京都大学防災研究所年報 第 58 号 B 平成 27 年 6 月
Annuals of Disas. Prev. Res. Inst., Kyoto Univ., No. 58 B, 2015

Abrasion Damage Estimation of Sediment Bypass Tunnels:
Validation and Comparison of two Prediction Models
Christian AUEL, Robert M. BOES(1) and Tetsuya SUMI
(1) Laboratory of Hydraulics, Hydrology and Glaciology, ETH Zurich, Switzerland
Synopsis
Two abrasion prediction models developed for hydraulic structures prone to supercriticalflows are compared herein: The models by Ishibashi (1983) and Auel et al. (2015). Bothmodels are compared and validated using prototype data from the Asahi sediment bypass
tunnel in Japan. The prediction model by Ishibashi (1983) contains some inconsistences in
its grinding stress term being much higher than its particle impact term which contradicts
statements by other researchers. However, it is shown herein that the model leads to goodresults if only the particle impact term is considered. The prediction model by Auel et al.
(2015) shows excellent performance if the abrasion coefficient kv is adjusted to prototype
measurements. The analysis presented herein leads to the conclusion that kv for concrete islower than 106 being a widely accepted value for bedrock abrasion. Independent of theprediction model, utmost interest has to be laid on the estimation of the sediment transport.
The theoretical transport capacity may be much higher than the real transport and should
be calibrated using prototype field data such as reservoir bed topography surveys.
Keywords: Reservoir sedimentation, Sediment bypass tunnel, Abrasion prediction model,
Hydro-abrasive wear

1

Introduction
The use of sediment bypass tunnels (SBTs) is an effective strategy to counteract reservoir sedimentation. During flood events sediment-laden flows are
bypassed around the dam into the tailwater inhibiting
sedimentation in the reservoir. Due to high flow velocities in combination with bedload sedimenttransport, many SBTs worldwide are prone to hydroabrasive wear as shown in Fig. 1 (Jacobs et al. 2001,
Sumi et al. 2004, Auel and Boes 2011, Boes et al.
2014, Nakajima et al. 2015, Baumer and Radogna
2015, Müller and Walker 2015).
Abrasion is a wear phenomenon involving progressive material loss due to hard particles forced against
and moving along a solid surface. In bedrock rivers,
abrasion is the driving process for bed incision (Sklar
and Dietrich 2004, 2006, Lamb et al. 2008, Turowski2009) while at hydraulic structures such as spillways,
weirs, flushing channels and SBTs abrasion causes severe damage of the invert surface. In general, abrasive
damage can always be expected when particle bedloadtransport takes place. Particles are transported in sliding, rolling or saltation motion depending on the flow
conditions causing grinding, rolling or saltating impact
stress on the bed. The vertical component of a saltating
particle causes the so called deformation wear, whichis related to the particle impact, whereas the horizontal
component causes cutting wear, which is related togrinding stress (Bitter 1963a, 1963b). Engel (1978)
stated that erosion depends on the sine of the impact
angle because the magnitude of the peak tensile stressvaries with the normal component of the impact velocity, i.e. the vertical velocity component is the driving
factor. Cutting wear caused by the horizontal velocity
component is important in ductile materials and in case― 540 ―

of highly angular impacting particles, but is not significant when brittle materials are impacted byrounded grains as present in river systems (Head andHarr 1970, Sklar and Dietrich 2004).
A number of models exist to predict abrasion. While
the models for the prediction of bedrock incision rate
(Sklar and Dietrich 2004, Lamb et al. 2008) focus on
typical flow conditions in river systems in the suband low supercritical flow regime, the others for prediction of abrasion on hydraulic structure surfaces(Ishibashi 1983, Helbig and Horlacher 2007, Auel etal. 2015) have to account for highly supercriticalflows. In this paper, the abrasion prediction model of
Ishibashi (1983) is explained in detail and compared
to a model recently developed by Auel et al. (2015).

the brittle material concrete. As the original work is
written in Japanese, a short description of the developed equations is given hereafter. Ishibashi (1983) calculated the abraded invert volume Va as:
Va  C1 Ek  C2W f

[m3]

[1]

where Ek = total particle kinetic energy by saltating
particles, Wf = total friction work by grinding particles,
and C1 and C2 = material property constants [m2/(kgf)]
given in Table 1. Note that the unit kilogram force corresponds to Newton as 1 kgf = 9.806 N. The total kinetic energy Ek is given by:

Ek  1.5Vts  Ei N i ni

[kgf m][2]

W f  5.513 sVts   im Ei N i ni [kgf m][3]

and the total friction work Wf by:where Vts = amount of transported sediment [m3],μs = 0.3 = dynamic friction coefficient, Ei = kinetic energy of a single particle, γim = particle impact angle, Ni= L/Lp = impact frequency, with Lp = particle saltation
length and L = total invert length, and ni = amount ofparticles per volume. The value ni is calculated as:ni (1   p )  s[1/m3]

Mp[4]

where λp = 0.4 = air porosity, ρs = particle density and
Mp = 1/6πρsD3 = particle mass, with D = mean particle
diameter. The single impact energy Ei is calculated as:Ei   Fi 5/3[kgf m][5]where Fi = impact force, and β = auxiliary parameter
given as:1/ 3
   2.5n12/3  D 2  





1

[m/(kgf)2/3][6]

where n1 = auxiliary parameter given as:
Fig. 1

2

Concrete invert abrasion at Asahi SBT, Japan (photo: C. Auel).

Abrasion prediction models

2.1 Ishibashi model
A widely applied formula in Japan to predict abrasion
is given by Ishibashi (1983). He considered both deformation and cutting wear due to the fact that ductile
materials such as steel were also investigated besidesn1 

43  k1  k2 

[kgf/m2][7]

where k1 and k2= auxiliary parameter accounting for
the Young’s modulus and Poisson’s ratio of both theparticle and invert materials, respectively. Ishibashi(1983) proposed a constant value of n1 = 2.41×109
kgf/m2 for sediment gravel transported over concrete.
The abrasion prediction model is based on laboratory
data published in Ishibashi and Isobe (1968). Openchannel flow experiments were conducted in a 9 m― 541 ―

long, 0.2 m wide and 0.2 m high laboratory flume toinvestigate the impact forces and saltation trajectories of bedload particles at supercritical flow conditions. Correlations for the particle saltation length Lp,
impact force Fi, and impact angle γim were proposedby Ishibashi (1983) based on Ishibashi and Isobe(1968). Lp follows:

Lp
D 100    c 

1.21

Ar 

with θ = U*2/[(s−1)gD] = Shields parameter, U* =friction velocity, s = ρs/ρ with ρ = fluid density, andθc = critical Shields parameter. θc for fixed beds withlow relative roughness heights is given by Ishibashi(1983) based on Novak and Nalluri (1975) as: c  0.0907 D

1  U V   1  P 
I

Up
Vs

 19.2    c 

0.13

[-]

  k 2 
PR  8.5  10      D 



[10]

[1/m]

[13]

 0.554[14]with k = geometric roughness height and D = mean
particle diameter. The hop length Lp for supercriticalflows is given with (Auel 2014, Auel et al. 2015):

[11]

Lp

where Vs = vertical particle settling velocity in stillwater, and Up = horizontal particle velocity.

D 251

[15]

Auel et al. (2015) found a simple linear correlation of
Wim to the friction velocity U* as

Tab. 1 Material property constants (Ishibashi 1983)
Material
C1 [m2/(kgf)] C2 [m2/(kgf)]
Concrete
1.189×10-7
1.135×10-8-11
Steel (SM 41)
3.73×10
6.59×10-11
Steel (HT 80)
2.53×10-11
4.78×10-11-11
Steel (SUS 304)
2.04×10
3.25×10-11
Steel (SCMnH 11) 1.18×10-11
1.33×10-11

Wim  U 

[m/s]

[16]

Additionally, Auel et al. (2015) analyzed bedrockabrasion laboratory experiments and compared them toconcrete abrasion tests. It was suggested to use theabrasion coefficient value kv = 106 given by Sklar andDietrich (2004) for bedrock abrasion as well as forconcrete structures. Based on field data of Asahi SBT,
it is shown in the following that kv varies from theabove mentioned value and should be adjusted for concrete inverts accordingly.

2.2 Auel model

Based on open-channel flow experiments in a laboratory flume conducted by Auel (2014), an abrasion
prediction model applicable for hydraulic structuresprone to supercritical flows was proposed by Auel etal. (2015). The model is based on the state-of-the-art
saltation abrasion model by Sklar and Dietrich(2004) developed to estimate abrasion in bedrock
river systems. The abrasion rate Ar is calculated as:R

where PR = rolling probability. The numerator of thefirst term on the right hand side is proposed by Sklar
and Dietrich (2004) and accounts for the mode shift
from saltation to suspension. The rolling probability PR
follows (Auel 2014):

where Mp* = 1/6π(ρs−1000)D3 = submerged particle
weight in [kg]. The impact angle is approximated as: im 

s

Lpwhere D* = [(s−1)gD3]0.5/ν = dimensionless grain sizewith ν = kinematic viscosity. The impact force Fi isgiven as follows:
[kgf]

[12]

2 0.5



[9]

Fi  1.95  103 M p    c [m/s]

where kv = abrasion resistance coefficient, YM =Young’s Modulus of elasticity of the invert material[Pa], ft = invert material splitting tensile strength [Pa],Wim = mean vertical particle impact velocity [m/s], I =impact rate [1/m], and qs = specific gravimetric bedload rate [kg/(sm)]. The number of impacts per unit
length I is defined as the reciprocal value of the particle
saltation length Lp as:

[8]

2/ 5

YM
Wim2  I  qs
kv f t 2

3

Procedure of model comparison

Comparison of the two prediction models is done byanalyzing prototype data of the Asahi River, reservoir
and SBT in Kii Peninsula, Japan. The reservoir is operated by Kansai Electric Power Co. Inc. and is part of― 542 ―

the Okuyoshino pump storage hydropower scheme
together with the upper Seto dam. The reservoir has
been facing severe sedimentation since its inauguration in 1978 due to large flood events caused byheavy rainfall and typhoons (Harada et al. 1997,
Nakajima et al. 2015). To counter the ongoing sedimentation, a SBT was constructed in 1998 with a design discharge of Qd = 140 m3/s. The total tunnel
length of 2,383.5 m includes an 18.5 m long steellined inlet section, a 2,350 m long concrete-lined tunnel and a 15 m long concrete-lined outlet. The tunnel
is b = 3.80 m wide with a slope of Sb = 0.029. Since
its inauguration the tunnel faces severe abrasion up
to several decimeters. Kansai Electric conducts regular invert abrasion measurements on an annual basis(Fig. 2).

Fig. 3 Percentage of high performance concrete (fc =70 MPa) on tunnel invert (adopted from
Nakajima et al. 2015).
Fig. 4 shows the annual sediment amount bypassed
through the tunnel compared to the measured invert
abrasion. Note that the bypassed sediment volume
given in Fig. 4 is calculated from the given raw-dataand slightly differs from a similar figure given in
Nakajima et al. (2015) due to the following reasons:Firstly, Qs is calculated with a sediment diameter
D50 = 7.5 cm based on a recent survey (Awazu 2015)
whereas the latter is based on field survey data where
D50 = 5 cm was found. Secondly, herein only the mean
diameter is taken into account, whereas in Nakajima etal. (2015) a fraction-wise sediment transport is applied
accounting for the entire size distribution.

Fig. 2 Cumulative abrasion pattern of Asahi SBTfrom 1998 to 2011. Flow from right to left
(adapted from Nakajima et al. 2015).
Abrasion is measured manually every 2 m in thestreamwise and every 10 cm in the lateral direction.
In case of deep scour holes in between the 2 m sections, the streamwise distance is adapted accordingly.
The tunnel invert was originally lined with a concreteof compression strength fc = 30 MPa. During annual
maintenance works the invert was successively linedwith 70 MPa concrete. The annual ratio of old to new
concrete is given in Fig. 3 (Nakajima et al. 2015).
The discharge in the upstream Asahi River reach is
given on an hourly basis. The catchment is rather
steep with a river bed slope varying from Sb = 0.03 to
0.10 and river widths varying from about 20 to 40 m.
The transported sediment has been calculated by
Newjec Inc. by means of a 1D numerical model using
the bedload formula of Ashida and Michiue (1972)
and calibrated with data of annual reservoir bed topography surveys (Harada et al. 1997, Kataoka 2000,
2003). As a result Newjec provided a correlation of
the river discharge to the amount of transported sediment which is used in the presented calculations.

