# Effects of Two-Phase Flow of Water and Air on Shallow Slope Failures Induced by Rainfall: Insights from Slope Stability Assessment at a Regional Scale

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}. Figure 1 shows the elevation distribution of the study area based on 1:5000 digital maps produced by the National Geographic Information Institute of Korea [37] using the airborne laser terrain mapping system. The highest elevation is 295 m above sea level. The area around Umyeon Mountain consists mainly of Pre-Cambrian banded biotite gneiss and granitic gneiss [38]. According to Jeong et al. [15], the bedrock is primarily intensely weathered and substantially fractured plagioclase, quartz, biotite, feldspar, and amphibole. Umyeon Mountain is completely surrounded by urban development.

#### 2.2. Data

#### 2.2.1. Landslide Inventories and Spatial Data

#### 2.2.2. Material Properties

#### 2.3. Methods

#### 2.3.1. Rainfall Infiltration Analysis

_{w}is the actual saturation of the wetting fluid, and ${S}_{w}^{r}$ is the residual saturation.

#### 2.3.2. Slope Stability Analysis

#### 2.3.3. Slope Geometry and Boundary/Initial Conditions

#### 2.3.4. Performance Evaluation of Slope Stability Assessment

## 3. Results and Discussion

#### 3.1. Validation with Comparison to Field Measurements

#### 3.2. Infiltration Analysis

_{w}and matric suction with time at the five zones when the single-phase flow model was applied to the slope of 30°, which is approximately equal to the mean slope angle of the slope failure sites in the study area (i.e., 31.2°). The variations in P

_{w}, P

_{a}, and matric suction from the two-phase flow model are shown in Figure 8. In the early stages (less than 9 h), variations in P

_{w}in the single-phase flow and two-phase flow models are similar at zones 1 and 4; ponding effects are negligible because of a high infiltration capacity with relatively high saturated hydraulic conductivities (see Table 1). However, at zones 2, 3, and 5, the ponding caused by a rainfall intensity higher than the infiltration capacity resulted in an increase in P

_{a}under the two-phase flow model. The increases in P

_{a}caused P

_{w}to increase even at certain depths under the slope where the rainfall had not reached. Therefore, increases in P

_{w}in the two-phase flow model became higher than those in the single-phase flow model. The P

_{w}in both the models decreased at 9 h from the start owing to the negligible rainfall between 7 and 9 h (see Figure 2). After 12 h, the infiltration rates increased sharply in the single-phase flow model during heavy rainfall whereas those from the two-phase flow model increased relatively gradually during this period. This is because air pressure interrupted the water flow into the void spaces and was accompanied by ponding ([20,35,56]). This shows clearly that air pressure impacts the infiltration rate particularly in the low permeable soils where ponding effects occur frequently.

_{a}(see Figure 7). Meanwhile, in the two-phase flow model, the matric suction was marginal at shallow depths. However, it did not dissipate owing to P

_{a}being higher than P

_{w}(see Figure 8).

_{w}at the north side (zones 1 and 2) were slower than those at the south side (zones 3, 4, and 5) in both the models. An exception was zone 3, where the saturated hydraulic conductivity was the lowest (1.80 × 10

^{−6}m/s). The infiltration rate was the highest at zone 4 owing to the largest relative permeability and saturated hydraulic conductivity (9.7 × 10

^{–6}m/s). The ground here was almost completely saturated after 15 h (7:00 on 27 July, 2 h prior to the landslide events). The infiltration rate was the lowest at zone 2 owing to the marginal relative permeability and saturated hydraulic conductivity.

_{w}increased toward the surface and larger depths, whereas at the north side, it increased first at shallow depths and subsequently at larger depths as the wetting bands progressed gradually into the depths because of the marginal relative permeability. In the two-phase flow model, positive P

_{w}occurred at shallow depths because of ponding effects and heavy rainfall (after 12 h). However, the matric suction was not dissipated completely in the shallow areas because here, P

_{a}is higher than P

_{w}(i.e., not fully saturated).

