# Effect of Density and Total Weight on Flow Depth, Velocity, and Stresses in Loess Debris Flows

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Materials and Sample Preparation

#### 2.1. Similarity Principle

_{0}); characteristic depth of the fluid (h); characteristic diameter of the particles (d); channel length (L); channel width (W); slope angle (α); fluid density (ρ

_{f}); density of solid material (ρ

_{s}); dynamic viscosity coefficient (μ); gravitational acceleration (g); elastic modulus of grain (E

_{s}); coefficient of friction (c); initial flow velocity (v

_{0}). The distribution of the flow velocity can be expressed as follows:

_{f}, v

_{0}, and L as a unit system:

_{e}is the Reynolds number, which represents the ratio of inertial forces to viscosity; F

_{r}is the Froude number, which represents the ratio of inertial forces to gravitational forces; and ε represents the ratio of the fluid depth to channel length. The dimensionless function format of flow velocity can then be obtained by using similarity theory:

_{e}and F

_{r}appear in the distribution function for the dynamic characteristics, and must be equal in the model being developed and the actual process it represents. To achieve this goal, it’s necessary to build a model of the study system that has the same scale as the system, and this is unrealistic in practice [58]. To achieve accurate experimental results, similarity methods are used [58]. To meet this goal for the geometric similarity and kinematic similarity, R

_{e}and F

_{r}are often used [25,58]. Because the gravitational force is one of the most important driving forces in flume experiments [58], we adopted F

_{r}as the research object.

#### 2.1.1. Geometric Similarity

_{l}is the length ratio, l

_{a}is the length of the actual, and l

_{f}is the length of the flume.

#### 2.1.2. Material Similarity

#### 2.1.3. Velocity Similarity

_{0}similarity ratio.

#### 2.2. Experimental Apparatus

_{l}(m) is the length of the flume, and Δ

_{t}is the elapsed time (s).

#### 2.3. Sample Preparation

_{5}) was 0.36 mm. Prior to conducting the experiment, the mixtures were prepared according to the required densities (including 1500 kg/m

^{3}, 1650 kg/m

^{3}, 1800 kg/m

^{3}, 1900 kg/m

^{3}, 2050 kg/m

^{3}, 2100 kg/m

^{3}, and 2300 kg/m

^{3}) and total weights (including 200 kg, 300 kg, 400 kg, and 500 kg).

## 3. Results and Discussion

#### 3.1. General Flow Patterns

#### 3.2. Flow Depth

^{3}to 2.3 g/cm

^{3}. The flow depths increased to approximately 1.3 times and 1.5 times the minimum depth when the density increased from 1.5 g/cm

^{3}to 1.8 g/cm

^{3}and from 1.8 to 2.3 g/cm

^{3}, respectively.

#### 3.3. Flow Velocity

^{3}to 2.3 g/cm

^{3}. The flow velocity also decreased by approximately 1.2 times and 1.5 times as the density increased from 1.5 g/cm

^{3}to 1.8 g/cm

^{3}and from 1.8 g/cm

^{3}to 2.3 g/cm

^{3}, respectively. This indicated that low-density debris flow fastest, and that the high-density debris flowed slowly. When the mixture weight decreased from 500 kg to 400 kg, the flow velocity decreased by about 8%. Because of the inertial force increased with increasing mixture weight, it gradually increased for a given mixture density [67]. Our results therefore confirm the results of previous studies, which reported that debris flow density and mixture weight both strongly affected the debris flow velocity [5,38,72]. Combined with the flow depth analysis, these results let us calculate the impact force of the flowing debris.

#### 3.4. Stress and Pressure Measurements

#### 3.4.1. Total Normal Stress

^{3}, and at position S1 when the density was low (i.e., 1.5 or 1.65 g/cm

^{3}). These results illustrate that the maximum values moved upstream as the mixture density decreased (Figure 7a). For a given density and monitoring point, the peak total normal stress increased with increasing weight, and the peak position gradually occurred later (Figure 7b). The magnitude of this change increased as the weight of the mixture increased from 200 kg to 500 kg. An especially large difference in the stress was observed when the weight increased from 300 kg to 400 kg.

