# Permeability Prediction Model Modified on Kozeny-Carman for Building Foundation of Clay Soil

^{1}

^{2}

^{*}

## Abstract

**:**

_{eff}) and effective SSA (S

_{eff}) are proposed. Based on the e

_{eff}and S

_{eff}, the permeability prediction model modified on Kozeny-Carman is built. Then, seepage experiments are conducted on two types of clay samples to test this prediction model; at the same time, the MIP combining freeze-drying methods are used to obtain the S

_{eff}and e

_{eff}. Through the discussion of the test results, three main conclusions are obtained: (1) there are invalid pores in clay due to the influence of clay mineral, this is the reason for which K-C equation is unsuitable for clay; (2) the e

_{eff}and S

_{eff}can reflect the structural state of clay during seepage; (3) the results of the permeability prediction model in this paper agree well with the test results, which indicates that this prediction model is applicable to clay. The research results of this paper are significant to solve the academic problem that K-C equation is not applicable to clay and significant to ensure the safety of building foundation pits in clay areas.

## 1. Introduction

_{eff}using the mercury intrusion porosimeter (MIP), this method could determine the invalid pores [36]. Meegoda et al. [37] proposed a method by chemical tracer tests to measure the effective void. Koponen et al. [40] used the lattice-gas method and simulated the effective e as a function of e. Meanwhile, Singh and Wallender [41] gave the effective e considering the thickness of the adsorbed water layer and the SSA of the clay. Urumović [42] presented the effective e in the function of referential grain size based on literature data. After reviewing the definitions and calculation methods by many scholars, Dolinar and Trcek [39] emphasized that effective e depended mostly on the external SSA of the soils. Wang et al. [10] considered the influence of bound water, and established a relationship between the effective e and the total e.

^{2}/g would there be a good relationship between the consistency limit and SSA. Hong et al. [11] found that even though the relationship was fitted by a large number of LL and SSA data, the predicted hydraulic conductivity still ranged between 1/3 and three times to the laboratory test value. Additionally, after checking more than 500 datasets from the literature, Spagnoli and Shimobe [65] found that even the relations of SSA with LL were robust, but the estimated SSA values slightly overestimated the measured SSA up to 100 m

^{2}/g.

_{eff}) and effective SSA (S

_{eff}) are proposed by considering the influence of adsorbed water and unconnected pores under seepage state. Based on this, the permeability prediction model modified on K-C equation by e

_{eff}and S

_{eff}is proposed. In order to test the rationality of e

_{eff}, S

_{eff}and the applicability of this model to clay, S type of artificial clay samples with different porosity and Y type of natural clay samples with different natural clay content are designed for permeability tests. At the same time, the freeze-drying method and MIP test are used to obtain the e

_{eff}and S

_{eff}. Then the SSA, e and hydraulic conductivity by different method are compared, and the result showed that the prediction model proposed in this paper had an outstanding performance in predicting the hydraulic conductivity of clay. The research results of this paper have certain significance to solve the academic problem that K-C equation cannot be used in clay, and also are significant to ensure the safety of building foundation pits in clay areas.

## 2. Effective Pores and the Permeability Prediction Model Modified on K-C Equation

#### 2.1. Effective Pores of Clay Seepage

#### 2.2. The Permeability Prediction Model Modified on K-C Equation

_{ave}is shown as:

_{p}is the specific gravity, and i

_{h}is the hydraulic gradient. For a full-flow circular tube, the hydraulic radius is:

_{s}is introduced, then the former formula becomes as follows:

_{f}.

_{H}can also be expressed as:

_{w}is the volume of the water and V

_{s}is the volume of the solid, ρ

_{s}is the density of soil particles. According to the introduction above, for clay, the fluid can only flow through interconnected pores, so S

_{p}in this formula is the SSA of the pores, that is, the SSA of particles along the interconnected pores through which liquid can effectively flow. At this point, Equation (5) can be written as follows:

^{−2}mol/L and 0.83 × 10

^{−4}mol/L [66]. The thicknesses of DEL are 3.33 nm and 33.3 nm (Figure 5), but not all liquids in the DEL cannot flow.

