# The Effects of Slope Initialization on the Numerical Model Predictions of the Slope-Vegetation-Atmosphere Interaction

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Prototype Slope

_{p}′ = 5 kPa − ϕ

_{p}′ = 14°, to c

_{p}′ = 20 kPa − ϕ

_{p}′ = 18° (Figure 2), if a few higher-strength soil samples are excluded [3]. The maximum rate of the landsliding occurs in late winter and it may reach tens of centimetres per year [3]. Landslides have severely damaged both an Apulian Aqueduct pipeline (lying few metres below ground level) and a road, both located at the base of the slope (Figure 1).

## 3. Fully Coupled Hydromechanical Modelling

#### 3.1. Governing Equations

#### 3.2. Geometry, Discretisation and Hydromechanical Parameters of the Slope Model

## 4. Initialization of the FE Slope Model

_{0}later) conditions across a horizontal clay stratum, before running an excavation stage to generate the clay slope. The authors show that the k

_{0_initial}value (i.e., the ratio of the principal horizontal effective stress to the principal vertical effective stress in oedometric conditions) impacts the numerical prediction of progressive failure in the slope after excavation, in particular when using a strain-softening constitutive law. The k

_{0_initial}value not only affects the morphology of the shear bands forming after excavation, but also the timing of progressive failure.

_{0}procedure” was adopted in the initialization stage (Table 2; analyses C to H), in which the initial ground level is horizontal and the stress‒strain conditions are oedometric, with a value of k

_{0_initial}set by the user. Hence, the numerical integration has been run, in drained conditions, to reach equilibrium in the whole model, resulting in a final k

_{0}value depending on the selected soil constitutive law and parameter values. After the initialization stage, the profile of the slope (Figure 4) was obtained by running the drained excavation of nine soil layers, each of 25 m depth, in order to simulate the river erosion. In the excavation phases, the steady state seepage through the slope model was run, accounting for the following hydraulic boundary conditions: (i) zero pore water pressure at the horizontal ground surface upslope, coherent with the presence of springs and ponding; (ii) zero pore water pressure at the ground surface downslope, simulating the presence of the Ofanto River; (iii) a suction of 40 kPa along the sloping ground level, according to the average suction monitored in situ at the ground surface [3]; (iv) impervious condition along both the lateral vertical boundaries and the bottom horizontal one.

_{0_initial}values set in the analyses carried out adopting the “k

_{0}procedure” have been: 0.428 (C), 0.65 (E), 1 (F), 1.5 (G) and 2 (H), with ν’ = 0.3; k

_{0_initial}= 0.65, with ν’ = 0.39 (D). In particular, the analyses implementing high k

_{0_initial}values, F (k

_{0_initial}= 1), G (k

_{0_initial}= 1.5), and H (k

_{0_initial}= 2), account for the high horizontal stresses caused by both the overconsolidation of the clay and the lateral compression due to tectonics. The alternative procedure used in the initialization of the slope model was “gravity loading.” This procedure sets the initial stress states by applying the soil self-weight to the slope model, which is set to have the final geometry from the start. The initial ratio of the normal effective stress on the vertical plane to the normal effective stress on the horizontal plane (which are not principal planes) is automatically set to the k

_{0}value in oedometric condition for an elastic material:

_{0}higher than 0.65 cannot be achieved using a reasonable value of $\mathsf{\nu}\prime $, i.e., $\mathsf{\nu}\prime $ ≤ 0.4. Therefore, the gravity loading initialization could not be run for ${k}_{0\_el}$ higher than 0.65. The seepage conditions in analyses A and B were set to be equivalent to those set in the other analyses.

#### Discussion of the Initialization Stage Results

_{0}procedure, Figure 8c–h). The corresponding shear strain fields are shown in Figure 9.

