3. Methodology

Quantitatively, the risk (R) can be defined in terms of the hazard P[T], understood as the total probability of a threatening event that happens, and the vulnerability P[C|T], understood as the conditional probability of damage considering that a failure has already occurred and the cost of the consequences C, by the equation:

$$R = P[T] \times P[\mathbb{C}/T] \times \mathbb{C} \tag{1}$$

to the amount of rainfall through so-called failure thresholds or numerical models with phys-

Effect of the Rainfall Infiltration Processes on the Landslide Hazard Assessment of Unsaturated Soils…

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167

The probabilities of failure Pfs and Pfns are calculated by reliability index (β1), as the probability

<sup>β</sup><sup>1</sup> <sup>¼</sup> E FOS ½ �� <sup>1</sup>

Pf ¼ Φ �β<sup>1</sup>

where E[FOS] is the deterministic value of FOS calculated with the mean values of the independent associate variables, and σ[FOS] is the standard deviation of FOS, considering that the critical value of FOS is 1.0. Φ is the standardized normal probability distribution. The β<sup>1</sup> index is related to the probability of failure, allowing a more consistent stability assessment. The most common way to assess the slope stability is using limit equilibrium methods with planar or circular surfaces. Particularly, for regional analysis, the concept of infinite slope is often used [7]. The resulting expression for infinite slope model in this work

cos β<sup>2</sup> tan ϕ

where H is the thickness of the failure zone [m], Hw is the water height measured from the failure surface [m], c is the soil cohesion [kPa], φ is the angle of internal friction of the soil

FOS <sup>¼</sup> <sup>c</sup> <sup>þ</sup> <sup>γ</sup><sup>H</sup> � <sup>γ</sup>wγHw

<sup>σ</sup>½ � FOS (3)

(4)

<sup>γ</sup><sup>H</sup> sin <sup>β</sup> cos <sup>β</sup> : (5)

].

], γ<sup>w</sup> is the unit weight of water [kN/m<sup>3</sup>

ical base to estimate the probability of saturation [19].

Figure 1. Schematic methodology adopted for hazard assessment.

that the factor of safety (FOS) is less than unity:

is presented as below:

�], γ is the unit weight of soil [kN/m<sup>3</sup>

[

This paper emphasizes landslide hazard through a probabilistic methodology for hazard assessment, which uses methods as first order second method (FOSM) and failure thresholds.

#### 3.1. Landslide hazard assessment

The methodology for the landslide hazard assessment was developed by [10, 18] through a calculation model based on FOSM. The methodology shown graphically in Figure 1 allows calculating the total probability of failure (TPF) according to the theorem of total probability of failure of a slope by the equation:

$$TPF = P[T] = P\_{\circ} \times P\_s + P\_{\text{fus}} \times (1 - P\_s) \tag{2}$$

where Pfs is the probability of slope failure due to the action of the rainfall in saturated condition, Pfns is the probability of failure where condition is not saturated, Ps is the marginal probability which the soil is in a saturation condition, and (1 � Ps) is the marginal probability which the soil is not in this condition. The slope probability of failure in both saturated and unsaturated condition usually can be calculated in an independent way. However, determining the probability that the soil is in a saturated condition is complicated, especially due to the complexity of the phenomenon that considers variation of the conditions of soil water content. The effect of accumulated rainfall and that the occurrence of landslides is possible to be related Effect of the Rainfall Infiltration Processes on the Landslide Hazard Assessment of Unsaturated Soils… http://dx.doi.org/10.5772/intechopen.70821 167

Figure 1. Schematic methodology adopted for hazard assessment.

Hence, the current approach involves several fronts to solve the complexity and uncertainty of

• A numerical modeling to solve the system of equations describing the failure mechanism due to rainfall. In this part, it is included numerical modeling of slope stability, considering the effects of infiltration process and spatial variability of geotechnical and hydraulic

• An experimental work in a laboratory with controlled conditions to evaluate how the

• An instrumentation of tests field for coupling of proposed theoretical and experimental models. This part will permit the validation of models under realistic conditions of geo-

These approaches clarify the effects of rainfall and its consequent infiltration in slope stability

Quantitatively, the risk (R) can be defined in terms of the hazard P[T], understood as the total probability of a threatening event that happens, and the vulnerability P[C|T], understood as the conditional probability of damage considering that a failure has already occurred and the

This paper emphasizes landslide hazard through a probabilistic methodology for hazard assessment, which uses methods as first order second method (FOSM) and failure thresholds.

The methodology for the landslide hazard assessment was developed by [10, 18] through a calculation model based on FOSM. The methodology shown graphically in Figure 1 allows calculating the total probability of failure (TPF) according to the theorem of total probability of

where Pfs is the probability of slope failure due to the action of the rainfall in saturated condition, Pfns is the probability of failure where condition is not saturated, Ps is the marginal probability which the soil is in a saturation condition, and (1 � Ps) is the marginal probability which the soil is not in this condition. The slope probability of failure in both saturated and unsaturated condition usually can be calculated in an independent way. However, determining the probability that the soil is in a saturated condition is complicated, especially due to the complexity of the phenomenon that considers variation of the conditions of soil water content. The effect of accumulated rainfall and that the occurrence of landslides is possible to be related

R ¼ P T½ �� P C½ �� =T C (1)

TPF ¼ P T½ �¼ Pfs � Ps þ Pfns � ð Þ 1 � Ps (2)

technical, geomorphologic and hydraulic parameters, and rainfall patterns.

propagation of water flows inside an unsaturated soil.

of unsaturated deposits of tropical mountainous regions.

cost of the consequences C, by the equation:

3.1. Landslide hazard assessment

failure of a slope by the equation:

the addressed problem:

166 Engineering and Mathematical Topics in Rainfall

parameter of soil.

3. Methodology

to the amount of rainfall through so-called failure thresholds or numerical models with physical base to estimate the probability of saturation [19].