_{v}

^{′}is the effective pore volume, V

_{noc}is the volume of pores influenced by absorbed water and with no contribution to seepage, V

_{unc}is the volume of pores which are unconnected such as blind pores and isolated pores, these pores can be measured by MIP test. Then the effective e of the interconnected pores and shown as:

_{s}is the volume of soil particles, and V

_{s}

^{′}is the volume of soil particles plus the volume of adsorbed water. At this point, there is:

_{eff}). Under stable laminar flow, substituting Equations (5), (10) and (11) into Equation (8), and changing the V

_{s}to V

_{s}′, the hydraulic conductivity is obtained according to Darcy’s law:

_{s}is a coefficient describing the tortuosity of pores, and it is difficult to obtain the exact value. At present, this coefficient is mostly selected by experience. According to the works of Mitchell and Soga [66], in this model C

_{s}is taken as 5. When S

_{eff}and e

_{eff}in the formula are taken as the specific surface area S and total pore e, the formula becomes the classic K-C equation.

## 3. Experimental Scheme and Result

#### 3.1. Experimental Method and Scheme

_{max}is the maximum amount of mercury injected and S

_{p}is the surface area of interconnected pores. In the MIP experiment, each sample was repeated three times to ensure the precision.

#### 3.2. Experimental Results

_{eff}by MIP method are all shown in Table 3. At the same time, the SSA obtained by some classical estimated methods are also shown in the Table 3. The estimated methods are mainly proposed by scholars [7,63,74,75].

_{eff}by different methods

_{eff}is obtained and as shown in Figure 8.

_{eff}decreases with the decrease of the design porosity of the sample by all test methods. The e obtained by all test methods and estimated methods are smaller than the design e. Except the MIP method, the e

_{eff}curves by other methods are almost parallel. For the Y type samples, the e

_{eff}decreases with the increase of the bentonite content. Similarly, except the MIP method, the e

_{eff}curves by other methods are almost parallel and linear.

_{eff}obtained by various methods into Equation (12), the hydraulic conductivities can be obtained. In Figure 9, the hydraulic conductivities obtained by all methods increased with the increasing of e, and decreased with the increasing of bentonite content.

_{eff}and e

_{eff}by MIP method achieves better agreement with the experimental values. The largest difference between them occurred when n = 0.5. At this point, the values of these two methods were 6.14 × 10

^{−10}m/s (the modified model value) and 1.1 × 10

^{−9}m/s (experimental values). The ratio of these two values was only 1.79, while the ratios for the other points were less than 1.6. For the BET and Churchman (1991) methods, the average ratios of the experimental values to the calculated values of the two methods were 3.1 and 19.85, while the other methods are 100 times larger, such as the method of Smith et al. (1985), and the ratio is astonishing at about 10,000 times. Similarly, for the Y type samples in Figure 9b, it could be seen that this modified equation still has advantages for natural remoulded soil. The results of this modified K-C equation by MIP method and the experimental values at other points are coincident, except at the point when the bentonite ratio was 0.1. The average values of the experimental values to the calculated values of BET and Churchman are 1.73 and 8.5 respectively, and for the other methods, this value is much larger. At the same time, the hydraulic conductivity curves of the EGME, BET, Churchman, Yukselen-Aksoy and Ren clearly show almost linear changes, while the calculated values of MIP method are not linearly changed.

## 4. Discussion

#### 4.1. The SSA and e_{eff} by Different Methods

_{eff}and the S

_{eff}change.

_{p}or SWCC, these methods are based on the EGME and BET tests. So, in Table 3 the SSA by Smith et al., Yeliz Yukselen-Aksoy and Ren methods are close to the values of EGME method, and the SSA by Churchman and Burke is close to the method of BET. However, for the MIP test, the test samples were prepared by the freeze-drying method, so the SSA by MIP test is the SSA of the interconnected pores of the seepage samples. That is to say, the MIP test aims at the clay with seepage structure while the EGME and BET test aim at the particle, which are essentially different. As mentioned above, clay particles are more easily to formed into a cluster structure. In this structure, particles that stack with each other will occupy the particle surface and the unconnected pores also reduce the SSA of the interconnected pores. So, the S

_{eff}determined by MIP test is the smallest one.

