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Coastal Engineering 140 (2018) 257–271
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Coastal Engineering
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Empirical model for probabilistic rock stability on flat beds under waves
with or without currents
Shigeru Kawamataa,∗, Manabu Kobayashib, Norio Tanadac
National Research Institute of Fisheries Engineering, Japan Fisheries Research and Education Agency, Hasaki, Kamisu, Ibaraki, 314-0408, Japan
Choshi Branch, International Meteorological and Oceanographic Consultants Co. Ltd, Kawaguchicho, Choshi, Chiba, 288-0001, Japan
Fisheries Research Division, Tokushima Agriculture, Forestry and Fisheries Technology Support Center, Dounoura, Seto, Naruto, Tokushima, 771-0361, Japan
Rock stability
Asymmetric wave
Minimum stable mass
Friction coefficient
Mobility index
An empirical formula was developed for predicting the stability of isolated quarry rocks on relatively flat beds
under waves with or without current in terms of the damage ratio. The damage ratio was examined using 100
crushed stones (median mass M50 = 293 g and mass density ρs = 2.62 g/cm3) as scale models of quarry rocks
and replicas with a lower density (ρs = 1.38 g/cm3) by placing them on beds of different roughness in a wave
flume, not only under periodic waves with periods of 2–3.5 s, but also in symmetric and asymmetric oscillatory
flows (periods: 8–12 s), simulating waves without and with current in a circulating water channel. Comparison
between hydrodynamic forces expressed in terms of three non-dimensional mobility indices: the maximum
velocity, maximum semi-velocity amplitude (Ua), and maximum acceleration relative to the friction force (expressed in terms of the median coefficient of friction) suggested that the damage ratio was most closely related to
the Ua-based mobility index (ψ2). Nevertheless, significant differences remained between data from the wave
flume and circulating water channel tests. The variation in the damage ratio, which included the effects of the
oscillatory-velocity asymmetry, oscillation period, superimposed steady current, mass density of stones, and
bottom friction, was reasonably well described via the product of ψ2 and a function of a Keulegan–Carpenter
number. The results of the field tests on quarry rocks (with M50 = 2.04 t) placed on a thin sand layer overlaying
hard substrate show that the minimum stable mass is consistent with the prediction.
1. Introduction
Quarry rocks are commonly used as a cost-effective material for
artificial reefs in civil engineering (Deysher et al., 2002; Seaman, 2007;
Bohnsack and Sutherland, 1985; Grant et al., 1982). Quarry rock reefs
have been often constructed at sites having a thin layer of sand overlaying a flat hard substrate, aiming to attract more fish in hard bottom
habitats or to create macroalgal beds without being buried by sand
(Grant et al., 1982). The proper placement of low-relief (thus smallsized) rock substrates or rocks at regular intervals on sandy bottoms can
aid the development and persistence of macroalgal stands (Deysher
et al., 2002; Kawamata et al., 2011; Ohno et al., 1990). The hydrodynamic stability of rocks or stones on the seabed should be treated as
probabilistic in nature because of the high variabilities not only in the
rock shape but also in the roughness of the substratum. The stability of
rocks in coastal sites has been well studied with regard to the design of
coastal structures including rubble revetments, breakwaters (Van der
Meer, 1987, 1992; Kobayashi and Jacobs, 1985), near-bed rubble
mounds (Van Gent and Wallast, 2001; Tørum et al., 2010; Wallast and
van Gent, 2002), and the protection of rock slopes and gravel beaches
(van der Meer and Pilarczyk, 1987). However, studies on the stability of
isolated rocks on comparatively flat bottoms are lacking. Present
common design approaches based on stability criteria, such as the
stability number, Shields number, and mobility number, cannot be
applied to such conditions because they assume stone layers with a
particular weight, shape, and density and do not explicitly evaluate the
effect of frictional resistance on stability.
In the current design criteria for artificial reefs in Japan, the critical
stable mass of rocks in surf zones is given by the following formula from
Akeda et al. (1992).
Mcr = CUm6
where Mcr is the critical stable mass (in kg) of rocks, C is an experimental coefficient depending on the density of the rocks deployed, and
Um is the wave-induced maximum peak velocity (in m/s). This formula
assumes that the hydraulic load and friction with respect to the rocks
Corresponding author.
E-mail address: (S. Kawamata).
Received 10 December 2017; Received in revised form 12 July 2018; Accepted 2 August 2018
Available online 04 August 2018
0378-3839/ © 2018 Elsevier B.V. All rights reserved.
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S. Kawamata et al.
Considering the probabilistic nature of rock stability, the damage ratio
rd was defined as the relative number of quarry rocks ‘damaged’ under
given physical conditions.
The mobility of stones on a relatively flat bed can be expressed
simply by the ratio of the maximum drag force (proportional to
ρUm2Dn502) to the frictional force (μ50(ρs − ρ)gDn503), i.e.,
are proportional to the square of the velocity and the submerged
weight, respectively, similar to the equation proposed by Isbash (1936)
for steady flows. Akeda et al. (1992) found the C value to be 25 for
solitary rocks by performing a scale wave flume experiment with
rounded pebbles on a smooth, flat cement mortar bed. They determined
the C value considering the performance in terms of the cost of constructing an artificial rock reef by defining the motion threshold as the
maximum velocity at which 10% of the test stones begin to move
(Akeda et al., 1992). Equation (1) is convenient for practical design but
is too simplified to evaluate the effects of factors other than Um, such as
the mass density, friction, and velocity asymmetry. A critical problem is
that the calculated stable mass for isolated rocks is often considerable.
For example, for Um = 3 m/s, Eq. (1) with C = 25 gives Mcr ≈ 18 Mg
(or t). It is unclear whether such a large stable mass is reasonable in
The objective of this study was to develop an empirical formula for
predicting the probability of the stability of solitary quarry rocks on
relatively flat beds under the effects of waves in shallow waters, including the effects of the asymmetry of the wave oscillatory velocity,
the coexisting flow, the friction coefficient between the rocks and the
seabed, and the mass density of the rocks. To do so, three different
laboratory experiments were conducted. The first was a common scale
experiment using a wave flume, conducted to analyze the stability of
the solitary crushed stones placed on different roughness beds under
non-breaking and breaking wave conditions. The second was similar to
the first but was conducted with lightweight replicas of the crushed
stones. Finally, a circulating water channel (CWC) experiment was
conducted to examine the stability of the same crushed stones as in the
first case, under sinusoidal oscillatory flows with or without currents.
The periods of the oscillatory flow were the same as those of sea waves,
thus corresponding to conditions under which stones smaller than the
actual quarry rocks are placed under the effects of full-scale waves or
with Keulegan–Carpenter numbers (KC) higher than that in the field.
The laboratory experiments were conducted under the assumption that
rocks are placed on flat hard substrates, such as closely packed cobble
and boulder beds or flat bedrock, without overlying sand. This is the
least stable condition of the rocks because the presence of a sand layer
increases the stability of the rocks; however, waves frequently wash
away the overlying thin layer of sand. In addition, an empirical method
for predicting the friction coefficient was also developed to enable
practical use of the proposed formula. Finally, a field stability test was
performed on quarry rocks at shallow coastal sites to demonstrate
whether the developed method provides a reasonable prediction of the
minimum stable mass, compared to the previous formula based solely
on Um.
ψ1 =
μ50 ΔgDn50
where Dn50 = (M50/ρs)
with ρ and ρs being the mass densities of
water and stones, respectively, and where M50 is the median mass of the
stones; μ50 is the median friction coefficient between the stones and the
bed; Δ = (ρs/ρ − 1), defining the submerged specific density of the
stones; and g is the acceleration due to gravity. Note that if the friction
coefficient in Eq. (2) is omitted under an implicit assumption of the
constant friction coefficient, then the ψ1 is equivalent to the well-known
“mobility number” for sediment particles (Brebner, 1980; Nielsen,
1992) or the frequently used “mobility parameter” for rubble-mound
materials (Van Gent and Wallast, 2001; Tørum et al., 2010; Wallast and
van Gent, 2002). However, the hydrodynamic force is due to not only
the drag, which is proportional to the square of the velocity u, but also
the inertia force, which is proportional to the acceleration of the fluid a.