#### 3.3. Slope Stability Assessment

_{w}and P

_{a}. Figure 9 shows the variations in the minimum safety factors on infinite slopes whose angles are the maximum, average, and minimum of the actual slope failure sites at each zone. The initial safety factors at zone 3 are larger than those at the other zones, with values higher than 3 (Figure 9c) owing to the large internal friction angle (37.6°) and high cohesion (7.7 kPa). Initial safety factors on slopes with the average angles at zones 1, 2, and 5, where the internal friction angles and cohesions are marginal (see Table 1), are smaller than two (Figure 9a,b,e). At the north side (zones 1 and 2), the safety factors decreased gradually owing to gradual infiltration rates and reduced abruptly after 13 h. Meanwhile, at the south side (zones 3, 4, and 5), the safety factors reduced gradually from the initial values and decreased to below one more rapidly than those at the north side. This is owing to the rapid infiltration (with an exception of zone 3 because of the smallest saturated hydraulic conductivity (1.8 × 10

^{−6}m/s)). Therefore, hydrological properties (i.e., saturated hydraulic conductivity and relative permeability) strongly affect variations in the safety factor, thereby influencing infiltration rates.

_{w}in the single-phase flow model, the increases in both P

_{w}and P

_{a}in the two-phase flow model resulted in a higher decrease in the safety factor. This is because the shear strength decreased as P

_{w}and P

_{a}increased (referring to Equation (13)). However, the safety factor in the single-phase flow model became lower than that in the two-phase flow model during the final stage. This is because the slopes in the single-phase flow model became saturated more rapidly than those in the two-phase flow model (as shown in Figure 7 and Figure 8). These observations are comparable with the observations of Cho [11].

_{w}and P

_{a}obtained from infiltration analyses by applying the single-phase flow and two-phase flow models in FLAC (Figure 10a,b). The distribution patterns for the areas of predicted slope failure are similar because the same topographic maps and geotechnical properties were applied to generate both maps. However, applying different models resulted in differences in the total area predicted for slope failure, as shown in Figure 10c,d. On the north side (zones 1 and 2), the total area of potential slope failure in the two-phase flow model (234,600 m

^{2}) is smaller than that in the single-phase flow model (405,525 m

^{2}). This is because the rainfall infiltration at the north side is slower in the two-phase flow model than it is in the single-phase flow model.

^{2}, respectively) are similarly distributed because the ground approached saturated conditions owing to rapid infiltration with large relative permeabilities. However, under the application of the two-phase flow model, the potential slope failure areas are markedly marginal in Zone 3 because of the considerable matric suction there (as shown in Figure 8). This is comparable with the fact that there was no actual slope failure in zone 3 with a large representative internal friction angle of 37.6°.