The probabilities of failure Pfs and Pfns are calculated by reliability index (β1), as the probability that the factor of safety (FOS) is less than unity:

$$\beta\_1 = \frac{E[FOS] - 1}{\sigma[FOS]} \tag{3}$$

$$P\_f = \mathcal{O}\left(-\beta\_1\right) \tag{4}$$

where E[FOS] is the deterministic value of FOS calculated with the mean values of the independent associate variables, and σ[FOS] is the standard deviation of FOS, considering that the critical value of FOS is 1.0. Φ is the standardized normal probability distribution. The β<sup>1</sup> index is related to the probability of failure, allowing a more consistent stability assessment. The most common way to assess the slope stability is using limit equilibrium methods with planar or circular surfaces. Particularly, for regional analysis, the concept of infinite slope is often used [7]. The resulting expression for infinite slope model in this work is presented as below:

$$FOS = \frac{\varepsilon + \left(\gamma H - \gamma\_w \gamma H\_w\right) \cos \beta^2 \tan \phi}{\gamma H \sin \beta \cos \beta}. \tag{5}$$

where H is the thickness of the failure zone [m], Hw is the water height measured from the failure surface [m], c is the soil cohesion [kPa], φ is the angle of internal friction of the soil [ �], γ is the unit weight of soil [kN/m<sup>3</sup> ], γ<sup>w</sup> is the unit weight of water [kN/m<sup>3</sup> ].

In mountainous tropical regions, landslides occur most often in rainy seasons in which increased soil saturation with consequent decrease in their cohesion and increased pore pressure are presented. The process of decrease in the shear strength due to changes in water content is a highly complex process, which is not considered in the development of this study. Therefore, the effect of saturation is taken into consideration only in the increase of the water pressure, and for purposes of analysis in this study, two situations were considered for the water height measured from the failure surface (Hw), one where the water level presented in the most critical condition was considered, i.e., Hw = H to obtain Pfs and another favorable in which Hw = 0 to obtain Pfns. The eventual saturation condition of the soil is a random phenomenon that must be taken into consideration in the evaluation of the probability of landslides. In this case, it was considering the probability that the soil is saturated or not.

#### 3.2. Probability of soil saturation (PSS) by rainfall thresholds

Slope stability in tropical areas is highly affected by rainfall but also depends on soil properties such as shear strength and hydromechanical properties. It has been identified that highintensity rainfall affect slopes in well-drained residual soils (permeability coefficient of saturated soil ks ≥ 10–<sup>4</sup> m/s), while low intensity and long duration rains mainly affect poorly drained slopes (ks ≤ 10–<sup>6</sup> m/s), rainfall in which the intensity match ks [20, 21].

In the other hand, it has been identified that the relationship between rainfall and landslides is influenced by conditions of preceding rainfall or accumulated rain on the ground before the triggering event [22–32], named thresholds of failure. There are proposed thresholds of failure in terms of the intensity of rainfall and the accumulated rainfall of several days [25], but in areas where insufficient rainfall information is available, thresholds have been set in terms of accumulated rainfall with precedent rainfall of 15, 30, 60, and 90 days and different durations of triggering events as 1, 3, or 5 days [22, 23]. Jaiswal and Van Westen [3, 27] conducted research in which empirical thresholds were used to estimate the probability of failure in slopes of roadways of southern India using the Poisson distribution.

The thresholds of failure can be used to assess the triggering effect of rainfall on landslides and the landslide hazard. Eq. (6) shows the relationship between rainfall and landslides in natural slopes of "La Iguana" river basin in the city of Medellin, Colombia, for assessing the landslide hazard [29]:

$$A = R\_3 - 60 + 0.55R\_{15} \tag{6}$$

R<sup>3</sup> ¼ 75 � 0:5R<sup>15</sup> (7)

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169

The similarity between Eqs. (6) and (7) results logical because "La Iguaná" river basin is located in the department of Antioquia, but the differences are associated with the use of different periods of time and geological considerations in the analysis. "La Iguaná" basin has high human intervention due to its proximity to hard urbanized areas of the city of Medellin. The threshold proposed by [30], which is presented in Eq. (7), is used by the early warning system of the Aburra Valley, Colombia-SIATA, to define the level of hazard [31]. In other works, has been identified as the most important constraint for the occurrence of landslides in the Aburra Valley are the long-term accumulative rainfall of around 60 mm in 30 days, 160 mm

Effect of the Rainfall Infiltration Processes on the Landslide Hazard Assessment of Unsaturated Soils…

For landslide hazard on roadways, Hidalgo and Assis [32, 33] presented a threshold as a relationship between precedent rainfall of 15 days and antecedent rainfall of 5 days. For precedent rainfall of 30 days, they reported exceedance rates greater than in the cases of 15 and 60 days. Considering that accumulated rainfall for 15 and 30 days are easier to obtain, thresholds based on rainstorms of 5-days duration and previous rainfall of 30 and 15 days are proposed, as shown in Figure 2. It is important to highlight that this relationship was determined with rainfall data of days when there were landslides of the manmade slopes of the roadway. Due to this, this threshold is lower than those defined by Eqs. (6) and (7) despite

In a methodology presented by [1], the thresholds are used to asses Ps. It is accepted that the condition given by the failure threshold represents a saturation condition conducive to landslides, with the already mentioned reduction of shear strength of the material due to the decrease in suction and pressure generation of pores [1, 20, 32]. The probability of saturation is calculated as the probability that the ordered pair accumulated rainfall preceding of 15 days

Figure 2. Combinations of total accumulated rainfall of 5 and 15 days, for days with landslide events [35].

in 60 days, and 200 mm for 90 days by the seasonal rainfall.

being located in the same river basin "La Iguana" [34].

where A is the hazard of landslide triggered by rainfall, R<sup>3</sup> is the 3-days antecedent rainfall, and R<sup>15</sup> is the accumulated rainfall of 15 days preceding the R3.

