Shock and Vibration
Volume 2018 (2018), Article ID 5460362, 15 pages
https://doi.org/10.1155/2018/5460362

Research Article

Experimental Study of the Debris Flow Slurry Impact and Distribution

Haixin Zhao,¹ Lingkan Yao,1,2,3 Yong You,⁴ Baoliang Wang,¹ and Cong Zhang¹

¹School of Civil Engineering, Southwest Jiaotong University, Chengdu 610031, China
²MOE Key Laboratory of High-Speed Railway Engineering, Chengdu 610031, China
³National Engineering Laboratory for Technology of Geological Disaster Prevention in Land Transportation, Chengdu 610031, China
⁴Institute of Mountain Hazards and Environment, CAS, Chengdu 610041, China

Correspondence should be addressed to Lingkan Yao; yaolk@swjtu.edu.cn

Received 13 March 2018; Revised 3 July 2018; Accepted 5 August 2018; Published 5 September 2018

Academic Editor: Elena Dragomirescu

Copyright © 2018 Haixin Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this study, we present a new method to calculate debris flow slurry impact and its distribution, which are critical issues for designing countermeasures against debris flows. There is no unified formula at present, and we usually design preventive engineering according to the uniform distribution of the maximum impact force. For conducting a laboratory flume experiment, we arrange sensors at different positions on a dam and analyze the differences on debris flow slurry impact against various densities, channel slopes, and dam front angles. Results show that the force of debris flow on the dam distributes unevenly, and that the impact force is large in the middle and decreases gradually to the both sides. We systematically analyze the influence factors for the calculation of the maximum impact force in the middle point and give the quantitative law of decay from the middle to the sides. We propose a method to calculate the distribution of the debris flow impact force on the whole section and provide a case to illustrate this method.

1. Introduction

Debris flow impact force includes slurry impact and huge rock impact, which provide the mechanical bases for checking engineering structures, including calculations for antisliding and anti-overturning [1–4]. The magnitude of the slurry impact force is one of the most important parameters for the Sabo dam design determining the movement of debris flow and its impact on the dam, especially for debris flow without huge rocks. Many formulas have been proposed to calculate the debris flow impact. Fei and Shu [5] conducted theoretical analysis and many simulation experiments on impact force and established a calculation model of the debris flow impact, according to different particle movement types. He et al. [6] proposed several methods for calculating the impact force of the large rock mass on the basis of elastoplastic theory. Chen et al. [7, 8] simplified the debris flow as solid-liquid two-phase flow and established the calculation of the two phases of the debris flow based on the theory of phase flow velocity as well as debris flow impact time. Armanini and Scotton [9] proposed a formula for the impact force based on a momentum conservation analysis to study the impact force of debris flow. Zeng [10] systematically studied differences in the vertical distribution of the impact force of debris flow and evaluated building vulnerability.

The formulas to calculate the impact force of debris flow are given in Table 1.

Table 1: Debris flow slurry impact formulas.

Number
Formula for debris flow impact
Reference

1

Mizuyama [1]

2

Chen [11]

3

Hungr et al. [12]

4

Li [13] and Zhou [14]

5

Hübl and Holzinger [15]

6

Zeng [10]

is the debris flow impact force; is the debris flow density; is the mud depth; is the mean velocity of the debris flow; is the acceleration of gravity; is the shape correction coefficient; is the gravity density of the debris flow; is the impact angle; is the cross-sectional area; is the empirical coefficient; is the Froude number.

The formulas in Table 1 are modified by the theoretical formula based on the energy method. The correction coefficient which is empirically based varies according to the factors considered. The theory of the energy method is derived by hydraulics, which is suitable for the general uniform flow, such as water, but it is not suitable for nonuniform flow [16–18], such as debris flow. A debris flow is a rapid, gravity-driven mass movement that involves water-charged, predominantly coarse-grained inorganic and organic materials. A positive correlation does not necessarily exist between the impact force of debris flow and the square of the mean velocity. Simply adding the correction coefficient does not guarantee the accuracy of the formula; as such, further discussion and improvement of the current methods are required.

