Pilot study to explain runout distances of debris flow and immature debris flow considering depositing rate

. In Japan, many debris flows and sediment-laden floods cause serious damage to human life and property. Effective measures require high-accuracy reproduction and prediction of runout distance (bed variation reach) via numerical simulations. One possible method to increase the accuracy of numerical simulation results involves reviewing the methods used to evaluate depositing rate for flows that contain many different sizes of sediments. This study performed experiments using an artificial channel and high-speed cameras to identify parameters that govern depositing rate. The channel implemented exhibited a change in slope point; thus, the slope of the upper reach differed from the slope of the lower reach. The experiments showed that the equation representing depositing rate is closely related to flow velocity. Additionally, the equation of depositing rate can be determined if other parameters are consistently considered.


Introduction
In recent years, direct damage caused by debris flows and sediment-laden floods has become increasingly apparent, as exemplified by the torrential rain-related disaster in Hiroshima Prefecture in July 2018 and the disaster in northern Kyushu in July 2017.
The runout distance of debris flows and/or immature debris flows can be roughly estimated from the flow volume [1,2] or evaluated by numerical calculations that involve shallow water flow equations [3]; however, it is often underestimated [1]. In this study, we define debris flows as flows with hs/h = 1 and immature debris flows as flows with hs/h < 1, where hs is the depth of the sediment movement layer and h is the flow depth. The runout distance is defined by the downstream terminus of the deposited sediment after a debris flow or immature debris flow. Although the scientific community has not been in complete agreement regarding constitutive equations for debris flows and/or immature debris flows, it would be difficult to improve such equations because their overall validity has been confirmed by theoretical considerations and hydraulic experiments [3,4,5,6]. However, models of entrainment and deposition processes of debris flows and/or immature debris flows are derived with the assumption that the flow contains a unit size of sediment. Thus, there is potential for improvement in reproducing actual flows that contain many sizes of sediment [3,[7][8][9][10][11][12].
Therefore, we assumed that the entrainment and depositing rate equation could be reviewed and revised if needed. Takahashi [3], Egashira et al. [7], and Miyamoto et al. [12] estimated entrainment and depositing rates with the assumption that the rate is * Corresponding author: izumiyama-h92ta@mlit.go.jp proportional to flow velocity. Conversely, Iverson & Ouyang [10] proposed a model proportional to bed shear stress and inversely proportional to flow velocity. Although the proportionality of the entrainment and depositing rate equation to flow velocity can also be inferred from the Iverson & Ouyang model [10], the selection of a constitutive equation reveals a contradiction. If entrainment and depositing rates are assumed to be inversely proportional to flow velocity, the amount of deposition is lower when the flow velocity is higher and greater momentum can be retained downstream. Therefore, the validity of these equations must be confirmed by comparison with actual propensity.
In this study, we conducted experiments to confirm the depositing trend at a change in slope point. We measured the sediment depositing rate at the change in slope point from camera images of debris flow and immature debris flow that were descending to the gentler downstream slope section. The velocities of the debris and immature debris flows were also measured.

Experimental setup
The experimental flume consisted of a straight channel (length, 10 m; width, 50 cm; and height, 50 cm) owned by the National Institute for Land and Infrastructure Management ( Figure 1).
This channel can include a change in slope at two points along the longitudinal direction. In this experiment, the channel width was fixed at 10 cm to restrict lateral variations in flow.
The slope of the channel was varied at the upper change point. The channel gradient was set at 15° in the upper reach and 5° in the lower reach; these settings ensured that the debris flow could descend in the upper section and be transformed into immature debris flow in the lower section. Additionally, the channel gradient was set at 5° in the upper reach and 1° in the lower reach; these settings ensured that immature debris flow could descend in the upper reach and be transformed to bedload in the lower reach.
The experimental material was used in two conditions: mixed (polydisperse) sand and homogeneous (monodisperse) sand. The mixing ratios of the materials were adjusted to ensure that the average grain sizes of the mixed sand were 7 mm and 3.1 mm, respectively. The diameters of the sands used were 1.3 mm, 3.1 mm, 7 mm, 12 mm, and 17 mm. The experimental material was sieved in accordance with Japanese Industrial Standards (JIS A 1204: 2009 test method for particle size distribution of soils).
The experimental material was distributed over the bed with a thickness of 0.05 m. Subsequently, to erode bed material in the upper reach section, water was supplied from the upstream end at a constant flow rate of 3.5 L/s. The experimental conditions are shown in Table 1. * Q is discharge of water supply, u is upstream channel gradient, d is downstream channel gradient, d is grain diameter, dm is averaged grain diameter.

