# Analysis and Prediction of Stable Height of Capillary Water Rising under Unsaturated Subgrade Soil

Authors: Jianfeng Cao; Zhihao Mao; Liqiang Hu; Yunfei Dai; Yu Luo; Wenwei Zhou
Journal: Advances in Civil Engineering
 (2022)

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## Abstract

In order to study the capillary water height of the highway subgrade and balanced moisture condition of the subgrade affected by groundwater, the standpipe method has been used in this study to track the measured data. It involves the analysis of the factors affecting the capillary water stability height of the coarse-grained soil and fine-grained soil filler as well as establishing the corresponding prediction model. The comparative analysis with the experimental tracking results and related literature shows that the prediction model developed in this study is highly accurate and can be applied for the analysis of the balanced moisture condition of the highway subgrade.


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## Body

## 1. Introduction

The capillary phenomenon of soil refers to the movement of water in soil upwards and in other directions under surface tension (capillary force) along the fine pore fissures, which often occurs in sandy soil and fine-grained soil with the soil particle size less than 2 mm (silty soil, clayey soil, or organic soil). The groundwater-influenced road bed fill leads to water accumulation under the capillary action, thus altering the humidity levels of the road bed. It affects the strength and stress-strain characteristics of the road bed fill, which correspondingly affects the mechanical properties and durability of the pavement structure. As the supporting structure of the pavement, the subgrade is an important part of the road, bearing the load transmitted by the road. Its stiffness, strength, and stability directly affect and determine the thickness of the pavement structure and performance. Capillary water rises to the subgrade workspace, making the subgrade soil wet and soft, and reducing the strength of the subgrade. The seasonally frozen region will cause frost heaving boiling and slope collapse, resulting in the reduction of pavement flatness or even destruction, which seriously threatens the safety and stability of engineering buildings, causing safety hazards to the normal operation of highways and additionally increasing the cost of highway maintenance, bringing great harm to the highway engineering construction, and causing significant losses to the national economy [1–4]. In the highway design, the subgrade needs to maintain the minimum fill height to avoid the influence of the rising capillary water on the balanced moisture condition of the road bed [5], and the determination of the minimum fill height is closely related to the rising of capillary water height (hc). The classical groundwater dynamics considers that the maximum height of the rising water along the soil micropore under the action of the capillary force is called the capillary water height, and a theoretical approximation relation for hc can be derived based on the capillary force equal to the capillary height as follows [6]:(1)hc=0.03D=0.075dm,where D is the average diameter of soil pores (mm), and d is the average particle size of the soil particles (mm).The average sand particle size is > 0.075 mm [7], and hc determined from (1) is < 1 m, mostly <0.5 m, which is consistent with the observation reported from the indoor and outdoor tests [6, 8–13]. However, for the fine-grained soil [7], the hc value calculated using (1) lies in the range 5∼6 m [14], even reaching 15 m, which significantly contradicts the findings from the indoor tests and engineering practice [6, 8, 9]. Hu [10] et al. reported that the capillary action is the result of the joint action of the capillary forces of liquid, cohesion, adhesion, and gravity. Zhang [6] and Dong [8] suggested that the influence of the bonded water viscosity between the soil particles is relatively large, especially for the fine-grained soil.The subgrade fill belongs to the unsaturated soil. The hydrogeological zone of the subgrade soil affected by groundwater is generally divided into four parts: residual water-bearing zone (bound, pore capillary, and partial suspension capillary water), capillary fringe, capillary saturated zone, and saturation zone (below the phreatic surface). The water content of the soil in the residual water-bearing zone is basically a certain value (residual water contentW0), while the water content in the capillary saturation zone is close to the saturation water content Ws (saturation is 100% water content), and the water content of the supporting capillary water zone transitions is from W0 to Ws. To enable the highway road bed fill to generate the moisture accumulation, the supporting capillary water zone must rise to the road bed range, thus making it “obviously wet.” According to the Specifications for Design of Highway Subgrades [5], for determining the lower saturation limit of the fine-grained soil, the “obviously wet” state is defined as saturation (S) > 80% or moisture content exceeding the plastic limit WP. In addition, the distance from the apparently wet interface at the top of the groundwater capillary-moisture zone to the diving surface is defined as the subsurface capillary water stability height (hc), also known as the strong capillary water height.A number of studies have focused on thehc values of various soils. Dong [8] conducted an indoor capillary rise test by employing the standpipe method for different coarse and fine soil materials, followed by the analysis and establishment of the regression equation of the capillary water stability height of the coarse-grained soil with soil porosity (n) and effective size d10. In another study, Chen [9] obtained the moisture content and capillary water height of different soils by using the indoor standpipe and frost swelling tests. Likewise, Wang [15] et al. obtained the moisture content and capillary water height under the capillary action of the unsaturated soil subgrade.The process of groundwater capillary action in the unsaturated subgrade soil is complex and slow. The subgrade moisture generally requires 2∼3 years after the opening of the vehicle to reach a stable state of equilibrium with the surrounding environment (subgrade balanced moisture state). The accuracy and stability requirements of the field subgrade humidity tracking observation are relatively high, and the current mainstream method is to quantitatively analyze the capillary water height of the unsaturated subgrade soil by employing the indoor standpipe simulation test. In general, the capillary action time of the coarse-grained soil is short, and the capillary water height easily reached a stable height in a short period of time (<1 year), while the capillary action of the fine-grained soil is longer, with the capillary water rising to a stable height in 2∼4 years or even longer. Therefore, it is difficult to assess the balanced moisture condition of the designed subgrade when performing the subgrade pavement design by determining the capillary water height of the soil used for the subgrade filling through the indoor tests. This study adopts the measured values of the different coarse and fine-grained soil capillary water stability heights obtained by Dong [8] et al. by employing the indoor simulation test over 4 years. Subsequently, an in-depth analysis of the various coarse and fine-grained soil property indicators and hc has been determined based on the establishment of the corresponding hc prediction model.


