# Calculation of Capillary Rise Height of Soils by SWCC Model

Authors: Yuhan Li; Chen Zhang; Chenghao Chen; Hao Chen
Journal: Advances in Civil Engineering
 (2018)

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## Abstract

The maximum capillary rise height of soil is a complex system which is mainly determined by the distribution characteristics of soil pores. The tests of the rising height of capillary water on 8 kinds of soils by the method of vertical tube are widely conducted to measure the maximum capillary rise height. Based on the BCC model and principles of thermodynamics, the soil-water characteristic curve test is designed for the purpose of calculating the pore distribution of soil samples. A new method for calculating the maximum capillary rise height of soil is proposed by the author by using the distribution function of the soil pore. The coefficientβ which reflects the relationship between the maximum capillary rise and the average pore radius of soils is utilized during the calculation process, and then the reference range of β for different soils is obtained according to series of experiments corresponding. The proposed calculation method offers an effective way to calculate the maximum capillary rise height, which can be applied to analyze the capillary effect area of relevant engineering problems.


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## Body

## 1. Introduction

Capillary-driven liquid flow is the main transport mechanism in the soil system of which the water erodes continuously by capillary rise from a lower elevation to higher elevation. Such capillary rise phenomena lead to an increase of the saturation of the soil, which will not only decrease the strength of the soil but also alter the elastic modulus of substructure soils, thereby leading to the corresponding changes in stress and strain response under the external load, for example, the traffic load. Therefore, the defect of the roadbed is closely related to capillary rise erosion. Seasonally frozen ground has always been an important problem in highway construction and channel slope at high latitudes. In the research of frost heaving zone, it is inevitable to determine the height of capillary rise of the substructure. For channel slope and embankment close to the riverside, capillary action path is shortened as well as the effect of capillary action on the supply of water is accelerated. As a result, these infrastructures are more prone to defect and frost heave in such particular areas because of capillary rise. To sum up, study of capillary rise, particularly the maximum capillary rise height, is of great significance to the design of the substructure and channel because the maximum capillary rise height is tightly connected to the strength reduction region and frost area.A series of studies have been carried out on the capillary rise; LU offered a complete analytical solution for the relationship between the rate and time of capillary rise in soils [1]. However, in the equation, the maximum height of capillary rise hc is a known soil parameter. In other words, the accurate determination of the maximum height of capillary rise is the prerequisite to calculate the rate of capillary rise. The maximum height of capillary rise has an important influence on the overall engineering behavior of unsaturated soils and is a highly complex system of both the soil and pore water properties. It is extremely difficult to calculate the maximum capillary height accurately in real soils. In order to overcome this, scholars have done a lot of studies and have put forward some empirical formulas which established the correlation between the maximum capillary height and certain measured soil parameters. The earliest formula was proposed by Lane and Washburn [2, 3], after conducting the capillary rising test for 8 kinds of different soils, and the result shows that the maximum capillary height is linear with lnD10:(1)hc=−990lnD10−1540.On the basis of (1), Peak and Hansen put forward another empirical formula [4]:(2)hc=CeD10.In this formula, 10% particle sized10, void ratio e, and the coefficient CC=3×10−5m2∼8×10−5m2 are used to calculate the maximum capillary rise height. However, the coefficient C is difficult to choose because the range of C is relatively large, and the value of C depends hugely on the real condition of soil. Kumar and Malik by carrying out indoor tests summarized as follows [5]:(3)hc=ha+134.84−5.16r,where ha is the height corresponding to air-entry pressure value of soils and r is the equivalent of capillary radius of soils. In this formula, the unit of r is μm, so it has little influence on the result of the calculation. Therefore, ha is the only key computational parameter which can be determined by the soil-water characteristic curve. However, it requires an instrument to control the matric suction very precisely especially when the suction is less than 1 kPa [6].We summarized that according to previous research results, using the micropore distribution of soils and using the pore radius parameter instead of the gradation parameter and void ratio are effective methods to accurately calculate the maximum capillary rise height. In this paper, a new method is proposed to calculate the maximum capillary rising height by using the radius of microscopic pore distribution. A large number of capillary rise tests as well as soil-water characteristic (SWCC) tests have been done for different kinds of soils, from which the parameters of different soils required for the calculation are obtained, and the feasibility of this method will be verified.


## 2. Theoretical Framework

### 2.1. Capillary Rise Equilibrium Equation in Soil System

In a single capillary when water column reaches the maximum height, the gravity of the water column is balanced by the surface tension along water-solid interface, as shown in Figure1. The balance equation is as follows:(4)hcρwgπr2=TSπdcosα,where α is the liquid-solid contact angle; ρw is the density of water; r is the radius of the capillary channel (r=d/2); and TS is the surface tension of water.Figure 1
Mechanical equilibrium of capillary rise in the small diameter tube.For soils, we regard it as homogenous porous media, which means the pores inside are similar in diameter and the dispersion is small. In the interior of soils, tiny pores which are connected with each other fulfill the precondition of capillary rise [7, 8]. The total capillary gas-liquid area A in each cross section can be expressed as follows:(5)A=ϕAπ∫0∞rfrdr2,where ϕA is the number of gas-liquid interfaces in total capillary gas-liquid area A of each cross section and fr is pore size distribution (PSD) function in the porous system of soils [9, 10]. When we established the gravity-tension balance equations assuming that soil pores are interconnected with each other, the capillary migration path of water is not like a single capillary tube, meaning that channel interleaving was bent and most of the gas-liquid interfaces on the same section are connected. As a simplified model, we argue that the total capillary route is equal to the maximum capillary rise height hc multiplied by a path retardation coefficient β; thereafter, the capillary equilibrium equation for any gas-liquid interface can be modified from (4) to(6)hcρwgπ∫0∞rfrdr2β=2TSπ∫0∞rfrdr,(7)hc=2TScosαρωgβ∫0∞rfrdr.Whenρw=1 g/cm3, g=980 cm/m2, TS=72 mN/m, and α=0° (dehydration process), (7) can be simplified as follows:(8)hc=0.15β∫0∞rfrdr.