Conversely, we usually take the debris flow impact force on the prevention engineering as a uniform distribution according to the maximum impact force. This makes the predicted overall structure resistance much larger than the actual overall structure resistance. Meanwhile, to ensure the safety of the weakest part, we also waste the construction material on the safer part such as the both sides of the top edge of the dam. Some scholars have studied the distribution of the impact force of the debris flow through an inhomogeneous distribution of flow velocity [19, 20]. Han et al. [21] presented a new approach for exploring the debris flow velocity distribution in a cross section that used an iteration algorithm based on the Riemann integral method to search an approximate solution to the unknown flow surface. In fact, their work emphasized the effect of terrain on the distribution of velocity and the impact force but did not consider internal interactions of debris flow and external interactions to the both sides of the channel.

In this paper, we arrange sensors at different positions on the dam and analyze the different debris flow slurry impact on the dam with various densities, channel slopes, and dam front angles through the laboratory flume experiment test. This study shows that the force of debris flow on the dam is unevenly distributed, and the impact force is large in the middle and small at both ends. We systematically analyzed the factors influencing the calculation of the maximum impact force in the middle point and give the quantitative law of decay from the middle to the sides. We propose a method for calculating the distribution of the debris flow impact force on the whole section and provide a case to illustrate this method.

2. Physical Model

We conducted our experiments in a debris flow simulation laboratory at the Institute of Mountain Hazards and Environment, which belongs to the Chinese Academy of Sciences (CAS). The flume consists of seven parts: hopper, flume, tailings poll, dam model, camera, gate, and acquisition instrument. Figure 1 is a schematic illustration of the model experiment. The cube hopper is 60 cm in length, 60 cm in width, and 80 cm in height. It has the capacity of 0.1 m³, and it is affixed to the flume and controlled the discharge from the gate. We conducted our experiments in a flume with a length of 400 cm, a width of 30 cm, and a height of 40 cm. The gradient of the flume was adjustable from 0° to 20°. We placed a dam model at the bottom of the flume. The dam modeled has a surface slope of 63°. We used the tailings pool to recycle the debris flow.

(a)

(b)

Figure 1: Experimental apparatus. 1: hopper; 2: flume; 3: tasilings poll; 4: dam model; 5: camera; 6: gate; 7: acquisition instrument.

The distance between the front edge of the dam model and the gate is 6 m, which helps to stabilize the debris flow velocity and prevents rapid changes at the impact location. We installed eight sensors on the surface of the model dam. Figure 2 shows the size of the surface of the dam, the surface size of the sensor, and the location of the sensor arrangement.

(a)

(b)

Figure 2: Size of the surface of the dam, surface size of the sensor, and location of the sensor arrangement.

In this study, we used a piezoelectric-type sensor whose range is 0–100 kPa with a sensitivity of 0.1%. The acquisition frequency is 2 kHz, and the sampling duration is 32 s. We opened the gate, and the debris quickly flowed out and affected the model dam under the action of gravity. Sensors installed on the dam can quickly collect the impact force of debris flow on the dam. We obtained impact force data by computer processing. Figure 3 shows the impact of debris flow on the dam. The experimental process does have errors, including electromagnetic interference and improper operation. We completed three repetitions to obtain intermediate data to minimize these errors.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 3: The phenomenon of the impact of debris flow on the dam. Time-series photos of (a) debris flow accelerating; (b) the front of the debris flow touching the dam; (c)–(f) debris flow affecting the dam and the dam changing the debris flow pattern; (g) the impact mitigation; (h) the swing back of silt.

We collected experimental material samples from sites located in Gan Gully, which is the Level 1 tributary canal of the Yinchang Gully in the upper stream of the Qian River in the Longmen mountain town of Pengzhou, Sichuan Province. This location experienced an outbreak of large-scale debris flow at the night of August 18, 2012. The specific sampling sites were located on the eastside of the alluvial fan. The maximum particle size of the experimental materials was 20 mm. We took five random samples for grain-size distribution analysis before conducting the experiments. Figure 4 shows the grain-size distribution of the experimental materials.

Figure 4: Particle gradation of the materials used in the experiment.

We mainly considered the effect of debris flow densities and channel slopes on the impact force. Meanwhile, the mass volume of debris flow was a major factor for designing countermeasures against debris flows in the field. In addition, we explored whether the dam front angles affected the impact force. Therefore, we designed three tests groups. Group A mainly analyzed the influence of debris flow density and channel slope on impact force. Group B mainly analyzed the influence of dam front angles on the impact force combined with Group A. Group C mainly analyzed the influence of debris flow mass volume on the impact force combined with Group A. After setting up the experimental apparatus and preparing the experimental materials, we carried out 33 group tests according to the experimental scheme shown in Tables 2 and 3.

