**3. Computer programs and general recommendations for different types of analysis**

Numerical techniques are currently preferred because of their ability to solve complex problems in which Eqs. (2) and (5) can be generalized to heterogeneous media with anisotropic materials and boundary conditions of variable complexity [34, 35]. The general methodology for performing a steady- or transient-state water flow analysis is illustrated in **Figure 18**, which shows that some of the most important data for performing water flow analysis are the hydraulic parameters of the materials, which must be obtained from field or laboratory tests that unfortunately are not always carried out because of time or cost. In addition, some tests require specialized personnel and equipment, such as the tests to determine the hydraulic parameters of unsaturated materials, including the soil–water characteristic curve (SWCC). The necessary hydraulic functions of the soil for analyzing unsaturated soils are as follows:


**Figures 19** and **20** show several general considerations for different types of water flow analyses, which can be performed relatively easily and rapidly using any of the existing specialized algorithms. **Figure 21** summarizes some of the most popular programs that are used to numerically solve water flow problems. One benefit of computer programs is their ability to facilitate the study of transient flow and unsaturated soil conditions, which are difficult and laborious to solve analytically.

Several programs for water flow analysis consider soil classification systems that are different from the Unified Soil Classification System (USCS, which is commonly used in geology, soil mechanics, and geotechnical engineering) because they involve parameters that are used to study unsaturated soils, such as:


**Figure 17.** Standard deviation of the (dimensionless) hydraulic gradient under a spillway structure in a medium of two

**3. Computer programs and general recommendations for different types of**

Numerical techniques are currently preferred because of their ability to solve complex problems in which Eqs. (2) and (5) can be generalized to heterogeneous media with anisotropic materials and boundary conditions of variable complexity [34, 35]. The general methodology for performing a steady- or transient-state water flow analysis is illustrated in **Figure 18**, which shows that some of the most important data for performing water flow analysis are the hydraulic parameters of the materials, which must be obtained from field or laboratory tests that unfortunately are not always carried out because of time or cost. In addition, some tests require specialized personnel and equipment, such as the tests to determine the hydraulic parameters of unsaturated materials, including the soil–water characteristic curve (SWCC). The necessary hydraulic functions of the soil for analyzing unsaturated soils are as follows: **–** *The water retention curve*(**Figure 27**) is also known as the *soil–water characteristic curve*(SWCC) depending on whether the suction is expressed in terms of the degree of saturation or the volumetric water content, respectively. The SWCC is broadly defined as the relationship

**–** *The hydraulic conductivity function* (**Figure 28**) represents the suction as a function of the

**Figures 19** and **20** show several general considerations for different types of water flow analyses, which can be performed relatively easily and rapidly using any of the existing specialized algorithms. **Figure 21** summarizes some of the most popular programs that are used to numerically solve water flow problems. One benefit of computer programs is their

between the amount of water in the soil and soil suction.

stratified isotropic materials [24].

104 Groundwater - Contaminant and Resource Management

**analysis**

permeability.

These systems imply that it is not advisable to use only the standard parameters that the computer programs include for certain types of materials (e.g., sand, clay, silt) because of the variations in the characteristics of soils (European, USA, or Dutch). It is preferable to assign the necessary hydraulic parameters based on the type of analysis and use values that are obtained from laboratory or field tests of the materials of the earth structure or soil that is being studied. In recent years, a comprehensive database that contains the hydraulic parameters of different types of soils from around the world has been developed [36]. Additionally, some algorithms [27] use granulometric curves as well as the index properties of the materials in the flow region and various mathematical expressions to estimate the hydraulic functions that are needed for the analyses (**Figures 27** and **28**). Some of the main mathematical models that are used to obtain the soil hydraulic functions are as follows[37]:


**Figures 25**–**31** show a case of the analysis of water flow through a cofferdam for *La Yesca* dam in Mexico composed of graduated materials assuming that the soil in one part of the cofferdam is partially saturated [37].

The PLAXFLOW algorithm [25] solves transient-state flow problems using the finite element method (FEM) by means of an approximate solution to Eq. (5) and by representing the flow in unsaturated soils with the *Van Genuchten* model. This algorithm performs analyses of transientstate flow in two different ways (**Figure 22**): (a) with step-wise conditions, in which each phase is defined by constant boundary conditions; that is, each time period is associated with a certain water level and (b) with time-dependent conditions, in which the continuous variation of the water level is explicitly considered as a function of time, which can be represented by linear functions, harmonic functions, or data in tables.

**Figures 23** and **24** show the results of water drawdown analyses that were carried out using the SEEP/W [26] and PLAXFLOW [25] algorithms, respectively [35, 43].

**Figure 18.** General methodology for steady- or transient-state flow analyses.

**Figure 19.** Data for water analyses.

**Figure 20.** General considerations for different types of water flow analyses.

**Figure 21.** Some algorithms for the numerical modelling of groundwater flow.

**Figure 18.** General methodology for steady- or transient-state flow analyses.

106 Groundwater - Contaminant and Resource Management

**Figure 20.** General considerations for different types of water flow analyses.

**Figure 19.** Data for water analyses.

**Figure 22.** Types of transient flow analyses with the PLAXFLOW algorithm [25].

**Figure 23.** Results of a drawdown analysis [43] using the SEEP/W algorithm [26].

**Figure 24.** Lines of drawdown at different times during a water drawdown [35] obtained using the PLAXFLOW algo‐ rithm [25].

**Figure 25.** Geometry and materials of a cofferdam for La Yesca Dam in Mexico [37].

**Figure 26.** Boundary conditions for numerical analysis of the cofferdam for La Yesca dam in Mexico [37].

**Figure 23.** Results of a drawdown analysis [43] using the SEEP/W algorithm [26].

108 Groundwater - Contaminant and Resource Management

**Figure 25.** Geometry and materials of a cofferdam for La Yesca Dam in Mexico [37].

**Figure 26.** Boundary conditions for numerical analysis of the cofferdam for La Yesca dam in Mexico [37].

rithm [25].

**Figure 24.** Lines of drawdown at different times during a water drawdown [35] obtained using the PLAXFLOW algo‐

**Figure 27.** Characteristic curves for materials from the cofferdam for La Yesca Dam in Mexico, used in this analysis [37].

**Figure 28.** Hydraulic conductivity functions utilized in the numerical model of the cofferdam for La Yesca Dam in Mexico [37].

**Figure 29.** Finite element mesh [37] generated with the SVFlux algorithm [27].

**Figure 30.** Distribution of the pore water pressure [37] calculated using the SVFlux algorithm 27.

**Figure 31.** Flow net [37] obtained using the SVFlux algorithm [27].

#### **4. Conclusions**

An exact analytical solution of a water flow problem is generally only possible to obtain when the geometry of the flux domain and the hydraulic and boundary conditions are simple (isotropic and homogeneous media). Exact solutions are difficult and impractical to obtain in the case of the most complex flow problems, such as those in practical geotechnical engineer‐ ing. In consequence, approximate solutions are usually sought. Numerical techniques are currently preferred over other methods due to their ability to solve complex problems in which the equations for water flow analyses can be generalized to heterogeneous media with anisotropic materials and boundary conditions of varying complexity. However, more sophisticated analyses require the use of a greater number of material parameters, which involves laboratory and field testing that require specialized knowledge and personnel. A series of mathematical models is available to approximate these material parameters. How‐ ever, the results of numerical analysis should be compared to measurements from monitoring (field instrumentation) of the structures being studied.

Some important comments about the numerical analyses of water flow are as follows:

