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

Authors: Annan Zhou; Yang Fan; Wen-Chieh Cheng; Junran Zhang
Journal: Advances in Civil Engineering
 (2019)

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## 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 (Dd ≤ 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.


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## Body

## 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, Se, 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, Kr (the ratio between the unsaturated and saturated values, K/Ks), and the matric suction, s, or the effective degree of saturation, Se. 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 (v) 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 (Sr) can be caused by a change in the matric suction or a change in soil volume (v) [40, 41]. Masin [30] proposed a hydraulic model that can predict the dependency of the degree of saturation (Sr) 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 (ew). Very recently, Sheng and Zhou [43] and Zhou et al. [44] proposed an incremental relationship between the degree of saturation (Sr) 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., parametersϕ and ψ in Gallipoli et al. [28]; parameter λse in Sun et al. [45]; parameter λp0 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 (sae∗), the fractal dimension of pore volume (D), and the fractal dimension of drainable pore volume (Dd).


## 2. Theory and Presentation of the Model

### 2.1. Fractal Porous Medium

As shown in Figure1, a porous medium (V0) contains a broad range of pore sizes, which decrease in the mean radius from r0 to ru (u >> 1) and in pore volume from P0 (the volume of the maximum pore) to Pu (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 (Vm in Figure 1) in this research. The mean radius of the interparticle pores decreases from r0 to rm−1, and the pore volume decreases from P0 to Pm−1. The mean radius of the interparticle pores decreases from rm to ru, and the pore volume decreases from Pm to Pu.Figure 1
Characterization of pores.Following Rieu and Sposito [22], a real porous medium can be assumed to be a fractal medium in which there is self-similarity of the pore radius in the entire domain from r0 to ru. In terms of pore space, we assume that(1)ri+1=γri,Pi+1=γEPi,where E is the Euclidean dimension, which is equal to 2 for two-dimensional objects (such as the Sierpinski triangle and the Sierpinski carpet) and 3 for three-dimensional objects (such as the Menger sponge), respectively. γ is a linear similarity ratio that is introduced to describe the scaling property of a fractal medium. For example, γ is equal to 1/3 for the Sierpinski carpet (Figure 2) and 1/2 for the Sierpinski triangle, respectively. Vi is defined as the ith self-similar partial volume, which contains all pores that have a radius ≤ ri. Vm−1 is the self-similar partial volume that contains the smallest interparticle pores (radius rm−1) and Vm, which stands for the volume of a soil solid particle that is nondeformable and contains all of the intraparticle pores. If the ith self-similar partial volume can be represented by its mean radius Ri, similar to pores in the fractal medium [22], we have(2)Ri+1=γRi,Vi+1=γEVi.Figure 2
Schematic diagram of fractal porous medium (Sierpinski carpet).The self-similarity property of a fractal porous medium means that the (i + 1)th self-similar partial volume repeats the same pore property of the ith self-similar partial volume. For example, for the Sierpinski carpet (Figure 2), V0 contains 1 P0 and 8 (=32 − 1) V1, V1 contains 1 P1 and 8 V2, …, Vm−1 contains 1 Pm−1 and 8 Vm. In general, such a self-similarity can be written as(3)Vi=NVi+1+Pi,where a constant number N stands for the number of the (i + 1)th self-similar partial volume in the ith self-similar partial volume. For the Sierpinski carpet, as shown in Figure 2, N = 8. According to equation (3), the total volume V0 can be written as(4)V0=∑i=0m−1NiPi+NmVm,where NmVm stands for the total solid volume (Vs) in this porous medium. In addition, the pore coefficient Γ is defined as the ratio between the ith pore volume and the ith self-similar partial volume, i.e., Γ = Pi/Vi. Γ = (1/3)2 for the Sierpinski carpet. Vi+1/Vi can be written as a function of Γ, i.e.,(5)Vi+1Vi=1−ΓN.Combining equations (4) and (5) generates(6)VmV0=VmVm−1Vm−1Vm−2⋯V2V1V1V0=1−ΓmNm.According to equation (6), the porosity of the medium thus can be written as(7)ϕ=V0−VsV0=V0−NmVmV0=1−1−Γm.For a fractal medium, in terms of pore volume distribution, the number ofN and the linear similarity ratio γ can be related through the fractal dimension D [49] of pore space:(8)Dv≡logNlog1/γorN=γ−D.As mentioned in the study of Gimenez [50], the fractal dimension D of pore volume can be measured using two-dimensional images of cross sections of soils with either the box-counting technique or by the pore size count [51, 52].Combining equations (2), (5), and (8), we have(9)Vi+1Vi=1−ΓN=γE⟶Γ=1−γE−Dor1−Γ=γE−D.Therefore, the porosity of the medium can be rewritten by substituting equation (9) into equation (7):(10)ϕ=1−γmE−D.According to equation (1), γm can be replaced by rm/r0. Therefore, equation (10) can be rewritten as(11)ϕ=1−rmr0E−D0<D<E,where rm is the radius of maximum intraparticle pores, which is a constant because the soil particles are assumed as nondeformable. It also augured that equation (11) disagrees with the porosity equation proposed by Katz and Thompson [53] and results in zero porosity when D = E and, therefore, questioned its validity. However, Hunt [54] concluded that Katz and Thompson’s model and Rieu and Sposito’s model (equation (11)) can be precisely compatible if Katz and Thompson’s model involves solid structures and Rieu and Sposito’s model involves pore spaces. As mentioned in the study of Hunt [54]; Rieu and Sposito’s model is valid for the pore space and Katz and Thompson’s cannot be valid if D refers to the pore space. Therefore, it is important to note that the fractal dimension (D) here refers to the pore space other than the solid space in this research. In fact, the condition D = E does not happen for soils in the real world. Experimental evidence [50] shows that the fractal dimensions for various soils are less than the Euclidean dimension. Therefore, Perrier et al. [21] added a boundary condition for equation (11), which always requires 0 < D < E. This boundary condition is also adopted in this article.According to Xu and Xia [55], the fractal dimension can be assumed to be independent of the soil porosity. Therefore, the porosity change only changes the value of r0 in equation (11). The change of r0 will lead to the change of the entire interparticle pore system. The maximum interparticle pore radius is r0′ if the porosity changes from ϕ to ϕ′, and r0′ can be written as(12)r0′r0=1−ϕ1−ϕ′1/E−D.