_{eff}by MIP test increases with the designed porosity. This is because change of designed porosity cannot change the property of the particles, but can change the pore structure in the clay. Similarly, for Y type samples, even though the porosity is constant, the increase of bentonite content changes the surface properties of particles and the pore structure of the clay, so the SSA obtained by these methods all change with the bentonite content. Through the above discussion, it can be found that the S

_{eff}obtained by MIP test method can simultaneously reflect the change of clay pore structure caused by the change of soil porosity, mineral composition and other factors.

_{eff}is calculated by the SSA as show in Equation (11). The e

_{eff}can reflect the influence of the adsorbed water, the disconnected pores and small pores on the effective pores for seepage. First, because there are invalid pores and the adsorbed water in clay, so the e

_{eff}calculated by all methods are smaller than the designed e. Similarly, whether the mineral composition or the porosity of the sample changes, the amount of disconnected pores and small pores all change, thus the e

_{eff}is changed. At the same time, this can also explain why the soil e

_{eff}curves in Figure 9, as except for the MIP method, the curves obtained by other methods are almost parallel and linear.

#### 4.2. The Hydraulic Conductivity

_{eff}is bigger than the SSA by EGME and BET methods. The estimated methods for SSA, such as Smith et al. (1985), Churchman (1991), Yukselen-Aksoy (2010) and Ren (2016) are all based on EGME and BET test. Since they are all empirical equations, the value of SSA calculated by some of them is too large; Therefore, the volume of adsorbed water calculated is amazingly big. So, the hydraulic conductivity calculated is too small and unreasonable.

_{eff}, the e

_{eff}and the hydraulic conductivity.

_{eff}and e

_{eff}considered the actual flow of the fluid in clay and used only the effective pore volume. Therefore, this prediction model had outstanding performance over the classic K-C equation. Additionally, it could achieve a high degree of concordance with the experimental values.

## 5. Conclusions

_{eff}and e

_{eff}. In order to test the rationality of e

_{eff}, S

_{eff}and the applicability of this model to clay, seepage tests were conducted on two type of clay samples. At the same time, the freeze-drying method with MIP test were used to obtain the e

_{eff}and S

_{eff}. Then, the SSA, e and hydraulic conductivity by different methods were compared and discussed. Through the discussion, three main conclusions were obtained:

_{eff}and S

_{eff}by the MIP method can better reflect the real pore structure of soil during the seepage process. The SSA obtained by EGME, BET and the estimated methods are the SSA of the particles. But the e

_{eff}and S

_{eff}are determined by the freeze-drying method and MIP method, so the e

_{eff}and S

_{eff}aim at the effective pores in the clay during seepage. Therefore, e

_{eff}and S

_{eff}are influenced both by the clay density and composition.

_{eff}and S

_{eff}show a good agreement with the experimental values. For S type, the largest difference between the prediction value and the experimental one is by the ratio of 1.79, while other points are less than 1.6. For Y type, except the point bentonite ratio which was 0.1, the results of the modified K-C equation and the experimental values at other points are almost coincident. In addition, through comparison, it is found that no matter what method is used to obtain e and S in the original K-C equation, the results of the seepage prediction model proposed in this paper are obviously outstanding. The research results of this paper have certain significance to solve the academic problem that the K-C equation cannot be used in clay and also are significant to ensure the safety of building foundation pits in clay areas.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

