**3. Unsaturated seepage theory**

In geotechnical engineering, water flow through soil can be categorized in several ways: laminar or turbulent flow (determined by the Reynolds number); one-dimensional (1D), twodimensional (2D), or three-dimensional (3D) flows (depending on the number of planes); steady-state and transient seepage (constant and variable across time, respectively); and confined and unconfined flows (depending on the limits defining them).

#### **3.1. Steady-state seepage**

**Model Equation**

170 Groundwater - Contaminant and Resource Management

Burdine (1953) *k*(*ψ*)=*ks*

Mualem (1976) *k*(*ψ*)=*ks*

*ψ* soil suction

*ρ<sup>w</sup>* water density

*g* acceleration of gravity

Gardner (1958) *<sup>k</sup>*(*ψ*)= *ks*

{1 <sup>−</sup> (*avbψ*)

<sup>1</sup> <sup>+</sup> *ag* ( *<sup>ψ</sup> <sup>ρ</sup><sup>w</sup> <sup>g</sup>* )*ng*

*ks* ( *<sup>ψ</sup>aev*

> *∫* ln*ψ b*

*k*(*ψ*)=*ks Θ<sup>d</sup>* (*ψ*) *<sup>q</sup>*

*ks* saturated hydraulic conductivity coefficient

*ag* adjustment parameter related to the air-entry value

*avb*, *avm* adjustment parameter related to the inverse of the air-entry value

*mvb* adjustment parameter (1-2/*nvb*); value oscillates between zero and one *mvm* adjustment parameter (1-1/*nvm*); value oscillates between zero and one

*y* fictitious variable that describes soil suction (logarithm scale)

*ng* soil parameter that depends on the desaturation process once the air-entry value is

*θ* ' variable derived from the function of soil storage capacity (characteristic curve)

*nvb*, *nvm* adjustment parameter of the characteristic curve obtained from Van Genuchten's model

*∫*ln(*ψaev* ) *<sup>b</sup> <sup>θ</sup>*(*<sup>e</sup> <sup>y</sup>* ) <sup>−</sup> *<sup>θ</sup><sup>s</sup> <sup>e</sup> <sup>y</sup> <sup>θ</sup>* '

*k*(*ψ*)=*ks*

*k*(*ψ*) hydraulic conductivity function

*ψaev* soil suction at the air-entry value *λ* pore size distribution index

exceeded

(1980)

*b* upper integration limit (1,000,000 kPa)

**Table 2.** Estimation models for determining hydraulic conductivity function.

*<sup>k</sup>*(*ψ*)={ *ks <sup>ψ</sup>*≤*ψaev*

{1 <sup>−</sup> (*avmψ*)

*nvb*−<sup>2</sup> <sup>1</sup> <sup>+</sup> (*avbψ*)

*nvb* <sup>2</sup>*mvb*

1 + (*avbψ*)

*<sup>ψ</sup>* )2+3*<sup>λ</sup> <sup>ψ</sup>*≥*ψaev*

1 + (*avmψ*)

*<sup>θ</sup>*(*<sup>e</sup> <sup>y</sup>* ) <sup>−</sup> *<sup>θ</sup>*(*ψ*) *<sup>e</sup> <sup>y</sup> <sup>θ</sup>* '

*nvm*−<sup>1</sup> <sup>1</sup> <sup>+</sup> (*avmψ*)

*nvm mvm*/2

(*e <sup>y</sup>*)*dy*

(*e <sup>y</sup>*)*dy*

*nvb mvb*}<sup>2</sup>

*nvm mvm*}<sup>2</sup>

Van Genuchten and

Brooks and Corey (1964)

Fredlund and Xing (1994)

Van Genuchten and

Leong and Rahardjo

(1997)

where

**Figure 3.** One- and two-dimensional water flow through an unsaturated element [24].

Considering that **Figure 3(a)** represents a soil sample subject to an upward flow and taking into consideration the law of continuity, the following can be proposed:

$$
\left(\nu\_{\text{wy}} + \frac{d\nu\_{\text{wy}}}{d\text{y}}d\text{y}\right)d\text{x}d\text{z} - \nu\_{\text{wy}}d\text{x}d\text{z} = 0\tag{1}
$$

After applying Darcy's law [31], the one-dimensional water flow through an unsaturated soil can be expressed as

$$k\_{\text{\tiny u\text{\textgreater}}} \frac{d^2 h\_w}{d\text{y}^2} + \frac{dk\_{\text{\tiny uv}}}{d\text{y}} \frac{dh\_w}{d\text{y}} = 0 \tag{2}$$

where

d*h <sup>w</sup>* / d*y* is the hydraulic gradient

d*h <sup>w</sup>* is the hydraulic head

For the example of **Figure 3(b)**, which demonstrates a two-directional water flow, the following expression can be deduced from the continuity equation:

$$
\left(\nu\_{\rm ux} + \frac{\partial \nu\_{\rm ux}}{\partial \mathbf{x}} d\mathbf{x} - \nu\_{\rm ux}\right) d\mathbf{y} d\mathbf{z} + \left(\nu\_{\rm uy} + \frac{\partial \nu\_{\rm uy}}{\partial \mathbf{y}} d\mathbf{x} - \nu\_{\rm uy}\right) d\mathbf{x} d\mathbf{z} = \mathbf{0} \tag{3}
$$