## 2. Standpipe Method Based on Capillary Water Height

### 2.1. Test Equipment

A capillary tester includes a test stand, plexiglass test tube, plexiglass water container, special hanging spring, and hanging rope. The inner diameter of the plexiglass pipe is 4.5 cm, and the wall thickness is about 5 mm; a small hole with a diameter of 10 mm is opened every 10 cm. The hole is equipped with a small plexiglass plug that can be tightened. The lower end is connected to the aluminum base with a silk buckle, and an exhaust hole is opened 1 cm away from the zero point. The base is equipped with a rubber washer and copper wire mesh. The two pipes are connected by external joints. A special spring is used to ensure that the water surface height remains unchanged when the water is falling [7].


### 2.2. Test Principle

The experimental principle of the standpipe method is based on the stable height of the rising capillary water at the position where the matrix and gravitational potentials are in equilibrium [16]. In case, the matric potential is greater than the gravitational potential, and the water rises along the voids in the soil, thus creating a capillary phenomenon. The soil columns with different soil quality, air-dry moisture content, and density were prepared based on the volume and weight relationship of soil, and the test equipment, methods, and procedures strictly complied with the Test Methods of Soils for Highway Engineering [7]. Considering the unfavorable condition of the roadbed field fill and indoor specimen forming method, the compaction degree of the soil samples used in the indoor standpipe simulation test was 90%, except for 85% for the 7# high liquid limit clay. The initial water content of the soil samples for the test was controlled near the optimum water content of the soil [17], as the capillary water height of the soil samples with different initial water contents did not differ significantly. Based on the direct observation and measurement of the capillary water height over time (4 years) as well as the soil moisture content per 10 cm height of the stabilized soil column, hc was calculated corresponding to the plastic limit moisture content (WP).


### 2.3. The Criteria for Classification of Soils

Test Methods of Soils for Highway Engineering [7] classifies engineering soils according to the particle composition characteristics (gradation composition, coefficient of uniformity Cu, and coefficient of curvature Cc), plasticity index of soils (liquid limit WL, plastic limit WP, and plasticity index IP), and organic matter content in soils.(2)Cu=d60d10;Cc=d302d60⋅d10,where d10, d30, and d60 are the characteristic particle sizes of the soil, which respectively represent the particle sizes (mm) in which the mass of soil particles smaller than this particle size accounts for 10%, 30%, and 60% of the total mass on the particle size distribution curve of soil.


#### 2.3.1. The Coarse-Grained Soil

The coarse-grained soil refers to the soil particle size greater than 0.075 mm mass which is greater than 50% of the total soil mass and the soil particle size greater than 60 mm mass which is less than or equal to 15% of the total mass. The coarse-grained soils are divided into gravelly soil and sandy soil, where sandy soil is the soil where the mass of soil particles with particle size between 2 mm and 60 mm is less than or equal to the mass of particles with particle size between 0.075 mm and 2 mm. Classification of the sandy soils is shown in Table1 (F indicates the particles smaller than 0.075 mm).Table 1
Classification of the sandy soils.
Soil typeCode nameContent ofF (%)Judgment conditionsSandWell-graded sandSWF ≤ 5Cu ≥ 5 and 1≤Cc ≤ 3Poorly graded sandSPDoes not meet the above conditionsFine-grained sandSF5 <F ≤ 15—Fine-grained soil sandSilty sandSM15 <F ≤ 50The fine-grained soil is located below line A of the plasticity chartClayey sandSCThe fine-grained soil is located on the line A or above the line A of the plasticity chart


#### 2.3.2. The Fine-Grained Soil

The fine-grained soil refers to the soil particle size less than 0.075 mm mass which is greater than or equal to 50% of the total soil mass. Fine-grained soils are divided into silty soil (M), clayey soil (C), and histosol and organic soils (O), and are named according to the plasticity chart in Figure1.Figure 1
The plasticity chart. Low liquid limit (L): WL<50%; high liquid limit (H): WL≥50%.


