NON-LINEAR CHARACTERISTICS OF BED MATERIAL OF RIVER CHANNEL AND ITS APPLICATION

  

 Wang Xiekang1, Cao Shuyu2 and Fang Duo3

1 Postdoctor, Instructor, National Laboratory of Hydraulics, Sichuan University, 24 S. Yihuan Rd. Chengdu, Sichuan, China, 610065(Tel: 86-28-540-5706; Fax: 86-28-5405148).

2 Professor, President of Institute of Water and Electricity, Sichuan University, 24 S. Yihuan Rd. Chengdu, Sichuan, China, 610065(Tel: 86-28-540-1148; Fax: 86-28-5405148).

3 Professor, National Laboratory of Hydraulics, Sichuan University, 24 S. Yihuan Rd. Chengdu, 610065 Sichuan, China, (Tel: 86-28-540-5706; E-mail: fang @ public. sc.cninfo.net).

Abstract: based on analyzing the property of sediment size distribution of bed material in the river channel, the fractal characteristic of sediment particle has been defined by means of fractal theory. Furthermore, the sediment particle fractal property has been applied to judge the process of armoring and stabilizing in the armoring experiment of bed material. The relationship between the riverbed roughness and its fractal dimension was obtained according to the experimental data. Finally, the relationship between bed material roughness and fractal dimension and representative diameter (d50，d90) has been discussed respectively.

Keywords: sediment size distribution, fractal theory, armoring riverbed, roughness.

1    INTRODUCTION

Sediment size distribution of bed material is a preliminary variable for analyzing fluvial process, model design, riverbed armoring and the other related problems. The property of sediment size distribution depends on watershed characteristics, flow and sediment condition, composition of bed material, and other factors such as sediment sampling method, analyzing method, and so on. Nowadays, there are two parameters for describing sediment size distribution, one is mid-diameter d50，the other is a kind of parameter , to describe the uniformity of the bed material such as , , , . However, the parameters of d50 and  can not properly describe the change of sediment size distribution. Based on analysis of the property of sediment size distribution of bed material in the river, the definition of fractal dimension is introduced, and two kinds of new methods are put forward, that is, the stabilization degree of fractal dimension of coarsening layer and sediment size distribution of bedload transport rate. Based on analysis of stochastic distribution of nonuniform sediment and simplification of distribution of particles, fractal characteristics of roughness with armoring riverbed were studied by means of fractal theory. The relationship between riverbed roughness and its fractal dimension was obtained according to the experimental data; furthermore, the relationships between bed material roughness and fractal dimension and representative diameter (d50，d90) have been discussed.

2    FRACTAL PARTICLE CHARCTERISTIC OF SEDIMENT SIZE DISTRIBUTION

The term“fractal，”was introduced by Mandelbrot(1967). He was the first to describe the geometry or characteristic of objects with highly disorder by means of fractal theory. Mandelbrot(1960) found that the change of the price of the cotton with time，that is， the law of daily change was similar to that of the change in every month. Mandelbrot regarded the figure of self-similarity with characteristic length as fractal. Fractal dimension is an important parameter for describing fractals, its common definition is

                                    (1)

Where is the length, is number of the figure with length , is fractal dimension.

Sediment size distribution depends on the condition of flow, sediment and the boundary condition of river channel. The composition of particle has no structure figure with characteristic length, and its fractal structure depends on a kind of statistical distribution that is a self-similarity. Assuming particle diameter is r，the number N(r) could be calculated by means of r. Comparison fractal characteristics with definition of fractal dimension Eq. (1), Eq. (2) could be presented as

N(r)∝r -D                                                           (2)

Assuming is cumulative gravity. If G(r)/G∝rb，we get

dG(r)∝r b-1dr                                 (3)

From Eq. (2), dN(r)∝r –D-1dr, and then

dG∝r3dN                                 (4)

thus it can be seen that D=3-b，and power law is of equal value to fractal distribution.

According to above analysis, if log G(r)/G vs log r could be presented as a kind of linear relationship, the phenomena would appear that the sediment size distribution has fractal structure. And fractal dimension D=3-b, b is longtinal distance.

3    ARMOURING AND STABILIZING ANALYZING

3.1    Fractal characteristics of armoring layer

Bed material armoring is a complex process. In recent years, there are two kinds of armoring model. One is for gravel bed material, and the other is for sand bed material. The armoring characteristics and mechanisms are different. The first model, when armoring layer deformed, bedload transport will be zero with the condition of the same oncoming flow discharge; for the second model, the above mentioned phenomena can not be found. Researchers often judge the stabilizing process for the second armoring model by means of the variability of bedload transport rate. For the gravel bed material, when armoring stabilization appears, sediment size distribution would be contanst, and it is a useful method for analyzing armoring process by means of the property of sediment size distribution. Using the experiment data in the flume from Liu (1992)，particle fractal dimension of armoring layer has been obtained, as show in table 1. After comparing the trend of variation of particle fractal dimension of every set of experiment data, the degree of armoring stabilization can be determined.

