Erodibility

Journal of Hydraulic Research

ISSN: 0022-1686 (Print) 1814-2079 (Online) Journal homepage: http://www.tandfonline.com/loi/tjhr20

Erodibility G.W. Annandale To cite this article: G.W. Annandale (1995) Erodibility, Journal of Hydraulic Research, 33:4, 471-494, DOI: 10.1080/00221689509498656 To link to this article: http://dx.doi.org/10.1080/00221689509498656

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Erodibility Erodabilité G.W. ANNANDALE HDR Engineering, Inc., 303 East 17th Avenue, suite 300, Denver, CO 80203-1256 e-mail: gannanda®hdrinc.com

ABSTRACT Hydraulic erodibility of natural and engineered earth materials, including both soil and rock, can be evaluated in terms of a rational correlation between rate of energy dissipation of flowing water and an erodibility classification of the materials. Earth materials ranging on a continuum from cohesionless granular soil through massive hard rock can be characterized in terms of an erodibility index, Kh. The parameters of the index include key material and mass properties that can be measured in the field and at low cost. These properties include earth mass strength, block/particle size, discontinuity/inter-particle bond shear strength, shape of material units and their orientation relative to the flow. The relative magnitude of erosive power of flowing water is represented by its rate of energy dissipation for a variety of flow conditions, including head cuts, hydraulic jumps, grade changes, and open channel flow. A log-log plot of experienced rate of energy dissipation versus the erodibility index of the studied materials demonstrates a correlation from which the critical threshold to initiate erosion of a material can be predicted for any given set of hydraulic conditions. The method is based on an analysis of 137 field observations of spillway performance collected by the U.S. Department of Agriculture, observations at Bartlett Dam, Salt River Project, Arizona, at four South African dams, and published data pertaining to initiation of sediment motion. RÉSUMÉ L'érodabilité sur Ie plan hydraulique de matériaux de terrain naturels ou remaniés, comprenant a la fois de la terre et des roches, peut être évaluée en terme de correlation entre Ie taux de dissipation d'énergie de 1'eau qui percole et une classification des matériaux selon leur erodabilité. Ces matériaux a base de terre, formant un continuum allant du sol constitué de granulats sans cohesion jusqu'au rocher massif, peuvent être caractérisés par un indice d'érodabilité Kh. Les paramètres de eet indice comprennent des propriétés caractéristiques de la nature du matériau et de sa masse qui peuvent être mesurées in situ pour un cout minime. Ces propriétés comprennent la resistance interne du matériau, la taille des elements granulaires, la resistance au cisaillement inter-particules, la forme des granulats et leur orientation relativement a l'écoulement. L'intensité relative du pouvoir érosif de l'eau en mouvement est représentée par son gradient de perte de charge pour une variété de conditions d'écoulement, comprenant des coupures de l'écoulement, des ressauts hydrauliques, des changements de régime et des écoulements a surface libre. Une representation graphique log - log des mesures de perte de charge en fonction de l'indice d'érodabilité des matériaux étudiés montre une correlation a partir de laquelle Ie seuil critique de début d'érosion d'un matériau peut être prédit pour tout ensemble donné de conditions hydrauliques. La methode est basée sur l'analyse de 137 observations de terrain sur Ie fonetionnement d'un évacuateur de crues faites par Ie Département de 1'Agriculture des USA au barrage de Bartlett sur Ie projet de la Salt River en Arizona, a quatre barrages en Afrique du Sud et a des données de la littérature concernant Ie début d'érosion.

Introduction This paper presents a rational method for determining the erodibility of earth materials. The term earth material is considered to embrace the entire spectrum of soil and rock materials, ranging conRevision received February 21, 1995. Open for discussion till February 28, 1996.