Fig. 4 Abraded invert volume and bypassed sedimentvolume of Asahi SBT from 1998 to 2011.
4

Results and discussion

Calculations for both models are made using a mean
grain size diameter of D = 7.5 cm to estimate both thesediment transport and the abrasion volume. A fraction-wise calculation considering the grain size distribution was not carried out.
4.1 Ishibashi model

The results of the abrasion prediction model given in― 543 ―

Eq. (1) by Ishibashi (1983) are shown in Fig. 5. The
kinetic impact and grinding stress terms C1Ek andC2Wf, respectively, are separated to better understandtheir effects. The total abraded volume Va is the sumof the two terms. To compare the results the relativechange δ is calculated as 

V a , estim ate  V a , realV a , real

 100

[%]

[17]

The results show that the abrasion volume is largely
overestimated by δ = 190±118% considering bothterms. The grinding stress term is in average 2.6
times higher than the kinetic impact term. Considering the fact that Wf has a negligible effect in case ofbrittle materials (Head and Harr 1970) and is neglected by other researchers (Sklar and Dietrich
2001, 2004, Auel et al. 2015), the value should be atleast smaller than C1Ek. Additionally, Auel (2014)
showed for an exemplary calculation that the energy
induced by rolling and sliding motion of a particle ona concrete surface is about 10% and 0.1%, respectively, compared to the kinetic energy by saltating
particles. Hence, the conclusion may be drawn thatthe grinding stress term in Eq. (1) is only appropriate
in case of ductile materials and should not be considered in case of brittle inverts such as concrete. It is
proposed to omit the grinding stress term and rewrite
Eq. (1) as:Va  C1  1.5Vts  Ei N i ni

[m3]

[18]

Fig. 5 reveals a good agreement of the measured tothe estimated data applying only the kinetic energy
as given in Eq. (18). The average deviation over 14
years is only δ = 35±23%.
A disadvantage of the Ishibashi model is the fact thatthe material properties are only considered in the coefficients C1 and C2 (Table 1). These coefficients areconstant for concrete, however, the material strength
which largely affects the abrasion depth is not accounted for in the estimation of the abrasion volume.

Fig. 5 Measured abraded invert volume of Asahi SBTfrom 1998 to 2011 and comparison with abrasion volume prediction by Ishibashi (1983).
4.2 Sensitivity of abrasion resistance coefficient kv

Auel et al. (2015) discussed the sensitivity of kv statingthat the coefficient is probably not a constant but dependent on the material properties based on statementsfrom Chatanantavet and Parker (2009) and Turowski
et al. (2013). Furthermore, Sklar and Dietrich (2004,2012) showed that kv ranges from 1 to 9×106 for different materials. Fig. 8 shows kv as a function of ft for anumber of research studies including rock and concrete
samples. The splitting tensile strength for concrete is
calculated using a correlation to the cubed compression strength fc after Arioglu et al. (2006) and EN
1992-1-1 as:

f t  0.387  0.8 f c 0.63

 0.336 f c 0.63 [MPa][19]

Six rock and three concrete samples from Sklar andDietrich (2004) are plotted as well as two theoretical
bedrock values by Chatanantavet and Parker (2009).
By means of a scaling analysis, the latter proposed kv= 104 for weak and kv = 106 for hard rock, respectively.
Herein, ft = 1 MPa and 15.5 MPa is assumed for weak
and hard rock, respectively, based on values given in
Sklar and Dietrich (2001). The values from Auel
(2014) are obtained from laboratory experiments using
weak mortar as a substitute to allow for invert abrasionin short time. Values range from kv = 8(±5)×102 to6(±6)×103. Data of Helbig et al. (2012) were obtained
from rotating drum experiments. 30×30 cm concrete
sample plates were mounted in a drum filled with both
10 kg of steel spheres and 10 kg of water. The drum
rotates around a horizontal axis with a radius of 43 cm
where the concrete samples immerge into the watersphere mixture at every rotation. Abrasion is measured
after different time steps by both laser scanning and― 544 ―

