# The Development of a 1-D Integrated Hydro-Mechanical Model Based on Flume Tests to Unravel Different Hydrological Triggering Processes of Debris Flows

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## Abstract

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## 1. Introduction

- What type of hydro-mechanical triggering mechanisms for debris flows can we distinguish in upstream channels of debris flow prone gullies?
- What are the main parameters and in what way are they controlling the type and temporal sequence of these triggering processes?
- What is the influence of hydro-mechanical parameters and related triggering processes on the meteorological thresholds for debris flow initiation?

## 2. Flume Tests to Reveal Types of Debris Flow Triggering

#### 2.1. Setupp of the Flume Experiments

^{−3}for the coarse, medium, and fine material, respectively with a SD of 1.7, 3.0, and 2.4, respectively. We could vary the slope angle of the flume between 14° and 20°. The initial moisture content of the flume material was more or less dry. The initial moisture content is important for the infiltration capacity, but since we used in the laboratory a large influx of water from above into the coarse bed material, we ignored the effect of the sorptivity (related to the initial moisture content) on the infiltration capacity of the bed material.

#### 2.2. Observations on Different Types of Hydrological Triggering Mechanisms in Flume Tests

^{−1}) and higher slope angles (>16°) the time to saturation overland flow was immediately followed by failure or with a small delay until nine seconds. Also, one could clearly observe that the time to saturation overland flow (and thus failure) decreased with decreasing slope angle (Table 2).

^{−1}). Bed failure in this case occurred a certain time after the start of Hortonian overland flow with a time lag ranging between 35 and 160 s (Table 2), because in this case, due to the lower percolation rate, it took time to bring the groundwater in the bed material to a critical failure level.

## 3. Integrated Model (1D) for Debris Flow Initiation in Upstream Channels

_{f}is overland flow discharge per unit width (m

^{3}m

^{−1}s

^{−1}); q

_{s}is subsurface discharge per unit width (m

^{3}m

^{−1}s

^{−1}); h

_{f}is thickness of overland flow (m); h

_{s}(m) is thickness of subsurface flow, ∂x (m) is distance along the slope, $\partial $t is the time (s), and B

_{1–2}are terms (m s

^{−1}) describing the inflow or outflow of water from the flow system, which is defined as follows:

^{−1}) describes the external input of rain into and i

_{f}(m s

^{−1}) the outflow of water by infiltration out of the overland flow system (Equation (2a)) (see also Figure 1). When there is no supply of rain, like in our flume experiments, r = 0. In the case of subsurface flow, i

_{f}in Equation (2b) is considered now as an inflow term of the subsurface flow system. If h

_{f}/Δt is larger than the infiltration capacity, Ks (m s

^{−1}) of the bed material the latter one is the limiting factor. Therefore the infiltration term i

_{f}of Equation (2) is the minimum (min) value of the infiltration capacity Ks (m s

^{−1}) and the current water depth (h

_{f}), which can infiltrate in one time step Δt into the bed material:

_{0}the slope gradient of the bed material.

_{s}is the amount of subsurface flow water per unit width (m

^{3}m

^{−1}s

^{−1}), $\theta $ is slope angle (degrees), and h

_{s}is the height of the flowing water component in the soil matrix (m). By comparing Equation (6) with the general momentum Equation (4b) we can define the parameters ${\alpha}_{s}$ and ${\beta}_{s}$ for subsurface flow:

_{f}, q

_{s}) [33]:

_{x}, $\alpha $ and $\beta $ should be read as q

_{f}

_{,s}$\alpha $

_{f}

_{,s}and $\beta $

_{f}

_{,s}, respectively.

_{s}and ${\gamma}_{w}$ (k N m

^{−3}) are the saturated bulk density of the material and water, respectively; $\phi $ is friction angle of the material; z and h

_{s}are the thickness of the soil and the height of the groundwater layer, respectively; h

_{s}can be solved with Equation (6) and (9), respectively.

_{90}and D

_{30}are grain sizes at which 90% and 30%, respectively, by weight of the material are finer; d

_{s}is the mass density of the solids, S is the slope gradient, and q

_{c}is the critical flow discharge for bedload entrainment. The experimental slopes were in the range of 0.03 > S > 0.20 (1.7°–11.3°), and the D

_{90}of the material ranged between 0.9 and 2 cm and D

_{30}between 0.06 and 1 cm with inflow rates of 10–30 L s

^{−1}. In the section below, we will calibrate Equation (11) for the steeper slopes in our flumes.

