## Abstract

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.

## HIGHLIGHTS

An improved model of the soil particle-water system was proposed.

The surface area ratio

*δ*is introduced.We establish an improved method for estimating the effective void ratio.

The relationship between LL and As for clay was established.

The correlation between the parameters

*C*and the specific surface area of soils was established.

### Graphical Abstract

## INTRODUCTION

Saturated hydraulic conductivity represents the ability of soils to transmit water under saturated conditions (Jabro 1992) and is also related to the characteristics of both fluid and its transport medium (Najafzadeh *et al.* 2017). When resolving some geotechnical engineering problems, such as the settlement of saturated soils, slope stability, and design of earth dams (Onur & Shakoor 2015), a reliable value of the hydraulic conductivity is required. Similarly, the determination of saturated hydraulic conductivity plays a key role in the drinking water supply, the management of water resources, water contamination, and facilities engineered for waste storage (Chapuis 2012).

However, the hydraulic conductivity value of soils is generally difficult to determine since it depends on numerous parameters, i.e., particle size distribution, particle shape, porosity, void ratio, saturation, clay content, and Atterberg Limit (Mesri & Olson 1971a, 1971b; Tan 1989; Alyamani & Sen 1993; Benson & Trast 1995; Chapuis & Aubertin 2003; Chapuis 2004; Trani & Indraratna 2010; Urumović & Urumović 2016; Kango *et al.* 2019). In addition, it would be more useful to characterize the diameters of pores, rather than those of the grains, in hydromechanics (Salarashayeri & Siosemarde 2012). 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 2002; Xu & Yu 2008; Xiao *et al.* 2019).

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.* 1989), 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 2000).

*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 2007). 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 2003).

*k*is the hydraulic conductivity (cm/s),

*C*is a constant related to the shape and tortuosity of channels (Hansen 2004), with a value approximately 0.2 (Taylor 1948),

*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}=

*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.

The Kozeny–Carman equation is considered to predict hydraulic conductivity of sandy soils but cannot be applied to the prediction of clayey soils (Lambe & Whitman 1969) without considering the electrochemical reactions between the soil particles and water (Carrier 2003). Some investigators have also validated the Kozeny–Carman equation in clay. The results show that the Kozeny–Carman equation can obtain ideal calculation results for sandy soil and silty clay. However, when calculating the hydraulic conductivity of clay, the predicted value of the Kozeny–Carman equation differs greatly from the experimental value.

Taylor (1948) proposed that a thin surface film of water was bound to all clay particles. Because of the bound water, seepage occurs only through a part of the pore space. Therefore, an improved concept of void ratio should be established to extend the use of the equation.

Saturated clays contain free water in addition to strongly adsorbed water (Dolinar *et al.* 2007). For clayey soils, only the part of pores occupied by free water affects the hydraulic conductivity or is effective. Therefore, the essential use of the Kozeny–Carman equation to predict the hydraulic conductivity of fine-grained soils is to propose a method to estimate an effective void ratio.

At present, there is little research work for the specific surface area correction of the Kozeny–Carman equation. In past research, the surface area loss due to the contact of soil particles was usually ignored when deducing the equation of the soil hydraulic conductivity, and the reduction in the surface area of the seepage pipes caused by contacts between particles was also not considered. Thus, it cannot really reflect the pore structure characteristics of clay.

To modify the Kozeny–Carman equation to predict hydraulic conductivity for a wide range of soils and enhance its accuracy, a surface area ratio *δ* is introduced in this paper to obtain the reduced surface area accurately, which enhances the calculation accuracy of the Kozeny–Carman equation. We also establish an improved method for estimating the effective void ratio. The performance and applicability of the improved equation are validated with experimental data for clays, silts, and sands.

## EFFECTIVE VOID RATIO

The void ratio, a dominant parameter for soils, can be divided into effective void ratio and ineffective void ratio in soil. The effective void ratio contributes mainly to seepage, while ineffective void ratio has little influence on seepage. The ineffective void ratio consists mainly of unconnected pores, dead-end pores, and pores occupied by the bound water adsorbed to the surfaces. The quantity of immobile water depends on many factors such as the mineral composition and the physicochemical properties of the soils (specific surface and cation exchange capacity), geometric characteristics of the pores and particles, salinity of the flow, and temperatures of the pore water (Koponen *et al.* 1997; Ren *et al.* 2016; Dolinar & Trcek 2019). Therefore, all ineffective pores of clay can be considered to be occupied by the bound water adsorbed to the surfaces.

*n*

_{e}is defined as the ratio of the pore volume occupied by the mobile water to the total pore volume (Ahuja

*et al.*1984). The effective void ratio

*e*

_{e}and the ineffective void ratio

*e*

_{i}are the ratio of mobile water volume and 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.