### 2.2. Testing Method of PSDfr

One effective way to study the pore characteristics of homogenous porous media is to establish the microscopic pore structure model. Many conventional models for pore characteristics, liquid distribution, flow, and transport in porous media are based on representing pore space geometry as a bundle of cylindrical capillaries (BCC model) [11, 12]. In the literature, there have been some standardized experimental methods for directly measuring pore size distribution (MIP, mercury intrusion porosimetry; BM, bubble method), and these methods are based on the basic theory of the BCC model. In BCC model, according to the Young–Laplace equation, the relationship between the pore radius r and the suction S (or capillary pressure) is represented as follows:(9)S=2TScosαr.Based on the assumption of pore local equilibrium, the balance of soil is achieved in a given suctionS∗ inside the pore while one part of the pore with the pore radius greater than r∗r∗=2Tscosα/S∗ is filled with air and the other one with pore radius less than r∗ is filled with water. By defining gr as pore size distribution function, grdr represents the percentage of void volume with pore radius ranging from r to r+dr under the unit soil volume. When the suction of the soil reaches S, the maximum pore radius which is filled with water becomes C/SC=2TScosα. Therefore, the relationship between soil volumetric water content and pore distribution can be expressed as follows:(10)dθ=grdr.In this formula,θ indicates the volumetric water content of soil when soil suction equals S and dθ represents the ratio of pore water volume with soil particle radius between r and r+dr to the total pore water volume under a well-saturated condition. In fact, in this formula, the volume of water being evacuated is used as an alternative to that of void in the soil. When the calculated radius is in the range of the whole pore medium, gr can be equal to fr.In this paper, the pore size distribution of soil was measured, and the methods mentioned above have some limitations when being applied. Taking MIP as an example, sample preparation for MIP is difficult, and under high pressure, liquid mercury will jeopardize soil particles, resulting in the deformation of soil skeleton and pore during the test. Another important reason which restricts the widespread use of conventional methods is the low popularity of test instruments as well as high test costs, which forced us to look for other convenient testing methods. In this case, mercury is replaced by water, and the soil-water characteristic curve is utilized instead of the mercury injection curve. Pores among soil particles rather than the internal pore of the particle are mainly considered with regard to the soil capillary phenomenon and the pore water flow problem. When the soil-water characteristic curve is used, the pressure can be increased to 2 MPa, which means the pore radius can be attached to 1000 Å correspondingly. In other words, it can obtain the exact pore distribution of the above 1000 Å, allowing us to have enough pore radius distribution data to analyze soil capillary phenomenon.


### 2.3. Basic Theories and Calculation Method

The pore of soil can be regarded as a capillary in the Kelvin formula without violating original characteristics of soil system. Radius, contact angle between soil particles and pore water, and surface tension are denoted byr, α, and TS, respectively. When using measurable physical quantities, Kelvin formula is expressed as(11)2TSvωcosαr=−RTlnuv1uv0=μ1−μ0,where form of matric suction is described as(12)r=2TScosαua−uw,where R = 8.314 J/(mol·K) is the universal gas constant; T = 298 K is the thermodynamic temperature; μvo (kPa) is the free water saturated vapor pressure in equilibrium state under the condition of temperature T. μv1 (kPa) is the vapor pressure value of solution at the state of balance; vm = 18 × 10−6  m3/mol is the partial molar volume of water vapor. ua−uw is the matric suction, uv1/uv0 is defined as relative humidity (RH) and can be obtained from (13) which is another expression of (11):(13)RH=exp−ua−uwvwRT.When the relative humidity or matric suction is in the increment of Stepi, the volume change of the pore volume of soil mass which is replaced by the gas can be calculated as(14)∇vpi=∇wiρw.Kelvin radius can be estimated by (12), while the actual pore radius is expressed as(15)rpi=rki+ti,where the thickness of adsorbed water film ti corresponding to RH or matric suction at Step i can be expressed by Halsey equation as [13](16)ti=τ−5lnRHi1/3,where τ is the effective diameter of adsorbed water molecules. Assuming that the cross-sectional area occupied by the water molecules is approximately 10.8 Å and the Avogadro constant NA is 6.02 × 1023/mol, the value of τ can be calculated as(17)τ=vwANA=2.77Å.After terminating the calculation of pore radius which corresponds to different suctions, points can be selected in the SWCC to calculate the pore size distribution curve. Considering the nonuniform pore size and inhomogeneous pore distribution as well as the data of pore distribution curve obtained by point-selecting method,r0 as average pore radius of soil by using weighted average of the statistical theory is defined as(18)r0=∑vprpi∑rpi.Similar to the parameters ofd10, d60, and Cu found in the particle size distribution curve, the average capillary pore radius can be obtained by pore size distribution curve. According to the definition, r0 represents the average radius within the calculating range. For example, if suction value ranges from 1 kPa to 10000 kPa, then the r0 is the average radius of the pore which falls into the corresponding range.When the calculated aperture range relates to the whole soil (suction value ranges from the air-entry pressure valueSS to the residual suction Sr), we can approximate r0 as r0=∑i=1nri/n, by ∑i=1nri≈nEr where(19)Er=∫0∞rfrdr.We can easily argue that(20)∫0∞rfrdr≈r0.By conducting substitution of (20) to (8), (8) can be simplified as follows:(21)hc=0.15βr0.As a parameter, average pore radiusr0 characterizing the pore state can be obtained by SWCC, and path retardation coefficient β as an empirical parameter will be measured by the following tests.