Measurements
The measurement variables were water level, flow velocity, depositing height, and thickness of particle mixture layer. Measurements were obtained by recording flow conditions in the channel using three high-speed cameras that were installed on the sides of the channel, then analyzing the resulting video images (Figure 2). The high-speed cameras were installed at the slope change point and at approximately 1 m upstream and downstream of the slope change point (depending on experimental conditions). The flow velocity profile was obtained by tracking particles with a density of approximately 1 g/cm 3 (neutrally buoyant particles) that were descending with the debris/immature debris flow at several heights. The depth-averaged flow velocity U and flow depth h were estimated from the profile at intervals of 0.5-1 s, then averaged over intervals of 0.5-9 s (estimated errors were ± 0.5 cm/s and ± 1.1 cm, respectively). The depositing height was obtained by tracking the static layer surface. The depositing rate E was obtained from the linear regression line calculated from the relationship between the depositing height and time. Specifically, E was time averaged value (estimated error was ± 0.1 cm).  The high-speed cameras used were model MC031CG-SY-UB cameras (Ximea, Germany) equipped with a JHF25M lens (Space Inc., Tokyo, Japan). The camera frame rate was set at 200 fps, and the image sizes were 1032 × 772 pixels.

Expression of entrainment rate
In Japan, entrainment rate E is considered proportional to the depth-averaged flow velocity U (i.e., ∝ ), as indicated by Takahashi [3] and Egashira et al. [7]. We investigated the experimental data using Equation (1): where is the riverbed gradient.
According to Iverson & Ouyang [10], the entrainment rate E can be expressed by Equation (2) (in this study, the term representing the effect of dilatancy was ignored): where 1 is the shear stress exerted by flow on the bed, 2 is the shear stress exerted by the bed on flow, ̅ 1 is the mass density of debris flow or immature debris flow, and 1 ( ) is the velocity of debris flow or immature debris flow at the bed surface. Considering that 1 ( ) is proportional to the depth-averaged velocity U, the entrainment rate can be expressed as Equation (3): (3) This study used ̅ 1 ℎsin as 1 , where g is the acceleration of gravity and h is the flow depth of debris or immature debris flow. Because we did not measure volumetric sediment concentration, we were unable to estimate ̅ 1 . Therefore, we omitted the term ̅ 1 and examined the experimental results using the relationship in Equation (4): Upon selection of a model in which 1 is proportional to the square of U, Equation (3) assumes the form of an equation proportional to the flow velocity.

Results and Discussion
The relationship between the depositing rate obtained experimentally at the slope change point and Equation (1) is shown in Figure 3a. U averaged over intervals of 0.5-9 s was used for calculation of Equation (1) to eliminate deviations in U. Overall, E was positively correlated with U tan, and there was no clear difference between the relationship for monodisperse material and the relationship for polydisperse material.
The relationship between the depositing rate at the slope change point and the right-hand side of Equation (4) is shown in Figure 3b. U and h in Equation (4) are obtained from measured values at the upper reach. U and h were obtained from values averaged over 0.5-9 s, thus eliminating deviations in U. Overall, there was a positive correlation between the right-hand side of Equation (4) and the depositing rate. There was also no clear difference between the relationship for monodisperse material and the relationship for polydisperse material. Therefore, the entrainment ratio can be expressed using an equation such as Equation (1) The relationships between the depositing rate and the runout distances of debris flow and immature debris flow are shown in Figure 4. The runout distance

Conclusions
This study conducted experiments regarding the depositing processes of debris flows and immature debris flows using channels with different gradients upstream and downstream. Subsequently, it analyzed the depositing rate at the slope change point and the associated factors. The results showed that the depositing rate could be expressed using previously established formulae. To clarify the relationships among depositing rate, flow velocity, and runout distance, future experiments will be conducted under more coherent conditions of debris flows and immature debris flows.