# Comparing Different Coupling and Modeling Strategies in Hydromechanical Models for Slope Stability Assessment

^{1}

^{2}

^{*}

## Abstract

**:**

^{−1}) and low (1 mm h

^{−1}) intensities are considered. The simulation results of the simplified approaches are compared to a comprehensive, fully coupled poroelastic hydromechanical model with a two-phase flow system. It was found that the most significant difference from the comprehensive model occurs in areas experiencing the most transient changes due to rainfall infiltration in all three simplified models. Among these simplifications, the transformation of the two-phase flow system to a one-phase flow system showed the most pronounced impact on the simulated local factor of safety (LFS), with a maximum increase of +21.5% observed at the end of the high-intensity rainfall event. Conversely, using a rigid soil without poroelasticity or employing a sequential coupling approach with no iteration between hydromechanical parameters has a relatively minor effect on the simulated LFS, resulting in maximum increases of +2.0% and +1.9%, respectively. In summary, all three simplified models yield LFS results that are reasonably consistent with the comprehensive poroelastic fully coupled model with two-phase flow, but simulations are more computationally efficient when utilizing a rigid porous media and one-phase flow based on Richards’ equation.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Coupled Hydromechanical Model

#### 2.2. Evaluation of Stability Status

^{−1}T

^{−2}], and the current state of Coulomb stress, τ [ML

^{−1}T

^{−2}], at each point within a hillslope. LFS = 1, therefore, defines the stability threshold, where failure potentially occurs for values lower than 1.0. The Coulomb stress at the potential failure state, τ*, can be defined by

^{−1}T

^{−2}] is the effective cohesion, ${\mathsf{\sigma}}^{\prime}$[ML

^{−1}T

^{−2}] is the effective stress, and ϕ′ [°] is the effective internal friction angle of the soil. The LFS of each element within the hillslope is then calculated as

_{I}′[ML

^{−1}T

^{−2}] and σ

_{II}′[ML

^{−1}T

^{−2}] are obtained based on the maximum and minimum principal stresses, σ

_{1,3}[ML

^{−1}T

^{−2}], and the suction stress, ${\mathsf{\sigma}}^{\mathrm{s}}$[ML

^{−1}T

^{−2}], as

#### 2.3. Implementation of Different Coupling and Modeling Concepts

_{w}) and air saturation (S

_{a}). Hence, the right boundary below the groundwater table, characterized by full saturation (S

_{a}= 0) and a prescribed hydrostatic water pressure (p

_{w}), is treated as a Dirichlet boundary. Above the water table, we set a no-flow boundary. The top surface acts as a Neumann boundary with an infiltration rate matching the rainfall rate. Here, two constant rainfall intensities were employed. The first set of simulations used a low-intensity rainfall (LIR) of 1 mm h

^{−1}(20% of K

_{s}) for 20 h. The second set applied a high-intensity rainfall (HIR) of 4 mm h

^{−1}(80% of K

_{s}) for 5 h to ensure an equivalent total infiltration amount for both events. In the mechanical model, the top surface is a free boundary, while the left and right boundaries have no displacement in the direction normal to the boundary (roller boundary). The bottom is fixed.

## 3. Results

#### 3.1. Fully Coupled Two-Phase Flow Model with Variable and Constant Porosity

_{x}− X

_{FC})/X

_{FC}× 100), with a positive change indicating an increase relative to the equivalent value in the comprehensive fully coupled model (FC) and a negative change representing a decrease in the respective value.

_{w}) and stability, we compared results from the comprehensive fully coupled two-phase flow model with those from the same model employing a constant porosity (Figure 5). The most significant disparities in p

_{w}and LFS between the two model implementations were observed at the soil surface and at the end of the high-intensity rainfall event. Specifically, the maximum discrepancies reached approximately −10.1% for p

_{w}(decrease) and +2.0% for LFS (increase) compared to the fully coupled model with poroelasticity. In the case of the low-intensity rainfall, the variations amounted to a maximum of −2.2% for p

_{w}and +1.1% for LFS at the end of the event. When considering the entire cross section for the high-intensity rainfall event, the average differences were only −0.8% for p

_{w}(decrease) and +0.2% for LFS (increase). Similarly, at the end of the low-intensity rainfall, these values were −0.3% for p

_{w}and +0.1% for LFS.

