Hunt and Gee (2002) gave the values of *θ*s and *θ*r were 0.4 and 0.01, respectively, the air-entry value was 4.6kPa. Using the fractal dimension of the grain-size distribution (GSD), the soilwater characteristic curve (SWCC) and the relative hydraulic conductivity (RHC) can be determined. The parameters of the grain-size distribution (GSD) used to determine the hydraulic properties are listed in Table 3. Comparisons between the predictions of Eq. (29) using the fractal dimension of the grain-size distribution (GSD) and the experimental data of soil-water characteristic curve (SWCC) are shown in Fig. 26. It was seen that a good result is obtained to predict the soil-water characteristic curve (SWCC) using the fractal dimension of the grain-size distribution (GSD).

Hunt and Gee (2002) gave the values of saturated hydraulic conductivity was 0.001cm/s. Comparisons between the predictions of Eqs. (44) and (45) and experimental data were shown in Fig. 27. The prediction of Eqs. (44) and (45) nearly satisfied the experimental data of relative hydraulic conductivity in Fig. 27. In Fig.27, the predictions of Eq. (45) deviated from the experimental data near saturation, and extend to equal to the experimental results at low water content. The soil hydraulic conductivity can be approximately determined using the fractal dimension of the grain-size distribution (GSD).

**6. Comparison of the fractal model with the van Genuchten–Mualem (G-**

Unsaturated Hydraulic Conductivity of Fractal-Textured Soils

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van Genuchten (1980) derived an empirical relationship to describe the soil-water character‐

1 (1 ) *n m* ay

Burdine (1953) and Mualem (1976) presented a similar model to estimate the unsaturated hydraulic conductivity from pore size distributions inferred from soil water retention charac‐ teristics. The models share a number of similarities, allowing them to be written in a general

d

c

y

Q -

y

where *k*w is the unsaturated hydraulic conductivity at any water content, *k*s is the saturated hydraulic conductivity, *Θ* is the normalized water content, *ψ* is matric suction. Generally accepted parameter values for the parameters (*l*, *χ, γ*) are (0.5, 1, 2) and (2, 2, 1) for the Mualem

Using the Mualem model, van Genuchten (1980) derived a closed-form to determine the

( )

 ay

*m n*

*m n n*

2

2

*m*

( )

1 1

ay

1

1 1

ay

2 r 1 1

*m k* é ù æ ö ê ú =Q - -Q ç ÷ ê ú ç ÷ ê ú è ø ë û

1

c

g

d

(49)

105

(50)

(51)

(52)

+

where *α*, *n* and *m* are the van Genuchten curve-fitting parameters, and *m*=1-1/*n*.

w 0 r 1 s 0

æ ö Q ç ÷ = = Q ç ÷ ç ÷ Q è ø ò

ò

*l k k k*

Q =

**6.1. van Genuchten–Mualem (G-M) model for relative hydraulic conductivity (RHC)**

**M) model**

istic curve (SWCC):

form as (Hoffmann-Riem et al., 1999)

(1976) and the Burdine (1953) models, respectively.

relative hydraulic conductivity (RHC) at a degree of saturation,

r

*k*

**Figure 25.** Fractal dimension of GSD for McGee Ranch soil (Data from Hunt and Gee, 2002)

**Figure 26.** Comparisons between predictions and experiments of SWCC for McGee Ranch soil (Data from Hunt and Gee, 2002)

## **6. Comparison of the fractal model with the van Genuchten–Mualem (G-M) model**

#### **6.1. van Genuchten–Mualem (G-M) model for relative hydraulic conductivity (RHC)**

van Genuchten (1980) derived an empirical relationship to describe the soil-water character‐ istic curve (SWCC):

$$\Theta = \frac{1}{(1 + \left| \alpha \boldsymbol{\nu} \boldsymbol{\nu} \right|^{\boldsymbol{\mu}})^m} \tag{49}$$

where *α*, *n* and *m* are the van Genuchten curve-fitting parameters, and *m*=1-1/*n*.

**Figure 25.** Fractal dimension of GSD for McGee Ranch soil (Data from Hunt and Gee, 2002)

**Figure 26.** Comparisons between predictions and experiments of SWCC for McGee Ranch soil (Data from Hunt and

Gee, 2002)

104 Hydraulic Conductivity

Burdine (1953) and Mualem (1976) presented a similar model to estimate the unsaturated hydraulic conductivity from pore size distributions inferred from soil water retention charac‐ teristics. The models share a number of similarities, allowing them to be written in a general form as (Hoffmann-Riem et al., 1999)

$$k\_{\mathbf{r}} = \frac{k\_{\mathbf{w}}}{k\_{\mathbf{s}}} = \Theta^{l} \left( \frac{\int\_{0}^{\Theta} \boldsymbol{\nu}^{-\mathcal{X}} \mathbf{d} \Theta}{\int\_{0}^{1} \boldsymbol{\nu}^{-\mathcal{X}} \mathbf{d} \Theta} \right)^{\mathcal{Y}} \tag{50}$$

where *k*w is the unsaturated hydraulic conductivity at any water content, *k*s is the saturated hydraulic conductivity, *Θ* is the normalized water content, *ψ* is matric suction. Generally accepted parameter values for the parameters (*l*, *χ, γ*) are (0.5, 1, 2) and (2, 2, 1) for the Mualem (1976) and the Burdine (1953) models, respectively.

Using the Mualem model, van Genuchten (1980) derived a closed-form to determine the relative hydraulic conductivity (RHC) at a degree of saturation,

$$k\_r = \frac{\left[1 - \left|\alpha\nu\right|^{n-1} \left(1 + \left|\alpha\nu\right|^{\left|^{n}}\right)^{-m}\right]^2}{\left(1 + \left|\alpha\nu\right|^{\left|^{n}}\right)^m}\tag{51}$$

$$k\_{\rm r} = \Theta^{\frac{1}{2}} \left[ 1 - \left( 1 - \Theta^{\frac{1}{m}} \right)^{m} \right]^{2} \tag{52}$$

Equations (51) and (52) are the representations of the G-M model, which is the most widely and popularly used to predict the hydraulic conductivity.

The other models for relative hydraulic conductivity (RHC) were also introduced as follows.

#### **6.2. van Genuchten–Burdine (G-B) model for relative hydraulic conductivity (RHC)**

Hydraulic conductivity resulting from the Burdine capillary models (Burdine, 1953) was expressed as

$$k\_{\mathbf{r}} = \Theta^2 \left[ 1 - \left( 1 - \Theta^{\frac{1}{m}} \right)^m \right] \tag{53}$$

1 1

*x xx*

*y a b*

*Ba b*

(1 ) d

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107

= (58)

= + (59)

= + (60)

1 1 1 0

The results presented by Touma (2009) were obtained with *c*=0.5 and *a*1=1.

( , ) (1 ) d , ( , ) (, )

**6.4. Brooks & Corey-Brutsaert model for relative hydraulic conductivity (RHC)**

*y*

Even though the conductivity predicted by the capillary model of Fatt and Dykstra (1951) gave the best results compared with both the quasi-analytical solution and observations, the resulting expression was not easy to use because it was necessary to evaluate Incomplete Beta and Beta functions. In order to simplify G-FD model, *Θ* was expressed by Eq. (49) and *k*r was according to Eq. (31) with *ω*=2+2.5/*mn*. The fitted result in a value of *b* was close to -*mn*, and a

Brutsaert (2000) presented a capillary model expressed as Eq. (37). Combining this capillary model with the soil-water characteristic curve (SWCC) of BC model (Brooks and Corey, 1966), relative hydraulic conductivity was written as Eq. (31) in form, and the parameter *ω* was given

> 2 2 *b*

Combining of BC model for the soil-water characteristics with the Mualem and the Burdine capillary models, the relative hydraulic conductivity curves were also expressed as Eq. (31) in

> 2 3 *mm*

2 2.5 *mm*

Here we focused on the G-M model, as it was the most widely used in soil science and

van Genuchten (1980) found that the predictions of the G–M model were less accurate for Beit Netofa clay. The fractal dimension and the air-entry value evaluated from the fitted soil-water characteristic curve (SWCC) are 2.89 and 60kPa, respectively (Fig. 28) The fitted parameters of

w

w

w

0

conductivity curve close to that predicted by Eq. (56).

form, and the parameter *ω* was given by

when applied with the Mualem condition.

hydrology.

when BC is applied with the Burdine condition, and

by

*a b*

*Ba b x x x I a b*

Replacing Eq. (49) in Eq. (53) yielded

$$k\_r = \frac{\left[1 - \left|\alpha\nu\right|^{n-2} \left(1 + \left|\alpha\nu\right|^n\right)^{-m}\right]}{\left(1 + \left|\alpha\nu\right|^n\right)^{2m}}\tag{54}$$

where *m*=1-2/*n*.

Note that for both Eqs. (52) and (54), because *m* is less than 1, the derivative d*k*/d*θ* is infinite for *θ*s, whatever the value of *n*.

#### **6.3. van Genuchten–Fatt & Dykstra (G-FD) model for relative hydraulic conductivity (RHC)**

The capillary model proposed by Fatt and Dykstra (1951) was expressed as

$$k\_{\mathbf{r}} = \frac{\int\_0^{\Theta} (\Theta - \mathbf{x}) \boldsymbol{\nu}(\mathbf{x})^{-2 - \mathbf{c}} \mathbf{d} \mathbf{x}}{\int\_0^1 (1 - \mathbf{x}) \boldsymbol{\nu}(\mathbf{x})^{-2 - \mathbf{c}} \mathbf{d} \mathbf{x}} \tag{55}$$

Evaluating *ψ*(*x*) from Eq. (49), the integrals of Eq. (55) with the condition of *m*>-(2+*c*)/2*n* and *n*>2+*c* resulted in:

1 1/ 1 2 1/ 2 r 1 2 ( ,) ( ,) ( ,) ( ,) ( ,) ( ,) *Ba bI a b Ba bI a b m m k Ba b Ba b* Q Q Q - = - (56)

where *a*1=*m*+(2+*c*)/*n*, *a*2=2*m*+(2+*c*)/*n*, *b*=1-(2+*c*)/*n*, *I*x(*a*, *b*) and *B*(*a*, *b*) are the Incomplete Beta and Beta functions of positive arguments *a* and *b*, respectively, and given by:

$$B(a,b) = \int\_0^1 \mathbf{x}^{a-1} (1-\mathbf{x})^{b-1} \, \mathrm{d}x, \quad I\_y(a,b) = \frac{\int\_0^y \mathbf{x}^{a-1} (1-\mathbf{x})^{b-1} \, \mathrm{d}x}{B(a,b)}\tag{57}$$

The results presented by Touma (2009) were obtained with *c*=0.5 and *a*1=1.

Equations (51) and (52) are the representations of the G-M model, which is the most widely

The other models for relative hydraulic conductivity (RHC) were also introduced as follows.

Hydraulic conductivity resulting from the Burdine capillary models (Burdine, 1953) was

1

( )

*m n*

*m n n*

 ay

( )

ay

Note that for both Eqs. (52) and (54), because *m* is less than 1, the derivative d*k*/d*θ* is infinite

**6.3. van Genuchten–Fatt & Dykstra (G-FD) model for relative hydraulic conductivity (RHC)**

( ) () d

y

Q -

(1 ) ( ) d

Evaluating *ψ*(*x*) from Eq. (49), the integrals of Eq. (55) with the condition of *m*>-(2+*c*)/2*n* and

1 1/ 1 2 1/ 2

( ,) ( ,) ( ,) ( ,) ( ,) ( ,) *Ba bI a b Ba bI a b m m*

where *a*1=*m*+(2+*c*)/*n*, *a*2=2*m*+(2+*c*)/*n*, *b*=1-(2+*c*)/*n*, *I*x(*a*, *b*) and *B*(*a*, *b*) are the Incomplete Beta and

1 2

*Ba b Ba b*

y

2-

*xx x*

*xx x*

*c*

*c*

ò (55)

Q Q Q - = - (56)

2

r 2 1 1

The capillary model proposed by Fatt and Dykstra (1951) was expressed as

0

ò

Q - =

*k*

r

*k*

0

Beta functions of positive arguments *a* and *b*, respectively, and given by:

r 1 2

ay

1

*m*

(53)

(54)

**6.2. van Genuchten–Burdine (G-B) model for relative hydraulic conductivity (RHC)**

2 r 1 1

*m k* é ù æ ö ê ú =Q - -Q ç ÷ ê ú ç ÷ ê ú è ø ë û

and popularly used to predict the hydraulic conductivity.