^{3}to 1.8 g/cm

^{3}. Therefore, there were obvious changes in the total normal stress as the mixture density increased (Figure 7c). The changes with respect to density result from the increasing proportion of the weight accounted for by water as the mixture density decreased, since the water decreased the resistance to flow, resulting in a faster flow velocity and reduced total normal stress [75]. However, the results differed for the high-density debris flow; although the weight of the mixture (thus, the gravitational force) was higher due to the higher content of particles, the resistance to flow also increased due to the decreased water content.

^{3}to 2.3 g/cm

^{3}. The stress also increased to between 1.5 and 3 times the initial value when the density increased from 1.5 g/cm

^{3}to 1.8 g/cm

^{3}, versus approximately 1.8 times as the density increased from 1.8 g/cm

^{3}to 2.3 g/cm

^{3}. Thus, the total normal stress was strongly influenced by the mixture density. In addition, as the mixture weight increased from 200 kg to 500 kg at a given density, the stress increased to approximately 6 times the initial value. However, the change was largest when the weight increased from 300 kg to 400 kg.

#### 3.4.2. Pore Fluid Pressure

^{3}or 1.9 g/cm

^{3}. The maximum peak was upstream when the mixture had a density <1.8 g/cm

^{3}. These results illustrate that the maximum values shift from the top of the flume to the bottom as the mixture density decreases (Figure 8a). The peak became increasingly large as the mixture weight increased at a given density and measurement position, and the position of the peak occurred later in the flow (Figure 8b). The magnitude of change changed greatly as the weight of the mixture increased from 200 kg to 500 kg. The magnitude of increase was largest when the mixture weight increased from 200 kg to 300 kg. For a given mixture weight and monitoring position, the peak occurred later as the mixture density decreased. The magnitude of the decrease was largest when the density increased from 1.65 g/cm

^{3}to 1.8 g/cm

^{3}. These results indicate that the pore fluid pressure changed greatly as the density changed (Figure 8c). In addition, a low-density debris flow sustains high pore fluid pressure during the flow, thus increasing the mobility of the debris and increasing the likelihood that the debris will be liquefied [76], and our results agree with previous experimental results [6]. However, for the high-density debris flow, we found reduced mobility of the debris and a reduced likelihood that the debris will liquefy.

^{3}to 1.4 g/cm

^{3}. The magnitude of the pore fluid pressure increased by 2 to 4 times with the density was high. However, when the density ranged from 1.8 g/cm

^{3}to 1.5 g/cm

^{3}, the pore fluid pressure increased by only approximately 1.5 times. For a given mixture density, the increase in pore fluid pressure at 500 kg was to approximately 6 times the value with a mixture weight of 200 kg.

#### 3.5. Development of Predictive Functions Based on the Study Data

#### 3.5.1. Parameterization of the Fitting Functions

^{dγ})

_{g}m + s

_{h}) × sS

_{i}× lnγ + s

_{j})

_{l}m + p

_{n}) × (p

_{0}× lnγ + p

_{q}),

#### 3.5.2. Parameterized Functions

^{2}) to evaluate the fitting accuracy [77,78]. The results were all excellent, with an adjusted R

^{2}value >0.93. Thus, our functions did a good job of explaining the variation in flow depth, flow velocity, total normal stress, and pore fluid pressure as a function of the mixture density and weight, which agrees with previous results [58,67,75].