#### 3.2. Fully Coupled vs. Sequentially Coupled Models

_{w}) in Figure 6 for the upper 1 m of cross section A during both low- and high-intensity rainfall events. The results reveal a maximum variation of −16% in p

_{w}with the sequentially coupled (SC) model during the high-intensity rainfall event, occurring in the middle of the event. In Figure 6, we also illustrate the resulting differences in simulated LFS. As the initial conditions for all models were identical, encompassing the same initial pressure distribution and LFS, these aspects are not depicted in subsequent figures. The sequentially coupled model exhibited a +7.5% deviation compared to the fully coupled model, with the most significant difference also appearing in the middle of the high-intensity rainfall event. For the low-intensity rainfall event, the sequentially coupled model showed a maximum variation of −6.3% in p

_{w}and a +4.3% difference in LFS. Averaging the top 1 m of cross section A during the high-intensity rainfall event, we observed average differences of −1.5% for simulated p

_{w}and +0.3% for LFS. Corresponding averages during the low-intensity rainfall event were −0.4% for p

_{w}and +0.2% for LFS. These disparities predominantly occurred near the surface, where dynamic changes in pore water pressure were most pronounced. For depths exceeding 1 m, the effect of increased soil weight due to infiltration remained below 0.01% for both p

_{w}and LFS.

#### 3.3. Fully Coupled Two-Phase vs. One-Phase Flow Model (Richards’ Equation)

_{w}) and LFS for both model implementations in the upper 1 m of cross section A during the low- and high-intensity rainfall events, along with their relative differences. Once more, the most significant deviation between the two models arises at the end of the high-intensity rainfall event. Here, we observe a +97.2% shift in p

_{w}and a −21.5% shift in LFS compared to the fully coupled model. This maximum difference diminishes to roughly +53.7% in p

_{w}and −11.9% in LFS for the low-intensity rainfall event. Additionally, we notice that the disparities between the two model implementations primarily occur in a smaller region near the slope surface during the high-intensity rainfall event. Consequently, the average differences for the top 1 m of cross section A are relatively consistent (−9.5% for p

_{w}and 1.9% for LFS at the end of the high-intensity rainfall and −8.2% for p

_{w}and 1.9% for LFS at the end of the low-intensity rainfall event, relative to the fully coupled model).

## 4. Discussion

#### 4.1. Effect of Poroelasticity

**K**) due to higher porosity in the poroelastic model outweighs the displacement (

_{eff}**u**) effect, resulting in elevated pore pressure near the surface. In contrast, simulations with constant hydraulic conductivity in a poroelastic fully coupled model showed lower pore pressure (results not shown) and consequently higher LFS at those locations.

#### 4.2. Effect of Coupling Strategy

_{w}values in the fully coupled model are due to its consideration of the complete interaction between effective pore pressure and volumetric strain within each time step. In the sequentially coupled model with no iteration, the impact of variable pore pressure on stress and strain distribution occurs in the same time step, but the feedback of volumetric strain to pore pressure is accounted for in the subsequent time step. As previously discussed, differences are more prominent during high-intensity rainfall events due to steeper pore water pressure gradients. These results again align with the findings of Beck et al. [30], emphasizing that the maximum difference between fully coupled and sequentially coupled models occurs in regions with transient processes involving rapid pore water pressure changes within a single time step. Despite the anticipated greater reliability of the fully coupled model due to its comprehensive interaction between hydrological and mechanical components, the results presented here indicate relatively minor variations in pore water pressure and consequent instability assessment between these two coupling strategies (Table 2).

_{w}and LFS between fully and sequentially coupled models are more significant than those between the fully coupled model with and without poroelastic changes. This smaller difference in the latter simulations is attributed to the counteractive effects of displacement (

**u**) and effective hydraulic conductivity (

**K**) on pore pressure in the fully coupled poroelastic model. Specifically, increased effective porosity due to infiltration reduces pore pressure and elevates

_{eff}**K**. The latter increases pore pressure, partially offsetting the overall effect of larger pore size, resulting in a reduced pore pressure decrease or even an increase.