### 2.4. Basic Physical Indicators of the Test Soil

A total of 10 test soils (Table2 for the six coarse-grained soils and Table 3 for the four fine-grained soils) were selected for the capillary water height simulation test by employing the standpipe method, and 2 sets of parallel experiments were performed for each soil.Table 2
The physical indices of the coarse-grained soils.
Sample numberParticle size composition (%)Dry density/g.cm−3Pore ratio/ed10/mmCuCcSoil type10∼55∼22∼0.50.5∼0.250.25∼0.0750.075∼0.050.05∼0.010.01∼0.0050.005∼0.0021#——0.9827.8739.1915.6613.971.850.481.8270.4390.0336.060.55Silty sand2#—0.145.9241.7438.655.427.180.810.141.7010.5410.0595.021.29Fine-grained sand3#1.8765.9617.6561.2411.170.401.010.300.401.6200.5970.2002.201.14Poorly graded medium sand4#1.4404.5713.5246.918.564.7512.003.504.751.8860.3790.01427.865.42Silty sand5#1.7205.4816.2256.3110.271.904.801.401.901.7010.5240.0755.692.66Fine-grained sand6#1.26011.7054.9923.635.740.511.290.370.511.5840.6340.2704.740.66Poorly graded coarse sandTable 3
The physical indices of the fine-grained soils.
Sample numberParticle size composition (%)Dry density/g.cm−3WP (%)WL (%)IP (%)Soil type2∼0.50.5∼0.250.25∼ 0.0750.075∼ 0.050.05∼ 0.010.01∼ 0.0050.005∼ 0.002<0.0027#0.210.340.6411.8644.4617.7816.807.911.55626.352.726.4High liquid limit clay8#0.280.410.6218.7547.3813.8213.824.921.68322.240.318.1Low liquid limit clay9#1.859.0414.4118.7542.389.004.57-1.71920.733.512.8Low liquid limit clay10#—9.0524.5027.2434.823.061.33-1.75518.828.19.3Low liquid limit siltThe soil particle size, compaction, density, and liquid-plastic limit of the test soil were determined according to the Test Methods of Soils for Highway Engineering [7], and the test soil was named according to its particle composition characteristics and plasticity index, including the coefficient of uniformity (Cu), coefficient of curvature (Cc), liquid limit (WL), plastic limit (WP), and plasticity index (Ip).


### 2.5. Experimental Results and Prediction Model Analysis

#### 2.5.1. Stable Height and Prediction of Capillary Water Rise in Coarse-Grained Soil

Table4 lists the average values of hc for the six coarse-grained soils determined from the indoor standpipe simulation tests in parallel experiments at the corresponding plastic limit water content (WP). In order to make the test closer to engineering practice, the compaction degree of the six coarse-grained soil samples was 90%. As can be observed, hc decreases with the soil grain size, which is consistent with equation (1).Table 4
The comparison of the measured and predicted values ofhc of the capillary water rise in the coarse-grained soils.
Sample numberSoil typePore ratio (e)Effective sized10 (mm)Measured value (cm)Hessian formula C value (×10−5m2)Predicted value [8] (cm)Equation (4) predictive value (cm)Equation (4) predictive error (cm)1#Silty sand0.4390.033248.53.60208.6241.0-7.52#Fine-grained sand0.5410.059138.34.41126.7146.98.73#Poorly graded medium sand0.5970.20069.38.2765.966.0-3.34#Silty sand0.3790.014219.31.16207.6220.10.85#Fine-grained sand0.5240.075118.04.64118.1127.89.86#Poorly graded coarse sand0.6340.27063.510.8761.956.0-7.5Various factors affect the rising height of the capillary water in the coarse-grained soils. A number of studies have carried out the multiple correlation analysis with respect to the influencing factors and concluded that the rising height of the capillary water in the coarse-grained soils is closely related to the soil conditions and degree of filling density. Hazen–Williams employed the pore ratio of the soil (e) and effective size d10 of the soil particles to establish the famous Hazen formula as follows [18]:(3)hc=Ce⋅d10,where C is a coefficient related to the shape and surface cleanliness of the soil particles, generally taken as 1 × 10−5∼5 × 10−5 (m2).However, the Hazen formula suffers from the too large selection range of theC value (Table 4), and the specific engineering selection is even more difficult.Referring to the Hazen formula, the scatter plot ofhc and the combined index (e⋅d10)−1 consisting of the pore ratio (e) and effective size (d10) was plotted as a smooth curve (Figure 2). As can be observed, hc of the coarse-grained soil is quadratically related to the combined index (e⋅d10)−1.Figure 2
The stable height of the capillary water for the coarse-grained soil.The relation of the coarse-grained soilhc value with the combined index (e⋅d10)−1 was established by the regression analysis, as shown in (4). The regression parameter R2 was as high as 99%.The measured values from the indoor standpipe simulation test and predicted values from the equation in the literature [8] as well as the predicted values and errors from equation (4) developed in this study are summarized in Table 4. As can be observed, the regression equation developed in this study is superior to the analysis of Dong et al. [8]. In addition, the findings in this study are close to the long-term follow-up measurements, with the errors below 10 cm. Thus, equation (4) addresses the shortcomings of the Hazen formula in engineering applications.(4)hc=−0.017e⋅d10−2+4.2e⋅d10−1+32.Analysis of the data in Table4 shows that the height of capillary water rise in coarse-grained soils increases with finer particles and more mud content. This is because both conditions make the pore space of the soil smaller, and the test data also prove this. In addition, it can be concluded from Figure 2 and equation (4) that the peak of the coarse-grained soil hc value is around 300 cm, which is consistent with the results of the groundwater impact model for the subgrade balanced moisture in the subgrade, as described by Specifications for Design of Highway Subgrades [5].


