# A Modeling Platform for Landslide Stability: A Hydrological Approach

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Structure

#### 2.1. Primary Preparation Unit

#### 2.2. Vegetation Impacts Unit

#### 2.3. Hydrological Effects Unit

#### 2.4. Slope Stability Analysis Unit

## 3. Mathematical Formulation

#### 3.1. Vegetation Module

#### 3.1.1. Reinforcement of Soil by Roots

^{−1}T

^{−2}) is the soil cohesion, ${c}_{r}^{\prime}$ (ML

^{−1}T

^{−2}) is root contribution cohesion, and c′ (ML

^{−1}T

^{−2}) is the total cohesion in the root zone. The apparent cohesion due to existence of the roots may be considered as two main approaches: measurement approach and estimation approach. In the first, the increased cohesion is measured by field or laboratory shear tests (the difference between shear resistance of the root-permeated soil and bare soil shows root contribution cohesion). If this approach is considered, the SSHV-2D program gets a constant value as apparent cohesion caused by roots for each root-permeated soil layer.

^{−1}T

^{−2}) is the average of tensile stress in the root, ${A}_{r}$ (L

^{2}) is the total cross-sectional area of roots, A (L

^{2}) is the cross-sectional area of soil in considered section, A

_{r}/A is so-called the root area ratio (RAR) and ${k}^{\u2033}$ is the reduction factor. A wide range of reduction factors reported by researchers is listed in Table 2. Wu [51] suggested using a reduction factor of 0.3–0.5 according to practical approaches.

_{er}) increases the soil resistance by providing an effective tension force [60]. The tension in the root (shown with F in Figure 3a) increases resistance through two mechanisms. If the effective length of the root (L

_{er}) is less than a threshold length (L

_{t}), then the root pullout resistance is dominant. Otherwise, the tensile capacity of the root determines additional cohesion by the roots [60]. The threshold length can be determined by the division of tensile capacity by the pullout resistance (see Figure 3b). The root direction is not necessarily perpendicular to the slip circle; the tension force will have a tangent and a normal component. The root contribution to the cohesion due to reinforcement can be expressed as [60]:

_{arc slip}[L] is the length of the slip arc. The summation sign in the above equation shows that the additional shear force must be considered for all trees. This method can be used if the location, length, inclination, tensile capacity, pullout resistance, and shear capacity of individual roots are known. In SSHV-2D it is assumed that the trees are at equal distances. So, having the distance between trees is sufficient to specify the location of trees. Root density (RD) is defined as the number of roots (or trees) per unit length of the slope. Accordingly, the distance between trees is equal to the reverse of the RD and the distance of the first tree is half of this value from the start point of the slope.

#### 3.1.2. Increasing Permeability

#### 3.1.3. Root Water Uptake

_{P}) is the potential transpiration factor. In this equation, h is negative pore-water pressure, β is the root length density, and T

_{P}is the potential transpiration rate.

_{1}(L), h

_{2}(L), h

_{3}(L), h

_{4}(L) are constant values. It is assumed that these points are known based on plant type and entered as input data into the program. This factor is shown in Figure 4.

_{0}and the depth of z = z

_{0}and decreases exponentially in both radial and vertical directions (Figure 5). Therefore, the root length density can be expressed by [73]:

_{max}(L. L

^{−3}) is the maximum root density and k

_{1}(L

^{−1}) and k

_{2}(L

^{−1}) are two experimental coefficients. Then Fatahi, Khabbaz, and Indraratna [73] presented the following hyperbolic tangent function for the root density factor:

_{3}[L

^{2}] is an empirical coefficient and V [L

^{3}] is root zone volume.

_{4}(L

^{−1}) is an experimental coefficient and z

_{max}(L) is the maximum depth of the root zone.

_{P})) in Equation (4). This value is calculated for each cell in the root zone.

#### 3.1.4. Surcharge Load

#### 3.1.5. Effective Rainfall

^{−1}) is the gross rainfall rate, D

_{i,c}(LT

^{−1}), and D

_{t},

_{c}(LT

^{−1}) are the drip from the canopy and the trunk drainage respectively. The value of D

_{i},

_{c}is zero if the canopy is not saturated. After the saturation of the canopy, the excess water on the canopy storage capacity drains and a part of it (P

_{d}) is inputted to trunk. Similar to the canopy, when the trunk is saturated, the excess water flows as trunk drainage (D

_{t},

_{c}). Here, P

_{eff}(L) is the effective precipitation which is the rainfall minus the interception loss.

^{−1}) is average rainfall, ${\overline{E}}_{c}$ (LT

^{−1}) is the average evaporation from the saturated covered area, and ε (-) is the ratio of evaporation from saturated trunk to covered evaporation. Based on the definition of ε, the term of $(1-\epsilon ){\overline{E}}_{c}$ shows evaporation from the saturated canopy and the term of $\epsilon {\overline{E}}_{c}$ demonstrates the evaporation from the saturated trunks. The parameters S (L) and S

_{t}(L) are the canopy and the trunk storage capacities, respectively. Other parameters were defined before. Gash et al. [28] presented the following equations to estimate throughfall and stemflow.

#### 3.1.6. Wind Force

_{D}is the drag force (MLT

^{−2}), C

_{D}is the drag coefficient (empirical) (-), ρ

_{a}is the air density (ML

^{−3}), V

_{w}is the average wind velocity (LT

^{−1}), and A

_{t}is the projected area of the tree against the wind. However, a forested slope is covered by many trees and the drag force should be calculated for all of them.

#### 3.2. Hydrological Module

_{x}and K

_{z}are hydraulic conductivity (LT

^{−1}) in the horizontal and vertical directions respectively, h is pressure head (L), z is the vertical dimension (L) which is positive downwards, and S is the sink/source term (T

^{−1}) for the root water uptake. Water content (θ) is a function of pressure head (h) or vice versa. Equation (16) can be written in terms of pressure head alone (the h-based form), or in terms of water content (the θ-based form), or in terms of both θ and h (the mixed form). Because the θ-based form only is valid in the unsaturated zone, it is not used in the current model. Celia et al. [85] demonstrated that the mixed form of Richards’ equation is better at conserving the mass compared with the h-based form. Therefore, the mixed form is used in the current program.

_{s}and K = K

_{s}). It has been demonstrated that the behavior of soil in wetting and drying cycles is not similar and the SWCC has a hysteresis property [93,94]. In the this stage of the SSHV-2D model, the hysteresis of the SWCC was not considered and wetting and drying cycles happen on a similar curve. The future development of the model this factor will be included.

#### 3.3. Slope Stability Analysis

#### 3.3.1. Shear Strength in the Saturated and Unsaturated Zone

^{−1}T

^{−2})) is replaced by the Terzaghi theorem [98].

^{−1}T

^{−2}), c′ is the effective cohesion (ML

^{−1}T

^{−2}), σ is total normal stress (ML

^{−1}T

^{−2}), u

_{w}is pore water pressure (ML

^{−1}T

^{−2}), and φ′ (-) is the effective internal friction angle. In this viewpoint, it is assumed that only pore water pressure is defined under the water table (in the saturated zone) but the pore water pressure is zero above the water table (in the unsaturated zone).