Table 2: The experimental conditions for the 33 performed laboratory tests.

Tests group
Tests Tag number
Various experimental conditions

Debris flow density (kN/m³)
Debris flow mass volume (L)
Channel slope (°)
Dam model front angles (°)

A
Case 1 to 25
13, 15, 17, 19, 21
60
7, 10, 11, 13, 15
63

B
Case 26 to 29
13, 17
60
9
77,90

C
Case 30 to 33
17, 19
90, 120
13
63

Table 3: The experimental conditions and results for 33 laboratory tests.

Tests group
Tests Tag number
Density (kN/m³)
Flume slope (°)
Mass volume (L)
Dam model front slope (°)
(m/s)
Impact force 1# (kPa)
Impact force 2# (kPa)
Impact force 3# (kPa)
Impact force 4# (kPa)
Impact force 5# (kPa)
Impact force 6# (kPa)
Impact force 7# (kPa)
Impact force 8# (kPa)

A
1
13
7
60
64
2.321
4.558
5.022
4.625
2.963
1.055
1.507
2.704
0.669

2
15
7
60
64
2.115
2.880
3.216
2.904
1.026
0.740
0.384
1.672
0.332

3
17
7
60
64
2.000
2.005
2.301
2.265
1.950
0.506
0.003
1.290
0.180

4
19
7
60
64
1.146
1.096
1.211
1.200
0.263
0.000
0.103
0.713
0.000

5
21
7
60
64
1.058
1.382
1.509
1.497
1.151
0.000
0.260
0.898
0.000

6
13
9
60
64
2.500
3.610
4.750
4.000
2.993
1.140
1.456
2.200
0.710

7
15
9
60
64
2.080
2.490
3.610
3.050
2.238
0.830
0.850
1.640
0.529

8
17
9
60
64
1.786
3.770
4.150
3.850
2.656
0.913
1.564
2.520
0.643

9
19
9
60
64
1.563
3.870
4.030
3.900
2.620
0.846
1.639
2.090
0.492

10
21
9
60
64
1.441
3.610
4.180
3.800
2.550
0.000
1.436
2.430
0.000

11
13
11
60
64
2.625
5.060
5.980
5.550
3.767
1.435
2.783
3.030
0.434

12
15
11
60
64
2.386
4.150
4.570
4.200
3.879
1.005
1.833
2.240
0.194

13
17
11
60
64
2.188
3.600
4.120
3.880
1.896
1.056
1.520
2.078
0.150

14
19
11
60
64
1.750
4.280
4.550
4.300
4.067
0.741
1.884
2.330
0.211

15
21
11
60
64
1.710
4.130
4.550
4.440
2.821
1.171
1.792
2.424
0.205

16
13
13
60
64
3.250
7.380
8.180
8.100
4.908
1.550
3.569
4.310
1.427

17
15
13
60
64
2.954
5.140
6.480
5.820
3.823
1.716
2.279
2.932
1.029

18
17
13
60
64
2.500
5.200
5.420
5.320
3.415
1.142
2.313
2.622
0.944

19
19
13
60
64
1.204
5.810
6.060
5.990
3.515
1.573
2.650
3.014
1.018

20
21
13
60
64
1.548
6.360
6.450
6.430
3.806
1.419
2.943
3.268
1.113

21
13
15
60
64
4.063
8.340
9.840
9.680
6.199
2.045
3.987
4.998
1.666

22
15
15
60
64
3.250
7.410
8.300
8.040
5.229
2.326
3.466
4.004
1.387

23
17
15
60
64
2.954
6.460
7.240
6.900
4.561
2.193
2.933
3.310
0.653

24
19
15
60
64
2.708
12.700
13.800
13.400
8.556
3.936
6.415
7.200
2.278

25
21
15
60
64
2.167
14.480
15.720
15.650
9.904
4.458
5.964
8.540
2.861

B
26
13
13
60
77
3.250
7.520
8.950
8.000
6.739
2.238
3.336
3.980
1.560

27
17
13
60
77
2.500
3.056
3.909
3.472
3.563
0.977
0.961
1.833
0.690

28
13
13
60
90
3.250
7.440
8.645
8.400
6.546
1.988
3.192
4.230
1.228

29
17
13
60
90
2.500
4.500
5.523
5.000
4.579
1.160
2.275
2.760
0.750

C
30
17
15
90
64
3.611
9.885
12.191
10.191
9.980
4.682
4.597
5.535
1.782

31
19
15
90
64
3.250
12.854
14.119
13.090
11.395
4.106
6.720
7.164
2.265

32
17
15
120
64
4.063
13.650
17.068
15.380
14.053
4.755
7.248
8.508
2.645

33
19
15
120
64
3.611
15.520
18.514
17.010
14.964
5.573
7.956
9.356
2.912

3. Characteristics of Debris Flow Maximum Impact Force

3.1. Maximum Impact Force Compared with Former Formula

The impact force collected by sensors 1, 2, and 3 is obviously greater than the other sensors. This is mainly because sensors 1, 2, and 3 are located in the bottom of the dam (as seen in Figure 2), and these positions are directly affected by the impact of debris flow. The first three sensors are arranged within the depth of the mud, whereas the others are arranged upon the mud depth (as seen in Figure 2). Through theoretical analysis, we know that the greater the velocity in the range of the mud depth, the greater the impact force. Therefore, the impact at the surface of debris flow is supposed to be the maximum. Above this mud depth, the impact force is caused by the debris flow superelevation, which is obviously smaller than the force in the range of the mud depth [10]. Meanwhile, data from sensor 2 is the largest among the first three sensors, and we selected the data from sensor 2 to analyze the maximum impact force. We used a high-resolution camera to measure the average velocity of debris flow and applied formulas 2, 4, and 6 in Table 1. We compared the results with the data from sensor 2 (as shown in Figure 5). The red lines in Figure 5 represent the linear fitting of the experimental values and the calculated values of the impact force of debris flow.