Therefore, the hydrodynamic force is at its maximum before the velocity reaches its maximum. Thus, there might be a better alternative to
ψ1. In the study, the maximum semi-velocity amplitude Ua and the
maximum acceleration amax (Fig. 1) were compared to Um in terms of its
explanatory power in determining the damage ratio of stones on a given
substratum. Two additional mobility indices can be defined using Ua
and amax as the ratios of ρUa2Dn502 and ρamaxDn503, respectively, to
μ50(ρs − ρ)gDn503:
ψ2 =
μ50 ΔgDn50
ψ3 =
μ50 Δg
The successive trough and crest velocities in the wave cycle with
maximum peak-to-peak velocity amplitude are denoted as umin and
umax, respectively (thus Ua = (umax − umin)/2). In case of a sinusoidal
velocity variation, amax = 2πumax/T, and therefore, if Um = umax (as in
most test cases), the ratio of ψ1 to ψ3 is proportional to KC, which is
defined as umaxT/Dn50, where T is the period of an individual oscillation
cycle. However, when the velocity asymmetry increases, the above
equation with respect to amax may be invalid. Instead, umax/Tzp can be
used as a better index for amax, where Tzp is the zero-to-peak period.
Accordingly, the ratio of ψ1 to ψ3 can be assessed as follows.
2. Definition of damage and its governing variables
K C = 4u max Tzp/ Dn50
It is difficult to accurately define the motion threshold of stones
resting on bed roughness elements under the effects of waves with or
without currents. In this study, considering the process of movement
and the design and construction practices of artificial reefs, ‘significant
movement’ or ‘damage’ was defined as the shifting of a stone entirely
out of its initially occupied area after placing it on the bed at a random
position, but with the largest flat surface oriented downward to make
the stone more stable. Such an orientation can readily occur via toppling due to flows, even if the stones are initially placed in unstable
postures. As the oscillatory velocity increases, stones placed on the irregular surfaces of the bed shake at first, occasionally with a slight
slide, and then distinctly move in a sudden manner by either sliding,
rolling, or both. Most of the initial movements cease at once, indicating
that they are primarily attributed to the initially unstable positions after
placement. Additionally, natural flat beds have irregular surfaces.
Therefore, an unexpectedly low flow velocity may lead to a small shift
even in large rocks. Thus, the above definition of the movement relative
to the size of the stones is preferred to an absolute definition.
The mobility index among ψ1, ψ2, and ψ3 that is most closely related
to the damage ratio will be used to predict the stone stability. However,
if there are considerable systematic deviations in the relationship between the best explanatory index and the damage ratio, the residual
components may be a function of KC. An attempt will be then made to
establish a better predictor to determine the damage ratio by multiplying the candidate by possible functions of KC.
3. Laboratory model experiments
3.1. Experimental setups and procedures
Stone stability was examined under various physical conditions
(Table 1) using a wave flume and a CWC at the National Research Institute of Fisheries Engineering. First, scale model experiments were
conducted to determine the stability of the stones under the effects of
waves in a 70 m long, 0.7 m wide, and 2.2 m deep wave flume with a
smooth sloping bottom made of cement mortar (Fig. 2). A 210 cm long
Coastal Engineering 140 (2018) 257–271
S. Kawamata et al.
Fig. 1. Schematic example of velocity and acceleration time-series showing the definition of the characteristic velocity quantities. Although in this example the wave
cycle with Um differs from that with Ua, both waves were the same in most test cases (thus Um = umax).
stones. They were tested by placing them apart from each other on the
bed in slightly different positions but with the largest surface area oriented downward and the largest projected side area oriented normal to
the wave direction. Subsequently, 7–13 waves were generated using a
piston-type wavemaker with given periods and amplitudes, and ‘significantly moved’ stones were counted (Fig. 4). The orientation of the
stones helped in testing the lowest potential stability in the given direction, considering the uncontrolled horizontal direction of the rocks
relative to the variable wave directions. For wave periods (T) of 2.0,
2.5, 3.0, and 3.5 s, the wave height was increased by 1 or 2 cm from
values at which no stones were ‘damaged’, up to the wave maker capacity.