#### 2.5.2. Stable Height and Prediction of Capillary Water Rise in Fine-Grained Soil

The fine-grained soil pores are smaller, and numerous capillaries are formed in the soil microstructure. These capillaries produce a large matrix suction, which makes the capillary action process longer, however, with a greater rise in height. In addition, the surface electric field of the fine-grained soil is strong, and the bound water is influenced by the viscous force. The Hessian formula, which only considers the dense state and effective size, is not applicable. Liu et al. [19] related the capillary water height of the unsaturated soil to the soil-water characteristic curve and established the analytical equations for the capillary water height in relation with the initial volume water content, residual volume water content, void ratio, and soil-water characteristic curve through the fractal model of the soil pore distribution. The soil-water characteristic curve is mainly affected by the mineral composition of the soil particles, pore size and distribution, pore structure, contractibility of soil, stress history and temperature of the soil, etc., and the mathematical model of the soil-water characteristic curve differs for the different soils [20]. Therefore, to accurately obtain the soil-water characteristic curve of the subgrade, accurate experiments are needed, which are time consuming and cumbersome, thus hindering the use in the engineering applications.The plasticity index (Ip) is a vital characteristic index of the fine-grained soil, and its value is related to the particle composition of the soil, mineral composition of the soil particles, ionic composition and concentration of water in the soil, etc. Generally, the larger is the Ip value, the higher is the clay grain content, the higher are its specific surface area and possible bound water content, the greater is the matrix suction at the same water content, and the stronger is the water holding capacity of the soil. Li et al. [21] reported that the soil material composition and pore structure are the basic factors affecting the soil-water characteristic curve, and these factors are closely related to Ip.The mean values ofhc obtained from the indoor standpipe simulation test for four kinds of the fine-grained soils are presented in Table 5. Among them, the compaction degree of 7# soil sample is 85%, 8#, 9#, and 10# soil samples are 90%. Furthermore, the scatter plots of hc and Ip are plotted and connected to a smooth curve, as shown in Figure 3.Table 5
The comparison of the measured and predicted values ofhc of the capillary water rise in the fine-grained soils.
Sample numberSoil typePlasticity indexIpMeasured value (cm)Predicted value (cm)Relative error7#High liquid limit clay26.4195.0194.3-0.4%8#Low liquid limit clay18.1308.5320.53.9%9#Low liquid limit clay12.8368.5351.4-4.7%10#Low liquid limit silt9.3342.2350.52.4%Figure 3
The rising stable height of the capillary water for the fine-grained soils as a function ofIp.The regression analysis can allow the prediction ofIp and hc of the fine-grained soil, as shown in equation (5), which is a quadratic polynomial equation with the regression correlation coefficient R2 over 97% and absolute value of the relative error between the predicted and measured values of hc below 5%.(5)hc=−0.69Ip2+15.5Ip+266.The analysis of Figure3 and equation (5) reveals a peak hc value for the fine-grained soil. On the left side of the peak, hc increases with Ip, whereas hc decreases with the increasing Ip on the right side of the peak. This phenomenon is related to the surface-bonded water content of the fine-grained soil and particle size distribution. As the plasticity index of the fine-grained soil is large, the surface energy is noted to be high and bonded water film is thick and overlapping, thus blocking the original pore by the bonded water. The formation of the dead end of the pore blocks the water flow, and the pores that are not completely blocked also become finer. The series of changes lead to an increased water migration tortuosity and resistance to the capillary ascent, along with slowing the ascent velocity and reducing the ascent height. The peak of hc is reached between Ip = 10∼14, with the peak between 350∼380 cm, which is in agreement with the results of the model [5] predicting the influence of groundwater on the balanced moisture of the subgrade.


## 2.1. Test Equipment

A capillary tester includes a test stand, plexiglass test tube, plexiglass water container, special hanging spring, and hanging rope. The inner diameter of the plexiglass pipe is 4.5 cm, and the wall thickness is about 5 mm; a small hole with a diameter of 10 mm is opened every 10 cm. The hole is equipped with a small plexiglass plug that can be tightened. The lower end is connected to the aluminum base with a silk buckle, and an exhaust hole is opened 1 cm away from the zero point. The base is equipped with a rubber washer and copper wire mesh. The two pipes are connected by external joints. A special spring is used to ensure that the water surface height remains unchanged when the water is falling [7].


## 2.2. Test Principle

The experimental principle of the standpipe method is based on the stable height of the rising capillary water at the position where the matrix and gravitational potentials are in equilibrium [16]. In case, the matric potential is greater than the gravitational potential, and the water rises along the voids in the soil, thus creating a capillary phenomenon. The soil columns with different soil quality, air-dry moisture content, and density were prepared based on the volume and weight relationship of soil, and the test equipment, methods, and procedures strictly complied with the Test Methods of Soils for Highway Engineering [7]. Considering the unfavorable condition of the roadbed field fill and indoor specimen forming method, the compaction degree of the soil samples used in the indoor standpipe simulation test was 90%, except for 85% for the 7# high liquid limit clay. The initial water content of the soil samples for the test was controlled near the optimum water content of the soil [17], as the capillary water height of the soil samples with different initial water contents did not differ significantly. Based on the direct observation and measurement of the capillary water height over time (4 years) as well as the soil moisture content per 10 cm height of the stabilized soil column, hc was calculated corresponding to the plastic limit moisture content (WP).