## 2.1. Capillary Rise Equilibrium Equation in Soil System

In a single capillary when water column reaches the maximum height, the gravity of the water column is balanced by the surface tension along water-solid interface, as shown in Figure1. The balance equation is as follows:(4)hcρwgπr2=TSπdcosα,where α is the liquid-solid contact angle; ρw is the density of water; r is the radius of the capillary channel (r=d/2); and TS is the surface tension of water.Figure 1
Mechanical equilibrium of capillary rise in the small diameter tube.For soils, we regard it as homogenous porous media, which means the pores inside are similar in diameter and the dispersion is small. In the interior of soils, tiny pores which are connected with each other fulfill the precondition of capillary rise [7, 8]. The total capillary gas-liquid area A in each cross section can be expressed as follows:(5)A=ϕAπ∫0∞rfrdr2,where ϕA is the number of gas-liquid interfaces in total capillary gas-liquid area A of each cross section and fr is pore size distribution (PSD) function in the porous system of soils [9, 10]. When we established the gravity-tension balance equations assuming that soil pores are interconnected with each other, the capillary migration path of water is not like a single capillary tube, meaning that channel interleaving was bent and most of the gas-liquid interfaces on the same section are connected. As a simplified model, we argue that the total capillary route is equal to the maximum capillary rise height hc multiplied by a path retardation coefficient β; thereafter, the capillary equilibrium equation for any gas-liquid interface can be modified from (4) to(6)hcρwgπ∫0∞rfrdr2β=2TSπ∫0∞rfrdr,(7)hc=2TScosαρωgβ∫0∞rfrdr.Whenρw=1 g/cm3, g=980 cm/m2, TS=72 mN/m, and α=0° (dehydration process), (7) can be simplified as follows:(8)hc=0.15β∫0∞rfrdr.


## 2.2. Testing Method of PSDfr

One effective way to study the pore characteristics of homogenous porous media is to establish the microscopic pore structure model. Many conventional models for pore characteristics, liquid distribution, flow, and transport in porous media are based on representing pore space geometry as a bundle of cylindrical capillaries (BCC model) [11, 12]. In the literature, there have been some standardized experimental methods for directly measuring pore size distribution (MIP, mercury intrusion porosimetry; BM, bubble method), and these methods are based on the basic theory of the BCC model. In BCC model, according to the Young–Laplace equation, the relationship between the pore radius r and the suction S (or capillary pressure) is represented as follows:(9)S=2TScosαr.Based on the assumption of pore local equilibrium, the balance of soil is achieved in a given suctionS∗ inside the pore while one part of the pore with the pore radius greater than r∗r∗=2Tscosα/S∗ is filled with air and the other one with pore radius less than r∗ is filled with water. By defining gr as pore size distribution function, grdr represents the percentage of void volume with pore radius ranging from r to r+dr under the unit soil volume. When the suction of the soil reaches S, the maximum pore radius which is filled with water becomes C/SC=2TScosα. Therefore, the relationship between soil volumetric water content and pore distribution can be expressed as follows:(10)dθ=grdr.In this formula,θ indicates the volumetric water content of soil when soil suction equals S and dθ represents the ratio of pore water volume with soil particle radius between r and r+dr to the total pore water volume under a well-saturated condition. In fact, in this formula, the volume of water being evacuated is used as an alternative to that of void in the soil. When the calculated radius is in the range of the whole pore medium, gr can be equal to fr.In this paper, the pore size distribution of soil was measured, and the methods mentioned above have some limitations when being applied. Taking MIP as an example, sample preparation for MIP is difficult, and under high pressure, liquid mercury will jeopardize soil particles, resulting in the deformation of soil skeleton and pore during the test. Another important reason which restricts the widespread use of conventional methods is the low popularity of test instruments as well as high test costs, which forced us to look for other convenient testing methods. In this case, mercury is replaced by water, and the soil-water characteristic curve is utilized instead of the mercury injection curve. Pores among soil particles rather than the internal pore of the particle are mainly considered with regard to the soil capillary phenomenon and the pore water flow problem. When the soil-water characteristic curve is used, the pressure can be increased to 2 MPa, which means the pore radius can be attached to 1000 Å correspondingly. In other words, it can obtain the exact pore distribution of the above 1000 Å, allowing us to have enough pore radius distribution data to analyze soil capillary phenomenon.