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Hungr, O.; Evans, S.; Bovis, M.; Hutchinson, J. A review of the classification of landslides of the flow type. Environ. Eng. Geosci.
**2001**, 7, 221–238. [Google Scholar] [CrossRef] - Godt, J.W.; Coe, J.A. Alpine debris-flows triggered by a 28 July 1999 thunderstorm in the Central Front Range, Colorado. Geomorphology
**2007**, 84, 80–97. [Google Scholar] [CrossRef] - Pudasaini, S.P. A general two-phase debris flow model. J. Geophys. Res.
**2012**, 117. [Google Scholar] [CrossRef][Green Version] - Takahashi, T. Debris Flow: Mechanics, Prediction and Counter-Measures, 2nd ed.; CRC Press: Boca Raton, FL, USA, 2014; p. 574. [Google Scholar]
- Wang, F.; Chen, X.Q.; Chen, J.G.; You, Y. Experimental study on a debris-flow drainage channel with different types of energy dissipation baffle. Eng. Geol.
**2017**, 220, 43–51. [Google Scholar] [CrossRef] - Iverson, R.M. The physics of debris flows. Rev. Geophys.
**1997**, 35, 245–296. [Google Scholar] [CrossRef][Green Version] - Iverson, R.M.; Reid, M.E.; Logan, M.; Lahusen, R.G.; Godt, I.W.; Groswold, J.P. Positive feedback and momentum growth during debris-flow entrainment of wet bed sediment. Nat. Geosci.
**2011**, 4, 116–121. [Google Scholar] [CrossRef] - VanDine, D.F.; Bovis, M. History and goals of Canadian debris-flow research. Nat. Hazards
**2002**, 26, 67–80. [Google Scholar] [CrossRef] - Tang, C.; Zhu, J.; Li, W.L.; Liang, J.T. Rainfall-triggered debris flows following the Wenchuan earthquake. Bull. Eng. Geol. Environ.
**2009**, 68, 187–194. [Google Scholar] [CrossRef] - Xu, Q.; Zhang, S.; Li, W.L.; van Asch, T.W.J. The 13 August 2010 catastrophic debris flows after the 2008 Wenchuan earthquake, China. Nat. Hazards Earth Syst. Sci.
**2012**, 12, 201–216. [Google Scholar] [CrossRef][Green Version] - Cui, P.; Zhou, G.G.D.; Zhu, X.H.; Zhang, J.Q. Scale amplification of natural debris flows caused by cascading landslide dam failures. Geomorphology
**2013**, 182, 173–189. [Google Scholar] [CrossRef] - Han, Z.; Wang, W.D.; Li, Y.G.; Huang, J.L.; Su, B.; Tang, C.; Chen, G.Q.; Qu, X. An integrated method for rapid estimation of the valley incision by debris flows. Eng. Geol.
**2018**, 232, 34–35. [Google Scholar] [CrossRef] - Jakob, M.; Friele, P. Frequency and magnitude of debris flows on Cheekye River, British Columbia. Geomorphology
**2010**, 114, 382–395. [Google Scholar] [CrossRef] - Stoffel, M.; Mendlik, T.; Schneuwly-Bollschweiler, M.; Gobiet, A. Possible impacts of climate change on debris-flow activity in the Swiss Alps. Clim. Chang.
**2014**, 122, 141–155. [Google Scholar] [CrossRef] - Li, Y.; Ma, C.; Wang, Y. Landslides and debris flows caused by an extreme rainstorm on 21 July 2012 in mountains near Beijing, China. Bull. Eng. Geol. Environ.
**2017**, 1–16. [Google Scholar] [CrossRef] - Lyu, H.M.; Shen, J.S.; Arulrajah, A. Assessment of geohazards and preventative countermeasures using AHP incorporated with GIS in Lanzhou, China. Sustainability
**2018**, 10, 304. [Google Scholar] [CrossRef] - Jakob, M. Debris-Flow Hazard Analysis. In Debris-Flow Hazards and Related Phenomena; Jakob, M., Hungr, O., Eds.; Springer: Berlin, Germany, 2005; pp. 411–443. [Google Scholar]
- Hürlimann, M.; Copons, R.; Altimir, J. Detailed debris flow hazard assessment in Andorra: A multidisciplinary approach. Geomorphology