_{eff}#### 4.3. Effect of the Multiphase Flow Model

_{a}) in the one-phase flow model based on Richards’ equation. In a two-phase flow system involving water and air, the air must move or exit the domain during rainfall infiltration. Consequently, the downward progression of water is impeded by elevated pore air pressure, resulting in reduced flow and a slower increase in p

_{w}. This also explains the enhanced water accumulation near the surface over time, leading to higher p

_{w}near the surface in the two-phase flow model. To substantiate this, we analyzed the development of pore air pressure (p

_{a}) during both low- and high-intensity rainfall events. Figure 8 illustrates the rising trend of air pressure with infiltration. As depicted, air pressure increases with depth and infiltration. This pattern arises from soil compaction and the displacement of air from shallower layers due to increasing depth and infiltration, respectively. In the case of low-intensity rainfall, air movement is less restricted and can be released from the area, resulting in a less pronounced increase in p

_{a}with infiltration. The steeper curve for low-intensity rainfall represents the initially dominant influence of soil compaction, which is more prominently affected by infiltration during high-intensity rainfall.

## 5. Conclusions and Outlook

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Srinivasan, K.; Howell, B.; Anderson, E.; Flores, A. A low cost wireless sensor network for landslide hazard monitoring. In Proceedings of the International Geoscience and Remote Sensing Symposium, Munich, Germany, 22–27 July 2012; pp. 4793–4796. [Google Scholar]
- Greco, R.; Marino, P.; Bogaard, T.A. Recent advancements of landslide hydrology. Wiley Interdiscip. Rev. Water
**2023**, 10, e1675. [Google Scholar] [CrossRef] - Cuomo, S.; Della Sala, M. Rainfall-induced infiltration, runoff and failure in steep unsaturated shallow soil deposits. Eng. Geol.
**2013**, 162, 118–127. [Google Scholar] [CrossRef] - Cuomo, S.; Della Sala, M. Large-area analysis of soil erosion and landslides induced by rainfall: A case of unsaturated shallow deposits. J. Mt. Sci.
**2015**, 12, 783–796. [Google Scholar] [CrossRef] - Eichenberger, J.; Ferrari, A.; Laloui, L. Early warning thresholds for partially saturated slopes in volcanic ashes. Comput. Geotech.
**2013**, 49, 79–89. [Google Scholar] [CrossRef] - Tsaparas, I.; Rahardjo, H.; Toll, D.G.; Leong, E.C. Controlling parameters for rainfall-induced landslides. Comput. Geotech.
**2002**, 29, 1–27. [Google Scholar] [CrossRef] - Lu, N.; Sener-Kaya, B.; Wayllace, A.; Godt, J.W. Analysis of rainfall-induced slope instability using a field of local factor of safety. Water Resour. Res.
**2012**, 48, W09524. [Google Scholar] [CrossRef] - Griffiths, D.V.; 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] - Lu, N.; Godt, J. Infinite slope stability under steady unsaturated seepage conditions. Water Resour. Res.
**2008**, 44, W11404. [Google Scholar] [CrossRef] - Lehmann, P.; Or, D. Hydromechanical triggering of landslides: From progressive local failures to mass release. Water Resour. Res.