_{a}is the pore air pressure (ML

^{−1}T

^{−2}), (u

_{a}− u

_{w}) is matric suction (positive) (ML

^{−1}T

^{−2}), and χ (-) is an empirical parameter called the matric suction coefficient. Bishop stated that the matric suction coefficient (χ) is related to the degree of saturation and varies between 0 (for dry soil) and 1 (for fully saturated soil). In saturated conditions, χ = 1 and the effective stress equation becomes σ′ = σ − u

_{w}, as suggested by Terzaghi [98]. Fredlund et al. [99] proposed an equation for shear strength of partially saturated soil:

^{b}(-) is the friction angle relative to matric suction. By comparing Equation (19) with Equations (17) and (18):

^{b}) and χ have the same behavior. In many references, χ is correlated with degree of saturation (or matric suction) in the unsaturated zone (e.g., [33,100]) but in some references φ

^{b}has a constant value in the range of 0 to φ′ in unsaturated soil (e.g., [99,101]) and it is equal to φ′ in saturated medium.

^{b}and the other is χ = S

_{r}(degree of saturation). In both cases, φ

^{b}= φ′ when there is no matric suction (saturated zone). In the constant φ

^{b}case, if φ

^{b}= 0 is used, the classic formulation for shear strength is used for slope stability analysis. Accordingly, the factor of safety equations was expressed in terms of shear strength in the unsaturated zone but it is also valid in the saturated zone.

#### 3.3.2. Simplified Bishop Method

_{s}(ML

^{−1}T

^{−2}) is effective cohesion of soil, c′

_{r}(ML

^{−1}T

^{−2}) is root contribution cohesion, W (MLT

^{−2}) is the weight of the slice, q

_{sur}(ML

^{−1}T

^{−2}) is the uniform surcharge load due to vegetation weight, u

_{a}(ML

^{−1}T

^{−2}) is pore air pressure (usually u

_{a}= 0), φ′ (-) is the effective friction angle, u

_{w}(ML

^{−1}T

^{−2}) is pore water pressure, u

_{a}− u

_{w}(ML

^{−1}T

^{−2}) is the matric suction, φ

^{b}is the angle indicating the rate of change in shear strength relative to matric suction (in unsaturated zone ϕ

^{b}≤ φ′, and in saturated zone ϕ

^{b}= φ′), b (L) is the width of the slice, α (-) is the inclination of slice base, and SF (-) is the safety factor of the slope.

#### 3.3.3. Janbu’s Simplified Method

^{−2}) is a normal force on the base of the slice, F

_{wind}is the sum of drag forces acting to all trees, f

_{0}(-) is the correction factor, and SF (-) is the factor of safety. Other parameters are similar to the parameters of the simplified Bishop method. Janbu et al. [31] presented a chart for estimating f

_{0}from the depth-to-length ratio of slipped mass (refer to Figure 8). Abramson et al. [102] presented the following formula for f

_{0}as a function of d and L which are defined in Figure 8.

_{0}.

## 4. Numerical Formulation

#### 4.1. Geometry Discretization

#### 4.2. Transient Water Infiltration

**Discretization of equation:**The two-dimensional Richards’ equation in mixed form is the governing equation in the hydrology module. To discretization this equation in time, explicit, fully implicit, and Crank–Nicolson schemes are available. Shahraiyni and Ataie-Ashtiani [103] demonstrated that the fully implicit scheme is more stable compared with Crank–Nicolson and two-stage Runge–Kutta (predictor–corrector) schemes. Therefore, the fully implicit scheme is used in the next problems and is described in the following. For spatial discretization, the centered scheme is selected because of higher-order accuracy. The Richards’ equation is a nonlinear and must be solved by an iteration scheme. The Picard iteration method is used in SSHV-2D. The discretized form of Richards’ equation for the node of (i,j) can be written as below.

^{n}

^{+1,m+1}:

^{−1}) and is called the specific moisture capacity. If all nonlinear terms are ignored in the above equation, substituting in Equation (27) results in:

^{−1}) is the hydraulic conductivity in the vertical direction, and K

_{ratio}(-) is the ratio of hydraulic conductivity in the horizontal direction to the vertical direction (= K

_{x}/K

_{z}). Other parameters were already introduced.

**Averaging of hydraulic conductivity:**In the above formulation, the hydraulic conductivity terms are defined as an average of two adjacent nodes. For example, ${K}_{i-{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},j}^{n+1,m}$ is the average of ${K}_{i-1,j}^{n+1,m}$ and ${K}_{i,j}^{n+1,m}$. Three averaging approaches are considered in SSHV-2D: arithmetic averaging, geometric averaging, and harmonic averaging [104]. Table 5 shows averaging relationships for each method. Since Richards’ equation is a nonlinear equation, it seems that the non-arithmetic averaging can increase the rate of convergence.

**Specific moisture capacity:**Another important parameter in Richards’ equation is specific moisture capacity ($C={\scriptscriptstyle \frac{\partial \theta}{\partial h}}$). For the evaluation of C in any node, the pressure head is substituted in the first-order derivative of the SWCC relationship. This scheme is called tangent approximation. Rathfelder and Abriola [105] proposed an alternative named standard chord slope (SCS) that is expressed by Equation (15). They demonstrated that the SCS scheme is better at mass conservation than the tangent approximation in the h-based Richards’ equation. Both schemes for evaluation of C are available in the SSHV-2D program.

**Time stepping schemes:**The infiltration of rainfall in the soil is a transient problem and time stepping value (∆t) can be effective in achieving a faster solution to the problem. In the program, in addition to constant time-stepping, automatic time stepping [103,106] is also available which adaptively determines Δt in the next step.

_{max}is allowed maximum number of iterations, and ∆t

^{new}(T) and ∆t

^{old}(T) are time steps of the next and current step respectively. If the number of iterations (N) was greater than N

_{max}, then ∆t is converted to ∆t/3 and the current iteration is repeated again. In automatic time-stepping, the time increment is increased if the solution condition is good (the number of iterations is low) and decreases if the solution condition is bad (the number of iterations is high).

**Boundary conditions (BC):**For boundaries of the domain, two main conditions are considered in the SSHV-2D program: constant head (Dirichlet BC) and flux (Neumann BC). In the former, the value of the head (unknown of the problem) in boundary nodes is definite and given by the user. But, in the second case, the flux on the boundary is given and the rate of the head is definite. According to Darcy’s law:

#### 4.3. Slope Stability Analysis

^{b}, W, u

_{w}, etc.) are related to soil and vegetation properties or the result of Richards’ equation. The user should determine the position of the slip surface so that SF can be obtained. In the SSHV-2D program, the slip circles are defined by center and radius. The centers are located above the slope and the interval between, along with the range of radius changes, and the increment of the radius is determined by the user. Figure 10 demonstrates the position of centers and the radii for a slip circle. The critical SF is the minimum of SF for all defined circles which is determined by the program. In Janbu’s method, the slip surface can be noncircular, and the noncircular failure surface is manually defined by the user.

## 5. Verifications and Results

#### 5.1. Example 1: 1-D Infiltration in the Vertical Soil Column

^{3}/cm

^{3}(or water pressure head of −61.5 cm) was maintained at the bottom of the column. The water content in the entire soil column was imposed θ = 0.1 cm

^{3}/cm

^{3}at the start time.

_{S}= 34 cm/h, θ

_{S}= 0.287, θ

_{r}= 0.075 and by using the least-square fit on the data, the empirical coefficients A = 1.175 × 10

^{6}, B = 4.74, α = 1.611 × 10

^{6}, and β = 3.96 were determined. In both equations, h is water pressure head in cm.