### 2.2. Porosity-Dependent SWRC

Water in the interparticle pores can be drained by applying soil matric suction. If the fractal porous medium containing interparticle pores with radius >ri is completely dried due to the capillary flow process, the volumetric water content θi can be written as(13)θi=∑i=im−1NiPiV0.However, some interparticle pores with radius >ri may be isolated by solids and pores with a radius less than ri. As drying occurs, not all pores of a given size drain at the appropriate suction because of incomplete pore connectivity [56, 57]. The number of drained pores, Nd, is assumed to be fractal and proportional to the power of (1/γ), as expressed by(14)Nd=γ−Dd≤γ−D=N,where Dd (≤D) is the fractal dimension for the drainable pore space defined by Perfect [57] and Cihan et al. [56]. Theoretically, Dd is equal to D − log (Pd)/log (γ), where Pd is the scale-invariant drainage probability for the pore network [57]. Experimentally, Dd can be estimated from the water retention curve; for example, Crawford et al. [58] reported Dd values ranging from 2.90 to 2.97 (E = 3) along with corresponding D values (obtained from thin section analysis) of between 2.94 and 2.98, for eight Japanese soils.Therefore, volumetric water contentθi can be revised as(15)θi=∑i=0m−1NiPi−∑i=0i−1NdiPiV0.Equation (13) is a special case of equation (15) when Nd = N. The porosity ϕ, which is related to D rather than Dd, can also be expressed similarly:(16)ϕ=V0−VsV0=∑i=0m−1NiPiV0.Comparing equations (15) and (16) yields(17)θiϕ=P01+γE−D+…+γm−1E−D−P01+γE−Dd+…+γi−1E−DdP01+γE−D+…+γm−1E−D,=1−γmE−D/1−γE−D−1−γiE−Dd/1−γE−Dd1−γmE−D/1−γE−D=1−1−γE−D1−γE−Dd1−γiE−Ddϕ.Consideringγi = ri/r0, equation (17) can be simplified as(18)θi=ϕ−1−γE−D1−γE−Dd1−rir0E−Ddrm≤ri≤r0,when ri is set to r0, which means pores with a radius larger than r0 are drained (i.e., no pores are drained), the water content is equal to the porosity which coincides with the fully saturated condition. The residual volumetric water constant (θm) can be calculated when ri is set to rm:(19)θm=ϕ−1−γE−D1−γE−Dd1−1−ϕE−Dd/E−D,when ri is set to rm, pores with a radius larger than rm are drained (i.e., the entire interparticle pores) except from those isolated pores. If D = Dd (i.e., no isolated pores), θm = 0, which coincides with the dried condition (i.e., only intraparticle water exists).Therefore, the effective degree of saturation (Se) can be written by combining equations (18) and (19):(20)Se=θi−θmϕ−θm=ri/r0E−Dd−1−ϕE−Dd/E−D1−1−ϕE−Dd/E−D.Applying the Young–Laplace equation (i.e.,s = σlg/r, σlg is the liquid-gas interfacial tension) to equation (18) yields(21)Se=1,s≤sae,s/saeDd−E−1−ϕa1−1−ϕa,sae1−ϕ1/D−E>s>sae,0,s≥sae1−ϕ1/D−E,where a is equal to (Dd − E)/(D − E), s is the matric suction or capillary potential, and sae is the air-entry value, which is inversely proportion to r0. If sae is associated with r0 and sae∗ with r0∗, then according to (12), we have(22)sae=sae∗1−ϕ∗1−ϕ1/D−E.Substituting equation (22) into equation (21) produces(23)Se=1,s≤sae∗1−ϕ∗1−ϕ1/D−E,s/sae∗Dd−E−1−ϕ∗a1−ϕ∗/1−ϕa−1−ϕ∗a,sae∗1−ϕ∗1/D−E>s>sae∗1−ϕ∗1−ϕ1/D−E,0,s≥sae∗1−ϕ∗1/D−E,if Dd = D and ϕ∗ = ϕ, equation (23) degenerates back to the renown fractal SWRC model proposed by Rieu and Sposito [22]:(24)Se=1,s≤sae,1ϕssaeD−E−1ϕ+1,sae1−ϕ1/D−E>s>sae,0,s≥sae1−ϕ1/D−E.