**2012**, 48, W03535. [Google Scholar] [CrossRef] - von Ruette, J.; Lehmann, P.; Or, D. Effects of rainfall spatial variability and intermittency on shallow landslide triggering patterns at a catchment scale. Water Resour. Res.
**2014**, 50, 7780–7799. [Google Scholar] [CrossRef] - Moradi, S.; Huisman, J.; Class, H.; Vereecken, H. The effect of bedrock topography on timing and location of landslide initiation using the local factor of safety concept. Water
**2018**, 10, 1290. [Google Scholar] [CrossRef] - Lanni, C.; McDonnell, J.; Hopp, L.; Rigon, R. Simulated effect of soil depth and bedrock topography on near-surface hydrologic response and slope stability. Earth Surf. Process. Landf.
**2013**, 38, 146–159. [Google Scholar] [CrossRef] - von Ruette, J.; Lehmann, P.; Or, D. Rainfall-triggered shallow landslides at catchment scale: Threshold mechanics-based modeling for abruptness and localization. Water Resour. Res.
**2013**, 49, 6266–6285. [Google Scholar] [CrossRef] - Andrés, S.; Dentz, M.; Cueto-Felgueroso, L. Multirate Mass Transfer Approach for Double-Porosity Poroelasticity in Fractured Media. Water Resour. Res.
**2021**, 57, 27. [Google Scholar] [CrossRef] - Mehrabian, A.; Abousleiman, Y.N. Generalized Biot’s theory and Mandel’s problem of multiple-porosity and multiple-permeability poroelasticity. J. Geophys. Res. Solid Earth
**2014**, 119, 2745–2763. [Google Scholar] [CrossRef] - Dugan, B.; Stigall, J. Origin of overpressure and slope failure in the Ursa region, Northern Gulf of Mexico. In Submarine Mass Movements and Their Consequences; Mosher, D.C., Shipp, R.C., Moscardelli, L., Chaytor, J.D., Baxter, C.D.P., Lee, H.J., Urgeles, R., Eds.; Springer: Dordrecht, The Netherlands, 2010; Volume 28, pp. 167–178. [Google Scholar]
- Li, B.; Tian, B.; Tong, F.G.; Liu, C.; Xu, X.L. Effect of the Water-Air Coupling on the Stability of Rainfall-Induced Landslides Using a Coupled Infiltration and Hydromechanical Model. Geofluids
**2022**, 2022, 16. [Google Scholar] [CrossRef] - Urgeles, R.; Locat, J.; Sawyer, D.E.; Flemings, P.B.; Dugan, B.; Binh, N.T.T. History of pore pressure build up and slope instability in mud-dominated sediments of ursa basin, gulf of Mexico continental slope. In Submarine Mass Movements and Their Consequences; Springer: Dordrecht, The Netherlands, 2010; Volume 28. [Google Scholar]
- Kim, J. Sequential Methods for Coupled Geomechanics and Multiphase Flow. Ph.D. Thesis, Stanford University, Stanford, CA, USA, 2010. [Google Scholar]
- Cho, S.E. Stability analysis of unsaturated soil slopes considering water-air flow caused by rainfall infiltration. Eng. Geol.
**2016**, 211, 184–197. [Google Scholar] [CrossRef] - Szymkiewicz, A. Modelling Water Flow in Unsaturated Porous Media: Accounting for Nonlinear Permeability and Material Heterogeneity; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Farthing, M.W.; Ogden, F.L. Numerical Solution of Richards’ Equation: A Review of Advances and Challenges. Soil Sci. Soc. Am. J.
**2017**, 81, 1257–1269. [Google Scholar] [CrossRef] - Oh, S.; Lu, N. Slope stability analysis under unsaturated conditions: Case studies of rainfall-induced failure of cut slopes. Eng. Geol.