Although numerous studies have examined rock stability under irregular waves, a model for predicting stability was developed in the
current work without irregular wave experiments by using a different
approach. The previous studies assumed that the probability of instability due to random waves could be represented with central location statistics, such as significant wave height and average wave period,
so that neither the highest wave height nor the detailed waveform of
velocity vs. time was taken into consideration. In the current research,
it is assumed that whether waves are random or not, the damage ratio
due to a wave train can be determined solely by a single wave exerting
a maximum hydrodynamic force on the rocks. Thus, only the maximum
and 70 cm wide test bed was set on top of a gentle slope (1:50) area
(i.e., cement mortar bed, referred to as bed CM) and then on a layer of
rounded coarse gravel (the median surface grain size B50 = 34.2 mm)
or medium gravel (B50 = 8.5 mm) was closely packed and glued using a
silicon sealant onto 5-mm thick polyvinyl chloride boards (referred to
as beds CG1 and MG1, respectively). The test beds helped to simulate
different surface roughnesses of the natural seabed. The still water
depth at the center of the test bed was 56 cm. The model scale was set to
1:15, providing the largest scale model that could generate waves large
enough to damage the scale model with a 1 t rock even on coarse gravel
beds with the highest friction.
One hundred crushed granite stones (Fig. 3) were used as scale
models of irregularly shaped quarry rock weighing approximately 1 t to
empirically determine the damage ratio under the given physical conditions. The main characteristics of the stones include masses ranging
from 232 to 358 g with M50 = 293 g, ρs = 2.62 g/cm3, and
Dn50 = 4.82 cm. In addition, a set of lightweight duplicates
(M50 = 161.5 g and ρs = 1.37 g/cm3) of the 100 crushed stones was
used, prepared by modeling each stone using gypsum and then casting a
mixture of ceramic mortar (Eagle 8, Eaglevision Co., Tokyo) and
lightweight foam aggregates (TS sand, Sekisui Plastics Co. Ltd., Osaka,
Japan) into the gypsum mold.
The 100 test stones were divided into 10 sets, each comprising 10
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S. Kawamata et al.
Table 1
Experimental conditions.
Case name
T (s)
Number of sub-cases
Wave flume experiment
Circulating water channel experiment
Um (cm/s)
umax − umin
Re = umaxDn50/ν (104)
Test bedb
Test stonesc
Hmax: Average of the highest wave heights measured within the test section.
CG1: large rounded coarse gravel bed (B50 = 34.2 mm), MG1: rounded medium gravel bed (B50 = 8.5 mm), CM: smooth cement mortar bed, MG2: subangular
medium gravel bed (B50 = 8.9 mm), CG2: small rounded coarse gravel bed (B50 = 26.5 mm).
CGS: 100 crushed granite stones (M50 = 293.0 g and ρs = 2.62 g/cm3), LWD: 100 lightweight duplicates (M50 = 161.5 g and ρs = 1.37 g/cm3).
μ50: median of friction coefficients shown in Fig. 8.
characteristic velocities of wave trains were used for analysis. Note that
the waves produced in this study were variable rather than regular (as
described below), so the maximum characteristic velocities of interest
differed from the mean wave velocities.
The near-bed velocity was recorded using three-dimensional Nortek
Vectrino acoustic Doppler velocimeters (ADVs). The sensing volumes of
the ADVs were set to be 7 cm above the bed surface to allow for stable
velocity measurement. In the preliminary ADV measurements, the
vertical velocity profile between 12.5 and 80 mm above the bottom was
nearly constant, even for the roughest bed (i.e., bed CG1), but with
more frequent contamination by spike noise closer to the bottom. The
velocity records were obtained at 50 Hz in three locations: in the midcross-section of the test-bed section, and 75 cm offshore and inshore
from there. Each velocity time series was low-pass filtered at a cutoff
frequency of 2 Hz to determine Um as the maximum velocity in the
wave train and three characteristic quantities of the oscillatory velocity,
being Ua, Tzp, and amax, for the individual wave (defined by the zero
down-crossing method) with the maximum peak-to-peak velocity amplitude. Despite a regular piston motion, the resultant wave trains were
not completely regular and, in many cases, showed significant
Fig. 3. The set of 100 crushed granite stones used for laboratory experiments.