#### 5.2. Example 2: 2-Dimensional Infiltration and Water Table Recharge

#### 5.3. Example 3: Stability Analysis of a Slope

**Case 1—A homogenous slope:**A homogeneous slope which was used in the studies of Fredlund and Krahn [97] and Xing [109] is shown in Figure 13. The properties of soil in this example are γ = 19.2 kN/m

^{3}, C′ = 29.3 kPa, and φ′ = 20° (the values used in various references are slightly different). They reported safety factors of the slope for different methods. Other researchers applied their proposed methods on this problem and reported the safety factor of the slope (Table 7).

**Case 2—A homogenous slope with a weak layer:**After the analysis of the circular slip surface, Fredlund and Krahn [97] and Xing [109] added a weak layer to the slope as shown in Figure 14. This layer was so weak so that it changed the slip surface from circular to composite slip surface. The different researchers computed the SF of this problem for a given center point and radius, but the position of the weak layer was different in these references. Accordingly, the elevation of the weak layer from the bottom of the model is defined by D (see Figure 14). Here, we used Janbu’s simplified method and calculated SFs for the determined center point and radius which are compared with other researcher’s results in Table 8. It is notable that, similar to the first case, the critical slip circle is identified different from the given circle, and the minimum safety factors are 1.366, 1.413, and 1.434 for D = 6.1 (oblique), 4 and 4.55 m, which is slightly different from values in Table 8.

#### 5.4. Example 4: The Effect of Matric Suction on the Stability of a Slope

^{3}, C’ = 10 kPa, and φ’ = 20°. To investigate the matric suction effect, two conditions of pore pressure were considered. In the first condition, the matric suction was ignored and in the other case, stability was analyzed by considering matric suction.

^{b}as a function of the effective degree of saturation (or matric suction) as Equation (20). The matric suction coefficient ($\chi $) with the Van-Genuchten model can be written as:

_{r}is the residual degree of saturation. The Van-Genuchten fitting parameters were assumed as α = 0.005 kPa

^{−1}and n = 1.7 in the original reference.

#### 5.5. Example 5: Improvement of Slope Stability by Vegetation

_{center}= Δz

_{center}= ΔR = 0.1 m. The safety factor of the slope and the maximum depth of the slip surface (h in Figure 16) are presented in Figure 17. The results demonstrate that the increase in root density enhances the stability of the slope. For instance, when the surface layer thickness is D = 1 m, the safety factor increases from 1.13 (in bare slope) to 1.66 (in RD = 0.8 m

^{−1}) which means 47% improvement in stability. Also, increasing the slip surface depth with root density indicates that the vegetation can make the surface layer more cohesive and prevent shallow landslides. Specifically, in the case that the thickness of the surface layer is low (D = 1 m) and the root density is high (RD > 0.4), the surface soil has become so strong that the slip surface is removed from the surface layer and is placed in a very stiff layer (see Figure 17b). In other thicknesses of the surface layer, this does not happen because the length of the roots is not sufficient to strengthen the whole depth of the surface layer.