JOURNAL OF HYDRAULIC RESEARCH, VOL. 33, 1995, NO. 4

471

tinuously from cohesionless granular soil through extremely hard, massive roek. The method is based on an analysis of 150 field observations, ineluding 137 made by the SCS and ARS at emergeney spillways that experieneed flow in Kansas and Arkansas (SCS, 1984 and 1991); the plunge pool of Bartlett Dam (Annandale, 1992; Cohen and Von Thun, 1994); seour of rock downstream of four South African dams (van Schalkwyk, 1992); and published data on the initiation of sediment motion by Tison (1953), Gilbert (1914), Kramer (1935), U.S. Army Corps of Engineers, Water ways Experiment Station (WES, 1935), and Vanoni (1964). The method presented herein for determining the credibility of rock and other earth materials com bines basic principles of hydraulics with techniques used in engineering geology for classifying earth materials. A comprehensive survey of 242 references (Bureau of Reclamation, 1993) indi cates that previous efforts to characterize erodibility of materials either concentrated more on engi neering geology and less on hydraulics (e.g. Cameron, et. al., 1988) or focussed more on hydraulics and less on engineering geology (e.g. Mason & Arumugam, 1985 and Mason, 1989). The approach reported in this paper gives equal weighing to the importance of both disciplines in determining the erodibility of rock and other earth materials. Conceptual framework Hydraulic erosion entails a ground-stream interactive problem that can be determined alternatively in terms of physical modelling, rigorous constitutive simulation, or rational correlation between material and flow properties. The basic laws of Newtonian mechanics and the constitution of the materials are to be satisfied in all three approaches. The proposed approach is based on a rational correlation between the rate of energy dissipation of flow and earth mass erodibility using field observations of spillways performance and published data on initiation of sediment motion. A conceptual model of the process of progressive dislodgment is summarized in three stages: - jacking (Figure la) - dislodgment (Figure lb), and - displacement (Figure le). Figure 1 illustrates the three stages in a stratum of jointed rock with a dip against the direction of flow. Models representing the erodibility process for other earth materials arc similar to Figure 1. Flowing water is subject to turbulence which, in turn, is associated with a loss in energy. Turbu lence at the boundary causes pressure fluctuations that result in an action that progressively jacks out a material unit from its position of rest. Once jacked out, a material unit is then dislodged by the power of the flow, and finally displaced. The correlation between rate of energy dissipation (P) and a material's resistance to erosion can be expressed as the function:

P = f(K„)

(1)

at the erodibility threshold. If P >f(Kh), the erodibility threshold is exceeded, and the material would be expected to erode. Conversely, if P Y

(9) 2

'

where, (Henderson, 1966b):

-Mi* + A z )

(10)

Hydraulic jumps Hydraulic jumps are considered in terms of Hagar's (1988) four types that can occur in the vicinity of abrupt slope changes, from steep to near horizontal (Figure 5). The classic hydraulic jump, Type A, forms downstream of steep slopes on the flatter section of the channel. The Type B jump forms where the slope changes from steep to flat with the roller located partly on the steep slope and partly on the flatter reaches. The Type C jump forms where the end of the roller occurs at the beginning of the flatter reach. The Type D jump forms where the roller occurs entirely on the steeper slope of the channel.

s s s s s s s s—.

s/ssssssssss

TYPE A

TYPEB

'7-y ss s s / / s s / / / / s TYPEC

TYPED

Fig. 5. Hydraulic jump types according to Hagar (1988).


477

The energy loss for the Type A jump, also known as the classical hydraulic jump, can be estimated with (e.g. Henderson, 1966a):

A£ =

^+A-i(^^-0-f^nHb >

(11)

where Frt = upstream Froude number, y, = upstream depth, and g = acceleration of gravity. The rate of energy dissipation per unit width of flow in a Type A hydraulic jump is determined by com bining equations (2) and (11):

V

zgy

'

s

yiyfJ\+SFr]-\)

)