3.2    Fractal dimension for particle size of bedload transport rate during armoring process

For a kind of oncoming flow, sediment and bed material, with the increasing of experiment time, sediment size distribution of armoring layer and bedload transport rate will change. Considering the sediment size distribution of initial bed material is constant, and the relationship between sediment size distribution of armoring layer and bedload transport rate is closely related. From the above analysis of fractal characteristics of armoring layer, if fractal dimension of sediment size distribution of armoring layer become constant, that of bedload transport rate will be constant. That is, the judge of armoring stabilization can be made by fractal dimension of sediment size distribution of bedload transport rate.

Table 1    Calculation of fractal™ dimension of armored layer

No.

Equation

Correlation coefficient

Remark

1-1

G(r)/G∝r0.70

0.70

0.90

2.30

Approximated stabilization

1-2

G(r)/G∝r0.75

0.75

0.91

2.25

2-1

G(r)/G∝r0.74

0.74

0.96

2.26

 

2-2

G(r)/G∝r1.00

1.00

0.93

2.00

armored stabilization

2-3

G(r)/G∝r1.01

1.01

0.96

1.99

 

3-1

G(r)/G∝r1.07

1.07

0.96

1.93

 

3-2

G(r)/G∝r1.06

1.06

0.97

1.94

armored stabilization

4-1

G(r)/G∝r1.09

1.09

0.95

1.91

 

4-2

G(r)/G∝r1.08

1.08

0.97

1.92

armored stabilization

5-1

G(r)/G∝r1.27

1.27

0.94

1.73

 

5-2

G(r)/G∝r1.01

1.01

0.98

1.99

armored stabilization

5-3

G(r)/G∝r1.01

1.02

0.98

1.98

 

6-1

G(r)/G∝r1.10

1.10

0.91

1.90

 

6-2

G(r)/G∝r1.09

1.09

0.93

1.91

armored stabilization

7-1

G(r)/G∝r1.10

1.10

0.92

1.90

 

7-2

G(r)/G∝r1.15

1.15

0.98

1.85

armored stabilization

7-3

G(r)/G∝r1.15

1.15

0.99

1.85

 

 

Table 2    Calculation of fractal dimension of bedload transport rate

No.

Equation

b

D=3-b

No.

Equation

b

D=3-b

1-1

G(r)/G∝r2.16

2.16

0.84

4-1

G(r)/G∝r2.00

2.00

1.00

1-2

G(r)/G∝r1.92

1.92

1.08

4-2

G(r)/G∝r1.67

1.67

1.33

1-3

G(r)/G∝r1.90

1.90

1.10

4-3

G(r)/G∝r1.66

1.66

1.34

2-1

G(r)/G∝r2.20

2.20

0.80

5-1

G(r)/G∝r2.10

2.10

0.90

2-2

G(r)/G∝r1.62

1.62

1.38

5-2

G(r)/G∝r1.36

1.36

1.64

2-3

G(r)/G∝r1.60

1.60

1.40

5-3

G(r)/G∝r1.34

1.34

1.66

3-1

G(r)/G∝r2.12

2.12

0.88

6-1

G(r)/G∝r2.20

2.20

0.80

3-2

G(r)/G∝r1.94

1.94

1.06

6-2

G(r)/G∝r1.57

1.57

1.43

3-3

G(r)/G∝r1.91

1.91

1.09

6-3

G(r)/G∝r1.56

1.56

1.44

Using experiment data from Tang (1996), calculation of fractal dimension of bedload transport rate has been obtained, as show in Table 2. Table 2 indicates that with the increasing of experiment time, the difference of fractal dimension of bedload transport rate become very small. This method is helpful to judge the degree of armoring stabilization and compare with different data.

4    FRACTAL DIMENSION OF RIVERBED ROUGHNESS

Because of complexity of sediment characteristics and its distribution in natural channel, the study on the particle exposure of bed material of riverbed becomes very difficlut. Assuming particle exposure belongs to uniform distribution, and according to the exposure concept of MERCER, a calculating method of sediment incipient motion that taking into account the exposure effect and its distribution has been provided by A.S.Paintal (1971). Assuming uniform distribution, a relative exposure concept has been defined by Hang (1982). However, Liu Xinnian believed particle exposure is not uniform distribution according to observation data (1986). Though particle exposure concept has been used very often for describing bed material roughness, a verified particle exposure formula has not get until now. Another parameter for presenting particle exposure is ks for uniform sediment. For nonuniform sediment, some presentative diameter often be selected, such as d65，d75，d90. Considering bed material roughness depends on bed material characteristics, and the latter is relative with flow and sediment condition. Then, the fractal dimension of bed material roughness would be provided to study particle distribution and relationship between fractal dimension and roughness coefficient.

