**4. Analysis of seepage forces and their effect in slope stability**

There are several practical cases where it is necessary to consider the forces produced by the water flow for the slope stability analysis of an earth structure. In the case of earth dams and levees, the water flow conditions that might occur and have to be consider for slope stability analysis are the following: a) transient flow that occurs during the first filling or a rapid drawdown conditions; b) constant flow which occurs sometimes after the reservoir is operating under regular water flow conditions; c) anisotropic water flow when the horizontal permeability is different than the vertical one. These three conditions might be more complicated when seismic forces have to be considered.

The water flow effects on the stability of an earth structure are the following: a) internal soil erosion o piping by removing and transporting soil particles, starting a duct that might increase rapidly, producing a complete failure; b) water pressure increase that will decrease the effective stresses and therefore decrease in the shear strength of the soil; c) increment on the water flow forces due to gravity might significantly decrease the safety factor and produce a slope failure.

Using either the graphical or numerical analysis, as it is explained later in this chapter, it is possible to obtain the hydraulic gradient at any point of the flow region.

For the most common practical cases that exist in earth dams and levees, Flores Berrones et al. (2003) have demonstrated that the water flow analysis can be reduced to a two dimensional system, so equation (4) is the one that must be considered for steady-state conditions:

$$\frac{\partial^2 \hbar}{\partial \mathbf{x}^2} + \frac{\partial^2 \hbar}{\partial y^2} = 0 \tag{4}$$

Internal Erosion Due to Water Flow Through Earth Dams and Earth Structures 291

Where *V* is the soil volume through which the water flow is taking place. If the hydraulic gradient is not constant but is a point function, the seepage force is the vector sum of all the forces applied in the volume of each element; for this case such seepage force is given by:

To illustrate the effect of the seepage force on soil erosion and slope stability, in Figure 7a are given the total, neutral and effective stress distribution of a saturated soil sample, where there is not a flow, whereas Figure 7b shows those stresses under the existence of an upward water flow. In the last figure (Fig. 7b), it can be observed the decrease of the effective stresses as a function of the water head *h*. It is important to notice that such stresses are nullified when:

> *m*'/

this case is known as critical hydraulic gradient, and when it takes place, the resistance friction forces of the soil particles against erosion also become zero. Under such conditions,

Fig. 7a. Total (), neutral (u) and effective () stress distribution in a soil sample without

Fig. 7b. Total (), neutral (u), and effective () stress distribution in a soil sample with an

the probability of soil erosion, particularly for fine cohesionless soils, is very high.

*<sup>w</sup>*, the effective stresses become null when *h*/*D* = 1. As it was mentioned before,

 *h* = *D*

any water flow (Flores et al., 2003)

upward water flow (Flores et al., 2003)

*m*' is the submerged weight of soil, and

Where 

soils *<sup>m</sup>*'  *<sup>V</sup> J idV* (8)

*<sup>w</sup>* (9)

*<sup>w</sup>* is the unit weight of water. Since for most

Fig. 6. Water flow forces over a soil element from the flow net (Flores et al., 2003)

This expression is the so-called Laplace's equation, which can be solved by different methods. The most common technique is the graphical solution to such equation, which is represented by two families of curves that intersect at right angles to form a pattern of square figures known as a *flow net*. In hydromechanics one set of these lines is called the *streamlines* or *flow lines*, and the other *equipotentials*. The flow net is constructed by setting first the boundary flow and equipotential conditions, and later on some additional flow lines are drawn in such a way that there will be, between each pair of flow lines, the same amount of water volume ∆*q*. The equipotential lines are constructed in such a way that there exist equal head losses, ∆*h*, between adjacent equipotential lines. Most flow nets are composed of curves that form curvilinear squares. A detail description to construct a flow net for any particular problem is given by Cedergren (1989) and Flores-Berrones (2000). From a flow net it is possible to obtain the total volume of water per unit of length at any part of the flow region, and also the water pressure, hydraulic gradient and flow velocity at any point of the studied domain.

On the other hand, the force over a soil element of a flow net, produced by a water flow, is analyzed in Figure 6. It can be observed in such figure that the force *J* per unit of length over the soil element is given by:

$$I = \Delta l \text{\textquotedbl{}\chi\_wA \textquotedbl{} (1) = \Delta l \text{\textquotedbl{}\chi\_wA \textquotedbl{}}\tag{5}$$

Where *A* is the cross-sectional area of the soil element, and *<sup>w</sup>* is the volumetric unit weight of water. The seepage force per unit of volume is:

$$\mathbf{j} = \frac{\Delta \mathbf{l} \boldsymbol{\gamma}\_w A}{cA} = \frac{\Delta \mathbf{l} \mathbf{l}}{c} \boldsymbol{\gamma}\_w = \mathbf{i} \boldsymbol{\gamma}\_w \tag{6}$$

Where *i* is the hydraulic gradient.

For regions in which there exists a uniform water flow, with a constant hydraulic gradient, the seepage force is given by:

$$\mathbf{J} = \mathbf{i} \ \mathbf{y}\_w \mathbf{V} \tag{7}$$

Fig. 6. Water flow forces over a soil element from the flow net (Flores et al., 2003)

any point of the studied domain.