After applying Darcy's law [31], the following equation can be proposed to describe steadystate seepage in two directions for an unsaturated anisotropic soil:

$$k\_{\rm uv} \frac{\partial^2 h\_w}{\partial \mathbf{x}^2} + k\_{\rm uv} \frac{\partial^2 h\_w}{\partial \mathbf{y}^2} + \frac{\partial k\_{\rm uv}}{\partial \mathbf{x}} \frac{\partial h\_w}{\partial \mathbf{x}} + \frac{\partial k\_{\rm uv}}{\partial \mathbf{y}} \frac{\partial h\_w}{\partial \mathbf{y}} = \mathbf{0} \tag{4}$$

Under isotropic conditions, the previous equation can be expressed as follows:

$$\mathcal{L}\_w \left( \frac{\partial^2 h\_w}{\partial \mathbf{x}^2} + \frac{\partial^2 h\_w}{\partial \mathbf{y}^2} \right) + \frac{\partial k\_w}{\partial \mathbf{x}} \frac{\partial h\_w}{\partial \mathbf{x}} + \frac{\partial k\_w}{\partial \mathbf{y}} \frac{\partial h\_w}{\partial \mathbf{y}} = \mathbf{0} \tag{5}$$

**Figure 4.** Three-dimensional water flow through an unsaturated element [24].

For three-directional water flow, considering the unsaturated soil sample subjected to the water flow conditions indicated in **Figure 4**, where the hydraulic conductivities vary in all directions, the following expression can be deduced based on flow continuity:

$$
\left(\mathbf{v}\_{wx} + \frac{\partial \mathbf{v}\_{wx}}{\partial \mathbf{x}} d\mathbf{x} - \mathbf{v}\_{wx}\right) d\mathbf{y} d\mathbf{z} + \left(\mathbf{v}\_{wy} + \frac{\partial \mathbf{v}\_{wy}}{\partial \mathbf{y}} d\mathbf{y} - \mathbf{v}\_{wy}\right) d\mathbf{x} d\mathbf{z} + \left(\mathbf{v}\_{wz} + \frac{\partial \mathbf{v}\_{wz}}{\partial \mathbf{z}} d\mathbf{z} - \mathbf{v}\_{wz}\right) d\mathbf{x} d\mathbf{y} = \mathbf{0} \tag{6}
$$

Once again, after applying Darcy's law, the equation that describes the steady-state seepage in three directions in anisotropic conditions is detailed as follows:

$$k\_{uv}\frac{\partial^2 h\_w}{\partial x^2} + k\_{uy}\frac{\partial^2 h\_w}{\partial y^2} + k\_{uz}\frac{\partial^2 h\_w}{\partial z^2} + \frac{\partial k\_{uv}}{\partial x}\frac{\partial h\_w}{\partial x} + \frac{\partial k\_{uy}}{\partial y}\frac{\partial h\_w}{\partial y} + \frac{\partial k\_{uz}}{\partial z}\frac{\partial h\_w}{\partial z} = 0 \tag{7}$$

For isotropic soils, the previous equation can be simplified to

$$k\_w \left(\frac{\partial^2 h\_w}{\partial \mathbf{x}^2} + \frac{\partial^2 h\_w}{\partial \mathbf{y}^2} + \frac{\partial^2 h\_w}{\partial \mathbf{z}^2}\right) + \frac{\partial k\_w}{\partial \mathbf{x}} \frac{\partial h\_w}{\partial \mathbf{x}} + \frac{\partial k\_w}{\partial \mathbf{y}} \frac{\partial h\_w}{\partial \mathbf{y}} + \frac{\partial k\_w}{\partial \mathbf{z}} \frac{\partial h\_w}{\partial \mathbf{z}} = \mathbf{0} \tag{8}$$

#### **3.2. Transient seepage (the Richards equation)**

In the transient seepage analysis and in contrast to steady state seepage, a variable hydraulic head exists over time. Variation occurs due to the changes in the boundaries of the system (due to variation in water levels over time).