Similarly, equations to express the rate of energy dissipation for other jump types can be deter mined by inserting estimates of energy loss into equation (2). Methods to determine energy loss for other jump types are often obtained from graphs and are not repeated here. The reader is referred to publications by Bakhmeteff and Matzke (1938), Argyropoulos (1962), Kindsvater (1944), Peterka (1983), Rajaratnam (1967), Hager (1988), Hager and Li (1992), Khalifa and McCorquodale (1992) and Hager and Bremen (1989). The validity of Equation (12) is demonstrated by making use of experimental findings by Fiorotto and Rinaldo (1992) as presented in Figure 2. The rate of energy dissipation increases with increas ing pressure fluctuations, the premise of the conceptual model in Figure 1. Changes in bed slope Figure 6 illustrates the model used to describe discharge characteristics in a channel with a change in bed slope from 8 degrees to a degrees. Depending on the magnitude of the slope change, separa tion of flow may occur. As water flows over the location of the slope change it impinges on the bot tom slope at an angle (9 - a) if flow separation occurs. The recirculation in the flow separation region is maintained by the trapped water at unit discharge q3. The unit discharge downstream of the point of impingement (3(1

(24)

For analysis of cohesionless granular material Kh can be less than 1. Table 3. Mass strength number for rock (Mv) Hardness

Very soft rock

Soft rock

Identification in Profile

Material crumbles under firm (moderate) blows with sharp end of geological pick and can be peeled off with a knife; is too hard to cut triaxial sample by hand. Can just be scraped and peeled with a knife; indentations 1 mm to 3 mm show in the specimen with firm (moderate) blows of the pick point.

Unconflned Compressive Strength (MPa)

Mass Strength Number (M.)

Less than 1,7

0,87

1,7 - 3,3

1,86

3,3 - 6,6

3,95

6,6 - 13,2

8,39

13,2 - 26,4

17,70

Hard rock

Cannot be scraped or peeled with a knife; hand-held specimen can be broken with hammer end of geological pick with a single firm (moderate) blow.

Very hard rock

Hand-held specimen breaks with hammer end of pick under more than one blow.

26,4 - 53,0 53,0 - 106,0

35,0 70,0

Extremely hard rock

Specimen requires many blows with geological pick to break through intact material.

Larger than 212,0

280,0

The discontinuity or inter-particle shear strength number, Klb is determined by the proportion J/J„ (Barton, 1988), where J, = joint roughness number and Ja = joint alteration number. Joint roughness refers to the roughness condition of the facing walls of a discontinuity. The joint alteration number reflects the weathering condition of the joint face material. Shear strength of a discontinuity is

JOURNAL OF HYDRAULIC RESEARCH, VOL. 33, 1995. NO. 4

483

directly proportional to the shear strength of the gouge and inversely proportional to the degree of alteration of the joint wall material. For granular materials, the quotient JrIJa crudely approximates tan (()),), where ()),. is the equivalent residual (minimum) friction angle (Barton, et. al., 1974). Values for the joint roughness and joint alteration numbers are found in tables 8 and 9. In addition to representing the effective dip of the least favorable discontinuity with respect to the flow, the relative ground structure number, Js, accounts for the shape of the material units that affect the ease with which the stream can penetrate the ground and dislodge individual units (Kirsten, 1982 and 1988). The effective dip is the apparent dip of a discontinuity adjusted for the slope of the stream channel relative to the direction of flow. Table 7 contains values of the relative ground structure number for various ratios of joint spacing. Table 4. Mass strength number for detritus (M() Consistency

Identification in Profile

In Situ Deformation Modulus (MPa)

Mass Strength Number (M.)

Very loose

Particles very loosely packed. High percentage voids and very easily dislodged by hand. Matrix crumbles very easily when scraped with geological pick. Ravelling often occurs in excavated faces.

0-4

0,02

Loose

Particles loosely packed. Some resistance to being dislodged by hand. Large number of voids. Matrix shows small resistance to penetration by sharp end of geological pick.

4-10

0,05

Medium dense

Particles closely packed. Difficult to dislodge individual particles by hand. Voids less apparent. Matrix has considerable resistance to penetration by sharp end of geological pick.

10-30

0,10

Dense

Particles very closely packed and occasionally very weakly cemented. Cannot dislodge individual particles by hand. The mass has a very high resistance to penetration by sharp end of geological pick - requires many blows to dislodge particles.