### 2.3. Porosity-Dependent HCF

The hydraulic conductivity function (HCF) plays a pivotal role in the flow and transport processes under both saturated and unsaturated conditions. The HCF for unsaturated soils is usually represented by the relative hydraulic conductivityKr, which is defined as the ratio between the unsaturated hydraulic conductivity K and the corresponding saturated hydraulic conductivity Ks. A number of methods have been proposed to quantify the relative hydraulic conductivity Kr for unsaturated soils [7, 12, 19, 59–61], and most of them express Kr as a function of the effective degree of saturation (Se), volumetric water content (θ), and matric suction (s). Among the proposed HCFs for unsaturated soils, the most cited HCF is that proposed by Mualem [7]:(25)Kr=Se∫0Se1/sdSe∫011/sdSe2.When using the model of Mualem [7], the resulting closed-form analytical expression for HCF depends on the model that is used to describe the SWRC. If the SWRC involved is porosity dependent, the HCF is able to reflect the porosity dependency of the unsaturated hydraulic conductivity. Substituting equation (23) into equation (25) generates the fractal model for a soil’s Kr and s relationship considering the influence of various soil porosities:(26)Kr=1,s≤sae∗1−ϕ∗1−ϕ1/D−E,s/sae∗Dd−E−1−ϕ∗a1−ϕ∗/1−ϕa−1−ϕ∗a0.5s/sae∗Dd−E−1−1−ϕ∗b1−ϕ∗/1−ϕb−1−ϕ∗b2,sae∗1−ϕ∗1/D−E>s>sae∗1−ϕ∗1−ϕ1/D−E,0,s≥sae∗1−ϕ∗1/D−E,where b is equal to (Dd − E − 1)/(D − E).