## 4. Conclusions

- Considering that air flow changes the rate of increase in P
_{w}on slopes with a low infiltration capacity when ponding occurs during heavy rainfall, and that the saturated hydraulic conductivity is relatively marginal in the mountainous areas of Korea (i.e., less than 2.78 × 10^{−5}m/s according to the National Disaster Management Institute [57]), it is necessary to apply the two-phase flow model to accurately interpret rainfall infiltration. - The initial safety factor prior to rainfall depends on the shear strength parameters (i.e., internal friction angle and cohesion); however, variations in the safety factor are strongly dependent on an increase in the P
_{w}rate. Slopes in the single-phase flow model rapidly become saturated during heavy rainfall because of the high rainfall infiltration rates without the interruption of air. Thus, safety factors decrease rapidly compared to those in the two-phase flow model. - Landslide susceptibility maps change depending on an increase in P
_{w}, which depends on relative permeability. It is also sensitive to the model type. The MSR and confusion matrix yield the highest performance for the two-phase flow model with an appropriate range of stable cell coverage for the best simulation. Thus, it is concluded that infiltration and slope stability analyses using the two-phase flow model have good applicability in evaluating landslide events in Umyeon Mountain. However, it is necessary to additionally evaluate the applicability of this model for other landslide cases. - We performed infiltration and slope stability analyses focusing on the geotechnical characteristics of unsaturated soils in the upper layer. Besides, we excluded geological characteristics given that most of the slope failure in the study area occurred at the colluvium layer with shallow depths up to 2 m. The two-phase flow model can be usefully applied to the region where shallow slope failure occurs primarily. However, further study is required to examine the effects of the geological structure for improving the applicability of the two-phase flow model to the region where deep slope failure occurs primarily, which is affected not only by geotechnical characteristics but also by geological structure.
- The performance of slope stability assessments at a regional scale is greatly affected by the uncertainty and variability of geotechnical and hydrological input parameters when physically based models are applied. Geotechnical and hydrological properties have probabilistically or statistically been characterized based on types of soil and lithology to properly consider the uncertainty and variability of them [58,59], whereas we used representative constant soil properties at each of the five zones as averaged from field measurements. The applicability of the two-phase flow model to a regional scale can be improved by further study applying the probabilistic approach for characterizing the uncertainty and variability of geotechnical and hydrological input parameters.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Chen, H.-E.; Tsai, T.-L.; Yang, J.-C. Threshold of Slope Instability Induced by Rainfall and Lateral Flow. Water
**2017**, 9, 722. [Google Scholar] [CrossRef] [Green Version] - Haque, U.; da Silva, P.F.; Devoli, G.; Pilz, J.; Zhao, B.; Khaloua, A.; Wilopo, W.; Andersen, P.; Lu, P.; Lee, J.; et al. The human cost of global warming: Deadly landslides and their triggers (1995–2014). Sci. Total Environ.
**2019**, 682, 673–684. [Google Scholar] [CrossRef] [PubMed] - Cho, S.E. Study on the characteristics of infinite slope failures by probabilistic seepage analysis. J. Korean Geotech. Soc.
**2014**, 30, 5–18. [Google Scholar] [CrossRef] [Green Version] - McGuire, L.A.; Rengers, F.K.; Kean, J.W.; Coe, J.A.; Mirus, B.B.; Baum, R.L.; Godt, J.W. Elucidating the role of vegetation in the initiation of rainfall-induced shallow landslides: Insights from an extreme rainfall event in the Colorado Front Range. Geophys. Res. Lett.
**2016**, 43, 9084–9092. [Google Scholar] [CrossRef] [Green Version] - Johnson, J.M.; Loáiciga, H.A. Coupled Infiltration and Kinematic-Wave Runoff Simulation in Slopes: Implications for Slope Stability. Water
**2017**, 9, 327. [Google Scholar] [CrossRef] [Green Version] - Ehlers, W.; Avci, O.; Markert, B. Computation of Slope Movements Initiated by Rain–Induced Shear Bands in Small–Scale Tests and In Situ. Vadose Zone J.
**2011**, 10, 512–525. [Google Scholar] [CrossRef] - Germer, K.; Braun, J. Effects of Saturation on Slope Stability: Laboratory Experiments Utilizing External Load. Vadose Zone J.
**2011**, 10, 477–486. [Google Scholar] [CrossRef] - 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] - Cho, S.E.; Lee, S.R. Instability of unsaturated soil slopes due to infiltration. Comput. Geotech.
