Prediction Method for Hydraulic Conductivity considering the Effect of Sizes of Ellipsoid Soil Particles from the Microscopic Perspective
In the existing research studies of hydraulic conductivity, most of them assume that soil consists of spheroidal particles and the value of hydraulic conductivity can be designated by the particle size. In the actual soil layers, the shape of soil particles is mostly ellipsoid or rod-like rather than ideal sphere. Therefore, the prediction of soil permeability using current method often deviates from the actual situation and cannot capture the anisotropy nature of soil without consideration of the effect of the axis size of soil particles in two different directions. To solve this problem, a new theoretical model with three different soil particle arrangements is introduced to derive a new hydraulic conductivity-particle size relationship considering the size difference in two directions. This model, from a microscopic perspective, divides pores into numerous pore units and obtains hydraulic conductivity of each tube unit, eventually predicting the permeability of soil layer based on an equivalence principle. The proposed equation is validated in comparison with experimental data from the existing literature and is proved to have a satisfied accuracy to predict hydraulic conductivity for a wide range of soils, from bentonite-silt mixed soils to sandy soils. The proposed model provides a new perspective for accurately predicting the hydraulic conductivity.
Hydraulic conductivity is one of the most important and useful parameters in the study of the seepage process of porous media  and plays a vital role in the research of consolidation, settlement of soils and foundation, rheology [2–5], dewatering design [6–8], migration of gas and liquid [9–11], sandstone storage [12–15], storage of CO2 and nuclear waste [16–20], and other geological and geotechnical engineering problems. Generally, it is difficult to measure dynamic variation of the hydraulic conductivity by experimental means. For example, during the process of consolidation, the hydraulic conductivity gradually reduces with the decrease of void ratio. However, it is obviously difficult to measure the hydraulic conductivity of soil at each consolidation stage. Hence, many researchers propose theoretical equations to predict the change of the hydraulic conductivity. Unfortunately, there is lack of accuracy using current method due to the various simplifications and weaknesses. Thereby, it is important to propose a predictive model which can take into account essential factors that control the accuracy of the prediction of hydraulic conductivity.
It has been well known that hydraulic conductivity is influenced by several factors, such as porosity [21–23], saturation , clay content , quantity of electric charge among soil particles , and soil particles geometry which includes grain size distribution [27–29], grain shape , and so on. Based on lots of experiments and theories, many theoretical equations have been proposed by studying the essence of percolation process [31–41]. These theoretical equations take into account some geotechnical parameters and, to some extent, can predict the value of hydraulic conductivity. For example, Ren et al.  introduced the concept of effective void ratio which can help to reduce the calculation deviation caused by relatively small pores and dead-end pores. Kozeny-Carman relation is the most common method used to predict hydraulic of soil, which is proposed by Kozeny and improved by Carman. Based on the Poiseuille law, the K-C equation reflects the possible relationship between hydraulic and void ratio:where k is the hydraulic conductivity (m/s) and e represents void ratio which is dimensionless; is the unit weight of liquid (); is a dimensionless shape parameter which is estimated about 0.2 ; is the fluid viscosity (); is the specific surface area of soil (); and represents particle density of soil ().
The K-C equation was deduced from the assumption that the cross section of tubes where water flows is constant and regular and has been experimentally verified for coarse-grained soils such as sands. Because of extremely small pore size, irregular shape of pore, and electrochemical reactions between solid particles and liquid, the K-C relation is not appropriate for clay soils or silty soils which are composed of soil particles with small size. In addition, the K-C equation uses as the key parameter to reflect the influences of pore geometry on permeability rather than the particle size that cannot capture the effect of size effects [43–47]. According to the K-C relation, Wemaere  suggested that the notion of maximum soil particle size as well as minimum soil particle size should be introduced to account for the influence of particle size on the hydraulic conductivity of soil layer. Hence, equation (1) can be written as follows:where is a test coefficient without dimension; is the volume components of particles with a certain size which is dimensionless as well; represent maximum soil particle and minimum soil particle size (mm), respectively; and SF is the shape coefficient. The effects of soil particle size and shape can be directly shown in this equation. However, the studied subjects of equation (2) are spherical soil particles that are not common in practical engineering. In fact, the natural soil layer is composed of countless soil particles; most of these are ellipsoidal or rod-like solids, rather than ideal spherical particles. The axial lengths of these nonspherical particles are not the same in different directions that would influence the shape and size of pores between particles. The differences of geometrical features of pores result in the soil anisotropy, especially anisotropic permeability. Hence, it is inappropriate to ignore the hydraulic anisotropy of soils by assuming perfect sphere particles of soil. Besides, equation (2) is not convenient to be used because several parameters cannot be easily obtained such as . To obtain these parameters, a lot of experiments need to be conducted; this is time-consuming and may cause larger system error.