Figure 5: Maximum impact force compared with former formula.

Figure 5 shows that the correlation coefficient between our experimental results and the calculation formulas of Chen [11], Li [11] and Zhou [14], and Zeng [10] are 0.57, 0.56, and 0.79, respectively. The results of the calculation by Li [11] and Zhou [14] are obviously larger than the experimental results. This difference is due to their correction being based on the integration of actual measurement data in the field, including the impact of large stones. The results of Chen’s (1983) calculation are in the same range as the experimental results, but the correlation coefficient is poor. The calculated results by Zeng [10] are best fitting the experimental results, but Zeng still used average velocity. In fact, the velocity distribution is not uniform, and the maximum impact force should be related to surface velocity, which is the maximum velocity. Therefore, this paper tries to find the relationship between the surface maximum velocity and the maximum impact force and proposes a new method to calculate the impact force of the maximum debris flow.

3.2. The Influence of Debris Flow Density on Impact Force

We used the same dam model for the upstream slope that we used to determine the maximum impact force. In this case, we analyzed the various debris flow impact force with various debris flow densities under the same scale of debris flow (quality) and the same channel characteristics (flume slope). The experimental results show that when the channel characteristics (flume slope), the dam model upstream slope ratio and the debris flow scale (quality), are the same, the value of the debris impact force does not necessarily increase along with the debris flow density. Figure 6 shows that when the flume slope is 15°, the debris flow impact force decreases gradually, with debris flow density changing from 13 kN/m³ to 17 kN/m³. Although the debris flow impact force increases gradually, the debris flow density changes from 17 kN/m³ to 21 kN/m³. The main cause for this phenomenon is that the velocity of debris flow is obviously reduced as debris flow density increases from 13 kN/m³ to 17 kN/m³, which can be observed in the experiment. When the density changes from 17 kN/m³ to 21 kN/m³, the effect of density on impact force is more significant than that of velocity, although it is still reducing. As the flume slope is bigger, bending becomes more obvious and looks like a “hook,” as shown in Figure 6.

Figure 6: The relationship between maximum impact force and density.

3.3. The Influence of Flume Slope on Impact Force

Similarly, we used the same method to analyze the influence of the channel slope on impact force. We used the same dam model for the upstream slope. In this case, we analyzed various debris flow impact forces with various flume slopes under the same scale of debris flow (mass volume) and the same property of debris flow (density). The experimental results show that the steeper the flume slope, the bigger the impact force. This is because the flume slope directly affects the velocity of debris flow. Figure 7 shows the relationship between the maximum impact force and the flume slope, where is the density of the debris flow. Corresponding to the density for all five slopes, the greater the slope, the greater the impact value. As density increases, this increasing trend becomes more obvious.