Eq. (6) was determined with 40 records of landslides and rainfall data of the meteorological station "San Cristobal" of the de Medellin public services company in the period 1980–2001. This threshold was exceeded by 95% of data processed. Later, [30] studied the relationship between rainfall and landslides in the department of Antioquia, Colombia, for the time period 1974–1998. With a total of 283 landslides, a threshold has been determined according to the equation:

Effect of the Rainfall Infiltration Processes on the Landslide Hazard Assessment of Unsaturated Soils… http://dx.doi.org/10.5772/intechopen.70821 169

$$R\_3 = 75 - 0.5R\_{15} \tag{7}$$

The similarity between Eqs. (6) and (7) results logical because "La Iguaná" river basin is located in the department of Antioquia, but the differences are associated with the use of different periods of time and geological considerations in the analysis. "La Iguaná" basin has high human intervention due to its proximity to hard urbanized areas of the city of Medellin. The threshold proposed by [30], which is presented in Eq. (7), is used by the early warning system of the Aburra Valley, Colombia-SIATA, to define the level of hazard [31]. In other works, has been identified as the most important constraint for the occurrence of landslides in the Aburra Valley are the long-term accumulative rainfall of around 60 mm in 30 days, 160 mm in 60 days, and 200 mm for 90 days by the seasonal rainfall.

In mountainous tropical regions, landslides occur most often in rainy seasons in which increased soil saturation with consequent decrease in their cohesion and increased pore pressure are presented. The process of decrease in the shear strength due to changes in water content is a highly complex process, which is not considered in the development of this study. Therefore, the effect of saturation is taken into consideration only in the increase of the water pressure, and for purposes of analysis in this study, two situations were considered for the water height measured from the failure surface (Hw), one where the water level presented in the most critical condition was considered, i.e., Hw = H to obtain Pfs and another favorable in which Hw = 0 to obtain Pfns. The eventual saturation condition of the soil is a random phenomenon that must be taken into consideration in the evaluation of the probability of landslides. In

Slope stability in tropical areas is highly affected by rainfall but also depends on soil properties such as shear strength and hydromechanical properties. It has been identified that highintensity rainfall affect slopes in well-drained residual soils (permeability coefficient of saturated soil ks ≥ 10–<sup>4</sup> m/s), while low intensity and long duration rains mainly affect poorly

In the other hand, it has been identified that the relationship between rainfall and landslides is influenced by conditions of preceding rainfall or accumulated rain on the ground before the triggering event [22–32], named thresholds of failure. There are proposed thresholds of failure in terms of the intensity of rainfall and the accumulated rainfall of several days [25], but in areas where insufficient rainfall information is available, thresholds have been set in terms of accumulated rainfall with precedent rainfall of 15, 30, 60, and 90 days and different durations of triggering events as 1, 3, or 5 days [22, 23]. Jaiswal and Van Westen [3, 27] conducted research in which empirical thresholds were used to estimate the probability of failure in

The thresholds of failure can be used to assess the triggering effect of rainfall on landslides and the landslide hazard. Eq. (6) shows the relationship between rainfall and landslides in natural slopes of "La Iguana" river basin in the city of Medellin, Colombia, for assessing the landslide

where A is the hazard of landslide triggered by rainfall, R<sup>3</sup> is the 3-days antecedent rainfall,

Eq. (6) was determined with 40 records of landslides and rainfall data of the meteorological station "San Cristobal" of the de Medellin public services company in the period 1980–2001. This threshold was exceeded by 95% of data processed. Later, [30] studied the relationship between rainfall and landslides in the department of Antioquia, Colombia, for the time period 1974–1998. With a total of 283 landslides, a threshold has been determined according to the

A ¼ R<sup>3</sup> � 60 þ 0:55R<sup>15</sup> (6)

this case, it was considering the probability that the soil is saturated or not.

drained slopes (ks ≤ 10–<sup>6</sup> m/s), rainfall in which the intensity match ks [20, 21].

slopes of roadways of southern India using the Poisson distribution.

and R<sup>15</sup> is the accumulated rainfall of 15 days preceding the R3.

hazard [29]:

equation:

3.2. Probability of soil saturation (PSS) by rainfall thresholds

168 Engineering and Mathematical Topics in Rainfall

For landslide hazard on roadways, Hidalgo and Assis [32, 33] presented a threshold as a relationship between precedent rainfall of 15 days and antecedent rainfall of 5 days. For precedent rainfall of 30 days, they reported exceedance rates greater than in the cases of 15 and 60 days. Considering that accumulated rainfall for 15 and 30 days are easier to obtain, thresholds based on rainstorms of 5-days duration and previous rainfall of 30 and 15 days are proposed, as shown in Figure 2. It is important to highlight that this relationship was determined with rainfall data of days when there were landslides of the manmade slopes of the roadway. Due to this, this threshold is lower than those defined by Eqs. (6) and (7) despite being located in the same river basin "La Iguana" [34].

In a methodology presented by [1], the thresholds are used to asses Ps. It is accepted that the condition given by the failure threshold represents a saturation condition conducive to landslides, with the already mentioned reduction of shear strength of the material due to the decrease in suction and pressure generation of pores [1, 20, 32]. The probability of saturation is calculated as the probability that the ordered pair accumulated rainfall preceding of 15 days

Figure 2. Combinations of total accumulated rainfall of 5 and 15 days, for days with landslide events [35].

(R15) and, accumulated rainfall antecedent of 3 days (R3), is above the failure threshold line for study area, i.e., soil is considered to be saturated if the relationship of the equation is true:

$$R\_{3m} \cong R\_3 \tag{8}$$

recommended only for preliminary assessments, and for special cases, when a good amount of

Effect of the Rainfall Infiltration Processes on the Landslide Hazard Assessment of Unsaturated Soils…

In recent years, slope stability analyses have been expanded to include coupled hydromechanical processes under variably saturated conditions. These analyses incorporate the variation of saturation, leading to more accurate assessments of slopes stability (under infiltration conditions), and demonstrate that a better physical representation of water flow and stress can be attained in unsaturated soils [17]. Hence, the analysis of seepage and coupled stressdeformation should be linked simultaneously [13]. Some recent studies are specifically focused on infiltration-induced landslides. However, most of these studies only consider slope failure below the groundwater table, overlooking the contribution of effective stress (suction stress) to the strength of the soil under transient unsaturated flow conditions [17]. Generally, these

K hð Þ <sup>∂</sup><sup>h</sup>

where α is the angle between flux direction and horizontal plane, h is the pressure head, z is the coordinate of the parallel position to the flow direction, K(h) is the hydraulic conductivity for a given pressure head, which in turn is a function of soil volumetric moisture content ϴ, and t is the time. The relationship between the suction and the degree of saturation, or moisture content, is established by means of the soil water characteristic curve (SWCC), as shown in

There are several empirical models to describe the characteristic curves [43]. However, in this work, it is proposed to use the equation of Fredlund and Xing [44]. This is a model of three

<sup>∂</sup><sup>z</sup> � sen<sup>α</sup>

(9)

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171

measurements of the involved parameters is possible [40].

works use the Richards equation for water flux:

Figure 4. Zones of the soil water characteristic curve [46].

