Calculation Methods of Seepage Coefficient for Clay Based on the Permeation Mechanism

,e existing empirical methods of the seepage coefficient for clay have difficulty in parameter acquisition and are poor in calculation accuracy, and it is possible to find reasonable ones based on the mature empirical methods of coarse-grained soil. Firstly, the permeation mechanism of clay was analysed including influencing factors, spatial process, and parameter calculation, and the ineffective and effective voids were defined; then the empirical methods of clay and coarse-grained soil were summarised and listed. Secondly, clay was put equivalent to coarse-grained soil according to bound-water consolidation, and the parameter “equivalent void ratio” was introduced to establish the equivalent calculation methods. Finally, the liquid-plastic limit method determining the volume of ineffective voids was explained, and the feasibility of equivalent calculation methods was evaluated by three examples. ,e results show that seepage, permeation, and effective permeation are three different concepts; groundwater permeates in clay via the effective void. For certain soil, the increasing sequence of permeability coefficients is the seepage coefficient, permeation coefficient, and effective permeation coefficient. Introduction of the “equivalent void ratio” unifies the calculation of the seepage coefficient for clay and coarse-grained soil. ,e maximum of the bound-water content is about 0.9 time the size of liquid limit. ,e Mesri equation and the equivalent calculation method of Terzaghi or Curson–Karman are suitable for calculating the seepage coefficient of clay for the closest results to experimental values.


Introduction
Seepage and permeation are two different concepts; the strict definitions in this study are as follows.Seepage process has the characteristics of actual current, such as flow direction, flow amount, water pressure, and friction resistance; water fills all soil space continually, so it is called "virtual current."However, spatial coordinate functions are continuous, and a seepage field can be formed.Permeation is a motion process that occurs only via soil voids, and the spatial coordinate functions are noncontinuous.As a result, the corresponding academic terms are called the seepage coefficient, seepage velocity, permeation coefficient, and permeation velocity.
Seepage coefficient is often used for indicating the soil permeability and evaluating stability in geotechnical engineering, such as slope engineering, subgrade engineering, and tunnel and underground engineering; the correlative subjects are groundwater dynamics, fluid mechanics, seepage mechanics, etc. [1][2][3][4][5][6][7][8][9].Currently, seepage coefficient is often obtained by experiment or consolidation inversion [10], while it is hard to be determined correctly for clay, and the difference between different methods is obvious [11].Especially, disturbance to undisturbed soil is inevitable.Seepage coefficient is related to basic physical parameters, such as particle size and porosity [3,4,7,10,11], so the empirical calculation methods including the above two main parameters for clay and coarse-grained soil have already been obtained based on massive experiment data.However, methods for clay, such as obtaining seepage coefficients by CM and Taylor, are limited to parameter acquisition and calculation accuracy [10,11].Inversely, empirical methods for coarse-grained soil are relatively correct, such as Terzaghi, China Institute of Water Resources and Hydropower Research (abbreviated as "IWHR"), Curson-Karman, and KC [11].erefore, it is possible to nd simple and correct approaches to calculate the seepage coe cient of clay based on coarse-grained soil.
Reddi and angavadivelu suggested one method to calculate the seepage coe cient of clay based on random network and percolation theory [2].Liang and Fang analysed the in uencing factors on seepage property for tiny clay and explained them in terms of "microelectric eld" and "microsize" e ects [4].Deng found the linear positive correlation between natural logarithm of the seepage coe cient and void ratio for clay, and the gradient is about 0.5e 0 (e 0 is the initial void ratio) [10].Dang et al. established the e ective calculation methods of the seepage coe cient for clay based on the e ective void ratio [11].He et al. obtained the modi ed calculation method of the saturated seepage coe cient for bentonite based on the Poiseuille law, which includes void ratio and solution concentration [3,12,13].However, the above methods are feasible under certain conditions, and they have limits in speci c applications.erefore, it is still necessary to nd simple and correct calculation methods.e academic thought, research idea, and method in this study are derived from Dang's study [11].