Figure 7: The relationship between the maximum impact force and the flume slope.

3.4. The Influence of Dam Model Upstream Slope on Impact Force

We used the same mass volume of debris flow (60 L) and flume slope (13°) and selected the density of 13 kN/m³ and 17 kN/m³ to analyze the relationship between the maximum impact and the dam model upstream slopes, which are 63°, 76°, and 90°, respectively. Figure 8 shows the experimental result. Our study indicates that the effect of the dam model upstream slope on impact force is not obvious. When the debris flow density was 13 kN/m³, the maximum impact force was 8.2 kPa, 9 kPa, and 8.7 kPa for the upstream slope ratios of 63°, 76°, and 90°, respectively. When the debris flow density was 17 kN/m³, the maximum impact force was 5.4 kPa, 5.3 kPa, and 5.5 kPa for the upstream slopes of 64°, 77°, and 90°, respectively. The change of the impact value is within 10%. This phenomenon is mainly related to the inhomogeneity of the material composition of the debris flow, which results in a different impact direction of the internal particles. These particles are randomly distributed, and a relationship does not necessarily exist between debris flow impact value and dam model upstream slope. Therefore, we do not use the model dam upstream slope in the calculation of the maximum impact force.

Figure 8: The relationship between the maximum impact force and the model dam upstream slope.

3.5. The Influence of Debris Mass Volume on the Impact Force

We used the model dam upstream slope of 64°, selected the debris flow density as 17 kN/m³ and 19 kN/m³, and used the flume slope of 15°. We changed only the debris flow mass volume (60 L, 90 L, and 120 L) and analyzed the impact force. Figure 9 shows the experimental result. Experimental data shows that the impact force of debris flow has a linear positive correlation with the mass volume of debris flow, and the greater the mass volume, the greater the impact. As shown in Figure 9, when the debris flow density was 17 kN/m³, the debris flow impact force increased by 5 kPa and 5.5 kPa, respectively, and the debris flow mass volume increased from 60 L to 90 L and then to 120 L. When the debris flow density was 19 kN/m³, the debris flow impact force increased by 2 kPa and 4 kPa, respectively, and the debris flow mass volume increased from 60 L to 90 L and then to 120 L. The reason for this phenomenon is that the flow mass volume would change the mud depth, which is related to the velocity of the debris flow. Table 4 provides the measured results of the mud depth of the debris flow varying with the mass volume.

Figure 9: The relationship between the maximum impact force and the mass volume.

Table 4: Mud depth varies with the mass volume.

Debris flow mass volume (L)
Mud depth with a density of 17 kN/m³ (cm)
Mud depth with a density of 19 kN/m³ (cm)

60
3.93
4.05

90
5.28
5.32

120
6.15
6.18

3.6. The Influence of Debris Flow Velocity on the Impact Force

Compared with the analysis of the influence of debris flow density, flume slope, model dam upstream slope, and debris flow mass volume, we found that the effects of these factors on impact force are all related to debris flow velocity. Therefore, Zeng [10] used the Froude number to calculate the impact, which had good compatibility with the experimental data, and used mean velocity to calculate the Froude number. The velocity of debris flow is unevenly distributed in the range of mud depth, and surface velocity is much larger, which may introduce an error in the calculation. Furthermore, we studied the relationship between impact force and debris flow surface velocity using this experimental method. Figure 10 shows the experimental results.

Figure 10: The relationship between maximum impact force and debris flow velocity.

The experimental results show that the larger the debris flow surface velocity, the larger the impact force. As density increases, this trend becomes more obvious. As shown in Figure 10, a bigger density corresponds to a larger debris flow surface velocity and a greater increase in the impact force. This finding is different from that given in Section 3.2; changing the density not only changes the impact force but also changes the debris flow surface velocity. Additionally, both the debris flow surface velocity and density will affect the impact force. Furthermore, we define the following three dimensionless parameters in Equations (1)–(3) to show this trend of change:

Based on this dimensionless analysis, the relationship between these 3 dimensionless parameters is defined in Equation (4):

Using this method, we established the empirical equation using nonlinear multiple regression analysis. We indicated that the relationship among these three dimensionless numbers depends on a series of experimental data, as shown in Equation (5):

The correlation coefficient between the measured value and predicted value is 0.87, and the maximum deviation is ±29%. Figure 11 shows the comparison between the measured and predicted by Equation (5). The fitting results show that using the debris flow surface velocity for the dimensionless parameter analysis, the positive proportional relationship between the impact and the surface velocity squared is obvious and the scale coefficient is related to debris flow density. The new formula fully embodies the characteristics of debris flow and is directly related to the debris flow surface velocity, which is the reference value of the maximum debris flow impact force.