**2006**, 78, 359–372. [Google Scholar] [CrossRef] - Cavalli, M.; Marchi, L. Characterisation of the surface morphology of an alpine alluvial fan using airborne LiDAR. Nat. Hazards Earth Syst. Sci.
**2008**, 8, 323–333. [Google Scholar] [CrossRef][Green Version] - Jeong, S.; Kim, Y.; Lee, J.K.; Kim, J. The 27 July 2011 debris flows at Umyeonsan, Seoul, Korea. Landslides
**2015**, 12, 799–813. [Google Scholar] [CrossRef] - Bai, S.; Xu, Q.; Wang, J.; Zhou, P. Pre-conditioning factors and susceptibility assessments of Wenchuan earthquake landslide at the Zhouqu segment of Bailongjiang basin, China. J. Geol. Soc. India
**2013**, 82, 575–582. [Google Scholar] [CrossRef] - Fan, X.; Xu, Q.; Scaringi, G.; Li, S.; Peng, D. A chemo-mechanical insight into the failure mechanism of frequently occurred landslides in the Loess Plateau, Gansu Province, China. Eng. Geol.
**2016**, 228, 337–345. [Google Scholar] [CrossRef] - Yuan, B.; Chen, W.W.; Tang, Y.Q.; Li, J.P.; Yang, Q. Experimental study on gully-shaped mud flow in the loess area. Environ. Earth Sci.
**2015**, 74, 759–769. [Google Scholar] [CrossRef] - Derbyshire, E. Geological hazards in loess terrain, with particular reference to the loess regions of China. Earth Sci. Rev.
**2001**, 54, 231–260. [Google Scholar] [CrossRef] - Cui, P.; Zeng, C.; Lei, Y. Experimental analysis on the impact force of viscous debris flow. Earth Surf. Process. Landf.
**2015**, 40, 1644–1655. [Google Scholar] [CrossRef][Green Version] - Chen, J.G.; Chen, X.Q.; Li, Y.; Wang, F. An experimental study of dilute debris flow characteristics in a drainage channel with an energy dissipation structure. Eng. Geol.
**2015**, 193, 224–230. [Google Scholar] [CrossRef] - Hubl, J.; Suda, J.; Proske, D.; Kaitna, R.; Scheidl, C. Debris flow impact estimation. In Proceedings of the 11th International Symposium on Water Management and Hydraulic Engineering, Ohrid, Macedonia, 8–12 September 2009; Volume 1, pp. 137–148. [Google Scholar]
- Armanini, A.; Larcher, M. Rational criterion for designing opening of slit-check dam. J. Hydraul. Eng.
**2001**, 127, 94–104. [Google Scholar] [CrossRef] - Huebl, J.; Fiebiger, G. Debris-Flow Mitigation Measures. In Debris-Flow Hazards and Related Phenomena; Jakob, M., Hungr, O., Eds.; Springer: Berlin, Germany, 2005; pp. 445–466. [Google Scholar]
- Hassanli, A.M.; Nameghi, A.E.; Beecham, S. Evaluation of the effect of porous check dam location on fine sediment retention (a case study). Environ. Monit. Assess.
**2009**, 152, 319–326. [Google Scholar] [CrossRef] [PubMed] - Takahisa, M. Structural countermeasures for debris flow disasters. Int. J. Eros. Control Eng.
**2008**, 1, 8–43. [Google Scholar] [CrossRef] - You, Y.; Pan, H.L.; Liu, J.F.; Ou, G.Q. The optimal cross-section design of the “Trapezoid-V” shaped drainage channel of viscous debris flow. J. Mt. Sci.
**2011**, 8, 103–107. [Google Scholar] [CrossRef] - Wendeler, C.; McArdell, B.; Volkwein, A.; Denk, M.; Gröner, E. Debris flow mitigation with flexible ring net barriers—Field tests and case studies. WIT Trans. Eng. Sci.
**2008**, 60, 23–31. [Google Scholar] [CrossRef] - Volkwein, A.; Baumann, R.; Rickli, C.; Wendeler, C. Standardization for Flexible Debris Retention Barriers; Lollino, G., Giordan, D., Crosta, G.B., Corominas, J., Azzam, R., Wasowski, J., Sciarra, N., Eds.; Springer: Berlin, Germany, 2011; Volume 2, pp. 193–196. [Google Scholar]
- Liu, J.F.; Nakatani, K.; Mizuyama, T. Effect assessment of debris flow mitigation works based on numerical simulation by using Kanako 2D. Landslides