**2001**, 28, 185–208. [Google Scholar] [CrossRef] - Ng, C.W.W.; Shi, Q. A numerical investigation of the stability of unsaturated soil slopes subjected to transient seepage. Comput. Geotech.
**1998**, 22, 1–28. [Google Scholar] [CrossRef] - 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] - Yeh, H.F.; Lee, C.H.; Lee, C.C. A rainfall–infiltration model for unsaturated soil slope stability. Sustain. Environ. Res.
**2008**, 18, 271–278. [Google Scholar] - Thomas, M.A.; Mirus, B.B.; Collins, B.D. Identifying Physics-Based Thresholds for Rainfall-Induced Landsliding. Geophys. Res. Lett.
**2018**, 45, 9651–9661. [Google Scholar] [CrossRef] - Richards, L.A. Capillary conduction of liquids through porous mediums. Physics
**1931**, 1, 318–333. [Google Scholar] [CrossRef] - Jeong, S.; Lee, K.; Kim, J.; Kim, Y. Analysis of Rainfall-Induced Landslide on Unsaturated Soil Slopes. Sustainability
**2017**, 9, 1280. [Google Scholar] [CrossRef] [Green Version] - Tran, T.V.; Lee, G.; An, H.; Kim, M. Comparing the performance of TRIGRS and TiVaSS in spatial and temporal prediction of rainfall-induced shallow landslides. Environ. Earth Sci.
**2017**, 76, 315. [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] - Borja, R.I.; White, J.A.; Liu, X.; Wu, W. Factor of safety in a partially saturated slope inferred from hydro–mechanical continuum modeling. Int. J. Numer. Anal. Methods Geomech.
**2012**, 36, 236–248. [Google Scholar] [CrossRef] [Green Version] - Hu, R.; Hong, J.-M.; Chen, Y.-F.; Zhou, C.-B. Hydraulic hysteresis effects on the coupled flow–deformation processes in unsaturated soils: Numerical formulation and slope stability analysis. Appl. Math. Model.
**2018**, 54, 221–245. [Google Scholar] [CrossRef] - Liu, G.; Tong, F.-G.; Zhao, Y.-T.; Tian, B. A force transfer mechanism for triggering landslides during rainfall infiltration. J. Mt. Sci.
**2018**, 15, 2480–2491. [Google Scholar] [CrossRef] - Sun, D.M.; Zang, Y.G.; Semprich, S. Effects of airflow induced by rainfall infiltration on unsaturated soil slope stability. Transp. Porous Media
**2015**, 107, 821–841. [Google Scholar] [CrossRef] - Wu, L.Z.; Zhang, L.M.; Zhou, Y.; Li, B.E. Analysis of multi-phase coupled seepage and stability in anisotropic slopes under rainfall condition. Environ. Earth Sci.
**2017**, 76, 469. [Google Scholar] [CrossRef] - Zhang, X.Y.; Zhu, Y.M.; Fang, C.H. The role for air flow in soil slope stability analysis. J. Hydrodyn.
**2009**, 21, 640–646. [Google Scholar] [CrossRef] - Chen, P.; Mirus, B.; Lu, N.; Godt, J.W. Effect of Hydraulic Hysteresis on Stability of Infinite Slopes under Steady Infiltration. J. Geotech. Geoenviron. Eng.
**2017**, 143, 04017041. [Google Scholar] [CrossRef] [Green Version] - Ebel, B.A.; Loague, K.; Borja, R.I. The impacts of hysteresis on variably saturated hydrologic response and slope failure. Environ. Earth Sci.
**2010**, 61, 1215–1225. [Google Scholar] [CrossRef] - Simoni, S.; Zanotti, F.; Bertoldi, G.; Rigon, R. Modelling the probability of occurrence of shallow landslides and channelized debris flows using GEOtop-FS. Hydrol. Process.
**2008**, 22, 532–545. [Google Scholar] [CrossRef] - Rigon, R.; Bertoldi, G.; Over, T.M. GEOtop: A Distributed Hydrological Model with Coupled Water and Energy Budgets. J. Hydrometeorol.
**2006**, 7, 371–388. [Google Scholar] [CrossRef] - Chen, L.-K.; Chang, C.-H.; Liu, C.-H.; Ho, J.-Y. Application of a Three-Dimensional Deterministic Model to Assess Potential Landslides, a Case Study: Antong Hot Spring Area in Hualien, Taiwan. Water
**2020**, 12, 480. [Google Scholar] [CrossRef] [Green Version] - Reid, M.E.; Christian, S.B.; Brien, D.L.; Henderson, S.T. Scoops3D—Software to Analyze Three-Dimensional Slope Stability Throughout a Digital Landscape (Version 1.0); U.S. Geological Survey: Reston, VA, USA, 2015.
- Park, D.W.; Lee, S.R.; Vasu, N.N.; Kang, S.H.; Park, J.Y. Coupled model for simulation of landslides and debris flows at local scale. Nat. Hazards
**2016**, 81, 1653–1682. [Google Scholar] [CrossRef] - Tran, T.V.; Alvioli, M.; Lee, G.; An, H.U. Three-dimensional, time-dependent modeling of rainfall-induced landslides over a digital landscape: A case study. Landslides
**2018**, 15, 1071–1084. [Google Scholar] [CrossRef] - GEO-SLOPE International. User’s Manual: SEEP/W and SLOPE/W; Geo-Slope International Ltd.: Calgary, AB, Canada, 2012. [Google Scholar]
- Baum, R.L.; Savage, W.Z.; Godt, J.W. TRIGRS—A Fortran Program for Transient Rainfall Infiltration and Grid-based Regional Slope-stability Analysis, Version 2.0; US Geological Survey Open-File Report 2008-1159; US Geological Survey: Reston, VA, USA, 2008; p. 75.
- An, H.; Viet, T.T.; Lee, G.; Kim, Y.; Kim, M.; Noh, S.; Noh, J. Development of time-variant landslide-prediction software considering three-dimensional subsurface unsaturated flow. Environ. Model. Softw.
