Advances in Civil Engineering

Volume 2019, Article ID 3965803, 13 pages

https://doi.org/10.1155/2019/3965803

## A Fractal Model to Interpret Porosity-Dependent Hydraulic Properties for Unsaturated Soils

^{1}School of Engineering, Royal Melbourne Institute of Technology (RMIT), Melbourne, VIC 3001, Australia^{2}School of Transportation Science and Engineering, Beihang University, Beijing 100191, China^{3}School of Civil Engineering, Xi’an University of Architecture and Technology, Xi’an 710055, China^{4}Henan Province Key Laboratory of Geomechanics and Structural Engineering, North China University of Water Resources and Electric Power, Zhengzhou, Henan 450045, China

Correspondence should be addressed to Annan Zhou; ua.ude.timr@uohz.nanna

Received 14 June 2018; Accepted 12 November 2018; Published 8 January 2019

Academic Editor: Castorina S. Vieira

Copyright © 2019 Annan Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a simple fractal model to quantify the effects of initial porosity on the soil-water retention curve and hydraulic conductivity of unsaturated soils. In the proposed conceptual model, the change of maximum pore radius, which largely determines the change of the air-entry value, is directly related to the fractal dimension of pore volume (*D*) and porosity change. The hydraulic properties of unsaturated soils are then governed by the maximum pore radius, the fractal dimension of pore volume (*D*), and the fractal dimension of drainable pore volume (*D*_{d} ≤ *D*). The new fractal model removes the empirical fitting parameters that have no physical meaning from existing models for porosity-dependent water retention and hydraulic behaviour and employs parameters of fractal dimensions that are intrinsic to the nature of the fractal porous materials. The proposed model is then validated against experimental data from the literature on soil-water retention behaviour and unsaturated conductivity.

#### 1. Introduction

Hydraulic properties usually refer to the properties that are related to the water retention behaviour and the hydraulic conductivity of soil, which have numerous applications in geotechnical engineering [1–6]. Soil-water retention behaviour is usually described by the soil-water retention curve (SWRC or the soil-water characteristic curve, SWCC), which is defined as the relationship between the effective degree of saturation, *S*_{e}, and the matric suction, *s*. Conversely, the hydraulic conductivity of soil is commonly described using the hydraulic conductivity function (HCF), which is defined as the relationship between the relative coefficient of conductivity, *K*_{r} (the ratio between the unsaturated and saturated values, *K*/*K*_{s}), and the matric suction, *s*, or the effective degree of saturation, *S*_{e}. It is generally recognised that the hydraulic conductivity for unsaturated soils can be effectively estimated using the soil-water retention curve, which is one of the most important applications of SWRC [7].

Numerous equations have been proposed to model SWRCs for partially saturated soils [7–11] and for HCFs [7, 11–14]. Some of these equations are based on a functional regression of the experimental data, while others are based on empirical correlations with other soil properties, such as particle or pore-size distribution, porosity, and specific surface area. However, concerns are often raised about the empirical nature of those models because they do not shed any light on the fundamental physical principles that govern the processes of unsaturated flow and drainage [15]. Several physical models for soil hydraulic properties based on the concept of fractal geometry for soil texture and pore structure have been developed [15–24]. The most important motivation to develop fractal SWRC models and fractal HCF models is that these models are able to remove the empirical fitting parameters that have no physical meaning and employ parameters of fractal dimensions that are intrinsic to the nature of the fractal porous materials [15].

One specific factor that affects the SWRC and HCF is the porosity (*ϕ*) or void ratio (*e* = *ϕ*/(1 − *ϕ*)) of the soil. A change in soil porosity can lead to a significant change in the SWRC and HCF (experimental evidence can be found in the studies of Croney and Coleman [25] and Laliberte et al. [26]); such a change is a common feature of natural soils [27]. However, it is difficult to justify that samples of a given soil with different porosities should be treated as entirely different soils for modelling. Nevertheless, most of the empirical and fractal SWRC and HCF models mentioned above omit the porosity dependency of soil hydraulic properties.