## 2.1. Fractal Porous Medium

As shown in Figure1, a porous medium (V0) contains a broad range of pore sizes, which decrease in the mean radius from r0 to ru (u >> 1) and in pore volume from P0 (the volume of the maximum pore) to Pu (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 (Vm in Figure 1) in this research. The mean radius of the interparticle pores decreases from r0 to rm−1, and the pore volume decreases from P0 to Pm−1. The mean radius of the interparticle pores decreases from rm to ru, and the pore volume decreases from Pm to Pu.Figure 1
Characterization of pores.Following Rieu and Sposito [22], a real porous medium can be assumed to be a fractal medium in which there is self-similarity of the pore radius in the entire domain from r0 to ru. In terms of pore space, we assume that(1)ri+1=γri,Pi+1=γEPi,where E is the Euclidean dimension, which is equal to 2 for two-dimensional objects (such as the Sierpinski triangle and the Sierpinski carpet) and 3 for three-dimensional objects (such as the Menger sponge), respectively. γ is a linear similarity ratio that is introduced to describe the scaling property of a fractal medium. For example, γ is equal to 1/3 for the Sierpinski carpet (Figure 2) and 1/2 for the Sierpinski triangle, respectively. Vi is defined as the ith self-similar partial volume, which contains all pores that have a radius ≤ ri. Vm−1 is the self-similar partial volume that contains the smallest interparticle pores (radius rm−1) and Vm, which stands for the volume of a soil solid particle that is nondeformable and contains all of the intraparticle pores. If the ith self-similar partial volume can be represented by its mean radius Ri, similar to pores in the fractal medium [22], we have(2)Ri+1=γRi,Vi+1=γEVi.Figure 2
Schematic diagram of fractal porous medium (Sierpinski carpet).The self-similarity property of a fractal porous medium means that the (i + 1)th self-similar partial volume repeats the same pore property of the ith self-similar partial volume. For example, for the Sierpinski carpet (Figure 2), V0 contains 1 P0 and 8 (=32 − 1) V1, V1 contains 1 P1 and 8 V2, …, Vm−1 contains 1 Pm−1 and 8 Vm. In general, such a self-similarity can be written as(3)Vi=NVi+1+Pi,where a constant number N stands for the number of the (i + 1)th self-similar partial volume in the ith self-similar partial volume. For the Sierpinski carpet, as shown in Figure 2, N = 8. According to equation (3), the total volume V0 can be written as(4)V0=∑i=0m−1NiPi+NmVm,where NmVm stands for the total solid volume (Vs) in this porous medium. In addition, the pore coefficient Γ is defined as the ratio between the ith pore volume and the ith self-similar partial volume, i.e., Γ = Pi/Vi. Γ = (1/3)2 for the Sierpinski carpet. Vi+1/Vi can be written as a function of Γ, i.e.,(5)Vi+1Vi=1−ΓN.Combining equations (4) and (5) generates(6)VmV0=VmVm−1Vm−1Vm−2⋯V2V1V1V0=1−ΓmNm.According to equation (6), the porosity of the medium thus can be written as(7)ϕ=V0−VsV0=V0−NmVmV0=1−1−Γm.For a fractal medium, in terms of pore volume distribution, the number ofN and the linear similarity ratio γ can be related through the fractal dimension D [49] of pore space:(8)Dv≡logNlog1/γorN=γ−D.As mentioned in the study of Gimenez [50], the fractal dimension D of pore volume can be measured using two-dimensional images of cross sections of soils with either the box-counting technique or by the pore size count [51, 52].Combining equations (2), (5), and (8), we have(9)Vi+1Vi=1−ΓN=γE⟶Γ=1−γE−Dor1−Γ=γE−D.Therefore, the porosity of the medium can be rewritten by substituting equation (9) into equation (7):(10)ϕ=1−γmE−D.According to equation (1), γm can be replaced by rm/r0. Therefore, equation (10) can be rewritten as(11)ϕ=1−rmr0E−D0<D<E,where rm is the radius of maximum intraparticle pores, which is a constant because the soil particles are assumed as nondeformable. It also augured that equation (11) disagrees with the porosity equation proposed by Katz and Thompson [53] and results in zero porosity when D = E and, therefore, questioned its validity. However, Hunt [54] concluded that Katz and Thompson’s model and Rieu and Sposito’s model (equation (11)) can be precisely compatible if Katz and Thompson’s model involves solid structures and Rieu and Sposito’s model involves pore spaces. As mentioned in the study of Hunt [54]; Rieu and Sposito’s model is valid for the pore space and Katz and Thompson’s cannot be valid if D refers to the pore space. Therefore, it is important to note that the fractal dimension (D) here refers to the pore space other than the solid space in this research. In fact, the condition D = E does not happen for soils in the real world. Experimental evidence [50] shows that the fractal dimensions for various soils are less than the Euclidean dimension. Therefore, Perrier et al. [21] added a boundary condition for equation (11), which always requires 0 < D < E. This boundary condition is also adopted in this article.According to Xu and Xia [55], the fractal dimension can be assumed to be independent of the soil porosity. Therefore, the porosity change only changes the value of r0 in equation (11). The change of r0 will lead to the change of the entire interparticle pore system. The maximum interparticle pore radius is r0′ if the porosity changes from ϕ to ϕ′, and r0′ can be written as(12)r0′r0=1−ϕ1−ϕ′1/E−D.