**2016**, 85, 172–183. [Google Scholar] [CrossRef] - Dong, H.; Huang, R.; Gao, Q.-F. Rainfall infiltration performance and its relation to mesoscopic structural properties of a gravelly soil slope. Eng. Geol.
**2017**, 230, 1–10. [Google Scholar] [CrossRef] - Touma, J.; Vauclin, M. Experimental and Numerical Analysis of Two Phase Infiltration in a Partially Saturated Soil. Transp. Porous Media
**1986**, 1, 27–55. [Google Scholar] [CrossRef] - National Geographic Information Institute of Korea. Available online: http://map.ngii.go.kr/mn/mainPage.do (accessed on 15 August 2019).
- Kang, S.; Lee, S.-R. Debris flow susceptibility assessment based on an empirical approach in the central region of South Korea. Geomorphology
**2018**, 308, 1–12. [Google Scholar] [CrossRef] - Kang, S.; Lee, S.-R.; Vasu, N.N.; Park, J.-Y.; Lee, D.-H. Development of an initiation criterion for debris flows based on local topographic properties and applicability assessment at a regional scale. Eng. Geol.
**2017**, 230, 64–76. [Google Scholar] [CrossRef] - Korea Meteorological Administration. Available online: http://www.weather.go.kr/weather/observation/aws_table_popup.jsp (accessed on 7 March 2019).
- Seoul Metropolitan Government. Causes Survey and Restoration Work of Mt. Umyeon Landslides; Seoul Metropolitan Government: Seoul, Korea, 2014; p. 435.
- Korean Geotechnical Society. Research Contract Report: Addition and Complement Causes Survey of Mt. Umyeon Landslides; Society, K.G., Ed.; Korean Geotechnical Society: Seoul, Korea, 2011; p. 268. [Google Scholar]
- van Genuchten, M.T. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J.
**1980**, 44, 892–898. [Google Scholar] [CrossRef] [Green Version] - Fredlund, D.G.; Rahardjo, H. Soil Mechanics for Unsaturated Soils; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 1993. [Google Scholar]
- Korea Institue of Geoscience and Mineral Resources. Available online: https://mgeo.kigam.re.kr/ (accessed on 3 March 2020).
- Itasca. User’s Manual: Fluid–Mechanical Interaction, FLAC 7.0; Itasca Consulting Group, Inc.: Minneapolis, MN, USA, 2011. [Google Scholar]
- 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] - Lenhard, R.J.; Parker, J.C. A model for hysteretic constitutive relations governing multiphase flow: 2. Permeability-saturation relations. Water Resour. Res.
**1987**, 23, 2197–2206. [Google Scholar] [CrossRef] - Bishop, A.W. The principle of effective stress. Teknisk Ukeblad
**1959**, 106, 859–863. [Google Scholar] - Chateau, X.; Dormieux, L. Micromechanics of Unsaturated Porous Media. In IUTAM Symposium on Theoretical and Numerical Methods in Continuum Mechanics of Porous Materials; Solid Mechanics and Its Applications, Vol 87; Ehlers, W., Ed.; Springer: Berlin/Heidelberg, Germany, 2001; pp. 125–130. [Google Scholar]
- Chateau, X.; Dormieux, L. Micromechanics of saturated and unsaturated porous media. Int. J. Numer. Anal. Methods Geomech.
**2002**, 26, 831–844. [Google Scholar] [CrossRef] - Zhang, L.; Wu, F.; Zheng, Y.; Chen, L.; Zhang, J.; Li, X. Probabilistic calibration of a coupled hydro-mechanical slope stability model with integration of multiple observations. Georisk Assess. Manag. Risk Eng. Syst. Geohazards
**2018**, 12, 169–182. [Google Scholar] [CrossRef] - Kim, J.; Jeong, S.; Park, S.; Sharma, J. Influence of rainfall-induced wetting on the stability of slopes in weathered soils. Eng. Geol.
**2004**, 75, 251–262. [Google Scholar] [CrossRef] - Beguería, S. Validation and Evaluation of Predictive Models in Hazard Assessment and Risk Management. Nat. Hazards
**2006**, 37, 315–329. [Google Scholar] [CrossRef] [Green Version] - Huang, J.C.; Kao, S.J. Optimal estimator for assessing landslide model performance. Hydrol. Earth Syst. Sci.
**2006**, 10, 957–965. [Google Scholar] [CrossRef] [Green Version] - Hu, R.; Chen, Y.; Zhou, C. Modeling of coupled deformation, water flow and gas transport in soil slopes subjected to rain infiltration. Sci. China Technol. Sci.
**2011**, 54, 2561. [Google Scholar] [CrossRef] - National Disaster Management Institute. Study on the Secure Plan for the Stability of Slope Based on Ground Characteristics; Report no.: 11-1311526-000053-01; National Disaster Management Institute: Seoul, Korea, 2012; p. 138.
- Salciarini, D.; Fanelli, G.; Tamagnini, C. A probabilistic model for rainfall–induced shallow landslide prediction at the regional scale. Landslides
**2017**, 14, 1731–1746. [Google Scholar] [CrossRef] - Tofani, V.; Bicocchi, G.; Rossi, G.; Segoni, S.; D’Ambrosio, M.; Casagli, N.; Catani, F. Soil characterization for shallow landslides modeling: A case study in the Northern Apennines (Central Italy). Landslides
**2017**, 14, 755–770. [Google Scholar] [CrossRef] [Green Version]