## 2.3. The Criteria for Classification of Soils

Test Methods of Soils for Highway Engineering [7] classifies engineering soils according to the particle composition characteristics (gradation composition, coefficient of uniformity Cu, and coefficient of curvature Cc), plasticity index of soils (liquid limit WL, plastic limit WP, and plasticity index IP), and organic matter content in soils.(2)Cu=d60d10;Cc=d302d60⋅d10,where d10, d30, and d60 are the characteristic particle sizes of the soil, which respectively represent the particle sizes (mm) in which the mass of soil particles smaller than this particle size accounts for 10%, 30%, and 60% of the total mass on the particle size distribution curve of soil.


### 2.3.1. The Coarse-Grained Soil

The coarse-grained soil refers to the soil particle size greater than 0.075 mm mass which is greater than 50% of the total soil mass and the soil particle size greater than 60 mm mass which is less than or equal to 15% of the total mass. The coarse-grained soils are divided into gravelly soil and sandy soil, where sandy soil is the soil where the mass of soil particles with particle size between 2 mm and 60 mm is less than or equal to the mass of particles with particle size between 0.075 mm and 2 mm. Classification of the sandy soils is shown in Table1 (F indicates the particles smaller than 0.075 mm).Table 1
Classification of the sandy soils.
Soil typeCode nameContent ofF (%)Judgment conditionsSandWell-graded sandSWF ≤ 5Cu ≥ 5 and 1≤Cc ≤ 3Poorly graded sandSPDoes not meet the above conditionsFine-grained sandSF5 <F ≤ 15—Fine-grained soil sandSilty sandSM15 <F ≤ 50The fine-grained soil is located below line A of the plasticity chartClayey sandSCThe fine-grained soil is located on the line A or above the line A of the plasticity chart


### 2.3.2. The Fine-Grained Soil

The fine-grained soil refers to the soil particle size less than 0.075 mm mass which is greater than or equal to 50% of the total soil mass. Fine-grained soils are divided into silty soil (M), clayey soil (C), and histosol and organic soils (O), and are named according to the plasticity chart in Figure1.Figure 1
The plasticity chart. Low liquid limit (L): WL<50%; high liquid limit (H): WL≥50%.


## 2.3.1. The Coarse-Grained Soil

The coarse-grained soil refers to the soil particle size greater than 0.075 mm mass which is greater than 50% of the total soil mass and the soil particle size greater than 60 mm mass which is less than or equal to 15% of the total mass. The coarse-grained soils are divided into gravelly soil and sandy soil, where sandy soil is the soil where the mass of soil particles with particle size between 2 mm and 60 mm is less than or equal to the mass of particles with particle size between 0.075 mm and 2 mm. Classification of the sandy soils is shown in Table1 (F indicates the particles smaller than 0.075 mm).Table 1
Classification of the sandy soils.
Soil typeCode nameContent ofF (%)Judgment conditionsSandWell-graded sandSWF ≤ 5Cu ≥ 5 and 1≤Cc ≤ 3Poorly graded sandSPDoes not meet the above conditionsFine-grained sandSF5 <F ≤ 15—Fine-grained soil sandSilty sandSM15 <F ≤ 50The fine-grained soil is located below line A of the plasticity chartClayey sandSCThe fine-grained soil is located on the line A or above the line A of the plasticity chart


## 2.3.2. The Fine-Grained Soil

The fine-grained soil refers to the soil particle size less than 0.075 mm mass which is greater than or equal to 50% of the total soil mass. Fine-grained soils are divided into silty soil (M), clayey soil (C), and histosol and organic soils (O), and are named according to the plasticity chart in Figure1.Figure 1
The plasticity chart. Low liquid limit (L): WL<50%; high liquid limit (H): WL≥50%.


## 2.4. Basic Physical Indicators of the Test Soil

A total of 10 test soils (Table2 for the six coarse-grained soils and Table 3 for the four fine-grained soils) were selected for the capillary water height simulation test by employing the standpipe method, and 2 sets of parallel experiments were performed for each soil.Table 2
The physical indices of the coarse-grained soils.
Sample numberParticle size composition (%)Dry density/g.cm−3Pore ratio/ed10/mmCuCcSoil type10∼55∼22∼0.50.5∼0.250.25∼0.0750.075∼0.050.05∼0.010.01∼0.0050.005∼0.0021#——0.9827.8739.1915.6613.971.850.481.8270.4390.0336.060.55Silty sand2#—0.145.9241.7438.655.427.180.810.141.7010.5410.0595.021.29Fine-grained sand3#1.8765.9617.6561.2411.170.401.010.300.401.6200.5970.2002.201.14Poorly graded medium sand4#1.4404.5713.5246.918.564.7512.003.504.751.8860.3790.01427.865.42Silty sand5#1.7205.4816.2256.3110.271.904.801.401.901.7010.5240.0755.692.66Fine-grained sand6#1.26011.7054.9923.635.740.511.290.370.511.5840.6340.2704.740.66Poorly graded coarse sandTable 3
The physical indices of the fine-grained soils.
Sample numberParticle size composition (%)Dry density/g.cm−3WP (%)WL (%)IP (%)Soil type2∼0.50.5∼0.250.25∼ 0.0750.075∼ 0.050.05∼ 0.010.01∼ 0.0050.005∼ 0.002<0.0027#0.210.340.6411.8644.4617.7816.807.911.55626.352.726.4High liquid limit clay8#0.280.410.6218.7547.3813.8213.824.921.68322.240.318.1Low liquid limit clay9#1.859.0414.4118.7542.389.004.57-1.71920.733.512.8Low liquid limit clay10#—9.0524.5027.2434.823.061.33-1.75518.828.19.3Low liquid limit siltThe soil particle size, compaction, density, and liquid-plastic limit of the test soil were determined according to the Test Methods of Soils for Highway Engineering [7], and the test soil was named according to its particle composition characteristics and plasticity index, including the coefficient of uniformity (Cu), coefficient of curvature (Cc), liquid limit (WL), plastic limit (WP), and plasticity index (Ip).