## 4. Calibration and Validation of the Theoretical Model on the Basis of Flume Test Results

_{f}reaches the lower end of the bed material, while the bed material is still not saturated (h

_{s}< Zs). Saturation overland flow and the time towards it is declared when h

_{s}= Zs over the entire bed. Pore pressure is calculated each time step according to Equation (10b). The discharge of h

_{f}+ h

_{s}is reported each time step at the end of the flume. Bed failure is declared as mentioned earlier when the average safety factor F over the bed length reaches the value of 1.

_{90}> 2 and 0.05 > d

_{30}> 1 cm, and with flow rates 0.5 > q

_{f}> 15 L s

^{−1}m

^{−1}. Figure 8 shows the best linear fit between q

_{solid}/q

_{f}and (d

_{90}/d

_{30})

^{0.2}S

^{2}, which delivered the following modified equation for slopes between 14° and 20°:

_{s}= 2.6:

_{c}in Equation (11) becomes zero or practically zero in Equation (12). At slopes larger than 15° the down slope component of the grain weight may reduce the critical shear stress τ

_{c}which in our case obviously reduced to nearly zero.

## 5. Hydro-Mechanical Triggering Patterns for Debris Flows in Relation to Hydrologic Conductivity of Bed Materials and Channel Gradient

#### 5.1. The Design of a Schematic Source Area at the Field Scale

^{−1}) is the lateral inflow of overland flow water from the slopes along the channel (Figure 9), r direct rain intensity input to the channel bed, and i

_{f}infiltration rate into the bed (see Equation (3)). The lateral inflow is calculated for these sensitivity analyses in a simple way, assuming steady state conditions in the mass balance equation for overland flow:

_{cn}(m s

^{−1}) is calculated using the curve number method [36], L is the length of the lateral slope, and W the width of the channel (see Figure 9). In our simulations we selected overland flow supplying slopes with soils with moderate to slow infiltration rates and a poor condition grass cover, which corresponds to a curve number (CN) of about 80. The CN number, reflecting the hydrological soil characteristics, land use, and antecedent soil moisture conditions that we can expect in high mountainous areas, was chosen arbitrarily and was kept constant in our simulations. The overland flow water that flows into the upper end of the channel bed is given by q

_{up}(m

^{2}s

^{−1}) (Figure 9).

#### 5.2. The Influence of the Hydraulic Conductivity (Ks) and Slope (θ) of the Channel Bed on the Type and Sequence of Hydrologic Triggering Processes for Debris Flows

^{−1}, the debris flow is initiated in the first stage by Hortonian overland flow erosion (R

_{h}E-I). The overland flow discharge reaches a steady state after a certain relatively short time. During the steady state, groundwater will rise by infiltration of run-on water until failure of the bed material, which happens between 1.7 and 11.2 minutes depending on the slope θ and Ks. In Table 4a, we see a dramatic drop in discharge between slopes with Ks = 0.001 and 0.005. The last Ks value reaches a significant boundary which determines whether or not a debris flow can be initiated by Hortonian overland flow transport.

^{−1}slope failure does not occur. The debris flow is initiated by overland flow; first by Hortonian overland flow and later when the groundwater has reached the surface by saturation overland flow. Discharge is relatively low when there is Hortonian overland flow, while obviously discharge dramatically increases at saturation overland flow. However, due to the lower slope angles, the volumetric sediment concentration is low (0.21 at 16° and 0.13 at 12°) (Table 4a, last column), which means the flow changes from a hyperconcentrated flow into a water flood with conventional suspended load and bed load.

^{−1}and θ = 28°–20°, bed failure seems the most dominant process (Table 4a). Due to the larger Ks values, infiltration into the bed is more important than overland flow discharge. The bed material turns out to be partly saturated in the upper part due to the larger upstream inflow, creating partly saturation overland flow and Hortonian overland flow. However, within one minute after the runoff discharge reached the lower end of the bed, failure of the bed material occurred already. Therefore, the contribution of overland flow to the transport of debris by overland flow can be ignored.

^{−1}and lower slope gradients (θ = 16°–12°), there is no slope failure but only saturation overland flow, (Table 4a) with low sediment concentrations in most cases not enough to call it a debris flow.