Figure 4.

∂θ <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>∂</sup> ∂z

where R3m is the accumulated rainfall from 3 days calculated from records of rainfall gauges, and R<sup>3</sup> is the accumulated rainfall from 3 days calculated using the equation for threshold calculation.

Based on the concepts presented above, records for each rainfall station were organized, and mobile windows from accumulated rainfall of 15 and 3 days were calculated for each date. Likewise, for each date, 3-days rainfall value was calculated using a threshold value defined by the threshold equation. The comparison between 3-days rainfall values was established as shown in Eq. (8). In order to establish the likelihood that the threshold was exceeded, the number of times which the threshold was exceeded along the records was determined, and then, this value of occurrences was divided by the total number of rainfall records, which for the methodology used in this work represents that soil reached the condition of critical saturation (Figure 3). The return period for these events is determined using a Gumbel distribution for the accumulated rainfall. After determining the probability that the soil is saturated according to data from meteorological stations, a geostatistical interpolation process to estimate the probability of saturation in each of the cells, its mean Ps is estimated spatially.

#### 3.3. Probability of soil saturation (PSS) by physically based methods

The effect of rainfall on slope stability is due to a complex physical process involving the advance of the wet front and the consequent increase in pore pressure and reduction of soil cohesion. Coupled models that consider the gradual advance of the wet front have been developed [24, 33, 35–42]; however, obtaining the input parameters is difficult and expensive, so these are not yet in common use. Consequently, the most rigorous methods are still

Figure 3. Schematic methodology adopted for exceedance probability and rainfall of analysis.

recommended only for preliminary assessments, and for special cases, when a good amount of measurements of the involved parameters is possible [40].

(R15) and, accumulated rainfall antecedent of 3 days (R3), is above the failure threshold line for study area, i.e., soil is considered to be saturated if the relationship of the equation is true:

where R3m is the accumulated rainfall from 3 days calculated from records of rainfall gauges, and R<sup>3</sup> is the accumulated rainfall from 3 days calculated using the equation for threshold

Based on the concepts presented above, records for each rainfall station were organized, and mobile windows from accumulated rainfall of 15 and 3 days were calculated for each date. Likewise, for each date, 3-days rainfall value was calculated using a threshold value defined by the threshold equation. The comparison between 3-days rainfall values was established as shown in Eq. (8). In order to establish the likelihood that the threshold was exceeded, the number of times which the threshold was exceeded along the records was determined, and then, this value of occurrences was divided by the total number of rainfall records, which for the methodology used in this work represents that soil reached the condition of critical saturation (Figure 3). The return period for these events is determined using a Gumbel distribution for the accumulated rainfall. After determining the probability that the soil is saturated according to data from meteorological stations, a geostatistical interpolation process to esti-

mate the probability of saturation in each of the cells, its mean Ps is estimated spatially.

The effect of rainfall on slope stability is due to a complex physical process involving the advance of the wet front and the consequent increase in pore pressure and reduction of soil cohesion. Coupled models that consider the gradual advance of the wet front have been developed [24, 33, 35–42]; however, obtaining the input parameters is difficult and expensive, so these are not yet in common use. Consequently, the most rigorous methods are still

3.3. Probability of soil saturation (PSS) by physically based methods

Figure 3. Schematic methodology adopted for exceedance probability and rainfall of analysis.

calculation.

170 Engineering and Mathematical Topics in Rainfall

R3<sup>m</sup> ≥ R<sup>3</sup> (8)

In recent years, slope stability analyses have been expanded to include coupled hydromechanical processes under variably saturated conditions. These analyses incorporate the variation of saturation, leading to more accurate assessments of slopes stability (under infiltration conditions), and demonstrate that a better physical representation of water flow and stress can be attained in unsaturated soils [17]. Hence, the analysis of seepage and coupled stressdeformation should be linked simultaneously [13]. Some recent studies are specifically focused on infiltration-induced landslides. However, most of these studies only consider slope failure below the groundwater table, overlooking the contribution of effective stress (suction stress) to the strength of the soil under transient unsaturated flow conditions [17]. Generally, these works use the Richards equation for water flux:

$$\frac{\partial \Theta}{\partial t} = \frac{\partial}{\partial z} \left[ K(h) \left( \frac{\partial h}{\partial z} - \text{sen}\alpha \right) \right] \tag{9}$$

where α is the angle between flux direction and horizontal plane, h is the pressure head, z is the coordinate of the parallel position to the flow direction, K(h) is the hydraulic conductivity for a given pressure head, which in turn is a function of soil volumetric moisture content ϴ, and t is the time. The relationship between the suction and the degree of saturation, or moisture content, is established by means of the soil water characteristic curve (SWCC), as shown in Figure 4.