30-80

0,21

Very dense

Particles very densely packed and usually cemented together. The mass has a high resistance to repeated blows of geological pick - requires power tools for excavation.

80-200

0,44

Note: Determined by plate bearing test of diameter 760 mm.

484

JOURNAL DB RECHERCHES HYDRAULIQUES, VOL. 33, 1995, NO. 4

Table 5. Joinl count number (J(.)

Number of Joints Per Cubic Metre

(J.)

Ground Quality Designation (RQD)

Number of joints Per Cubic Metre (J.)

Ground Quality Designation (RQD)

33

5

18

55

32

10

17

60

30

15

15

65

29

20

14

70

27

25

12

75

26

30

11

80

24

35

9

85

23

40

8

90

21

45

6

95

20

50

5

100

Table 6. Joint set number (J„)

Number of Joint Sets

Joint Set Number

(J.) Intact, no or few joints/fissures

1,00

One joint/fissure set

1,22

One joint/fissure set plus random

1,50

Two joint/fissure sets

1,83

Two joint/fissure sets plus random

2,24

Three joint/fissure sets

2,73

Three joint/fissure sets plus random

3,34

Four joint/fissure sets

4,09

Multiple joint/fissure sets

5,00


485

Table 7. Relative ground structure number (/,.)

Dip Direction of Closer Spaced Joint Set (degrees)

Dip Angle of Closer Spaced Joint Set (degrees)

180/0 In direction of stream flow

0/180 Against direction of stream flow

180/0 Notes:

486

1. 2.

Ratio of Joint Spacing, r 1:1

1:2

1:4

1:8

90

1,14

1,20

1,24

1,26

89 85 80 70 60 50 40 30 20 10 5 1

0,78 0,73 0,67 0,56 0,50 0,49 0,53 0,63 0,84 1,25 1,39 1,50

0,71 0,66 0,60 0,50 0,46 0,46 0,49 0,59 0,77 1,10 1,23 1,33

0,65 0,61 0,55 0,46 0,42 0,43 0,46 0,55 0,71 0,98 1,09 1,19

0,61 0,57 0,52 0,43 0,40 0,41 0,45 0,53 0,67 0,90 1,01 1,10

0

1,14

1,09

1,05

1,02

-1 -5 -10 -20 -30 -40 -50 -60 -70 -80 -85 -89

0,78 0,73 0,67 0,56 0,50 0,49 0,53 0,63 0,84 1,26 1,39 1,50

0,85 0,79 0,72 0,62 0,55 0,52 0,56 0,68 0,91 1,41 1,55 1,68

0,90 0,84 0,78 0,66 0,58 0,55 0,59 0,71 0,97 1,53 1,69 1,82

0,94 0,88 0,81 0,69 0,60 0,57 0,61 0,73 1,01 1,61 1,77 1,91

-90

1,14

1,20

1,24

1,26

For intact material take K, = 1,0 For values of r greater than 8 take K, as for r = 8

JOURNAL DE RECHERCHES HYDRAUÜQUES, VOL. 33, 1995, NO. 4

Table 8. Joint roughness number (Jr)

Condition of Joint

Joint Roughness Number

Joints/fissures tight or closing during excavation

Discontinuous joints/fissures Rough or irregular, undulating Smooth undulating Slickensided undulating Rough or irregular, planar Smooth planar Slickensided planar

4,0 3,0 2,0 1,5 1,5 1,0 0,5

Joints/fissures open and remain open during excavation

Joints/fissures either open or containing relatively soft gouge of sufficient thickness to prevent joint/fissure wall contact upon excavation. Shattered or micro-shattered clays

1,0

Joint Separation

1,0

Table 9. Joint alteration number (JJ Description of Gouge

Joint Alteration Number (J.) for Joint Separation (mm) 1,0'

1,0 - 5-0!