## 6. Sensitivity Analyses for Parameters Influencing the Rain Intensity-Duration (I-D) Threshold Curves for Different Initiation Processes of Debris Flows

_{h}E-I) and bed failure (BF-I). As we have seen in Table 4 and Table 5, debris flow initiation by saturation overland flow (R

_{s}E-I) can only take place around 16 degrees. At lower slope angles, sediment concentrations are too low to call it a debris flow (Table 4). At higher slope angles, we have bed failure before saturation overland flow can take place.

^{−1}. Figure 10a shows the influence of the Ks value on the I-D threshold curves for runoff erosion initiation (R

_{h}E-I). The range of Ks values was chosen between 0.0005 and 0.005 m s

^{−1}. The figure shows that for Ks values lower than 0.001 m s

^{−1}there is nearly no effect of Ks on the position of the I-D curve, but there is a difference in the minimum intensity values (dotted lines) below which no debris flow can occur. A slight difference can be observed for lower intensities (<60 mm h

^{−1}). Higher Ks values (>0.001 m s

^{−1}) have a larger influence on the I-D curves. (Figure 10a).

^{−1}with a default value of 0.01 m s

^{−1}. The effect of the selected range in geometric values R&L, Lx, W, θ, and Zs (Figure 11b,d,f–h, respectively) seems to be more or less the same. Less effect has the porosity (Por) of the bed material and the φ values (Figure 11c,e, respectively). No effect had the hydraulic conductivity Ks (Figure 11a), which is related to the simplicity of the model describing instantaneous downward percolation for these high permeable bed materials. Interesting was to see that at lower Ks values (around 0.001 m s

^{−1}) and higher rain intensities, the rate of groundwater storage, and therefore the critical duration for failure, was nearly the same (Figure 11a).

## 7. Discussion

_{s}), and therefore after failure much additional overland flow water is needed to maintain the movement further down slope. Important is also the mechanism of erosion and erosive power of both types of debris flows further downstream in order to grow to a mature debris flow [6,43,44,45,46,47].

## 8. Conclusions

_{h}-I), saturation overland flow (RO

_{s}-I), and by infiltrating water causing failure of channel bed material (BF-I). On the basis of these flume tests, an integrated hydro-mechanical model was developed, which was calibrated and validated with a number of process indicators measured during the flume tests. We were able to assess by means of this model the influence of important parameters on the mode of debris flow initiation. The hydraulic conductivity of the bed sediments is an important factor controlling the type and sequence of processes triggering debris flows. At lower Ks values, Hortonian overland will be the first process to start debris flows followed by bed failure or saturation overland flow. At higher Ks values, triggering by Hortonian overland flow is not possible anymore in this relatively coarse bed material, and triggering by bed failure will be the dominant process if the slope gradient is steep enough (>16°). Therefore, the slope gradient of the channel bed is a second important factor controlling the type of hydro-mechanical triggering. On gentler slopes which remain stable under saturated conditions, saturation overland flow might create debris flows if slope gradient is not too gentle, and therefore sediment concentration too low to call it a debris flow.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Observed and calculated time to saturation overland flow (black symbols) and Hortonian overland flow (open symbols).

**Figure 5.**Maximum pore pressure measured during flume tests in relation to calculated pore pressures.

**Figure 6.**Example of the rise in pore pressure (measured/calculated) due to infiltration of run-on water in the bed material (test: medium grain size/20°).

**Figure 8.**Calibration of Rickenmann’s bedload equation for steeper slopes in our flume tests between 14 and 20 degrees.

**Figure 9.**Morphometric and hydro-mechanical parameters, which were used for model simulations of debris flow initiation. For an explanation see Table 3 and text. D90/D30: 90% and 30% lower than grain size D90 and D30, respectively. φ: Friction angle; Ks: Hydraulic conductivity; Por: Porosity; Zs: Depth of material; θ: Slope angle; q

_{up}: Water that flows into the upper end of the channel bed; R: Radius of source area above the channel; Lx and W: Length and width of the channel bed; L: Length of lateral contributing slope; latin: Lateral inflow of water to the channel; CN: Curve number value for the soil hydrological and land use characteristics of the contributing slopes.

**Figure 10.**I-D curves for debris flow initiation by Hortonian overland flow in relation to different geometrical and hydrological parameters. The regression formula belong to the curves from top to bottom. For the definition of parameters see Table 3.