- Chang, W.; Wang, P.; Wang, H.; Chai, S.; Yu, Y.; Xu, S. Simulation of the Q(2) loess slope with seepage fissure failure and seismic response via discrete element method. Bull. Eng. Geol. Environ.
**2021**, 80, 3495–3511. [Google Scholar] [CrossRef] - Dong, H.; Huang, R.; Gao, Q. Rainfall infiltration performance and its relation to mesoscopic structural properties of a gravelly soil slope. Eng. Geol.
**2017**, 230, 1–10. [Google Scholar] [CrossRef] - Jie, Y.X.; Jie, G.; Mao, Z.; Li, G. Seepage analysis based on boundary-fitted coordinate transformation method. Comput. Geotech.
**2004**, 31, 279–283. [Google Scholar] [CrossRef] - Yao, Z.; Chen, Z.; Fang, X.; Wang, W.; Li, W.; Su, L. Elastoplastic damage seepage-consolidation coupled model of unsaturated undisturbed loess and its application. Acta Geotech.
**2020**, 15, 1637–1653. [Google Scholar] [CrossRef] - Ba-Phu, N.; Kim, Y. An analytical solution for consolidation of PVD-installed deposit considering nonlinear distribution of hydraulic conductivity and compressibility. Eng. Comput.
**2019**, 36, 707–730. [Google Scholar] - Pane, V.; Croce, P.; Znidarcic, D.; Ko, H.-Y.; Olsen, H.W.; Schiffman, R.L. Effects of consolidation on permeability measurements for soft clay. Géotechnique
**1983**, 33, 67–72. [Google Scholar] [CrossRef] - Ren, X.; Zhao, Y.; Deng, Q.; Kang, J.; Li, D.; Wang, D. A relation of hydraulic conductivity—Void ratio for soils based on Kozeny-carman equation. Eng. Geol.
**2016**, 213, 89–97. [Google Scholar] [CrossRef] - Chapuis, R.P. Predicting the saturated hydraulic conductivity of soils: A review. Bull. Eng. Geol. Environ.
**2012**, 71, 401–434. [Google Scholar] [CrossRef] - Zhang, F.; Wang, T.; Liu, F.; Peng, M.; Bate, B.; Wang, P. Hydro-mechanical coupled analysis of near-wellbore fines migration from unconsolidated reservoirs. Acta Geotech.
**2022**, 17, 3535–3551. [Google Scholar] [CrossRef] - Wang, M.; Wang, J.; Xu, G.; Zheng, Y.; Kang, X. Improved model for predicting the hydraulic conductivity of soils based on the Kozeny–Carman equation. Hydrol. Res.
**2021**, 52, 719–733. [Google Scholar] [CrossRef] - Hong, B.; Li, X.; Wang, L.; Li, L.; Xue, Q.; Meng, J. Using the effective void ratio and specific surface area in the Kozeny-Carman Equation to predict the hydraulic conductivity of loess. Water
**2020**, 12, 24. [Google Scholar] [CrossRef] - Jang, J.; Narsilio, G.A.; Santamarina, J.C. Hydraulic conductivity in spatially varying media—A pore-scale investigation. Geophys. J. Int.
**2015**, 184, 1167–1179. [Google Scholar] [CrossRef] [Green Version] - Chai, J.C.; Agung, P.M.A.; Hino, T.; Igaya, Y.; Carter, J.P. Estimating hydraulic conductivity from piezocone soundings. Geotechnique
**2011**, 8, 699–708. [Google Scholar] [CrossRef] - Costa, A. Permeability-porosity relationship: A reexamination of Kozeny-Carman equation based on a fractal pore-space geometry assumption. Geophys. Res. Lett.
**2006**, 33, 87–94. [Google Scholar] [CrossRef] - Asger, M.N.; Fridolin, O.; Henrik, B. Reexamination of Hagen-Poiseuille flow: Shape dependence of the hydraulic resistance in microchannels. Phys. Rev. E
**2005**, 71, 057301. [Google Scholar] - Chapuis, R.P.; Legare, P.P. A simple method for determining the surface area of fine aggregates and fillers in bituminous mixtures. In Proceedings of the Effects of Aggregates and Mineral Fillers on Asphalt Mixture Performance, San Diego, CA, USA, 10 December 1991; pp. 177–186. [Google Scholar]
- Samarasinghe, A.M.; Huang, Y.H.; Drnevich, V.P. Permeability and consolidation of normally consolidated soils. J. Geotech. Eng. Div.
**1982**, 108, 835–850. [Google Scholar] [CrossRef] - Childs, E.C. Dynamics of fluids in Porous Media. Eng. Geol.
**1972**, 7, 174–175. [Google Scholar] [CrossRef] - Mesri, G.; Olson, R.E. Consolidation characteristics of montmorillonite. Geotechnique
**1971**, 4, 341–352. [Google Scholar] [CrossRef] - Taylor, D.W. Fundamentals of soil mechanics. Soil Sci.
**1948**, 66, 161. [Google Scholar] [CrossRef] - Carman, P.C. Permeability of saturated sands, soils and clays. J. Agric. Sci.
**1939**, 2, 12. [Google Scholar] [CrossRef] - Carman, P.C. Fluid flow through granular beds. Chem. Eng. Res. Des.
**1937**, 75, S32–S48. [Google Scholar] [CrossRef] - Kozeny, J. Uber Kapillare Leitung des Wassers im Boden. Sitzungsber. Akad. Wiss.
**1927**, 136, 271–306. [Google Scholar] - Li, M.; Chen, H.; Li, X.; Liu, L.; Lin, J. Permeability of granular media considering the effect of grain composition on tortuosity. Int. J. Eng. Sci.
**2022**, 174, 103658. [Google Scholar] [CrossRef] - Xu, P.; Zhang, L.; Rao, B.; Qiu, S.; Shen, Y.; Wang, M. A fractal scaling law between tortuosity and porosity in porous media. Fractals
**2020**, 28, 2050025. [Google Scholar] [CrossRef] - Xiao, B.; Wang, W.; Zhang, X.; Long, G.; Fan, J.; Chen, H.; Deng, L. A novel fractal solution for permeability and Kozeny-Carman constant of fibrous porous media made up of solid particles and porous fibers. Powder Technol.
**2019**, 349, 92–98. [Google Scholar] [CrossRef] - Bayesteh, H.; Mirghasemi, A.A. Numerical simulation of porosity and tortuosity effect on the permeability in clay: Microstructural approach. Soils Found.
**2015**, 55, 1158–1170. [Google Scholar] [CrossRef] [Green Version] - Luo, L.; Yu, B.; Cai, J.; Zeng, X. Numerical simulation of tortuosity for fluid flow in two-dimensional pore fractal models of porous media. Fractals
**2014**, 22, 1450015. [Google Scholar] [CrossRef] - Matyka, M.; Khalili, A.; Koza, Z. Tortuosity-porosity relation in porous media flow. Phys. Rev. E
**2008**, 78, 026306. [Google Scholar] [CrossRef] [Green Version] - Scholes, O.N.; Clayton, S.A.; Hoadley, A.F.A.; Tiu, C. Permeability anisotropy due to consolidation of compressible porous media. Transp. Porous Media
**2007**, 68, 365–387. [Google Scholar] [CrossRef] - Ichikawa, Y.; Kawamura, K.; Fujii, N.; Kitayama, K. Microstructure and micro/macro-diffusion behavior of tritium in bentonite. Appl. Clay Sci.
**2004**, 26, 75–90. [Google Scholar] [CrossRef] - Mathavan, G.N.; Viraraghavan, T. Coalescence/filtration of an oil-in-water emulsion in a peat bed. Water Res.
**1992**, 26, 91–98. [Google Scholar] [CrossRef] - Bear, J. Dynamics of Fluids in Porous Media; Elsevier: New York, NY, USA, 1972. [Google Scholar]
- Ahuja, L.R.; Naney, J.W.; Williams, R.D. Estimating soil water characteristics from simpler properties or limited Data. Soil Sci. Soc. Am. J.
**1985**, 49, 1100–1105. [Google Scholar] [CrossRef] - Ahuja, L.R.; Naney, J.W.; Green, R.E.; Nielsen, D.R. Macroporosity to characterize spatial variability of hydraulic conductivity and effects of land management. Soil Sci. Soc. Am. J.
**1984**, 48, 699. [Google Scholar] [CrossRef] - Horton, R.; Thompson, M.L.; Mcbride, J.F. Estimating transit times of noninteracting pollutants through compacted soil materials. Soil Sci. Soc. Am. J.
**1985**, 51, 48–53. [Google Scholar] [CrossRef] - Meegoda, N.J.; Knodel, P.C.; Gunasekera, S.D. A new method to measure the effective porosity of clays. ASTM Geotech. Test. J.
**1992**, 15, 12. [Google Scholar] - Antognozzi, M.; Humphris, A.; Miles, M.J. Observation of molecular layering in a confined water film and study of the layers viscoelastic properties. Appl. Phys. Lett.