5,0'

0,75

-

-

Unaltered joint walls, surface staining only

1,0

-

-

Slightly altered, non-softening, noncohesive rock mineral or crushed rock filling

2,0

2,0

4,0

Non-softening, slightly clayey non-cohesive filling

3,0

6,0*

10,0*

Non-softening, strongly over-consolidated clay mineral filling, with or without crushed rock

3,0*

6,0**

10,0

Softening or low friction clay mineral coatings and small quantities of swelling clays

4,0

8,0*

13,0*

Softening moderately over-consolidated clay mineral filling, with or without crushed rock

4,0*

8,0**

13,0

Shattered or micro-shattered (swelling) clay gouge, with or without crushed rock

5,0*

10,0**

18,0

Tightly healed, hard, non-softening impermeable filling

Note: 1. 2. 3. 4. 5.

Joint walls effectively in contact. Joint walls come into contact after approximately 100 mm shear. Joint walls do not come into contact at all upon shear. * Values added to Barton et al's data. ** Also applies when crushed rock occurs in clay gouge without rock wall contact.

JOURNAL OF HYDRAULIC RKSLARCH, VOL. 33, 1995, NO. 4

487

Erodibility threshold The erodibility threshold of earth materials was established by relating the erosive power of water and the relative ability of the earth materials to resist erosion for each of the 150 field observations and the published data on initiation of sediment motion (Tison, 1953; Gilbert, 1914; Kramer, 1935; Waterways Experiment Station, 1935 and Vanoni, 1964). The earth materials encountered in the field observations included intact rocklike material, jointed and fractured rock, weathered rock, vegetated soils and cohesive soils. The characteristics of cohesionless granular earth materials, ranging from gravel to silt, were obtained from the published data and from information in Bureau of Reclamation (1977). The materials were indexed by using the Erodibility Index and the erosive power of the water was estimated for each condition by making use of Equations (I) through (23), as appropriate. The following three sub-sections present the relationship between the Erodibility Index and the rate of energy dissipation for (1) the comprehensive data, (2) cohesionless granular materials and (3) rock and other complex earth materials respectively. The set of field data is comprehensive, making representation in tabular form difficult. This data is therefore not repeated in this paper. A summary of the remainder of the data that was used in the investigation is presented in Tables 10 and 11. Table 10 contains the pertinent information on initiation of sediment motion from the literature, whereas Table 11 shows analyzed data that were obtained from the graphs in Figures H-13 and H-14 in Bureau of Reclamation (1977). Table 10. Published data on initiation of sediment motion and erodibility index calculation Data Set

Casey

Grand Lab

Gilbert

Kramer

iThijse Tison

Vanoni USWES

Particle size (mm! 0.170 0.680 0.940 0.800 2.000 4.000 5.000 1.710 3.170 4.938 7.010 0.510 0.510 0.510 0.510 0.530 0.530 0.530 0.530 0.550 0.550 0.550 0.550 0.280 0.150 0.250 0.300 1.000 2.000 0.102 0.205 0.205

Settling velocity (m/s) 0.020 0.100 0.150 0.140 0.250 0.360 0.390 0.210 0.320 0.390 0.450 0.080 0.080 0.080 0.080 0.085 0.085 0.085 0.085 0.088 0.088 0.088 0.088 0.040 0.015 0.030 0.035 0.150 0.250 0.010 0.025 0.025

Shear Vcr/Vss velocity Re Number 2.200 10.530 15.800 12.930 51.000 187.000 269.000 55.690 166.750 339.810 757.040 6.400 6.550 6.660 6.760 7.340 7.370 8.170 8.340 7.930 8.140 7.840 7.530 3.040 1.610 2.640 3.430 19.330 65.000 1.500 2.470 2.670

8.350 2.671 2.230 3.280 2.606 2.444 2.502 1.843 1.892 1.920 2.208 4.667 3.656 3.550 3.577 3.781 3827 3.120 3.206 4.073 3.869 3.890 3.557 7.010 12.583 7385 6.268 2.913 2.573 27.673 8.165 9.005