## 2.2. Porosity-Dependent SWRC

Water in the interparticle pores can be drained by applying soil matric suction. If the fractal porous medium containing interparticle pores with radius >ri is completely dried due to the capillary flow process, the volumetric water content θi can be written as(13)θi=∑i=im−1NiPiV0.However, some interparticle pores with radius >ri may be isolated by solids and pores with a radius less than ri. As drying occurs, not all pores of a given size drain at the appropriate suction because of incomplete pore connectivity [56, 57]. The number of drained pores, Nd, is assumed to be fractal and proportional to the power of (1/γ), as expressed by(14)Nd=γ−Dd≤γ−D=N,where Dd (≤D) is the fractal dimension for the drainable pore space defined by Perfect [57] and Cihan et al. [56]. Theoretically, Dd is equal to D − log (Pd)/log (γ), where Pd is the scale-invariant drainage probability for the pore network [57]. Experimentally, Dd can be estimated from the water retention curve; for example, Crawford et al. [58] reported Dd values ranging from 2.90 to 2.97 (E = 3) along with corresponding D values (obtained from thin section analysis) of between 2.94 and 2.98, for eight Japanese soils.Therefore, volumetric water contentθi can be revised as(15)θi=∑i=0m−1NiPi−∑i=0i−1NdiPiV0.Equation (13) is a special case of equation (15) when Nd = N. The porosity ϕ, which is related to D rather than Dd, can also be expressed similarly:(16)ϕ=V0−VsV0=∑i=0m−1NiPiV0.Comparing equations (15) and (16) yields(17)θiϕ=P01+γE−D+…+γm−1E−D−P01+γE−Dd+…+γi−1E−DdP01+γE−D+…+γm−1E−D,=1−γmE−D/1−γE−D−1−γiE−Dd/1−γE−Dd1−γmE−D/1−γE−D=1−1−γE−D1−γE−Dd1−γiE−Ddϕ.Consideringγi = ri/r0, equation (17) can be simplified as(18)θi=ϕ−1−γE−D1−γE−Dd1−rir0E−Ddrm≤ri≤r0,when ri is set to r0, which means pores with a radius larger than r0 are drained (i.e., no pores are drained), the water content is equal to the porosity which coincides with the fully saturated condition. The residual volumetric water constant (θm) can be calculated when ri is set to rm:(19)θm=ϕ−1−γE−D1−γE−Dd1−1−ϕE−Dd/E−D,when ri is set to rm, pores with a radius larger than rm are drained (i.e., the entire interparticle pores) except from those isolated pores. If D = Dd (i.e., no isolated pores), θm = 0, which coincides with the dried condition (i.e., only intraparticle water exists).Therefore, the effective degree of saturation (Se) can be written by combining equations (18) and (19):(20)Se=θi−θmϕ−θm=ri/r0E−Dd−1−ϕE−Dd/E−D1−1−ϕE−Dd/E−D.Applying the Young–Laplace equation (i.e.,s = σlg/r, σlg is the liquid-gas interfacial tension) to equation (18) yields(21)Se=1,s≤sae,s/saeDd−E−1−ϕa1−1−ϕa,sae1−ϕ1/D−E>s>sae,0,s≥sae1−ϕ1/D−E,where a is equal to (Dd − E)/(D − E), s is the matric suction or capillary potential, and sae is the air-entry value, which is inversely proportion to r0. If sae is associated with r0 and sae∗ with r0∗, then according to (12), we have(22)sae=sae∗1−ϕ∗1−ϕ1/D−E.Substituting equation (22) into equation (21) produces(23)Se=1,s≤sae∗1−ϕ∗1−ϕ1/D−E,s/sae∗Dd−E−1−ϕ∗a1−ϕ∗/1−ϕa−1−ϕ∗a,sae∗1−ϕ∗1/D−E>s>sae∗1−ϕ∗1−ϕ1/D−E,0,s≥sae∗1−ϕ∗1/D−E,if Dd = D and ϕ∗ = ϕ, equation (23) degenerates back to the renown fractal SWRC model proposed by Rieu and Sposito [22]:(24)Se=1,s≤sae,1ϕssaeD−E−1ϕ+1,sae1−ϕ1/D−E>s>sae,0,s≥sae1−ϕ1/D−E.


## 2.3. Porosity-Dependent HCF

The hydraulic conductivity function (HCF) plays a pivotal role in the flow and transport processes under both saturated and unsaturated conditions. The HCF for unsaturated soils is usually represented by the relative hydraulic conductivityKr, which is defined as the ratio between the unsaturated hydraulic conductivity K and the corresponding saturated hydraulic conductivity Ks. A number of methods have been proposed to quantify the relative hydraulic conductivity Kr for unsaturated soils [7, 12, 19, 59–61], and most of them express Kr as a function of the effective degree of saturation (Se), volumetric water content (θ), and matric suction (s). Among the proposed HCFs for unsaturated soils, the most cited HCF is that proposed by Mualem [7]:(25)Kr=Se∫0Se1/sdSe∫011/sdSe2.When using the model of Mualem [7], the resulting closed-form analytical expression for HCF depends on the model that is used to describe the SWRC. If the SWRC involved is porosity dependent, the HCF is able to reflect the porosity dependency of the unsaturated hydraulic conductivity. Substituting equation (23) into equation (25) generates the fractal model for a soil’s Kr and s relationship considering the influence of various soil porosities:(26)Kr=1,s≤sae∗1−ϕ∗1−ϕ1/D−E,s/sae∗Dd−E−1−ϕ∗a1−ϕ∗/1−ϕa−1−ϕ∗a0.5s/sae∗Dd−E−1−1−ϕ∗b1−ϕ∗/1−ϕb−1−ϕ∗b2,sae∗1−ϕ∗1/D−E>s>sae∗1−ϕ∗1−ϕ1/D−E,0,s≥sae∗1−ϕ∗1/D−E,where b is equal to (Dd − E − 1)/(D − E).