**Figure 11.**I-D curves for debris flow initiation by bed failure in relation to different geometrical and hydro-mechanical parameters. The regression formula belong to the curves from top to bottom. For the definition of parameters see Table 3.

**Figure 12.**Ultimate variation in I-D curves as a result of our sensitivity analyses compared to the maximum difference in I-D curves found worldwide.

**Table 1.**Hydro-mechanical characteristics of three types of bed material used in the flume tests. Friction means friction angle of the material in degrees. D30/50 means that 30/50% of the sample had a lower diameter than what is indicated in the column.

Particle Size Class | Friction φ ( ^{o}) | Density k Nm ^{−3} | SD k Nm ^{−3} | Hydraulic Conductivity (m s ^{−1}) | D_{30}(mm) | D_{50}(mm) | D_{90}(mm) |
---|---|---|---|---|---|---|---|

Coarse | 34.6 | 15.4 | 1.7 | 4.91E-03 | 9 | 11 | 18 |

Medium | 33.7 | 16.3 | 3.0 | 3.28E-03 | 4 | 6 | 16 |

Fine | 29.2 | 19.5 | 1.4 | 0.54E-03 | 0.7 | 1.6 | 8 |

**Table 2.**Observed time to overland flow and bed failure, overland flow type and failure mode in flume experiments for three types of bed material and for different bed slope angles: (a) saturation overland flow; (b) Hortonian overland flow; (c) slow continuous bed failure; (d) rapid failure; nf: no failure.

Grain size | Slope | Inflow L s ^{−1} m^{−1} | Time to Run-Off Initiation sec | Time to Bed Failure sec | ||
---|---|---|---|---|---|---|

Coarse 0.0049 | 20° | 0.18 | 402 | 411 | ||

18° | 0.22 | 103 | 110 | |||

16° | 0.29 | 63 | 72 | |||

15° | 0.40 | 73 | nf | |||

14° | 0.41 | 59 | ||||

Medium 0.0033 | 20° | 0.11 | 168 | 168 | ||

18° | 0.13 | 140 | 140 | |||

16° | 0.16 | 54 | 106 | |||

15° | 0.18 | 27 | nf | |||

14° | 0.19 | 46 | ||||

Fine 0.00054 | 20° | 0.03 | 30 | 270 | ||

18° | 0.06 | 220 | 613 | |||

16° | 0.09 | 90 | 270 | |||

15° | 0.10 | 110 | nf | |||

14° | 0.11 | 102 |

**Table 3.**Default values (bold italic) and maximum and minimum values of input parameters for overland flow erosion (RE-I) and bed failure (BF-I) triggering debris flows. Ks: Saturated hydraulic conductivity; Zs: Thickness of bed material; φ friction angle of material; θ: Slope angle of channel bed; W and Lx: width and length of channel bed respectively; Por: Available volumetric pore space; R: Radius of source area; L: Length of lateral slopes; n: Manning’s n of bed material.

Ks for RE-I | 0.001–0.0025–0.005 m s^{−1} | Lx | 50–100–200 m |

Ks for BF-I | 0.001–0.01–0.1 m s^{−1} | Por | 0.4–0.3–0.2 |

Zs | 2–4–6m | R | 250–350–450 m |

φ | 28–32–36 | L | 250–350–450 m |

θ | 16°–20–28° | n bed | 0.08 |

W | 2–4–6-m |

**Table 4.**Time sequence of different initiation processes R

_{h}E-I and R

_{s}E-I, (erosion by Hortonian and saturation overland flow respectively) and BF-I (bed failure) in relation with hydraulic conductivity (Ks) and slope angle of bed material. Further are given the discharge (discharg) and solid concentration (concent) during R

_{h}E-I and R

_{s}E-I. Table 4a and Table 4b: simulated rain intensities of 80 mm and 40 mm, respectively.

a | ||||||||||

80 mm | Ks | 0.001 m s^{−1} | 0.005 m s^{−1} | 0.01 m s^{−1} | 0.1 m s^{−1} | |||||

Slope Degrees | Initiat Proc. | Timemin. | DischarL s m^{−1} | Timemin. | DischarL s m^{−1} | Timemin. | DischarL s m^{−1} | Timemin. | DischarL s m^{−1} | Vol ConcentL L^{−1} |