Data from Table 1 in: Yang, C.T. (1973), Incipient motion of sediment transport, Journal of the Hydraulics Div, ASCE, HY10, p 10067

488

Kinematic viscosity (m"2/s) 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06

Shear velocity (m/s) 0.0129 0.0155 0.0168 0.0162 0.0255 0.0468 0.0538 0.0326 0.0526 0.0688 0.1080 0.0125 0.0128 0.0131 0.0133 0.0138 0.0139 0.0154 0.0157 0.0144 0.0148 0.0143 0.0137 0.0109 0.0107 0.0106 0.0114 0.0193 0.0325 0.0147 0.0120 0.0130

Shear Stress (N/mA2) 0.167 0.240 0.283 0.261 0.650 2.186 2.894 1.061 2.767 4.736 11.663 0.157 0.165 0.171 0.176 0.192 0.193 0.238 0.248 0.208 0.219 0.203 0.187 0.118 0.115 0.112 0.131 0.374 1.056 0.216 0.145 0.169

Critical Velocity (m/s) 0.167 0.267 0.334 0.459 0.651 0.880 0.976 0.387 0.605 0.749 0.994 0.373 0.292 0.284 0.286 0.321 0.325 0.265 0.273 0.358 0.340 0.342 0.313 0.280 0.189 0.222 0.219 0.437 0.643 0.277 0.204 0.225

Critical S/P (kW/m) 2.80E-05 6.40E-05 9.45E-05 1.20E-04 4.24E-04 1.92E-03 2.82E-03 4.10E-04 1.68E-03 3.55E-03 1.16E-02 5.88E-05 4.82E-05 4.84E-05 5.03E-05 6.16E-05 6.29E-05 6.30E-05 6.75E-05 7.45E-05 7.46E-05 6.96E-05 5.87E-05 3.31 E-05 2.17E-05 2.47E-05 2.87E-05 1.63E-04 6.79E-04 5.98E-05 2.95E-05 3.81 E-05 Km = Kd = Ks =

Kh

5.7E-11 3.6E-09 9.6E-09 5.9E-09 9.2E-08 7.4E-07 1.4E-06 5.8E-08 3.7E-07 1.4E-06 4.0E-06 1.5E-09 1.5E-09 1.5E-09 1.5E-09 1.7E-09 1.7E-09 1.7E-09 1.7E-09 1.9E-09 1.9E-09 1.9E-09 1.9E-09 2.5E-10 3.9E-11 1.8E-10 3.1E-10 1.2E-08 9.2E-08 1.2E-11 1.0E-10 1.0E-10 0.02 0.5774 1

JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 33. I995. NO. 4

Table 11. Indexing of coarse granular material with information from Figures H-12 and H-14 in Bureau of Reclamation (1977) Particle Diameter (mm)

Ms

Kb

2 4 6 8 10 20 40 60 80 100

0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09

Kd

8.00E-06 6.40E-05 2.16E-04 5.12E-04 1.00E-03 8.00E-03 6.40E-02 2.16E-01 5.12E-01 1.00E+00

Kh

Js

0.577 0.577 0.601 0.625 0.675 0.700 0.727 0.754 0.781 0.810

4.15E-07 3.32E-06 1.17E-05 2.88E-05 6.07E-05 5.04E-04 4.19E-03 1.47E-02 3.60E-02 7.29E-02

Rate of Energy D ssipation (Watt/m) 0.84 1.29 1.88 2.71 4.01 13.56 41.35 68.36 125.08 184.28

Comprehensive data The relationship between the rate of energy dissipation and the Erodibility Index for the comprehensive set of data is shown in Figure 7. The lower index values represent very fine, cohesionless material (particle diameters of the order of 0.1 to 0.2 mm), whereas the high values represent good quality rock. Two symbol types appear on the graph, one set represents conditions where erosion occurred and the other set where erosion did not occur. The symbols in the lower left part of the graph represent initiation of motion of cohesionless granular material (Tables 10 and 11). The upper right of the graph shows that the symbols representing "erosion" are generally separated from the symbols representing "no erosion". The zone of separation between the two symbol types is a continuation of the trend of the lower line that represents initiation of motion for cohesionless, granular material. The separation zone in the upper right of the graph and the progression of the relationship between the rate of energy dissipation and the Erodibility Index in the lower left of the graph represents the erodibility threshold for materials ranging from 0.1 mm diameter, through gravel, cohesive soils, vegetated soils, weathered rock, and jointed and fractured rock.