**2013**, 10, 161–173. [Google Scholar] [CrossRef] - Okano, K.; Suwa, H.; Kanno, T. Characterization of debris flows by rainstorm condition at a torrent on the Mount Yakedake volcano, Japan. Geomorphology
**2012**, 136, 88–94. [Google Scholar] [CrossRef] - Navratil, O.; Liébault, F.; Bellot, H.; Travaglini, E.; Theule, J.; Chambon, G.; Laigle, D. High-frequency monitoring of debris-flow propagation along the Réal Torrent, Southern French Prealps. Geomorphology
**2013**, 201, 157–171. [Google Scholar] [CrossRef] - Rickenmann, D. Empirical relationships for debris flows. Nat. Hazards
**1999**, 19, 47–77. [Google Scholar] [CrossRef] - Song, E.; Sang, J.I.; Dongyeob, K.D.; Kun, W.C. Flow and deposition characteristics of sediment mixture in debris flow flume experiments. For. Sci. Technol.
**2017**, 13, 61–65. [Google Scholar] [CrossRef] - Suwa, H.; Okud, S.; Yokoya, K. Observation system on rocky mud-flow. Bull. Dis. Prev. Res. Inst.
**1973**, 23, 59–73. [Google Scholar] - Hu, K.H.; Wei, F.Q.; Li, Y. Real-time measurement and preliminary analysis of debris-flow impact force at Jiangjia Ravine, China. Earth Surf. Process. Landf.
**2011**, 36, 1268–1278. [Google Scholar] [CrossRef] - Wendeler, C.; Volkwein, A.; Roth, A.; Denk, M.; Wartmann, S. Field Measurements Used for Numerical Modelling of Flexible Debris Flow Barriers. In Debris-Flow Hazards Mitigation, Mechanics, Prediction, and Assessment; Chen, C.I., Major, J.J., Eds.; Millpress: Rotterdam, The Netherlands, 2007; pp. 681–687. [Google Scholar]
- Hürlimann, M.; Abancó, C.; Moya, J.; Vilajosana, I. Results and experiences gathered at the Rebaixader debris-flow monitoring site, Central Pyrenees, Spain. Landslides
**2013**, 11, 939–953. [Google Scholar] [CrossRef] - Morino, C.; Conway, S.J.; Balme, M.R.; Hillier, J.; Jordan, C.; Saemundsson, Þ.; Argles, T. Debris-flow release processes investigated through the analysis of multi-temporal LiDAR datasets in north-western Iceland. Earth Surf. Process. Landf.
**2018**. [Google Scholar] [CrossRef] - Bossi, G.; Cavalli, M.; Crema, S.; Frigerio, S.; Quan Luna, B.; Mantovani, M.; Marcato, G.; Schenato, L.; Pasuto, A. Multi-temporal LiDAR-DTMs as a tool for modelling a complex landslide: A case study in the Rotolon catchment (eastern Italian Alps). Nat. Hazards Earth Syst. Sci.
**2015**, 15, 715–722. [Google Scholar] [CrossRef] - Cavalli, M.; Goldin, B.; Comiti, F.; Brardinoni, F.; Marchi, L. Assessment of erosion and deposition in steep mountain basins by differencing sequential digital terrain models. Geomorphology
**2017**, 291, 4–16. [Google Scholar] [CrossRef] - D’Agostino, V.; Cesca, M.; Marchi, L. Field and laboratory investigations of runout distances of debris flows in the Dolomites (Eastern Italian Alps). Geomorphology
**2010**, 115, 294–304. [Google Scholar] [CrossRef] - Scheidl, C.; Chiari, M.; Kaitna, R.; Müllegger, M.; Krawtschuk, A.; Zimmermann, T.; Proske, D. Analysing debris-flow impact models, based on a small scale modelling approach. Surv. Geophys.