## 6. Conclusions

^{−1}increased the safety factor from 1.13 (in bare slope) to 1.66, which means about 50% improvement of stability. Future works will be focused on sensitivity analysis of vegetation characteristics and the effect of hydrological parameters on the stability of slopes in real-world problems.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Kirschbaum, D.B.; Adler, R.; Hong, Y.; Hill, S.; Lerner-Lam, A. A global landslide catalog for hazard applications: Method, results, and limitations. Nat. Hazards
**2010**, 52, 561–575. [Google Scholar] [CrossRef] - Petley, D. Global patterns of loss of life from landslides. Geology
**2012**, 40, 927–930. [Google Scholar] [CrossRef] - Froude, M.J.; Petley, D. Global fatal landslide occurrence from 2004 to 2016. Nat. Hazards Earth Syst. Sci.
**2018**, 18, 2161–2181. [Google Scholar] [CrossRef][Green Version] - Aleotti, P.; Chowdhury, R. Landslide hazard assessment: Summary review and new perspectives. Bull. Eng. Geol. Environ.
**1999**, 58, 21–44. [Google Scholar] [CrossRef] - Haque, U.; Blum, P.; Da Silva, P.F.; Andersen, P.; Pilz, J.; Chalov, S.R.; Malet, J.-P.; Auflič, M.J.; Andres, N.; Poyiadji, E. Fatal landslides in Europe. Landslides
**2016**, 13, 1545–1554. [Google Scholar] [CrossRef] - Klose, M.; Maurischat, P.; Damm, B. Landslide impacts in Germany: A historical and socioeconomic perspective. Landslides
**2016**, 13, 183–199. [Google Scholar] [CrossRef] - Yavari-Ramshe, S.; Ataie-Ashtiani, B. Numerical modeling of subaerial and submarine landslide-generated tsunami waves—Recent advances and future challenges. Landslides
**2016**, 13, 1325–1368. [Google Scholar] [CrossRef] - Panizzo, A.; De Girolamo, P.; Di Risio, M.; Maistri, A.; Petaccia, A. Great landslide events in Italian artificial reservoirs. Nat. Hazards Earth Syst. Sci.
**2005**, 5, 733–740. [Google Scholar] [CrossRef][Green Version] - Dilley, M.; Chen, R.S.; Deichmann, U.; Lerner-Lam, A.L.; Arnold, M. Natural Disaster Hotspots: A Global Risk Analysis; The World Bank: Washington, DC, USA, 2005. [Google Scholar]
- Sepúlveda, S.A.; Petley, D.N. Regional trends and controlling factors of fatal landslides in Latin America and the Caribbean. Nat. Hazards Earth Syst. Sci.
**2015**, 15, 1821–1833. [Google Scholar] [CrossRef][Green Version] - Wienhöfer, J.; Lindenmaier, F.; Zehe, E. Challenges in Understanding the Hydrologic Controls on the Mobility of Slow-Moving Landslides. Vadose Zone J.
**2011**, 10, 496–511. [Google Scholar] [CrossRef] - Strauch, A.M.; MacKenzie, R.A.; Giardina, C.P.; Bruland, G.L. Climate driven changes to rainfall and streamflow patterns in a model tropical island hydrological system. J. Hydrol.
**2015**, 523, 160–169. [Google Scholar] [CrossRef] - Forbes, K.; Broadhead, J.; Bischetti, G.; Brardinoni, F.; Dykes, A.; Gray, D.; Imaizumi, F.; Kuriakose, S.; Osman, N.; Petley, D.; et al. Forests and Landslides The Role of Trees and Forests in the Prevention of Landslides and Rehabilitation of Landslide-Affected Areas in Asia; Forbes, K., Broadhead, J., Eds.; FAO Regional Office for Asia and the Pacific: Bangkok, Thailand, 2011. [Google Scholar]
- Meehan, W.R. Influences of Forest and Rangeland Management on Salmonid Fishes and Their Habitats; American Fisheries Society: Bethesda, MD, USA, 1991. [Google Scholar]
- Gaillard, J.; Liamzon, C.; Villanueva, J. ‘Natural’ disaster? A retrospect into the causes of the late-2004 typhoon disaster in Eastern Luzon, Philippines. Environ. Hazards
**2007**, 7, 257–270. [Google Scholar] [CrossRef] - Lee, I. A review of vegetative slope stabilization. Hong Kong Inst. Eng.
**1985**, 13, 9–12. [Google Scholar] - Greenway, D. Vegetation and slope stability. In Slope Stability: Geotechnical Engineering and Geomorphology; Anderson, M.G., Richards, K.S., Eds.; Wiley: Hoboken, NJ, USA, 1987. [Google Scholar]
- Waldron, L.J. The Shear Resistance of Root-Permeated Homogeneous and Stratified Soil1. Soil Sci. Soc. Am. J.
**1977**, 41, 843. [Google Scholar] [CrossRef] - Wu, T.H.; McKinnell Iii, W.P.; Swanston, D.N. Strength of tree roots and landslides on Prince of Wales Island, Alaska. Can. Geotech. J.
**1979**, 16, 19–33. [Google Scholar] [CrossRef] - Gardner, W.R. Relation of Root Distribution to Water Uptake and Availability. Agron. J.
**1964**, 56, 41–45. [Google Scholar] [CrossRef] - Feddes, R.A.; Kowalik, P.; Kolinska-Malinka, K.; Zaradny, H. Simulation of field water uptake by plants using a soil water dependent root extraction function. J. Hydrol.
**1976**, 31, 13–26. [Google Scholar] [CrossRef] - Prasad, R. A linear root water uptake model. J. Hydrol.
**1988**, 99, 297–306. [Google Scholar] [CrossRef] - Li, K.Y.; Boisvert, J.B.; Jong, R.D. An exponential root-water-uptake model. Can. J. Soil Sci.
**1999**, 79, 333–343. [Google Scholar] [CrossRef][Green Version] - Rutter, A.J.; Kershaw, K.A.; Robins, P.C.; Morton, A.J. A predictive model of rainfall interception in forests, 1. Derivation of the model from observations in a plantation of Corsican pine. Agric. Meteorol.
**1971**, 9, 367–384. [Google Scholar] [CrossRef] - Rutter, A.J.; Morton, A.J.; Robins, P.C. A Predictive Model of Rainfall Interception in Forests. II. Generalization of the Model and Comparison with Observations in Some Coniferous and Hardwood Stands. J. Appl. Ecol.
**1975**, 12, 367–380. [Google Scholar] [CrossRef] - Gash, J. An analytical model of rainfall interception by forests. Q. J. R. Meteorol. Soc.
**1979**, 105, 43–55. [Google Scholar] [CrossRef] - Valente, F.; David, J.S.; Gash, J.H.C. Modelling interception loss for two sparse eucalypt and pine forests in central Portugal using reformulated Rutter and Gash analytical models. J. Hydrol.
**1997**, 190, 141–162. [Google Scholar] [CrossRef] - Gash, J.H.; Lloyd, C.; Lachaud, G. Estimating sparse forest rainfall interception with an analytical model. J. Hydrol.
**1995**, 170, 79–86. [Google Scholar] [CrossRef] - Bishop, A.W. The use of the Slip Circle in the Stability Analysis of Slopes. Géotechnique
**1955**, 5, 7–17. [Google Scholar] [CrossRef] - Fellenius, W. Calculation of stability of earth dam. In Proceedings of the Transactions: 2nd Congress Large Dams, Washington, DC, USA, 7 September 1936; pp. 445–462. [Google Scholar]
- Janbu, N.; Bjerrum, L.; Kjaernsli, B. Soil Mechanics Applied to Some Engineering Probl Ems; Norwegian Geotechnical Institute: Oslo, Norway, 1956. [Google Scholar]
- Spencer, E. A Method of analysis of the Stability of Embankments Assuming Parallel Inter-Slice Forces. Géotechnique
**1967**, 17, 11–26. [Google Scholar] [CrossRef] - Bishop, A.W. The principle of effective stress. Tek. Ukebl.
**1959**, 39, 859–863. [Google Scholar] - Darcy, H.P.G. Les Fontaines Publiques de la Ville de Dijon. Exposition et Application des Principes à Suivre et des Formules à Employer Dans les Questions de Distribution D’eau, Etc; Victor Dalamont: Paris, France, 1856. [Google Scholar]
- Richards, L.A. Capillary Conduction of Liquids through Porous Mediums. Physics
**1931**, 1, 318–333. [Google Scholar] [CrossRef] - Anderson, M.; Lloyd, D. Using a combined slope hydrology-stability model to develop cut slope design charts. Proc. Inst. Civ. Eng. Part 2
**1991**, 91, 705–718. [Google Scholar] [CrossRef] - Wilkinson, P.L.; Anderson, M.G.; Lloyd, D.M.; Renaud, J.-P. Landslide hazard and bioengineering: Towards providing improved decision support through integrated numerical model development. Environ. Model. Softw.
**2002**, 17, 333–344. [Google Scholar] [CrossRef] - Greenwood, J.R. SLIP4EX–A program for routine slope stability analysis to include the effects of vegetation, reinforcement and hydrological changes. Geotech. Geol. Eng.
**2006**, 24, 449. [Google Scholar] [CrossRef] - Chok, Y.H. Modelling the Effects of Soil Variability and Vegetation on the Stability of Natural Slopes. Ph.D. Thesis, University of Adelaide, Adelaide, Australia, 2009. [Google Scholar]
- Anagnostopoulos, G. Hydrological Modelling of Slope Stability; ETH Zurich: Zürich, Switzerland, 2014. [Google Scholar]