Erodibility Rate of Energy Dissipation (kW/m)

Comprehensive Data Base 100000 |

!

,

1000 -

—i

,

1

f r ^ —

Erosion

o.i 0.001

-

.00001 I

1.0E-12

^yy ^ ^ - J —

1.0E-09

-&&*^—|— —'

-4^-

O No Erosion

—:—

—i

1.0E-06 0.001 Erodibility Index

1

'

1

1000

1000000

Fig. 7. Erodibility threshold for comprehensive data set.

JOURNAL OF HYDRAULIC RESEARCH. VOL. 33, 1995. NO. 4

Cohesionless granular materials Figure 8 represents the erodibility threshold for cohesionless granular material (Tables 10 and 11). The relationship represents the threshold of motion for particle diameters ranging from 100 mm to 0.1 mm and can be used concomitantly with other relationships such as the Shields diagram or Yang's criterium for incipient motion (Yang, 1973).

Erodibility Rate of Energy Dissipation (kW/m) p

Cohesionless Granular Material: Incipient Motion

1 0.1

!

0.01

s

0.001

I

I

,

I

,

1

I ■

.

1-—"~t~a~*f^

'

L_«l ■

0.0001

K

.00001 1.000E-12

1.000E-10

1.000E-08 1.000E-06 Erodibility Index

0.0001

0.01

Fig. 8. Erodibility threshold for cohesionless granular material. Rock and other complex earth materials The erodibility threshold for rock and other complex earth materials is defined by the region in the immediate vicinity of the dashed line in Figure 9. Symbols above the dashed line represent events where erosion were observed, whereas symbols below the dashed line represent events where erosion did not occur. The erodibility of rock and complex earth materials can be determined by calculating the Erodibility Index and relating it on the graph to the calculated rate of energy dissi pation (using Equations (2) to (23)). When the point on the graph is located below the region repre senting the erodibility threshold, erosion is unlikely to occur. Erosion is likely to occur if the point is located above the erodibility threshold.

490

JOURNAL DE RECHERCHES HYDRAULIQUES, VOL. 33, 1995, NO. 4

Erodibility

$ Erosion

Rock and Complex Earth Materials

Rate of Energy Dissipation (kW/m)

10000 j=

1

O No Erosion

1

;

1000 =

1

1

♦

r

---]

:

-; —f

* *

^--g-oo—^_

>

/ »

♦

....---■ o o — ^ T

# c*

D

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Fig. 9. Erodibility threshold for rock and other complex earth materials. Conclusion A rational method is presented for evaluating the erodibility of earth materials, ranging on a contin uum from cohesionless granular soil through massive hard rock. The method is based on an analy sis of the erodibility of materials for 150 field observations and published data on initiation of sediment motion. An erodibility index is used to characterize the ability of earth material to resist erosion. The rate of energy dissipation is used to characterize the relative magnitude of the erosive power of water. The erodibility index is a dimensionless number representing several geological parameters, including the mass strength of the material, block size or particle diameter, discontinu ity or inter-particle shear strength, shape of material units and orientation relative to the flow. The rate of energy dissipation is calculated for a variety of flow conditions that can lead to erosion, including headcuts, hydraulic jumps, grade changes, and open channel flow. A graphic relationship between the erodibility index of earth materials and the rate of energy dissipation for the complete data set exhibits an erodibility threshold that applies continuously across a broad range of materials. Acknowledgements The author thanks Dr. Ted Yang, US Bureau of Reclamation, Denver, CO., for reviewing the hydraulics portion of an earlier draft of the paper. Dr. Steve Abt, Colorado State University, Fort


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Collins reviewed the final draft of the paper. Their comments and suggestions were useful and included in the final paper. Mr. John Moore, Soil Conservation Service, USDA made a significant contribution. The author also wishes to thank Dr. Hendrik Kirsten for introducing him to the con cepts associated with the ripability index. The author thanks Mr. Darrel Temple, USDA Agri cultural Research Service, Stillwater, OK., for his assistance in making available unpublished earth spillway performance data his agency collected in cooperation with the USDA Soil Conservation Service.