*k*

expressed as

106 Hydraulic Conductivity

where *m*=1-2/*n*.

*n*>2+*c* resulted in:

for *θ*s, whatever the value of *n*.

Replacing Eq. (49) in Eq. (53) yielded

Even though the conductivity predicted by the capillary model of Fatt and Dykstra (1951) gave the best results compared with both the quasi-analytical solution and observations, the resulting expression was not easy to use because it was necessary to evaluate Incomplete Beta and Beta functions. In order to simplify G-FD model, *Θ* was expressed by Eq. (49) and *k*r was according to Eq. (31) with *ω*=2+2.5/*mn*. The fitted result in a value of *b* was close to -*mn*, and a conductivity curve close to that predicted by Eq. (56).

#### **6.4. Brooks & Corey-Brutsaert model for relative hydraulic conductivity (RHC)**

Brutsaert (2000) presented a capillary model expressed as Eq. (37). Combining this capillary model with the soil-water characteristic curve (SWCC) of BC model (Brooks and Corey, 1966), relative hydraulic conductivity was written as Eq. (31) in form, and the parameter *ω* was given by

$$
\alpha = 2 \cdot \frac{2}{b} \tag{58}
$$

Combining of BC model for the soil-water characteristics with the Mualem and the Burdine capillary models, the relative hydraulic conductivity curves were also expressed as Eq. (31) in form, and the parameter *ω* was given by

$$
\alpha \alpha = 3 + \frac{2}{mm} \tag{59}
$$

when BC is applied with the Burdine condition, and

$$
\alpha \alpha = 2.5 + \frac{2}{mm} \tag{60}
$$

when applied with the Mualem condition.

Here we focused on the G-M model, as it was the most widely used in soil science and hydrology.

van Genuchten (1980) found that the predictions of the G–M model were less accurate for Beit Netofa clay. The fractal dimension and the air-entry value evaluated from the fitted soil-water characteristic curve (SWCC) are 2.89 and 60kPa, respectively (Fig. 28) The fitted parameters of the van Genuchten model, *α* and *n* are 0.0015 and 1.17, respectively from Fig.28. The parameters for the determination of the relative hydraulic conductivity (RHC) are listed in Table 5.

**Soil type**

conductivity.

1980)

**Table 5.** Parameters obatined from SWCC for prediction of RHC

**Fractal model G-M model**

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109

**δ** *D* **ξ ξ/δ α (kPa***-1) n*

Beit Netofa clay -0.11 2.89 -2.33 21.2 0.004 1.17 Hanford glacial soil 0-099 -0.62 2.38 -3.86 6.23 0.015 1.71 Hanford glacial soil 2-1637 -0.82 2.18 -4.46 5.44 0.076 1.89 Grenoble sand -0.64 2.36 -3.92 6.13 0.4 2.17

The saturated and residual water contents of Beit Netofa clay are 0.446 and 0, respectively. Comparisons between the predictions of both the fractal model (Eq. (44)) and the G–M model (Eq. (51)) and the experimental data of the relative hydraulic conductivity (RHC) for Beit Netofa clay are shown in Fig. 29. It is seen that predictions of the fractal model (Eq. (44)) satisfactorily agree with the experimental data, especially at high matric suction, while the predictions of the G–M model (Eq. (51)) are found to deviate from the experimental data. However, the fractal model shows a sharp corner at the air-entry value point, which is disagree with the measured data that usually show smooth transition after the air-entry value point. This default does not reduce the advantages to predict the relative hydraulic conductivity (RHC) using the fractal dimension obtained from the pore-size distribution (PSD), because the hydraulic conductivity near the point of the air-entry value nearly equal to the saturated

**Figure 29.** Comparison between the predictions of the RHC using fractal model and G-M model (van Genuchten,

**Figure 27.** Comparisons between predictions and experimental data of RHC for McGee Ranch soil (Data from Hunt and Gee, 2002)

**Figure 28.** Fractal dimension and the parameters of G-M model obtained from SWCC for Beit Netofa clay (Data from van Genuchten, 1980)

**Table 5.** Parameters obatined from SWCC for prediction of RHC

the van Genuchten model, *α* and *n* are 0.0015 and 1.17, respectively from Fig.28. The parameters

**Figure 27.** Comparisons between predictions and experimental data of RHC for McGee Ranch soil (Data from Hunt

**Figure 28.** Fractal dimension and the parameters of G-M model obtained from SWCC for Beit Netofa clay (Data from

and Gee, 2002)

108 Hydraulic Conductivity

van Genuchten, 1980)

for the determination of the relative hydraulic conductivity (RHC) are listed in Table 5.

The saturated and residual water contents of Beit Netofa clay are 0.446 and 0, respectively. Comparisons between the predictions of both the fractal model (Eq. (44)) and the G–M model (Eq. (51)) and the experimental data of the relative hydraulic conductivity (RHC) for Beit Netofa clay are shown in Fig. 29. It is seen that predictions of the fractal model (Eq. (44)) satisfactorily agree with the experimental data, especially at high matric suction, while the predictions of the G–M model (Eq. (51)) are found to deviate from the experimental data. However, the fractal model shows a sharp corner at the air-entry value point, which is disagree with the measured data that usually show smooth transition after the air-entry value point. This default does not reduce the advantages to predict the relative hydraulic conductivity (RHC) using the fractal dimension obtained from the pore-size distribution (PSD), because the hydraulic conductivity near the point of the air-entry value nearly equal to the saturated conductivity.

**Figure 29.** Comparison between the predictions of the RHC using fractal model and G-M model (van Genuchten, 1980)

**Figure 30.** Fractal dimension and G-M parameters obtained from SWCC for Hanford glacial (Khaleel and Relyea, 1995)

**Figure 31.** Comparison between the predictions using fractal model and G-M model (Data from Khaleel and Relyea,

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111

The combinations of Eq. (29) with fractal capillary model were tested on Grenoble sand. Data points for the soil-water characteristic curve (SWCC) and hydraulic conductivity were taken from Touma (2009). The infiltration experiment was conducted under a constant head of 2.3cm (Touma & Vauclin, 1986). The solid lines in Figure 37 were the fitted curves for Eq. (29), and

1995)

**Figure 32.** Fitted SWCC curves

The measured soil–water characteristic curves of Hanford glacial soil by Khaleel and Relyea (1995) were shown in Fig. 30. The saturated water contents (*θ*s) are 0.338 and 0.303 for the samples 0–099 and 2–1637, respectively. The residual water contents (*θ*r) are 0.039 and 0.025 for the samples 0–099 and 2–1637, respectively. From Fig. 30, it is seen that the soil-water characteristic curves (SWCC) of Hanford glacial soil are in good accord with the fractal model. The fractal dimensions (*D*), according to Eq. (29), are 2.35 and 2.18 obtained from the soil-water characteristic curves for the samples 0–099 and 2–1637 of Hanford glacial soil, respectively.

From the experimental data of the soil-water characteristic curve (SWCC), the parameters for determining relative hydraulic conductivity (RHC) using the fractal dimension are listed in Table 5. The larger the fractal dimension, the larger the complexity of the pore connectivity will be. The larger the fractal dimension, the larger the change in the water content with the same change in matric suction will occur. From the fittings of the soil–water characteristic curve in Fig. 30, the parameters (*a* and *n*) in the van Genuchten model (1980) for the soil–water characteristic curve (SWCC) are obtained, and are also listed in Table 5. Comparisons between the predictions of the fractal model (Eq. (45)) and the experimental data of the relative hydraulic conductivity (RHC) for Hanford glacial soil are shown in Fig. 31. The predictions of the fractal model (Eq. (45)) nearly agree with the experimental data of the relative hydraulic conductivity (RHC) for Hanford glacial soil in Fig. 31. The predictions of the G–M model (Eq. (52)) deviate from the experimental data of the relative hydraulic conductivity (RHC) in Fig. 31. It is seen that the predictions of the fractal model are better than those of the G–M model for Hanford glacial soil in Fig. 31.

**Figure 31.** Comparison between the predictions using fractal model and G-M model (Data from Khaleel and Relyea, 1995)

**Figure 32.** Fitted SWCC curves

**Figure 30.** Fractal dimension and G-M parameters obtained from SWCC for Hanford glacial (Khaleel and Relyea, 1995)

The measured soil–water characteristic curves of Hanford glacial soil by Khaleel and Relyea (1995) were shown in Fig. 30. The saturated water contents (*θ*s) are 0.338 and 0.303 for the samples 0–099 and 2–1637, respectively. The residual water contents (*θ*r) are 0.039 and 0.025 for the samples 0–099 and 2–1637, respectively. From Fig. 30, it is seen that the soil-water characteristic curves (SWCC) of Hanford glacial soil are in good accord with the fractal model. The fractal dimensions (*D*), according to Eq. (29), are 2.35 and 2.18 obtained from the soil-water characteristic curves for the samples 0–099 and 2–1637 of Hanford glacial soil, respectively.

From the experimental data of the soil-water characteristic curve (SWCC), the parameters for determining relative hydraulic conductivity (RHC) using the fractal dimension are listed in Table 5. The larger the fractal dimension, the larger the complexity of the pore connectivity will be. The larger the fractal dimension, the larger the change in the water content with the same change in matric suction will occur. From the fittings of the soil–water characteristic curve in Fig. 30, the parameters (*a* and *n*) in the van Genuchten model (1980) for the soil–water characteristic curve (SWCC) are obtained, and are also listed in Table 5. Comparisons between the predictions of the fractal model (Eq. (45)) and the experimental data of the relative hydraulic conductivity (RHC) for Hanford glacial soil are shown in Fig. 31. The predictions of the fractal model (Eq. (45)) nearly agree with the experimental data of the relative hydraulic conductivity (RHC) for Hanford glacial soil in Fig. 31. The predictions of the G–M model (Eq. (52)) deviate from the experimental data of the relative hydraulic conductivity (RHC) in Fig. 31. It is seen that the predictions of the fractal model are better than those of the G–M model

for Hanford glacial soil in Fig. 31.

110 Hydraulic Conductivity

The combinations of Eq. (29) with fractal capillary model were tested on Grenoble sand. Data points for the soil-water characteristic curve (SWCC) and hydraulic conductivity were taken from Touma (2009). The infiltration experiment was conducted under a constant head of 2.3cm (Touma & Vauclin, 1986). The solid lines in Figure 37 were the fitted curves for Eq. (29), and the resulting predicted hydraulic conductivity were shown in Fig. 32. Fractal model gave good results for the Grenoble sand. The parameters obtained through the fitting results of soil-water characteristic curves (SWCC) were used to predict hydraulic conductivity curve. Comparison between the fractal model and the G-M model were shown in Figs. 33. It was found that the prediction of hydraulic conductivity curve using Eq. (45) closed to the experiments for Grenoble sand.

and the development of seepage forces which both depend on the evolution of the pore water

Here, an individual soil slice can be treated as a one-dimensional column with vertical infiltration. Suppose that any lateral flow between the adjacent slices will be equal on the upslope and down-slope boundaries. Then considering vertical infiltration only could meet the flow continuity requirement. These one-dimensional infiltration analyses are addressed by the saturated/unsaturated seepage finite element software SEEP/W (GEO-SLOPE International Ltd.2007) coupled with the air flow software AIR/W (GEO-SLOPE International Ltd.2007).