**2015**, 184, 96–103. [Google Scholar] [CrossRef] - Borja, R.I.; White, J.A. Continuum deformation and stability analyses of a steep hillside slope under rainfall infiltration. Acta Geotech.
**2010**, 5, 1–14. [Google Scholar] [CrossRef] - Settari, A.; Walters, D.A. Advances in coupled geomechanical and reservoir modeling with applications to reservoir compaction. SPE J.
**2001**, 6, 334–342. [Google Scholar] [CrossRef] - Koch, T.; Gläser, D.; Weishaupt, K.; Ackermann, S.; Beck, M.; Becker, B.; Burbulla, S.; Class, H.; Coltman, E.; Emmert, S.; et al. DuMux 3—An open-source simulator for solving flow and transport problems in porous media with a focus on model coupling. Comput. Math. Appl.
**2021**, 81, 423–443. [Google Scholar] [CrossRef] - White, J.A.; Castelletto, N.; Tchelepi, H.A. Block-partitioned solvers for coupled poromechanics: A unified framework. Comput. Meth. Appl. Mech. Eng.
**2016**, 303, 55–74. [Google Scholar] [CrossRef] - Preisig, M.; Pervost, J.H. Coupled multi-phase 1036 thermo-poromechanical effects. Case study: CO
_{2}injection at In Salah, Algeria. Int. J. Greenh. Gas Control**2011**, 5, 1055–1064. [Google Scholar] [CrossRef] - Beck, M.; Rinaldi, A.P.; Flemisch, B.; Class, H. Accuracy of fully coupled and sequential approaches for modeling hydro- and geomechanical processes. Comput. Geosci.
**2020**, 24, 1707–1723. [Google Scholar] [CrossRef] - Kim, J.; Tchelepi, H.A.; Juanes, R. Stability and convergence of sequential methods for coupled flow and geomechanics: Fixed-stress and fixed-strain splits. Comput. Meth. Appl. Mech. Eng.
**2011**, 200, 1591–1606. [Google Scholar] [CrossRef] - Darcis, M.Y. Coupling Models of Different Complexity for the Simulation of CO
_{2}Storage in Deep Saline Aquifers. Ph.D. Thesis, University of Stuttgart, Stuttgart, Germany, 2013. [Google Scholar] - Kim, J.M. A fully coupled finite element analysis of water-table fluctuation and land deformation in partially saturated soils due to surface loading. Int. J. Numer. Methods Eng.
**2000**, 49, 1101–1119. [Google Scholar] [CrossRef] - Bao, J.; Xu, Z.J.; Fang, Y.L. A coupled thermal-hydromechanical simulation for carbon dioxide sequestration. Environ. Geotech.
**2016**, 3, 312–324. [Google Scholar] [CrossRef] - Abdollahipour, A.; Marji, M.F.; Bafghi, A.Y.; Gholamnejad, J. Time-dependent crack propagation in a poroelastic medium using a fully coupled hydromechanical displacement discontinuity method. Int. J. Fract.
**2016**, 199, 71–87. [Google Scholar] [CrossRef] - Della Vecchia, G.; Jommi, C.; Romero, E. A fully coupled elastic-plastic hydromechanical model for compacted soils accounting for clay activity. Int. J. Numer. Anal. Methods Geomech.
**2013**, 37, 503–535. [Google Scholar] [CrossRef] - Freeman, T.; Chalaturnyk, R.; Bogdanov, I. Fully coupled thermo-hydro-mechanical modeling by COMSOL Multiphysics, with applications in reservoir geomechanical characterization. In Proceedings of the COMSOL Conference, Boston, MA, USA, 9–11 October 2008. [Google Scholar]
- Tang, Y.; Wu, W.; Yin, K.; Wang, S.; Lei, G. A hydro-mechanical coupled analysis of rainfall induced landslide using a hypoplastic constitutive model. Comput. Geotech.