References/Bibliographie American Society for Testing and Materials, 1988, Rock classification systems for engineering purposes: ASTM STP 984, L. Kirkaldie (Ed.), 167 p. ANNANDALE, G.W., 1992, Bartlett Dam, Salt River Project, Arizona: Determination of the erodibility of the plunge pool under Probable Maximum Flood conditions: Bureau of Reclamation, Denver, CO, October. ARGYROPOULOS, P.A., 1962, General solution of the hydraulic jump in sloping channels, Proc. ASCE, Jnl. of the Hydr. Div., Vol. 88, HY4, pp. 61-75. BAKHMETEFF, B.A. and A.E. MATZKE, 1938, The hydraulic jump in sloped channels: Trans. ASME, Vol. 60, pp. 111-118. BARTON, N., 1988, Rock mass classification and tunnel reinforcement selection using the Q-System: in Rock Classification Systems for Engineering Purposes: ASTM STP-984, Kirkaldie, L.(Ed.), American Society for Testing and Materials, Philadelphia, pp. 59-88. BARTON, N., R. LIEN, and J. LUNDE, 1974, Engineering classification of rock masses for the design of tunnel support: Rock Mechanics, Vol. 6, No. 4, pp. 189-236. Bureau of Reclamation, 1993, Dam foundation erosion: survey of literature: Denver, CO, 118 p. Bureau of Reclamation, 1977, Design of small dams: U.S. Government Printing Office, Washington, D.C. COHEN, E.A. and J.L. VON THUN, 1994, Dam safety assessment of the erosion potential of the service spillway at Bartlett Dam: Q. 68, 18th Congress, ICOLD, Durban, South Africa, November. CAMERON, C.P., D.M. PATRICK, K.D. CATO, and J.H. MAY, 1988, Geotechnical aspects of rock erosion in

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of overflow erosion on embankments, II: Hydraulic and Design Considerations, Journal of Hydraulic Engineering, ASCE, Vol. 115, No. 8, August, pp. 1056-1075. RAJARATNAM, N., 1963, Discussion to Argyropoulos, P.A., General solution of the hydraulic jump in sloping channels: Journal Hydr. Div., ASCE, Vol. 89, HY1, pp. 258-261. RAJARATNAM, N., 1966, The hydraulic jump in sloping channels: J. Central Board of Irrigation and Power, India, Vol. 23, 2, pp. 137-149. RAJARATNAM, N., 1967, Hydraulic jumps, Advances in Hydroscience, V.T. Chow (Ed.), Vol. 4, Academic Press, New York. RAJARATNAM, N. and V. MURAHARI, 1974, Flow characteristics of sloping channel jumps: Jnl. Hydr. Div., ASCE, Vol. 100, HY6, pp. 731-740. Soil Conservation Service, USDA, 1984, National Bulletin No. 210-4- 1, Arkansas, December 1982, Spillway Performance Report, Washington, D.C. Soil Conservation Service, USDA, 1991, National Bulletin No. 210-1- 13, Kansas, June 1989, Spillway Per formance Report, Washington, D.C. STEIN, O.R., P.Y. JULIEN, and C.V. ALONSO, 1993, Mechanics of jet scour downstream of a headcut: Journal of Hydraulic Research, IAHR, Vol. 31, No. 6, pp. 723-738. TEMPLE, D.M., J.A. BREVARD, J.S. MOORE, G.J. HANSON, E.H. GRISSINGER, and J.M. BRADFORD, 1993,

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