## 3. Experimental Validations

### 3.1. SWRCs with Different Initial Porosities

A series of drying tests on a compacted till were reported by Vanapalli et al. [62]. The soil specimens were compacted to different initial void ratios at the optimum water content. Figures 3(a)–3(d) show the drying test results for these specimens, by different symbols for different initial void ratios (e) or initial porosities (ϕ). The test data with an initial void ratio of 0.517 (ϕ∗ = 0.341) were adopted to calibrate the parameters using the method of least squares. The values of D and Dd can be determined by maximising the value of R2 when the air-entry value sae∗ is set to 8 kPa according to the data points:(27)R2=1−∑1jSeCal−SeExp2∑1jSeCal−S¯eCal2,where j is the number of experimental data points (in this case, j = 11), SeExp is the measured (experimental) effective degree of saturation for a data point, and SeCal is the calculated effective degree of saturation for a data point for the same data point by equation (23). S¯eCal is the mean of the calculated effective degree of saturation for all data points. For all the cases in this study, the Euclidean dimension E is set to 2. Thus, we have the range of Dd and D as 1 < Dd ≤ D < 2. The initial values of Dd and D are set to 1, and the increment for Dd and D is set to 10−3 (i.e., ΔDd = ΔD = 10−3). For each set of Dd and D, a value R2 is calculated and recorded. Finally, the maximum value of R2 can be achieved, which is equal to 0.9837, when Dd = 1.915 and D = 1.972. The other 3 data sets (corresponding to e = 0.514, e = 0.474, and e = 0.444) were used to validate the proposed equation. The predicted SWRCs were calculated using equation (23) and are shown as solid curves in Figure 3. The value of R2 for rest three data sets is computed to illustrate the validity of the model predictions. The predicted SWRCs in Figures 3(b)–3(d) match the experimental data well with R2 varying from 0.9364 to 0.9726, indicating that the proposed fractal approach captures the effects of the initial void ratio on the SWRCs well.Figure 3
Measured and predicted SWRCs for specimens compacted at optimum water content (data after Vanapalli et al. [62]): (a) e = 0.517; (b) e = 0.514; (c) e = 0.474; (d) e = 0.444.
(a)
(b)
(c)
(d)A series of laboratory experiments were conducted by Huang et al. [63] to investigate the water retention behaviour of a deformable unsaturated soil. The soil tested was silty sand from the Saskatchewan Department of Highway borrow pit. This silty sand consisted of 52.5% sand, 37.5% silt, and 10% clay. The specific gravity, liquid limit, and plastic limit were 2.68, 22.2%, and 16.6%, respectively. The air-dried silty sand was mixed with distilled water to prepare slurried specimens for the experimental program. Six initially slurried specimens (grouped as PPCT2) were one-dimensionally preconsolidated under different pressures to obtain different initial void ratios for the water retention tests. The test results are replotted in the Se-s plane together with the predictions of equation (23) in Figures 4(a)–4(f). The water retention test data for an initial void ratio of 0.525 (ϕ∗ = 0.344) were used to calibrate the parameters. The values of D (=1.883) and Dd (=1.883) were determined by maximising the value of R2 (=0.9966) when the air-entry value sae∗ set to 24 kPa according to the data points. As shown in Figures 4(b)–4(f), the predicted SWRCs agree very well with the experimental SWRCs (R2 = 0.9821∼0.9968).Figure 4
Measured and predicted SWRCs for silty sand with different initial void ratios (data after Huang et al. [63]): (a) e = 0.525; (b) e = 0.513; (c) e = 0.490; (d) e = 0.474; (e) e = 0.454; (f) e = 0.426.
(a)
(b)
(c)
(d)
(e)
(f)Very recently, Salager et al. [64] conducted a series of tests on a clayey sand to investigate the water retention behaviour of a granular soil. The tested soil consisted of 72% sand, 18% silt, and 10% clay. The water retention behaviour of the clayey sand compacted at five different void ratios (e) is replotted in the Se-s plane in Figures 5(a)–5(e). The variation in the initial void ratios ranges from 0.44 to 1.01. The water retention data for the loosest specimen (e∗ = 1.01, i.e., ϕ∗ = 0.502) were adopted to calibrate the parameters. The values of D (=1.951) and Dd (=1.935) were determined by maximising the value of R2 (=0.9834) when the air-entry value sae∗ is set to 0.2 kPa according to the data points. The SWRCs predicted by equation (23) are shown in Figures 5(b)–5(e). Again, the comparison between the experimental data and predictions is of acceptable accuracy with the value of R2 between 0.8865 and 0.9794.Figure 5
Measured and predicted family of SWRCs for a clayey sand with different initial void ratios (data after Salager et al. [64]): (a) e = 1.01; (b) e = 0.86; (c) e = 0.68; (d) e = 0.55; (e) e = 0.44.
(a)
(b)
(c)
(d)
(e)