Permeation Mechanism of Clay
In order to study the calculation method of the seepage coe cient for clay, it is very important to cognise the permeation mechanism rst.
en, in uencing factors, spatial process, and parameter calculation have to be analysed theoretically, thus providing fundamental works for the following study.

In uencing Factors.
A soil void is interconnected, semiconnected, or sealed, and the seepage coe cient reduces with the decreasing porosity for certain soil.Generally, the seepage coe cient of clay is low for low porosity, but it is not because porosity is negatively correlated to the compactness and it is in uenced by particle gradation, compaction power, and so on but not by the absolute particle size.
Here, presuming clay, silt, sand, and roundstone as round balls, their equivalent sizes are 0.002 mm, 0.02 mm, 0.2 mm, and 2.0 mm.Subsequently, they are put uniformly into four cubic boxes (50 * 50 * 50 mm 3 ), respectively, and then the spatial porosities are calculated, which are shown in Figure 1.
Figure 1 shows that porosities are equal for di erent soil particles, which are about 47.67%, so they are not correlated to the absolute particle size.
erefore, the seepage coe cient of clay may be in uenced by the absolute particle size under certain porosity.
Although particle size does not in uence the porosity, it decides the speci c surface area (shape coe cient), which are shown in Figure 2.
Figure 2 shows that the di erence in speci c surface area is large for di erent soil particles.For example, it is thousand times the value of particle size, that is, 2.0 mm for 0.002 mm.It is negatively correlated with speci c surface area and particle size, and the variation rate is larger when the particle size is small.
Large speci c surface areas will cause a variety of problems, such as long permeation route, more pressure loss, high surface energy, strong adsorptivity, and large bound-water volume (shown in Table 1).However, bound water has no solvent ability, owability, transmission of water pressure, and permeability, so it occupies partial permeation space, thus reducing the void opening perpendicular to the permeation direction.erefore, particle size is one of the main in uencing factors.
Similarly, under certain particle size and gradation, the speci c surface area is a constant.Porosity can be reduced by vibration or compaction, so the permeation space will also be compressed, thus decreasing the permeability.erefore, porosity is also one of the main in uencing factors.
In summary, particle size and porosity are the main in uencing factors for clay permeability.Under certain porosity, the speci c surface area and bound-water volume become large when the particle size is small, so the permeation route is lengthened and void opening is reduced.On the contrary, under certain particle size, the void opening is still reduced when the porosity becomes low.erefore, permeability of clay is in uenced by the coupling of particle size and porosity.

Spatial Process.
Soil consists of the solid phase, liquid phase, and gas phase, in terms of permeation space, soil particles, and permeation voids contained.It is mentioned that speci c surface area and bound-water volume are low for coarse-grained soil but large for clay.In order to present the bound-water volume directly, presuming colloid, clay, silt, and silty sand as round balls, their equivalent sizes are 0.002 mm, 0.005 mm, 0.05 mm, and 0.1 mm.en, putting them uniformly into four cubic boxes (50 * 50 * 50 mm 3 ), respectively, bound-water porosity and residual porosity are calculated, which are shown in Figure 3.
Figure 3 shows that bound-water porosity increases with the decreasing particle size, and the variation rate becomes large when the particle size is small.It is 7.54% for the particle size 0.005 mm, while it is only 0.38% for the particle size 0.1 mm.erefore, bound water in clay cannot be neglected.Based on the characteristics of bound water and e ective stress principle, bound-water voids can be regarded 2 Advances in Civil Engineering as ine ective voids, and the residual is the e ective void [11], which is shown in Figure 4.
Figure 4 shows that groundwater ows in clay via the e ective void mainly, according to the concepts of seepage and permeation, and it can be called the "e ective permeation," and the spatial coordinate function is also noncontinuous.Besides, the corresponding academic terms are the e ective permeation coe cient and e ective permeation velocity.For clay, e ective permeation is the actual current, and it is not mandatory to regard permeation as the actual current without considering bound water.