## 2.5. Experimental Results and Prediction Model Analysis

### 2.5.1. Stable Height and Prediction of Capillary Water Rise in Coarse-Grained Soil

Table4 lists the average values of hc for the six coarse-grained soils determined from the indoor standpipe simulation tests in parallel experiments at the corresponding plastic limit water content (WP). In order to make the test closer to engineering practice, the compaction degree of the six coarse-grained soil samples was 90%. As can be observed, hc decreases with the soil grain size, which is consistent with equation (1).Table 4
The comparison of the measured and predicted values ofhc of the capillary water rise in the coarse-grained soils.
Sample numberSoil typePore ratio (e)Effective sized10 (mm)Measured value (cm)Hessian formula C value (×10−5m2)Predicted value [8] (cm)Equation (4) predictive value (cm)Equation (4) predictive error (cm)1#Silty sand0.4390.033248.53.60208.6241.0-7.52#Fine-grained sand0.5410.059138.34.41126.7146.98.73#Poorly graded medium sand0.5970.20069.38.2765.966.0-3.34#Silty sand0.3790.014219.31.16207.6220.10.85#Fine-grained sand0.5240.075118.04.64118.1127.89.86#Poorly graded coarse sand0.6340.27063.510.8761.956.0-7.5Various factors affect the rising height of the capillary water in the coarse-grained soils. A number of studies have carried out the multiple correlation analysis with respect to the influencing factors and concluded that the rising height of the capillary water in the coarse-grained soils is closely related to the soil conditions and degree of filling density. Hazen–Williams employed the pore ratio of the soil (e) and effective size d10 of the soil particles to establish the famous Hazen formula as follows [18]:(3)hc=Ce⋅d10,where C is a coefficient related to the shape and surface cleanliness of the soil particles, generally taken as 1 × 10−5∼5 × 10−5 (m2).However, the Hazen formula suffers from the too large selection range of theC value (Table 4), and the specific engineering selection is even more difficult.Referring to the Hazen formula, the scatter plot ofhc and the combined index (e⋅d10)−1 consisting of the pore ratio (e) and effective size (d10) was plotted as a smooth curve (Figure 2). As can be observed, hc of the coarse-grained soil is quadratically related to the combined index (e⋅d10)−1.Figure 2
The stable height of the capillary water for the coarse-grained soil.The relation of the coarse-grained soilhc value with the combined index (e⋅d10)−1 was established by the regression analysis, as shown in (4). The regression parameter R2 was as high as 99%.The measured values from the indoor standpipe simulation test and predicted values from the equation in the literature [8] as well as the predicted values and errors from equation (4) developed in this study are summarized in Table 4. As can be observed, the regression equation developed in this study is superior to the analysis of Dong et al. [8]. In addition, the findings in this study are close to the long-term follow-up measurements, with the errors below 10 cm. Thus, equation (4) addresses the shortcomings of the Hazen formula in engineering applications.(4)hc=−0.017e⋅d10−2+4.2e⋅d10−1+32.Analysis of the data in Table4 shows that the height of capillary water rise in coarse-grained soils increases with finer particles and more mud content. This is because both conditions make the pore space of the soil smaller, and the test data also prove this. In addition, it can be concluded from Figure 2 and equation (4) that the peak of the coarse-grained soil hc value is around 300 cm, which is consistent with the results of the groundwater impact model for the subgrade balanced moisture in the subgrade, as described by Specifications for Design of Highway Subgrades [5].