weighing. All hydraulic parameters required to estimate kv from Eq. (12) can be calculated using the rotational velocity and the time the sample is submerged. However, one parameter, the saltation
length Lp, turns out to be difficult to estimate. Whenthe sample immerges into the water, the spheres start
to impact the sample surface. Depending on thesphere size (D = 4.4 to 8 mm) about 4,782 to 28,744
spheres are present in the water corresponding to 0.3
to 2.1 spheres/cm3. Additionally, due to the rotation,some spheres are transported towards a higher location inside the drum and drop back into the water.
Both the large amount and the falling spheres lead toa very high impact rate. Hence it is assumed that Lp
is in the range of a millimeter. In Fig. 8, three datagroups of Helbig et al. (2012) are plotted corresponding to Lp = 0.1, 1 and 5 mm, resulting in corresponding kv values of 3.1(±0.8)×106, 3.1(±0.8)×105 and6.2(±1.7) ×104, respectively. A precise estimate of Lp
is difficult, but due to the large amount of particles inthe drum, the values of 0.1 and 1 mm seem to be mostadequate. Finally, the prototype data from Asahi SBT
are calculated and additionally presented in Fig. 7.
The data fit into the range of Helbig’s data with Lp =1 mm.

ft, and the data set of Helbig et al. (2012) largely depends on the estimation of Lp and does not allow for
precise analysis. Further investigation is needed to adequately define the correlation of kv to the invert material properties.
4.3 Auel model

Applying the prediction model given in Eq. (12) leads
to the results given in Fig. 8. As the tunnel invert concrete strength varied over time (Fig. 3), Eq. (12) wasapplied for both material strengths, i.e. fc = 30 MPa and
70 MPa, and averaged using the ratio given in Fig. 3.
Auel et al. (2015) proposed an abrasion coefficient kv =106 if no additional abrasion measurement data are
available. Applying this value leads to an underestimation of the real abrasion volume of about δ = 80±7% inaverage over 14 years (Fig. 8).

Fig. 8 Measured abraded invert volume of Asahi SBTfrom 1998 to 2011 and comparison with abrasion volume predictions by Auel et al. (2015)
based on kv values calibrated from the 1998 to2006 data.

Fig. 7 Abrasion coefficient kv as a function of splitting tensile strength ft.
The data do not follow a clear trend. The sets of Auel
(2014), Chatanantavet and Parker (2009), the rock setof Sklar and Dietrich (2004), and the prototype dataof Asahi indicate an increase of kv with increasing
tensile strength. However, the concrete data of Sklar
and Dietrich (2004) show high kv values even for low

However, a calibration is possible, as the real abrasion
at Asahi SBT is known. The total measurement period
from 1998 to 2011 has been divided into a calibration
period (1998 to 2006) and a validation period (2007 to
2011). The abrasion coefficient has been calculated forthe calibration period, leading to a value of
kv = 2.0×105. Applying this value to the validation period leads to the results given in Fig. 8. The abrasion
volumes in 2008 and 2010 are of minor extent leading
to an overestimation of 54% and 26%, respectively.
However, neglecting these two values, thus taking into
account only the three years with considerable abrasion in 2007, 2009, and 2011 leads to a deviation of
only δ = 12±4%.

― 545 ―

5data as calibration. This large difference underlines theneed for precise estimation of the real sedimenttransport impacting the hydraulic structure.

Conclusion

Herein, the two abrasion prediction models by Ishibashi (1983) and Auel et al. (2015) developed for hydraulic structures prone to supercritical flows arecompared. Both models are validated using prototype
data from the Asahi sediment bypass tunnel (SBT) in
Japan showing good agreement if adjusted and calibrated.
The prediction model by Ishibashi (1983) reveals
certain inconsistencies in its grinding stress term accounting for the cutting wear. It is proposed to omit
this term in case of brittle materials such as concrete.
The model leads to good results if the grinding stressis not considered.
The prediction model by Auel et al. (2015) shows excellent performance if adjusted to the prototype
measurements. However, if prototype abrasion measurements are not available, a kv value has to be assumed. The analysis presented herein leads to theconclusion that kv is lower for concrete than the
widely applied value of kv = 106 valid for bedrockabrasion. A value around 2.0×105 seems to be more
adequate in case of Asahi SBT. However, more research on prototype hydraulic facilities as well as laboratory experiments should be conducted to amplify
the knowledge of the variation of kv. Hagmann et al.
(2015) conduct research at three Swiss SBTs whereabrasion volumes as well as sediment transport aremeasured. These results are expected to complement
and enhance the present results.
Independent of the prediction model, utmost focus
has to be put on the estimation of the sediment bedload transport as both models are directly correlated
to it. The theoretical transport capacity is not always
adequate, as the real transport may be much lower.
The transport has to be calibrated using prototype
data such as periodical sedimentation survey data of
the reservoir or direct measurements such as geophones or plate microphones (Hagmann et al. 2015,
Koshiba 2015). Furthermore the upstream riverbed
slope, the river cross-sections as well as the particle
size distribution should be analyzed as precisely as
possible. In the case of the Asahi River, the theoretical capacity applying local values of bed slope andriver width surveyed directly upstream of the SBT intake (Awazu 2015) lead to about 9 times higher theoretical sediment transport capacity as the data evaluated by Newjec Inc. using the real reservoir survey