The objective of this study is to present a theoretical method for predicting the hydraulic conductivity considering the influences of soil particles geometry from a microscopic view. Based on a series of assumptions, we link the hydraulic conductivity with the size of the major and minor axis and further propose a theoretical equation for the hydraulic conductivity that is suitable for a wide range of soils. The proposed model reflects the influence of soil particle size on hydraulic conductivity and embodies the characteristics of soil anisotropy.
2. Theoretical Method
In the existing research studies, many scholars approximately assumed that the actual shape of pore is equivalent to an ideal circular and then calculated the value of hydraulic conductivity by relative simplified method. However, from the microscopic view, the outer contour of pore tightly fits to the surfaces of soil particles and distributes periodically along the major axle of particles. Hence, the seepage flow pore actually is a channel of variable cross section, and it is impractical to simplify the pore as a cylinder.
In addition, the size and shape of pore is also affected by the arrangement of soil particles. By means of scanning electron microscopy (SEM), Sun et al.  analyzed the geometrical characteristics of solid particles in sandy soils. It is found that the geometrical shape of sandy soils is closer to that of regular ellipsoids. The spatial distribution of soil particles, according to the approximate range of void ratio (0.6–1.5), is divided into 3 different groups: three-particle combination, four-particle combination, and five-particle combination (as shown in Figure 1). In spite of the same particles size, the size and shape of the pore could vary with different combinations. As a result, hydraulic conductivity may be completely different. Based on these assumptions, the paper proposes a theoretical equation to predict the hydraulic conductivity under 3 different arrangements of particles.
2.1. Theoretical Hypothesis
The assumptions of soil properties of the present model can be summarized as follows:(i)Soils consist of even and equal-sized particles which orient in the major axle direction (Figure 2). This assumption limits the spatial arrangement of particles and makes the model idealized. According to the assumption, the gradation of soil is neglected, and the hydraulic conductivity is directly determined by the size of soil particles.(ii)Soil particles arrange periodically. The gap between particles, along the long axis, is small and negligible. Compared with the particle size, the gap size along the long axis between soil particles is too small, which shows little effect on hydraulic conductivity. In order to simplify the calculation and reduce the amount of unimportant parameters, the influence of gap is not considered in this paper.(iii)Soil particles are ellipsoids with unequal axes length in two different directions. The size of soil particles along the Z and Y axis is equal and is noted by minor half-axis a, while the size of soil particles along the X is noted by semimajor axis b.(iv)Soil particles are rigid bodies without deformation and displacement when fluid flows in the pores of soil particles. Hence, this paper does not consider the phenomenon of particles deformation and fragmentation and assumes that the deformation of soil layer is entirely caused by pore compression.(v)Do not consider the effect of hydrated films surrounding surfaces of soil particles. This paper ignores relatively small pores and dead-end pores and assumes that all the pores are effective pores where free water can flow without obstructs.
In addition, there are several assumptions of fluid properties as follows:(i)The basic state of the fluid is continuous, full, and laminar.(ii)Fluid properties (including the unit weight and volumetric flow rate Q) remain constant(iii)Fluid only flows in the pore tubes which are surrounded by particle systems(iv)The theory neglects surface effects and body forces such as gravity, centrifugal force, Coriolis force, and electromagnetic force
2.2. Horizontal Hydraulic Conductivity
This study establishes the theoretical model based on the relationship between the cross-sectional size of pore and hydraulic conductivity. Hence, along the surfaces of soil particles, hydraulic conductivity constantly changes with the variable cross-sectional size of tubes. The derivation of this chapter takes the four-particle arrangement as an example (Figures 2–4). In order to clearly describe the process of derivation, some basic definitions of the pore have been introduced as shown in Figure 3. We defined the pore among a four-particle system as the pore period. The pore period can be divided into n () pieces along the horizontal seepage direction. Each piece is represented as one unit tube, and length of that is designated dx. Countless connected pore periods constitute a complete microchannel which is used to denote the seepage channel between the upper and lower layer of soil particles. Therefore, the solution of this paper is as follows: firstly, based on the relative theories, this study proposes a predictive equation of hydraulic conductivity for each unit tube. Then, according to principle of flow equivalence, we can further obtain the predictive formula for the pore period. Finally, the theoretical model for predicting hydraulic conductivity of the microchannel will be established according to the periodicity of soil particles, which can show the permeability of whole soil layer.
Because the present model is derived from Darcy’s law, it is necessary to analyze the flow state in the pores. In this section, we seek to restrict the relationship between the geometric parameters and the velocity of the cross section in order to keep fluid conforming to laminar flow.
