**2. General theoretical foundations of unsaturated soils**

#### **2.1. Soil suction**

Several definitions exist that define soil suction and its significance [3–5]; however, for practical engineering applications, proposed definitions are either unsuitable or complex because they are largely based on thermodynamic concepts. In simpler terms, suction can be defined as a state of negative water pressure in the soil, which is influenced by several factors [6], including temperature, gravity, capillary effects, salt content, and electrical charge (van der Waals forces), among others.

Total suction, or simply suction, is composed of two variables: (a) matric suction and (b) osmotic suction. Matric suction describes the difference between air pressure and water pressure [6], whereas osmotic suction is defined as the negative pressure resulting from the effect of the dissolved salts in the water of the soil matrix. Osmotic suction is commonly disregarded due to its lesser influence on total suction in addition to the difficulty of separating these two variables.

#### **2.2. Water storage function or SWCC**

are not available because of the high costs of these tests or the requirements of

**•** 3D models have important advantages over 2D models because they can include irregular geometries of the structure under study, topographical configuration of soil, unsaturated flow conditions, more realistic boundary conditions representing the

**•** To get more representative numerical analyses, the unsaturated soil theory should

**•** In this chapter, it is demonstrated that the considerations exposed for modelling, estimation, and fitting of hydraulic functions of soil provide the necessary elements

**•** Computer programs facilitate the study of transient-state flow and unsaturated soil condition, whose analytical solutions are generally complicated and laborious.

**Keywords:** water flow, unsaturated soils, numerical analysis, coupled water flowslope stability analysis, soil-water characteristic curve, hydraulic conductivity func‐

Unsaturated soils are characterized by negative pore-water pressure. In practical terms, partially saturated and unsaturated are synonymous: both terms indicate a degree of satura‐ tion lower than one; however, in specific terms, unsaturated implies the introduction of a third phase (gaseous) to the two-phase system already present in the saturated soils (liquid and solid

Currently, computer programs are helpful for solving problems of water flow and facilitate both the study of transient-state flow and the characterization of unsaturated soils, which, from

To demonstrate the application of the theoretical foundations presented in this chapter, an analysis of a tailings dam is presented as a case study. These structures are generally found in an unsaturated state. Thus, the primary goals of this analysis are to describe the applied methodology and to establish criteria and recommendations that can be followed to solve

Several definitions exist that define soil suction and its significance [3–5]; however, for practical engineering applications, proposed definitions are either unsuitable or complex because they are largely based on thermodynamic concepts. In simpler terms, suction can be defined as a

an analytical standpoint, are complicated and laborious tasks [2].

**2. General theoretical foundations of unsaturated soils**

be considered for all those situations where the material is in this state.

specialized equipment and personnel to perform them.

environment, among others.

164 Groundwater - Contaminant and Resource Management

tion

**1. Introduction**

phases only) [1].

related problem types.

**2.1. Soil suction**

to carry out these analyses in a simple way.

The storage function, which is also defined by the soil-water characteristic curve (SWCC), describes the relationship between the water content of the soil (degree of saturation, gravi‐ metric, or volumetric water content) and the soil suction. The nature of the SWCC is directly associated with grain-size distribution and soil structure. Therefore, the water content-suction ratio varies as a function of the material type (**Figure 1**).

**Figure 1.** Soil-water characteristic curve (SWCC) for different soil types [7].

The SWCC represents a fundamental relationship that can be used to describe the behaviour of unsaturated soils [8]. Similarly, the SWCC can be used to determine other properties, such as the permeability and shear strength of soil and the volumetric changes of soil [9].

Several methods and empirical relationships have been proposed to describe the SWCC. Several of these models use relationships that depend on the air entry value, the residual water content, or the slope of the curve [10, 11]. Other procedures are based on the grain-size distribution curve in addition to other soil properties (volume-mass properties) or the use of statistical correlations between soil data and soil water content [9, 12, 13]. The reliability of the models depends on the quality and quantity of data used for the statistical correlations [14]. Other methods consider the pore-size distribution within the soil, which, in certain cases, can be determined from the grain-size distribution curve of the material [15–17].

#### *2.2.1. Interpretation of the SWCC*

The SWCC can be associated with three zones that describe the desaturation process of a soil (**Figure 2**). In addition, the SWCC allows for the saturated and residual water content to be determined, as well as their respective suction values, which are the main parameters that represent the SWCC as an input into one of the fitting methods mentioned in this chapter:

**Figure 2.** Zones corresponding to the soil-water characteristic curve (SWCC).

**•** *Saturated capillary zone*. The soil zone that is maintained in a saturated state, whose defining limit coincides with the air-entry value [18], which can be described as the value that the matric suction must exceed before air enters into the soil macropores.