**2001**, 78, 300–302. [Google Scholar] [CrossRef] [Green Version] - Dolinar, B.; Trcek, B. A new relationship between the mobile and the adsorbed water in fine-grained soils using an effective void-ratio estimation. Bull. Eng. Geol. Environ.
**2019**, 78, 4623–4631. [Google Scholar] [CrossRef] - Koponen, A.; Kataja, M.; Timonen, J. Permeability and effective porosity of porous media. Phys. Rev. E
**1997**, 3, 3319–3325. [Google Scholar] [CrossRef] - Singh, P.N.; Wallender, W.W. Effects of adsorbed water layer in predicting saturated hydraulic conductivity for clays with Kozeny–Carman equation. J. Geotech. Geoenvironmental Eng.
**2008**, 134, 829–836. [Google Scholar] [CrossRef] - Urumović, K.; Urumović, S.K. The effective porosity and grain size relations in permeability functions. Hydrol. Earth Syst. Sci. Discuss.
**2014**, 11, 6675–6714. [Google Scholar] - Sanzeni, A.; Colleselli, F.; Grazioli, D. Specific surface and hydraulic conductivity of fine-grained soils. J. Geotech. Geoenvironmental Eng.
**2013**, 139, 1828–1832. [Google Scholar] [CrossRef] - Indraratna, B.; Trani, L. The use of particle size distribution by surface area method in predicting the saturated hydraulic conductivity of graded granular soils. Géotechnique
**2010**, 60, 957–962. [Google Scholar] - Carrier, W.D., III. Goodbye, Hazen; Hello, Kozeny-Carman. J. Geotech. Geoenvironmental Eng.
**2003**, 129, 1054–1056. [Google Scholar] [CrossRef] [Green Version] - Gregg, S.; Sing, K. Adsorption Surface Area and Porosity; Academic Press: London, UK, 1967. [Google Scholar]
- Santamarina, J.C.; Klein, K.A.; Wang, Y.H.; Prencke, E. Specific surface: Determination and relevance. Can. Geotech. J.
**2002**, 39, 233–241. [Google Scholar] [CrossRef] - Phelps, G.W.; Harris, D.L. Specific surface and dry strength by methylene blue adsorption. Ceram. Bull.
**1968**, 12, 1146–1150. [Google Scholar] - Churchman, G.J.; Burke, C.M.; Parfitt, R.L. Comparison of various methods for the determination of specific surfaces of sub soils. Eur. J. Soil Sci.
**2010**, 42, 449–461. [Google Scholar] [CrossRef] - Cerato, A.B.; Lutenegger, A.J. Determination of surface area of fine-grained soils by the Ethylene Glycol Monoethyl Ether (EGME) method. Geotech. Test. J.
**2002**, 25, GTJ11087J. [Google Scholar] - Brunauer, S.; Emmett, P.H.; Teller, E. Adsorption of gases in multimolecular layers. J. Am. Chem. Soc.
**1938**, 60, 309–319. [Google Scholar] [CrossRef] - Garzón, E.; Sánchez-Soto, P.J. An improved method for determining the external specific surface area and the plasticity index of clayey samples based on a simplified method for non-swelling fine-grained soils. Appl. Clay Sci.
**2015**, 115, 97–107. [Google Scholar] [CrossRef] - Akin, I.D.; Likos, W. Single-point and multi-point water-sorption methods for specific surface areas of clay. Geotech. Test. J.
**2016**, 39, 20150117. [Google Scholar] [CrossRef] - Locat, J.; Lefebvre, G.; Ballivy, G. Mineralogy, chemistry, and physical properties interrelationships of some sensitive clays from Eastern Canada. Can. Geotech. J.
**1984**, 21, 530–540. [Google Scholar] [CrossRef] - Cerato, A. Influence of Specific Surface Area on Geotechnical Characteristics of Fine-Grained Soils. Master’s Thesis, Civil Engineering, College of Engineering, University of Massachusetts, Amherst, MA, USA, 2001. [Google Scholar]
- Macek, M.; Mauko, A.; Mladenovic, A.; Majes, B.; Petkovsek, A. A comparison of methods used to characterize the soil specific surface area of clays. Appl. Clay Sci.