### 3.2. HCFs with Different Initial Porosities

Laliberte et al. [26] measured the SWRCs (Se versus s) and HCFs (Kr versus s) of a silt loam, which is referred to as the Touchet silt loam, with different initial porosities. A Touchet silt loam is coarse silt, consisting of 32% sand, 53% silt, and 15% clay, with a particle density of 2.599 g/cm3. The initial porosities for the water retention tests are 0.493 (e∗ = 0.972), 0.463 (e = 0.862), and 0.430 (e = 0.754). The initial porosities for the hydraulic conductivity tests are 0.503 (e = 1.012), 0.478 (e = 0.916), 0.449 (e = 0.815), 0.423 (e = 0.733), and 0.395 (e = 0.653). The data set for the loosest specimen (ϕ∗ = 0.493) of the water retention test is used for calibration. The parameters for SWRC are set to sae∗  = 4.5 kPa, D = 1.7, and Dd = 1.05, which result in a value of R2 = 0.9909 for all data sets with three different initial porosities (Figure 6(a)). As shown in Figures 6(b)–6(f), the measured relative hydraulic conductivities for five different initial porosities (0.503∼0.395) are replotted in the Kr-s plane (double logarithmic scales), which are the predictions obtained using equation (26). The predicted Kr-s curves agree with the experimental data reasonably well, which confirms that the proposed porosity-dependent fractal HCF model (i.e., equation (26)) captures well the effects of the initial porosity on the soil’s unsaturated permeability.Figure 6
Comparison between measured and predicted SWRCs and HCFs for the Touchet silt loam: (a) SWRCs, (b) HCF (ϕ = 0.503), (c) HCF (ϕ = 0.478), (d) HCF (ϕ = 0.449), (e) HCF (ϕ = 0.423), and (f) HCF (ϕ = 0.395).
(a)
(b)
(c)
(d)
(e)
(f)


## 3.1. SWRCs with Different Initial Porosities

A series of drying tests on a compacted till were reported by Vanapalli et al. [62]. The soil specimens were compacted to different initial void ratios at the optimum water content. Figures 3(a)–3(d) show the drying test results for these specimens, by different symbols for different initial void ratios (e) or initial porosities (ϕ). The test data with an initial void ratio of 0.517 (ϕ∗ = 0.341) were adopted to calibrate the parameters using the method of least squares. The values of D and Dd can be determined by maximising the value of R2 when the air-entry value sae∗ is set to 8 kPa according to the data points:(27)R2=1−∑1jSeCal−SeExp2∑1jSeCal−S¯eCal2,where j is the number of experimental data points (in this case, j = 11), SeExp is the measured (experimental) effective degree of saturation for a data point, and SeCal is the calculated effective degree of saturation for a data point for the same data point by equation (23). S¯eCal is the mean of the calculated effective degree of saturation for all data points. For all the cases in this study, the Euclidean dimension E is set to 2. Thus, we have the range of Dd and D as 1 < Dd ≤ D < 2. The initial values of Dd and D are set to 1, and the increment for Dd and D is set to 10−3 (i.e., ΔDd = ΔD = 10−3). For each set of Dd and D, a value R2 is calculated and recorded. Finally, the maximum value of R2 can be achieved, which is equal to 0.9837, when Dd = 1.915 and D = 1.972. The other 3 data sets (corresponding to e = 0.514, e = 0.474, and e = 0.444) were used to validate the proposed equation. The predicted SWRCs were calculated using equation (23) and are shown as solid curves in Figure 3. The value of R2 for rest three data sets is computed to illustrate the validity of the model predictions. The predicted SWRCs in Figures 3(b)–3(d) match the experimental data well with R2 varying from 0.9364 to 0.9726, indicating that the proposed fractal approach captures the effects of the initial void ratio on the SWRCs well.Figure 3
Measured and predicted SWRCs for specimens compacted at optimum water content (data after Vanapalli et al. [62]): (a) e = 0.517; (b) e = 0.514; (c) e = 0.474; (d) e = 0.444.
(a)
(b)
(c)
(d)A series of laboratory experiments were conducted by Huang et al. [63] to investigate the water retention behaviour of a deformable unsaturated soil. The soil tested was silty sand from the Saskatchewan Department of Highway borrow pit. This silty sand consisted of 52.5% sand, 37.5% silt, and 10% clay. The specific gravity, liquid limit, and plastic limit were 2.68, 22.2%, and 16.6%, respectively. The air-dried silty sand was mixed with distilled water to prepare slurried specimens for the experimental program. Six initially slurried specimens (grouped as PPCT2) were one-dimensionally preconsolidated under different pressures to obtain different initial void ratios for the water retention tests. The test results are replotted in the Se-s plane together with the predictions of equation (23) in Figures 4(a)–4(f). The water retention test data for an initial void ratio of 0.525 (ϕ∗ = 0.344) were used to calibrate the parameters. The values of D (=1.883) and Dd (=1.883) were determined by maximising the value of R2 (=0.9966) when the air-entry value sae∗ set to 24 kPa according to the data points. As shown in Figures 4(b)–4(f), the predicted SWRCs agree very well with the experimental SWRCs (R2 = 0.9821∼0.9968).Figure 4
Measured and predicted SWRCs for silty sand with different initial void ratios (data after Huang et al. [63]): (a) e = 0.525; (b) e = 0.513; (c) e = 0.490; (d) e = 0.474; (e) e = 0.454; (f) e = 0.426.
(a)
(b)
(c)
(d)
(e)
(f)Very recently, Salager et al. [64] conducted a series of tests on a clayey sand to investigate the water retention behaviour of a granular soil. The tested soil consisted of 72% sand, 18% silt, and 10% clay. The water retention behaviour of the clayey sand compacted at five different void ratios (e) is replotted in the Se-s plane in Figures 5(a)–5(e). The variation in the initial void ratios ranges from 0.44 to 1.01. The water retention data for the loosest specimen (e∗ = 1.01, i.e., ϕ∗ = 0.502) were adopted to calibrate the parameters. The values of D (=1.951) and Dd (=1.935) were determined by maximising the value of R2 (=0.9834) when the air-entry value sae∗ is set to 0.2 kPa according to the data points. The SWRCs predicted by equation (23) are shown in Figures 5(b)–5(e). Again, the comparison between the experimental data and predictions is of acceptable accuracy with the value of R2 between 0.8865 and 0.9794.Figure 5
Measured and predicted family of SWRCs for a clayey sand with different initial void ratios (data after Salager et al. [64]): (a) e = 1.01; (b) e = 0.86; (c) e = 0.68; (d) e = 0.55; (e) e = 0.44.
(a)
(b)
(c)
(d)
(e)