There are several empirical models to describe the characteristic curves [43]. However, in this work, it is proposed to use the equation of Fredlund and Xing [44]. This is a model of three

Figure 4. Zones of the soil water characteristic curve [46].

continuous parameters for the entire suction domain. The parameters of the model are related to the air inlet pressure (a), the distribution of pore sizes (n), and the symmetry of the curve (m). The model is based on the possibility of describing the distribution of soil pore sizes from statistical functions [45]. The proposed equation, obtained from integrating a law of frequency distribution in the suction domain, corresponds to:

$$\theta = \frac{1}{\left[\ln\left(\varepsilon + \left(\frac{\psi}{a}\right)^n\right)\right]^m} \tag{10}$$

$$a = \psi\_I \tag{11}$$

• To calculate the infiltration, process by which water penetrates from the surface into the soil, using the Horton, Green-Ampt, and the curve number (CN) models [49]. In the study of infiltration processes, a particular problem is to determine the variation of the soil infiltration capacity, the variation of the wetting front, and the suction of the soil that occurred during a rainfall event, since they influence the magnitude of the torrential

Effect of the Rainfall Infiltration Processes on the Landslide Hazard Assessment of Unsaturated Soils…

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173

The study area is located in the northwest of Colombia, on the eastern central slope of the Aburra Valley, in the city of Medellin. Specifically, the area is located in the "Llanaditas"

The geology of the study area is predominantly characterized by the presence of dunites, slope deposits, and anthropic deposits. On the other hand, the statistical analyzes were carried out on soils of the predominant geological formations in the central area of the municipality of Medellin, the basement rocky is composed mainly of rocks corresponding to dunite from Medellin, which may be covered by slope deposits. Soil resistance parameters were obtained from the analysis of a database of 193 direct shear tests performed on unaltered samples located in the study area, of which 78 tests were performed on slope deposits, 56 on dunite residual soil, and the remaining 59 on saprolite of dunite from Medellin [49]. The results of the mechanical characterization of the materials and their variability are reported in Table 1.

Hydrological information was obtained from the "Villa Hermosa" meteorological station, which has 67 years of records that begin in July 1948 and end in July 2015. Different procedures

neighborhood on the northwestern flank of the Aburra Valley (Figure 6).

avenues associated with this event.

Figure 5. Schematic methodology adopted for infiltration assessment.

4.1. Soil and rainfall characterization

4. Application case

$$m = 3.67 \ln \left( \frac{\theta\_s}{\theta\_i} \right) \tag{12}$$

$$m = \frac{1.31^{m+1}}{m\*\Theta\_s} \* 3.72\*s\*\psi\_I \tag{13}$$

where Ψ is the matric suction, Ψ<sup>I</sup> and θ<sup>I</sup> are the coordinates of the inflection point, and θ<sup>s</sup> is the saturated water content.

Most of the time, the infiltration evaluations are done in a deterministic way, which ignores the uncertainty that is present in this flow process. As it was presented above, it is necessary to estimate the probability of saturation in order to calculate the total probability of failure. There is a probabilistic analysis which ignores the spatial variability of the unsaturated deposits of soil and underestimates the probability of slope failure. Due to this, the effects of soil spatial variability on unsaturated slope have been scarcely studied. In this work, a probabilistic methodology that uses the FOSM method and the Richards' equation to obtain the probability of saturation is proposed. Similarly, in the β<sup>1</sup> index for FOS, for the saturation probability in terms of Z, a reliability index β<sup>2</sup> as a function of the hydraulic properties of the soil in addition to the wetting front progress modeled is defined as:

$$
\beta\_2 = \frac{\mathbb{E}(Z\_c) - Z\_c}{\sigma\_Z} \tag{14}
$$

where Zc is the deep (m) of the wet front, E(Zc) is the Zc mean, and σ<sup>Z</sup> is the standard deviation of Zc obtained using the FOSM method described in Eqs. (3) and (4) taking as a function the Richards' equation.

To solve the Richards' equation, this methodology uses the CHEMFLO-2000 software [47], which is based on the finite difference method. Soil parameters (characteristic curve and saturated permeability) are required as input data. In this case, the Fredlund and Xing model is used and some borders conditions are defined as a flow rate, infiltration rate, or hydraulic load. Any of these boundary conditions requires that the rainfall characteristics of the zone be determined. To do this, it can use the following procedure (Figure 5).

• Getting rainfall information from meteorological stations near the study area. These rainfalls can be accumulated daily or with a higher resolution and must have records for at least 20 years.

Effect of the Rainfall Infiltration Processes on the Landslide Hazard Assessment of Unsaturated Soils… http://dx.doi.org/10.5772/intechopen.70821 173

Figure 5. Schematic methodology adopted for infiltration assessment.

• To calculate the infiltration, process by which water penetrates from the surface into the soil, using the Horton, Green-Ampt, and the curve number (CN) models [49]. In the study of infiltration processes, a particular problem is to determine the variation of the soil infiltration capacity, the variation of the wetting front, and the suction of the soil that occurred during a rainfall event, since they influence the magnitude of the torrential avenues associated with this event.

## 4. Application case

continuous parameters for the entire suction domain. The parameters of the model are related to the air inlet pressure (a), the distribution of pore sizes (n), and the symmetry of the curve (m). The model is based on the possibility of describing the distribution of soil pore sizes from statistical functions [45]. The proposed equation, obtained from integrating a law of frequency

� � � �<sup>n</sup> h i<sup>m</sup> (10)

a ¼ ψ<sup>I</sup> (11)

∗ 3:72 ∗ s ∗ψ<sup>I</sup> (13)

(12)

(14)

<sup>θ</sup> <sup>¼</sup> <sup>1</sup> ln <sup>e</sup> <sup>þ</sup> <sup>ψ</sup> a

<sup>n</sup> <sup>¼</sup> <sup>1</sup>:31<sup>m</sup>þ<sup>1</sup> m ∗ θ<sup>s</sup>

<sup>m</sup> <sup>¼</sup> <sup>3</sup>:67 ln <sup>θ</sup><sup>s</sup>

where Ψ is the matric suction, Ψ<sup>I</sup> and θ<sup>I</sup> are the coordinates of the inflection point, and θ<sup>s</sup> is the

Most of the time, the infiltration evaluations are done in a deterministic way, which ignores the uncertainty that is present in this flow process. As it was presented above, it is necessary to estimate the probability of saturation in order to calculate the total probability of failure. There is a probabilistic analysis which ignores the spatial variability of the unsaturated deposits of soil and underestimates the probability of slope failure. Due to this, the effects of soil spatial variability on unsaturated slope have been scarcely studied. In this work, a probabilistic methodology that uses the FOSM method and the Richards' equation to obtain the probability of saturation is proposed. Similarly, in the β<sup>1</sup> index for FOS, for the saturation probability in terms of Z, a reliability index β<sup>2</sup> as a function of the hydraulic properties of the soil in addition