For practical applications, Darcy's law [31] can be generalized to unsaturated water flow problems by considering hydraulic conductivity to be a function of the soil suction or suction head [32, 33]:

$$\left\|q\_{\boldsymbol{x}}=-k\_{\boldsymbol{x}}(h\_{\boldsymbol{m}})^{\hat{\boldsymbol{\alpha}}}\right\|\_{\widehat{\mathcal{O}}\mathbb{X}} ; \; q\_{\boldsymbol{y}}=-k\_{\boldsymbol{y}}(h\_{\boldsymbol{m}})^{\hat{\boldsymbol{\alpha}}}\big|\_{\widehat{\mathcal{O}}\mathbb{Y}} ; \; q\_{\boldsymbol{z}}=-k\_{\boldsymbol{z}}(h\_{\boldsymbol{m}})^{\hat{\boldsymbol{\alpha}}}\big|\_{\widehat{\mathcal{O}}\mathbb{X}}\tag{9}$$

where

For the example of **Figure 3(b)**, which demonstrates a two-directional water flow, the following

0 *wx wy*

After applying Darcy's law [31], the following equation can be proposed to describe steady-

2 2 <sup>0</sup> *<sup>w</sup> w wx w wy <sup>w</sup>*

2 2 0 *w w ww ww*

For three-directional water flow, considering the unsaturated soil sample subjected to the water flow conditions indicated in **Figure 4**, where the hydraulic conductivities vary in all

0 *wx wy wz*

æ ö ¶ ¶ æ ö ¶ æ ö ç ÷ + - ++ - ++ - = ç ÷ ç ÷ è ø ¶¶¶ è ø è ø (6)

directions, the following expression can be deduced based on flow continuity:

*wx wx wy wy wz wz v v <sup>v</sup> v dx v dydz v dy v dxdz v dz v dxdy xyz*

+

*h h kh h k*

*x y xx yy* ¶ ¶ ¶¶ ¶ ¶ ++ + =

Under isotropic conditions, the previous equation can be expressed as follows:

*h h kh kh <sup>k</sup> x y xx yy* ¶ ¶ ¶¶ ¶¶

+ + æ ö ç ÷

ç ÷ + - ++ - = ç ÷ è ø ¶ ¶ è ø (3)

¶ ¶ ¶¶ ¶¶ (4)

<sup>ø</sup> ¶ ¶ <sup>=</sup> ¶ ¶ ¶ ¶ (5)

*wx wx wy wy <sup>v</sup> <sup>v</sup> v dx v dydz v dx v dxdz x y* æ ö ¶ æ ö ¶

state seepage in two directions for an unsaturated anisotropic soil:

2 2

2 2

**Figure 4.** Three-dimensional water flow through an unsaturated element [24].

*wx wy*

*k k*

*w*

è

expression can be deduced from the continuity equation:

172 Groundwater - Contaminant and Resource Management

*h <sup>m</sup>* is the suction head

*k*(*h <sup>m</sup>*) is the hydraulic conductivity function.

If the osmotic pressure head is disregarded, then the total head of an unsaturated soil can be expressed as the sum of the matric suction head and the elevation head (*h* =*h <sup>m</sup>* + *z*). Thus, if this consideration is substituted in the equation of the law of conservation of matter and a constant water density is assumed, then the following expression can be obtained:

$$\frac{\partial}{\partial \mathbf{x}} \left[ k\_x(h\_m) \frac{\partial h\_m}{\partial \mathbf{x}} \right] + \frac{\partial}{\partial \mathbf{y}} \left[ k\_y(h\_m) \frac{\partial h\_m}{\partial \mathbf{y}} \right] + \frac{\partial}{\partial \mathbf{z}} \left[ k\_z(h\_m) (\frac{\partial h\_m}{\partial \mathbf{z}} + \mathbf{l}) \right] = \frac{\partial \theta}{\partial t} \tag{10}$$

where the additional term in the direction of the *z*-axis is due to the elevation head.

The term on the right side of Eq. (10) can also be expressed as a function of the matric suction head:

$$\frac{\partial \hat{\theta}}{\partial t} = \frac{\hat{\sigma}\theta}{\hat{\sigma}h\_m} \frac{\partial h\_m}{\partial t} \tag{11}$$

where ∂*θ* / ∂*h <sup>m</sup>* is the slope of the relationship between the volumetric water content and the suction head, which can be directly determined from the SWCC. The slope denotes the specific moisture capacity, which is typically denoted as *C*. As the soil storage function is not linear, it is necessary to describe the specific moisture capacity as a function of the suction or suction head:

$$C(h\_m) = \frac{\partial \theta}{\partial h\_m} \tag{12}$$

If Eqs. (11) and (12) are substituted in Eq. 10, the expression that describes transient seepage in unsaturated soils can be expressed as follows:

$$\frac{\partial}{\partial \mathbf{x}} \left[ k\_{\boldsymbol{x}}(h\_{m}) \frac{\partial h\_{m}}{\partial \mathbf{x}} \right] + \frac{\partial}{\partial \mathbf{y}} \left[ k\_{\boldsymbol{y}}(h\_{m}) \frac{\partial h\_{m}}{\partial \mathbf{y}} \right] + \frac{\partial}{\partial \mathbf{z}} \left[ k\_{\boldsymbol{z}}(h\_{m}) (\frac{\partial h\_{m}}{\partial \mathbf{z}} + \mathbf{l}) \right] = C(h\_{m}) \frac{\partial \theta}{\partial t} \tag{13}$$

Eq. (13) is known as the Richards [33] equation, where given the boundary limits and initial conditions specific to a system, the equation provides the values for suction across space and time. It is highlighted that in using this equation, it is necessary to have data on the SWCC and the hydraulic conductivity function that are specific to the material being studied.

