Estimation of wetting hydraulic conductivity function for unsaturated sandy soil

doi:10.1016/j.enggeo.2021.106034

Engineering Geology

Volume 285, May 2021, 106034

https://doi.org/10.1016/j.enggeo.2021.106034 Get rights and content

Highlights

•
Estimation of wetting hydraulic conductivity, k_w, from SWCC.
•
Reduction in wetting k_w due to the “rain-drop” effect.
•
Reduction in wetting k_w due to the “ink-bottle” effect.
•
Proposed method is applicable for the sandy soil.

Abstract

The occurrence of rainfall-induced slope failures is mainly due to the infiltrated rainwater which reduces the shear strength of soil. The slope is initially observed in an unsaturated condition and it becomes wetted during rainfall. In other words, the infiltration of rainwater into the soil slope is a wetting process. As a result, the amount of the infiltrated rainwater is mainly governed by the hydraulic conductivity of the unsaturated soil under the wetting process. Direct measurement of the hydraulic conductivity of unsaturated soil under the wetting process is time-consuming and costly. On the other hand, there are various available methods or models for the estimation of hydraulic conductivity of unsaturated soil under the drying process. In this paper, the methodology for the estimation of hydraulic conductivity function (HCF) of unsaturated soil under the wetting process based on the drying soil-water characteristic curve (SWCC) is proposed. The proposed method is based on the concept of the pore size distribution function. Both “ink-bottle” and “rain-drop” effects on the hysteresis of SWCC are incorporated in the estimation of the HCF of unsaturated soil under the wetting process. The estimated results were verified with the experimental data from published literature.

Introduction

Seepage analyses are commonly carried out for the evaluation of rainwater infiltration for assessment of rainfall-induced slope instability. The hydraulic properties of soil such as the soil-water characteristic curve (SWCC) and the hydraulic conductivity function (HCF) are the essential information required in the seepage analysis. The function defines the relationship between the hydraulic conductivity of soil and the soil suction is commonly named as the HCF. As the direct measurement of the hydraulic conductivity of unsaturated soil is time consuming and costly, the indirect method such as the statistical method is commonly adopted. As a result, the HCF of the unsaturated soil under the wetting process is commonly estimated from the wetting SWCC using the statistical method for incorporation in the seepage analysis. However, based on the work from Zhai and Rahardjo (2015), Zhai et al. (2019a) and Zhai et al. (2020a), the assumptions adopted in the statistical method may be inappropriate for the estimation of the wetting HCF. It is known that the contact angle between the air-water interface and the soil particle under the wetting process is different from that under the drying process. Kelvin's capillary law, the equation that relates suction to pore radius under the drying process, needs to be modified (i.e., by adopting a different contact angle) to define the pore radius corresponding to a given suction in the wetting process. The pore size distribution function (PSDF) defines the geometrical pore space distribution in soil and it does not change if there is no soil volume change regardless in a drying process or in a wetting process. As the drying SWCC has been commonly considered to be analogous to PSDF, the wetting SWCC (which is different from the drying SWCC) should not be analogous to the PSDF. Zhai et al., 2019c, Zhai et al., 2020a explained that the difference between the drying SWCC and wetting SWCC were mainly resulted from the “rain-drop” and “ink-bottle” effect. Therefore, the conventional statistical method cannot be used for the estimation of the wetting HCF from the wetting SWCC.

In this study, both the “rain-drop” and “ink-bottle” effects on the water flow in the wetting process are incorporated in the estimated wetting HCF. The drying SWCC is referred to as the PSDF in the calculation. The conventional statistical method is modified and a new equation is proposed for the estimation of the wetting permeability function. The estimated results are verified with the experimental data from published literature.

Section snippets

Statistical model for the permeability function and hysteresis model for wetting SWCC

Both the theories of the statistical models for the calculation of the drying HCF and the hysteresis model for the estimation of the wetting SWCC are reviewed in this section. Consequently, the theories of the statistical model and the hysteresis model are combined to develop a new method for the estimation of the wetting HCF.

Model for the estimation of the wetting HCF

The effects of the “rain-drop” and “ink-bottle” on the HCF of soil under the wetting process are explained in this section. Consequently, a new mathematical equation for the estimation of the wetting HCF from the drying SWCC is proposed.