For a homogenous, isotropic soil, one-dimensional water flow equation (GEO-SLOPE Inter‐

w w a w w w w w *H H P m k mQ ty t t*

where *H*w is the total water energy potential comprised of both pressure and elevation potentials; *P*a is the pore air pressure; *kw* is hydraulic conductivity; *y* is y coordinate; *m*w is the

For the air conversation of mass, we can arrive at the pore air general mass balance equation

0a 0a

where *k*a is air permeability; *θ*a is volumetric air content; *ρ*a is air density; *γ*oa is initial air unit weight; *ρ*oa is initial air density; *R*¯is ideal air constant and *R*¯=287J/(kg K) for dry air; *T* is temperature; *H*w is total head and *P*a is pore air pressure. These two governing equations can be used to determine total pressure head profile for the water and the pore air pressure, and

The relationships relating volumetric water content and hydraulic conductivity to suction must be known to solve the Eq. (61). And similarly to calculate the pore air pressure the pore air permeability function is also wanted. To find the effect of different fractal dimensions and air-entry values of different soil types, series of parameter studies have conducted. Different types of Soil-water characteristic curves and hydraulic conductivity curves were shown in Figs. 34 and 35. In the legends of these pictures, Soil 10, 2.1,-5, for example, means a type of soil with air-entry value, *ψ*e=10kPa, the fractal dimension, *D*=2.1 and saturated hydraulic conductivity *k*s=1×10-5m/s. It is assumed that *θ*r=0. The pore air permeability function curve used in these studies was shown in Fig.36, and it remained a constant (GEO-SLOPE Interna‐

*RT t y y t*

w w aw w

*m m*

é ù æ ö ¶ ¶ ¶ ¶ ç + = ÷ ê + +ú ¶¶ ¶ ê ú ¶ è ø ë û

r

g

2 a a aa a a a w

r l (62)

*P kP k H*

 r

 r

¶¶ ¶ ¶ è ø (61)

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113

¶ ¶ æ ö ¶ ¶ = ç ÷ + +

pressure profile must been addressed in detail.

g

(GEO-SLOPE International Ltd.2007) as like:

r

then suction is attained.

tional Ltd.2007).

q

slope of SWCC; *t* is time; and *Q*w has units of length per time.

national Ltd.2007) is given by:

**Figure 33.** Comparison between fractal model and G-M model

## **7. Rainfall infiltration of unsaturated soils**

Rainfall induced slope failures often occur as relatively shallow failure surfaces orientated parallel to the slope surface and are observed and analyzed by different mechanisms. The effect of seepage on slope stability is introduced in the analyses by calculating the critical depth for an infinite slope with rainfall infiltration, while the effect of both negative and positive pore water pressures on the stabilities of initially unsaturated slopes are explained and coupled with infinite slope analysis and pore-air flow analysis methods in order to present a predictive formulation of slope failures that occurs in rainfall evens and is derived from a fractal model on unsaturated soil. The formulation serves as a baseline analysis method for evaluating potentially unstable slopes.

The pore water pressure pattern that develops in the initially unsaturated soil will occur as a transient process as the infiltration moves downward into the soil profile. Several factors should been taken into account in unsaturated analyses. The shear strength of the soil mass and the development of seepage forces which both depend on the evolution of the pore water pressure profile must been addressed in detail.

the resulting predicted hydraulic conductivity were shown in Fig. 32. Fractal model gave good results for the Grenoble sand. The parameters obtained through the fitting results of soil-water characteristic curves (SWCC) were used to predict hydraulic conductivity curve. Comparison between the fractal model and the G-M model were shown in Figs. 33. It was found that the prediction of hydraulic conductivity curve using Eq. (45) closed to the experiments for

Rainfall induced slope failures often occur as relatively shallow failure surfaces orientated parallel to the slope surface and are observed and analyzed by different mechanisms. The effect of seepage on slope stability is introduced in the analyses by calculating the critical depth for an infinite slope with rainfall infiltration, while the effect of both negative and positive pore water pressures on the stabilities of initially unsaturated slopes are explained and coupled with infinite slope analysis and pore-air flow analysis methods in order to present a predictive formulation of slope failures that occurs in rainfall evens and is derived from a fractal model on unsaturated soil. The formulation serves as a baseline analysis method for evaluating

The pore water pressure pattern that develops in the initially unsaturated soil will occur as a transient process as the infiltration moves downward into the soil profile. Several factors should been taken into account in unsaturated analyses. The shear strength of the soil mass

Grenoble sand.

112 Hydraulic Conductivity

**Figure 33.** Comparison between fractal model and G-M model

**7. Rainfall infiltration of unsaturated soils**

potentially unstable slopes.

Here, an individual soil slice can be treated as a one-dimensional column with vertical infiltration. Suppose that any lateral flow between the adjacent slices will be equal on the upslope and down-slope boundaries. Then considering vertical infiltration only could meet the flow continuity requirement. These one-dimensional infiltration analyses are addressed by the saturated/unsaturated seepage finite element software SEEP/W (GEO-SLOPE International Ltd.2007) coupled with the air flow software AIR/W (GEO-SLOPE International Ltd.2007).

For a homogenous, isotropic soil, one-dimensional water flow equation (GEO-SLOPE Inter‐ national Ltd.2007) is given by:

$$m\_{\rm w} \gamma\_{\rm w} \frac{\partial H\_{\rm w}}{\partial t} = \frac{\partial}{\partial y} \left( k\_{\rm w} \frac{\partial H\_{\rm w}}{\partial t} \right) + m\_{\rm w} \frac{\partial P\_{\rm a}}{\partial t} + Q\_{\rm w} \tag{61}$$

where *H*w is the total water energy potential comprised of both pressure and elevation potentials; *P*a is the pore air pressure; *kw* is hydraulic conductivity; *y* is y coordinate; *m*w is the slope of SWCC; *t* is time; and *Q*w has units of length per time.

For the air conversation of mass, we can arrive at the pore air general mass balance equation (GEO-SLOPE International Ltd.2007) as like:

$$
\left(\rho\_{\rm w}m\_{\rm w} + \frac{\theta\_{\rm a}}{\overline{R}T}\right)\frac{\partial P\_{\rm a}}{\partial t} = \frac{\partial}{\partial y}\left[\frac{\rho\_{\rm a}k\_{\rm a}}{\gamma\_{0\rm a}}\frac{\partial P\_{\rm a}}{\partial y} + \frac{\rho\_{\rm a}^{2}k\_{\rm a}}{\rho\_{0\rm a}}\right] + \rho\_{\rm a}\lambda\_{\rm w}m\_{\rm w}\frac{\partial H\_{\rm w}}{\partial t}\tag{62}
$$

where *k*a is air permeability; *θ*a is volumetric air content; *ρ*a is air density; *γ*oa is initial air unit weight; *ρ*oa is initial air density; *R*¯is ideal air constant and *R*¯=287J/(kg K) for dry air; *T* is temperature; *H*w is total head and *P*a is pore air pressure. These two governing equations can be used to determine total pressure head profile for the water and the pore air pressure, and then suction is attained.

The relationships relating volumetric water content and hydraulic conductivity to suction must be known to solve the Eq. (61). And similarly to calculate the pore air pressure the pore air permeability function is also wanted. To find the effect of different fractal dimensions and air-entry values of different soil types, series of parameter studies have conducted. Different types of Soil-water characteristic curves and hydraulic conductivity curves were shown in Figs. 34 and 35. In the legends of these pictures, Soil 10, 2.1,-5, for example, means a type of soil with air-entry value, *ψ*e=10kPa, the fractal dimension, *D*=2.1 and saturated hydraulic conductivity *k*s=1×10-5m/s. It is assumed that *θ*r=0. The pore air permeability function curve used in these studies was shown in Fig.36, and it remained a constant (GEO-SLOPE Interna‐ tional Ltd.2007).

1.E-7

**Figure 36.** Pore air permeability function used in study

**Figure 37.** Sketch map of the one-dimensional infiltration model

0 0.5 1

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115

*θ*

A 4m-deep soil column was modeled for the one-dimensional analyses, as shown in Fig. 37, and infiltration occurs from the top of the column, while the bottom was set to be a pervious boundary. A linear hydrostatic suction distribution presented for the initial conditions. Analyses for the rainfall infiltration into different types of soils were performed using the unsaturated characteristic curves shown in Figs. 34-36. In applying a top boundary condition to simulate the infiltration of rainfall, it is important to realize that an influx boundary depends on the relationship between the saturated hydraulic conductivity and the rainfall intensity. In the following analyses, if a rainfall intensity greater than or equal to *k*s with the non-infiltrating

1.E-6

1.E-5

1.E-4

*k*a (m/s)

1.E-3

1.E-2

**Figure 34.** Soil-water characteristic curves for soils

**Figure 35.** Hydraulic conductivity curves f

**Figure 36.** Pore air permeability function used in study

**Figure 34.** Soil-water characteristic curves for soils

114 Hydraulic Conductivity

**Figure 35.** Hydraulic conductivity curves f

**Figure 37.** Sketch map of the one-dimensional infiltration model

A 4m-deep soil column was modeled for the one-dimensional analyses, as shown in Fig. 37, and infiltration occurs from the top of the column, while the bottom was set to be a pervious boundary. A linear hydrostatic suction distribution presented for the initial conditions. Analyses for the rainfall infiltration into different types of soils were performed using the unsaturated characteristic curves shown in Figs. 34-36. In applying a top boundary condition to simulate the infiltration of rainfall, it is important to realize that an influx boundary depends on the relationship between the saturated hydraulic conductivity and the rainfall intensity. In the following analyses, if a rainfall intensity greater than or equal to *k*s with the non-infiltrating rainfall running off the slope, the top boundary condition is set to a total head equal to 4m; while if field results indicate that the rainfall intensity is less than *k*s, then a flux type boundary condition is more appropriate.

Since initial failures often have small depth-to-length ratios and form failure planes parallel to the slope surface, the use of infinite slope analysis in modeling the infiltration process by vertical one-dimensional analysis makes it justified in describing the physical process of failure initiation. However, the methods used in traditional infinite slope analysis must be modified to take into account the variation of the suction profile that results from the infiltration process. There are two distinct failure mechanisms can be initiated by the infiltration process. The failure takes place due to positive pore pressure and it takes place while the suction still exists.

The shear strength of unsaturated soil can be represented by fractal model, and written as follows (Xu, 2004):

3 2 a e ( )tan tan *D D* t s fy y f*c u* - - =+ - + ¢¢ ¢ (63)

comprehensive method for determining the location and time of failure for a slope if the soil, slope, and rainfall parameters are given. The methodology of the coupled analysis involves plotting a stability envelope as defined by Eq. (65) for specified soil and slope parameters, over a given infiltration profile generated from the particular rainfall and unsaturated characteristic curves. Coulomb failure and the initiation of slope mobilization are defined at the points where the infiltration trace intersects the stability envelope. Intersections of the infiltration trace and the stability envelope indicate points at which the slope is unstable and can be thought of as ''critical depths'' of failure. It must therefore be assumed that if the slope is initially stable, the point at which the stability envelope inter sects the initial suction distribution line will not contribute any further information about

Here, factors controlling the unsaturated soil slope under infiltration are studied and the results which combine infiltration analyses and stability analyses are shown in Figs.43-47. Infiltration analyses were performed by software SEEP/W coupled with AIR/W in order to calculate the suction file and the stability envelope as defined by Eq.(65) for slope parame‐

Soil column was saturated due to rainfall infiltration from the initial hydrostatic suction profile. It was shown that infiltration took place in unsaturated soil much faster for the soil with higher fractal dimension. Three fractal dimensions (*D*=2.1, 2.35 and 2.78) were calculated respectively and the results were shown in Fig.38. It was seen from Fig. 38 that the higher initial hydraulic conductivity existed on the top of soil column which had a higher fractal dimension. On the other hand, the stability envelope for the soil with higher fractal dimension was steeper. There was an obvious inflection point in the stability envelope of the soil with the low fractal dimension. The results also indicated that it was the relative shape of the unsaturated characteristic curves (Fig.34 and Fig. 35) that has a

Three air-entry values (*ψ*e=1kPa, 10kPa and 20kPa) are calculated respectively and the results as shown in Figs.39. Compared with the results of three different air-entry values, both the infiltration trace and the stability envelope have changed with the different air-entry values. When the air-entry value is relatively small (*ψ*e=1kPa), as shown in Fig.39a, passive pore water pressure was generated on the upper soil column at the time *t*=32h. Due to the low initial permeability on the top of soil column and a state of lower saturation, fewer pores initially filled with water and there are fewer channels available for fluid transport and consequently the flow of water is hindered. With the commencement of infiltration at the top boundary, water is forced into a soil which is not capable of transporting it efficient‐

, and shear strength parameters *c*'=5kPa, *φ*'=15°. Other

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117

the failure mechanisms from infiltration.

parameters are studied in a systematic way.

controlling effect on the suction distribution.