#### **4. Slope stability analyses**

One of the most common methods for evaluating slope stability is the general limit equilibrium (GLE) method. A series of equations has been proposed by several authors, who agree in dividing the slide zone into slices. The primary differences are related to the equations that these seek to satisfy and to the differential forces influencing each slice, including the existing relationship between shear and normal forces. The foundation of this method is based on two equations that determine the factor of safety and evaluate the relationship between normal and shear forces [34, 35]. Thus, one equation provides the factor of safety with respect to the equilibrium of moments (*Fm*) and the other with respect to horizontal forces (*Ff* ).

For the GLE, shear forces are determined according to the equation proposed by Morgenstern and Price [36]:

$$X = E\mathcal{X}f(\mathbf{x})\tag{14}$$

where

*m*

<sup>=</sup> ¶¶¶ (11)

<sup>=</sup> ¶ (12)

q

).

( ) (14)

*h*

where ∂*θ* / ∂*h <sup>m</sup>* is the slope of the relationship between the volumetric water content and the suction head, which can be directly determined from the SWCC. The slope denotes the specific moisture capacity, which is typically denoted as *C*. As the soil storage function is not linear, it is necessary to describe the specific moisture capacity as a function of the suction or suction

*m*

*h* ¶q

If Eqs. (11) and (12) are substituted in Eq. 10, the expression that describes transient seepage

( ) ( ) ( )( 1) ( ) *mm m x m y m z m m hh h k h k h k h C h x xy yz z t* ¶ ¶¶ ¶¶ ¶ ¶ é ù é ù é ù

the hydraulic conductivity function that are specific to the material being studied.

equilibrium of moments (*Fm*) and the other with respect to horizontal forces (*Ff*

Eq. (13) is known as the Richards [33] equation, where given the boundary limits and initial conditions specific to a system, the equation provides the values for suction across space and time. It is highlighted that in using this equation, it is necessary to have data on the SWCC and

One of the most common methods for evaluating slope stability is the general limit equilibrium (GLE) method. A series of equations has been proposed by several authors, who agree in dividing the slide zone into slices. The primary differences are related to the equations that these seek to satisfy and to the differential forces influencing each slice, including the existing relationship between shear and normal forces. The foundation of this method is based on two equations that determine the factor of safety and evaluate the relationship between normal and shear forces [34, 35]. Thus, one equation provides the factor of safety with respect to the

For the GLE, shear forces are determined according to the equation proposed by Morgenstern

*X E fx* = l

<sup>+</sup> ê ú + += ê ú ê ú ¶ ¶¶ ¶¶ ¶ ¶ ë û ë û ë û (13)

*m*

*t ht* ¶ ¶¶ q

( ) *<sup>m</sup>*

*C h*

in unsaturated soils can be expressed as follows:

174 Groundwater - Contaminant and Resource Management

**4. Slope stability analyses**

and Price [36]:

head:

 q


For the GLE, the factor of safety is determined with respect to the method of moments and is determined by

$$F\_{\mu} = \frac{\sum [c'\beta R + (N - \mu\beta)R\tan\phi']}{\sum W\chi - \sum Nf - Dd} \tag{15}$$

However, the factor of safety with respect to the balance of forces is determined by

$$F\_f = \frac{\sum \left[ c' \beta \cos \alpha + (N - \mu \beta) \tan \varphi' \cos \alpha \right]}{\sum N \sin \alpha - \sum D \cos \alpha} \tag{16}$$

where

*c* ' is the effective cohesion

*φ*' is the effective friction angle

*u* is the pore pressure

*N* is the normal force at base of slice

*W* is the weight of slice

*D* is the load line

*β*, *R*, *x*, *f* , *d*, *ω* are the geometric parameters

*α* is the slope.

One important factor that is relevant for the previous equations is the normal force *N* , which is a term defined as

$$N = \frac{W = X\_R - X\_L - \frac{\alpha^\prime \mathcal{J} \sin \alpha + u \mathcal{J} \sin \alpha \tan \varphi^\prime}{F}}{\cos \alpha + \frac{\sin \alpha \tan \varphi^\prime}{F}} \tag{17}$$