**2019**, 112, 284–292. [Google Scholar] [CrossRef] - Tufano, R.; Formetta, G.; Calcaterra, D.; De Vita, P. Hydrological control of soil thickness spatial variability on the initiation of rainfall-induced shallow landslides using a three-dimensional model. Landslides
**2021**, 18, 3367–3380. [Google Scholar] [CrossRef] - Chen, X.; Zhang, L.; Zhang, L.; Zhou, Y.; Ye, G.; Guo, N. Modelling rainfall-induced landslides from initiation of instability to post-failure. Comput. Geotech.
**2021**, 129, 103877. [Google Scholar] [CrossRef] - Wu, L.; Huang, R.; Li, X. Hydro-Mechanical Analysis of Rainfall-Induced Landslides; Springer: Beijing, China, 2020. [Google Scholar] [CrossRef]
- Pedone, G.; Tsiampousi, A.; Cotecchia, F.; Zdravkovic, L. Coupled hydro-mechanical modelling of soil–vegetation–atmosphere interaction in natural clay slopes. Can. Geotech. J.
**2022**, 59, 272–290. [Google Scholar] [CrossRef] - Moradi, S. Stability Assessment of Variably Saturated Hillslopes Using Coupled Hydromechanical Models. Ph.D. Thesis, University of Stuttgart, Stuttgart, Germany, 2021. [Google Scholar]
- Lu, N.; Godt, J.W. Hillslope Hydrology and Stability; Cambridge University Press: Cambridge, UK, 2013. [Google Scholar]
- Huang, Y.H. Slope Stability Analysis by the Limit Equilibrium Method; ASCE Press: Reston, VA, USA, 2014. [Google Scholar] [CrossRef]
- Moradi, S.; Heinze, T.; Budler, J.; Gunatilake, T.; Kemna, A.; Huisman, J.A. Combining Site Characterization, Monitoring and Hydromechanical Modeling for Assessing Slope Stability. Land
**2021**, 10, 423. [Google Scholar] [CrossRef] - Flemisch, B.; Darcis, M.; Erbertseder, K.; Faigle, B.; Lauser, A.; Mosthaf, K.; Muthing, S.; Nuske, P.; Tatomir, A.; Wolff, M.; et al. DuMu(x): DUNE for multi-{phase, component, scale, physics, …} flow and transport in porous media. Adv. Water Resour.
**2011**, 34, 1102–1112. [Google Scholar] [CrossRef] - Fetzer, T.; Becker, B.; Flemisch, B.; Glaser, D.; Heck, K.; Koch, T.; Schneider, M.; Scholz, S.; Weishaupt, K. Dumux, 2.12.0; Zenodo: Geneve, Switzerland, 2017. [Google Scholar] [CrossRef]
- Blatt, M.; Burchardt, A.; Dedner, A.; Engwer, C.; Fahlke, J.; Flemisch, B.; Gersbacher, C.; Graeser, C.; Gruber, F.; Grueninger, C.; et al. The Distributed and Unified Numerics Environment, Version 2.4. Arch. Numer. Softw.
**2016**, 4, 13–17. [Google Scholar] [CrossRef] - Bastian, P.; Blatt, M.; Dedner, A.; Engwer, C.; Klofkorn, R.; Kornhuber, R.; Ohlberger, M.; Sander, O. A generic grid interface for parallel and adaptive scientific computing. Part II: Implementation and tests in DUNE. Computing
**2008**, 82, 121–138. [Google Scholar] [CrossRef] - Bastian, P.; Blatt, M.; Dedner, A.; Dreier, N.-A.; Engwer, C.; Fritze, R.; Gräser, C.; Grüninger, C.; Kempf, D.; Klöfkorn, R.; et al. The Dune framework: Basic concepts and recent developments. Comput. Math. Appl.
**2021**, 81, 75–112. [Google Scholar] [CrossRef] - Zhang, K.; Cao, P.; Liu, Z.Y.; Hu, H.H.; Gong, D.P. Simulation analysis on three-dimensional slope failure under different conditions. Trans. Nonferrous Met. Soc. China
**2011**, 21, 2490–2502. [Google Scholar] [CrossRef] - Kristo, C.; Rahardjo, H.; Satyanaga, A. Effect of hysteresis on the stability of residual soil slope. Int. Soil Water Conserv. Res.
**2019**, 7, 226–238. [Google Scholar] [CrossRef]