Figure 11: Comparison between measured and predicted computed with Equation (5).

4. The Lateral Distribution of Debris Flow Impact and Surface Velocity

After the analysis of the calculation of the maximum impact force of debris flow, we focus on the difference on the collected data between sensor 2 and sensors 1 and 3, namely, the lateral distribution of debris flow impact. We set one end of the dam as the origin of the coordinate and assume that the distribution of the impact force of the debris flow is symmetrical in the center of the dam model. Figure 12 shows the experimental results for debris flow density of 13 kN/m. The horizontal ordinate is the ratio of the distance between the measuring points to the sidewall and the width of the whole dam. The longitudinal coordinate is the relative impact value compared with the maximum impact force. The experimental results show that the force of debris flow on the dam is distributed unevenly. Debris flow velocity is greater in the middle and decreases gradually to the both sides. At the same time, the velocity of debris flow at the sidewall is 0, and the impact force of the debris flow at the sidewall is 0. For the magnitude of the impact force values of different debris flow positions, we fit the curve equation of the relative impact force between the relative position and the relative impact force and found that the logarithmic fitting curve better reflected the experimental results. Therefore, we propose a logarithmic model to calculate the lateral distribution of the impact force of the debris flow as follows:

where is the dimensionless reduction coefficient, namely, , and is the dimensionless calculating distance .

Figure 12: The lateral distribution of debris flow impact.

Using the analysis in Equation (6), we find that it is an increasing function, and the range of the independent variable is 0 to 0.5. This indicates the lateral impact force of debris flow and the reduction characteristic of the middle to the sidewall. The velocity quickly decreases to zero when approaching the both sidewalls. The defect of this model is that when calculating the location of the sidewall, the independent variable cannot reach 0 and only can obtain the minimum value approaching 0. At the same time, when the independent variable is 0.5, the calculation result may be slightly larger than that of 1, but the error is quite small, which can meet the needs of actual debris flow prevention and control engineering.

Furthermore, we analyzed the coefficients and , which are the two essential parameters in Equation (6). We established an empirical equation using nonlinear multiple regression analysis and analyzed the relationship between , and three dimensionless parameters (, , and ) on a series of experimental data as shown in the following equations:

where is a dimensionless parameter obtained by dividing mud depth by unit length. The correlation coefficient between the measured and predicted is 0.94, and the maximum deviation is ±12%. Figure 13 shows the comparison between the measured and predicted by Equation (6). Our method provides a basis for quantifying the lateral distribution of the impact force.

Figure 13: Comparison between measured and predicted computed with Equation (6).

We assumed that impact and surface velocity are positively correlated and that the scale coefficient is related to the debris flow density from Equations (1) and (5). Therefore, we used a reduction coefficient to show the decreasing trend of debris flow surface velocity and obtained the distribution of debris flow surface velocity as Equation (9) from Equations (6)–(8), as follows:

Note that we obtained the surface velocity distribution derived from Equation (6) without considering the variation of mud depth. In Section 5, we will analyze the distribution of debris flow velocity at any point in the debris flow section.

5. The Calculation Method of the Distribution of Debris Flow Impact on a Section

In this paper, we used the Bingham model to calculate the velocity of debris flow [22–26]. In the Bingham model, it is understood that velocity is not evenly distributed vertically; however, a uniform section of velocity is referred to as “plug.” Plug is the scope of in Figure 14. Below the plug, the velocity of the debris flow decreases quickly and the bottom velocity is zero. The shear force in the Bingham model is

where is the coefficient of the viscosity and is Bingham limit shear stress.

Figure 14: The Bingham model.