## 2.3. Basic Theories and Calculation Method

The pore of soil can be regarded as a capillary in the Kelvin formula without violating original characteristics of soil system. Radius, contact angle between soil particles and pore water, and surface tension are denoted byr, α, and TS, respectively. When using measurable physical quantities, Kelvin formula is expressed as(11)2TSvωcosαr=−RTlnuv1uv0=μ1−μ0,where form of matric suction is described as(12)r=2TScosαua−uw,where R = 8.314 J/(mol·K) is the universal gas constant; T = 298 K is the thermodynamic temperature; μvo (kPa) is the free water saturated vapor pressure in equilibrium state under the condition of temperature T. μv1 (kPa) is the vapor pressure value of solution at the state of balance; vm = 18 × 10−6  m3/mol is the partial molar volume of water vapor. ua−uw is the matric suction, uv1/uv0 is defined as relative humidity (RH) and can be obtained from (13) which is another expression of (11):(13)RH=exp−ua−uwvwRT.When the relative humidity or matric suction is in the increment of Stepi, the volume change of the pore volume of soil mass which is replaced by the gas can be calculated as(14)∇vpi=∇wiρw.Kelvin radius can be estimated by (12), while the actual pore radius is expressed as(15)rpi=rki+ti,where the thickness of adsorbed water film ti corresponding to RH or matric suction at Step i can be expressed by Halsey equation as [13](16)ti=τ−5lnRHi1/3,where τ is the effective diameter of adsorbed water molecules. Assuming that the cross-sectional area occupied by the water molecules is approximately 10.8 Å and the Avogadro constant NA is 6.02 × 1023/mol, the value of τ can be calculated as(17)τ=vwANA=2.77Å.After terminating the calculation of pore radius which corresponds to different suctions, points can be selected in the SWCC to calculate the pore size distribution curve. Considering the nonuniform pore size and inhomogeneous pore distribution as well as the data of pore distribution curve obtained by point-selecting method,r0 as average pore radius of soil by using weighted average of the statistical theory is defined as(18)r0=∑vprpi∑rpi.Similar to the parameters ofd10, d60, and Cu found in the particle size distribution curve, the average capillary pore radius can be obtained by pore size distribution curve. According to the definition, r0 represents the average radius within the calculating range. For example, if suction value ranges from 1 kPa to 10000 kPa, then the r0 is the average radius of the pore which falls into the corresponding range.When the calculated aperture range relates to the whole soil (suction value ranges from the air-entry pressure valueSS to the residual suction Sr), we can approximate r0 as r0=∑i=1nri/n, by ∑i=1nri≈nEr where(19)Er=∫0∞rfrdr.We can easily argue that(20)∫0∞rfrdr≈r0.By conducting substitution of (20) to (8), (8) can be simplified as follows:(21)hc=0.15βr0.As a parameter, average pore radiusr0 characterizing the pore state can be obtained by SWCC, and path retardation coefficient β as an empirical parameter will be measured by the following tests.


## 3. Materials and Methods

### 3.1. Test Materials

The test soils are distinguished as two different kinds of soils: fine-grained soils and coarse-grained soils. Fine-grained soils include low liquid limit clay (CL), high liquid limit clay (CH), low liquid limit silt (ML), and high liquid limit expansive clay (CHE), while coarse-grained soils include silt sand (SS), fine sand (FS), medium sand (MS), and coarse sand (CS). The parameters of the soil samples are obtained by the conventional tests and are shown in Tables1 and 2.Table 1
Main physical indexes of fine-grained soils.
Soil samples
γ
d
max
ω
P
ω
L
I
p
Particle size (mm) and content (%)
≥0.5 mm
0.25∼0.05 mm
0.05∼0.01 mm
0.01∼0.002 mm
≤0.002 mm
CL
1.88
22.4
41.8
19.4
0.8%
13.4%
21.7%
34.5%
29.6%
CH
1.84
27.9
55.6
27.7
0.3%
9.5%
17.5%
37.1%
35.6%
ML
1.93
18.2
28.3
8.1
3.9%
29.6%
31.3%
25.4%
7.8%
CHE
1.83
28.3
65.2
36.9
0.2%
7.3%
16.5%
34.8%
41.2%Table 2
Main physical indexes of coarse-grained soils.
Soil samples
γ
d
max
Particle size (mm) and content (%)
≥5 mm
5∼2 mm
2∼0.5 mm
0.5∼0.25 mm
0.2∼0.075 mm
0.07∼0.01 mm
0.0∼0.005 mm
≤0.005 mm
SS
2.08
0%
0.3%
1.2%
26.5%
41.8%
27.8%
2.1%
0.3%
FS
1.91
0%
1.6%
8.7%
34.4%
39.8%
13.8%
1.5%
0.2%
MS
1.84
1.9%
7.6%
17.5%
48.1%
19.9%
4.6%
0.3%
0.1%
CS
1.78
2.1%
11.6%
48.3%
32.7%
2.1%
1.2%
0.1%
0.1%


### 3.2. Capillary Rise Test

The method of vertical tube (a transparent plexiglass tube with diameter of 4.5 cm and height of 500 cm) is applied when performing soil capillary rise test. 20 capillary plexiglass tubes with 8 kinds of soil samples are tested and the compaction degree of soils is controlled by the total density of soils during the sample loading process. The test results of the steady rising height of capillary water in different soil samples are shown in Table3.Table 3
The test results of the steady rising height of capillary water.
Sample
Compaction degree
Maximum height (cm)
CL
0.9
331
0.85
317
0.8
301
CH
0.9
355
0.85
334
0.8
315
ML
0.9
293
0.85
267
0.8
239
CHE
0.9
391
0.85
358
0.8
339
SS
0.9
194
0.85
181
0.8
159
FS
0.85
140
0.8
121
MS
0.85
91
0.8
81
CS
0.85
73


### 3.3. SWCC Test

SWCC is mainly determined by pressure plate method, in which four samples of each kinds of soil and the as-fabricated samples were placed in a vacuum saturator for 48 h to ensure the full saturation. PF moisture characteristic curve instrument (DIK-4303) is used for measuring soil-water characteristic curve. First of all, samples are weighted by using electronic balance to ensure that the mass is recorded. And then, samples are put into a determinator (4 samples at a time) which connects a water channel between the instrument and the ceramic plate. The upper cover should be well tightened in case of air leakage. The maximum pressure is set to 1200 kPa, which is controlled by the increment of each step of the 100 kPa, and before applying the pressure, each sample’s gravimetric mass is recorded. The suction balance will be achieved if no water flows out from the pipe within 48 hours.