## *2.2.2. Estimation methods of SWCC*

The SWCC represents a fundamental relationship that can be used to describe the behaviour of unsaturated soils [8]. Similarly, the SWCC can be used to determine other properties, such

Several methods and empirical relationships have been proposed to describe the SWCC. Several of these models use relationships that depend on the air entry value, the residual water content, or the slope of the curve [10, 11]. Other procedures are based on the grain-size distribution curve in addition to other soil properties (volume-mass properties) or the use of statistical correlations between soil data and soil water content [9, 12, 13]. The reliability of the models depends on the quality and quantity of data used for the statistical correlations [14]. Other methods consider the pore-size distribution within the soil, which, in certain cases, can

The SWCC can be associated with three zones that describe the desaturation process of a soil (**Figure 2**). In addition, the SWCC allows for the saturated and residual water content to be determined, as well as their respective suction values, which are the main parameters that represent the SWCC as an input into one of the fitting methods mentioned in this chapter:

**•** *Saturated capillary zone*. The soil zone that is maintained in a saturated state, whose defining limit coincides with the air-entry value [18], which can be described as the value that the

as the permeability and shear strength of soil and the volumetric changes of soil [9].

be determined from the grain-size distribution curve of the material [15–17].

**Figure 2.** Zones corresponding to the soil-water characteristic curve (SWCC).

matric suction must exceed before air enters into the soil macropores.

*2.2.1. Interpretation of the SWCC*

166 Groundwater - Contaminant and Resource Management

When laboratory data on the relationship between suction and water content are unavailable or insufficient, estimation models are one method used to determine the SWCC as a function of the index properties of soil (volume-mass and grain-size distribution relationships). Currently, there are several models that allow for the SWCC to be determined, including the following:


## *2.2.3. Fitting methods of SWCC*

Fitting methods are based on experimental or empirical equations that aim to define the SWCC according to data obtained from laboratory tests. Fitting methods are used when laboratory data are available, yet given the dispersion of the values, it is necessary to apply models to adjust or define the data trend to generate a representative curve for the material considered in the study.

Several models have been developed to define the SWCC [23], which consider fitting param‐ eters that provide a range of flexibility to represent distinct materials. In **Table 1**, several of the most common fitting methods for SWCC are listed. These equations consider gravimetric water content as the primary variable; however, the equations can also use volumetric water content or the degree of soil saturation as inputs [24].



**Table 1.** Experimental fitting models of the SWCC.

**Model Equation**

Gardner (1958) *ww* <sup>=</sup>*wr* <sup>+</sup> (*ws* <sup>−</sup>*wr*){ <sup>1</sup>

168 Groundwater - Contaminant and Resource Management

*ww* <sup>=</sup>*wr* <sup>+</sup> (*ws* <sup>−</sup>*wr*){ <sup>1</sup>

*ww* <sup>=</sup>*wr* <sup>+</sup> (*ws* <sup>−</sup>*wr*) *ac*

*ww* <sup>=</sup>*wr* <sup>+</sup> (*ws* <sup>−</sup>*wr*){ <sup>1</sup>

*ww* <sup>=</sup>*wr* <sup>+</sup> (*ws* <sup>−</sup>*wr*){ <sup>1</sup>

*ln*(1 <sup>+</sup> *<sup>ψ</sup> ψr* )

*ln*(1 <sup>+</sup> <sup>10</sup><sup>6</sup> *ψr*

*ws* saturated gravimetric water content *wr* residual gravimetric water content

value has been exceeded

value has been exceeded

value has been exceeded

*mvm*=1-1/*nvm*

*mvb*=1-2/*nvb*

1 + (*avbψ*)

1 + (*avmψ*)

1 + (*avgψ*)

) { <sup>1</sup> ln(*e* + ( *ψ a f* ) *nf* ) *mf* }

*avb* adjustment parameter that depends on the air-entry value of the soil

*ag* adjustment parameter derived from the air-entry value of the soil

*avm* adjustment parameter that depends on the air-entry value of the soil

*avg* adjustment parameter that depends on the air-entry value of the soil

*ag<sup>ψ</sup> ng* }

*ψ nc* *nvb mvb* }

*nvm mvm* }

*nvg mvg* }

*nvb* adjustment parameter that depends on the desaturation velocity of the soil once the air-entry

*mvb* adjustment parameter related to the residual water content of soil, which is considered to be

*ng* adjustment parameter that depends on the desaturation velocity of the soil once the air-entry

*nvm* adjustment parameter that depends on the desaturation velocity of the soil once the air-entry

*mvm* adjustment parameter related to the residual water content of soil, which is considered to be

Van Genuchten (1980)

Brooks and Corey (1964)

Van Genuchten and Mualem (1976)

Van Genuchten (1980)

Fredlund and

where

Xing (1994) *ww* =*ws* 1−

*ψ* soil suction

*ac* air pressure

*nc* soil pore size index

#### **2.3. Hydraulic conductivity function**

The hydraulic conductivity function represents the relationship between hydraulic conduc‐ tivity and soil suction and can be expressed as a function of the degree of saturation or volumetric water content of soil.

Several estimation methods are based on the SWCC and the saturated hydraulic conductivity at distinct suction intervals. These techniques can be classified into the following categories:


A summary of the main estimation methods expressed as a function of hydraulic conductivity is presented in **Table 2**.


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