Effect of the “ink-bottle” effect on the permeability function

Zhai and Rahardjo (2015) explained that the HCF was calculated from the effective areas which allow water to flow through the section. Zhai et al. (2019a) adopted the “valve model” to explain the effect of the effective areas on flow rates in soil. If the pore size densities of pores r_i and r_j (r_i > r_j) are f(ψ_i) and f(ψ_j), respectively, then the effective area that pore r_i connecting to r_j is f(ψ_i)f(ψ_j)πr_j². As a result, if suction increases to ψ_i, the largest saturated tube is r_i. The

Mathematical equation for the estimation of the wetting HCF

As illustrated in Section 3.1, the “rain-drop” effect can be incorporated by introducing the additional parameter “k” while the “ink-bottle” effect can be incorporated by multiplying by the factor of S(ψ). Zhai et al. (2020a) indicated that value of “k” can be obtained from measured drying and wetting SWCC data. Therefore, a new mathematical equation can be obtained for the estimation of the wetting HCF by the summation of the effective area as follows: $k (ψ_{x, w}) = \{\begin{array}{c} \begin{array}{l} k (ψ_{ref, w}) when ψ_{x, w} > \frac{ψ_{ref, w}}{k} \\ Otherwise, \end{array} \\ k (ψ_{}) \end{array}$

Comparison between the estimated wetting HCF and the measured results

Experimental data on both drying and wetting SWCCs for the porous body I from Poulovassilis (1970) and the natural deposit sand from Liakopoulos (1965) were selected in this study. Both measured data and estimated results obtained using Zhai et al., 2020a, Zhai et al., 2020b’s model are illustrated in Fig. 3.

The comparisons between the estimated wetting HCFs from the proposed equation (i.e., Eq. 5) and the measured results for both soils are illustrated in Fig. 4.

Fig. 4 indicates that the

Conclusions and recommendations

A new mathematical equation (i.e., Eq. (5)) for the estimation of the wetting HCF from the drying SWCC was proposed in this paper. The Eq. (5) is extended from the work from Zhai et al. (2020a). Both “rain-drop” and “ink-bottle” effects have been incorporated in the proposed method. It should be noted that the proposed equation in this paper is based on the assumption that soil does not experience significant soil volume change during the wetting process. The proposed equation has been verified

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The first and fifth authors would like to acknowledge the financial supports he received from the National Natural Science Foundation of China (No. 51878160,52078128) and the Research Funding from China Huaneng Group Co. Ltd. (No. HNKJ19-H17). tyt

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Technical noteEstimation of wetting hydraulic conductivity function for unsaturated sandy soil

Highlights

Abstract

Introduction

Section snippets

Statistical model for the permeability function and hysteresis model for wetting SWCC

Model for the estimation of the wetting HCF

Effect of the “ink-bottle” effect on the permeability function

Mathematical equation for the estimation of the wetting HCF

Comparison between the estimated wetting HCF and the measured results

Conclusions and recommendations

Declaration of Competing Interest

Acknowledgements

Eng. Geol.

Eng. Geol.

About permeability of subsurface soils in case of incomplete saturation

Hydraulic Properties of Porous Medium

Relative Permeability Calculations from Pore-Size Distribution Data

Pore Pressures and Volume Changes in Unsaturated Soils

The permeability of porous materials

Proc. R. Soc. Lond. Ser. London, Math. Phys. Sci.

The interrelation between gas and oil relative permeabilities

Producer’s Mon.

Salt effects on hydraulic properties of a swelling soil

Soil Sci. of Am. J.

A general approach to hysteresis – part 3: a formal treatment of the independent domain model of hysteresis

Trans. Faraday Soc.

A general approach to hysteresis – part 4: an alternative formulation of the domain model

Trans. Faraday Soc.

Equations for the soil – water characteristic curve

Can. Geotech. J.

Predicting the permeability function for unsaturated soils using the soil – water characteristic curve

Can. Geotech. J.

Some steady state solutions of the unsaturated moisture flow equation with application to evaporation from a water table

Soil Sci.

Relative permeabilities of California cores by the capillary – pressure method, drilling and production practice

Am. Pet. Inst. Q.

The hysteresis effect in capillary properties and the modes of moisture distribution associated therewith

J. Agric. Sci.

Introduction to Soil Physics

On the Hydraulic Conductivity of Unsaturated Soils

Fundamentals of Continuum Mechanics of Soils

Factors important in the calculation of hydraulic conductivity

Proc. Soil Sci. Soc. Am.

Permeability function for unsaturated soils

J. Geotech. Geoenviron. Eng.

Technical note
Estimation of wetting hydraulic conductivity function for unsaturated sandy soil