2.2.1. Analysis of Flow State
Darcy  held that seepage velocity is directly proportional to hydraulic gradient under laminar flow conditions. When the flow rate increases to a certain value, flow state would change from laminar flow to turbulence. At this point, the regularity of percolation shows nonlinear characteristics and deviates from Darcy’s law. Most scholars use the Reynolds number as a critical condition for laminar; Bahrami et al.  suggested that the Reynolds number can be calculated by the following equation:where is the density of liquid (); is the average velocity of flow (m/s); A is the cross-sectional area of the tube (); and is the fluid viscosity (). Many scholars agree that Darcy’s law is suitable when ( represents critical Reynolds number which usually is a certain value between 1 and 10) [52–54]. Equation (3) shows a strong relevance between the geometric properties of tube and flow state. Accordingly, the following parts of this section will analyze the influences of soil particle size on the variety of soil parameters, such as viscosity of the particle-liquid system, and eventually give the relationship between the particle size and velocity under Darcy’s law.
(1) Viscosity of the Solid-Liquid System. Taking the four-particle system as an example, the geometric shape of pores in a cycle is shown in Figure 4. The square formed by the dotted line represents the maximum permeable area of the pore, while the black shaded part indicates the minimum permeable area. According to equation (3), represents the fluid viscosity which can show the frictional resistance during the flow of liquid. However, for the soil-liquid system, due to the existence of soil particles, actually indicates the viscosity of the solid-liquid system rather than that of the pure liquid. Many researchers proposed various theoretical equations related to the calculation of viscous coefficient of the solid-liquid system. Einstein  argued that there was a strong relevance between volume fraction of solids with viscosity coefficient based on the negligence of contact force among soil particles. On the basis of the study of Einstein, Batchelor  considered the interaction forces between particles and further obtained a more accurate calculation method. However, the abovementioned methods are mainly suitable for spherical soil particles which are uncommon in the practical soil layer. The method proposed by Brenner and Condiff  considering the ellipsoidal particles is adopted in this study, to obtain viscosity of the solid-liquid system. The viscosity coefficient can be expressed aswhere is the viscous coefficient of the sole liquid, which is a parameter only related to the temperature and the pressure; is a dimensionless variable, expressed as ; is the volume fraction of solids; and represents the higher orders of . We ignore during calculation due to the fact that they have little effect on the results. Compared to viscosity coefficient of pure liquid, equation (4), considers the influences of particle size and aspect ratio on viscosity coefficient and more conforms to actual situation.
(2) Cross-Sectional Areas of the Pore. For convenience of indication and calculation, we take the pore surrounded by four particles as a period and establish the Cartesian coordinate system in this pore period as shown in Figure 4(b), where the origin sets at the pore center. Hence, a complete sequence of periods can be seemed as a connected microchannel of arbitrary cross section. For one pore period, along the seepage direction of fluid, the cross-sectional radius of ellipsoid particles varies uniformly with the change of X coordinates. It can be expressed as follows:where a and b are the sizes of half minor and major axis, respectively. The section area of the pore is determined entirely by the arrangement of soil particles and the size of soil particles. The formula of the section area of pore can therefore be written as
Furthermore, the average area of pore section in a period will be obtained as
(3) Critical Velocity. The abovementioned geometrical parameters of the pore are only related to soil particles. The average areas of pore sections as well as viscosity are determined when particles sizes are constant. At the same time, fluid velocity is the only factor to affect the average Reynolds number of the whole microchannel. Accordingly, the relationship between and can be described as the following equation:
When the particle size and velocity satisfy certain relationship, the seepage state of fluid can adhere to Darcy’s law and the rest of the deduction in this paper will be effective and valid. Therefore, in this paper, 6 groups of soil particles with different sizes are selected to analyze the relationship between critical velocity and soil particle size. Figure 5 shows the trial calculation with different b. It is found out that the size of the major and minor axis has different effects on the critical velocity. Only when and satisfy a certain relationship (as shown in the shaded area of Figure 5), fluid state can be seemed as laminar.