> <sup>β</sup><sup>2</sup> <sup>¼</sup> E Zð Þ� <sup>c</sup> Zc σZ

where Zc is the deep (m) of the wet front, E(Zc) is the Zc mean, and σ<sup>Z</sup> is the standard deviation of Zc obtained using the FOSM method described in Eqs. (3) and (4) taking as a function the

To solve the Richards' equation, this methodology uses the CHEMFLO-2000 software [47], which is based on the finite difference method. Soil parameters (characteristic curve and saturated permeability) are required as input data. In this case, the Fredlund and Xing model is used and some borders conditions are defined as a flow rate, infiltration rate, or hydraulic load. Any of these boundary conditions requires that the rainfall characteristics of the zone be

• Getting rainfall information from meteorological stations near the study area. These rainfalls can be accumulated daily or with a higher resolution and must have records for at

determined. To do this, it can use the following procedure (Figure 5).

θi � �

distribution in the suction domain, corresponds to:

172 Engineering and Mathematical Topics in Rainfall

to the wetting front progress modeled is defined as:

saturated water content.

Richards' equation.

least 20 years.

The study area is located in the northwest of Colombia, on the eastern central slope of the Aburra Valley, in the city of Medellin. Specifically, the area is located in the "Llanaditas" neighborhood on the northwestern flank of the Aburra Valley (Figure 6).

#### 4.1. Soil and rainfall characterization

The geology of the study area is predominantly characterized by the presence of dunites, slope deposits, and anthropic deposits. On the other hand, the statistical analyzes were carried out on soils of the predominant geological formations in the central area of the municipality of Medellin, the basement rocky is composed mainly of rocks corresponding to dunite from Medellin, which may be covered by slope deposits. Soil resistance parameters were obtained from the analysis of a database of 193 direct shear tests performed on unaltered samples located in the study area, of which 78 tests were performed on slope deposits, 56 on dunite residual soil, and the remaining 59 on saprolite of dunite from Medellin [49]. The results of the mechanical characterization of the materials and their variability are reported in Table 1.

Hydrological information was obtained from the "Villa Hermosa" meteorological station, which has 67 years of records that begin in July 1948 and end in July 2015. Different procedures

Figure 6. Study zone location [48].

were performed, which have as a fundamental principle, the processing of the rainfall data of the station. The average daily rainfall values are presented in Figure 7. Through the average annual cycle of monthly rainfall, two annual peaks were identified in the analysis period, corresponding to the months of May and October, months with average rainfall greater than 180 mm [19]. The daily rainfall was determined for the analysis period, and the calculation of the rainfall of the previous 3 and 15 days was carried out for the subsequent classification of the events, according to the thresholds defined by [31].

Subsequently, a day of analysis was selected that coincided with some database record within the historical records of landslides in the Aburra Valley [19]. The disaster that occurred on November 13, 2010, in "Villa Tina" neighborhood, urban area of the city of Medellin, was selected for the analysis, because it is located in the same area where "Villa Hermosa" station is located. In the area, there was a mass movement which was detonated by high-accumulated rainfall in the previous days [50] and that resulted in the death of one person, one person injured, one house destroyed, and two more houses were affected. Figure 8 shows daily rainfall data for 18 days preceding the event, which according to the documented information

Geologic unit Parameter n x ̅ σ CV (%)

) 78 14.42 1.47 8.45

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175

) 78 11.72 2.11 18.03

) 56 17.75 1.21 6.84

) 56 12.04 1.59 13.19

) 59 17.29 1.41 8.16

) 59 10.96 1.95 17.79

c (kPa) 78 19.91 10.42 52.34

Effect of the Rainfall Infiltration Processes on the Landslide Hazard Assessment of Unsaturated Soils…

c (kPa) 56 20.76 14.66 70.62

c (kPa) 59 15.56 12.11 77.8

) 59 24.04 7.07 29.39

) 56 23.87 7.31 30.88

) 78 24.11 5.52 22.89

Mudflow γ<sup>h</sup> (kN/m<sup>3</sup>

Residual soil γ<sup>h</sup> (kN/m<sup>3</sup>

Saprolite γ<sup>h</sup> (kN/m<sup>3</sup>

γ<sup>d</sup> (kN/m<sup>3</sup>

γ<sup>d</sup> (kN/m<sup>3</sup>

γ<sup>d</sup> (kN/m<sup>3</sup>

Figure 7. Average annual cycle of monthly rainfall for the period 1948–2015.

ϕ (

ϕ (

ϕ (

Table 1. Mechanical characterization of the materials [49].

Effect of the Rainfall Infiltration Processes on the Landslide Hazard Assessment of Unsaturated Soils… http://dx.doi.org/10.5772/intechopen.70821 175


Table 1. Mechanical characterization of the materials [49].

Figure 7. Average annual cycle of monthly rainfall for the period 1948–2015.

were performed, which have as a fundamental principle, the processing of the rainfall data of the station. The average daily rainfall values are presented in Figure 7. Through the average annual cycle of monthly rainfall, two annual peaks were identified in the analysis period, corresponding to the months of May and October, months with average rainfall greater than 180 mm [19]. The daily rainfall was determined for the analysis period, and the calculation of the rainfall of the previous 3 and 15 days was carried out for the subsequent classification of

the events, according to the thresholds defined by [31].

Figure 6. Study zone location [48].

174 Engineering and Mathematical Topics in Rainfall

Subsequently, a day of analysis was selected that coincided with some database record within the historical records of landslides in the Aburra Valley [19]. The disaster that occurred on November 13, 2010, in "Villa Tina" neighborhood, urban area of the city of Medellin, was selected for the analysis, because it is located in the same area where "Villa Hermosa" station is located. In the area, there was a mass movement which was detonated by high-accumulated rainfall in the previous days [50] and that resulted in the death of one person, one person injured, one house destroyed, and two more houses were affected. Figure 8 shows daily rainfall data for 18 days preceding the event, which according to the documented information

Figure 8. Daily rainfall of 18 days antecedent to the mass movement of day 13 of November 2010.

is the cause of the mass movement, also this accumulative rainfall is 296.4 mm, it has a return period of approximately 20 years; therefore, the probability of the event being exceeded is 5%.