## 3.1. Test Materials

The test soils are distinguished as two different kinds of soils: fine-grained soils and coarse-grained soils. Fine-grained soils include low liquid limit clay (CL), high liquid limit clay (CH), low liquid limit silt (ML), and high liquid limit expansive clay (CHE), while coarse-grained soils include silt sand (SS), fine sand (FS), medium sand (MS), and coarse sand (CS). The parameters of the soil samples are obtained by the conventional tests and are shown in Tables1 and 2.Table 1
Main physical indexes of fine-grained soils.
Soil samples
γ
d
max
ω
P
ω
L
I
p
Particle size (mm) and content (%)
≥0.5 mm
0.25∼0.05 mm
0.05∼0.01 mm
0.01∼0.002 mm
≤0.002 mm
CL
1.88
22.4
41.8
19.4
0.8%
13.4%
21.7%
34.5%
29.6%
CH
1.84
27.9
55.6
27.7
0.3%
9.5%
17.5%
37.1%
35.6%
ML
1.93
18.2
28.3
8.1
3.9%
29.6%
31.3%
25.4%
7.8%
CHE
1.83
28.3
65.2
36.9
0.2%
7.3%
16.5%
34.8%
41.2%Table 2
Main physical indexes of coarse-grained soils.
Soil samples
γ
d
max
Particle size (mm) and content (%)
≥5 mm
5∼2 mm
2∼0.5 mm
0.5∼0.25 mm
0.2∼0.075 mm
0.07∼0.01 mm
0.0∼0.005 mm
≤0.005 mm
SS
2.08
0%
0.3%
1.2%
26.5%
41.8%
27.8%
2.1%
0.3%
FS
1.91
0%
1.6%
8.7%
34.4%
39.8%
13.8%
1.5%
0.2%
MS
1.84
1.9%
7.6%
17.5%
48.1%
19.9%
4.6%
0.3%
0.1%
CS
1.78
2.1%
11.6%
48.3%
32.7%
2.1%
1.2%
0.1%
0.1%


## 3.2. Capillary Rise Test

The method of vertical tube (a transparent plexiglass tube with diameter of 4.5 cm and height of 500 cm) is applied when performing soil capillary rise test. 20 capillary plexiglass tubes with 8 kinds of soil samples are tested and the compaction degree of soils is controlled by the total density of soils during the sample loading process. The test results of the steady rising height of capillary water in different soil samples are shown in Table3.Table 3
The test results of the steady rising height of capillary water.
Sample
Compaction degree
Maximum height (cm)
CL
0.9
331
0.85
317
0.8
301
CH
0.9
355
0.85
334
0.8
315
ML
0.9
293
0.85
267
0.8
239
CHE
0.9
391
0.85
358
0.8
339
SS
0.9
194
0.85
181
0.8
159
FS
0.85
140
0.8
121
MS
0.85
91
0.8
81
CS
0.85
73


## 3.3. SWCC Test

SWCC is mainly determined by pressure plate method, in which four samples of each kinds of soil and the as-fabricated samples were placed in a vacuum saturator for 48 h to ensure the full saturation. PF moisture characteristic curve instrument (DIK-4303) is used for measuring soil-water characteristic curve. First of all, samples are weighted by using electronic balance to ensure that the mass is recorded. And then, samples are put into a determinator (4 samples at a time) which connects a water channel between the instrument and the ceramic plate. The upper cover should be well tightened in case of air leakage. The maximum pressure is set to 1200 kPa, which is controlled by the increment of each step of the 100 kPa, and before applying the pressure, each sample’s gravimetric mass is recorded. The suction balance will be achieved if no water flows out from the pipe within 48 hours.