### 2.5.2. Stable Height and Prediction of Capillary Water Rise in Fine-Grained Soil

The fine-grained soil pores are smaller, and numerous capillaries are formed in the soil microstructure. These capillaries produce a large matrix suction, which makes the capillary action process longer, however, with a greater rise in height. In addition, the surface electric field of the fine-grained soil is strong, and the bound water is influenced by the viscous force. The Hessian formula, which only considers the dense state and effective size, is not applicable. Liu et al. [19] related the capillary water height of the unsaturated soil to the soil-water characteristic curve and established the analytical equations for the capillary water height in relation with the initial volume water content, residual volume water content, void ratio, and soil-water characteristic curve through the fractal model of the soil pore distribution. The soil-water characteristic curve is mainly affected by the mineral composition of the soil particles, pore size and distribution, pore structure, contractibility of soil, stress history and temperature of the soil, etc., and the mathematical model of the soil-water characteristic curve differs for the different soils [20]. Therefore, to accurately obtain the soil-water characteristic curve of the subgrade, accurate experiments are needed, which are time consuming and cumbersome, thus hindering the use in the engineering applications.The plasticity index (Ip) is a vital characteristic index of the fine-grained soil, and its value is related to the particle composition of the soil, mineral composition of the soil particles, ionic composition and concentration of water in the soil, etc. Generally, the larger is the Ip value, the higher is the clay grain content, the higher are its specific surface area and possible bound water content, the greater is the matrix suction at the same water content, and the stronger is the water holding capacity of the soil. Li et al. [21] reported that the soil material composition and pore structure are the basic factors affecting the soil-water characteristic curve, and these factors are closely related to Ip.The mean values ofhc obtained from the indoor standpipe simulation test for four kinds of the fine-grained soils are presented in Table 5. Among them, the compaction degree of 7# soil sample is 85%, 8#, 9#, and 10# soil samples are 90%. Furthermore, the scatter plots of hc and Ip are plotted and connected to a smooth curve, as shown in Figure 3.Table 5
The comparison of the measured and predicted values ofhc of the capillary water rise in the fine-grained soils.
Sample numberSoil typePlasticity indexIpMeasured value (cm)Predicted value (cm)Relative error7#High liquid limit clay26.4195.0194.3-0.4%8#Low liquid limit clay18.1308.5320.53.9%9#Low liquid limit clay12.8368.5351.4-4.7%10#Low liquid limit silt9.3342.2350.52.4%Figure 3
The rising stable height of the capillary water for the fine-grained soils as a function ofIp.The regression analysis can allow the prediction ofIp and hc of the fine-grained soil, as shown in equation (5), which is a quadratic polynomial equation with the regression correlation coefficient R2 over 97% and absolute value of the relative error between the predicted and measured values of hc below 5%.(5)hc=−0.69Ip2+15.5Ip+266.The analysis of Figure3 and equation (5) reveals a peak hc value for the fine-grained soil. On the left side of the peak, hc increases with Ip, whereas hc decreases with the increasing Ip on the right side of the peak. This phenomenon is related to the surface-bonded water content of the fine-grained soil and particle size distribution. As the plasticity index of the fine-grained soil is large, the surface energy is noted to be high and bonded water film is thick and overlapping, thus blocking the original pore by the bonded water. The formation of the dead end of the pore blocks the water flow, and the pores that are not completely blocked also become finer. The series of changes lead to an increased water migration tortuosity and resistance to the capillary ascent, along with slowing the ascent velocity and reducing the ascent height. The peak of hc is reached between Ip = 10∼14, with the peak between 350∼380 cm, which is in agreement with the results of the model [5] predicting the influence of groundwater on the balanced moisture of the subgrade.


## 2.5.1. Stable Height and Prediction of Capillary Water Rise in Coarse-Grained Soil

Table4 lists the average values of hc for the six coarse-grained soils determined from the indoor standpipe simulation tests in parallel experiments at the corresponding plastic limit water content (WP). In order to make the test closer to engineering practice, the compaction degree of the six coarse-grained soil samples was 90%. As can be observed, hc decreases with the soil grain size, which is consistent with equation (1).Table 4
The comparison of the measured and predicted values ofhc of the capillary water rise in the coarse-grained soils.
Sample numberSoil typePore ratio (e)Effective sized10 (mm)Measured value (cm)Hessian formula C value (×10−5m2)Predicted value [8] (cm)Equation (4) predictive value (cm)Equation (4) predictive error (cm)1#Silty sand0.4390.033248.53.60208.6241.0-7.52#Fine-grained sand0.5410.059138.34.41126.7146.98.73#Poorly graded medium sand0.5970.20069.38.2765.966.0-3.34#Silty sand0.3790.014219.31.16207.6220.10.85#Fine-grained sand0.5240.075118.04.64118.1127.89.86#Poorly graded coarse sand0.6340.27063.510.8761.956.0-7.5Various factors affect the rising height of the capillary water in the coarse-grained soils. A number of studies have carried out the multiple correlation analysis with respect to the influencing factors and concluded that the rising height of the capillary water in the coarse-grained soils is closely related to the soil conditions and degree of filling density. Hazen–Williams employed the pore ratio of the soil (e) and effective size d10 of the soil particles to establish the famous Hazen formula as follows [18]:(3)hc=Ce⋅d10,where C is a coefficient related to the shape and surface cleanliness of the soil particles, generally taken as 1 × 10−5∼5 × 10−5 (m2).However, the Hazen formula suffers from the too large selection range of theC value (Table 4), and the specific engineering selection is even more difficult.Referring to the Hazen formula, the scatter plot ofhc and the combined index (e⋅d10)−1 consisting of the pore ratio (e) and effective size (d10) was plotted as a smooth curve (Figure 2). As can be observed, hc of the coarse-grained soil is quadratically related to the combined index (e⋅d10)−1.Figure 2
The stable height of the capillary water for the coarse-grained soil.The relation of the coarse-grained soilhc value with the combined index (e⋅d10)−1 was established by the regression analysis, as shown in (4). The regression parameter R2 was as high as 99%.The measured values from the indoor standpipe simulation test and predicted values from the equation in the literature [8] as well as the predicted values and errors from equation (4) developed in this study are summarized in Table 4. As can be observed, the regression equation developed in this study is superior to the analysis of Dong et al. [8]. In addition, the findings in this study are close to the long-term follow-up measurements, with the errors below 10 cm. Thus, equation (4) addresses the shortcomings of the Hazen formula in engineering applications.(4)hc=−0.017e⋅d10−2+4.2e⋅d10−1+32.Analysis of the data in Table4 shows that the height of capillary water rise in coarse-grained soils increases with finer particles and more mud content. This is because both conditions make the pore space of the soil smaller, and the test data also prove this. In addition, it can be concluded from Figure 2 and equation (4) that the peak of the coarse-grained soil hc value is around 300 cm, which is consistent with the results of the groundwater impact model for the subgrade balanced moisture in the subgrade, as described by Specifications for Design of Highway Subgrades [5].