**Figure 2.**Hourly rainfall and cumulative rainfall recorded at the Namhyun station during the two days from 26 July to 27 July 2011 (obtained from KMA).

**Figure 3.**(

**a**) Locations where slope failure occurred in the area of Umyeon Mountain in 2011. (

**b**) Geology map of the study area produced by the Korea Institute of Geoscience and Mineral resources [45] and locations of three sites (BL-1, BL-2, and BL-3) at which boring investigations were performed. (

**c**) Boring logs and groundwater levels obtained at three sites shown in (b) (modified from the Seoul Metropolitan Government [41]). (

**d**) Locations of the sites’ selected borehole for the investigation conducted by the Seoul Metropolitan Government [41] and the distribution of the five zones proposed by Park et al. [30].

**Figure 4.**(

**a**) van Genuchten soil-water characteristic curves (SWCCs) for the north side of Umyeon Mountain (zones 1 and 2) and its south side (zones 3, 4, and 5) (obtained from Seoul Metropolitan Government [41]). (

**b**) Relative permeability (K

_{r}) curves computed from the two SWCCs shown in Figure 4a. The constants of the two van Genuchten SWCCs used to compute Mualem–van Genuchten relative permeability curves [43] are presented in Table 1.

**Figure 5.**An example of the infinite slope model and finite difference discretization for infiltration analysis with a depth of 2 m, a length of 10 m, and an angle of 31°, which is approximately equal to the mean slope angle of the slope failure sites in the study area (i.e., 31.2°). Various angles were considered based on different cells of the GIS-based slope raster. ${q}_{w}$, $I\left(t\right)$, ${P}_{a}$, ${P}_{w}$, ${u}_{x}$, and ${u}_{y}$ denote water flux, variations in rainfall intensity with time, pore water and air pressures, displacements in the horizontal and vertical directions, respectively.

**Figure 6.**Comparison between measured and simulated matric suction for 140 h (from 0:00 on 29 June 2012 to 20:00 5 July 2012).

**Figure 7.**Variations in pore water pressure and matric suction with time when the single-phase flow model was applied to a slope of 30°. The vertical axis of each graph represents the depth from the ground surface. The first and second columns represent the ranges of the pore water pressure and matric suction, respectively.

**Figure 8.**Variations in pore water/air pressures and matric suction with time when the two-phase flow model was applied to a slope of 30°. The vertical axis of each graph represents the depth from the ground surface. The columns represent the ranges of the pore water pressure, pore air pressure, and matric suction from left to right.

**Figure 9.**Variations in minimum safety factors with time when the two- and single-phase flow models were applied to slopes with maximum, average, and minimum slope angles for actual failure slopes: (

**a**)–(

**e**) for zones 1–5. We used the maximum, average, and minimum slope angles for the entire actual slope failure sites in the study area only for zone 3, where an actual slope failure did not occur. The maximum, average, and minimum slope angles at zones 1–5 are 38.8°, 30.7°, and 26.5°; 37.4°, 31°, and 23.3°; 40.5°, 31.2°, and 23.3°; 40.5°, 31.8°, and 24.8°; and 40.3°, 31.1°, and 24.7°, respectively.