## 4. Analysis and Calculation

The results of the SWCC test are shown in Figures2 and 3, but due to the design restriction in pressure plate method which means that a pressure over 1200 kPa cannot be exerted and maintained in the experiment, points exceeding 1200 kPa in SWCC are obtained by fitting with VG model [14]. To be precise, the datum acquired from the test is injected into Van-Genuchten model in ORIGIN to obtain soil-water characteristic curve in a wider range of suction. The fitting equation is(22)θ=θs−θr11+anua−uwnm+θr,where θ is the volumetric water content; θs is the saturated volumetric water content; θr is the residual volumetric water content; and a, m, and n are independent coefficients (satisfy the condition m=1−1/n as much as possible). The fitting results from (22) for the soil samples are shown in Table 4.Figure 2
Soil-water characteristic curve test result for fine-grained soils.Figure 3
Soil-water characteristic curve test result for coarse-grained soils.Table 4
Fitting results of SWCC parameters for soil samples.
Samples
r
d
θ
s (%)
θ
r (%)
a
n
m
CL
0.9
38.2163
1.7842
0.0012
1.7798
0.6021
0.85
39.9741
2.0259
0.0013
1.9266
0.5988
0.8
40.5377
1.9624
0.0014
2.3215
0.5002
CH
0.9
41.3376
1.0621
0.0012
1.6198
0.5549
0.85
42.2937
1.2063
0.0013
1.8688
0.4889
0.8
43.2986
1.4079
0.0014
2.1398
0.4766
ML
0.9
30.6177
0.8823
0.0016
1.8423
0.4501
0.85
31.7787
0.7213
0.0018
1.9932
0.4821
0.8
33.1944
0.5056
0.0020
1.9161
0.4533
CHE
0.9
43.6997
2.2268
0.0010
2.0968
0.5001
0.85
44.4501
0.9385
0.0012
2.0152
0.4896
0.8
46.7998
1.3176
0.0013
1.9668
0.5102
SS
0.9
23.2396
0.7603
0.0014
1.9339
0.5983
0.85
23.9630
0.5382
0.0017
1.9117
0.6021
0.8
24.8004
0.6886
0.0017
2.0129
0.5891
FS
0.85
21.7180
0.2819
0.0018
1.6740
0.6213
0.8
23.9116
0.5884
0.0022
1.6523
0.6011
MS
0.85
21.4904
0.3096
0.0026
1.3913
0.6542
0.8
22.2219
0.2342
0.0027
1.5630
0.7311
CS
0.85
18.5616
0.1443
0.0033
1.5067
0.7001When selecting suction calculation range, the value range should be considered to represent the entity of soil as well as the measurement range of suction. After many attempts, we propose that fine-grained soils matric suction range be controlled at 200 kPa∼10000 kPa and that the range of coarse-grained soils be controlled at 50 kPa∼4000 kPa. (The external values of the test range are calculated by fitting the results of VG model.)According to the theory introduced above, soil-water characteristic curve is used to calculate the radiusr0. Calculation results of soil sample of CL with compaction degree 0.9 are listed in Table 5 as an example while the numerical integration calculation steps are as follows:(1)
Get the SWCC of the soil and select the data of 200 kPa∼20000 kPa (the data outside the range of test are obtained by fitting with VG model).(2)
Calculate RH (relative humidity) of matrix suction according to (13).(3)
Translate mass water content into per unit mass of soil-water filled pore volume (volumetric water content).(4)
Use (12) to calculate the Kelvin radius rki under the suction of different substrates with which the contact angle α is 0 (soil in dehydration process).(5)
Calculate the thickness of the water filmti under the suction of different substrates by (16).(6)
Calculate actual pore radiusrpi by results (4) and (5) applying (15).(7)
Calculate the reduction of pore volume waterΔvp in per unit mass of soil when the matrix suction is changed form one substrate to another. (Here, Δvp is the pore volume corresponding to the water that is discharged.)(8)
Calculate the average pore radiusrpavg in the process of evacuation of pore water by averaging the result of (6) for per step. (Here rpavg is the average pore radius corresponding to the pore volume vp.)(9)
Add the pore volume to calculate the cumulative pore volume∑Vp of per unit mass, and draw relationship diagram between the pore volume Δvp and the average pore radius rpavg.(10)
Calculate ther0 by (18).Table 5
Equivalent radius calculated form of soil sample of CL with compaction degree 0.9.
u
a
−
u
w (kPa)
W(g/g)
RH (%)
V
p (cm3/g)
r
k (Å)
t (Å)
r
p (Å)
Δ
v
p (cm3/g)
r
p
avg (Å)
∑
V
p (cm3/g)
200
0.176
99.85
0.176
7200
41.827
7241.8
—
—
—
300
0.168
99.78
0.168
4800
36.539
4836.5
0.008
6039.2
0.008
400
0.152
99.71
0.152
3600
33.198
3633.2
0.016
4234.9
0.024
500
0.137
99.64
0.137
2880
30.818
2910.8
0.015
3272.0
0.039
600
0.124
99.57
0.124
2400
29.001
2429.0
0.013
2669.9
0.052
700
0.114
99.49
0.114
2057
27.549
2084.7
0.010
2256.8
0.062
800
0.101
99.42
0.101
1800
26.349
1826.3
0.013
1955.5
0.076
900
0.089
99.35
0.089
1600
25.335
1625.3
0.011
1725.8
0.087
1000
0.081
99.28
0.081
1440
24.461
1464.5
0.008
1544.9
0.095
1100
0.072
99.20
0.072
1309
23.696
1332.8
0.010
1398.6
0.104
1200
0.063
99.13
0.063
1200
23.018
1223.0
0.009
1277.9
0.113
2000
0.054
98.56
0.054
720
19.414
739.4
0.009
981.2
0.122
4000
0.048
97.14
0.048
360
15.409
375.4
0.006
557.4
0.128
6000
0.036
95.74
0.036
240
13.461
253.5
0.012
314.4
0.140
10000
0.025
93.00
0.025
144
11.354
155.4
0.011
204.4
0.150
20000
0.018
86.48
0.018
72
9.011
81.011
0.007
118.2
0.158The average pore radiusr0 of soil sample is calculated by the method above. After determining the radius r0, the retardation coefficient β of different soil samples will be obtained. And the results of coefficient β are shown in Table 6.Table 6
Calculation result of coefficientβ for soil samples.
Sample
Compaction degree
Maximum height (cm)
Capillary radius (Å)
Coefficientβ
CL
0.9
331
2128
21.3
0.85
317
2212
21.4
0.8
301
2336
21.3
CH
0.9
355
2014
21.0
0.85
334
2107
21.3
0.8
314
2234
21.4
ML
0.9
293
2359
21.7
0.85
267
2650
21.2
0.8
239
2881
21.8
CHE
0.9
391
1811
21.2
0.85
358
1989
21.1
0.8
339
2090
21.2
SS
0.9
194
3305
23.4
0.85
181
3399
24.4
0.8
159
3841
24.6
FS
0.85
140
4499
23.8
0.8
121
4950
25.0
MS
0.85
91
6374
25.9
0.8
81
7116
26.0
CS
0.85
73
7854
26.2As clearly observed from the above chart, the coefficient of the fine-grained soils was about 21 cm−2 while the range barely fluctuates with value relatively fixed. But for coarse-grained soils, β changes from 23 cm−2 to 26 cm−2. One main reason we argue for coarse-grained soils is that the proportion of large pore size increased significantly, contributing greatly to a maximum capillary rise. As a result, for (21) the numerical value of the coefficient β is considered to be 21 cm−2 and 25 cm−2 for fine-grained soils and coarse-grained soils, respectively.