Subsequently, a rainfall event was established which exceeded the failure thresholds and that

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To validate the water flow through the soil by infiltration processes on the formations of the study area, it was necessary to quantify this phenomenon. For this reason, the Horton, Curve Number, and Green & Ampt models were used, which allow to calculate the amount of infiltrated water for an event of 18-days duration. Table 2 presents the uniform infiltration rate for the analysis time (cm/h). The results obtained from the infiltration tests allowed to determine the permeability of soils. In the case of soils derived from the dunite from Medellin, at the level of residual soil and saprolite, the infiltration rate was 0.8 mm/min, and for the mudflow, it

Suction tests with filter paper were also carried out in order to obtain the characteristic curves of the soil (Figure 10) and to determine the adjustment parameters for each of the models

Residual soil Mudflow

considering the theoretical characteristics curves presented in the literature by [51, 52].

Curve Number CN (zone 1) 0.041 0.029 Curve Number CN (zone 2) 0.019 0.013 Horton 0.068 0.049 Green & Ampt 0.058 0.035

Infiltration method Infiltration rate (cm/h)

had been documented. For this event, the intensity of the rainfall was determined.

Figure 9. Probability of event exceedance for different return periods and different PDFs.

was 0.2 mm/min [19].

Table 2. Uniform infiltration rate [19].

In order to evaluate the probability of exceedance of the rainfall threshold, a frequency analysis of the data was performed, where the rainfall series of 18 days that exceed the threshold were selected, the annual maximum records were selected, and with these results, the probability of recurrence was determined for different return periods and confidence intervals according to the distribution functions proposed by Gumbel, Log-Normal, and Frechet (Figure 9). Alternately, field and laboratory tests were performed to measure soil hydraulic capacity. A double ring infiltration tests and suction tests with filter paper (in the laboratory) were performed for each geological formation of interest, with the aim to calculate the characteristic curve of these soils. The infiltration rate, which is the rate at which the water penetrates the soil through its surface, is expressed in mm/min, and its maximum value coincides with the hydraulic conductivity of the saturated soil.

Once defining the mechanical and hydraulic properties of the soil, the boundary conditions and hydrogeological properties of the soil were determined. Rainfall data were collected, allowing to validate the failure thresholds and the amount of water infiltrated in this punctual zone. For this purpose, rainfall data were collected with daily resolution in the analysis period. This information was processed in order to characterize the rainfall regimes of the area.

Figure 9. Probability of event exceedance for different return periods and different PDFs.

Subsequently, a rainfall event was established which exceeded the failure thresholds and that had been documented. For this event, the intensity of the rainfall was determined.

To validate the water flow through the soil by infiltration processes on the formations of the study area, it was necessary to quantify this phenomenon. For this reason, the Horton, Curve Number, and Green & Ampt models were used, which allow to calculate the amount of infiltrated water for an event of 18-days duration. Table 2 presents the uniform infiltration rate for the analysis time (cm/h). The results obtained from the infiltration tests allowed to determine the permeability of soils. In the case of soils derived from the dunite from Medellin, at the level of residual soil and saprolite, the infiltration rate was 0.8 mm/min, and for the mudflow, it was 0.2 mm/min [19].

Suction tests with filter paper were also carried out in order to obtain the characteristic curves of the soil (Figure 10) and to determine the adjustment parameters for each of the models considering the theoretical characteristics curves presented in the literature by [51, 52].


Table 2. Uniform infiltration rate [19].

is the cause of the mass movement, also this accumulative rainfall is 296.4 mm, it has a return period of approximately 20 years; therefore, the probability of the event being exceeded is 5%. In order to evaluate the probability of exceedance of the rainfall threshold, a frequency analysis of the data was performed, where the rainfall series of 18 days that exceed the threshold were selected, the annual maximum records were selected, and with these results, the probability of recurrence was determined for different return periods and confidence intervals according to the distribution functions proposed by Gumbel, Log-Normal, and Frechet (Figure 9). Alternately, field and laboratory tests were performed to measure soil hydraulic capacity. A double ring infiltration tests and suction tests with filter paper (in the laboratory) were performed for each geological formation of interest, with the aim to calculate the characteristic curve of these soils. The infiltration rate, which is the rate at which the water penetrates the soil through its surface, is expressed in mm/min, and its maximum value coincides with the hydraulic con-

Figure 8. Daily rainfall of 18 days antecedent to the mass movement of day 13 of November 2010.

Once defining the mechanical and hydraulic properties of the soil, the boundary conditions and hydrogeological properties of the soil were determined. Rainfall data were collected, allowing to validate the failure thresholds and the amount of water infiltrated in this punctual zone. For this purpose, rainfall data were collected with daily resolution in the analysis period. This information was processed in order to characterize the rainfall regimes of the area.

ductivity of the saturated soil.

176 Engineering and Mathematical Topics in Rainfall

conditions described in [19], were entered into the numerical model (CHEMFLO-2000). Modeling the wetting front progress was identified considering the relationship between the moisture content and the soil suction, for a previously characterized rainfall event. To determine the

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For this, the probability distribution of the wetting front progress was determined, in function of the random variables that condition this function (α, n, θs, θr). A modeling was done in the software, using the infiltration model of the curve method as the most conservative, the random variables were slightly modified by a parameter of variation α, which according to

In order to determine the standard deviation of the selected parameters, a statistical base of 44 data was collected, which were collected from the literature and laboratory tests executed in the development of this work [19]. It reports the expected value and the reliability index for each formation of the hydraulic properties that allow to finally determine the probability of

The landslide hazard assessment was calculated as the probability of landslides occurring in the area using the ArcGIS software. The area was divided into a grid of 25 m, generating 7766 calculation elements. Each cell was assigned the shear strength parameters shown in Table 1. Likewise, each cell was assigned a saturation probability according to those determined in the previous sections for different depths of the wetting front progress and rainfall threshold. Figure 13 presents the results of the hazard assessment in "Llanaditas" neighborhood, taking into account the failure threshold and the wetting front progress, respectively, considering

probability of saturation, the first order second order method (FOSM) was used.

the literature can be assumed 10 (ten).