28 | R_{h}E-I | 1.0 | 912 | 1.3 | 139 | 0.47 | ||||

BF-I | 5.4 | 1.7 | 2.4 | 3.0 | ||||||

24 | R_{h}E-I | 1.0 | 783 | 1.3 | 119 | 0.39 | ||||

BF-I | 8.4 | 2.3 | 2.8 | 3.1 | ||||||

20 | R_{h}E-I | 1.1 | 683 | 1.4 | 104 | 0.30 | ||||

BF-I | 11.2 | 2.9 | 3.1 | 3.1 | ||||||

16 | R_{h}E-I | 1.1 | 606 | 1.5 | 92 | 0.21 | ||||

R_{s}E-I | 14.2 | 732 | 3.7 | 733 | 3.7 | 732 | 3.6 | 705 | ||

12 | R_{h}E-I | 1.2 | 550 | 1.5 | 83 | 0.13 | ||||

R_{s}E-I | 14.8 | 666 | 3.9 | 665 | 3.8 | 664 | 3.6 | 646 | ||

b | ||||||||||

40 mm | Ks | 0.001 m s^{−1} | 0.005 m s^{−1} | 0.01 m s^{−1} | 0.1 m s^{−1} | |||||

Slope Degrees | Initiat Proc. | Timemin. | DischarL s m^{−1} | Timemin. | DischarL s m^{−1} | Timemin. | DischarL s m^{−1} | Timemin. | DischarL s m^{−1} | Vol ConcentL L^{−1} |

28 | R_{h}E-I | 1.5 | 162 | 0.47 | ||||||

BF-I | 5.8 | 7.2 | 7.9 | 8.0 | ||||||

24 | R_{h}E-I | 1.6 | 139 | 0.39 | ||||||

BF-I | 8.8 | 7.8 | 8.2 | 8.3 | ||||||

20 | R_{h}E-I | 1.7 | 121 | 0.30 | ||||||

BF-I | 11.7 | 8.5 | 8.5 | 8.4 | ||||||

16 | R_{h}E-I | 1.8 | 108 | 0.21 | ||||||

R_{s}E-I | 14.8 | 234 | 9.5 | 235 | 9.4 | 233 | 9.3 | 208 | ||

12 | R_{h}E-I | 1.9 | 98 | 0.13 | ||||||

R_{s}E-I | 15.0 | 214 | 9.6 | 213 | 9.5 | 212 | 9.2 | 170 |

**Table 5.**Sequence of different initiation processes for debris (hyperconcentrated) flows in relation to the hydraulic conductivity and slope of the channel bed material. Simulated rain intensity is 40 mm.

40 mm | Ks Values | 0.001 m s^{−1} | 0.005 m s^{−1} | 0.01 m s^{−1} | 0.05 m s^{−1} | 0.1 m s^{−1} |
---|---|---|---|---|---|---|

Gradient | Flow type | Initiation process | Initiation process | |||

28^{o} | Debris flow | t1: Hortonian overland flow t2: Bed failure | t1: Bed failure | |||

24^{o} | ||||||

20^{o} | ||||||

16^{o} | Hyper -concentratedflow | t1: Hortonian overland flow t2: Saturation overland flow | t1: Saturation overland flow | |||

12^{o} | No debris flow | t1: Hortonian overland flow t2: Saturation overland flow | t1: Saturation overland flow |

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**MDPI and ACS Style**

Van Asch, T.W.J.; Yu, B.; Hu, W. The Development of a 1-D Integrated Hydro-Mechanical Model Based on Flume Tests to Unravel Different Hydrological Triggering Processes of Debris Flows. *Water* **2018**, *10*, 950.
https://doi.org/10.3390/w10070950

**AMA Style**

Van Asch TWJ, Yu B, Hu W. The Development of a 1-D Integrated Hydro-Mechanical Model Based on Flume Tests to Unravel Different Hydrological Triggering Processes of Debris Flows. *Water*. 2018; 10(7):950.
https://doi.org/10.3390/w10070950

**Chicago/Turabian Style**

Van Asch, Theo W. J., Bin Yu, and Wei Hu. 2018. "The Development of a 1-D Integrated Hydro-Mechanical Model Based on Flume Tests to Unravel Different Hydrological Triggering Processes of Debris Flows" *Water* 10, no. 7: 950.
https://doi.org/10.3390/w10070950