**2013**, 34, 21–140. [Google Scholar] [CrossRef] - Haas, T.; Braat, L.; Leuven, J.R.F.W.; Lokhorst, I.R.; Kleinhans, M.G. Effects of debris flow composition on runout, depositional mechanisms, and deposit morphology in laboratory experiments. J. Geophys. Res. Earth Surf.
**2015**, 120, 1949–1972. [Google Scholar] [CrossRef][Green Version] - Hürlimann, M.; Rickenmann, D.; Graf, C. Field and monitoring data of debris-flow events in the Swiss Alps. Can. Geotech. J.
**2003**, 40, 161–175. [Google Scholar] [CrossRef] - Takahashi, T. A review of Japanese debris flow research. Int. J. Eros. Control Eng.
**2009**, 2, 1–14. [Google Scholar] [CrossRef] - McCoy, S.W.; Kean, J.W.; Coe, J.A.; Staley, D.M.; Wasklewicz, T.A.; Tucker, G.E. Evolution of a natural debris flow: In situ measurements of flow dynamics, video imagery, and terrestrial laser scanning. Geology
**2010**, 38, 735–738. [Google Scholar] [CrossRef] - Marchi, L.; Tecca, P.R. Dating Torrential Processes on Fans and Cones. In Debris-Flow Monitoring in Italy; Schneuwly-Bollschweiler, M., Stoffel, M., Rudolf-Miklau, F., Eds.; Springer: Berlin, Germany, 2013; pp. 309–318. [Google Scholar]
- Moriguchi, S.; Borja, R.; Yashima, A.; Sawada, K. Estimating the impact force generated by granular flow on a rigid obstruction. Acta Geotech.
**2009**, 4, 57–71. [Google Scholar] [CrossRef] - Iverson, R.M. Scaling and design of landslide and debris-flow experiments. Geomorphology
**2015**, 244, 9–20. [Google Scholar] [CrossRef] - Arattano, M.; Franzi, L. On the evaluation of debris flows dynamics by means of mathematical models. Nat. Hazards Earth Syst. Sci.
**2003**, 3, 539–544. [Google Scholar] [CrossRef][Green Version] - Armanini, A.; Larcher, M.; Odorizzi, M. Dynamic impact of a debris flow front against a vertical wall. In Proceedings of the 5th International Conference on Debris-flow Hazard Mitigation, Rome, Italy, 14–17 June 2011; Genevois, R., Douglas, L., Eds.; Casa Editrice Università La Sapienza: Roma, Italy, 2011; pp. 1041–1049. [Google Scholar]
- Wang, D.; Chen, Z.; He, S.; Liu, Y.; Tang, H. Measuring and estimating the impact pressure of debris flows on bridge piers based on large-scale laboratory experiments. Landslides
**2018**, 15, 1331–1345. [Google Scholar] [CrossRef] - Egashira, S.; Honda, N.; Itoh, T. Experimental study on the entrainment of bed material into debris flow. Phys. Chem. Earth Part C
**2001**, 26, 645–650. [Google Scholar] [CrossRef] - Acharya, G.; Cochrane, T.; Davies, T.; Bowman, E. Quantifying and modeling postfailure sediment yields from laboratory-scale soil erosion and shallow landslide experiments with silty loess. Geomorphology
**2011**, 129, 49–58. [Google Scholar] [CrossRef] - Peng, J.; Huo, A.; Cheng, Y.; Dang, J.; Wei, H.; Wang, X.; Li, C. Submersion simulation in a typical debris flow watershed of Jianzhuangchuan catchment, Loess Plateau. Environ. Earth Sci.
**2017**, 76. [Google Scholar] [CrossRef] - Peng, J.; Fan, Z.; Wu, D.; Zhuang, J.; Dai, F.; Chen, W.; Zhao, C. Heavy rainfall triggered loess-mudstone landslide and subsequent debris flow in Tianshui, China. Eng. Geol.