2.2.2. Horizontal Hydraulic Conductivity of Unit Tube
As mentioned earlier, the pore in one period is divided into n unit tubes. As a result, the derivation process of horizontal hydraulic conductivity of any unit tubes is as follows:
Akbari et al.  proposed an equation for calculating the flow resistance R, which is the ratio of pressure drop and volumetric flow rate Q, using the perturbation solution .where with is designated as the specific polar moment of cross-sectional inertia  and and are the pore cross-sectional area at locations, respectively. The first item on the right side of the equation represents the flow resistance due to friction and the second represents inertia term. The inertial pressure drop in periodic channel will be negligible for low Reynolds numbers. Hence, equation (9) is reformulated as
Equation (10) can be rewritten in the differential form. Accordingly, the flow resistance equation of the arbitrary unit tube can be furthermore expressed as the following equation:
Then, the volumetric flow rate can be expressed as
Thus, according to Darcy’s law, the rate of flow in the unit tube could be indicated in the following equation, where is the total area of the cross section including the tube walls:where along the direction of seepage flow, the hydraulic gradient can be described as
Equation (15) shows that is mainly related to three parameters, namely, , , and . When the soil layer is entirely composed of four-particle systems as shown in Figure 2, equation (6) can be used to calculate the area of arbitrary cross section of unit tubes , while can be expressed by the sum of and which means the cross-sectional areas of tube walls:
In addition, the polar moment of inertia can be expressed as
For the sake of simplification, equation (15) is used for calculation in this study instead of equation (18). Based on the above derivation, the horizontal hydraulic conductivity of arbitrary unit tube can be predicted by the major axis and minor axis size of soil particles. Combining with the equivalent flow method which will be described in the next Section 2.2.3, we can get the horizontal hydraulic conductivity of whole soil layer .
2.2.3. Equivalent Horizontal Hydraulic Conductivity
Based on the equivalent flow method, we propose a formula to compute the whole horizontal hydraulic conductivity which can reflect horizontal permeability of soil.
Different pore elements are parallel to each other in seepage direction as shown in Figure 6. In addition, the volumetric flow rate of each pore unit is equal. The velocity of flow, at location, can be expressed as follows:
In a similar way, the pressure drop at both ends of the microchannel can be defined aswhere the pressure drop at both ends of the microchannel (Pa); L = channel length (m); and the equivalent cross-sectional area of the channel, which is represented by average cross-sectional area as equation (7). The relationship between the pressure drop of whole channel and unit tube can be expressed as follows:
To implement the proposed equation in practice, a further simplification of equation (23) is suggested to calculate horizontal hydraulic conductivity. First, it assumes , where means the viscosity of the solid-liquid system, which can be obtained from equation (4). When the total length of microchannel is designated L, according to the theoretical hypothesis, there are m periods of pore. Hence, microchannel L can be expressed as . Moreover, . Due to the periodicity of the microchannel and the parity of , we can obtain that . Hence, equation (23) can be rewritten as
The equivalent hydraulic conductivity of the whole soil layer in horizontal direction is expressed as equation (24). In practice, can be regarded as the horizontal hydraulic conductivity of whole layer in order to reflect seepage law. Various microfactors such as particle shape and particle size distribution are incorporated in equation (4). Therefore, the proposed equation can well capture the microscopic mechanism of soil seepage.
2.2.4. Vertical Hydraulic Conductivity
The derivation process of formula of the vertical hydraulic conductivity is the same as the horizontal counterpart in this paper. However, compared with horizontal pores, the shape and size of vertical microchannels are quite different as shown in Figure 7. The shaded area and dotted rectangle represent the minimum seepage area and the maximum seepage area, respectively, in the vertical direction. The cross-sectional area of pore can be expressed as the following equation:where , .
Vertical average sectional area of pore can be obtained as follows:
Then, in the vertical direction, the area of the cross section of tube walls can be computed as
Using and , we can further obtain total area of the cross section including the tube walls which is designated :
Similarly, the polar moment of inertia of arbitrary vertical unit tube can be computed as
Based on the same equivalence principle, the equivalent hydraulic conductivity of the whole soil layer in vertical direction is given by
2.3. Other Particle Combinations
According to the hypothesis in Section 2.1, the soil layer is composed of numerous “particle systems.” These particle systems can be divided into three-particle combination, four-particle combination, and five-particle combination according to void ratio e. Similar to the proposed method for four-particle combination, the computation equation of hydraulic conductivity for soil under two other combinations can be obtained. However, the arrangement of particles for three or five particles in the vertical plane is not as regular as that of four particles. For three-particle system as well as five-particle system, in the YZ plane, there is a certain gap between the upper and lower rows of soil particles, which means the seepage route is not connected along the Z axis as shown in Figure 8. As a result, the calculation method proposed in this paper is not applicable for calculating the vertical hydraulic conductivity of these arrangements. However, for the sake of clarity, we did not deduce new formulas for the other two cases. Equation (30) is recommended to calculate the vertical hydraulic conductivity for soil with all particle systems. This means the effect of soil particle arrangement on hydraulic conductivity is ignored during calculation; only the size of soil particles is taken into consideration. As a result, the size of pore, especially for three-particle combination, could be overestimated, and the predicted results will be inaccurate. Therefore, the vertical calculation results for three-particle and five-particle combination would merely be the reference resources. More precise calculation methods still need to be further studied.