Figure 1 shows a commonly used model of pore water distribution. Head losses occur when the flow of water through seepage pipes of soils is due to the friction between soil particles and water and the friction inside the fluid. In a fluid undergoing laminar flow, variations in velocity from point to point are accompanied by friction losses. The surface of seepage pipes is composed of the surface of the soil particles. The surface area of the voids is usually considered equal to the surface area of the particles based on the assumption that the amount of surface area lost to the contacts between particles is negligibly small. However, the area lost to the contacts between particles cannot be ignored because of differences in the shape and size of soil particles, resulting in a corresponding reduction in the surface area of the seepage pipes (Figure 2).

*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*=

*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 bywhere

*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.

*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},

*L*, then the hydraulic radius

*R*

_{H}can be expressed aswhere

*V*

_{w}is the volume of flow tubes and

*S*

_{wl}is the surface of flow tubes.

*S*

_{wl}can be expressed aswhere

*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}.

*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.

## EVALUATION AND VALIDATION

The improved equation is evaluated using a database comprised of 450 experimental values of *k*_{sat} and *e* from 10 publications containing clayey soils, silty soils, and sandy soils (Table 1). The hydraulic conductivity test methods are represented in the database: falling-head test, constant-head test, consolidation test, and oedometer test. Only data for saturated hydraulic conductivity measured with water as the permeant are included. The data plotted in this study include disturbed and undisturbed specimens.

Sources . | Materials . | n
. | Experimental method . | Liquid limit . | Plastic limit . | . | USCS . |
---|---|---|---|---|---|---|---|

Lambe & Whitman (1969) | Boston silt | 6 | Laboratory permeability test | / | / | / | Silt |

Mesri & Olson (1971b) | Kaolinite, smectite | 72 | Consolidation test | 40–1,160 | 27–47 | 2.65–2.8 | CH, ML |

Tavenas et al. (2011) | Champlain sea clays, Canadian clays | 118 | Permeability test and oedometer test | 20–74 | 14–28 | / | CH, CL |

Towhata et al. (1993) | Bentonite | 9 | Consolidation test | 450 | 29 | 2.785 | CH |

Nagaraj et al. (1994) | Red soil, Brown soil, Black cotton soil, Marine soil | 72 | Falling-head test | 50–106 | 27–47 | 2.64–2.67 | CH, MH |

Terzaghi et al. (1996) | Berhierville clay, St. Hilaire clay, Batiscan clay, Vasby clay | 113 | Falling-head and constant-head test | 46–122 | 22–41 | / | CL, CH |

Dolinar (2009) | Crystallized kaolinite, Calcium montmorillonite | 25 | Falling-head test | 40.1–129 | 25.9–68.2 | / | ML, MH |

Kim et al. (2013) | Marine fine-grained sediments | 15 | Consolidation test | 38.8–77 | 11.3–27.4 | 2.45–2.74 | MH |

Wang et al. (2018) | Calcareous soil | 20 | Constant-head test | / | / | 2.78–2.80 | Sand |

Sources . | Materials . | n
. | Experimental method . | Liquid limit . | Plastic limit . | . | USCS . |
---|---|---|---|---|---|---|---|

Lambe & Whitman (1969) | Boston silt | 6 | Laboratory permeability test | / | / | / | Silt |

Mesri & Olson (1971b) | Kaolinite, smectite | 72 | Consolidation test | 40–1,160 | 27–47 | 2.65–2.8 | CH, ML |

Tavenas et al. (2011) | Champlain sea clays, Canadian clays | 118 | Permeability test and oedometer test | 20–74 | 14–28 | / | CH, CL |

Towhata et al. (1993) | Bentonite | 9 | Consolidation test | 450 | 29 | 2.785 | CH |

Nagaraj et al. (1994) | Red soil, Brown soil, Black cotton soil, Marine soil | 72 | Falling-head test | 50–106 | 27–47 | 2.64–2.67 | CH, MH |

Terzaghi et al. (1996) | Berhierville clay, St. Hilaire clay, Batiscan clay, Vasby clay | 113 | Falling-head and constant-head test | 46–122 | 22–41 | / | CL, CH |

Dolinar (2009) | Crystallized kaolinite, Calcium montmorillonite | 25 | Falling-head test | 40.1–129 | 25.9–68.2 | / | ML, MH |

Kim et al. (2013) | Marine fine-grained sediments | 15 | Consolidation test | 38.8–77 | 11.3–27.4 | 2.45–2.74 | MH |

Wang et al. (2018) | Calcareous soil | 20 | Constant-head test | / | / | 2.78–2.80 | Sand |

The liquid limit and plasticity index are used for classification of fine-grained soils based on a Plasticity Chart (Figure 5) given in the standard by ASTM (2017). The database consists of 119 lean clays, 193 fat clays, 41 silts, and 71 elastic silts.

The liquid limit and plasticity index are used for the classification of fine-grained soils based on a Plasticity Chart (Figure 5) given in the standard by ASTM (2017). The database consists of 119 lean clays, 193 fat clays, 41 silts, and 71 elastic silts.

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, the database consists of 312 (69.4%) clayey soils, 118 (26.2%) silty soils, and 20 (4.4%) sandy soils.

