# Role of unsaturated soil mechanics in geotechnical engineering

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## Abstract

The understanding of unsaturated soil mechanics principles is of interest to a wide spectrum of geotechnical problems associated with soils above water table and compacted soils. This paper describes the stress state variables and constitutive equations based on the unsaturated soil mechanics principles. In addition, the basic concepts for characterization of unsaturated soils and measurements of matric suction (or negative pore-water pressures) are also explained. The application of unsaturated soil mechanics theories is illustrated through the use of capillary barrier system for minimizing rain infiltration into residual soil slopes.

## Keywords

Unsaturated soil Stress state variables Constitutive equations Soil–water characteristic curve Numerical analysis## Introduction

Significant changes in climatic conditions in past decades are affected mostly by the increase in global temperature. Many countries experienced longer days of rainfall with higher intensities. Despite numerous debates about the existence of global warming, over 97% of climatologists reached an agreement that the changes in global climatic conditions are affected mainly by human activities [9]. A study on one aspect of climate change by Kim et al. [24], Rahardjo et al. [46] and Strauch et al. [51] indicated that climate change resulted in higher rainfall intensity with lesser frequency of rainy days. In addition, more water vapour is retained in the atmosphere due to the increase in average air temperature, which causes an exponential increase in the water vapour carrying capacity of the air [37].

The magnitude of rainwater infiltration and the variations in the groundwater table during dry and rainy periods are affected significantly by soil properties. Hence, the fluctuations in pore-water pressures, and stability of residual soil slopes are greatly influenced by the mechanical and hydraulic behaviour of the unsaturated soil [25, 26, 28, 41]. Therefore, it is necessary to understand the unsaturated soil mechanics principles, characterization and analyses of unsaturated soils and measurements of matric suction or negative pore-water pressures. This paper presents the principles of the unsaturated soil mechanics, the related apparatuses for measurement of unsaturated soil properties and numerical analyses incorporating unsaturated soil mechanics as well as the application of the unsaturated soil mechanics in engineering practise.

## Unsaturated soil mechanics

Matric suction can be explained as a pore-water pressure with negative value with respect to the pore-air pressure. The changes in the water content within soil pores result in the changes in matric suction. Hence, the water flow within the unsaturated soil is affected by the less water-filled spaces among solid particles [55]. As a result, the permeability of the unsaturated soil varies with the changes in matric suction (called permeability function). The contractile skin generates a tension force on the soil particle when matric suction increases, resulting in the additional shear strength of the soil. In the opposite, the shear strength of the soil decreases when the water infiltration occurs since the matric suction decreases [14, 23].

Equations for saturated and unsaturated soil mechanics.

Summarized from Fredlund and Rahardjo [13]

Principle or equation | Saturated soil | Unsaturated soil | ||
---|---|---|---|---|

Stress state variables | \((\sigma - u_{w} )\) | (3) | \((\sigma - u_a) \ {\text{and}} \ (u_a - u_w)\) | (4) |

Shear strength | \(\tau = c^{\prime} \, + (\sigma - u_{w} ){ \tan }\phi^{\prime}\) | (5) | \(\tau = c^{\prime} + (u_{a} - u_{w} ){ \tan }\phi^{b} + (\sigma - u_{a} ){ \tan }\phi^{\prime}\) | (6a) |

\(c = c^{\prime} + (u_{a} - u_{w} ){ \tan }\phi^{b}\) | (6b) | |||

Flow law for water (Darcys’ law) | \(\begin{aligned} v_{w} &= - k_s (\partial h_{w} /\partial y) \hfill \\ h_{w} &= y + (u_{w} /\rho_{w} g) \hfill \\ \end{aligned}\) | (7) | \(\begin{aligned} v_{w} &= - k_{w} (u_{a} - u_{w} )(\partial h_{w} /\partial y) \hfill \\ h_{w} &= y + (u_{w} /\rho_{w} g) \hfill \\ \end{aligned}\) | (8) |

Unsteady state seepage | \(k_{s} = \left( {\frac{{\partial^{2} h_{w} }}{{\partial x^{2} }}} \right) + k_{s} \left( {\frac{{\partial^{2} h_{w} }}{{\partial y^{2} }}} \right) = m_{v} \rho_{w} g\frac{{\partial h_{w} }}{\partial t}\) | (9) | \(\frac{\partial }{\partial x}\left( {k_{w} \frac{{\partial h_{w} }}{\partial x}} \right) + \frac{\partial }{\partial y}\left( {k_{w} \frac{{\partial h_{w} }}{\partial y}} \right) = m_{2}^{w} \rho_{w} g\frac{{\partial h_{w} }}{\partial t}\) | (10) |