Acknowledgments

The support of Sohei Kobayashi and Takahiro Koshiba
regarding the translation of Ishibashi’s work is gratefully acknowledged. Furthermore, the first author
acknowledges the financial support of the Japanese Society for the Promotion of Science (JSPS).
NotationAr

vertical abrasion rate

[m/s]

B

tunnel width[m]

C1,C2 material property constants[m2/(kgf)]

D

particle diameter

[m]

D*

dimensionless grain size

[-]

Ei

kinetic energy of a single particle

[kgf]

Ek

kinetic energy by saltating particle [kgf m]

Fi

impact force

[kgf]

fc

cubed compression strength

[Pa]

ft

splitting tensile strength

[Pa]

g

gravitational acceleration[m/s2]

hflow depth

[m]

Hp

particle saltation height[m]I

number of particle impacts

[1/m]

k

bed roughness height[m]

k1, k2 auxiliary parameter[m2/kgf]

kv

rock resistance coefficient[-]Ltotal invert length

[m]

Lp

particle saltation length

[m]

Mp

particle mass

[kg]

Mp*

submerged particle weight

[kg]

Ni

L/Lp = impact frequency

[-]

ni

amount of particles per volume[1/m3]

n1

auxiliary parameter[kgf/m2]PRrolling probability

[-]

Qd

design discharge[m3/s]― 546 ―

Qs

gravimetric bedload rateqs

specific gravimetric bedload rate [kg/(sm)]sdensity ratio s = ρs/ρ

[-]

Sb

bed slope[-]U

uniform flow velocity

[m/s]

Up

horizontal particle velocity

[m/s]

U

friction velocity U = (gRhS)0.5

[m/s]

Va

abraded volume

[m3]

Vts

amount of transported sediment [m3]Vsparticle settling velocity

Wimvertical particle impact velocity [m/s]Wffriction work due to grinding stress [kgf m]

YMYoung’s Modulus of elasticity

[Pa]

β

auxiliary parameter[m/(kgf)2/3]

γimparticle impact angle[-]δrelative change[%]θShields parameter θ = U2/[(s-1)gD] [-]θccritical Shields parameter

[-]

λp

air porosity

[-]

μs

dynamic friction coefficient[-]ν

kinematic viscosity[m2/s]

ρ

fluid density[kg/m3]ρsparticle density[kg/m3]

Boes, ed.), ETH Zurich, Switzerland, pp. 101121.

[kg/s]

Auel, C. (2014). Flow characteristics, particle motion
and invert abrasion in sediment bypass tunnels.
PhD thesis 22008, also published as VAW-Mitteilung 229 (R. Boes, ed.), ETH Zurich, Switzerland.
Auel, C. and Boes, R.M. (2011). Sediment bypass tunnel
design – review and outlook. Proc. ICOLD Symposium „Dams under changing challenges“(A.J. Schleiss & R.M. Boes, eds.), 79th Annual
Meeting, Lucerne. Taylor & Francis, London,
UK, pp. 403-412.

[m/s]

Awazu, Y. (2015). Effects of sediment bypass operationon the riverbed environment downstream ofdams. Master thesis, Department of Urban Management, Graduate School of Engineering,
Kyoto University, 72 p. (in Japanese).
Baumer, A. and Radogna, R. (2015). Rehabilitation of
the Palagnedra sediment bypass tunnel (20112013). Proc. Int. Workshop on Sediment Bypass
Tunnels, VAW-Mitteilung 232 (R. Boes, ed.),ETH Zurich, Switzerland, pp. 235-245.
Bitter, J. G. A. (1963a). A study of erosion phenomena,
part I. Wear 6, pp. 5–21.
Bitter, J. G. A. (1963b). A study of erosion phenomena,
part II. Wear 6, pp. 169–190.

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