_{0}procedure and using the same parameter values of either analysis A or B, the numerical modelling achieved convergence. This was the case even if the predicted fields of plastic points and of shear strain in the analyses C and D were rather similar to those resulting from the corresponding analyses A and B, respectively. However, in the analyses using the k

_{0}initialization procedure, by the end of the initialization stage the mobilized shear stresses were lower than those mobilized in analyses A and B in large portions of the slope model. By the end of the initialization stage, the slope achieved a safety factor of F = 1.135 in analysis C and F = 1.140 in analysis D (Table 3). All the reported safety factor values (Table 3) have been computed through the shear strength reduction technique [76,77]. It can be concluded that the strain fields predicted through the analysis using the Mohr‒Coulomb model are not so sensitive to the differences in initialization procedure, as already reported by Griffiths et al. [78]. Nonetheless, the size of the mobilized shear stress depends on the initialization procedure. Evidently, the c’ and ϕ’ values representative for the reference slopes (Table 1) are too low to guarantee equilibrium in the slope, once the initialization procedure disregards the loading history of the slope, as is the case with gravity loading.

_{0}procedure (i.e., analyses E, F, G and H), whose factors of safety are reported in Table 3. For all the analyses initialized through the k

_{0}procedure (analyses C to H), the values of the factor of safety are slightly higher than 1.1.

_{0_initial}<1, the propagation of failure is characterized by an advancing mode. It starts at the top of the slope and propagates downslope, but does not reach the toe by the end of the initialization (Figure 8c–e and Figure 9c–e). On the contrary, when k

_{0_initial}≥1, failure starts at the toe and retrogresses.

_{0_initial}= 1 (i.e., analysis F), the shear band appears to acquire a morphology compatible with the progressive development of a deep roto-translational sliding mechanism. For k

_{0_initial}>1, instead, the shear band tends to deepen towards the bedrock of the slope model (Figure 9g,h). Moreover, the bigger the k

_{0_initial}, the higher the deepening of the shear band. The deepening of the shear bands has been previously observed by Potts et al. [74], although when using a strain softening Mohr‒Coulomb model.

_{0_initial}values, the highest shear strains are achieved in the upper portion of the slope, where the yielding may be largely a tension yield (i.e., controlled by tension cutoff).

_{0_initial}>0.65 (i.e., analyses F, G and H), the stress paths at N follow similar trends. They are all characterized by an initial increase in deviatoric stress associated with a decrease in mean normal effective stress, which corresponds to a first stage of the valley excavation. After the stress paths approach the yield envelope, they move along it, during the unloading, while remaining at yield. It is worth clarifying that the stress paths in Figure 10 seem not to join the yield envelope, but rather follow lines at a small distance from it, merely due to a numerical tolerance adopted in the calculation, since point N in Figure 4 is predicted always to reach yield. Furthermore, the analysis predicts that the onset of yielding at N delays with reducing k

_{0_initial}.

_{0_initial}equal to 0.428 and to 0.65 (analyses C and E), the first part of the stress path complies with an initial reduction in deviatoric stress upon the vertical unloading due to excavation. Thereafter, the approach of the isotropic stress state, the stress path starts following a trend similar to that of the stress paths for k

_{0_initial}>0.65 described before.

_{0}= 1.5 and k

_{0}= 2, the shear strains localized in the shear band are much larger than for 0.65 ≤ k

_{0}≤ 1, and the yielding is more diffuse. However, the morphology of the shear band predicted by the analyses in these cases is not such as to predispose the mobilization of a roto-translational landslide body. Conversely, the analysis with k

_{0_initial}= 1 predicts the generation of a shear band predisposing the onset of roto-translational landsliding.

## 5. The Predicted SLVA Interaction

_{0_initial}= 1) in Figure 12. Point V is at 15 m depth and point W at 36 m depth. The results of analysis F are very close to those achieved in all the other analyses initialized with the k

_{0}procedure (C to H).

_{0_initial}= 1 (analysis F), the shear strain increments localize in a shear band that represents the further development of the shear band formed by the slope toe at the end of the river erosion stage. In this case, the maximum shear strain increment is 16.3%; in the other cases, instead, the shear strain increments are much lower. Evidently, for k

_{0_initial}values equal to 0.65, 1.5 and 2, the stress strain history of the soils across the slope (modelled through the initialization stage) has caused major straining and yielding in portions of the slope that do not appear to have significant further loading as an effect of the SLVA interaction. Hence, this interaction is not capable of generating important new shear straining.