Slope stability based on limit equilibrium | ||||

Moment equilibrium | \(F_{m} = \frac{{\sum {\left[ {c^{\prime}\beta R + \{ N - u_{w} \beta \} R\tan \phi^{\prime}} \right]} }}{{\sum {Wx - \sum {Nf} } }}\) | (11) | \(F_{m} = \frac{{\sum {\left[ {c^{\prime}\beta R + \left\{ {N - u_{w} \beta \frac{{\tan \phi^{b} }}{{\tan \phi^{\prime}}}} \right\}R\tan \phi^{\prime}} \right]} }}{{\sum {Wx - \sum {Nf} } }}\) | (12) |

Force equilibrium | \(F_{f} = \frac{{\sum {\left[ {c^{\prime}\upbeta \cos \upalpha + \{ N - u_{w} \upbeta \} \tan \upphi^{\prime}\cos \upalpha } \right]} }}{{\sum {N\sin \upalpha } }}\) | (13) | \(F_{f} = \frac{{\sum {\left[ {c^{\prime}\upbeta \cos\upalpha + \left\{ {N - u_{w} \upbeta \frac{{\tan \upphi^{b} }}{{\tan \upphi^{\prime}}}} \right\}\tan \upphi^{\prime}\cos \upalpha } \right]} }}{{\sum {N\sin \upalpha } }}\) | (14) |

The variation of soil water content with respect to suction is defined by soil–water characteristic curve (SWCC). It is commonly presented in a graph of either gravimetric water content (\(w\)), volumetric water content (\(\theta_{w}\)) or degree of saturation (\(S\)) in the vertical axis against matric suction in the horizontal axis in a logarithmic scale [14]. Among the three possible representations of SWCC, SWCC in the form of degree of saturation (\(S\)-SWCC) incorporates soil volume change calculations and it is commonly assumed to be equal to pore-size distribution function. Thus, \(S\)-SWCC is normally adopted for representing probability of random pore connections [57], since the region within the pore-suction distribution function defines the degree of saturation.

An important parameter of the SWCC for unsaturated soil mechanics is the air-entry value (AEV), associated with matric suction where water starts to be drained out from the biggest soil pores [5]. Alternatively, AEV could be equivalently defined as the matric suction that breaks the meniscus formed by water surface tension in the largest pores [15]. The AEV has been found to depend on the grain size distribution of the soil. A larger proportion of fine particles implies smaller intra-particle pore spaces between soil particles, resulting in a higher AEV [4]. On the other hand, the residual suction refers to the suction beyond which there is no significant removal of water from the soil with the increase in suction.

^{6}kPa and zero water content, and \(\psi_{r}\) = Suction at which residual water content occurs.

Leong and Rahardjo [33] suggested that \(C\left( \psi \right) = 1\) in Eq. 15 to obtain a better SWCC fitting in the low-suction range (less than 500 kPa), although compromising the fitting accuracy at the higher suction range. In addition, due to the close relationship between SWCC and coefficient of permeability, \(k_{s}\) [7], it is possible to estimate the permeability function of unsaturated soils from the available SWCC and \(k_{s}\). Mualem [36] classified various prediction models for the permeability functions by different researchers into three groups: empirical, macroscopic, and statistical models. Leong and Rahardjo [34] later found that the statistical model is the most rigorous and yields the most accurate permeability function. The permeability function can be estimated statistically using the equation developed by Kunze et al. [32] and best-fitted using the equation developed by Leong and Rahardjo [34] to form a continuous permeability function curve.

## Measurement of unsaturated soil properties

Unsaturated soil properties play important roles in affecting the rate of rainwater infiltration and factor of safety variations with time. Therefore, it is important to characterize the hydraulic and mechanical properties of unsaturated soil with the appropriate apparatuses. The hydraulic properties, i.e., a soil–water characteristic curve (SWCC) and a permeability function are required for seepage analyses or analyses of water flow throughout soil pores with respect to variations in matric suctions. The mechanical properties, i.e. unsaturated shear strength is required for stability analyses of soil slope varying matric suctions.