- Aristizábal, E.; Vélez, J.I.; Martínez, H.E.; Jaboyedoff, M. SHIA_Landslide: A distributed conceptual and physically based model to forecast the temporal and spatial occurrence of shallow landslides triggered by rainfall in tropical and mountainous basins. Landslides
**2016**, 13, 497–517. [Google Scholar] [CrossRef] - Pasculli, A.; Calista, M.; Sciarra, N. Variability of local stress states resulting from the application of Monte Carlo and finite difference methods to the stability study of a selected slope. Eng. Geol.
**2018**, 245, 370–389. [Google Scholar] [CrossRef] - Vu-Bac, N.; Lahmer, T.; Zhuang, X.; Nguyen-Thoi, T.; Rabczuk, T. A software framework for probabilistic sensitivity analysis for computationally expensive models. Adv. Eng. Softw.
**2016**, 100, 19–31. [Google Scholar] [CrossRef] - Pollen, N.; Simon, A.; Collison, A. Advances in assessing the mechanical and hydrologic effects of riparian vegetation on streambank stability. In Riparian Vegetation and Fluvial Geomorphology; American Geophysical Union: Washington, DC, USA, 2004; pp. 125–139. [Google Scholar] [CrossRef]
- Pollen, N.; Simon, A. Estimating the mechanical effects of riparian vegetation on stream bank stability using a fiber bundle model. Water Resour. Res.
**2005**, 41. [Google Scholar] [CrossRef] - Schwarz, M.; Cohen, D.; Or, D. Root-soil mechanical interactions during pullout and failure of root bundles. J. Geophys. Res.
**2010**, 115. [Google Scholar] [CrossRef][Green Version] - Schwarz, M.; Giadrossich, F.; Cohen, D. Modeling root reinforcement using a root-failure Weibull survival function. Hydrol. Earth Syst. Sci.
**2013**, 17, 4367–4377. [Google Scholar] [CrossRef][Green Version] - Lu, N.; Godt, J.W. Hillslope Hydrology and Stability; Cambridge University Press: Cambridge, UK, 2013. [Google Scholar]
- Mickovski, S.B.; Hallett, P.D.; Bransby, M.F.; Davies, M.C.R.; Sonnenberg, R.; Bengough, A.G. Mechanical Reinforcement of Soil by Willow Roots: Impacts of Root Properties and Root Failure Mechanism. Soil Sci. Soc. Am. J.
**2009**, 73, 1276. [Google Scholar] [CrossRef] - Operstein, V.; Frydman, S. The influence of vegetation on soil strength. Proc. Inst. Civ. Eng. Ground Improv.
**2000**, 4, 81–89. [Google Scholar] [CrossRef] - Wu, T.H. Root reinforcement of soil: Review of analytical models, test results, and applications to design. Can. Geotech. J.
**2013**, 50, 259–274. [Google Scholar] [CrossRef] - Wu, T.H.; Watson, A. In situ shear tests of soil blocks with roots. Can. Geotech. J.
**1998**, 35, 579–590. [Google Scholar] [CrossRef] - Bischetti, G.B.; Chiaradia, E.A.; Epis, T.; Morlotti, E. Root cohesion of forest species in the Italian Alps. Plant Soil
**2009**, 324, 71–89. [Google Scholar] [CrossRef] - De Baets, S.; Poesen, J.; Reubens, B.; Wemans, K.; De Baerdemaeker, J.; Muys, B. Root tensile strength and root distribution of typical Mediterranean plant species and their contribution to soil shear strength. Plant Soil
**2008**, 305, 207–226. [Google Scholar] [CrossRef] - Docker, B.B.; Hubble, T.C.T. Quantifying root-reinforcement of river bank soils by four Australian tree species. Geomorphology
**2008**, 100, 401–418. [Google Scholar] [CrossRef] - Fan, C.-C.; Su, C.-F. Role of roots in the shear strength of root-reinforced soils with high moisture content. Ecol. Eng.
**2008**, 33, 157–166. [Google Scholar] [CrossRef] - Mao, Z.; Saint-André, L.; Genet, M.; Mine, F.-X.; Jourdan, C.; Rey, H.; Courbaud, B.; Stokes, A. Engineering ecological protection against landslides in diverse mountain forests: Choosing cohesion models. Ecol. Eng.
**2012**, 45, 55–69. [Google Scholar] [CrossRef] - Adhikari, A.R.; Gautam, M.R.; Yu, Z.; Imada, S.; Acharya, K. Estimation of root cohesion for desert shrub species in the Lower Colorado riparian ecosystem and its potential for streambank stabilization. Ecol. Eng.
**2013**, 51, 33–44. [Google Scholar] [CrossRef] - Meijer, G.J.; Bengough, A.G.; Knappett, J.A.; Loades, K.W.; Nicoll, B.C. In situ measurement of root reinforcement using corkscrew extraction method. Can. Geotech. J.
**2018**, 55, 1372–1390. [Google Scholar] [CrossRef][Green Version] - Zhu, H.; Zhang, L.M.; Xiao, T.; Li, X.Y. Enhancement of slope stability by vegetation considering uncertainties in root distribution. Comput. Geotech.
**2017**, 85, 84–89. [Google Scholar] [CrossRef] - Collison, A.; Anderson, M. Using a combined slope hydrology/stability model to identify suitable conditions for landslide prevention by vegetation in the humid tropics. Earth Surf. Process. Landf.
**1996**, 21, 737–747. [Google Scholar] [CrossRef] - Radcliffe, D.; Hayden, T.; Watson, K.; Crowley, P.; Phillips, R. Simulation of Soil Water within the Root Zone of a Corn Crop 1. Agron. J.
**1980**, 72, 19–24. [Google Scholar] [CrossRef] - Ali, N. The Influence of Vegetation Induced Moisture Transfer on Unsaturated Soils. Ph.D. Thesis, Cardiff University, Cardiff, UK, 2007. [Google Scholar]
- Rees, S.W.; Ali, N. Seasonal water uptake near trees: A numerical and experimental study. Geomech. Geoengin.
**2006**, 1, 129–138. [Google Scholar] [CrossRef] - Kokutse, N.; Fourcaud, T.; Kokou, K.; Neglo, K.; Lac, P. 3D numerical modelling and analysis of the influence of forest structure on hill slopes stability. In Interpraevent; Universal Academy Press: Tokyo, Japan, 2006; pp. 561–567. [Google Scholar]
- Fatahi, B. Modelling of Influence of Matric Suction Induced by Native Vegetation on Sub-Soil Improvement. Ph.D. Thesis, University of Wollongong, Wollongong, Australia, 2007. [Google Scholar]
- Levin, A.; Shaviv, A.; Indelman, P. Influence of root resistivity on plant water uptake mechanism, part I: Numerical solution. Transp. Porous Media
**2007**, 70, 63–79. [Google Scholar] [CrossRef] - Novák, V. Estimation of soil-water extraction patterns by roots. Agric. Water Manag.
**1987**, 12, 271–278. [Google Scholar] [CrossRef] - Perrochet, P. Water uptake by plant roots—A simulation model, I. Conceptual model. J. Hydrol.
**1987**, 95, 55–61. [Google Scholar] [CrossRef] - Feddes, R.A. Simulation of field water use and crop yield. In Simulation of Plant Growth and Crop Production; Penning de Vries, F.W.T., Laar, H.H.V., Eds.; Pudoc: Wageningen, The Netherlands, 1982; pp. 194–209. [Google Scholar]
- Dobson, M.; Moffat, A. A re-evaluation of objections to tree planting on containment landfills. Waste Manag. Res.
**1995**, 13, 579–600. [Google Scholar] [CrossRef] - Landsberg, J. Tree water use and its implications in relation to agroforestry systems. In The Ways Trees Use Water; Landsberg, J., Ed.; Rural Industries Research and Development Corporation (RIRDC): Canberra, Australia, 1999; pp. 1–27. [Google Scholar]
- Fatahi, B.; Khabbaz, H.; Indraratna, B. Bioengineering ground improvement considering root water uptake model. Ecol. Eng.
**2010**, 36, 222–229. [Google Scholar] [CrossRef] - Bishop, D.M.; Stevens, M.E. Landslides on Logged Areas in Southeast Alaska; Northern Forest Experiment Station, Forest Service, U.S. Dept. of Agriculture: Alaska, AL, USA, 1964.
- Kokutse, N.K.; Temgoua, A.G.T.; Kavazović, Z. Slope stability and vegetation: Conceptual and numerical investigation of mechanical effects. Ecol. Eng.