, *γ*w=10kN/m3

ters, *β*=40°, *γ*=19.6kN/m3

**7.1. Effect of fractal dimension**

**7.2. Effect of air-entry value**

where *τ* is the shear strength; *c*' is the effective cohesion; (*σ*-*u*a) is the total normal stress; *φ*' is effective friction angle; *ψ* is the suction; *ψ*e is the air-entry value. The stability envelope can be represented as (Collins and Znidarcic, 2004):

$$d\_{\rm cr} = \frac{c' + \mathcal{Y}\_{\rm w} h\_{\rm c} \tan \phi' \cdot \mathcal{Y}\_{\rm w} h\_{\rm p} \tan \phi'}{\gamma \cos^2 \beta (\tan \beta - \tan \phi')} \tag{64}$$

where *d*cr is the critical depth; *β* is slope angle; *h*p is the pore passive pressure and *h*c is the pore negative pressure.

We can get suction with analyses of both pore water pressure and pore air pressure in the soil slope under rainfall infiltration. Combining Eq. (63) with Eq. (64), the new stability envelope can be derived and written as follows:

$$d\_{\rm cr} = \frac{c' + \nu\_{\rm e}^{\prime -D} \nu^{D-2} \tan \phi' \cdot \gamma\_{\rm w} h\_{\rm p} \tan \phi'}{\gamma \cos^2 \beta (\tan \beta - \tan \phi')} \tag{65}$$

The critical depth for infinite slope failure is now a function of the suction, *ψ*, pressure head *h*p, the given material and slope characteristics, *c*', *φ*', *γ*, *ψ*e, *ξ*, *γ*w, and *β*. In this way, the slope stability issues related to the decrease in shear strength from a loss of soil suction and the development of seepage forces from positive pressure head generation can be clearly under‐ stood.

Because the infiltration results and the slope stability results were both presented in terms of the pressure head and the suction profile, the two analyses can be coupled to yield a comprehensive method for determining the location and time of failure for a slope if the soil, slope, and rainfall parameters are given. The methodology of the coupled analysis involves plotting a stability envelope as defined by Eq. (65) for specified soil and slope parameters, over a given infiltration profile generated from the particular rainfall and unsaturated characteristic curves. Coulomb failure and the initiation of slope mobilization are defined at the points where the infiltration trace intersects the stability envelope. Intersections of the infiltration trace and the stability envelope indicate points at which the slope is unstable and can be thought of as ''critical depths'' of failure. It must therefore be assumed that if the slope is initially stable, the point at which the stability envelope inter sects the initial suction distribution line will not contribute any further information about the failure mechanisms from infiltration.

Here, factors controlling the unsaturated soil slope under infiltration are studied and the results which combine infiltration analyses and stability analyses are shown in Figs.43-47. Infiltration analyses were performed by software SEEP/W coupled with AIR/W in order to calculate the suction file and the stability envelope as defined by Eq.(65) for slope parame‐ ters, *β*=40°, *γ*=19.6kN/m3 , *γ*w=10kN/m3 , and shear strength parameters *c*'=5kPa, *φ*'=15°. Other parameters are studied in a systematic way.

#### **7.1. Effect of fractal dimension**

rainfall running off the slope, the top boundary condition is set to a total head equal to 4m; while if field results indicate that the rainfall intensity is less than *k*s, then a flux type boundary

Since initial failures often have small depth-to-length ratios and form failure planes parallel to the slope surface, the use of infinite slope analysis in modeling the infiltration process by vertical one-dimensional analysis makes it justified in describing the physical process of failure initiation. However, the methods used in traditional infinite slope analysis must be modified to take into account the variation of the suction profile that results from the infiltration process. There are two distinct failure mechanisms can be initiated by the infiltration process. The failure takes place due to positive pore pressure and it takes place while the suction still exists. The shear strength of unsaturated soil can be represented by fractal model, and written as

3 2

 f*c u* - - =+ - + ¢¢ ¢ (63)

 y

a e ( )tan tan *D D*

where *τ* is the shear strength; *c*' is the effective cohesion; (*σ*-*u*a) is the total normal stress; *φ*' is effective friction angle; *ψ* is the suction; *ψ*e is the air-entry value. The stability envelope can be

wc wp

 fg

*ch h*

gbb

3 2

 y

*D D c h*

gbb

cr 2

y

*d*

cos (tan tan )

where *d*cr is the critical depth; *β* is slope angle; *h*p is the pore passive pressure and *h*c is the pore

We can get suction with analyses of both pore water pressure and pore air pressure in the soil slope under rainfall infiltration. Combining Eq. (63) with Eq. (64), the new stability envelope

e w p

cos (tan tan )

The critical depth for infinite slope failure is now a function of the suction, *ψ*, pressure head *h*p, the given material and slope characteristics, *c*', *φ*', *γ*, *ψ*e, *ξ*, *γ*w, and *β*. In this way, the slope stability issues related to the decrease in shear strength from a loss of soil suction and the development of seepage forces from positive pressure head generation can be clearly under‐

Because the infiltration results and the slope stability results were both presented in terms of the pressure head and the suction profile, the two analyses can be coupled to yield a

 fg

tan - tan

 f

 f

tan - tan

 f

¢ ¢¢ + = - ¢ (64)

 f

 fy

condition is more appropriate.

116 Hydraulic Conductivity

follows (Xu, 2004):

negative pressure.

stood.

t

represented as (Collins and Znidarcic, 2004):

can be derived and written as follows:

 s

cr 2

g

*d*

Soil column was saturated due to rainfall infiltration from the initial hydrostatic suction profile. It was shown that infiltration took place in unsaturated soil much faster for the soil with higher fractal dimension. Three fractal dimensions (*D*=2.1, 2.35 and 2.78) were calculated respectively and the results were shown in Fig.38. It was seen from Fig. 38 that the higher initial hydraulic conductivity existed on the top of soil column which had a higher fractal dimension. On the other hand, the stability envelope for the soil with higher fractal dimension was steeper. There was an obvious inflection point in the stability envelope of the soil with the low fractal dimension. The results also indicated that it was the relative shape of the unsaturated characteristic curves (Fig.34 and Fig. 35) that has a controlling effect on the suction distribution.

#### **7.2. Effect of air-entry value**

Three air-entry values (*ψ*e=1kPa, 10kPa and 20kPa) are calculated respectively and the results as shown in Figs.39. Compared with the results of three different air-entry values, both the infiltration trace and the stability envelope have changed with the different air-entry values. When the air-entry value is relatively small (*ψ*e=1kPa), as shown in Fig.39a, passive pore water pressure was generated on the upper soil column at the time *t*=32h. Due to the low initial permeability on the top of soil column and a state of lower saturation, fewer pores initially filled with water and there are fewer channels available for fluid transport and consequently the flow of water is hindered. With the commencement of infiltration at the top boundary, water is forced into a soil which is not capable of transporting it efficient‐

ly, which results in the development of positive pressure heads as the infiltration front progresses downward. While the air-entry value is relatively larger, as shown in Fig.39c, infiltration moves downward much faster and suction has reached to the air-entry value. Passive pore water pressure occurs at the bottom of the soil column which means that there is little unsaturated zone in the middle of soil column during the rainfall infiltration. Besides, the stability envelope is much smooth when the soil has a relatively low air-entry value, and consequently the critical depth is smaller.

**Figure 38.** Infiltration results for soils with different fractal dimension and the new stability envelope

26

28

The dominate factors which control the stability of unsaturated soil slope under rainfall infiltration are the soil saturated hydraulic conductivity and rainfall intensity. While the soil saturated hydraulic conductivity determines the water transportation ability of soil and the infiltration quantity most depends on the rainfall intensity. In this parameter study series, three different saturated hydraulic conductivity values (*k*s=1×10-5m/s, 1×10-6m/s, 1×10-7m/s) are taken

When the rainfall intensity is much larger than the saturated hydraulic conductivity, soil in the top region of the column has been saturated soon after the infiltration begins and its suction almost reaches to zero at the time *t*=1.5h. It can be seen that the suction at the base is increasing over time as a positive number, which means that the soil is actually de-saturating at its base while wetting up above. This is caused by the air which becomes trapped when the top has saturated and starts to resist water infiltration. As the situation that the rainfall intensity is much smaller than the saturated hydraulic conductivity as shown in Fig.40a, the suction of the

**c**

soil 20,2.35,-5 rain 36mm/h

4h 5h

4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

*z* /m -10 0 10 20 30 40 /kPa

> 32h 40h 44h

> > **a**

soil 1,2.35,-5 rain 36mm/h

4h 12h

4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

**7.3. Effect of saturated hydraulic conductivity**

into account and the results are shown in Fig.40.

**Figure 39.** Infiltration results for soils with different air-entry value and the new stability envelope

*z* /m

Eq.(65)

4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

Eq.(65)

0h 1h 2h 3h

*z* /m

0 10 20 30 40 /kPa

Unsaturated Hydraulic Conductivity of Fractal-Textured Soils

**b**

12h

soil 10,2.35,-5,rain 36mm/h

4h 6h

Eq.(65)

1h 0h

http://dx.doi.org/10.5772/56716

Unsaturated Hydraulic Conductivity of Fractal-Textured Soils http://dx.doi.org/10.5772/56716 119

**Figure 39.** Infiltration results for soils with different air-entry value and the new stability envelope

#### **7.3. Effect of saturated hydraulic conductivity**

ly, which results in the development of positive pressure heads as the infiltration front progresses downward. While the air-entry value is relatively larger, as shown in Fig.39c, infiltration moves downward much faster and suction has reached to the air-entry value. Passive pore water pressure occurs at the bottom of the soil column which means that there is little unsaturated zone in the middle of soil column during the rainfall infiltration. Besides, the stability envelope is much smooth when the soil has a relatively low air-entry value,

26

and consequently the critical depth is smaller.

0 10 20 30 40 /kPa

> 8h 4h

> > **a**

12h

soil 10,2.1,-5 rain 36mm/h

Eq.(65)

4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

**Figure 38.** Infiltration results for soils with different fractal dimension and the new stability envelope

*z* /m

4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

0.5h 1.5h 2.5h

0 10 20 30 40 /kPa

**c**

4h

soil 10,2.78,-5 rain 36mm/h

*z* /m

0 10 20 30 40 /kPa

**b**

12h

Eq.(65)

0h

soil 10,2.35,-5,rain 36mm/h

4h 6h 1h 0h

Eq.(65)

1h 0h

4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

*z* /m

118 Hydraulic Conductivity

The dominate factors which control the stability of unsaturated soil slope under rainfall infiltration are the soil saturated hydraulic conductivity and rainfall intensity. While the soil saturated hydraulic conductivity determines the water transportation ability of soil and the infiltration quantity most depends on the rainfall intensity. In this parameter study series, three different saturated hydraulic conductivity values (*k*s=1×10-5m/s, 1×10-6m/s, 1×10-7m/s) are taken into account and the results are shown in Fig.40.