**Figure 10.**Locations of slope failures and landslide susceptibility maps, which present distributions of safety factors: (

**a**,

**b**) from the two- and single-phase flow models using FLAC. (

**c**) Zoomed-in views of Figure 10a,b. (

**d**) Distribution of the slope failure area at each zone predicted by the two- and single-phase flow models.

Zone No. | $\mathit{\varphi}$ (°) | c (kPa) | ${\mathit{\gamma}}_{\mathit{t}}(\mathbf{kN}/{\mathbf{m}}^{3})$ | ${\mathit{\gamma}}_{\mathit{d}}(\mathbf{kN}/{\mathbf{m}}^{3})$ | k_{s} (m/s) | Van Genuchten SWCC Coefficient | |
---|---|---|---|---|---|---|---|

1/P_{0} | a | ||||||

1 | 25.3 | 9.6 | 18.1 | 15 | 7.15 × 10^{−6} | 0.051 | 0.35 |

2 | 28.5 | 5.8 | 17.7 | 14.9 | 3.87 × 10^{−6} | 0.051 | 0.35 |

3 | 37.6 | 7.7 | 17 | 14 | 1.80 × 10^{−6} | 0.038 | 0.63 |

4 | 30.9 | 7.6 | 17.3 | 14.5 | 9.70 × 10^{−6} | 0.038 | 0.63 |

5 | 28.2 | 6.3 | 18.2 | 15 | 3.69 × 10^{−6} | 0.038 | 0.63 |

_{t}: Total unit weight of soil; γ

_{d}: Dry unit weight of soil; k

_{s}: Saturated hydraulic conductivity; P

_{0}: Parameter based on the air-entry value; and a: van Genuchten m constant; SWCC: soil-water characteristic curve.

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

Viscosity ratio, ${\mu}_{w}/{\mu}_{a}$ | 56 |

Water density, ${\rho}_{w}$ | 1000 kg/m^{3} |

Air density, ${\rho}_{a}$ | 1.25 kg/m^{3} |

Bulk modulus of water, K_{w} | 2 × 10^{9} Pa |

Bulk modulus of air, K_{a} | 1 × 10^{5} Pa |

Total Probability (Total Cells: 63,157) | TP-Flow Actual (Observed) Class | SP-Flow Actual (Observed) Class | |||
---|---|---|---|---|---|

Positive | Negative | Positive | Negative | ||

Predicted class | Positive | True positive 148 (0.23%) | False positive 12,498 (19.79%) | True positive 150 (0.24%) | False positive 15,913 (25.2%) |

Negative | False negative 13 (0.02%) | True negative 50,498 (79.96%) | False negative 11 (0.02%) | True negative 47,083 (74.55%) | |

Efficiency (= (TP + TN)/N) | 80.19% | 74.79% | |||

Positive predictive power (= TP/(TP + FP)) | 1.17% | 0.93% | |||

Sensitivity (= TP/(TP + FN)) | 91.93% | 93.17% | |||

Specificity (= TN/(FP + TN)) | 80.16% | 74.74% |

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**MDPI and ACS Style**

Kang, S.; Cho, S.-E.; Kim, B.; Go, G.-H.
Effects of Two-Phase Flow of Water and Air on Shallow Slope Failures Induced by Rainfall: Insights from Slope Stability Assessment at a Regional Scale. *Water* **2020**, *12*, 812.
https://doi.org/10.3390/w12030812

**AMA Style**

Kang S, Cho S-E, Kim B, Go G-H.
Effects of Two-Phase Flow of Water and Air on Shallow Slope Failures Induced by Rainfall: Insights from Slope Stability Assessment at a Regional Scale. *Water*. 2020; 12(3):812.
https://doi.org/10.3390/w12030812

**Chicago/Turabian Style**

Kang, Sinhang, Sung-Eun Cho, Byungmin Kim, and Gyu-Hyun Go.
2020. "Effects of Two-Phase Flow of Water and Air on Shallow Slope Failures Induced by Rainfall: Insights from Slope Stability Assessment at a Regional Scale" *Water* 12, no. 3: 812.
https://doi.org/10.3390/w12030812