Fig. 2. Wave flume dimensions (in meters) and location of measurement instruments in the test section on the slope. WHM: resistance type wave gauge.
Coastal Engineering 140 (2018) 257–271
S. Kawamata et al.
Fig. 4. Examples of top views of 10 crushed stones deployed over a test bed before and after wave action in the wave flume experiment. The crushed stones indicated
by circles are judged to be ‘damaged’.
patterns of the oscillatory flows: symmetric (umax ≈ –umin) and asymmetric (umax ≈ –2umin) oscillatory flows with a constant oscillation
period of 10 s on bed MG2; and asymmetric oscillatory flows with
constant oscillation periods of 8 s and 12 s, and a stepwise increase in
only umax but nearly constant umin on bed CG2. The asymmetric oscillatory flows had a net transport, which helped in simulating the combined effect of the waves and currents. The freestream velocity was
measured 20 cm above the bed at the center of the test section using an
ADV at 10 Hz.
fluctuations due to long-period oscillations (Fig. 5). Thus, the wave
with Um did not always coincide with that with Ua or amax. The values of
Um, Ua, amax, and Tzp obtained using the three ADVs were averaged for
The stone stability in an oscillatory flow with or without current
was examined using the same set of crushed stones as in the CWC
(Fig. 6). The CWC is equipped with a computer-controlled, reversible
impeller, using which an arbitrarily time-varying velocity was produced
in a fully enclosed 3 m (length) × 0.4 m (width) × 0.4 m (height) test
section. However, unlike the wave flume, near-sinusoidal oscillatory
flows with stable velocity amplitudes of > 1 m/s can be produced for
longer wave periods (≥8 s) but not for similarly shorter periods because of the cavitation at the impeller. The crushed stones in such
periods of oscillatory flows relative to the orbital amplitude of the fluid
are smaller than the actual quarry rocks, thus obtaining a KC value
higher than expected without increasing the relative effect of viscosity
on the hydrodynamic forces. Therefore, if an empirical formula can
describe our experimental data overall, it can be expected to be applicable over a wide range of field conditions through interpolation.
The inner bottom surface of the test section was covered using subangular medium gravel (MG2) or rounded coarse gravel (CG2) bed. The
critical velocities at which crushed stones were initially ‘damaged’ were
examined for all the crushed stones (selected in sets of six or fewer) as
follows: (1) The stones were arranged in the same manner as in the
wave flume experiment in a test-bed area located 50 cm from both ends
and 10 cm from the side walls of the test section; (2) Oscillatory flows
were generated with predetermined time variations and a constant
period but with an increase in the velocity amplitude and mean flow
velocity (Fig. 7); and (3) The initial displacement time of each stone
was recorded to later determine the characteristic flow quantities (Um,
Ua, amax, and Tzp) of the oscillation cycle at the time of 'damage'. Lowpass filtering at a cutoff frequency of 0.5 Hz was applied to the velocity
data prior to the determination of the quantities. Unlike the wave flume
experiment, once the stones began to displace, they travelled much
further compared to their sizes. Consequently, some stones came into
contact or collided with other displaced stones before moving with the
flow. In such cases, the stones were re-examined. The stability of the
crushed stones was examined in four cases with different time-variation
3.2. Determination of friction coefficients
The friction coefficients for the test stones on each test bed were
determined from the friction angles θ as tan θ. The friction angles were
measured with a portion of the test bed fixed on a tilting frame. The 100
test stones were each individually placed on the sample bed at random
positions but with the same orientation as in the above experiments.
Subsequently, the frame was tilted very slowly until the stones moved
more than one stone diameter. The maximum tilting angle before
movement was measured to the nearest 0.1° using a digital inclinometer, which was attached to the tilting frame. The measurement
was repeated three times for each stone by changing its position.