2.3.1. Three-Particle Arrangement
The soil layer consisting of three-particle system usually characterizes of small pores and low hydraulic conductivity. As shown in Figure 9(a), the shaded part represents the minimum seepage area and the dashed triangle indicates the maximum pore area. Under this condition, the cross-sectional area and the polar moment of inertia can be expressed as
Furthermore, can be obtained as
Finally, substituting formulas of , , and into equation (24), the equivalent hydraulic conductivity in horizontal direction with three particles can be obtained.
2.3.2. Five-Particle Arrangement
The size of pore formed by five particles is relatively large. The maximum seepage cross-sectional area can be simplified as regular pentagon while the minimum area is surrounded by five tangent circles, as the shaded area shown in Figure 9(b). Similarly, the cross-sectional area and the polar moment of inertia are given bywhere .
Thus, equation (33) yields
2.4. Hydraulic Conductivity vs Void Ratio e
In addition to particle size, void ratio is another essential factor influencing the permeability of soil. In practice, void ratio will constantly change with the process of consolidation; as a result, the hydraulic conductivity will also vary with the change of void ratio. Based on the large number of experiments, several theoretical equations have been proposed to describe the relationship between void ratio and hydraulic conductivity. For example, Lambe and Whitman  argued that there is a logarithmic relationship between the hydraulic conductivity and the void ratio:where denotes the coefficient of permeability change and and denote the initial void ratio and the initial hydraulic conductivity, respectively. This equation only considers the relationship between hydraulic conductivity and void ratio and cannot reflect the coupling effect of particle size and void ratio from the microscopic and macro perspective, respectively.
This section takes the four-particle combination as an example and further derives the relation between hydraulic conductivity and pore ratio. As shown in Figure 10, in the initial state, the upper and lower particles are not closely connected, and there is a certain size of gap designated as which is determined by the void ratio e of soil. During the process of consolidation, the gap between particles is gradually compressed. As a result, hydraulic conductivity changed with the decline of and e.
Hence, the area of arbitrary cross section of unit tubes can be rewritten as
Furthermore, the cross-sectional areas of tube walls can be expressed as follows:
According to the assumption that the soil layer is composed of innumerable particle arrangement systems, void ratio in four-particle combination can represent the void ratio of the whole soil layer. Thus,
Then, the size of gap can be expressed by void ratio e:
Thus, the equivalent cross-sectional area of the microchannel and the polar moment of inertia can be rewritten as
Substituting equations (41) and (42) into equation (24), we can finally obtain the predictive equation of horizontal hydraulic conductivity considering the change of void ratio. According to the same principle, we can further derive the theoretical model for vertical hydraulic conductivity and other particle combinations.
3. Results and Analysis
3.1. Hydraulic Conductivity vs. Particle Size a, b
The change of hydraulic conductivity of soil with the parameter of soil particle size is listed in Tables 1 and 2. Due to the fact that the model presented in this paper is not suitable for calculating the vertical hydraulic conductivity of soil under the combination of three and five particles, this section does not analyze the change of vertical hydraulic conductivity of soil layer under the two different soil particle arrangements. From the theoretical point of view, 10 groups of theoretical particles within the range of sand soil are studied. The detailed information is shown in Tables 1 and 2.
3.1.1. Four-Particle Arrangement
Figure 11 shows the relationship between critical velocity and particle size for four-particle arrangement under laminar flow via equation (8). Taking b = 2.0 mm as an example, when the coordinates of particle size and critical velocity fall within the shaded area, the flow pattern can be considered as laminar flow.
Moreover, the variations of hydraulic conductivity with the axial length of particles in two directions are shown in Figures 12 and 13, respectively. Figure 12 shows that the hydraulic conductivity of soil gradually linearly decreases with the increase of particle size. For higher a, the decrease trend of hydraulic conductivity is more significant. Figure 13 shows that the relation between a and hydraulic conductivity can be expressed as a power function. Generally, minor axis size a has a greater influence on hydraulic conductivity. This is because the hydraulic conductivity predicted by this study is mainly related to the cross-sectional area of pore, which is mainly influenced by the size of minor axis.
Furthermore, comparing the variation of hydraulic conductivity in horizontal and vertical directions, it can be seen that is generally about 1.54 times of under the same particle size. This behavior is due to fact that when fluid moves along the vertical direction, the cross-sectional area of pore is determined by the major and minor axis of particles. Accordingly, the cross-sectional area of vertical pore is larger than that of horizontal tube, resulting in a higher permeability of soil layer vertically. It should be pointed out that the overall trends for vertical and horizontal permeability are basically the same. From formulas (24) and (30), it can be seen that the hydraulic conductivity in this model is only related to the area and shape of the pore. In horizontal and vertical directions, when the size of soil particles changes, only the size of voids between soil particles changed. Hence, the changes of hydraulic conductivity are consistent without considering the slip between particles.
