# 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

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**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.
https://doi.org/10.3390/w11102146

**AMA Style**

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