Parameter Calculation.
Based on the above context, seepage, permeation, and e ective permeation are three di erent concepts.Under the same preconditions of cross section, ow direction, ow amount, water pressure, particle volume, and friction for soil, they have di erent explanations and applications of permeability.Although seepage is the virtual current, the value is actual.For coarse-grained soil, e ective permeation is equivalent to the permeation.
e calculation formulas of the permeability coe cient for the above three concepts are shown in Table 2.
Table 2 shows that k nk p n e k e for certain soil, that is, k ≤ k p ≤ k e .Hence, di erent calculation methods of the permeability coe cient can be selected for di erent needs, and the seepage coe cient is widely used for continuous variables in the seepage eld.

Empirical Calculation Methods
Particle size and porosity are the main in uencing factors for clay permeability; accordingly, the existing empirical calculation methods [1,10] including these two variables are shown in Table 3.
It can be found that the parameter particle size is not included in the CM or Taylor equation because it is certain when established, so the limit is inevitable.Furthermore, parameters in the formulas are di cult to be acquired.Although there is no particle size parameter in the Mesri equation, C F and A c are included, so it still indicates the signi cant in uence caused by clay and colloid particles.
For coarse-grained soil, the empirical formulas [11] are often used to calculate the seepage coe cient in engineering practice, which are shown in Table 4.

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k is the seepage coefficient (cm/s) Q is the general flow amount within t (cm 3 ) i is the hydraulic gradient A is the area of the soil section (cm 2 ) t is the permeation time (s) Q′ is the general flow amount per second (cm 3 /s) Δh is the water-height difference between infall and outfall (cm) L is the soil length along the permeation direction (cm) v is the seepage velocity (cm/s) k p is the permeation coefficient (cm/s) n is the porosity A v is the equivalent void area (cm 2 ) V v is the void volume (cm 3 ) V is the general volume (cm 3 ) v p is the permeation velocity (cm/s) k e is the effective permeation coefficient (cm/s) n e is the effective porosity, 0 ≤ n e ≤ n A e is the equivalent effective void area (cm 2 ) V e is the effective void volume (cm 3 ) V e is the effective permeation velocity (cm/s) Permeation coefficient   d 10 is the effective particle size (mm) k 10 is the seepage coefficient when the water temperature is 10 °C (cm/s) d 20 is the particle size in which sieving accumulated mass is twenty percent of the whole (mm) c 2 is the influencing coefficient decided by particle shape and flow direction (0.125) ρ WZ is the density of water (1.0 g/cm 3 ) s is the specific surface area (mm −1 ) η is the viscosity of water (10 c WZ is the density of water (10 kN/m 3 ) a v is the compressive coefficient (MPa −1 ) R is the radius of the capillary (mm) β is the globular coefficient of the soil particle; it is π/6 when the particle is a ball d 50 is the mean particle size (mm) λ′ is the influencing coefficient caused by the neighboring soil particle; it is 3π for the ball particle in infinite water e others are the same as given above Expressed by consolidation degree Advances in Civil Engineering Table 4 shows that the empirical calculation methods for coarse-grained soil include particle size and porosity simultaneously and also the other influencing factors.Hence, some parameters are still difficult to be obtained, so the widely used five methods are Terzaghi, IWHR, Curson-Karman, KC, and Stokes void flow [11].According to the actual situation, the former four methods and Mesri equation of the seepage coefficient for clay are selected for use.
In summary, empirical calculation methods of the seepage coefficient for clay are limited for problems on influencing factors and parameter acquisition, while those for the coarse-grained soil are considerable and relatively easy.erefore, it is significant to find calculation methods for clay based on the methods for coarse-grained soil.Dang et al. explored the effective calculation methods for clay based on "effective void ratio," and they found the calculation value and experimental result had the same order of magnitude.is is one new approach [11].In this study, equivalent calculation methods of the seepage coefficient for clay will be studied based on bound-water "consolidation" and "equivalent void ratio."