When the rainfall intensity is much larger than the saturated hydraulic conductivity, soil in the top region of the column has been saturated soon after the infiltration begins and its suction almost reaches to zero at the time *t*=1.5h. It can be seen that the suction at the base is increasing over time as a positive number, which means that the soil is actually de-saturating at its base while wetting up above. This is caused by the air which becomes trapped when the top has saturated and starts to resist water infiltration. As the situation that the rainfall intensity is much smaller than the saturated hydraulic conductivity as shown in Fig.40a, the suction of the

top region decreases with the infiltration developing, however, the top region has remained unsaturated and the infiltration depth is also small due to the small rainfall quantity into the soil.

**Figure 41.** The time of failure vs. rainfall intensity Figure 41.The time of failure vs. rainfall intensity

4.0

**Figure 42.** Infiltration results without air phrase consideration

3.5

3.0

2.5

2.0

z /m

1.5

1.0

0.5

0.0

Figure 42.Infiltration results without air phrase consideration

> 0.5h 1.5h

> > 2.5h 3h

> > > 4h

soil 10,2.35,-5 rain 100mm/h

**7.5. Infiltration analyses coupled with the air flow analyses**

Eq.(65)

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121

**8. Slope stability analyses due to rainfall infiltration**

table depth and slope geometry only played a secondary role.

In the infiltration analyses, air flow movement in the soil pore is also conducted. The effect of the pore air flow movement is especially obvious in the situation when the rainfall intensity is much larger than the soil saturated hydraulic conductivity. Compared with the results of infiltration analyses coupled with air flow analyses, the results without air phrase consideration are shown in Fig.42. In this condition, the negative pore water pressure gradually reaches zero at the top of soil column. While the pore water pressure remains unchanged at the bottom of the column. If we take the pore air flow movement into consideration, the results are different as shown in Fig.40c. Besides, the suction could be calculated more precisely when we attain both the pore

It was widely recognized that the rainfall infiltration took a great role in causing landslides, while the relative importance of soil properties, rainfall intensity, initial water table depth and slope geometry in inducing instability of a homogenous unsaturated soil slope under different rainfall was investigated through a series of parametric studies (Rahardjo, et al, 2007). Soil properties and rainfall intensity were found to be the primary factors controlling the instability of slopes due to rainfall, while the initial water

The factors affecting the stability of a slope were considered to be the soil properties, rainfall intensity, initial depth of the groundwater table, and the slope geometry (i.e., slope angle and slope height). To assess the effects and relative contribution of controlling factors, a series of parametric studies were performed on a typical geometry of a homogeneous soil slope shown in Fig. 43. It had the boundary conditions as follows: ab, bc, cd=*q*=*I*r(rainfall intensity); ah, de, fg=*q*=0m3/s (i.e. no flow boundary); and ef, gh=*h*t(total head at the side). Four slope heights, *Hs*,(3m, 4m, 5m and 6m), three slope angles, *α*,(26.6, 33.7 and 45.0), eight initial depths of groundwater table (GWT), *Hw*,(0.5m, 1m, 2m, 3m, 5m, 7.5m, 10m and 15m), three fractal dimensions *D* (2.1, 2.35 and

water pressure and the pore air pressure by simulating infiltration analyses coupled with air flow analyses.

**Figure 40.** Infiltration results for soils due to different rainfall intensities

#### **7.4. Effect of rainfall intensity**

The rainfall intensity for a certain saturated hydraulic conductivity has taken into considered to simulate the infiltration process. The range is changed from 0.3mm/h to 100mm/h and the relationship between the time of failure and rainfall intensity for the three soil types of saturated hydraulic conductivity has plotted in Fig.41. These curves decrease as exponent forms similarly while the magnitude of different curves has changed dramatically. The time at which the soil slope failure occurs can be about ten hours for the soil saturated hydraulic conductivity *k*s equals to 1×10-5m/s, and for the situation that *k*s equals to 1×10-6m/s, the time can range from 10 to 100 hours. While much more time is needed for the lower saturated hydraulic conductivity soil type.

**Figure 41.** The time of failure vs. rainfall intensity Figure 41.The time of failure vs. rainfall intensity

top region decreases with the infiltration developing, however, the top region has remained unsaturated and the infiltration depth is also small due to the small rainfall quantity into the

> 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

0.5h 0h

Eq.(65)

*z* /m

0 10 20 30 40 /kPa

> 1.5h 3.5h 4h

> > **c**

The rainfall intensity for a certain saturated hydraulic conductivity has taken into considered to simulate the infiltration process. The range is changed from 0.3mm/h to 100mm/h and the relationship between the time of failure and rainfall intensity for the three soil types of saturated hydraulic conductivity has plotted in Fig.41. These curves decrease as exponent forms similarly while the magnitude of different curves has changed dramatically. The time at which the soil slope failure occurs can be about ten hours for the soil saturated hydraulic conductivity *k*s equals to 1×10-5m/s, and for the situation that *k*s equals to 1×10-6m/s, the time can range from 10 to 100 hours. While much more time is needed for the lower saturated

soil 10,2.35,-5 rain 100mm/h

0 10 20 30 40 /kPa

**b**

12h

soil 10,2.35,-5,rain 36mm/h

4h 6h 1h 0h

Eq.(65)

30

soil.

120 Hydraulic Conductivity

4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

*z* /m

0 10 20 30 40 /kPa

12h

soil 10,2.35,-5,rain 15mm/h

10h 5h

**a**

1h

Eq.(65)

4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

*z* /m

**Figure 40.** Infiltration results for soils due to different rainfall intensities

**7.4. Effect of rainfall intensity**

hydraulic conductivity soil type.

Figure 42.Infiltration results without air phrase consideration

**7.5. Infiltration analyses coupled with the air flow analyses**

**8. Slope stability analyses due to rainfall infiltration**

table depth and slope geometry only played a secondary role.

In the infiltration analyses, air flow movement in the soil pore is also conducted. The effect of the pore air flow movement is especially obvious in the situation when the rainfall intensity is much larger than the soil saturated hydraulic conductivity. Compared with the results of infiltration analyses coupled with air flow analyses, the results without air phrase consideration are shown in Fig.42. In this condition, the negative pore water pressure gradually reaches zero at the top of soil column. While the pore water pressure remains unchanged at the bottom of the column. If we take the pore air flow movement into consideration, the results are different as shown in Fig.40c. Besides, the suction could be calculated more precisely when we attain both the pore

It was widely recognized that the rainfall infiltration took a great role in causing landslides, while the relative importance of soil properties, rainfall intensity, initial water table depth and slope geometry in inducing instability of a homogenous unsaturated soil slope under different rainfall was investigated through a series of parametric studies (Rahardjo, et al, 2007). Soil properties and rainfall intensity were found to be the primary factors controlling the instability of slopes due to rainfall, while the initial water

The factors affecting the stability of a slope were considered to be the soil properties, rainfall intensity, initial depth of the groundwater table, and the slope geometry (i.e., slope angle and slope height). To assess the effects and relative contribution of controlling factors, a series of parametric studies were performed on a typical geometry of a homogeneous soil slope shown in Fig. 43. It had the boundary conditions as follows: ab, bc, cd=*q*=*I*r(rainfall intensity); ah, de, fg=*q*=0m3/s (i.e. no flow boundary); and ef, gh=*h*t(total head at the side). Four slope heights, *Hs*,(3m, 4m, 5m and 6m), three slope angles, *α*,(26.6, 33.7 and 45.0), eight initial depths of groundwater table (GWT), *Hw*,(0.5m, 1m, 2m, 3m, 5m, 7.5m, 10m and 15m), three fractal dimensions *D* (2.1, 2.35 and

water pressure and the pore air pressure by simulating infiltration analyses coupled with air flow analyses.

**Figure 42.** Infiltration results without air phrase consideration

## **7.5. Infiltration analyses coupled with the air flow analyses**

In the infiltration analyses, air flow movement in the soil pore is also conducted. The effect of the pore air flow movement is especially obvious in the situation when the rainfall intensity is much larger than the soil saturated hydraulic conductivity. Compared with the results of infiltration analyses coupled with air flow analyses, the results without air phrase considera‐ tion are shown in Fig.42. In this condition, the negative pore water pressure gradually reaches zero at the top of soil column. While the pore water pressure remains unchanged at the bottom of the column. If we take the pore air flow movement into consideration, the results are different as shown in Fig.40c. Besides, the suction could be calculated more precisely when we attain both the pore water pressure and the pore air pressure by simulating infiltration analyses coupled with air flow analyses.

> In the seepage analysis, the governing partial differential equation (GEO-SLOPE International Ltd.2007) for a two-dimensional transient water flow used in the finite element seepage model

**g f**

**c**

Groundwater Table

*H*w

b

*H*s

w w w w

g

a

**h**

calculate the factor of the slope safety, *F*s.

**8.1. Effect of soil properties**

( ) *H y H P m k mQ tyt t*

where *H* is the total head; *kx* are *ky* ars hydraulic conductivity in x, y direction respectively; *Q* is the applied boundary flux; *m*w is the slope of the SWCC and *γ*w is the unit weight of water. Eq. (66) was solved by SEEP/W software (Geo-Slope 2007). The boundary conditions used in the transient seepage analysis were shown in Fig. 43. And the initial condition for the analyses was a hydrostatic condition with a limiting pore water pressure of -75kPa. Then the SEEP/W software generated the negative pore water pressure as the initial condition. The pore-water pressures obtained from the seepage analysis were then used in the slope stability analyses to

The equation for unsaturated shear strength was used as Eq. (63). In the study, we chose *c*'=10kPa, *φ'*=26°. The Morgenstern-Price Method was adopted in the slope stability analysis. This method considered both shear and normal inter-slice forces, satisfied both moment and force equilibrium, and allowed for a variety of user-selected inter-slice force function. The slope stability analysis using Morgenstern-Price Method was performed using SLOPE/W software (Geo-Slope 2007). The pore-water pressures, *u*w, obtained from the transient seepage analyses using SEEP/W were added to SLOPE/W to be incorporated in the slope stability analyses.

The results were presented with attention on the effects of the factors which impacting on the soil slope stability under rainfall for 24h with a combination of various controlling factors.

The variation in factor of safety with time for a homogeneous soil slope of constant slope height *H*s=6m, initial groundwater table depth *H*w*=*2m, subjected to rainfall intensities of 3, 9 and

¶ - ¶ æ ö ¶ ¶

w a

= ++ ç ÷ ¶¶ ¶ ¶ è ø (66)

**d**

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123

**e**

was written as follows:

**Figure 43.** Sketch map of the slope in study

## **8. Slope stability analyses due to rainfall infiltration**

It was widely recognized that the rainfall infiltration took a great role in causing landslides, while the relative importance of soil properties, rainfall intensity, initial water table depth and slope geometry in inducing instability of a homogenous unsaturated soil slope under different rainfall was investigated through a series of parametric studies (Rahardjo, et al, 2007). Soil properties and rainfall intensity were found to be the primary factors controlling the instability of slopes due to rainfall, while the initial water table depth and slope geometry only played a secondary role.

The factors affecting the stability of a slope were considered to be the soil properties, rainfall intensity, initial depth of the groundwater table, and the slope geometry (i.e., slope angle and slope height). To assess the effects and relative contribution of controlling factors, a series of parametric studies were performed on a typical geometry of a homogeneous soil slope shown in Fig. 43. It had the boundary conditions as follows: ab, bc, cd=*q*=*I*r(rainfall intensity); ah, de, fg=*q*=0m3 /s (i.e. no flow boundary); and ef, gh=*h*t (total head at the side). Four slope heights, *Hs*,(3m, 4m, 5m and 6m), three slope angles, *α*,(26.6, 33.7 and 45.0), eight initial depths of groundwater table (GWT), *Hw*,(0.5m, 1m, 2m, 3m, 5m, 7.5m, 10m and 15m), three fractal dimensions *D* (2.1, 2.35 and 2.78),four air-entry values *ψ*e (1kPa, 10kPa, 20kPa and 50kPa), five values of saturated hydraulic conductivity, *k*s (10-4m/s, 10-5m/s, 10-6m/s, 10-7m/s and 10-8m/ s) and five rainfall intensities *Ir*(3mm/h, 9mm/h, 15mm/h, 36mm/h and 100mm/h each for 24h duration) were used in six cases of parametric studies. The shear strength parameters of the soils used in the parametric study were *c*'=10kPa, effective angle of internal friction, *φ'*=26°, and unit weight of soil, *γ*=20kN/m3 .