## 3.2. HCFs with Different Initial Porosities

Laliberte et al. [26] measured the SWRCs (Se versus s) and HCFs (Kr versus s) of a silt loam, which is referred to as the Touchet silt loam, with different initial porosities. A Touchet silt loam is coarse silt, consisting of 32% sand, 53% silt, and 15% clay, with a particle density of 2.599 g/cm3. The initial porosities for the water retention tests are 0.493 (e∗ = 0.972), 0.463 (e = 0.862), and 0.430 (e = 0.754). The initial porosities for the hydraulic conductivity tests are 0.503 (e = 1.012), 0.478 (e = 0.916), 0.449 (e = 0.815), 0.423 (e = 0.733), and 0.395 (e = 0.653). The data set for the loosest specimen (ϕ∗ = 0.493) of the water retention test is used for calibration. The parameters for SWRC are set to sae∗  = 4.5 kPa, D = 1.7, and Dd = 1.05, which result in a value of R2 = 0.9909 for all data sets with three different initial porosities (Figure 6(a)). As shown in Figures 6(b)–6(f), the measured relative hydraulic conductivities for five different initial porosities (0.503∼0.395) are replotted in the Kr-s plane (double logarithmic scales), which are the predictions obtained using equation (26). The predicted Kr-s curves agree with the experimental data reasonably well, which confirms that the proposed porosity-dependent fractal HCF model (i.e., equation (26)) captures well the effects of the initial porosity on the soil’s unsaturated permeability.Figure 6
Comparison between measured and predicted SWRCs and HCFs for the Touchet silt loam: (a) SWRCs, (b) HCF (ϕ = 0.503), (c) HCF (ϕ = 0.478), (d) HCF (ϕ = 0.449), (e) HCF (ϕ = 0.423), and (f) HCF (ϕ = 0.395).
(a)
(b)
(c)
(d)
(e)
(f)


## 4. Conclusions

A simple physical model based on fractal geometry was proposed to quantify the effects of initial porosity on the soil-water retention curve (SWRC) and the hydraulic conductivity function (HCF) for unsaturated porous media. The proposed fractal model involves three parameters: (1) the air-entry value that is related to the size of the maximum pores, (2) the fractal dimension of pore volume (D), and (3) the fractal dimension of drainable pore volume (Dd). The range of D and Dd is limited to between E − 1 and E (i.e., Euclidean dimension). The difference between D and Dd (D ≥ Dd) implies that incomplete pore connectivity may exist in the porous medium. The values of D and Dd can be calibrated conveniently using one data set of the water retention tests at initial porosity. A regression analysis using the method of least squares indicates that the proposed model is valid to reproduce the porosity-dependent SWRCs and HCFs for a variety of unsaturated soils.



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*Source: 3965803-2019-01-08.xml*