Fig. 8 shows the cumulative frequencies of the friction coefficients
for the test stones resting on the test beds for all cases. The median and
standard deviation of the coefficient of friction on the even surface of
bed CM in W3 were the lowest, as expected. However, for a small
percentage of stones (e.g., 10%), the coefficients of friction on beds
MG1 and MG2 in W2, W4, R1, and R2, which had higher surface
roughnesses, were lower than that measured on bed CM.
3.3. Analysis of explanatory variables for damage ratio
The relationships described by (1) Um vs. rd, (2) Ua vs. rd, and (3)
amax vs. rd were compared to determine the best explanatory variable
for the damage ratio. It should be noted that the damage ratio calculated from the 100 samples varies to some extent because of the relatively small number of samples. For example, when 10 stones are ‘damaged’ among 100 samples, rd = 10%; however, assuming the same
Coastal Engineering 140 (2018) 257–271
S. Kawamata et al.
Fig. 5. Examples of the velocity time series observed in the wave flume experiment, where x is the inshore distance from the center of the test-bed section.
4. Development of prediction model
probability of damage among the individual stones, a binomial distribution suggests that the 95% confidence interval of the true rd ranges
from 4.9% to 17.6% (Zar, 1999). Therefore, the 95% binomial confidence intervals were calculated to check the variability of the damage
ratio. Furthermore, for all the cases, the three dimensionless mobility
indices ψ1, ψ2, and ψ3 were compared. If even the best-correlated mobility index failed to adequately account for the damage ratio, a new
dimensionless quantity was established by modifying the mobility
index using an empirically determined function of KC.
4.1. Model for stability of rocks
Fig. 9 shows the damage ratio with respect to the characteristic flow
quantities Um, Ua, and amax for each case. The solid line represents a
flexible and robust Bayesian spline regression (Ishiguro and Arahata,
1982) to show a trend expected to monotonically increase. Although
any characteristic flow quantities did not have a regression line that
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Fig. 6. Schematic of the circulating water channel.
Fig. 8. Cumulative frequency of the friction coefficient μ for each combination
of test stones and test beds. The abbreviations in the legend are listed in Table 1.
passed through all the 95% confidence intervals of the data points, the
damage ratio, overall, was closely related to Um and Ua, and less so to
amax. On closer inspection, however, it was apparent that in case W1
with highest velocity skew umax/(umax − umin) in the wave flume experiment (Table 1), the damage ratio remained closely related to Ua but
not to Um.
Fig. 10 shows the variation in the damage ratio using the mobility
indices ψ1, ψ2, and ψ3 for all cases. There were large differences between
cases, particularly between the wave flume and CWC data for ψ3. In
contrast, the differences were minimal for ψ2. Furthermore, the slopes
of the imaginary trend lines in the individual cases for ψ2 are similar to
those of a sigmoid curve, whereas the slopes of the curves in the individual cases for ψ1 and ψ3 decrease with the increase in the magnitude
Fig. 7. Time variations in the velocity generated in the four cases of the CWC
experiment. Graphs on the right show enlarged views of the rectangular portions indicted in the graphs on the left.
Coastal Engineering 140 (2018) 257–271
S. Kawamata et al.
factor for ψ2 that fit the data, using the Marquardt nonlinear leastsquares algorithm. Consequently, it was found that the damage ratio
could be even better represented by the following sigmoid function of
the dimensionless predictor ϕ, which is a function of KC and ψ2:
of the indices. In terms of a predictor for the damage ratio, however, the
remaining deviations in ψ2 are still considerable, especially between the
wave flume and CWC data. The deviations in the correlation between ψ2
and rd might be attributed to the incomplete description of the maximum hydrodynamic force with ρUa2Dn502. A correction factor was
therefore introduced to establish a better predictor than ψ2. From previous studies (Chakrabarti and Armbrust, 1987; Grace and Zee, 1981;
Keulegan and Carpenter, 1958), the ratio of the maximum wave force to
ρUa2Dn502 can be expected to be a decreasing function of KC. This is
consistent with an apparent tendency of the damage threshold for ψ2 in
each case to be larger with larger values of KC (see Table 1 and Fig. 10).