**2016**, 86, 146–153. [Google Scholar] [CrossRef] - Van Asch, T.W.J.; Deimel, M.S.; Haak, W.J.C.; Simon, J. The viscous creep component in shallow clayey soil and the influence of tree load on creep rates. Earth Surf. Process. Landf.
**1989**, 14, 557–564. [Google Scholar] [CrossRef] - Simon, A.; Collison, A.J.C. Quantifying the mechanical and hydrologic effects of riparian vegetation on streambank stability. Earth Surf. Process. Landf.
**2002**, 27, 527–546. [Google Scholar] [CrossRef] - Abernethy, B.; Rutherfurd, I.D. Does the weight of riparian trees destabilize riverbanks? Regul. Rivers Res. Manag.
**2000**, 16, 565–576. [Google Scholar] [CrossRef] - Waldron, L.J.; Dakessian, S. Effect of Grass, Legume, and Tree Roots on Soil Shearing Resistance1. Soil Sci. Soc. Am. J.
**1982**, 46, 894–899. [Google Scholar] [CrossRef] - Kim, D.; Im, S.; Lee, C.; Woo, C. Modeling the contribution of trees to shallow landslide development in a steep, forested watershed. Ecol. Eng.
**2013**, 61, 658–668. [Google Scholar] [CrossRef] - Chiaradia, E.A.; Vergani, C.; Bischetti, G.B. Evaluation of the effects of three European forest types on slope stability by field and probabilistic analyses and their implications for forest management. For. Ecol. Manag.
**2016**, 370, 114–129. [Google Scholar] [CrossRef] - Bischetti, G.B.; Bassanelli, C.; Chiaradia, E.A.; Minotta, G.; Vergani, C. The effect of gap openings on soil reinforcement in two conifer stands in northern Italy. For. Ecol. Manag.
**2016**, 359, 286–299. [Google Scholar] [CrossRef] - Horton, R.E. Rainfall interception. Mon. Weather Rev.
**1919**, 47, 603–623. [Google Scholar] [CrossRef] - Muzylo, A.; Llorens, P.; Valente, F.; Keizer, J.J.; Domingo, F.; Gash, J.H.C. A review of rainfall interception modelling. J. Hydrol.
**2009**, 370, 191–206. [Google Scholar] [CrossRef][Green Version] - Celia, M.A.; Bouloutas, E.T.; Zarba, R.L. A general mass-conservative numerical solution for the unsaturated flow equation. Water Resour. Res.
**1990**, 26, 1483–1496. [Google Scholar] [CrossRef] - Brooks, R.H.; Corey, A.T. Hydraulic properties of porous media and their relation to drainage design. Trans. ASAE
**1964**, 7, 26–28. [Google Scholar] - Fredlund, D.; Xing, A. Equations for the soil-water characteristic curve. Can. Geotech. J.
**1994**, 31, 521–532. [Google Scholar] [CrossRef] - Fredlund, D.; Xing, A.; Huang, S. Predicting the permeability function for unsaturated soils using the soil-water characteristic curve. Can. Geotech. J.
**1994**, 31, 533–546. [Google Scholar] [CrossRef] - Gardner, W. Some steady-state solutions of the unsaturated moisture flow equation with application to evaporation from a water table. Soil Sci.
**1958**, 85, 228–232. [Google Scholar] [CrossRef] - Haverkamp, R.; Vauclin, M.; Touma, J.; Wierenga, P.; Vachaud, G. A comparison of numerical simulation models for one-dimensional infiltration 1. Soil Sci. Soc. Am. J.
**1977**, 41, 285–294. [Google Scholar] [CrossRef] - Mualem, Y. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res.
**1976**, 12, 513–522. [Google Scholar] [CrossRef][Green Version] - Van Genuchten, M.T. A Closed-form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils1. Soil Sci. Soc. Am. J.
**1980**, 44, 892. [Google Scholar] [CrossRef] - Ma, K.-C.; Tan, Y.-C.; Chen, C.-H. The influence of water retention curve hysteresis on the stability of unsaturated soil slopes. Hydrol. Process.
**2011**, 25, 3563–3574. [Google Scholar] [CrossRef] - Pasculli, A.; Sciarra, N.; Esposito, L.; Esposito, A.W. Effects of wetting and drying cycles on mechanical properties of pyroclastic soils. CATENA
**2017**, 156, 113–123. [Google Scholar] [CrossRef] - Duncan, J.M.; Wright, S.G.; Brandon, T.L. Soil Strength and Slope Stability; John Wiley & Sons: Hoboken, NJ, USA, 2014. [Google Scholar]
- Janbu, N. Slope stability computations. In Embankment-Dam Engineering; Hirschfeld, R., Poulos, S., Eds.; John Wiley and Sons Inc.: NewYork, NY, USA, 1973. [Google Scholar]
- Fredlund, D.; Krahn, J. Comparison of slope stability methods of analysis. Can. Geotech. J.
**1977**, 14, 429–439. [Google Scholar] [CrossRef] - Terzaghi, K. Theoretical Soil Mechanics; John Wiley & Sons: New York, NY, USA, 1943. [Google Scholar]
- Fredlund, D.; Morgenstern, N.R.; Widger, R. The shear strength of unsaturated soils. Can. Geotech. J.
**1978**, 15, 313–321. [Google Scholar] [CrossRef] - Skempton, A.W. Effective stress in soils, concrete and rocks. In Proceedings of the Conference on Pore Pressure and Suction in Soils, London, UK, 30–31 March 1960; pp. 4–16. [Google Scholar]
- Kristo, C.; Rahardjo, H.; Satyanaga, A. Effect of variations in rainfall intensity on slope stability in Singapore. Int. Soil Water Conserv. Res.
**2017**, 5, 258–264. [Google Scholar] [CrossRef] - Abramson, L.W.; Lee, T.S.; Sharma, S.; Boyce, G.M. Slope Stability and Stabilization Methods; John Wiley & Sons: Hoboken, NJ, USA, 2001. [Google Scholar]
- Shahraiyni, H.T.; Ataie-Ashtiani, B. Mathematical Forms and Numerical Schemes for the Solution of Unsaturated Flow Equations. J. Irrig. Drain. Eng.
**2012**, 138, 63–72. [Google Scholar] [CrossRef] - Srivastava, R.; Guzman-Guzman, A. Analysis of Hydraulic Conductivity Averaging Schemes for One-Dimensional, Steady-State Unsaturated Flow. Ground Water
**1995**, 33, 946–952. [Google Scholar] [CrossRef] - Rathfelder, K.; Abriola, L.M. Mass conservative numerical solutions of the head-based Richards equation. Water Resour. Res.
**1994**, 30, 2579–2586. [Google Scholar] [CrossRef] - Vogel, T. SWMII-Numerical Model of Two-Dimensional Flow in a Variably Saturated Porous Medium; Wageningen Agricultural University: Wageningen, The Netherlands, 1988. [Google Scholar]
- Vauclin, M.; Khanji, D.; Vachaud, G. Experimental and numerical study of a transient, two-dimensional unsaturated-saturated water table recharge problem. Water Resour. Res.
**1979**, 15, 1089–1101. [Google Scholar] [CrossRef] - Clement, T.; Wise, W.R.; Molz, F.J. A physically based, two-dimensional, finite-difference algorithm for modeling variably saturated flow. J. Hydrol.
**1994**, 161, 71–90. [Google Scholar] [CrossRef] - Xing, Z. Three-Dimensional Stability Analysis of Concave Slopes in Plan View. J. Geotech. Eng.
**1988**, 114, 658–671. [Google Scholar] [CrossRef] - Chen, Z.; Wang, X.; Haberfield, C.; Yin, J.-H.; Wang, Y. A three-dimensional slope stability analysis method using the upper bound theorem: Part I: Theory and methods. Int. J. Rock Mech. Min. Sci.
**2001**, 38, 369–378. [Google Scholar] [CrossRef] - Chen, Z.; Mi, H.; Zhang, F.; Wang, X. A simplified method for 3D slope stability analysis. Can. Geotech. J.
**2003**, 40, 675–683. [Google Scholar] [CrossRef] - Sultan, N.; Gaudin, M.; Berne, S.; Canals, M.; Urgeles, R.; Lafuerza, S. Analysis of slope failures in submarine canyon heads: An example from the Gulf of Lions. J. Geophys. Res.
**2007**, 112. [Google Scholar] [CrossRef][Green Version] - Ge, X.-R. The vector sum method: A new approach to calculating the safety factor of stability against sliding for slope engineering and dam foundation problems. In Advances in Environmental Geotechnics; Springer: Berlin/Heidelberg, Germany, 2010; pp. 99–110. [Google Scholar]
- Sun, G.; Zheng, H.; Jiang, W. A global procedure for evaluating stability of three-dimensional slopes. Nat. Hazards
**2011**, 61, 1083–1098. [Google Scholar] [CrossRef] - Liu, G.; Zhuang, X.; Cui, Z. Three-dimensional slope stability analysis using independent cover based numerical manifold and vector method. Eng. Geol.
**2017**, 225, 83–95. [Google Scholar] [CrossRef] - Li, K.S.; White, W. Rapid evaluation of the critical slip surface in slope stability problems. Int. J. Numer. Anal. Methods Geomech.
**1987**, 11, 449–473. [Google Scholar] [CrossRef] - Hungr, O.; Salgado, F.; Byrne, P. Evaluation of a three-dimensional method of slope stability analysis. Can. Geotech. J.
**1989**, 26, 679–686. [Google Scholar] [CrossRef] - Lam, L.; Fredlund, D. A general limit equilibrium model for three-dimensional slope stability analysis. Can. Geotech. J.
**1993**, 30, 905–919. [Google Scholar] [CrossRef] - Huang, C.-C.; Tsai, C.-C. New Method for 3D and Asymmetrical Slope Stability Analysis. J. Geotech. Geoenviron. Eng.
**2000**, 126, 917–927. [Google Scholar] [CrossRef] - Kim, J.; Salgado, R.; Lee, J. Stability Analysis of Complex Soil Slopes using Limit Analysis. J. Geotech. Geoenviron. Eng.
**2002**, 128, 546–557. [Google Scholar] [CrossRef] - Zheng, H. Eigenvalue Problem from the Stability Analysis of Slopes. J. Geotech. Geoenviron. Eng.
**2009**, 135, 647–656. [Google Scholar] [CrossRef] - Zheng, H. A three-dimensional rigorous method for stability analysis of landslides. Eng. Geol.
**2012**, 145–146, 30–40. [Google Scholar] [CrossRef] - Griffiths, D.; Lu, N. Unsaturated slope stability analysis with steady infiltration or evaporation using elasto-plastic finite elements. Int. J. Numer. Anal. Methods Geomech.
**2005**, 29, 249–267. [Google Scholar] [CrossRef]