3.1.2. Three-Particle Arrangement and Five-Particle Arrangement
Figure 14 shows the relationship between critical velocity and particle size for laminar flow under three-particle arrangement and five-particle arrangement. For soils consisting of the three-particle system and five-particle system, the effect of particle size on permeability is analyzed only in horizontal direction. With the increase of particle size, the variation of hydraulic conductivity under these two types of particle arrangements is basically the same as that of four-particle arrangement. Generally, the microchannel enclosed by three particles has smaller cross-sectional area; accordingly, its hydraulic conductivity is low (Figures 15(a) and 15(b)). Under the same size, the hydraulic conductivity of the soil layer that consists of three-particle system is about 15% of that of the four-particle system. For the soil consisting of five-particle arrangement, the permeability is obviously higher because of its greater cross-sectional area. As a result, the hydraulic conductivity computed by five-particle model is 4.57 times that of four-particle model (Figures 15(c) and 15(d)). According to the above analysis, it can be concluded that different arrangements of soil particles will exert a significant impact on the permeability of the soil layer.
3.2. Effects of Void Ratio e
To analyze the relationship between void ratio and hydraulic conductivity based on current model, the sand soil with different particle arrangement conditions are analyzed and the results are shown in Figures 16 and 17. For three kinds of particle combinations, the values of void ratio are smallest when the upper and lower particles closely contact (). In that case, void ratios for different combinations are
Hence, we can obtain the curve of three-particle combination from e = 0.654, that of four-particle combination from e = 0.910, and that of five-particle combination from e = 1.539. It is clear that the changes of hydraulic conductivity are not same with the increase of void ratio under the three combination conditions as shown in Figure 16. Generally, the hydraulic conductivity gradually rises with the increase of void ratio. Comparing three kinds of combination conditions, it is obvious that the hydraulic conductivity increases more rapidly and the slope of curve is lager under the five-particle arrangement. Specifically, when e = 2.0, the size of is 0.417 cm/s. With the process of consolidation, the void ratio decreases gradually. When e = 1.5, is only 0.106 cm/s, which decreases by 74.6%. In contrast, the hydraulic conductivity changes slowly under the three-particle arrangement, and the overall growth rate is low.
As can be seen from Figure 17, with the increase of void ratio, the changes of hydraulic conductivity are both smooth curved, and the curvature radii of curves in such two directions are different. When void ratio falls into different ranges, the slopes of the curves are totally different. Because the derivative equation of hydraulic conductivity is very complex, the change rate of hydraulic conductivity will change gradually, rather than keep constant with the increase of void ratio. According to the slopes of the curves, we divide the curves into three parts: ① when 0.95 < e < 1.55, ( is the angularity of tangent line which can describe the slope of curve; subscript V and H mean vertical direction and horizontal direction, respectively). In that case, with the increase of void ratio, the growth rate of hydraulic conductivity in horizontal direction is greater than that in vertical direction; ② when 1.55 < e < 1.85, and the growth rates of hydraulic conductivity in different directions are basically the same; ③ when 1.85 < e < 2.5, . The growth rate of hydraulic conductivity in vertical direction is greater than that in horizontal direction, and the gap between two directions rises gradually with the increase of void ratio.
In addition, the vertical hydraulic conductivity is always greater than the horizontal hydraulic conductivity with the increase of void ratio. This is because, from the perspective of the geometry of the soil layer, when the coordinate values of the two directions are equal (X = Y), the pore cross-sectional area in the vertical direction is always larger than that in the horizontal direction, so the vertical permeability is better than that in the horizontal direction; from the perspective of the model equation, the values of hydraulic conductivity in equations (24) and (30) are mainly affected by the higher order terms and . Due to that , the vertical hydraulic conductivity is larger than the horizontal hydraulic conductivity. Furthermore, when remains unchanged, the increment of vertical pore section area is larger than that of horizontal direction , which also affects the permeability of soil layer.
4. Verification and Applications
In previous studies, the shape of soil particles is deemed as spheres rather than ellipsoids. Hence, the relative experimental data concerning the influences of a and b on permeability are insufficient. Some scholars [61, 62] adopted average diameter of pore and specific surface to indicate the influence of soil particle size. There is a strong correlation between , , and soil particle size. Thereby, the theoretical model proposed by this study can be, to some extent, validated by , . First, based on the volume equivalent principle, we assume the variable cross-sectional pore as the cylinder with constant cross section. Then, the relationship between mean cross-sectional area ( can be expressed by particle size a, b) and mean pore diameter can be expressed as follows:and it can be rewritten as follows:
However, the specific surface area is not reported in most studies. We estimate the specific surface area by using other published soil descriptions. In the case of fine-particle soils, estimates were based on liquid limits :
For sandy soils made of rotund grains, the specific surface area can be estimated from the cumulative grain size distribution:where the coefficient of uniformity is and (mm) are the grain diameters for 10%, 50%, and 60% cumulative passing fractions; indicates the specific weight of soil; and is the density of soil. By equation (46) or equation (47), we can calculate the values of by a series of geotechnical parameters. Furthermore, the method for computing particle sizes a and b proposed by Santamarina  is adopted in this study, expressed aswhere .