## 5. Verification and Discussion

In this study, in order to verify the accuracy of the maximum capillary rise height formula proposed by the author, capillary rise tests are carried out on 44 kinds of soil samples, and thereafter, a comparison is made between the capillary rise estimated from our equation (21) and the estimations from (1)–(3). Here, the coefficient C for (2) takes 5 × 10−5 m2The test results and calculation results of the maximum capillary rise height for 44 kinds of soil samples are shown in Table7.Table 7
Test and calculation results of maximum capillary rise height for 44 kinds of soils.
Test number
Soil sample
Void ratio e
d
10 (cm)
Coefficientβ (cm−2)
h
a (cm)
Capillary radius (Å)
Measured height (cm)
Results by formula (e/er)/cm
Results by (1) (e/er)/cm
Results by (2) (e/er)/cm
Results by (3) (e/er)/cm
1
CL
0.89
0.001
21
178
2253
309
317 (8/2.5%)
530 (221/72%)
281 (−28/9.1%)
312 (3/1.0%)
2
CL
0.92
0.001
21
202
2158
325
331 (6/1.8%)
530 (205/63%)
271 (−54/17%)
336 (11/3.4%)
3
CL
1.05
0.0008
21
194
2070
324
345 (21/6.5%)
552 (228/70%)
297 (−27/8.3%)
328 (4/1.2%)
4
CL
0.88
0.0007
21
184
2268
319
315 (−4/1.3%)
565 (246/77%)
404 (85/27%)
318 (−1/0.3%)
5
CL
0.88
0.0009
21
192
2001
341
357 (16/4.7%)
540 (199/58%)
317 (−24/7.0%)
326 (−15/4.4%)
6
CL
0.81
0.001
21
199
2171
322
329 (7/2.2%)
530 (208/65%)
309 (−13/4.0%)
333 (11/3.4%)
7
CL
0.98
0.0011
21
195
2089
318
340 (22/6.9%)
520 (202/64%)
231 (−87/27%)
329 (11/3.5%)
8
CL
1.01
0.0007
21
180
2035
344
351 (7/2.0%)
565 (221/64%)
350 (6/1.7%)
314 (−30/8.7%)
9
CH
0.96
0.0008
21
214
2312
317
309 (−8/2.5%)
552 (235/74%)
327 (10/3.2%)
348 (31/9.8%)
10
CH
0.84
0.0006
21
231
2158
328
331 (3/0.9%)
580 (252/77%)
494 (166/51%)
365 (37/11%)
11
CH
0.73
0.0007
21
209
2165
343
330 (−13/3.8%)
565 (222/65%)
492 (149/43%)
343 (0/0%)
12
CH
0.96
0.0006
21
218
2232
329
320 (−9/2.7%)
580 (251/76%)
433 (104/32%)
352 (23/7.0%)
13
CH
1.01
0.0007
21
199
2158
315
331 (16/5.1%)
565 (250/79%)
355 (40/13%)
333 (18/5.7%)
14
CH
0.89
0.0009
21
233
2191
347
326 (−21/6.1%)
540 (193/56%)
310 (−37/11%)
367 (20/5.8%)
15
ML
0.74
0.0018
21
145
2421
288
295 (7/2.4%)
472 (184/64%)
186 (−102/35%)
279 (−9/3.1%)
16
ML
0.90
0.0025
21
165
2524
276
283 (7/2.5%)
439 (163/59%)
110 (−166/60%)
299 (23/8.3%)
17
ML
0.86
0.0029
21
137
2655
259
269 (10/3.9%)
424 (165/64%)
100 (−159/61%)
271 (12/4.6%)
18
ML
0.78
0.0031
21
131
2716
247
263 (16/6.5%)
418 (171/69%)
104 (−143/58%)
265 (18/7.3%)
19
ML
0.74
0.0043
21
158
2636
260
271 (11/4.2%)
385 (125/48%)
78 (−182/70%)
292 (32/12%)
20
ML
0.82
0.0019
21
122
2758
248
259 (11/4.4%)
466 (218/88%)
161 (−87/35%)
256 (8/3.2%)
22
MH
0.72
0.003
21
101
2892
236
247 (11/4.7%)
421 (185/78%)
116 (−120/51%)
235 (−1/0.4%)
23
MH
0.78
0.0022
21
95
2869
257
249 (−8/3.1%)
452 (195/76%)
146 (−111/43%)
229 (−28/11%)
24
MH
0.69
0.0037
21
88
3161
221
226 (5/2.3%)
400 (179/81%)
97 (−124/56%)
222 (1/0.5%)
25
MH
0.81
0.0042
21
94
2737
259
261 (2/0.8%)
388 (129/50%)
73 (−186/72%)
228 (−31/12%)
26
MH
0.72
0.0035
21
88
3079
240
232 (−8/3.3%)
406 (166/69%)
99 (−141/59%)
222 (−18/7.5%)
27
SM
0.62
0.009
25
78
4167
199
180 (−19/9.5%)
312 (113/57%)
40 (−159/72%)
211 (12/6.0%)
29
SM
0.60
0.0085
25
83
4121
204
182 (−22/10.8%)
318 (114/56%)
49 (−155/59%)
216 (12/5.9%)
30
SM
0.62
0.0081
25
80
4190
185
179 (−6/3.2%)
323 (138/75%)
50 (−135/80%)
213 (28/15%)
34
SS
0.42
0.019
25
58
4098
197
183 (−14/7.1%)
238 (41/21%)
31 (−166/84%)
191 (−6/3.0%)
35
SS
0.53
0.012
25
71
4167
184
180 (−4/2.2%)
284 (100/54%)
39 (−145/79%)
204 (20/10%)
36
SS
0.46
0.021
25
61
4491
178
167 (−11/6.2%)
228 (50/28%)
26 (−152/85%)
194 (16/9.0%)
37
SS
0.43
0.015
25
58
4190
180
179 (−1/0.6%)
262 (82/46%)
39 (−141/78%)
191 (11/6.1%)
38
FS
0.52
0.031
25
43
5474
139
137 (−2/1.4%)
190 (51/37%)
16 (−123/88%)
176 (37/27%)
39
FS
0.42
0.052
25
41
6356
110
118 (8/7.3%)
139 (29/26%)
11 (−99/90%)
174 (64/58%)
40
FS
0.49
0.047
25
39
6466
118
116 (−2/1.4%)
149 (31/26%)
11 (−107/91%)
172 (54/46%)
41
FS
0.42
0.055
25
42
7353
98
102 (4/4.1%)
133 (35/36%)
11 (−87/89%)
175 (77/79%)
42
MS
0.38
0.092
25
34
7732
