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

Figure 5: Three-dimensional structure of four size washed sand.

Table 1: Fractal dimension and mean porosity of washed sand

Particles size (mm) Parameters

Fractal dimension Mean porosity (%)

0.15–0.22 1.42 39.00.30–0.35 1.32 42.00.40–0.45 1.25 41.00.50–0.60 1.17 36.5

Table 2: B–C model parameters of four interval size washed sand

B–C parameters Particles size (mm)

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 0.830Reciprocal of air entry value (mm−1) 0.014 0.033 0.053 0.083Saturated water content (m3/m3) 0.390 0.420 0.410 0.365Residual water content (m3/m3) 0.020 0.020 0.020 0.020

Figure 6: Three-dimensional reconstruction of solid–liquid phases.

Prediction of capillary suction in porous media based on micro-CT technology and B–C model 911

particle size and pore characteristics but also the distri-bution phase of the fluid medium. Therefore, the experi-mental results explain the necessity of segmented functionin the saturated-larger saturation range.

The location wetted by liquid at a time is defined asthe height of wetting front. Particle size has obvious effecton wetting front (Figure 9), which lowers with the

increase in grain size at the same time. The moisturegradient changes from steeply to gently along withtime. The capillary height of the washed sand size of0.15–0.22 mm is 320mm after 10 h, but of 0.50–0.60mmis only 130mm. Moister of sand column near water sourceis stable latterly and presents a kind of overlap state. Sucha region adsorbs little water and plays a role of watertransfer in later stages.

5.3 Comparison of the results of capillaryabsorption test and calculation with theparameters

The capillary rise results of calculation and testing arein good agreement (Figure 10). The change of heightincreases with time and later closes to stability gradually.The deviation degree of specimen composed of smallgrains is higher than that with large grains, and thetested value of small grains specimen is greater thanthe calculated one, which accounts for the cohesivenessforce caused from tiny clay particle residual and the smallpassageway between grains. Capillary rise height increasedwith reducing passageway diameter. Generally, model

0 10 20 30 40 50 60 70 80 90 1000.0

0.1

0.2

0.3

0.4

Inte

rfac

ial A

rea

(m2 /m

3

(

Saturation (%)

0.5-0.6mm 0.2-0.3mm

Figure 7: Relationship between air–water interfacial area andsaturation.

Figure 8: Relationship between moisture distribution and time: (a) h = 20mm, (b) h = 50mm, (c) h = 100mm, (d) h = 150 mm, and(e) h = 200mm.

912 Min Li et al.

deduced in this article can reflect actually the phenom-enon of capillary rise for porous media.

6 Conclusion

Micro-CT technology is introduced to predict the capillarysuction in porous media like sand. It indicates that para-meters air entry pressure and pore size distribution indexcan be obtained acceptably by the micro-CT scanning,and the model can be well used to predict and analyzethe dynamic capillary rise process of porous mediatogether with theoretical deduction with experimentalverification.

Capillary pressure is balanced by the inertial forces,the viscous forces, and the hydrostatic pressure, andcapillary rise depends on the dominant force. Particlesize has obvious effect on capillarity, and the wetting

front lowers with the increase in grain size and thedecrease in rising rate. Region near water source iswetted quickly and then saturated and stable latterly.This region presents a kind of overlap state and plays arole of water transfer. At the early capillary rise, themigration rate does not entirely correspond to the particlesize. From the statistical analysis of air–water interfacialarea obtained by tomography for two different sizeswashing sand, the values were cross under the conditionsof lager saturation. It is also proved that the capillarysuction is decided by the aperture size and phasemorphology simultaneously.

Generally, dynamic capillary process of porous mediacan be well predicted with micro-CT technology and B–Cmodel. The model suggested has explicit meaning andconvenient implementation. The achievements of thisresearch can provide some reference value for quantizingthe capillary rise along with different saturations in thefuture study.

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.

Prediction of capillary suction in porous media based on micro-CT technology and B–C model 913

Acknowledgments: This project was supported by theNational Natural Science Foundation of China (51978235),the Natural Science Foundation of Hebei (E2018202274),Technology Innovation Strategy Foundation of HebeiProvince (405211), and the Natural Science Foundation ofTianjin (17JCZDJC39200). The authors are grateful to theHebei Research Center of Civil Engineering Technology forits support.

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Prediction of capillary suction in porous media based on micro-CT technology and B–C model 915