Parameters *C* and *δ* were obtained by fitting a series of *k* and *e* values using nonlinear regression. Then, the hydraulic conductivity was predicted by substituting the fitted parameters *C* and *δ* into the proposed equation. It is relatively accurate if a predicted *k* value of a soil lies between one-third and three times the measured *k* value, which is within the expected margin of variation for the usual laboratory permeability test results. The predicted values are plotted on a log–log scale against the measured values (Figure 6). The result shows that the accuracies of clayey soils, silty soils, and sandy soils with two fitted parameters are 94.6, 96.6 and 100%, respectively, while the accuracy of all samples is 95.3%, which means that the improved Kozeny–Carman equation based on the surface area ratio 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.

Various techniques are used to measure the specific surface of solids. The gas adsorption method and the methylene blue absorption method are the most commonly used techniques (Santamarina *et al.* 2002). However, these techniques are not commonly used in soil mechanics and hydrogeology because several of them require the use of high-tech equipment (Chapuis 2012). Alternatively, the specific surface area can be estimated based on complete particle size distribution or empirical relationships with the Atterberg Limit.

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 (Sanzeni *et al.* 2013). The specific surface area of sands is a function of particle size and shape (Fair *et al.* 1933). Many later researchers have studied and proposed some formulas for the calculation of the specific surface area of sands (Chapuis & Legare 1992; Carrier 2003).

*et al.*(2002) proposed the correlation for estimating the specific surface area of sands made of spherical or cubical particles: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 (Nishida & Nakagawa 1969; Sridharan & Honne 2005; Farrar & Coleman 2006; Feng & Vardanega 2019; Spagnoli & Shimobe 2019).

*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.

*a*,

*b*, and

*c*are shown in Figure 7. Then, the volume of the particle isand the mass of the particle is

*t*and

^{2}*t*tend to 0 because adsorbed water at the corners of the clay particle is negligibly small. Then, Equation (38) is simplified to

^{3}*fc*is the force required to break all the bonds or links per unit volume. Muhunthan (1991) proposed

*F*is approximately a constant at the liquid limit. Equation (44) can be written as follows:

*et al.*(1984), Sridharan

*et al.*(1986), Sridharan

*et al.*(1988), and Yukselen-Aksoy & Kaya (2010) have been plotted in Figure 8. Linear regression analysis was used to solve for parameters

*λ*and

*η*. The best-fitted line (

*R*

^{2}= 0.7054) of Figure 8 corresponds to the equation as follows:

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.

*C*and the specific surface area on a log–log scale. The best-fitted straight line (

*R*

^{2}= 0.8429) is

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.

*C*and the specific surface area with one fitted parameter

*δ*. The best-fitted line (

*R*

^{2}= 0.9401) is

## DISCUSSION

We can consider that the proposed model has a satisfactory capability to predict hydraulic conductivity for a wide range of soils from clays to sands. The deviation of the prediction results may be caused by such factors as input parameters, specimen preparation methods, and unsaturated specimens.

The specimens used in the database include disturbed and undisturbed specimens. The test methods contain the falling-head test, the constant-head test, the consolidation test, and the oedometer test. In addition, the temperatures during the tests are not constant. All the aforementioned factors will give rise to the uncertainty in the true *k* values. In particular, the compaction method during specimen preparation is critical for the measured values. Heavy compaction should be avoided for the creation of high pore pressure and local internal erosion or clogging, and otherwise large errors of the measured values may result (Chapuis & Aubertin 2003).

The standard for the permeability test (ASTM 2002) indicates that the specimens should be saturated by using a vacuum pump. However, Chapuis (2004) 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 proposed equation is used to predict the vertical *k*. The results show that the equation provides a good prediction of hydraulic conductivity, ranging mostly within 1/3–3 times for a wide range of soils from clays to sands. The equation can be used for a simple estimate of the vertical *k* of soils after determining *A*_{s} and *e* and a check on the accuracy of laboratory permeability test results.

The surface area ratio *δ* is a key parameter for predicting hydraulic conductivity, but it cannot be determined through theoretical derivation. To obtain a more accurate value of the parameter *δ*, the size, distribution, shape, contact, and other properties of soil particles should be further studied. The anisotropy of hydraulic conductivity is beyond the scope of this paper, because the proposed method uses only scalar parameters such as void ratio and specific surface area to predict the vertical *k* values. Therefore, the improved equation cannot represent the anisotropy of hydraulic conductivity correctly.

## AUTHOR CONTRIBUTIONS

G.X. initiated and led this research. M.W. designed the analytical framework of this study, produced maps and ﬁgures, performed the data analysis and interpretation, and wrote the paper. J.W. and Y.Z. contributed the sections. X.K. reviewed and edited the paper.

## COMPETING INTERESTS

The authors declare that they have no conflict of interest.

## ACKNOWLEDGEMENTS

This work was supported by the National Natural Science Foundation of China (Grant No. 41772314).

## DATA AVAILABILITY STATEMENT

Data cannot be made publicly available; readers should contact the corresponding author for details.