Improved model for predicting the hydraulic conductivity of soils based on the Kozeny–Carman equation

The saturated hydraulic conductivity of soils is a critical concept employed in basic calculation in the geotechnical engineering field. The Kozeny–Carman equation, as a well-known relationship between hydraulic conductivity and the properties of soils, is considered to apply to sands but not to clays. To solve this problem, a new formula was established based on Hagen–Poiseuille’s law. To explain the influence on the seepage channel surface caused by the interaction of soil particles and partially viscous fluid, the surface area ratio was introduced. A modified framework for determining the hydraulic radius was also proposed. Next, the relationship between the effective void ratio and the total void ratio was established for deriving the correlation of hydraulic conductivity and total void ratio. The improved equation was validated using abundant experimental results from clays, silts, and sands. According to the results, the accuracies of the proposed model with two fitted multipliers for clays, silts, and sands are 94.6, 96.6, and 100%, respectively, but with only one fitted parameter, the accuracies are 97.1, 91.5, and 100%, respectively. The proposed model can be considered to have a satisfactory capability to predict hydraulic conductivity for a wide variety of soils, ranging from clays to sands.

it would be more useful to characterize the diameters of pores, rather than those of the grains, in hydromechanics (Salarashayeri & Siosemarde ). Hence, some researchers have studied the pore geometry about the control permeability of porous media, including pore size distribution, tortuousness of capillaries, the coordination number of the pore, and pore shape (Fauzi ; Xu & Yu ; Xiao et al. ).
Generally, the saturated hydraulic conductivity value of soils can be measured by both field and laboratory tests or predicted. In real situations, accurate estimates of hydraulic conductivity in the field are limited to the deficient comprehension of the aquifer geometry and hydraulic boundaries (Uma et al. ), and the cost of the field tests is quite high.
However, the samples used in laboratory tests cannot be totally representative in most cases, and the time limitation also makes laboratory tests defective, as well (Boadu ).
where k is the hydraulic conductivity (cm/s), C H is the Hazen empirical coefficient, and D 10 is the particle size for which 10% of the soil is finer (mm).
The Hazen formula was originally developed for the determination of hydraulic conductivity of uniformly graded sand but is also useful for the fine sand to gravel range, provided that the sediment has a uniformity coefficient less than 5 and an effective grain size between 0.1 and 3 mm (Odong ). The published value of C H varies from 1 to 1,000. A change of three orders of magnitude makes the prediction result inevitably produce a certain deviation due to the difficulty of including all possible variables in porous media by effective diameter D 10 . The formula has continued to be used for its simplicity and ease of memorization (Carrier ).

The Kozeny-Carman equation was proposed by Kozeny
and improved by Carman: where k is the hydraulic conductivity (cm/s), C is a constant related to the shape and tortuosity of channels (Hansen ), with a value approximately 0.2 (Taylor ), g is the gravitational constant (m/s 2 ), ρ w is the density of water (kg/m 3 ), ρ s is the density of solids (kg/m 3 ), G s is the specific gravity of solids (G s ¼ ρ s /ρ w ), μ w is the dynamic viscosity of water (N·s/m 2 ), S S is the specific surface (m 2 /kg), and e is the void ratio.

EFFECTIVE VOID RATIO
The void ratio, a dominant parameter for soils, can be immobile water volume to the solid volume, respectively.
where V mw is mobile water volume, V iw is immobile water volume, and V s is solid volume.  the contacts between particles. Therefore, this part of the surface area has no actual effect on seepage. Poiseuille's law proposed that the velocity at any point in the round capillary tube (Figure 3) can be described as follows: where R is the radius of a capillary tube, r is the radius of any concentric cylinder of liquid within the tube, γ w is the unit weight of water, i is the hydraulic gradient, μ is the dynamic viscosity of water, and v is the velocity of a fluid at a distance r from the centre of the capillary tube. In laminar flow, the fluid in contact with the inside surface of the pipe (r ¼ R) has been shown to have zero velocity, which means that there is a part of the viscous water occupying the corresponding surface area of the soil particles. This part of the surface area also has no contribution to the head losses of the fluid. A surface area ratio δ is introduced, then given by where S s is the surface area of all soil particles, and S D is the reduced surface area.
Following the above analysis, an improved model has been established ( Figure 4). If the immobile water can be regarded as part of the solid soil, then a new equivalent soil particle consisting of the soil particle and immobile water is proposed. The surface area is lost due to the contacts between new equivalent particles. A part of the thin viscous water is adsorbed on the surface, which is considered to play a role in reducing surface area only without counting as immobile water.
The expression for flow through a circular tube is given by Taylor (): We can conclude that the flow through any given geometrical shape of cross-section is given by where C s is a shape constant that has a definite value for any specific shape of the cross-section. The shape constant can be evaluated mathematically for a simple shape. a is the area of the effective tube and R H is the hydraulic radius, which is defined as the ratio of the area of the cross-section available for flow S w to the wetted perimeter P w , For a cross-section of a soil, the area of the effective tube a is equal to the area of the mobile water. The total area of the cross-section is S t , the effective porosity is n e . Then   (11) into Equation (8) gives