The shear force is

where is the bulk density, is the mud depth, and is the slope of the gully. By combining Equations (10) and (11), we were able to obtain the velocity below the plug, based on vertical integration, as follows:

The thickness of the plug is

The velocity at the top of the ”nonplug” is equal to the velocity in the plug section, and the surface velocity is the velocity in the plug section, as follows:

Furthermore, we put forward a calculated method for the debris flow impact distribution, which uses mass conservation [20] and takes the difference of velocity in the horizontal direction into account. A debris flow section is shown in Figure 15. We set the lowest point as O point and the horizontal line at O point as the baseline and the elevation of O point is 0. Take the appropriate number of points as with transverse coordinates of , and the elevation values are . These known point can fit polynomial or piece-wise linear function , whose independent variable span is .

Figure 15: Cross section and coordinate representation of debris flow.

Furthermore, we established the following equation:

Then, we are able to get two points () and () as shown in Figure 15.

We divide the section into n parts along the x direction. When n tends to infinity, each part is nearly a rectangular section whose elevation is , length is , and mud depth is . For the position of the maximum mud depth, it is generally the center of the debris flow section, and the longitudinal velocity distribution using the Bingham model is

By combing Equations (9) and (16), we are able to obtain the velocity at any position on the debris flow section:

For a microunit of the section (), the debris flow discharge can be calculated as follows:

Sum these microunits, and we obtain the whole section discharge as follows:

The relationship between section discharge and maximum mud depth can be deduced by Equation (19), Bingham limit shear stress , debris bulk density , and the slope of the gully ; we measured the coefficient of the viscosity by a field investigation and rheological experiment of debris flow.

By combining Equations (16)–(19), we obtain the velocity distribution on a section by the following equation:

We take the value of in Equation (5) to calculate the impact force at any point on the section and obtain the distribution of impact force by the following equation:

We selected the data of the impact force of the debris flow in the range of 19 kN/m³ and 21 kN/m³ and the flume slope of 7° to 13°. The debris flow in this range was more consistent with the Bingham model [22–26]. Table 5 provides the results of the rheological properties of the debris flow [27–28].

Table 5: Analysis of the rheological properties of the debris flow.

Density (kN/m³)
Coefficient of viscosity (Pa·s)
Bingham limit shear stress (Pa)

19.00
0.49
2.10

20.00
0.62
4.90

21.00
1.00
15.00

We used Equations (14)–(21) to calculate the impact force of debris flow at three locations in Figure 2 (numbers 1, 2, and 3) and compared it with the actual measurement results. Figure 16 shows the results of the comparison between measured and predicted , and its correlation coefficient is 0.91.

Figure 16: Comparison between measured and predicted .

6. Case Study

The considered debris flow gully is a valley with an altitude of 1820 m∼3500 m and a basin area of 12.2 km². Debris flows have been produced many times, which seriously threatens the local people’s lives and property. To reduce the debris flow disaster, we proposed to build a debris dam in the circulation area of the debris flow. We used the following steps to design the spatial distribution of the debris flow impact load at the proposed dam:

(1)By measuring a large-scale topographic map, we determined that the debris flow gully had an average longitudinal slope of 0.09.(2)According to the actual sample measurements, we determined that the bulk density of the debris flow was 21 kN/m³.(3)Based on the rheological experiments on the debris flow and considering the effect of large particles in the field, we determined that the debris flow viscous coefficient was 50 Pa. We also determined that the debris flow yield stress was 190 Pa.(4)We selected the computed section, which is perpendicular to the movement direction and the ground. The design discharge of the debris flow was 132 m³/s.(5)We used a trial calculation method to determine the maximum mud depth : assume  = 1 m; then, on the debris flow section (in Figure 17), we set the lowest point as O point with the horizontal line at O point as the baseline, and the elevation of O point is 0. Take the appropriate number of points as with transverse coordinates of −8 m, −16 m, 8 m, and 16 m, and the elevation values are 0.80 m, 0.40 m, 0.40 m, and 0.80 m.

Figure 17: Cross section of one debris flow.

We used the five points measured previously and fit the piece-wise linear function , and , , whose independent variable span is −20 m ≤  ≤ 20 m.

The thickness of the plug is 0.1 m according to Equation (13), and the corresponding is −18 and 18.

Use Equations (6)–(9) to determine the form of as follows:

Next calculate the section discharge of 132 m³/s by Equations (16)–(19) and make the calculated equal to the design discharge . Then, determine the maximum mud depth  =  = 1 m.

(6)Taking the obtained from step 5 into Equation (20), we obtained the velocity at any point on the section.