## 2.5.2. Stable Height and Prediction of Capillary Water Rise in Fine-Grained Soil

The fine-grained soil pores are smaller, and numerous capillaries are formed in the soil microstructure. These capillaries produce a large matrix suction, which makes the capillary action process longer, however, with a greater rise in height. In addition, the surface electric field of the fine-grained soil is strong, and the bound water is influenced by the viscous force. The Hessian formula, which only considers the dense state and effective size, is not applicable. Liu et al. [19] related the capillary water height of the unsaturated soil to the soil-water characteristic curve and established the analytical equations for the capillary water height in relation with the initial volume water content, residual volume water content, void ratio, and soil-water characteristic curve through the fractal model of the soil pore distribution. The soil-water characteristic curve is mainly affected by the mineral composition of the soil particles, pore size and distribution, pore structure, contractibility of soil, stress history and temperature of the soil, etc., and the mathematical model of the soil-water characteristic curve differs for the different soils [20]. Therefore, to accurately obtain the soil-water characteristic curve of the subgrade, accurate experiments are needed, which are time consuming and cumbersome, thus hindering the use in the engineering applications.The plasticity index (Ip) is a vital characteristic index of the fine-grained soil, and its value is related to the particle composition of the soil, mineral composition of the soil particles, ionic composition and concentration of water in the soil, etc. Generally, the larger is the Ip value, the higher is the clay grain content, the higher are its specific surface area and possible bound water content, the greater is the matrix suction at the same water content, and the stronger is the water holding capacity of the soil. Li et al. [21] reported that the soil material composition and pore structure are the basic factors affecting the soil-water characteristic curve, and these factors are closely related to Ip.The mean values ofhc obtained from the indoor standpipe simulation test for four kinds of the fine-grained soils are presented in Table 5. Among them, the compaction degree of 7# soil sample is 85%, 8#, 9#, and 10# soil samples are 90%. Furthermore, the scatter plots of hc and Ip are plotted and connected to a smooth curve, as shown in Figure 3.Table 5
The comparison of the measured and predicted values ofhc of the capillary water rise in the fine-grained soils.
Sample numberSoil typePlasticity indexIpMeasured value (cm)Predicted value (cm)Relative error7#High liquid limit clay26.4195.0194.3-0.4%8#Low liquid limit clay18.1308.5320.53.9%9#Low liquid limit clay12.8368.5351.4-4.7%10#Low liquid limit silt9.3342.2350.52.4%Figure 3
The rising stable height of the capillary water for the fine-grained soils as a function ofIp.The regression analysis can allow the prediction ofIp and hc of the fine-grained soil, as shown in equation (5), which is a quadratic polynomial equation with the regression correlation coefficient R2 over 97% and absolute value of the relative error between the predicted and measured values of hc below 5%.(5)hc=−0.69Ip2+15.5Ip+266.The analysis of Figure3 and equation (5) reveals a peak hc value for the fine-grained soil. On the left side of the peak, hc increases with Ip, whereas hc decreases with the increasing Ip on the right side of the peak. This phenomenon is related to the surface-bonded water content of the fine-grained soil and particle size distribution. As the plasticity index of the fine-grained soil is large, the surface energy is noted to be high and bonded water film is thick and overlapping, thus blocking the original pore by the bonded water. The formation of the dead end of the pore blocks the water flow, and the pores that are not completely blocked also become finer. The series of changes lead to an increased water migration tortuosity and resistance to the capillary ascent, along with slowing the ascent velocity and reducing the ascent height. The peak of hc is reached between Ip = 10∼14, with the peak between 350∼380 cm, which is in agreement with the results of the model [5] predicting the influence of groundwater on the balanced moisture of the subgrade.


## 3. Conclusions

(1)
The rising of the capillary water in the unsaturated subgrade soil to a stable height is a slow process, and the soil particle size of the coarse-grained and fine-grained soil fillers as well as the time and rise height differences are relatively large. For example, in the 10 soil samples analyzed in this thesis, the highesthc and lowest hc of the coarse-grained soils were 248.5 cm and 63.5 cm, respectively, while the highest hc of the fine-grained soils reached 368.5 cm and the lowest hc was 195 cm.(2)
Simulation of the rise of the capillary water in the subgrade soil by employing the indoor standpipe method as well as the determination of its stable rise height by the long-term observation has been proven to be an effective method to accurately obtain the stable height of the underground capillary water rise under the balanced moisture condition of the subgrade.(3)
The capillary water stability height (hc) for the coarse-grained soil is closely correlated with the combined index (e⋅d10)−1 consisting of the pore ratio e and effective size d10, while the hc value of the fine-grained soil has the highest correlation with the plasticity index (Ip). The regression analysis provides the prediction equation of hc. The relative errors of hc for the coarse-grained soils ranged from 0.8% to 9.8% and those for the fine-grained soils ranged from 0.4% to 4.7%, and the regression equation exhibits a high degree of correlation, which can be used for estimating the capillary stability height.



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*Source: 5993984-2022-03-23.xml*