**Figure 1.**The geometry, boundary conditions, and discretization of the 2D, homogeneous silty slope used to compare the four hydromechanical model implementations. Please note that the same color and line type were used for the hydrological infiltration and mechanical free boundary conditions at the surface because they are both of the Neumann-type.

**Figure 2.**The spatial and temporal variability of LFS for the 2D silty slope simulated with the FC two-phase flow model for the LIR at (

**a**) t = 0 h, (

**b**) t = 8 h, and (

**c**) t = 20 h, and the HIR at (

**d**) t = 0 h, (

**e**) t = 2 h, and (

**f**) t= 5 h.

**Figure 3.**The simulated change in effective stress at cross section A of the 2D silty slope using the two-phase FC model for the LIR at (

**a**) t = 0 h, (

**b**) t = 8 h, and (

**c**) t = 20 h and the HIR at (

**d**) t = 0 h, (

**e**) t = 2 h, and (

**f**) t = 5 h.

**Figure 4.**The dynamics of simulated vertical effective stress and porosity at cross section A of the 2D silty slope using the two-phase FC model for the LIR at (

**a**) t = 0 h, (

**b**) t = 8 h, and (

**c**) t = 20 h and the HIR at (

**d**) t = 0 h, (

**e**) t = 2 h, and (

**f**) t = 5 h.

**Figure 5.**The simulated p

_{w}and LFS distribution at cross section A of the 2D silty slope using the two-phase fully coupled (2P-FC) model with variable and constant porosity for the LIR at (

**a**) t = 0 h, (

**b**) t = 8 h, and (

**c**) t = 20 h and the HIR at (

**d**) t = 0 h, (

**e**) t = 2 h, and (

**f**) t = 5 h. The difference between the two model implementations for low- and high-intensity rainfall are shown in panels (

**g**–

**i**).

**Figure 6.**The simulated p

_{w}and LFS distribution at cross section A for the 2D silty slope using fully and sequentially coupled two-phase flow models (2P-FC and 2P-SC, respectively) with the LIR at (

**a**) t = 8 h and (

**b**) t = 20 h and the HIR at (

**c**) t = 2 h and (

**d**) t = 5 h. The differences between the two model implementations for low- and high-intensity rainfall are shown in panels (

**e**,

**f**).

**Figure 7.**The simulated p

_{w}and LFS distribution at cross section A for the 2D silty slope using the fully coupled two-phase and Richards’ models (2P-FC and 1P-FC, respectively) with the LIR at (

**a**) t = 8 h, and (

**b**) t = 20 h and the HIR at (

**c**) t = 2 h, and (

**d**) t = 5 h. The differences between the two model implementations for low- and high-intensity rainfall are shown in panels (

**e**,

**f**).

**Figure 8.**The simulated pa distribution at cross section A for the 2D silty slope using fully coupled two-phase models (2P-FC and 1P-FC, respectively) for LIR and HIR at different time steps.

Model | Abbreviation |
---|---|

Fully coupled two-phase flow model with variable porosity | 2P-FC-var.Por. |

Fully coupled two-phase flow model with constant porosity | 2P-FC-const.Por. |

sequentially coupled two-phase flow model | 2P-SC |

One-phase flow model (Richards’ equation) | 1P-FC |

**Table 2.**Comparative analysis of simulated pore water pressure and Local Factor of Safety of the simplified models relative to the comprehensive fully coupled model (in percentage) under the two investigated rainfall intensities.

2P-FC-var.Por. vs. … | Parameter | HIR (4 mm h^{−1}) (%) | LIR (1 mm h^{−1}) (%) |
---|---|---|---|

2P-FC- const-Por. | p_{w} | −10.1 | −2.2 |

LFS | +2.0 | +1.1 | |

2P-SC | p_{w} | −16.0 | −6.3 |

LFS | +7.5 | +4.3 | |

1P-FC | p_{w} | +97.2 | +53.7 |

LFS | −21.5 | −11.9 |

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**MDPI and ACS Style**

Moradi, S.; Huisman, J.A.; Vereecken, H.; Class, H.
Comparing Different Coupling and Modeling Strategies in Hydromechanical Models for Slope Stability Assessment. *Water* **2024**, *16*, 312.
https://doi.org/10.3390/w16020312

**AMA Style**

Moradi S, Huisman JA, Vereecken H, Class H.
Comparing Different Coupling and Modeling Strategies in Hydromechanical Models for Slope Stability Assessment. *Water*. 2024; 16(2):312.
https://doi.org/10.3390/w16020312

**Chicago/Turabian Style**

Moradi, Shirin, Johan Alexander Huisman, Harry Vereecken, and Holger Class.
2024. "Comparing Different Coupling and Modeling Strategies in Hydromechanical Models for Slope Stability Assessment" *Water* 16, no. 2: 312.
https://doi.org/10.3390/w16020312