_{0}procedure, assuming k

_{0_initial}= 1. This is consistent with a geological history in which the soils have been overconsolidated, then subjected to significant tectonic-induced lateral loading and, later, to lateral unloading due to river erosion.

## 6. Conclusions and Future Research Perspectives

_{0}procedure, using k

_{0_initial}= 1, in order to account for the important lateral loading that the slope soils have been subjected to in their history, and for the following unloading due to the excavation of the valley caused by the river erosion.

_{0}initialization procedure, the SLVA interaction has been shown to cause seasonal piezometric fluctuations down to a large depth. For k

_{0_intial}= 1, these have been shown to determine a slope progressive failure capable of generating, in the long term, the base portion of the shear band of a possible deep roto-translational landslide.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Geological map of the Southern Italy, together with both the limited grey area representing the Southern-Eastern Apennine and the location of the Pisciolo slope. Geological map (

**b**) and sections (

**c**) of the Pisciolo slope (adapted from [3]); section I‒II is the geological section, whereas section III‒IV is the geotechnical one, whose traces are reported; Key: 1) debris and alluvial deposits; 2) N sandstones; 3) PD clays (a; fractured rock strata), 4) R clays; 5) stratigraphic contact (i), fault (ii) and anticline axis (iii); 6) landslide crown (i) and body (ii); 7) borehole with piezometers (P), or with inclinometer (I), GPS sensor (S2); 8) line of section; 9) inclinometer shear bending (i), disturbed soil (ii) and piezometer cell (iii); 10) labelled landslide—crown (i) and slip surface (ii) (Cotecchia et al. 2019).

**Figure 2.**Undrained shear stress paths of Pisciolo clay samples (revisited after [3]) and shear strength envelopes: 1) minimum recorded strength; 2) maximum recorded strength; 3) used in the modelling.

**Figure 3.**The 180-day cumulative rainfall (weather station located in Melfi, 3 km far from the Pisciolo slope), piezometric levels at both 15 m and 36 m depths (vertical P7 in Figure 1) and displacement rates measured at 19 m depth (inclinometer I12 in Figure 1) and at the ground level (GPS S2 in Figure 1); data from [3] and new data.

**Figure 4.**Mesh of the numerical model, together with the final geometry of the slope after the initialization stage (bold contour). Both the red and the blue points (M, N) are 5 m below ground level.

**Figure 5.**Schematic outline of the response of overconsolidated clays to free drying, according to the framework proposed by Cafaro and Cotecchia [69]: (

**a**) void ratio—mean effective stress path of the clay under either isotropic compression, or free-drying; (

**b**,

**c**) Cafaro and Cotecchia [69] model. The dashed line represents the response to external loading; the continuous thin line represents the response to drying when S = 1 for s < s

_{des}; the dash and dot line represents the response to drying when 0.9 < S < 1 for s < s

_{des}.

**Figure 7.**Soil water retention curve (

**a**) and hydraulic conductivity function (

**b**) used in the modelling (Table 1).

**Figure 8.**Plastic points at the end of the initialization stage of the numerical analyses listed in Table 2. Figures from

**a**) to

**h**) correspond to the analyses from A to H respectively.

**Figure 9.**Shear strains at the end of the initialization stage of the numerical analyses listed in Table 2. Figures from

**a**) to

**h**) correspond to the analyses from A to H respectively.

**Figure 10.**Stress paths during the excavation phases at the stress points M and N (

**a**); zoom of the stress paths at point M (

**b**). The black thin line represents the Mohr-Coulomb strength envelope.

**Figure 13.**Cumulated shear strain after 10 years of net rainfall at the ground surface;

**a**) analysis E,

**b**) analysis F,

**c**) analysis G and

**d**) analysis H.

**Figure 14.**Analysis F: evolution of the progressive failure due to the SLVA interaction in terms of cumulated shear strain during the 10 years of net rainfall; the strain fields refer to the end of the first, second, third, fourth, fifth and 10th years. Figures from