Then, by referring to the experimental data in other literatures, we can obtain the values of , and other parameters of different soils. Combining equations (45) and (48), , will be transformed to the sizes of particles (a, b). By comparing the theoretical calculation results with the actual data [64–66] and other existing calculation models, such as Indraratna model, Hazen model, Kozeny-Carman model, we can verify the rationality of the model. The calculation models of the three-particle system presented in this paper have good reliability in calculating the hydraulic conductivity with different sizes of soil particles (see Tables 3 and 4). It can be concluded from Table 3 that for most of the size of soil particles, the four-particle model is more suitable for calculation; for example, when particle size a falls within 0∼0.5 mm, the error between predicted and measured hydraulic conductivity calculated by the four-particle model is less than 50%. However, for soils with larger particle size (1 mm < a < 5 mm), the calculated results using the three-particle model are closer to the measured values. Generally, although there is a certain disparity between the predicted results and the measured values, the proposed model still has high reliability.
In order to validate the rationality of the model, we use the three-particle model and four-particle model to describe the relationship between horizontal hydraulic conductivity and void ratio for sandy soil and smectite soil, respectively, as shown in Figure 18. It is clear that the predicted results have high accuracy and coincide with the test data. From Figure 18(a), we can conclude that the three-particle model has high accuracy for predicting hydraulic conductivity of sand soil. When the values of void ratio are high (e > 0.85), the predicted results of K-C model obviously deviate from the test data, while the predicted values still keep good agreement with the test data. This result also indirectly validates the previous conclusion: three-particle model is more suitable for soils with large size particles. Figure 18(b) shows that for smectite soil, predicted results by four-particle model are in good agreement with the measured values when the void ratio is small (e = 1∼4). In contrast, when the void ratio falls within the range of 4 to 10, the calculated results deviated from the measured value. Especially, when the void ratio is large enough (e > 6), the calculated results are obviously lower than the measured values and the accuracy is not as good as equations (1) and (37). Therefore, the model proposed by this paper is not suitable for the soil with high void ratio.
To further examine the suitability of the proposed equation, referring to the data of geotechnical parameters (such as void ratio, specific surface area, and equivalent particle size) in many literatures, this study calculates abundant predicted values of hydraulic conductivity and compares them with the measured values in corresponding literatures, as shown in Figure 19 [9, 61, 69–77].
On the basis of a large number of calculations, when the measured results of a certain kind of soil are close to the values predicted by one of the three models, we think that this kind of soil is mainly composed of the corresponding particle combination of the model and assume that this model is suitable for calculating the hydraulic conductivity of this kind of soil. Based on the comparison between computed result and experimental data (as shown in Figure 19), it can be conducted that the three prediction models (three-particle model, four-particle model, and five-particle model) proposed in this study are suitable for sandy soils, clay-silt mixed soils, and bentonite-silt mixed soils, respectively. There are three findings that can be drawn from Figure 19:(1)For the sandy soil, the theoretical values computing by three-particle arrangement model is closer to the actual measured value. Most predicted values fall within . However, for clayey-silty mixed soils and bentonite-silt mixed soils, the predicted values are obviously lower than the measured values. This observation also verifies the above conclusion that three-particle model is more suitable to predict for large granular soil. In addition, it can be speculated that for sandy soil, the arrangement of three particles may be dominant in quantity.(2)The theoretical model of four-particle arrangement can calculate the hydraulic conductivity for the soil layer with clayey and silty mixtures more accurately. Despite certain predicted results with a large error, most calculated values still distribute in reasonable interval.(3)Comparing with other two kinds of soil, the five-particle model can reliably predict the hydraulic conductivity of bentonite-silty mixed soil that when the size of particles is smallest, the results are of high accuracy. However, is far greater than when calculating sandy soil with larger particle size.
5. Discussion and Limitations
Although this paper verifies the reliability of the model based on the substantial experimental data, there are still many shortcomings and limitations due to certain idealizations and assumptions. This section will discuss the limitations of the model and propose corresponding remedies. More appropriate optimization needs to be further studied.
5.1. Inhomogeneous Particle Geometry
Practically, the soil layer is composed of soil particles with different sizes, and the shape of soil particles is not limited to ellipsoid. The assumption in this paper is only applicable to the ideal soil layer. However, there are still some shortcomings when it is used in practice. Therefore, it is suggested that the parameters such as uniformity coefficient Cu should be used to link the actual particle size with the even particle size in the computation model so that it can be applied to the actual condition circumstance.