Substitution of Equation
According to Darcy's law, Combining Equations (12) and (13) gives If the length of the tube is designated L, then the hydraulic radius R H can be expressed as where V w is the volume of flow tubes and S wl is the surface of flow tubes. S wl can be expressed as where δ is the surface area ratio and S es is the surface area of new equivalent soil particles. If the soil consists of homogeneous spherical particles with a radius of R s , then the new equivalent soil particles have a radius of R es .
The volume of the new equivalent soil particle can be written as Then, the R es can be expressed as Substitution of Equations (16) and (19) into Equation (15) gives The relationship between the effective void ratio and the total void ratio can be established based on the influence of surface area ratio δ on pores. The relationship between the effective void ratio and the total void ratio can be assumed as follows: where a and b are constant parameters that can be calculated according to the boundary conditions, n is porosity.
Two extreme boundary conditions are considered to solve a and b, when the porosity tends to 0 and 1, respectively, then The values of a and b are À1 and 1, respectively. Then, Equation (21) can be expressed as Then, the effective void ratio and the ineffective void ratio can be computed as Substitution of Equations (20), (25), and (26) into Equation (14) gives

EVALUATION AND VALIDATION
The improved equation is evaluated using a database com-  Six specimens in the database without providing liquid limit and plastic limit were classified as silty soils in the literature, and there are 20 sandy soils in the database.
According to the above classification of fine-grained soils, Simplifying the equation gives Then, Equation (27) can be generalized as has a satisfactory capability to predict hydraulic conductivity for a wide range from clayey soils to sandy soils.

DETERMINATION OF THE PARAMETERS C AND δ
The empirical or semi-empirical relationships between C and δ and some easily available parameters are established for facilitating the use of the improved equation. The value of parameter C s , which depends on porosity, microstructures of pores, and capillaries, is not a constant. Due to the lack of available test temperature information from most of the literature, temperature-related parameters such as dynamic viscosity μ w cannot be accurately evaluated. In summary, the parameter C s cannot be obtained simply by calculation. Therefore, we studied the correlation between the parameters and the specific surface area of soils. The method to determine A s from the complete particle size distribution is applicable to soils in which particle behaviour is governed by gravimetric-skeletal forces rather than surface-related forces such as nonplastic soils ( where C u is the coefficient of uniformity, and D 50 is the particle diameter corresponding to 50% passing.
The specific surface area of plastic soils can be estimated empirically based on the plasticity index properties. Numerous studies have shown that A s of plastic soils is related to the liquid limit, the plastic limit, the shrinkage limit, and the plasticity index ( Fine-grained soils contain free water in addition to strongly adsorbed water. Then, the total water content can be expressed as where w t is the total water content, w iw is the adsorbed water content, and w mw is the free water content.
The three primary ways in which clay particles can be arranged are: edge to edge, edge to face, and face to face.
Dawson () observed that clay particles become parallel as they shear past each other. Goodeve () proposed where F is the shearing force per unit area, f is the critical force required to break an interparticle bond or link, c is the number of bonds or links per unit volume, and z is the distance between two particles.
Two parallel clay particles with length, width, and height of a, b, and c are shown in Figure 7. Then, the volume of the particle is and the mass of the particle is The specific surface area of particle can be expressed as With the assumption that the thickness of the adsorbed water is t, the adsorbed water content is The free water content and total water content can be expressed as The parameters t 2 and t 3 tend to 0 because adsorbed water at the corners of the clay particle is negligibly small.
Then, Equation (38) is simplified to Neglecting the contribution of edges to A s , the specific surface area is By substituting Equation (40) into Equation (39), the results are expressed as Equation (42) can be expressed in the form below at the liquid limit: where fc is the force required to break all the bonds or links per unit volume. Muhunthan () proposed Different fine-grained soils have an approximately equal undrained shear strength at the liquid limit (Casagrande ). Hence, F is approximately a constant at the liquid limit. Equation (44) can be written as follows: Combining with Equation (45), Equation (43) can be modified as follows: The  Figure 9 shows that there is no obvious relationship between δ and the specific surface area that is related to the type of soil. The values of parameter δ for clays, silts, and sands are in the ranges of 1-3, 1-6 and 5-6.5, respectively.
Following the above analysis, with the assumption that the values of δ are equal to 2, 3, and 5 for clays, silts, and sands, the hydraulic conductivity was computed with one fitted parameter C. The predicted values were plotted on a log-log scale against the measured values ( Figure 11).
The result shows that the accuracies of clayey soils, silty soils, and sandy soils with one fitted parameter are 97.1, 91.5, and 100%, respectively, while the accuracy of all samples is 95.8%. The difference in accuracy between one fitted parameter and two fitted parameters is quite limited. However, values computed with one fitted parameter of silty soils produce a larger error, probably because the value of δ has a wider range of variation causing relatively large scatter. Figure 12 shows that there is a stronger correlation between the parameter C and the specific surface area with one fitted parameter δ. The best-fitted line (R 2 ¼ 0.9401) is The standard for the permeability test (ASTM ) indicates that the specimens should be saturated by using a vacuum pump. However, Chapuis () proposed that if a rigid-wall permeameter is used without precautions that are not mandatory in the standard, the degree of saturation is usually in a range of 75-85%. Then, the unsaturated k  value is only 15-30% of the fully saturated k value. In other words, if a specimen is not fully saturated, the true k value may be multiple the measured k value, in which case a large error between k p and k m will be induced.
The values of parameter δ also have a definite region for any specific type of soils. For prediction of the hydraulic conductivity, we assumed that the parameter δ is constant, which leads to a larger scatter and inaccuracy. Most previous publications did not provide the information of specific surface area. The value of the specific surface area, which is empirically estimated by the correlation with other available parameters such as liquid limit and the coefficient of uniformity, is the main cause of the inaccuracy of C and the prediction values.

CONCLUSIONS
An improved model of the soil particle-water system is proposed, which presents an explanation for the reduction of the seepage area. A new equation for predicting hydraulic conductivity is derived based on the surface area ratio δ and the total void ratio e. By using published data, the pro-