4.1    Exposure of particles of bed material

The elevation of riverbed is always uneven, and the degree of particle exposure is also variable. The function of exposure of particle of bed material can be defined from

  (5)

where z(x，y) is the bed elevation function，V(Δx,Δy) is the second-order structure function of bed elevation function, Δx=nδx，Δy=mδy，δx andδy are the sampling intervals, N and M are the total numbers of measuring points of bed elevations in directions x and y, respectively. In order to simplifying Eq.(5), the longitudinal and transverse structure functions have been discussed, separately. Using the relationship Eq.(5), two equations have been obtained

                (6)

And also can be expressed as from R.F.Voss (1985)

                   (7)

where〈 〉is mean value of sample,z(Δx=0,Δy)，z(Δx,Δy=0)，and fractal dimension from K.J.Falconer (1990) could be calculated as Dx=2-Hx，Dy=2-Hy. And then, Eq. (6) can be also presented as

                      (8)

where Ax，Ay are coefficient，βx，βy are the slope angle of plot between V (Δx=0，Δy)，V(Δx，Δy=0) and Δy，Δx, respectively. Ax，Ay are longitudinal distance. The roughness fractal characteristics are presented as

Dx-2-βx/2，Dy-2-βy/2                                (9)

4.2    Calculation of fractal dimension of riverbed roughness

Considering the difficult of acquiring the field bed elevation data, a group experimental flume data has been used to analyze roughness fractal characteristics. The bed roughness can be described by means of location of sediment particle. Assuming the shape of all sediment particles is spherical and the bed roughness is the function of sediment particles. If the distribution of bed material has several layers, and sediment particle elevation is related to its diameter，Z(di)-di，and the longtinal and transverse variance have the relationship,Δx=Δy =di， the exposure V(di) of the Eq.(6) can be presented as

                   (10)

where N(di) is the number of sediment particle，N(di)=6p(di)/(π(rs-r)di3)，p(di) is the volume gravity. Assuming the shape of sediment particle is spherical，the Eq.(10) will be simplified as

                (11)

The exposure V(di) can be calculated using sediment size distribution, and roughness fractal dimension Dx，Dy can be obtained according to the log-log curve between V(di) and di

LnV(di)=B-DLndi                           (12)

where B is the longitudinal distance，D is the roughness fractal dimension.

Using the analysis of 12 set of armoring experiment data from Tang Zhaozhao (1996), calculation value of roughness fractal dimension of armoring bed material has been obtained, as showin Table 3.

Table 3    Roughness n, presentative diameter d50，d90 and fractal dimension D

n

D50(m)

D90(m)

D

0.0315

0.009

0.0198

0.46

0.0303

0.0085

0.0175

0.44

0.034

0.0068

0.0188

0.50

0.036

0.0072

0.0189

0.48

0.0288

0.006

0.015

0.54

0.0297

0.0032

0.0082

0.62

0.0301

0.0038

0.011

0.48

0.0324

0.0035

0.0089

0.44

0.0359

0.0047

0.0089

0.50

0.032

0.0091

0.02

0.48

0.0329

0.0075

0.0175

0.41

0.0323

0.006

0.0161

0.45

 

4.3    Comparison between roughness n, presentative diameter d50，d90 and fractal dimension D

From the Eq. (12), V(di)∝di-D can describe the relationship between roughness fractal characteristics and sediment diameter. Some researchers believed that roughness of movable bed material is in proportion to sediment diameter, the common relationship is

                                (13)

Where the unit of d is meter, and there are some related formulas, as follows:

, ,                         (14)

Where the unit of d50，d90 is meter.

Considering Eq. (13) and Eq. (14) describe the bed roughness by means of the presentative diameter, and bed roughness strongly depended on the location of sediment particle, and so, there must be limitation of this method. In verse, roughness fractal dimension can describe the impact of every sediment particle, therefore the relationship between roughness coefficient and roughness fractal dimension has been studied. Making use of data from Table 3, the relationships of roughness coefficient, presentative diameter and roughness fractal dimension have been studied, and three kinds of function relationships have been obtained as follows:

                          (15)

5    Conclusion

(1) After analyzing sediment size distribution, the parameters d50 and Ψ can not properly described the nature of sediment size distribution, but the fractal dimension can better express the its property, and can be helpful to study process of armoring and stabilizing of riverbed.

(2) Based on analysis of particle distribution of bed material, the exposure function has been founded, and fractal dimension of roughness characteristics can be calculated using simplified method.

(3) Relationships between bed material roughness, fractal dimension and representative diameter (d50，d90) have been discussed, respectively. The preliminary result appears that the fractal characteristics of roughness can be better for describing bed material roughness. 

Acknowledgements

This study was sponsored by the National Nature Science Foundation of China (Grant No. 59890200 & 49831010). The authors wish to express their appreciation for the financial support.

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