(7)Taking obtained from step 6 into Equation (5), we obtained the impact force at any point on the section.

Figure 18 shows the calculated results of the distribution of debris flow impact on a section.

Figure 18: Calculated results of the distribution of debris flow impact on a section.

7. Discussion

7.1. The Weakness of Our Proposed Method

Although we carried out some experiments on impact force and drew some conclusions, the proposed method still has some weakness. First, there is an error in the measurement of the maximum impact force of debris flow. Through theoretical analysis, we can conclude that the maximum impact force of debris flow should occur at the mud depth where the surface velocity is the largest. Under different experimental combinations, the mud depth of debris flow will change, whereas the position of the sensors remains fixed. Therefore, it is difficult to ensure that the center point of the sensor surface remains in the mud depth position of the debris flow. Therefore, only the area near the mud depth position and the measured impact value deviate slightly. The result of the bottom sensor only can be approximated to be the maximum impact force.

Second, the calculation method of the distribution of debris flow impact on a section performs better for viscous debris flow, which obeys the rheology of the Bingham model. For other types of debris flow, a similar method can be used to calculate the distribution of debris flow impact on a section.

7.2. Future Plan to Improve the Current Study

In our future study, we plan to use a flat panel collector to measure the impact force of debris flow. The advantage of the flat plate collector is that it can collect the maximum impact force within the section area even though the mud depth is changed. The temporal and spatial distribution of the impact process can be well reconstructed.

In addition, we plan to set up the sensors used in the field to measure the impact force data and the distribution of the debris flow under actual conditions and to compare these findings with experimental results to optimize the calculation method of the debris flow slurry impact and distribution.

8. Conclusion

We proposed a new method to calculate the maximum slurry impact force and the distribution through a laboratory flume experiment with various debris flow densities, mass volume, flume slopes, and dam front angles. We summarize the following conclusions from the results of the study:

(1)Our new calculation method for the maximum impact force of debris flow considers the characteristics of debris flows and gullies, which are directly related to the debris flow instantaneous velocity and relative bulk density, and provides the basis for the calculation of the distribution of the impact force of debris flow.(2)We consider the influence of debris flow channel sidewall as well as internal interactions and deduce the transverse distribution under the same mud depth. We propose a new method to calculate the reduction from the middle to the ends.(3)We propose a calculation method for the distribution of the debris flow impact force on preventive engineering considering the sidewall effect and longitudinal mud depth and give an example to illustrate this calculation.(4)Our calculation method is based on the Bingham model, which is appropriate for the calculation of viscous debris flow but inadequate for other diluted debris flow. Additional methods and calculations will need to be developed to address this issue.

Our new calculation method is better to calculate the impact and velocity distribution of viscous debris flow. Moreover, our calculation method can be beneficial to improve preventive engineering efforts, resulting in improved safety and less loss of life and farmland.

Notations

: | Debris flow impact force (kPa)
: | Impact force at x position (kPa)
: | Impact force in the middle of the dam (kPa)
: | The distance from one point to the near end (m)
: | Debris flow density (kg/m³)
: | Water density (kg/m³)
: | Gravity density of debris flow (kN/m³)
: | Gravity density of water (kN/m³)
: | Mud depth (m)
: | The mean velocity of debris flow (m/s)
: | The surface velocity of debris flow (m/s)
: | Acceleration of gravity (m/s²)
: | Shape correction coefficient (–)
: | Impact angle (°)
: | Cross-sectional area (m²)
: | Empirical coefficient (–)
: | Reduction coefficient of impact force (–)
: | Reduction coefficient of velocity (–)
: | Froude number (–)
: | Dam width (m)
: | Bingham limit shear stress
: | Slope of the gully (–)
: | Coefficient of viscosity.

Data Availability

The acquisition of experimental data is obtained by the physical model experiment conducted in a debris flow simulation laboratory at the Institute of Mountain Hazards and Environment, which is part of the Chinese Academy of Sciences (CAS). The experimental results are repeatable. Relevant scholars can use similar experimental models or visit Institute of Mountain Hazards and Environment to further verify the reliability of the experimental data.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the National Program on Key Research Project of China (Grant no. 2016YFC0802206) and the National Natural Science Foundation of China (Grant no. 41571004). The authors would like to thank LetPub for providing linguistic assistance during the preparation of this manuscript.

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