**2013**, 83–84, 144–152. [Google Scholar] [CrossRef] - Kobayashi, I.; Owada, H.; Ishii, T.; Iizuka, A. Evaluation of specific surface area of bentonite-engineered barriers for Kozeny-Carman law. Soils Found.
**2017**, 57, 683–697. [Google Scholar] [CrossRef] - Ismeik, M.; Al-Rawi, O. Modeling soil specific surface area with artificial neural networks. Geotech. Test. J.
**2021**, 37, 20130146. [Google Scholar] [CrossRef] - Sharma, A.; Hazra, B.; Spagnoli, G.; Sekharan, S. Probabilistic estimation of specific surface area and cation exchange capacity: A global multivariate distribution. Can. Geotech. J.
**2021**, 58, 1077–1094. [Google Scholar] [CrossRef] - Meegoda, J.N.; Martin, L. In-situ determination of specific surface area of clays. Geotech. Geol. Eng.
**2019**, 37, 465–474. [Google Scholar] [CrossRef] - Dolinar, B. Predicting the hydraulic conductivity of saturated clays using plasticity-value correlations. Appl. Clay Sci.
**2009**, 45, 90–94. [Google Scholar] [CrossRef] - Dolinar, B.; Misic, M.; Trauner, L. Correlation between surface area and atterberg limits of fine-grained soils. Clays Clay Miner.
**2007**, 55, 519–523. [Google Scholar] [CrossRef] - Yukselen-Aksoy, Y.; Kaya, A. Method dependency of relationships between specific surface area and soil physicochemical properties. Appl. Clay Sci.
**2010**, 50, 182–190. [Google Scholar] [CrossRef] - Deng, Y.; Liu, Q.; Cui, Y.; Wang, Q.; Liu, S. Revisiting relationships among specific surface area, soil consistency limits, and group index of clays. J. Test. Eval.
**2019**, 47, 1392–1404. [Google Scholar] [CrossRef] - Spagnoli, G.; Shimobe, S. A statistical reappraisal of the relationship between liquid limit and specific surface area, cation exchange capacity and activity of clays. J. Rock Mech. Geotech. Eng.
**2019**, 11, 874–881. [Google Scholar] [CrossRef] - Mitchell, J.K.; Soga, K. Fundamentals of Soil Behavior, 3rd ed.; John Wiley & Sons: Hoboken, NJ, USA, 2005. [Google Scholar]
- Chen, J.; Fang, Y.; Gu, R.; Shu, H.; Ba, L.; Li, W. Study on pore size effect of low permeability clay seepage. Arab. J. Geosci.
**2019**, 12, 238. [Google Scholar] [CrossRef] - GB/T 50123-2019; Standard for Geotechnical Testing Method. Ministry of Housing and Urban-Rural Development: Beijing, China, 2019.
- BS ISO 9277-2010; Determination of the Specific Surface Area of Solids by Gas Adsorption. BET Method. BSI Group: London, UK, 2010.
- ASTM D2434-68(2000); Standard Test Method for Permeability of Granular Soils (Constant Head). ASTM: West Conshohocken, PA, USA, 2000.
- Zeng, Z.; Cui, Y.J.; Talandier, J. Evaluating the influence of soil plasticity on hydraulic conductivity based on a general capillary model. Eng. Geol.
**2020**, 278, 105826. [Google Scholar] [CrossRef] - Wang, Q.; Cui, Y.-J.; Tang, A.M.; Li, X.-L.; Ye, W.-M. Time- and density-dependent microstructure features of compacted bentonite. Soils Found.
**2014**, 54, 657–666. [Google Scholar] [CrossRef] [Green Version] - Chen, B.; Zhang, H.; Chen, P. Influence of hyper-alkaline solution infiltration on microscopic pore structure of compacted GMZ bentonite. J. Zhejiang Univ. Eng. Sci.
**2013**, 47, 602–608. [Google Scholar] - Smith, C.W.; Hadas, A.; Dan, J.; Koyumdjisky, H. Shrinkage and Atterberg limits in relation to other properties of principal soil types in Israel. Geoderma
**1985**, 35, 47–65. [Google Scholar] [CrossRef] - Churchaman, G.J.; Burke, C.M. Properties of sub soils in relation to various measures of surface area and water content. Eur. J. Soil Sci.
**1991**, 42, 463–478. [Google Scholar] [CrossRef]