### Soil–water characteristic curve

Matric suction is equivalent to the difference in pore-air and pore-water pressures. In the field, pore-air pressure is equivalent to zero at the atmospheric condition. Therefore, matric suction is also referred to as the negative pore-water pressure [47]. In SWCC tests, the matric suction is applied to the soil specimen through the application of pore-air pressure using the axis translation technique as introduced by Hilf [20]. Through this procedure, the applied pore-water pressure is kept at atmospheric pressure, resulting in pore-water pressure equal to zero. Therefore, the magnitude of the matric suction is equal to the applied air pressure. The conventional methods for measurement of SWCC commonly involve Tempe cell and pressure plate for a suction range up to 100 kPa and 1500 kPa, respectively [2].

^{3}).

### Permeability function

^{3}) over the mass change, \(q_{i}\) = the rate of water flow (m/s), \(A\) = the cross-sectional area (cm

^{2}) of the column, \(\Delta t_{i}\) = the interval of time between two measurements.

Based on Eq. 22, different values of coefficient of permeability can be calculated for each average soil suction obtained from the HYPROP at individual time intervals. The instantaneous value of permeability can be plotted against the corresponding soil suction to obtain the unsaturated permeability of the soil.

^{−1}.(kPa)

^{2}), \(T_{s}\) = surface tension of water (mN m

^{−1}), \(\rho_{w}\) = density of water (1000 kg/m

^{3}), \(g\) = value of gravitational acceleration (m/s

^{2}), \(\mu_{w}\) = absolute viscosity of water (Pa s), \(\theta_{s}\) = volumetric water content at saturation, \(p\) = a constant value to represent the relationship of pores under different sizes, \(N\) = the number of intervals calculated from zero volumetric water content to the saturated volumetric water content,\(j\) = a counter from “\({\text{i}}\)” to “\({\text{m}}\)”, \(m\) = the number of intervals from the saturated volumetric water content, \(\theta_{s}\) (where \({\text{i}}\) = 1) to the lowest volumetric water content, \(\theta_{L}\) (where \({\text{i}}\) = \({\text{m}}\)); \(u_{a} - u_{w}\) = matric suction corresponding to the \(j{\text{th}}\) interval.

### Unsaturated shear strength

^{3}/s [43]. The deviator stress is simultaneously maintained constant and therefore, there is no shear strain applied to the specimen. Nanyang Technological University (NTU) mini suction probes can be used to measure matric suctions in a soil specimen during the tests [43]. Note that during water injection to the specimen (i.e., infiltration), the pore-air pressure is under a drained condition.

## Application of unsaturated soil mechanics in slope stabilization

Numerous slope failures frequently occur in steep residual soil slopes with a deep groundwater table during rainfalls. A significant thickness of unsaturated soil zone above the groundwater table is a general characteristic of steep residual soil slopes. Negative pore-water pressures or matric suctions as a crucial part of the stability of residual soil slopes are needed to be maintained in a slope under varying climatic conditions and to be considered in the slope assessment. Infiltration of rainwater into the slope surface contributes to raising the groundwater table and decreasing matric suctions. The reduction of matric suction in unsaturated residual soils results in a decrease in shear strength of the soil along the potential slip surface [12]. As rainwater infiltration into soil slopes is the major cause of rainfall-induced landslides, it is important to protect the slope with preventive measures that can avert or minimize rainwater infiltration into the slope. One of the common applications of unsaturated soil mechanics to slope stabilization is in the design and construction of a capillary barrier system (CBS) as a slope cover to minimize rain infiltration into slopes.

### Capillary barrier system (CBS)

In order to ensure the effectiveness of CBS performance, the selection of CBS materials must be carried out carefully following three main criteria as studied by Rahardjo et al. [48]. Those controlling parameters in selecting CBS materials are: the water-entry value of the coarse-grained layer, the ratio between the water-entry value of the fine-grained layers and the coarse-grained layers (\(\psi_{w}\)-ratio), and the saturated coefficient permeability of the fine-grained layer. The \(\psi_{w}\)-ratio between fine- and coarse-grained layers should be higher than 10 in order to generate a good barrier effect and to reduce the rainwater infiltration into the coarse-grained layer. The water-entry value of the coarse-grained layer must be less than 1 kPa. The saturated permeability of the fine-grained layer must be higher than 10^{−5} m/s in order to ensure that water can be drained out rapidly from the fine-grained layer through lateral diversion.