The fractal model of the soil-water characteristic curves for unsaturated soil was written as Eq. (29). The function of relative hydraulic conductivity was used as Eqs. (44) and (45). The derived SWCC and RHC for the all six soils were shown in Figs. 34 and 35. In the legends of these pictures, the symbol S in the soil names represent "soil", the first number means the airentry value, and the second one means the fractal dimension.

**Figure 43.** Sketch map of the slope in study

**7.5. Infiltration analyses coupled with the air flow analyses**

**8. Slope stability analyses due to rainfall infiltration**

/s (i.e. no flow boundary); and ef, gh=*h*t

.

entry value, and the second one means the fractal dimension.

and unit weight of soil, *γ*=20kN/m3

coupled with air flow analyses.

secondary role.

122 Hydraulic Conductivity

fg=*q*=0m3

In the infiltration analyses, air flow movement in the soil pore is also conducted. The effect of the pore air flow movement is especially obvious in the situation when the rainfall intensity is much larger than the soil saturated hydraulic conductivity. Compared with the results of infiltration analyses coupled with air flow analyses, the results without air phrase considera‐ tion are shown in Fig.42. In this condition, the negative pore water pressure gradually reaches zero at the top of soil column. While the pore water pressure remains unchanged at the bottom of the column. If we take the pore air flow movement into consideration, the results are different as shown in Fig.40c. Besides, the suction could be calculated more precisely when we attain both the pore water pressure and the pore air pressure by simulating infiltration analyses

It was widely recognized that the rainfall infiltration took a great role in causing landslides, while the relative importance of soil properties, rainfall intensity, initial water table depth and slope geometry in inducing instability of a homogenous unsaturated soil slope under different rainfall was investigated through a series of parametric studies (Rahardjo, et al, 2007). Soil properties and rainfall intensity were found to be the primary factors controlling the instability of slopes due to rainfall, while the initial water table depth and slope geometry only played a

The factors affecting the stability of a slope were considered to be the soil properties, rainfall intensity, initial depth of the groundwater table, and the slope geometry (i.e., slope angle and slope height). To assess the effects and relative contribution of controlling factors, a series of parametric studies were performed on a typical geometry of a homogeneous soil slope shown in Fig. 43. It had the boundary conditions as follows: ab, bc, cd=*q*=*I*r(rainfall intensity); ah, de,

*Hs*,(3m, 4m, 5m and 6m), three slope angles, *α*,(26.6, 33.7 and 45.0), eight initial depths of groundwater table (GWT), *Hw*,(0.5m, 1m, 2m, 3m, 5m, 7.5m, 10m and 15m), three fractal dimensions *D* (2.1, 2.35 and 2.78),four air-entry values *ψ*e (1kPa, 10kPa, 20kPa and 50kPa), five values of saturated hydraulic conductivity, *k*s (10-4m/s, 10-5m/s, 10-6m/s, 10-7m/s and 10-8m/ s) and five rainfall intensities *Ir*(3mm/h, 9mm/h, 15mm/h, 36mm/h and 100mm/h each for 24h duration) were used in six cases of parametric studies. The shear strength parameters of the soils used in the parametric study were *c*'=10kPa, effective angle of internal friction, *φ'*=26°,

The fractal model of the soil-water characteristic curves for unsaturated soil was written as Eq. (29). The function of relative hydraulic conductivity was used as Eqs. (44) and (45). The derived SWCC and RHC for the all six soils were shown in Figs. 34 and 35. In the legends of these pictures, the symbol S in the soil names represent "soil", the first number means the air-

(total head at the side). Four slope heights,

In the seepage analysis, the governing partial differential equation (GEO-SLOPE International Ltd.2007) for a two-dimensional transient water flow used in the finite element seepage model was written as follows:

$$m\_{\rm w} \mathcal{I}\_{\rm w} \frac{\partial (H - y)}{\partial t} = \frac{\partial}{\partial y} \left( k\_{\rm w} \frac{\partial H\_{\rm w}}{\partial t} \right) + m\_{\rm w} \frac{\partial P\_{\rm a}}{\partial t} + Q \tag{66}$$

where *H* is the total head; *kx* are *ky* ars hydraulic conductivity in x, y direction respectively; *Q* is the applied boundary flux; *m*w is the slope of the SWCC and *γ*w is the unit weight of water. Eq. (66) was solved by SEEP/W software (Geo-Slope 2007). The boundary conditions used in the transient seepage analysis were shown in Fig. 43. And the initial condition for the analyses was a hydrostatic condition with a limiting pore water pressure of -75kPa. Then the SEEP/W software generated the negative pore water pressure as the initial condition. The pore-water pressures obtained from the seepage analysis were then used in the slope stability analyses to calculate the factor of the slope safety, *F*s.

The equation for unsaturated shear strength was used as Eq. (63). In the study, we chose *c*'=10kPa, *φ'*=26°. The Morgenstern-Price Method was adopted in the slope stability analysis. This method considered both shear and normal inter-slice forces, satisfied both moment and force equilibrium, and allowed for a variety of user-selected inter-slice force function. The slope stability analysis using Morgenstern-Price Method was performed using SLOPE/W software (Geo-Slope 2007). The pore-water pressures, *u*w, obtained from the transient seepage analyses using SEEP/W were added to SLOPE/W to be incorporated in the slope stability analyses.

The results were presented with attention on the effects of the factors which impacting on the soil slope stability under rainfall for 24h with a combination of various controlling factors.

#### **8.1. Effect of soil properties**

The variation in factor of safety with time for a homogeneous soil slope of constant slope height *H*s=6m, initial groundwater table depth *H*w*=*2m, subjected to rainfall intensities of 3, 9 and 15mm/h of respective soil for 24h with a combination of various soil types (different fractal dimensions and air-entry values) were analysized as follows.

of slope changes quickly leading unsaturated soil to become saturated. The passive pore water pressure generated by the increase of degree of soil saturation makes a negative impact on the

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The effects of air-entry value on the stability of soil slope due to rainfall infiltration are shown in Figs. 45. The air-entry value, *ψ*e is 1kPa, 10kPa, 20kPa and 50kPa respectively and the fractal dimension, *D* remains 2.35. The style of the slope safety curve changes a lot with different airentry values in Figs. 45. In Fig. 45a, the air-entry value, *ψ*e is 1kPa and the decrease of factor of safety occurs during 16~20h after rainfall began. As the air-entry value, *ψ*e is 10kPa, shown in Fig. 45b, the decrease of slope safety factor occurs at 8h after rainfall began. At the beginning of rainfall infiltration, the decrease happens in Figs. 45c-d, in which the air-entry value is 20kPa and 50kPa respectively. This means that a delay time appears which is depended with the airentry value. And it is obvious that small air-entry value decides a relative long delay time. Besides, the factor of slope safety remains minimum during another 24 hours after rainfall stops in Fig. 45a, which has a low air-entry value (*ψ*e=1kPa). While *ψ*e=50kPa, the recovery of safety factor curve in Fig. 45c occurs and dramatically. This is also due to the style of hydraulic conductivity function, as shown in Fig. 35. The initial hydraulic conductivity value becomes large when the air-entry value increases with fractal dimension, saturated hydraulic conduc‐

The effects of saturated hydraulic conductivity on stability of a homogenous soil slope are reflected through the relationship between *F*s and *k*s, shown in Fig. 46. The saturated hydraulic conductivity, *k*s, was varied with five different values of 10-8m/s, 10-7m/s, 10-6m/s, 10-5m/s and 10-4m/s for a homogeneous soil slope of constant soil type S10, 2.35 (*ψ*e=10kPa, *D*=2.35), *H*w=2m, *H*s=6m, *α*=33.7°, subjected to rainfall for 24h with six rainfall intensities of 3, 9, 15, 36, 100 and 360mm/h. All the plots in Fig. 47 shows that soil with *k*s values of 10-8m/s and 10-7m/s, respec‐ tively, are less affected by rainfall. Contrarily, soil with *k*s values of 10-5m/s and 10-4m/s are greatly affected by rainfall. It is suggested that soil slopes with a low saturated hydraulic conductivity are relatively safe under short-duration rainfall infiltration and for the soil slopes with a high saturated hydraulic conductivity the stability is affected by the short-duration rainfall greatly. This means that slopes with low saturated hydraulic conductivity need a long-

The *F*s(min)-*k*s critical curve was presented in the broken line in Fig. 47 as follows:

*a F*

s s (min) 1 *ck*

*be*- = +

where *F*s(min) is minimum factor of safety; *k*s is saturated hydraulic conductivity; *a*, *b* and *c* are the fitting parameters and *e* is natural number (i.e., 2.71…). Here, *a*=1.293, *b*=-0.492, *c*=1.2×105

is 0.9998.

(67)

,

instability of soil slope under rainfall.

tivity and saturation degree remain the same.

**3. Effect of saturated hydraulic conductivity**

duration rainfall to intrigue the instability.

and the corresponding coefficient of correlation, r2

**2. Effect of air-entry value**

#### **1. Effect of fractal dimension**

In Figs.44, the fractal dimension, *D* was 2, 2.35 and 2.78, respectively. The air-entry value, *ψ*e remained 10kPa. It is shown that the reduction of slope stability is larger in the condition of larger fractal dimension under the condition of the same rainfall intensities and the saturated hydraulic conductivity. This means that the soil slope is much safe if the fractal dimension is low while the saturated hydraulic conductivity remains constant. The factor of safety decreases in a short time under rainfall infiltration when the fractal dimension is relatively large, as shown in Fig.44c, while the decrease occurs at a longer time for the low fractal dimension soil type, as shown in Fig.44a.

**Figure 44.** Effect of fractal dimension on slope stability

If the air-entry value and the saturated hydraulic conductivity remain unchanged, the slope of the hydraulic conductivity function is small when the fractal dimension is large. Then the initial hydraulic conductivity value at the beginning of the rainfall infiltration is much big, which results in the speed of rainfall infiltration is great and the saturated degree of soil inside

of slope changes quickly leading unsaturated soil to become saturated. The passive pore water pressure generated by the increase of degree of soil saturation makes a negative impact on the instability of soil slope under rainfall.

#### **2. Effect of air-entry value**

15mm/h of respective soil for 24h with a combination of various soil types (different fractal

In Figs.44, the fractal dimension, *D* was 2, 2.35 and 2.78, respectively. The air-entry value, *ψ*e remained 10kPa. It is shown that the reduction of slope stability is larger in the condition of larger fractal dimension under the condition of the same rainfall intensities and the saturated hydraulic conductivity. This means that the soil slope is much safe if the fractal dimension is low while the saturated hydraulic conductivity remains constant. The factor of safety decreases in a short time under rainfall infiltration when the fractal dimension is relatively large, as shown in Fig.44c, while the decrease occurs at a longer time for the low fractal dimension soil

34

If the air-entry value and the saturated hydraulic conductivity remain unchanged, the slope of the hydraulic conductivity function is small when the fractal dimension is large. Then the initial hydraulic conductivity value at the beginning of the rainfall infiltration is much big, which results in the speed of rainfall infiltration is great and the saturated degree of soil inside

dimensions and air-entry values) were analysized as follows.

**1. Effect of fractal dimension**

124 Hydraulic Conductivity

type, as shown in Fig.44a.