**a**) to

**f**) correspond to the results at the end of the 1st, 2nd, 3rd, 4th, 5th, and 10th year of analysis.

**Figure 15.**Analysis F: evolution of the progressive failure due to the SLVA interaction in terms of displacements during the 10 years of net rainfall; the total displacements refer to the end of the first, second, third, fourth, fifth and 10th years. Figures from

**a**) to

**f**) correspond to the results at the end of the 1st, 2nd, 3rd, 4th, 5th, and 10th year of analysis.

Property | Symbol | Value |
---|---|---|

Saturated unit weight | γ_{sat} | 19 [kN/m^{3}] |

Coefficient of saturated permeability | k_{sat} | 3 × 10^{−9} [m/s] |

Effective Young modulus | E’ | 20,000 [kPa] |

Effective Poisson’s ratio | ν′ | 0.3 |

Effective cohesion intercept | c’ | 20 [kPa] |

Effective friction angle | ϕ’ | 23 [°] |

Dilation angle | ψ | 0 [°] |

Unsaturated unit weight | γ_{unsat} | 17 [kN/m^{3}] |

Saturated degree of saturation | S_{sat} | 1 |

Residual degree of saturation | S_{res} | 0.45 |

Van Genuchten parameter | g_{n} | 1.7 |

Van Genuchten parameter | g_{a} | 0.0095 [1/m] |

Van Genuchten parameter | g_{l} | 0.5 |

Analysis | Initialization Procedure | Ratio of the Horizontal to the Vertical Effective Stress (k_{0}) | Poisson’s Ratio (ν′) |
---|---|---|---|

A | Gravity loading | k_{0} = k_{ela} = 0.428 | ν′ = 0.3 |

B | Gravity loading | k_{0} = k_{ela} = 0.65 | ν′ = 0.39 |

C | k_{0} procedure | k_{0} = 0.428 | ν′ = 0.3 |

D | k_{0} procedure | k_{0} = 0.65 | ν′ = 0.39 |

E | k_{0} procedure | k_{0} = 0.65 | ν′ = 0.3 |

F | k_{0} procedure | k_{0} = 1 | ν′ = 0.3 |

G | k_{0} procedure | k_{0} = 1.5 | ν′ = 0.3 |

H | k_{0} procedure | k_{0} = 2 | ν′ = 0.3 |

Analysis | Initialization Procedure | Ratio of the Horizontal to the Vertical Effective Stress (k_{0}) | Safety Factor |
---|---|---|---|

A | Gravity loading | k_{0} = k_{ela} = 0.428 | No convergence |

B | Gravity loading | k_{0} = k_{ela} = 0.65 | No convergence |

C | k_{0} procedure | k_{0} = 0.428 | 1.135 |

D | k_{0} procedure | k_{0} = 0.65 | 1.140 |

E | k_{0} procedure | k_{0} = 0.65 | 1.142 |

F | k_{0} procedure | k_{0} = 1 | 1.136 |

G | k_{0} procedure | k_{0} = 1.5 | 1.138 |

H | k_{0} procedure | k_{0} = 2 | 1.136 |

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## Share and Cite

**MDPI and ACS Style**

Tagarelli, V.; Cotecchia, F.
The Effects of Slope Initialization on the Numerical Model Predictions of the Slope-Vegetation-Atmosphere Interaction. *Geosciences* **2020**, *10*, 85.
https://doi.org/10.3390/geosciences10020085

**AMA Style**

Tagarelli V, Cotecchia F.
The Effects of Slope Initialization on the Numerical Model Predictions of the Slope-Vegetation-Atmosphere Interaction. *Geosciences*. 2020; 10(2):85.
https://doi.org/10.3390/geosciences10020085

**Chicago/Turabian Style**

Tagarelli, Vito, and Federica Cotecchia.
2020. "The Effects of Slope Initialization on the Numerical Model Predictions of the Slope-Vegetation-Atmosphere Interaction" *Geosciences* 10, no. 2: 85.
https://doi.org/10.3390/geosciences10020085