Study by Tami et al. [52] indicated that additional criteria should be satisfied to ensure the effectiveness of CBS. The presence of fines content within the fine-grained layer should be low enough to maintain the steepness of SWCC and to ensure a large amount of water can be drained out from the fine-grained layer. Hence, the storage capacity of the fine-grained layer can return to normal quickly after the rain stops. In addition, the fine-grained layer can be prevented from the development of cracks if the presence of fines content is low, especially during dry period when matric suctions are high.

## Numerical analyses of slope stability with and without capillary barrier system

Comprehensive numerical analyses were carried out to illustrate the stability of unsaturated soil slopes subjected to a prolonged rainfall. Typical soil slopes with and without a slope cover CBS were compared to study their performance under the prolonged rainfall. The slope is composed of the residual soil from the sedimentary Jurong Formation. The CBS is comprised of fine recycled concrete aggregate (FRCA) and coarse recycled concrete aggregate (CRCA) as the capillary barrier components.

### Material properties of residual soil and CBS

Laboratory tests were carried out to determine saturated and unsaturated properties of the residual soil, fine recycled concrete aggregate (FRCA) and coarse recycled concrete aggregate (CRCA) used in this study.

Index properties of residual soil, fine RCA, and coarse RCA

Properties | Residual soil | FRCA | CRCA |
---|---|---|---|

USCS classification | CH | SP | GP |

Specific gravity, \(G_{s}\) | 2.61 | 2.57 | 2.66 |

Porosity, \(n\) | 0.440 | 0.390 | 0.437 |

Unit weight, \(\gamma_{t}\) (kN/m | 20.3 | 19.0 | 20.0 |

Natural water content, \(w\) (%) | 38 | 6.70 | 6.56 |

Optimum water content, \(w_{opt}\) (%) | 11 | NA | NA |

Hydraulic properties of residual soil, fine RCA, and coarse RCA

Properties | Residual soil | FRCA | CRCA |
---|---|---|---|

Saturated volumetric water content, \(\theta_{s}\) | 0.423 | 0.387 | 0.437 |

Air entry value, \(\psi_{a}\) (kPa) | 112.5 | 7.02 | 0.13 |

Water entry value, \(\psi_{w}\) (kPa) | 1500 | 50 | 0.8 |

Residual matric suction, \(\psi_{r}\) (kPa) | 1576 | 16.24 | 0.43 |

Residual volumetric water content, \(\theta_{r}\) | 0.042 | 0.001 | 0.031 |

Fredlund and Xing [15] parameters for predicting SWCC drying curve | |||

\({\text{a}}\) (kPa) | 1630 | 10 | 0.20 |

\({\text{n}}\) | 1.06 | 5 | 6 |

\({\text{m}}\) | 7 | 1.2 | 1.2 |

Saturated permeability, \(k_{s}\) (m/s) | 1.0 × 10 | 1.2 × 10 | 4.0 × 10 |

^{−6}m/s and 4.0 × 10

^{−3}m/s, respectively. The \(k_{s}\) of CRCA is higher than 1.0 × 10

^{−5}m/s to ensure that water can flow effectively through the fine-grained layer.

Mechanical properties of residual soils, fine RCA, and coarse RCA

Properties | Residual soil | FRCA | CRCA |
---|---|---|---|

Elastic modulus, \(E\) (kPa) | 6000 | 18,000 | 210,000 |

Poison’s ratio, \(\nu\) | 0.4 | 0.33 | 0.33 |

Effective cohesion, \(c'\) (kPa) | 5 | 0 | 0 |

Effective friction angle, \(\phi '\) (\(^\circ\)) | 28 | 34 | 35 |

Angle indicating the increase in shear strength due to the increase in suction, \(\phi^{b}\) (\(^\circ\)) | 15 | 0 | 0 |

### Seepage, deformation and stability analyses

Seepage analyses (i.e., SEEP/W) were conducted to compute the pore-water pressure changes due to rainfall infiltration. The computed pore-water pressure distributions were then used as an initial condition in both stress–strain analyses (i.e., SIGMA/W) to assess the slope deformation and slope stability analyses (i.e., SLOPE/W) to assess the factor of safety of the slope.

^{−6}m/s) was applied to the surface of the slope. Ponding was not allowed to occur on the ground surface, which means that the pore-water pressure on the ground surface would not be allowed to be higher than 0 kPa. The nodal flux, \(Q\), equal of zero was applied along the sides of the slope above the groundwater table and along the bottom of the slope to simulate no flow zone. The nodal flux, \(Q\), equal to zero with review was applied along the outer boundaries of the drain to drain out the collected water. A boundary condition equal to total head, \(h_{w}\), was applied along the sides of the slope below the groundwater table. The negative pore-water pressure in the slope was limited to 50 kPa to reflect the typical values measured in the field.