**Figure 44.** Effect of fractal dimension on slope stability

The effects of air-entry value on the stability of soil slope due to rainfall infiltration are shown in Figs. 45. The air-entry value, *ψ*e is 1kPa, 10kPa, 20kPa and 50kPa respectively and the fractal dimension, *D* remains 2.35. The style of the slope safety curve changes a lot with different airentry values in Figs. 45. In Fig. 45a, the air-entry value, *ψ*e is 1kPa and the decrease of factor of safety occurs during 16~20h after rainfall began. As the air-entry value, *ψ*e is 10kPa, shown in Fig. 45b, the decrease of slope safety factor occurs at 8h after rainfall began. At the beginning of rainfall infiltration, the decrease happens in Figs. 45c-d, in which the air-entry value is 20kPa and 50kPa respectively. This means that a delay time appears which is depended with the airentry value. And it is obvious that small air-entry value decides a relative long delay time. Besides, the factor of slope safety remains minimum during another 24 hours after rainfall stops in Fig. 45a, which has a low air-entry value (*ψ*e=1kPa). While *ψ*e=50kPa, the recovery of safety factor curve in Fig. 45c occurs and dramatically. This is also due to the style of hydraulic conductivity function, as shown in Fig. 35. The initial hydraulic conductivity value becomes large when the air-entry value increases with fractal dimension, saturated hydraulic conduc‐ tivity and saturation degree remain the same.

#### **3. Effect of saturated hydraulic conductivity**

The effects of saturated hydraulic conductivity on stability of a homogenous soil slope are reflected through the relationship between *F*s and *k*s, shown in Fig. 46. The saturated hydraulic conductivity, *k*s, was varied with five different values of 10-8m/s, 10-7m/s, 10-6m/s, 10-5m/s and 10-4m/s for a homogeneous soil slope of constant soil type S10, 2.35 (*ψ*e=10kPa, *D*=2.35), *H*w=2m, *H*s=6m, *α*=33.7°, subjected to rainfall for 24h with six rainfall intensities of 3, 9, 15, 36, 100 and 360mm/h. All the plots in Fig. 47 shows that soil with *k*s values of 10-8m/s and 10-7m/s, respec‐ tively, are less affected by rainfall. Contrarily, soil with *k*s values of 10-5m/s and 10-4m/s are greatly affected by rainfall. It is suggested that soil slopes with a low saturated hydraulic conductivity are relatively safe under short-duration rainfall infiltration and for the soil slopes with a high saturated hydraulic conductivity the stability is affected by the short-duration rainfall greatly. This means that slopes with low saturated hydraulic conductivity need a longduration rainfall to intrigue the instability.

The *F*s(min)-*k*s critical curve was presented in the broken line in Fig. 47 as follows:

$$F\_s(\text{min}) = \frac{a}{1 + be^{-ck\_s}}\tag{67}$$

where *F*s(min) is minimum factor of safety; *k*s is saturated hydraulic conductivity; *a*, *b* and *c* are the fitting parameters and *e* is natural number (i.e., 2.71…). Here, *a*=1.293, *b*=-0.492, *c*=1.2×105 , and the corresponding coefficient of correlation, r2 is 0.9998.

where *F*s(min) is minimum factor of safety; *I*r is rainfall intensity; and *a, b, c, d* are fitting parameters. The values for the fitting parameters and the corresponding coefficient of

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1E-8 1E-7 1E-6 1E-5 1E-4

*k*s(m/s)

3mm/h 9mm/h 15mm/h 36mm/h 100mm/h 360mm/h Fs-ks critical line

, for the sigmoid line in Fig.47.

1

**Figure 46.** Relationship between *K*s and minimum factor of safety

**Figure 47.** Effect of rainfall intensity on variation of minimum factor of safety

1.5

2

*F*s(min)

2.5

3

correlation, r2

**Figure 45.** Effect of air-entry value on the slope stability

#### **8.2. Effect of rainfall intensity**

The relationship between minimum factor of safety, *F*s(min), versus logarithmic of rainfall intensity, *I*r are plotted in Fig. 47. The semi log plot shows that generally the *F*s(min) and *I*r relationships follow a sigmoid shape in Fig. 47. There are two inflect points for the sigmoid shape line. The *Fs*(min) is almost constant at very low rainfall intensities for all soil types. And it starts to decrease rapidly when the first inflection point is reached. The trend in *F*s(min) versus *I*r relationship observed in Fig. 47 can be described by a sigmoid equation (MMF Line) as the form of

$$F\_{\rm s} \text{(min)} = \frac{ab + cI\_{\rm r}^{\
u}}{b + I\_{\rm r}^{\
u}} \tag{68}$$

where *F*s(min) is minimum factor of safety; *I*r is rainfall intensity; and *a, b, c, d* are fitting parameters. The values for the fitting parameters and the corresponding coefficient of correlation, r2 , for the sigmoid line in Fig.47.

**Figure 46.** Relationship between *K*s and minimum factor of safety

**Figure 47.** Effect of rainfall intensity on variation of minimum factor of safety

36

*b I*

+

s

(min)

*ab cI F*

+ =

r

*d*

*d*

(68)

r

The relationship between minimum factor of safety, *F*s(min), versus logarithmic of rainfall intensity, *I*r are plotted in Fig. 47. The semi log plot shows that generally the *F*s(min) and *I*r relationships follow a sigmoid shape in Fig. 47. There are two inflect points for the sigmoid shape line. The *Fs*(min) is almost constant at very low rainfall intensities for all soil types. And it starts to decrease rapidly when the first inflection point is reached. The trend in *F*s(min) versus *I*r relationship observed in Fig. 47 can be described by a sigmoid equation (MMF Line)

**Figure 45.** Effect of air-entry value on the slope stability

**8.2. Effect of rainfall intensity**

as the form of

126 Hydraulic Conductivity

## **8.3. Effect of initial water table location**

The effects of initial groundwater table location on stability of a homogenous soil slope are reflected through the relationship between *F*s(ini) and *F*s(min) with *H*w, shown in Fig. 48. The initial depth of water table, *H*w, was varied with five different values of 0.5, 1, 2, 2.5, 5, 7.5, 10 and15m for a homogeneous soil slope of constant soil type S10, 2.35, -5 (*ψ*e=10kPa, *D*=2.35, *k*s=10-5m/s), *H*s=6m, *α*=33.7°, subjected to rainfall for 24h with five rainfall intensities of 3, 9, 15, 36, and 100mm/h. The relationship between *F*s(ini), and *H*w shown in Fig. 48 appears to be linear up to a depth of 7.5m beyond which *F*s(ini) remains constant. This is because the initial pore-water pressure profiles generated for the slope at *H*w=7.5m as same as those generated when *H*w=7.5m. This is due to the limiting pore-water pressure of −75kPa adopted in the analyses. In Fig. 54 it also shows that the rainfall intensity is the dominated factor impacting on the reduction in factor of safety, *F*s. And the initial depth of water table, *H*w, mainly determines the value of initial factor of safety, *F*s(ini). The *F*s(ini) is smaller for slopes with a shallower *H*w which means that the soil slope stability is much lower. Therefore, slopes with a shallow *H*w are more likely to fail due to a rainfall compared with slopes which has a deep *H*w.

**Figure 49.** Effect of slope angle on factor of safety

The effect of slope geometry on the stability of a homogenous soil slope is evaluated in terms of slope angle (*α*) and slope height (*H*s). The effects of slope angel on the stability of a homo‐ genous soil slope are reflected through the relationship between *F*s(ini) and *F*s(min) with *α*, shown in Fig. 49. The slope angel, *α* was varied with three different values of 26°, 33.7°, 45°and 63° for a homogeneous soil slope of constant soil type S10, 2.35, -5 (*ψ*e=10kPa, *D*=2.35, *k*s=10-5m/ s), *H*s=6m, *H*w=2m, subjected to rainfall for 24h with five rainfall intensities of 3, 9, 15 and 36mm/ h. The relationship between *F*s(min) and *α* shown in Fig. 49 appears to be negative linear. In general, the higher the slope angle, the lower the initial factor of safety and the minimum factor of safety. Because a steep slope will yield a lower factor of safety compared with a flat slope.

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The effects of slope angel on the stability of a homogenous soil slope are reflected through the relationship between *F*s(ini) and *F*s(min) with *H*s, shown in Fig. 50. The slope height, *H*s, was varied with four different values of 3m, 4m, 5m and 6m for a homogeneous soil slope of

**8.4. Effect of slope angle**

**8.5. Effect of slope height**

**Figure 48.** Effect of initial groundwater table depth on factor of safety

**Figure 49.** Effect of slope angle on factor of safety

#### **8.4. Effect of slope angle**

**8.3. Effect of initial water table location**

128 Hydraulic Conductivity

**Figure 48.** Effect of initial groundwater table depth on factor of safety

The effects of initial groundwater table location on stability of a homogenous soil slope are reflected through the relationship between *F*s(ini) and *F*s(min) with *H*w, shown in Fig. 48. The initial depth of water table, *H*w, was varied with five different values of 0.5, 1, 2, 2.5, 5, 7.5, 10 and15m for a homogeneous soil slope of constant soil type S10, 2.35, -5 (*ψ*e=10kPa, *D*=2.35, *k*s=10-5m/s), *H*s=6m, *α*=33.7°, subjected to rainfall for 24h with five rainfall intensities of 3, 9, 15, 36, and 100mm/h. The relationship between *F*s(ini), and *H*w shown in Fig. 48 appears to be linear up to a depth of 7.5m beyond which *F*s(ini) remains constant. This is because the initial pore-water pressure profiles generated for the slope at *H*w=7.5m as same as those generated when *H*w=7.5m. This is due to the limiting pore-water pressure of −75kPa adopted in the analyses. In Fig. 54 it also shows that the rainfall intensity is the dominated factor impacting on the reduction in factor of safety, *F*s. And the initial depth of water table, *H*w, mainly determines the value of initial factor of safety, *F*s(ini). The *F*s(ini) is smaller for slopes with a shallower *H*w which means that the soil slope stability is much lower. Therefore, slopes with a shallow *H*w are more likely to fail due to a rainfall compared with slopes which has a deep *H*w.

> The effect of slope geometry on the stability of a homogenous soil slope is evaluated in terms of slope angle (*α*) and slope height (*H*s). The effects of slope angel on the stability of a homo‐ genous soil slope are reflected through the relationship between *F*s(ini) and *F*s(min) with *α*, shown in Fig. 49. The slope angel, *α* was varied with three different values of 26°, 33.7°, 45°and 63° for a homogeneous soil slope of constant soil type S10, 2.35, -5 (*ψ*e=10kPa, *D*=2.35, *k*s=10-5m/ s), *H*s=6m, *H*w=2m, subjected to rainfall for 24h with five rainfall intensities of 3, 9, 15 and 36mm/ h. The relationship between *F*s(min) and *α* shown in Fig. 49 appears to be negative linear. In general, the higher the slope angle, the lower the initial factor of safety and the minimum factor of safety. Because a steep slope will yield a lower factor of safety compared with a flat slope.

#### **8.5. Effect of slope height**

The effects of slope angel on the stability of a homogenous soil slope are reflected through the relationship between *F*s(ini) and *F*s(min) with *H*s, shown in Fig. 50. The slope height, *H*s, was varied with four different values of 3m, 4m, 5m and 6m for a homogeneous soil slope of constant soil type S10, 2.35, -5(*ψ*e=10kPa, *D*=2.35, *k*s=10-5m/s), *H*w=2m, *α*=33.7°, subjected to rainfall for 24h with four rainfall intensities of 3mm/h, 9mm/h, 15mm/h and 36mm/h. Fig. 50 shows that initial factor of safety decreases exponentially as the slope height increases. It also suggests that high slopes are generally easier to fail under rainfall due to the low initial factor of safety. The reduction in factor of safety for a high slope is smaller and occurs at a slower rate compared with a low slope.