**Figure 3.**(

**a**) Schematic view of soil–root interaction model; (

**b**) the dominant mechanism in root reinforcement (adapted from Zhu, et al. [60]).

**Figure 4.**The general form of soil suction factor (adapted from [70]).

**Figure 5.**Schematic image of root distribution (modified from [66]).

**Figure 9.**(

**a**) The required nodes to define slope geometry; (

**b**) the discretized domain in the finite difference method (FDM).

**Figure 13.**Homogeneous slope studied by Fredlund and Krahn [97].

**Figure 14.**Homogeneous slope with a weak layer, studied by Fredlund and Krahn [97].

**Figure 15.**(

**a**) The geometry of the slope, and (

**b**) safety factor (SF) for different elevations of the water table.

**Figure 16.**Schematic profile of slope used in example 5 [60].

**Figure 17.**(

**a**) The factor of safety and (

**b**) the maximum depth of the slip surface for the vegetated slope.

**Table 1.**Influences of vegetation on slope stability [17].

Mechanism | Result | Influence |
---|---|---|

Root (underground) portion | ||

Reinforcement and anchorage by root | Increasing soil shear strength | + |

Root extracts moisture from the soil | Lower pore water pressure | + |

Increasing hydraulic conductivity | Increased infiltration capacity | − |

Canopy (aboveground) portion | ||

Canopy intercepts and evaporates rainfall | Reduce rainfall for infiltration | + |

Weight of trees surcharges the slope | Increasing normal stress | +/− |

Vegetation exposed wind forces into the slope | Increasing driving force | − |

Reference | k″ | Method | Soil | Vegetation |
---|---|---|---|---|

Wu and Watson [52] | 0.33 | In-situ shear test | Silty sand | Pinus radiata |

Operstein and Frydman [50] | 0.21 | In-situ and laboratory tests | Chalky and clay | Alfalfa, Rosemary, Pistacia lentiscus, Cistus |

Pollen, et al. [44] | 0.34 | Direct shear-box test | Clayey-silt | Riparian vegetation (12 species) |

Pollen and Simon [45] | Trees: 0.6–0.82 Grass: 0.48 | In comparison with FBM | Silt | trees: Cottonwood, Sycamore, River birch, Pine, Black willow grass: Switchgrass |

Docker and Hubble [55] | 0.61–0.64 | In-situ shear test | Alluvial (loam and sandy loam) | Casuarina glauca, Eucalyptus amplifolia, Eucalyptus elata, and Acacia floribunda |

Fan and Su [56] | Peak: 0.325 Residual: 0.35 | In-situ shear test | Sands mixed with silts | Prickly sesban |

Bischetti, et al. [53] | 0.27–0.83 | Direct shear test | Various (gravel-sand mixture, clayey, silt, …) | European beech, Norway spruce, European larch, Sweet chestnut, European hop-hornbeam |

Mickovski, et al. [49] | ~0.75 | Direct shear test (laboratory) | Agricultural soil (71% sand, 19% silt, 10% clay) | Willow |

Mao, et al. [57] | 0.55–1.0 | In comparison with FBM | Various (silt, silty-clay, coarse elements) | Norway spruce, Silver fir, European beech |

Adhikari, et al. [58] | 0.35–0.56 | In comparison with FBM | Fine sand texture | A. lentiformis, A. occidentialis, L. andersonii, L. tridentata |

Meijer, et al. [59] | 0.08, 0.225 | Corkscrew test (field and lab.) | Slightly clayey sand, sandy silt | Blackcurrant (shrub), Sitka spruce (tree) |

Species | Surcharge (kPa) | Size Indicator | Density (tree/ha) | Reference |
---|---|---|---|---|

Sitka spruce | Estimated: 5.2 (Used: 3.8) | Avg. height = 6 m | Dense | Wu et al. [19] |

Sitka spruce | Average: 2.5 | Bishop and Stevens [74] | ||

Maritime pine | 0.6 | 350 | Kokutse et al. [75] | |

Conifer forest | up to 2 | Height = up to 80 m | Fully stocked | Greenway [17] |

Pinus sylvestris | 3.5 | Van Asch et al. [76] | ||

Riparian vegetation | 1.2 | Avg. height = 18 m | Simon and Collison [77] | |

Silver wattle | Average tree: 0.81 Large tree: 5.06 | 5000 | Abernethy and Rutherfurd [78] | |

Pine | 0.228, 0.135 | Age = 52-month | Waldron and Dakessian [79] | |

Korean pine | 2.94 | Age = 20-year-old | Fully stocked | Kim et al. [80] |

European beech | 0.309 | Avg. stem dia. = 14–42 cm | 308–2451 | Chiaradia et al. [81] |

Sweet chestnut | 0.070 | Avg. stem dia. = 13–31 cm | 2268–3764 | |

Norway spruce | 0.275 | Avg. stem dia. = 22–46 cm | 416–2066 | |

Mixed conifer forest | 0.275 | Height ≈ 30 m | Bischetti et al. [82] |

**Table 4.**The available soil-water characteristic curve (SWCC) and hydraulic conductivity function (HCF) relationships in the integrated two-dimensional slope stability model (SSHV-2D).