**2015**, 186, 79–90. [Google Scholar] [CrossRef] - Tu, X.B.; Kwong, A.K.L.; Dai, F.C.; Tham, L.G.; Min, H. Field monitoring of rainfall infiltration in a loess slope and analysis of failure mechanism of rainfall induced landslides. Eng. Geol.
**2009**, 105, 134–150. [Google Scholar] [CrossRef] - Shi, J.S.; Wu, L.Z.; Wu, S.R.; Li, B.; Wang, T.; Xin, P. Analysis of the causes of large-scale loess landslides in Baoji, China. Geomorphology
**2016**, 264, 109–117. [Google Scholar] [CrossRef] - Tan, Q.M. Dimensional Analysis: With Case Studies in Mechanics; Springer: Berlin, Germany, 2011. [Google Scholar]
- Vagnon, F.; Segalini, A. Debris flow impact estimation on a rigid barrier. Nat. Hazards Earth Syst. Sci.
**2016**, 16, 1–17. [Google Scholar] [CrossRef] - Choi, C.E.; Ng, C.W.W.; Au-Yeung, S.C.H.; Goodwin, G.R. Froude characteristics of both dense granular and water flows in flume modelling. Landslides
**2015**, 12, 1197–1206. [Google Scholar] [CrossRef] - Ditzler, C.; Scheffe, K.; Monger, H.C. Soil Survey Manua, Soil Science Division Staff; USDA Handbook 18; Government Printing Office: Washington, DC, USA, 2017.
- Zhang, D.X.; Wang, G.H. Study of the 1920 Haiyuan earthquake-induced landslides in loess (China). Eng. Geol.
**2007**, 94, 76–88. [Google Scholar] [CrossRef] - Xu, L.; Dai, F.C.; Tham, L.G.; Tu, X.B.; Min, H.; Zhou, Y.F.; Wu, C.X.; Xu, K. Field testing of irrigation effects on the stability of a cliff edge in loess, North-west China. Eng. Geol.
**2011**, 120, 10–17. [Google Scholar] [CrossRef] - Wang, G.H.; Zhang, D.X.; Furuya, G.; Yang, J. Pore-pressure generation and fluidization in a loess landslide triggered by the 1920 Haiyuan earthquake, China: A case study. Eng. Geol.
**2014**, 174, 36–45. [Google Scholar] [CrossRef][Green Version] - Chen, J.G.; Chen, X.Q.; Wang, T.; Zou, Y.H.; Zhong, W. Types and causes of debris flow damage to drainage channels in the Wenchuan earthquake area. J. Mt. Sci.
**2014**, 11, 1406–1419. [Google Scholar] [CrossRef] - Gaál, L.; Szolgay, J.; Kohnová, S.; Hlavčová, K.; Parajka, J.; Viglione, A.; Merz, R.; Blöschl, G. Dependence between flood peaks and volumes: A case study on climate and hydrological controls. Hydrol. Sci. J.
**2015**, 60, 968–984. [Google Scholar] [CrossRef] - Kean, J.W.; McCoy, S.W.; Tucker, G.E.; Staley, D.M.; Coe, J.A. Runoff-generated debris flows: Observations and modeling of surge initiation, magnitude, and frequency. J. Geophys. Res. Earth Surf.
**2013**, 118, 2190–2207. [Google Scholar] [CrossRef][Green Version] - Lyu, L.; Wang, Z.; Cui, P.; Xu, M. The role of bank erosion on the initiation and motion of gully debris flows. Geomorphology
**2017**, 285, 137–151. [Google Scholar] [CrossRef] - Ilstad, T.; Marr, J.G.; Elverhøi, A.; Harbitz, C.B. Laboratory studies of subaqueous debris flows by measurements of pore-fluid pressure and total stress. Mar. Geol.
**2004**, 213, 403–414. [Google Scholar] [CrossRef] - Gujarati, D.N. Basic Econometrics; Tata McGraw-Hill Education: New Yrok, NY, USA, 2009. [Google Scholar]
- Song, F.; Wang, H.; Jiang, M.J. Analytically-based simplified formulas for circular tunnels with two liners in viscoelastic rock under anisotropic initial stresses. Constr. Build. Mater.
**2018**, 175, 746–767. [Google Scholar] [CrossRef]