4.4. Results obtained from landslide hazard assessment

Figure 12. Probability of saturation according to depth, for each surface formation.

saturation (Figure 12).

different depths 0.2, 0.5, and 1 m.

Figure 10. Soil water characteristic curve for tested soils.

#### 4.2. Results obtained from probability of saturation using rainfall thresholds

Using the failure threshold defined in Eq. (8) and "Villa Hermosa" meteorological station data, a threshold exceedance probability of 17.81% was determined for the 67-year analysis; however, for a particularly rainy year such as 2010, the threshold exceedance was 39.72% (Figure 11).

#### 4.3. Results obtained from probability of saturation using the physically based model

In order to evaluate the infiltration using the Richards' equation, the parameters corresponding to the characteristic curve, as well as the initial conditions and boundary

Figure 11. Probability of saturation through threshold exceedance (year 2010).

conditions described in [19], were entered into the numerical model (CHEMFLO-2000). Modeling the wetting front progress was identified considering the relationship between the moisture content and the soil suction, for a previously characterized rainfall event. To determine the probability of saturation, the first order second order method (FOSM) was used.

For this, the probability distribution of the wetting front progress was determined, in function of the random variables that condition this function (α, n, θs, θr). A modeling was done in the software, using the infiltration model of the curve method as the most conservative, the random variables were slightly modified by a parameter of variation α, which according to the literature can be assumed 10 (ten).

In order to determine the standard deviation of the selected parameters, a statistical base of 44 data was collected, which were collected from the literature and laboratory tests executed in the development of this work [19]. It reports the expected value and the reliability index for each formation of the hydraulic properties that allow to finally determine the probability of saturation (Figure 12).

#### 4.4. Results obtained from landslide hazard assessment

4.2. Results obtained from probability of saturation using rainfall thresholds

Figure 11. Probability of saturation through threshold exceedance (year 2010).

Figure 10. Soil water characteristic curve for tested soils.

178 Engineering and Mathematical Topics in Rainfall

(Figure 11).

Using the failure threshold defined in Eq. (8) and "Villa Hermosa" meteorological station data, a threshold exceedance probability of 17.81% was determined for the 67-year analysis; however, for a particularly rainy year such as 2010, the threshold exceedance was 39.72%

4.3. Results obtained from probability of saturation using the physically based model

In order to evaluate the infiltration using the Richards' equation, the parameters corresponding to the characteristic curve, as well as the initial conditions and boundary The landslide hazard assessment was calculated as the probability of landslides occurring in the area using the ArcGIS software. The area was divided into a grid of 25 m, generating 7766 calculation elements. Each cell was assigned the shear strength parameters shown in Table 1. Likewise, each cell was assigned a saturation probability according to those determined in the previous sections for different depths of the wetting front progress and rainfall threshold. Figure 13 presents the results of the hazard assessment in "Llanaditas" neighborhood, taking into account the failure threshold and the wetting front progress, respectively, considering different depths 0.2, 0.5, and 1 m.

Figure 12. Probability of saturation according to depth, for each surface formation.

5. Conclusions

complex models.

mountainous regions.

Author details

References

Rainfall is the main trigger of mass movements in tropical regions, so the evaluation of its effect on stability becomes increasingly important, leading to the generation of increasingly

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181

There are several proposals to consider the effect of rainfall on slope stability, empirical models such as failure thresholds, and physically based analytical models. The approximate solution of these analytical models is more used today. A probabilistic approach such as the one presented in this work allows to incorporate in the analysis the different sources of uncertainty that affect the behavior of the infiltration in soils and permit evaluating the effect of the rainfall infiltration processes on the landslide hazard assessment of unsaturated soils in tropical

It was observed that in the soils considered, the variation of the wetting front, in rain conditions such as those shown, only affects the most superficial layer of the soil, reason why the

The rainfall threshold approach is an efficient methodology to evaluate shallow landslide with respect to physically based approach (wetting front progress), because it requires less processing

[1] Hidalgo C, Vega J. Estimation of the threat of landslides triggered by earthquakes and

[2] Intrieri E, Gigli G, Casagli N, Nadim F. Landslide Early Warning System: Toolbox and

[3] Jaiswal P, Van Westen C. Rainfall-based temporal probability for landslide initiation along transportation routes in Southern India, de Landslide processes: From geomorphologic

[4] Lari S, Frattini P, Crosta G. A probabilistic approach for landslide hazard analysis.

[5] Chen H, Zhang L. A physically-based distributed cell model for predicting regional

rainfall-induced shallow slope failures. Engineering Geology. 2014;176:79-92

general concepts. Natural Hazards and Earth System Sciences. 2013;13:85-90

time, characterization soils, laboratory tests, and elaborated methodological approaches.

Cesar Augusto Hidalgo, Johnny Alexander Vega\* and Melissa Parra Obando

rainfall (Valle de Aburrá-Colombia). Revista EIA. 2014;11(22):103-117

mapping to dynamic modelling. Strasbourg, France; 2009

Engineering Geology. 2014;182:3-14

effect of the rains mainly generates faults of the shallow type.

\*Address all correspondence to: javega@udem.edu.co

School of Engineering, University of Medellin, Colombia

Figure 13. Total probability of failure (static condition) at different depths considering wetting front progress and rainfall. (A) TPF by rainfall threshold (0.2 m-depth), (B) TPF by wetting front progress (0.2 m-depth), (C) TPF by rainfall threshold (0.5 m-depth), (D) TPF by wetting front progress (0.5 m-depth), (E) TPF by rainfall threshold (1.0 m-depth), (F) TPF by wetting front progress (1.0 m-depth).

According to the stability analysis, it can be concluded that the stability condition of the slopes (in static condition) decreases with the increase of the probability of saturation, which occurs in the superficial strata (from 0.2 m), encouraging the configuration and/or evolution of instability phenomena of varying magnitude due to loss of shear strength. As shown in the Figure 13, between both empirical and physically based approaches, three evaluated scenarios present a difference approximately of 1% for the TPF intervals considered.