The issue of the effects of porosity on the hydraulic properties of soil was perhaps first raised by Croney and Coleman [25] and then followed by Laliberte et al. [26]. Recently, the study of the porosity effects on hydraulic properties of unsaturated soils has attracted much attention because of the rapid development of unsaturated soil mechanics involving hydromechanical coupling [28–38]. In the literature, a few approaches have been proposed to model the effect of soil porosity on SWRCs and HCFs. For example, Gallipoli et al. [28] suggested including a function of specific volume () in the SWRC equation proposed by Van Genuchten [11]. Assouline suggested an empirical approach based on regression that could model the effects of an increasing soil bulk density on the soil-water retention curve (SWRC) and the hydraulic conductivity function (HCF) [27, 39]. Sun et al. proposed a hydraulic model where a change in the degree of saturation (*S*_{r}) can be caused by a change in the matric suction or a change in soil volume () [40, 41]. Masin [30] proposed a hydraulic model that can predict the dependency of the degree of saturation (*S*_{r}) on the void ratio (*e*) using the effective stress principle. Tarantino [42] proposed a SWRC equation for deformable soils based on an empirical power function of the water ratio (). Very recently, Sheng and Zhou [43] and Zhou et al. [44] proposed an incremental relationship between the degree of saturation (*S*_{r}) and the void ratio (*e*) by realising that the SWRC is obtained under constant stress instead of constant volume.

However, almost all these approaches are based on phenomenological methodology. Therefore, the modelling parameters used to interpret the dependency of SWRC and HCF on the initial void ratio (e.g., parameters *ϕ* and *ψ* in Gallipoli et al. [28]; parameter *λ*_{se} in Sun et al. [45]; parameter *λ*_{p0} in Masin [30]; and parameter *ζ* in Zhou et al. [44]) lack any physical meaning and depend on experimental observations. In this paper, we propose a simple physical model based on fractal geometry to quantify the effects of initial porosity on the soil-water retention curve (SWRC) and the hydraulic conductivity function (HCF) of unsaturated soils. The proposed porosity-dependent SWRC and HCF models only require three parameters: the air-entry value at a specific initial void ratio (), the fractal dimension of pore volume (*D*), and the fractal dimension of drainable pore volume (*D*_{d}).

#### 2. Theory and Presentation of the Model

##### 2.1. Fractal Porous Medium

As shown in Figure 1, a porous medium (*V*_{0}) contains a broad range of pore sizes, which decrease in the mean radius from *r*_{0} to *r*_{u} (*u* >> 1) and in pore volume from *P*_{0} (the volume of the maximum pore) to *P*_{u} (the volume of the minimum pore). The pores are further divided into two categories [46, 47]: interparticle pores (including interaggregate macropores and intraaggregate micropores), which can be deformed via external loads and dewatered by the capillary process or heating, and intraparticle pores, which contain water that is strongly bounded with soil solids. Mercury intrusion porosimetry (MIP) test can be employed to determine the distribution of interparticle pores of soil [48]. During the test, mercury was compressed into pores with different radii at different intrusion pressures. The MIP technique has been widely used for geomaterials like soils. The major limitations of MIP technique include (1) it can only measure the largest available access to a pore (i.e., the size of the entrance towards a pore; for most cases, the entrance size to a pore can be substantially smaller than the inner pore size.) and (2) all the calculations are based on the assumption of cylinder pores. Intraparticle pores are nondeformable, and the intraparticle pore water cannot be dehydrated in the context of this research. In other words, strongly bounded water can be approximately considered a part of the soil solids (*V*_{m} in Figure 1) in this research. The mean radius of the interparticle pores decreases from *r*_{0} to *r*_{m−1}, and the pore volume decreases from *P*_{0} to *P*_{m−1}. The mean radius of the interparticle pores decreases from *r*_{m} to *r*_{u}, and the pore volume decreases from *P*_{m} to *P*_{u}.