**5. Graphical and numerical methods for analyzing the erosion problem**

Solving soil erosion problems involves the calculation of hydraulic gradients, seepage forces, water or pore pressure, flow velocities, flow rates, among other variables. The assessment of such properties is carried out by solving partial differential equations. For steady-state conditions, water flow is calculated by Laplace's equation (see eq. 4, applicable to homogeneous and isotropic soils). For transient flow conditions in a homogeneous and isotropic soil domain the following partial differential equation is used:

$$\operatorname{div}\Big[\operatorname{kg}\operatorname{rad}(h)\Big] + c\frac{\partial h}{\partial t} = Q \tag{10}$$

Where *k* is hydraulic conductivity of soil, *h* is hydraulic potential (also named hydraulic head), *c* is specific capacity of soil, *t* is elapsed time and *Q* is a discharge quantity corresponding to a possible source within the medium.

The above equations (eqs. 4 and 11) combine Darcy's law and continuity of flow. They can easily be generalized to the case of heterogeneous and anisotropic soils (Auvinet & Lopez-Acosta, 2010; Lam et al., 1987). In the case of partially saturated soils, specific capacity depends on porosity and degree of saturation. Deformability of soil skeleton is commonly ignored. At the same time, degree of saturation and permeability depend on local pressure (Van Genuchten, 1980).

The resolution of the above equations can be performed in an exact or an approximate form, by analytical or numerical techniques (Alberro, 2006; Cedergren, 1989; Lopez-Acosta et al., 2010; among others). Thus, the methods that can be used for evaluating steady and transient state flow conditions include:


In general, exact and analytical solutions are laborious when geometric, hydraulic and boundary conditions become complex. Approximate solutions are usually used. Nowadays, numerical methods are preferred with increasing frequency due to their easy adaptation and automation to widely varying conditions, and in general because of their capability for solving complex problems. Numerical methods have been applied by different authors (Auvinet & Lopez-Acosta, 2010; Freeze 1971; Huang & Jia, 2009; Lam & Fredlund 1984; Lam et al., 1987; Ng & Shi, 1998; among others). The present chapter focuses on the finite element method (FEM), using the *Plaxflow* algorithm (Delft University of Technology, 2007), a specialized computer program which is applied to solve steady and transient flow problems by means of the approximate solution of Laplace's equation and equation 11. This algorithm utilizes the previously mentioned Van Genuchten model to represent flow in unsaturated soils and allows carrying out steady-state analyses following the methodology indicated in

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

Solving soil erosion problems involves the calculation of hydraulic gradients, seepage forces, water or pore pressure, flow velocities, flow rates, among other variables. The assessment of such properties is carried out by solving partial differential equations. For steady-state conditions, water flow is calculated by Laplace's equation (see eq. 4, applicable to homogeneous and isotropic soils). For transient flow conditions in a homogeneous and

*<sup>h</sup> div kgrad h c Q <sup>t</sup>*

Where *k* is hydraulic conductivity of soil, *h* is hydraulic potential (also named hydraulic head), *c* is specific capacity of soil, *t* is elapsed time and *Q* is a discharge quantity

The above equations (eqs. 4 and 11) combine Darcy's law and continuity of flow. They can easily be generalized to the case of heterogeneous and anisotropic soils (Auvinet & Lopez-Acosta, 2010; Lam et al., 1987). In the case of partially saturated soils, specific capacity depends on porosity and degree of saturation. Deformability of soil skeleton is commonly ignored. At the same time, degree of saturation and permeability depend on local pressure

The resolution of the above equations can be performed in an exact or an approximate form, by analytical or numerical techniques (Alberro, 2006; Cedergren, 1989; Lopez-Acosta et al., 2010; among others). Thus, the methods that can be used for evaluating steady and transient


In general, exact and analytical solutions are laborious when geometric, hydraulic and boundary conditions become complex. Approximate solutions are usually used. Nowadays, numerical methods are preferred with increasing frequency due to their easy adaptation and automation to widely varying conditions, and in general because of their capability for solving complex problems. Numerical methods have been applied by different authors (Auvinet & Lopez-Acosta, 2010; Freeze 1971; Huang & Jia, 2009; Lam & Fredlund 1984; Lam et al., 1987; Ng & Shi, 1998; among others). The present chapter focuses on the finite element method (FEM), using the *Plaxflow* algorithm (Delft University of Technology, 2007), a specialized computer program which is applied to solve steady and transient flow problems by means of the approximate solution of Laplace's equation and equation 11. This algorithm utilizes the previously mentioned Van Genuchten model to represent flow in unsaturated soils and allows carrying out steady-state analyses following the methodology indicated in

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**5. Graphical and numerical methods for analyzing the erosion problem** 

isotropic soil domain the following partial differential equation is used:


corresponding to a possible source within the medium.

(Van Genuchten, 1980).

2009).

state flow conditions include:

the stability of earth dams and levees.

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, among others, as exposed below.

Fig. 8. Types of flow analyses performed with *Plaxflow* algorithm (Lopez-Acosta & Auvinet, 2010)