**References**

New York, NY. 1970

Drain Div 1966, 92:61–68.

Agr Engrs 1967;10:400-404.

1950; 201: 392-405.

37–44.

[1] Abramowitz M, Stegun I. Handbook of Mathematical Functions. Dover Publications,

Unsaturated Hydraulic Conductivity of Fractal-Textured Soils

http://dx.doi.org/10.5772/56716

131

[2] Avnir D, Jaroniec M, An isotherm equation for adsorption on fractal surfaces of het‐

[3] Brooks RH, Corey AT. Properties of porous media affecting fluid flow. ASCE J Irrig

[4] Brutsaert W. Some methods of calculating unsaturated permeability. Trans Am Soc

[5] Brutsaert W. The permeability of a porous medium determined from certain proba‐

[6] Brutsaert W. A concise parameterization of the hydraulic conductivity of unsaturated

[7] Burdine NT. Relative permeability calculations from pore-size distribution data.

[8] Burdine NT. Relative permeability calculation from pore size distribution data. Transactions of the American Institute of Mining Engineers, 1953, 198, 71–78.

[9] Carman PC. Flow of gases through porous media. New York: Academic Press, 1956.

[10] Childs EC, Collis-George N. The permeability of porous materials. Proc Roy Soc A

[11] Collins,B.D., Znidarcic. Stability analyses of rainfall induced landslides [J].Journal of

[12] Corey AT. Pore-size distribution. In: van Genuchten MTh, Leij FJ, Lund LJ, editors. Proceedings of the International Workshop, Indirect Methods for Estimating the Hy‐ draulic Properties of Unsaturated Soils, University of California, Riverside, 1992. p.

[13] Crawford JW. The relationship between structure and the hydraulic conductivity of

[14] Delage P, Audiguier M, Cui YJ, Howat MD. Microstructure of a compacted silt. Can

[15] Dirksen C. Unsaturated hydraulic conductivity. In: Smith KA, Mullins CE, editors.

[16] Fatt I, Dykstra H. Relative permeability studies. Transactions of the American Insti‐

Soil analysis physical methods. New York: Dekker; 1991. p. 209–69.

geotechnical and geoenvironmental engineering, 2004,4,362-372.

bility laws for pore size distribution. Water Resour Res 1968; 4: 425-34.

erogeneous porous materials. Langmuir, 1989, 5, 1431–1433.

soils. Advances in Water Resources, 2000, 23, 811-815.

Trans Am Inst Min Engrs 1953;198:71-7

soil. Eur J Soil Sci 1994, 45:493–501.

tute of Mining Engineers, 1951, 192: 249-255.

Geotech J 1996;33:150-158.

**Figure 50.** Effect of slope height on factor of safety

## **Acknowledgements**

The National Natural Science Foundation of China (Grant No. 41272318) and State Key Laboratory of Ocean Engineering are sincerely acknowledged for their financial support.

## **Author details**

Yongfu Xu\*

Address all correspondence to<EMAIL_ADDRESS>
Department of Civil Engineering, Shanghai Jiao Tong University, Shanghai, China

## **References**

constant soil type S10, 2.35, -5(*ψ*e=10kPa, *D*=2.35, *k*s=10-5m/s), *H*w=2m, *α*=33.7°, subjected to rainfall for 24h with four rainfall intensities of 3mm/h, 9mm/h, 15mm/h and 36mm/h. Fig. 50 shows that initial factor of safety decreases exponentially as the slope height increases. It also suggests that high slopes are generally easier to fail under rainfall due to the low initial factor of safety. The reduction in factor of safety for a high slope is smaller and occurs at a slower

The National Natural Science Foundation of China (Grant No. 41272318) and State Key Laboratory of Ocean Engineering are sincerely acknowledged for their financial support.

Department of Civil Engineering, Shanghai Jiao Tong University, Shanghai, China

rate compared with a low slope.

130 Hydraulic Conductivity

**Figure 50.** Effect of slope height on factor of safety

Address all correspondence to<EMAIL_ADDRESS>
**Acknowledgements**

**Author details**

Yongfu Xu\*

[17] Fuentes C, Vauclin M, Parlange J-I. A note on the soil–water conductivity of a fractal soil. Transp Porous Media 1996, 23:31–36.

[32] Matsuoka H, Soil Mechanics. Morikita Shuppan Co., Ltd., 1999, p. 61.

porous media. Water Resources Res 1976, 12:513–22.

JMR 1962).

1949;186:39-46.

2007,12,1532-1543.

2011, 2, 484-492.

251, 163-176.

2965.

W11510.

Sci Soc Am J 1991; 55: 1483–1489.

soil. J Hydro 1990, 117: 1-13.

[33] Mualem Y. A new model for predicting the hydraulic conductivity of unsaturated

Unsaturated Hydraulic Conductivity of Fractal-Textured Soils

http://dx.doi.org/10.5772/56716

133

[34] Polubarinova-Kochina PYa. Theory of ground water movement. Princeton, NJ: Princeton University Press, 1952. p. 613 (translated from the Russian by DeWiest,

[35] Purcell WR. Capillary pressures-their measurement using mercury and the calcula‐ tion of permeability therefrom. Trans Am Inst Min Met Engrs Petrol Devel Technol

[36] Rahardjo H, Ong TH, Rezaur RB. Factors controlling instability of homogeneous soil slopes under rainfall [J]. Journal of geotechnical and geoenvironmental engineering,

[37] Ravichandran N, Krishnapillai S. A Statistical Model for the Relative Hydraulic Con‐ ductivity of Water Phase in Unsaturated Soils. International Journal of Geosciences,

[38] Rieu M, Sposito G. Fractal fragmentation, soil porosity and soil water properties. Soil

[39] Schaap MG, Leij FJ, van Genuchten MT. Rosetta: A computer program for estimating soil hydraulic parameters with hierarchical pedotransfer functions, J. Hydrol., 2001,

[40] Simunek T, Wendroth O, van Genuchten MT, et al., Water Resources Res. 35 (1999)

[41] Smettem KRJ, Kirkby C. Measuring the hydraulic properties of a stable aggregate

[42] Stingaciu LR, Weihermüller L, Haber-Pohlmeier S et al. Determination of pore size distribution and hydraulic properties using nuclear magnetic resonance relaxometry: A comparative study of laboratory methods. Water Resources Research, 2009, 46:

[43] Sugii T, Uno T, Hayashi T, Proceedings of the 31st Japan National Conference on Ge‐

[44] Toledo PG, Novy RA, Davis HT, Scriven LE. Hydraulic conductivity of porous me‐

[45] Touma J, Vauclin M. Experimental and numerical analysis of two-phase infiltration

in a partially saturated soil. Transport in Porous Media, 1986, 1, 27-55.

otechnical Engineering, Kitami, Japan, 1996, pp. 2075

dia at low water content. Soil Sci Soc Am J 1990;54:673–9.

[32] Matsuoka H, Soil Mechanics. Morikita Shuppan Co., Ltd., 1999, p. 61.

[17] Fuentes C, Vauclin M, Parlange J-I. A note on the soil–water conductivity of a fractal

[18] Fuentes C, Haverkamp R, Parlange J-Y. Parameter constraints on closed-form soil

[19] Gates JI, Tempelaar-Lietz W. Relative permeabilities of California cores by the capil‐ lary pressure method. Drilling and Production Practice, Am Petrol Inst 1950:285-298.

[20] GEO-SLOPE International Ltd. Seepage Modeling with SEEP/W2007 [M].Third Edi‐

[21] GEO-SLOPE International Ltd. Air Flow Modeling with AIR/W2007 [M].Third Edi‐

[22] GEO-SLOPE International Ltd. Stability Modeling with SLOPE/W2007 [M].Third Ed‐

[23] Gimenez D, Perfect E, Rawls WJ, Pacheoaky Ya. Fractal models for predicting soil

[24] Hoffmann-Riem H, van Genuchten M.Th, Flühler H. A general model of the hydraul‐ ic conductivity of unsaturated soils, in: van Genuchten M.Th, Leij FJ, Wu L. (Eds.), Proceedings of International Workshop, Characterization and Measurements of Hy‐ draulic Properties of Unsaturated Porous Media. Riverside, CA. 22–24th Oct. 1997.

[25] Hunt AG, Gee GW. Application of critical path analysis to fractal porous media:com‐ parison with examples from the Hanford site. Adv in Water Resources 2002;25:129–

[26] Irmay S. On the hydraulic conductivity of unsaturated soils. Trans Am Geophys Un

[27] Jarvis NJ, Messing I. Near-saturated hydraulic conductivity in soils of contrasting texture as measured by tension infiltrometers. Soil Sci Soc Am J 1995;59:27–34.

[28] Kahr G., Kraehenbuehl F, Stoeckli HF, Muller-Vonmoos M. Study of the water –ben‐ tonite system by vapour adsorption, immersion colorimetry and X-ray technique: Part II. Heats of immersion, swelling pressures and thermodynamic properties. Clay

[29] Khaleel R, Relyea JF. Evaluation of van Genuchten–Mualem relationships to estimate unsaturated hydraulic conductivity at low water contents. Water Resour Res 1995;

[30] Kozeny J. Ueber kapillare leitung des wassers im boden, sitzungsberichte, akad der

[31] Mandelbrot BB, The Fractal Geometry of Nature, W.H. Freeman, New York, 1982.

wissensch. Wien, Math-Naturw Klass Abt IIa 1927;136:271-306.

water relationships. Journal of Hydrology, 1992, 134, 117-142.

hydraulic properties: a review. Eng Geo 1997;48:161-183.

University of California, Riverside, 1999, pp. 31–42.

soil. Transp Porous Media 1996, 23:31–36.

tion, Canada, 2008,3.

132 Hydraulic Conductivity

tion, Canada, 2008,3.

ition, Canada, 2008,3.

46.

1954;35:463-467.

Miner. 1990, 25, 499–506.

31: 2659–1668.

[46] Touma J. Comparison of the soil hydraulic conductivity predicted from its water re‐ tention expressed by the equation of van Genuchten and different capillary models. European Journal of Soil Science, 2009, 60: 671-680

**Chapter 5**

**Calibration of a New Device to Measure Water Content**

The vadose zone, which extends from the soil–atmospheric interface to the capillary fringe of the water table, is a fundamental part of the hydrologic cycle. It controls how the precipitation splits into infiltration, surface runoff, evapotranspiration, groundwater recharge; it regulates the storage, transfer, filtering, adsorption, retard and attenuation of solutes and contaminants

Until the last decades, the vadose zone was considered as a black box that merely connect surface water and groundwater. As consequence an incomplete understanding of the complex

Nowadays, instead, the monitoring of the hydrological processes that occur through the vadose zone, are receiving increased attention mainly with refer to contaminant transport processes. Fluxes of water and solutes strongly depend on water content thus its monitoring

A wide range of methods, sensors and technologies are available for the measure of soil water

The traditional standard method for direct measurement of soil water content is the gravi‐ metric method [1]. This method implies the sampling of the soil to carry out laboratory measurements, modifying the natural condition. As gravimetric method is destructive, labour intensive, not timely and costly, many alternative non-destructive methods for measuring water content have been developed. However, none of all methods measure the water content directly but only some properties, named "surrogate measure", that changes as the soil water content changes. By measuring the value of the surrogate parameter, it is possible to estimated

> © 2013 Caputo and Masciale; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Maria Clementina Caputo and Rita Masciale

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/56699

before these reach the groundwater.

dynamics of the vadose zone existed.

and estimation becomes an important issue.

content, mainly used in management of precision farming.

**1. Introduction**

**of Rocks**


**Author details**

Måns Almqvist1 , Gustav Mattsson1 \*, Robin Razmi1 and Peter Magnusson1,2

1 Centre for Research and Development, Uppsala University, Sweden

2 Cardiology Research Unit, Department of Medicine, Karolinska Institutet, Stockholm, Sweden

\*Address all correspondence to<EMAIL_ADDRESS>
© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.