86
97 (11/12.8%)
82 (−4/5%)
7 (−79/92%)
167 (81/94%)
43
MS
0.33
0.1
25
30
8152
83
92 (9/10.8%)
74 (−9/11%)
8 (−75/90%)
163 (80/96%)
44
MS
0.41
0.0864
25
29
8523
79
88 (9/11.4%)
88 (9/11%)
7 (−72/91%)
162 (83/105%)It can be seen from the table that the estimating error of the maximum height from the formula proposed by the author is always less than 10% and the maximum error is 22 cm (204 cm), indicating its relatively higher accuracy compared with other formulas mentioned in this paper. This also means that the results coincide better with the actual situation and have higher practical value.Accurately determining the maximum capillary rise height of the soils is a complicated issue. In this paper, we calculated the average pore radius by method of SWCC and then determined the maximum capillary rise height using the pore radius. Obtaining the pore distribution data from the SWCC data is the core of the calculation method. We can easily get the pore distribution curve through the soil-water characteristic curve. However, similar to the particle size curve, we need to find a set of parameters to evaluate this curve as well as calculate the maximum capillary rise height. At first, we tried to user50 as the evaluation parameter but later found when fine-soil samples are at different degrees of compaction, the r50 values are very close. According to the definition, as long as the soils have the same matrix suction at the saturation of 50%, the values of r50 still remain the same. It is difficult to use a specific pore radius to represent the equivalent capillary radius in the calculation of the maximum capillary rise height of the soil; therefore, we chose the weighted average of the pore radius r0 as the calculation parameter. Whether there is a better way to evaluate the distribution of soil pore and to analyze the behavior of soils is a project that we still need to explore.Another important parameter isβ which is an empirical one obtained from the test. We use r0 to distinguish different soil samples, so we hope different soil samples would obtain the same values of β as much as possible. Test results show that β of fine-grained soil is very close and can be directly taken to 21, but β also has the tendency to augment with the increase of average pore radius r0 and this trend is particularly evident in coarse-grained soils. Therefore, there is an issue about how to pick the value of β in the actual calculation process and we argue that β of fine-grained soils can be fixed to 21, but for coarse-grained soil the value of β is not a settled figure, ranging from 24 to 26. Moreover, the value rises with the increase of the content of the coarse particles. Here the author suggests taking an average of 25.


## 6. Conclusion

After theoretical calculation and large numbers of indoor experiments, the following conclusions can be summarized as follows:(1)
The average pore radius formula is deduced on summary of experiments and statistical theory, and the calculation theory can be used to distinguish the pore distribution characteristics of different compaction degree of soil samples with different particle size distribution. Therefore, it can offer references for similar research and also can be used for estimating the maximum height of capillary. In particular, for fine-grained soils the calculation results for the maximum capillary height are relatively accurate.(2)
Using the soil pore radius to calculate the maximum capillary rise height is an accurate and feasible approach. During the calculation process, the coefficientβ of the fine-grained soil was about 21 to 22, while β of the coarse-grained soil was about 24 to 26. The greater the number of coarse particles is, the greater the retardation coefficient β is.(3)
The maximum capillary rise height of the soil is mainly determined by the distribution of the large size pore, and meanwhile, the effect of the small size pore is quite insignificant.



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*Source: 5190354-2018-08-15.xml*