**Figure 6.**Permeability instruments: (1) Overall drawing of test equipment; (2) Permeameter; (3) Sealing method of specimen.

Sample No. | Quartz | Feldspar | Illite | Chlorite | Kaolinite | Others |
---|---|---|---|---|---|---|

N1 | 19.9 | 25.7 | 23.8 | 30.6 | ||

N2 | 24.0 | 7.2 | 24.3 | 24.3 | 22.1 | |

N3 | 25.8 | 27.8 | 23.8 | 22.6 | ||

N4 | 24.8 | 23.4 | 26.3 | 23.3 | 2.2 | |

N5 | 21.4 | 16.9 | 30.0 | 31.8 | ||

N6 | 22.5 | 26.3 | 24.4 | 26.9 | ||

N7 | 24.0 | 24.9 | 19.9 | 29.3 | 1.9 | |

N8 | 22.2 | 27.4 | 25.9 | 24.5 | ||

N9 | 19.7 | 19.0 | 30.0 | 31.3 | ||

N10 | 23.2 | 26.5 | 25.3 | 23.2 | 1.9 |

No. | Component | Porosity | Test Items | No. | Component | Porosity | Test Items |
---|---|---|---|---|---|---|---|

(m_{bent}:m_{kao}) | (m_{natural}:m_{bent}) | ||||||

S1 | bentonite+ kaolinite(0.134:1) | 0.45 | PL; I _{p};e; SSA k | Y1 | 0.9:0.1 | 0.48 | PL; I _{p};e; SSA k |

S2 | 0.5 | Y2 | 0.85:0.15 | ||||

S3 | 0.55 | Y3 | 0.8:0.2 | ||||

S4 | 0.6 | Y4 | 0.75:0.25 | ||||

S5 | 0.65 | Y5 | 0.7:0.3 | ||||

Y6 | 0.65:0.35 |

No. | n | Mass Ratio | EGME | BET | Smith et al. (1985) [74] | Churchman and Burke (1991) [75] | Yeliz Yukselen-Aksoy (2010) [63] | Ren (2016) [7] | S_{eff} |
---|---|---|---|---|---|---|---|---|---|

S1 | 0.45 | 0.143:1 | 79.8 | 19.8 | 411.1 | 45.7 | 135.9 | 106.0 | 13.0 |

S2 | 0.5 | 0.143:1 | 79. 8 | 19.8 | 411.1 | 45.7 | 135.9 | 106.0 | 12.5 |

S3 | 0.55 | 0.143:1 | 79. 8 | 19.8 | 411.1 | 45.7 | 135.9 | 106.0 | 12.4 |

S4 | 0.6 | 0.143:1 | 79. 8 | 19.8 | 411.1 | 45.7 | 135.9 | 106.0 | 12.2 |

S5 | 0.65 | 0.143:1 | 79. 8 | 19.8 | 411.1 | 45.7 | 135.9 | 106.0 | 12.2 |

Y1 | 0.48 | 0.1:1 | 108.2 | 26.1 | 388.2 | 41.7 | 124.1 | 100.2 | 21.5 |

Y2 | 0.48 | 0.15:1 | 124.1 | 27.1 | 431.0 | 49.1 | 132.9 | 111.0 | 21.8 |

Y3 | 0.48 | 0.2:1 | 138.7 | 21.4 | 470.3 | 55.9 | 140.9 | 121.0 | 22.0 |

Y4 | 0.48 | 0.25:1 | 152.1 | 28.0 | 506.4 | 62.1 | 148.4 | 130.2 | 22.9 |

Y5 | 0.48 | 0.3:1 | 164.5 | 29.5 | 539.8 | 67.8 | 155.2 | 138.6 | 23.6 |

Y6 | 0.48 | 0.35:1 | 175.9 | 30.2 | 570.7 | 73.1 | 161.6 | 146.5 | 24.4 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, J.; Tong, H.; Yuan, J.; Fang, Y.; Gu, R.
Permeability Prediction Model Modified on Kozeny-Carman for Building Foundation of Clay Soil. *Buildings* **2022**, *12*, 1798.
https://doi.org/10.3390/buildings12111798

**AMA Style**

Chen J, Tong H, Yuan J, Fang Y, Gu R.
Permeability Prediction Model Modified on Kozeny-Carman for Building Foundation of Clay Soil. *Buildings*. 2022; 12(11):1798.
https://doi.org/10.3390/buildings12111798

**Chicago/Turabian Style**

Chen, Jian, Huawei Tong, Jie Yuan, Yingguang Fang, and Renguo Gu.
2022. "Permeability Prediction Model Modified on Kozeny-Carman for Building Foundation of Clay Soil" *Buildings* 12, no. 11: 1798.
https://doi.org/10.3390/buildings12111798