**Figure 1.**Schematic diagram of the experimental apparatus. Total normal stress sensors (labeled “S”) and pore fluid-pressure sensors (labeled “P”) are installed at upstream (S1 and P1), midstream (S2 and P2), and downstream (S3 and P3) positions.

**Figure 3.**Video frames from the debris flow experiments with densities of (

**a**) 1650 kg/m

^{3}and (

**b**) 2050 kg/m

^{3}.

**Figure 4.**Example of a representative set of experimental results for the normal stress, pore pressure, and flow depth for a sample with a density of γ = 2200 kg/m

^{3}and a weight of w = 500 kg. Data are for sampling positions S2 and P2 in Figure 1.

**Figure 5.**Effect on flow depth of (

**a1**,

**a2**) mixture density (γ) for two representative weights and (

**b1**,

**b2**) total mixture weight (w) for two representative densities.

**Figure 7.**Effect of mixture density (γ) and total weight (w) on the total normal stress at monitoring points S1 to S3 in Figure 1: (

**a1**,

**a2**) hydrographs for mixtures with constant weight and density at the three monitoring points (whose positions are shown in Figure 1); (

**b1**,

**b2**) hydrographs for mixtures with different weights but constant density at the three monitoring points; and (

**c1**,

**c2**) hydrographs for mixtures with different densities but constant weight at the three monitoring points.

**Figure 8.**Changes in the pore fluid pressure as a function of mixture density and weight: (

**a1**,

**a2**) hydrographs for mixtures at the three monitoring points P1 to P3 (Figure 1) but with constant weight and density; (

**b1**,

**b2**) hydrographs for mixtures with different weights but with constant density at a given monitoring point; and (

**c1**,

**c2**) hydrographs for mixtures with different densities but with constant weight at a given monitoring point.

**Figure 9.**Fitting functions for the maximum flow depth as a function of the mixture weight and density.

**Figure 10.**Fitting functions for the maximum flow velocity as a function of the mixture weight and density.

**Figure 11.**Fitting function for the total normal stress: (

**a**) the mixture weight and total normal stress; (

**b**) the mixture density and total normal stress. The locations of positions S1 to S3 are shown in Figure 1.

**Figure 12.**Fitting function for the pore fluid pressure as a function of: (

**a**) the mixture weight and pore fluid; (

**b**) the mixture density and pore fluid pressure. The locations of positions P1 to P3 are shown in Figure 1.

**Figure 13.**Response surfaces for the (

**a**) maximum flow depth, and (

**b**) mean flow velocity as a function of mixture weight and density.

Index | a | B | c | d | Adj. R^{2} |
---|---|---|---|---|---|

Coefficient value | 0.0524 | 27.2243 | 0.1044 | 0.8405 | 0.9729 |

Index | u | K | y | z | Adj. R^{2} |
---|---|---|---|---|---|

Coefficient values | 0.0009 | 0.6280 | −1.5142 | 4.9659 | 0.9834 |

**Table 3.**Coefficients in Equation (19) for maximum total normal stress at monitoring positions S1 to S3 (Figure 1).

Position | s_{g} | s_{h} | S_{i} | s_{i} | Adj. R^{2} |
---|---|---|---|---|---|

S1 | 0.0022 | 0.5259 | 3.2798 | −0.4498 | 0.9603 |

S2 | 0.0019 | 0.5764 | 4.3714 | −0.9746 | 0.9370 |

S3 | 0.0016 | 0.6155 | 5.8129 | −1.8288 | 0.9412 |

**Table 4.**Coefficients in Equation (20) for the maximum pore fluid pressure at monitoring positions P1 to P3 (Figure 1).

Position | p_{l} | p_{n} | p_{o} | p_{q} | Adj. R^{2} |
---|---|---|---|---|---|

P1 | 0.0017 | 0.4085 | −6.0102 | 5.5407 | 0.9548 |

P2 | 0.0019 | 0.3766 | −4.6781 | 4.7382 | 0.9586 |

P3 | 0.0021 | 0.2782 | −3.3683 | 3.8258 | 0.9423 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Shu, H.; Ma, J.; Yu, H.; Hürlimann, M.; Zhang, P.; Liu, F.; Qi, S. Effect of Density and Total Weight on Flow Depth, Velocity, and Stresses in Loess Debris Flows. *Water* **2018**, *10*, 1784.
https://doi.org/10.3390/w10121784

**AMA Style**

Shu H, Ma J, Yu H, Hürlimann M, Zhang P, Liu F, Qi S. Effect of Density and Total Weight on Flow Depth, Velocity, and Stresses in Loess Debris Flows. *Water*. 2018; 10(12):1784.
https://doi.org/10.3390/w10121784

**Chicago/Turabian Style**

Shu, Heping, Jinzhu Ma, Haichao Yu, Marcel Hürlimann, Peng Zhang, Fei Liu, and Shi Qi. 2018. "Effect of Density and Total Weight on Flow Depth, Velocity, and Stresses in Loess Debris Flows" *Water* 10, no. 12: 1784.
https://doi.org/10.3390/w10121784