Second, stress–strain analyses were conducted using SIGMA/W [17] software for the load/deformation modelling of the slope. The finite element model was a two-dimensional plane strain model. The finite element mesh of the slope model in SEEP/W was imported to SIGMA/W. The pore-water pressure was computed by seepage analyses and used as initial pore-water pressure for load/deformation analyses. The fixed boundary condition of the slope model was assumed that horizontal displacement was fixed on both side boundaries and displacements in both directions were fixed on the bottom boundary of the model. The pore-water pressure distribution was selected for each time increment, and the corresponding deformation was calculated.

Third, slope stability analyses were conducted using SLOPE/W [17]. The finite element mesh and pore-water pressure distribution of the slope model in SEEP/W were imported to SLOPE/W. The typical saturated and unsaturated shear strengths for the residual soils were used in the slope stability analyses using Morgenstern-Price method. The pore-water pressure distribution was selected for each time increment and the corresponding factor of safety was calculated.

#### Results of numerical analysis

Figure 18 shows the relationship between the dimensionless displacement and elapsed time at different locations during the rainfall. To find the failure condition, two tangent lines were drawn on each curve in Fig. 18. Figure 18a shows that the dimensionless displacement of the original slope at three locations increased sharply after an elapsed time of 36 h, which can be considered as a condition near instability (corresponding FS was 1.12 as indicated in Fig. 15). On the other hand, the dimensionless displacement of the slope with CBS cover at three locations increased, but the rate of increase in the dimensionless displacement was not significant to find the failure condition.

The comparison between SLOPE/W and SIGMA/W results shows that time to failure from SIGMA/W analyses tends to be earlier than that from SLOPE/W analyses. This could be due to the fact that the limit equilibrium solution provides only one average FS for the critical slip surface without considering the stress–strain distribution in the slope due to rainfall infiltration, whereas the stress–strain analyses performed in this study considers the deformation characteristics from the stress–strain distribution in the slope at several points in the slope. Therefore, the stress–strain analyses can detect progressive failure and potential failure of the slope earlier. Therefore, the design criteria [21], require a specific FS of greater than 1.0 in the design of actual soil slopes. For example, the slopes in South Korea are designed using a national code with a FS of 1.3 [30].

The comparison between the original and the slope with CBS cover in SIGMA/W shows that the volumetric water content in unsaturated soils may increase under rainfall, inducing the change in permeability function rapidly. At the same time, the alteration of the hydraulic condition, in turn, changes the water flow within the slope and may also alter volumetric response of the soil. The CBS can control the rain water infiltration into the soil slope and SIGMA/W analyses can capture the coupling processes under the transient conditions. Consequently, unsaturated soil slopes are vulnerable to lose their overall stability due to the reduction in the shear strength caused by rain infiltration.

## Conclusions

- 1.
The principles and theories of the unsaturated soil mechanics are required for describing the behaviour of natural soils which are commonly observed in unsaturated conditions.

- 2.
Apparatuses, procedures and methodologies for unsaturated soil testing are available for characterization of unsaturated soil properties.

- 3.
Assessment of environmental changes on geotechnical structures such as slopes should incorporate the mechanics and properties of unsaturated soils in the numerical analyses.

- 4.
The capillary barrier system using the principles of unsaturated soil mechanics has been shown to be effective in minimizing rainwater infiltration into slopes and preventing slope failures.

## Recommendations

Unsaturated soil mechanics principles are also important in designing other slope preventive measures, such as: horizontal drains and vegetative cover. Future studies can be carried out to investigate the optimum design of horizontal drains in maintaining the positions of groundwater table during dry and rainy periods within different type of soil slopes. In addition, future research works can be carried out to study the best type of vegetations that are effective in preventing the rainfall-induced slope failures.

## Notes

### Acknowledgements

The work described in this paper is supported by the Housing and Development Board (HDB) and Nanyang Technological University, Singapore. The Authors acknowledge gratefully the research funding provided by Building and Construction Authority (BCA), Singapore for this presentation.

### Authors’ contributions

HR designed the study, YK carried out the numerical simulations and participated in the result analysis, AS conducted the laboratory experiments and prepared for the manuscript. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

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