Reference | SWCC | HCF | Parameters |
---|---|---|---|

Haverkamp et al. [90] | $\theta (h)=\left({\theta}_{s}-{\theta}_{r}\right)\frac{\alpha}{\alpha +{\left|h\right|}^{\beta}}+{\theta}_{r}$ | $K(h)={K}_{s}\frac{A}{A+{\left|h\right|}^{B}}$ | α, β: Fitting parameters A, B: Fitting parameters |

Van Genuchten [92] | $\theta (h)=\frac{{\theta}_{s}-{\theta}_{r}}{{\left[1+{\left(\alpha \left|h\right|\right)}^{n}\right]}^{m}}+{\theta}_{r}$ | $K(h)={K}_{s}\frac{{\left\{1-\frac{{\left(\alpha \left|h\right|\right)}^{n-1}}{{\left[1+{\left(\alpha \left|h\right|\right)}^{n}\right]}^{m}}\right\}}^{2}}{{\left[1+{\left(\alpha \left|h\right|\right)}^{n}\right]}^{\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$ | α, n, m: fitting parameters where: $m=1-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.$ |

Fredlund and Xing [87], Fredlund et al. [88] | $\theta (h)=C(\psi )\frac{{\theta}_{s}-{\theta}_{r}}{{\left[\mathrm{ln}\left(e+{\left(\frac{\psi}{a}\right)}^{n}\right)\right]}^{m}}+{\theta}_{r}$ where: $C(\psi )=1-\frac{\mathrm{ln}\left(1+\frac{\psi}{{\psi}_{r}}\right)}{\mathrm{ln}\left(1+\frac{{10}^{6}}{{\psi}_{r}}\right)}$ $\psi ={\gamma}_{w}\xb7\left|h\right|$ | $K(h)={K}_{s}\frac{{\displaystyle {\int}_{\mathrm{ln}(\psi )}^{\mathrm{ln}({10}^{6})}\frac{\theta ({e}^{y})-\theta (\psi )}{{e}^{y}}{\theta}^{\prime}({e}^{y})dy}}{{\displaystyle {\int}_{\mathrm{ln}({\psi}_{aev})}^{\mathrm{ln}({10}^{6})}\frac{\theta ({e}^{y})-{\theta}_{s}}{{e}^{y}}{\theta}^{\prime}({e}^{y})dy}}$ where: ${\theta}^{\prime}\left(\psi \right)=\frac{\partial \theta}{\partial \psi}$ | e: the natural number a, n, m: fitting parameters ψ: matric suction (varied between 0 to 10 ^{6} kPa)ψ _{r}: matric suction corresponding to residual water contentC(ψ): correction factor ψ _{aev}: matric suction at air entry valuey: dummy variable of integration |

_{s}is saturated water content, θ

_{r}is residual water content, K

_{s}is saturated hydraulic conductivity, and h is pressure head (h is negative in the unsaturated zone).

Parameter | Arithmetic Method | Geometric Method | Harmonic Method |
---|---|---|---|

K_{avg} | $\frac{1}{2}\left({K}_{1}+{K}_{2}\right)$ | $\sqrt{{K}_{1}\times {K}_{2}}$ | $\frac{{K}_{1}\times {K}_{2}}{{K}_{1}+{K}_{2}}$ |

**Table 6.**Summary of parameters used in the water table recharge problem [107].

Parameter | Value |
---|---|

Soil-Water Characteristic Curve (SWCC) parameters | |

θ_{S} (cm^{3}/cm^{3}) | 0.30 |

θ_{r} (cm^{3}/cm^{3}) | 0 |

α | 40,000 |

β | 2.90 |

Hydraulic conductivity function parameters | |

K_{s} (cm/h) (in both horizontal and vertical directions) | 35 |

A | 2.99 × 10^{6} |

B | 5.0 |

Numerical assumption | |

∆x (cm) | 10 |

∆z (cm) | 5 |

∆t (hr) | 0.1 |

Time stepping method | Automatic |

Hydraulic conductivity averaging method | Geometric |

Estimation method of specific moisture capacity | SCS |

Reference | Analysis Method | SF |
---|---|---|

Fredlund and Krahn [97] | Simplified Bishop Method | 2.080 |

Janbu’s Simplified Method | 2.041 | |

Xing [109] | Proposed 3D Method | 2.122 |

Chen et al. [110] | Upper bound method | 2.262 |

Chen et al. [111] | Proposed 3D Method (STAB-3D) | 2.188 |

Plain-Strain 3D Method | 2.073 | |

Sultan et al. [112] | Upper Bound Theorem (SAMU-3D Program) | 2.213 |

Ge [113] | Vector Sum Method (VSM) | 2.037 |

Sun et al. [114] | Proposed 3D Method | 2.000 |

Liu et al. [115] | 3D independent cover-based manifold method (ICMM3D) and vector sum method (VSM). | 2.061 |

This Study (SSHV-2D) | Simplified Bishop Method | 2.079 |

Janbu’s Simplified Method | 2.024 |

Reference | Analysis Method (Program Name) | D (m) | Condition of Weak Layer | SF |
---|---|---|---|---|

Fredlund and Krahn [97] | Simplified Bishop Method | 6.1 | Oblique * | 1.377 |

Janbu’s Simplified Method | 6.1 | Oblique * | 1.448 | |

Li and White [116] | New Proposed Method | 4 | Horizontal | 1.387 |

Xing [109] | Proposed 3D Method | 6.1 | Oblique * | 1.548 |

Hungr et al. [117] | 3D extension of the Bishop’s Simplified method (CLARA) | N/A | N/A | 1.62 |

Lam and Fredlund [118] | Janbu’s Simplified Method | 5 | Horizontal | 1.558 |

Huang and Tsai [119] | Modified Bishop Simplified Method | ~4.6 | Horizontal | 1.658 |

Kim et al. [120] | Lower-Bound Method | 4.6 | Horizontal | 1.25 |

Upper-Bound Method | 1.37 | |||

Chen et al. [111] | Proposed 3D Method (STAB-3D) | ~4.6 | Horizontal | 1.64 |

Plain-Strain 3D Method | 1.384 | |||

Ge [113] | Vector Sum Method (VSM) | 4.55 | Horizontal | 1.585 |

Zheng [121] | Proposed 3D Method | 4.55 | Horizontal | 1.707 |

Sun et al. [114] | Proposed 3D Limit Equilibrium Method | 4.55 | Horizontal | 1.68 |

Zheng [122] | Spencer’s Method (RMP3D) | 4.55 | Horizontal | 1.735 |

Corps of Engineers Assumption | 1.766 | |||

Liu et al. [115] | 3D independent cover-based manifold method (ICMM3D) and vector sum method (VSM). | 4.55 | Horizontal | 1.530 |

This Study | Janbu’s Simplified Method (SSHV-2D) | 6.1 | Oblique * | 1.391 |

5 | Horizontal | 1.446 | ||

4.55 | Horizontal | 1.489 |

**Table 9.**Root properties for modeling of soil–root interactions [60].

Definition | Parameter (Unit) | Value |
---|---|---|

Root tensile capacity | T (kN) | 12.5 |

Root pullout resistance | P (kN/m) | 2.5 |

Lateral bending strength of the root | Q (kN) | 6.25 |

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Emadi-Tafti, M.; Ataie-Ashtiani, B. A Modeling Platform for Landslide Stability: A Hydrological Approach. *Water* **2019**, *11*, 2146.
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Emadi-Tafti M, Ataie-Ashtiani B. A Modeling Platform for Landslide Stability: A Hydrological Approach. *Water*. 2019; 11(10):2146.
https://doi.org/10.3390/w11102146

**Chicago/Turabian Style**

Emadi-Tafti, Mohsen, and Behzad Ataie-Ashtiani. 2019. "A Modeling Platform for Landslide Stability: A Hydrological Approach" *Water* 11, no. 10: 2146.
https://doi.org/10.3390/w11102146