Where *i* is the hydraulic gradient.

the seepage force is given by:

 *J* = ∆*h* 

of water. The seepage force per unit of volume is:

 *J* = *i* 

Where *A* is the cross-sectional area of the soil element, and

the soil element is given by:

This expression is the so-called Laplace's equation, which can be solved by different methods. The most common technique is the graphical solution to such equation, which is represented by two families of curves that intersect at right angles to form a pattern of square figures known as a *flow net*. In hydromechanics one set of these lines is called the *streamlines* or *flow lines*, and the other *equipotentials*. The flow net is constructed by setting first the boundary flow and equipotential conditions, and later on some additional flow lines are drawn in such a way that there will be, between each pair of flow lines, the same amount of water volume ∆*q*. The equipotential lines are constructed in such a way that there exist equal head losses, ∆*h*, between adjacent equipotential lines. Most flow nets are composed of curves that form curvilinear squares. A detail description to construct a flow net for any particular problem is given by Cedergren (1989) and Flores-Berrones (2000). From a flow net it is possible to obtain the total volume of water per unit of length at any part of the flow region, and also the water pressure, hydraulic gradient and flow velocity at

On the other hand, the force over a soil element of a flow net, produced by a water flow, is analyzed in Figure 6. It can be observed in such figure that the force *J* per unit of length over

*w A* (1) = ∆*h*

*<sup>w</sup> w w h A <sup>h</sup> j i cA c* 

For regions in which there exists a uniform water flow, with a constant hydraulic gradient,

 (6)

*w V* (7)

*<sup>w</sup> A* (5)

*<sup>w</sup>* is the volumetric unit weight

Where *V* is the soil volume through which the water flow is taking place. If the hydraulic gradient is not constant but is a point function, the seepage force is the vector sum of all the forces applied in the volume of each element; for this case such seepage force is given by:

$$J = \int\_{V} i dV \tag{8}$$

To illustrate the effect of the seepage force on soil erosion and slope stability, in Figure 7a are given the total, neutral and effective stress distribution of a saturated soil sample, where there is not a flow, whereas Figure 7b shows those stresses under the existence of an upward water flow. In the last figure (Fig. 7b), it can be observed the decrease of the effective stresses as a function of the water head *h*. It is important to notice that such stresses are nullified when:

$$h = D \,\gamma\_m \,' \,\!/\gamma\_w \tag{9}$$

Where *m*' is the submerged weight of soil, and *<sup>w</sup>* is the unit weight of water. Since for most soils *<sup>m</sup>*' *<sup>w</sup>*, the effective stresses become null when *h*/*D* = 1. As it was mentioned before, this case is known as critical hydraulic gradient, and when it takes place, the resistance friction forces of the soil particles against erosion also become zero. Under such conditions, the probability of soil erosion, particularly for fine cohesionless soils, is very high.

Fig. 7a. Total (), neutral (u) and effective () stress distribution in a soil sample without any water flow (Flores et al., 2003)

Fig. 7b. Total (), neutral (u), and effective () stress distribution in a soil sample with an upward water flow (Flores et al., 2003)

Internal Erosion Due to Water Flow Through Earth Dams and Earth Structures 293

Figure 8a; and transient flow analyses in two different ways: 1) *Step-wise conditions* and, 2) *Time-dependent conditions* as illustrated in Figure 8b. The *Plaxflow* algorithm provides hydraulic potential field, flow velocity field, pore pressure, degree of saturation field,

> **-** Boundary conditions **-** Hydraulic conductivity

Hydraulic head, pore pressure, flow velocity, flow rate, but also degree of saturation, free surfaces, upper flow lines, etc.

(a)

introduced by tables.

Transient-state flow analysis

Step-wise conditions Time-dependent conditions

(b) Fig. 8. Types of flow analyses performed with *Plaxflow* algorithm (Lopez-Acosta & Auvinet,

**6. Procedures and practical recommendations for preventing damages due to** 

The design of an earth dam or a levee is based on analytical studies of the site of construction and on personal experience of the individual designer. At a given site, it is possible to design a variety of earth dams which would be both, economical and safe. The final design depends on the quantities, types and location of the soil available for constructing the embankment, as well as the size and shape of the valley and the nature of the foundation. Sherard et al. (1967) present several typical designs of earth and earth-rock

Nivel de agua (m)

Water level (m)

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44

It explicitly considers the continuous variation of water level, which can be represented by linear or harmonic functions of time, or by particular data of variation of water level

Time (day) Tiempo (días)

Steady-state flow analysis

**-** Geometry

of materials

among others, as exposed below.

Each stage is defined by constant

boundary conditions, that is, a specific water surface level corresponding to certain time interval is assumed in each phase.

2010)

**soil erosion** 

As it has been demonstrated by some authors (Cedergren, 1989; Flores-Berrones et al., 2003), seepage forces might decrease (or increase in some particular cases) the factor of safety on the stability of earth dams and levees.