5.2. Not considering Unavailable Pore
In this paper, it is assumed that the pore channels are all effective pore which means free water can flow in any pore and there is no blockage. However, because of the existence of “relatively small pores” and “dead-end pores” as well as the effect of clay surfaces and interlayers, not all of water is stored in fine-grained materials that participate in flow. Carman  defined water that does not circulate as immobile water and proposed the concept of effective void ratio . In the saturated soil, effective void ratio also represents the ratio of mobile water volume to the volume of solid [22, 78–80]. This paper ignores the effect of effective void ratio on hydraulic conductivity and assumes no existences of invalid pore which is different from the actual situation. In the next study, we can refer to the research methods of effective pores proposed by Ren et al. , assuming that there is a bound water film around the soil particles. This bound water film can affect the cross-sectional shape and size of the pore. We can establish the relationship between effective void ratio and hydraulic conductivity by observing this phenomenon.
5.3. Uncompleted Equation for Vertical Hydraulic Conductivity
In this paper, the formula for calculating the vertical hydraulic conductivity is proposed only for the case of four-particle arrangement, which has certain reliability. However, for the arrangement of three and five particles, because of the special arrangement form, there are dislocations among the layers of soil particles and the vertical pores are not completely connected. As a result, the proposed model cannot be applied to these two types of soil layers. In the future studies, we can introduce the concept of tortuosity. By establishing the relationship between hydraulic conductivity and tortuosity under the arrangement of three particles and five particles, we can discover the geometric characteristics of actual pore in these two cases and then obtain a revised equation for vertical hydraulic conductivity of soil layer.
(i)Based on Akbari’s formula for flow resistance, the paper proposes a new method for calculating soil hydraulic conductivity. Compared with the traditional calculation method, the proposed model considers the influence of ellipsoidal soil particles on the hydraulic conductivity, which is closer to the actual soil layer. At the same time, the proposed method can more accurately calculate the hydraulic conductivity in different directions, reflecting the anisotropy of the soil layer.(ii)According to the range of void ratio e (0.6–1.5), this paper assumes that the soil layer consists of three different particle systems (three-particle system, four-particle system, and five-particle system), in which the pore cross-sectional area of five particles is the largest. As a result, the corresponding hydraulic conductivity is the highest.(iii)The proposed model can reflect the influence of particle size in different directions on permeability of soil layer. When the size of minor axis of soil particles increases, the hydraulic conductivity of soil layer increases significantly. In addition, the hydraulic conductivity linearly decreases with the increase of major axis size. Compared with the major axis, the particle minor axis size has greater influence on the permeability of the soil layer.(iv)The models can be used to calculate the varied hydraulic conductivity with different void ratios. The hydraulic conductivity increases more rapidly, and the slope of curve is larger for the five-particle arrangement. In contrast, the hydraulic conductivity changes slowly under the three-particle arrangement, and the overall growth rate is low. For the four-particle combination, the vertical hydraulic conductivity is always greater than the horizontal hydraulic conductivity with the increase of void ratio and the variation trends of hydraulic conductivity in the two directions are basically the same.(v)The three models deduced in this paper are of high applicability when applied to different types of soil layers. The five-particle model is more accurate to calculate hydraulic conductivity for bentonite-silt mixed soil with relatively small particle size and most predicted values fall within . Four-particle model is more suitable for calculating silt-clay mixed soil layer with middle-size particle. When predicting the soil layer consisting of small particle, is often less than . The three-particle model has high reliability in calculating the hydraulic conductivity of large-sized particles and is suitable for predicting for sandy soil and some large-sized silt.
The Table 3 data used to support the findings of this study have been deposited in Table 1 of Reference . The data of Figures 4 and 9–12 used to support the findings of this study are included within the article, which are expressed as the pictures exported from “Origin Software”; therefore, if anyone wants to verify, replicate, or analyze these data, please open these pictures by “Origin 9.” A part of the Figure 12 data, which are related to measured values, used to support the findings of this study have been deposited in Figure 2 of Reference .
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This work was supported by the National Natural Science Foundation of China (Key Program, grant no. 41830110), the National Key Research and Development Program of China (grant no. 2017YFC1501102), the China Postdoctoral Science Foundation (grant no. 2015M571656), the Science and Technology Project of Zhejiang Provincial Department of Transportation (grant nos. 2015J09 and 2016008), the Science and Technology Project of Jiangsu Province Construction System (grant no. 2017ZD090), and the Science and Technology Project of Zhejiang Provincial Water Resources Department (grant no. RA1503). The authors gratefully acknowledge these supports.
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