Equivalent Calculation Methods of Seepage Coefficient for Clay
In order to regard clay as coarse-grained soil, it is important to make it have the same or similar soil structure and permeation mechanism.Groundwater permeates in clay via the effective void mainly, while it permeates through the whole in coarse-grained soil.erefore, if the bound water in ineffective voids can be regarded as "solid soil," particle size of clay will increase and the permeation mechanism will also be consistent with coarse-grained soil.Hence, the equivalent principle consolidates the bound water in clay, regarding it as "solid soil."

Equivalent Calculation Methods.
e equivalent calculation method has to be established under bound-water "consolidation," so the main parameter values will vary.Change of specific surface area after "consolidation" is neglected for the small thickness of the water film; simultaneously, semiconnected and sealed voids are also neglected.In Dang's study [11], void ratio in empirical calculation methods for coarse-grained soil is replaced by effective void ratio, and the calculation accuracy is improved, but it does not consider the equivalent principle; variation of void ratio is also neglected, so the difference is still inevitable.erefore, when void ratio is replaced by "equivalent void ratio," the equivalent calculation methods of the seepage coefficient for clay are obtained.
, then e ′ � e λ(1 where e is the initial void ratio, e e is the effective void ratio, e 0 is the ineffective void ratio, V e is the volume of effective voids (mm 3 ), V 0 is the volume of ineffective voids (mm 3 ), V s is the volume of the soil particle (mm 3 ), e ′ is the "equivalent void ratio," and λ is the ratio between ineffective and effective void ratios or volumes and is a constant for certain soil.
Substituting equation ( 1) into the former four formulas in Table 4, the equivalent calculation methods of the seepage coefficient for clay can be obtained preliminarily, which are shown in Table 5.
Table 5 shows that the parameter λ is included in the equivalent calculation methods.erefore, it is very important to determine λ.When parameter λ is low, it indicates the ineffective void is low, and the calculation method is also suitable for coarse-grained soil.Especially, equivalent calculation methods of the seepage coefficient for clay are the same as those for coarsegrained soil when λ � 0, so it unifies different calculation methods.

Determination of Parameter λ.
Parameter λ depends on the method to determine the volume of ineffective voids, which is equal to that of bound water.e liquid-plastic limit method of macroscopic soil mechanics is often adopted to determine the content of bound water [11,14], which is shown in Figure 5 and in the following:

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where w 0 is the content of bound water, w s is the content of strong bound water, w is the moisture content, w P is the plastic limit, w w is the content of weak bound water, α is a constant (0 < α < 1), and w L is the liquid limit.
Under permeation condition, the moisture content is not the initial one but the saturated moisture content.In Dang's study [11], the relation between saturated moisture content and αw L is not considered and it is not strict to some extent.
where m w0 is the mass of ine ective void water (g), ρ w is the density of water (1.0 g/cm 3 ), m s is the mass of the soil particle (g), ρ s is the density of the soil particle (g/cm 3 ), w sat is the saturated moisture content, d s is the relative density of the soil particle, and the others are the same as given above.
It can be found that e 0 e when 0 < w sat ≤ αw L ; namely, bound water lls the soil void, the e ective void is inexistent, and the seepage coe cient is 0. In Dang's study [11], α is 0.9.In order to verify its suitability, comparison calculations are made in the next section, and the selected values are 0.80, 0.85, 0.95, and 1.0; the results show they are appropriate when α is 0.9.

Evaluation of Equivalent Calculation Methods
In order to verify the feasibility and accuracy of equivalent calculation methods, three types of clay soils are selected-compacted reconstruction loess (No. 1), soft clay from South China Sea (No. 2), and coastal saline soil from Kuwait (No. 3) [5,6,11,15,16]; the correlative parameters are shown in Tables 6 and 7. Based on the calculation formulas in Table 5 (formula 1: Terzaghi; formula 2: IWHR; formula 3: Curson-Karman; formula 4: KC), the calculation results   8. Table 8 shows that seepage coe cients under di erent calculation methods have the coherence for every soil, the di erence between calculation method for coarse-grained soil and experimental value is the largest, the e ective calculation method for clay takes the second place, and the equivalent calculation method for clay is the closest; hence, the superiority of the equivalent calculation method is apparent.In order to evaluate them quantitatively, the    Advances in Civil Engineering multiples between them are calculated; the results show that, for every soil, (1) the di erence between calculation method for coarse-grained soil and experimental value is a little larger, and the minimal and maximal di erence in orders of magnitude is one and three; (2) the di erence between seepage coe cient of IWHR in every method and experimental value is also a little larger, and the minimal and maximal di erence in orders of magnitude is two and three; and (3) seepage coe cients of Terzaghi and Curson-Karman in every method are closer to experimental values, and the coe cients of the equivalent calculation method are the closest, that is, they are equal approximately.
erefore, the equivalent calculation method of the seepage coe cient for clay is feasible, and it is accurate to some extent.
From Table 8, it can be found that seepage coe cients of KC are all low under di erent methods for every soil, and the di erence between results calculated by the equivalent method and experimental value is three orders of magnitude, so it is not suitable for calculating the seepage coe cient for clay.In addition, results (shown in Table 9) calculated by the Mesri equation in Table 3 are also close to experimental values, they have the same order of magnitude as well, and they are also equal approximately.
erefore, the recommended methods to calculate the seepage coe cient for clay are the Mesri equation and equivalent calculation method of Terzaghi or Curson-Karman.

Conclusions
e following conclusions can be obtained: (1) Particle size and porosity are the main in uencing factors for clay permeability.Under certain porosity, speci c surface area and bound-water volume are large when the particle size is small, thus lengthening the permeation route along the permeation direction and reducing the void opening perpendicular to the permeation direction.Similarly, under certain particle size, void opening is also small when the porosity is low.ese e ects consume the permeation energy and compress the permeation space, thus reducing the clay permeability.
(2) Seepage, permeation, and e ective permeation are three di erent concepts; groundwater permeates in    clay via the effective void mainly and the effective permeation is the actual current, while it permeates via voids in coarse-grained soil.For certain soil, the increasing sequence of permeability coefficients is the seepage coefficient, permeation coefficient, and effective permeation coefficient.(3) Based on bound-water "consolidation," clay can be equivalent to coarse-grained soil in terms of soil structure and permeation mechanism.e void ratio after "consolidation" is not the simple subtraction of initial and ineffective void ratios, and introduction of "equivalent void ratio" unifies the calculation methods of the seepage coefficient for clay and coarse-grained soil.(4) Bound-water content can be calculated by the liquidplastic limit method, the maximal value is about 0.9 time the size of liquid limit, and the saturated moisture content needs to be considered certainly.

Figure 2 :
Figure 2: Speci c surface areas for di erent soil particles with particle sizes of (a) 0.002 mm and 0.02 mm and (b) 0.2 mm and 2.0 mm.
, and soft clay k � ce y (CM equation) c and C ′ are the dimensionless coefficients e is the void ratio y and m are the empirical exponents, and m is 5.0 C F is the content of clay particles A c is the activity coefficient e others are the same as given above k � C′(e m /1 + e) (Taylor equation) Soft clay k � 6.54 × 10 −11 ((e/C F )/A c + 1) 4 (Mesri equation)
(5) e feasible methods to calculate the seepage coefficient of clay are the Mesri equation and the equivalent Terzaghi or Curson-Karman method because their calculation results are the closest to experimental values.

Table 1 :
Bound-water volumes for di erent soil particles.

Table 2 :
Calculation formulas of the permeability coefficient for different concepts.

Table 3 :
Existing empirical calculation methods of the seepage coefficient for clay.

Table 4 :
Existing empirical calculation methods of the seepage coefficient for coarse-grained soil.

Table 5 :
Calculation methods of the seepage coe cient for coarse-grained soil and clay.

Table 6 :
e parameter λ and equivalent void ratio.

Table 9 :
Seepage coe cients of clay based on the Mesri equation.