Therefore, several functions of KC were tested to determine a correction
rd = exp { −exp[−b (ϕ − c )]}
ϕ = (α − ln K C ) ψ2
where α = 8.66, b = 0.608, and c = 4.17 are experimental constants
determined using the nonlinear least squares regression (R2 = 0.849, 1
standard error (SE) = 9.5%, and n = 183). Fig. 11 shows the scatter
Fig. 9. Variation in the damage ratio rd with respect to Umax, Ua, and amax for each case. Error bars indicate the 95% confidence intervals, and solid lines indicate
spline regression lines.
Coastal Engineering 140 (2018) 257–271
S. Kawamata et al.
Fig. 9. (continued)
plots of rd vs. ϕ along with the regression Equation (6). The KC value
ranges from 6.0 to 364.5 so that the KC correction factor, 8.66 – ln KC,
varies from 6.87 to 2.76. The correction factor decreases in an unbounded manner for higher KC (smaller stones or longer period waves);
however, it should be noted that the KC value at which the factor
reaches 0 is extremely high (approximately 5770). Overall, the regression results show a reasonable agreement with the experimental
results, with reduced discrepancy between the wave flume and CWC
The physical model assumes that the predominant forces determining rock stability are the gravity and the inertial force, which is
proportional to the velocity squared. In small physical models, however, the viscous force proportional to the velocity can be large relative
to the inertial force, leading to a scale effect. The scale effect may be
evaluated by a Reynolds number, which represents, in physical terms,
the ratio of the inertial force to the vicious force. The Reynolds number
Re is defined as UmDn50/ν, where ν is the kinematic viscosity of the
fluid, ranging from 8 × 103 to 7.2 × 104 (Table 1). As discussed later,
these Reynolds numbers are high enough to expect correct scaling of
the forces.
If the critical damage ratio rd, cr is considered as the maximum damage ratio for practical design, the corresponding critical value ϕcr can
be obtained as follows.
ϕcr = c − ln (−ln rd, cr )/ b
If rd,
cr = 0.1,
which has typically been applied in design in Japan,
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Fig. 10. Relationships between the three mobility indices and the damage ratio.
Fig. 12. Relationship between the ratio of the stone diameter to the median
surface bed diameter and the median friction angle. D = Dn50 in the present
study. The horizontal dashed line represents the value for the smooth bed CM,
which is assumed to be the asymptote of the regression model.
Fig. 11. Relationship between ϕ and damage ratio rd. The curved line indicates
the results obtained using the nonlinear least squares regression Equation (6a).
The red circle with an error bar indicates the result of the field test off the
Shiwagi coast, where the error bar indicates the 95% confidence interval. (For
interpretation of the references to colour in this figure legend, the reader is
referred to the Web version of this article.)
4.2. Model for friction coefficients
The friction coefficient needs to be estimated in order to apply the
above model to determine the stability of rocks. As it is difficult to
measure the friction coefficient of quarry rocks on the seabed in situ, it
was estimated instead from the ratio of the quarry rock size to the bed
grain size as follows. Kirchner et al. (1990) showed that the median
friction angle of a single stone on a rough bed could be described using
the following empirical relationship.
Eq. (7) gives ϕcr = 2.80. The minimum stable rock mass corresponding
to the critical damage ratio can be obtained from the following equation
by substituting Eqs. (3) and (5) into Eq. (6b).
⎛α − ln u max Tzp ⎞ Ua
= ϕcr
⎠ 50
θ50 = 55. 2(D / B50)−0.307
Mcr = ρs Dcr3
where θ50 is the median friction angle in degrees, D is the diameter of
the stone, and B50 is the median surface grain size of the bed. They
measured the intermediate axis of randomly selected grains using a
caliper to determine B50. However, such a method cannot be applied in
a field environment. Alternatively, B50 was determined from the cumulative area-weighted frequency of the surface grains with short axis
diameters, which can be easily determined by processing photographic
where Dcr and Mcr are the nominal diameter and mass of the rocks at the
critical damage ratio, respectively. The value of Dcr can be obtained
from Eq. (8) using an iterative calculation procedure, such as the
Newton's method.
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tide station, Awayuki (33°46′N, 134°36′E), was 1.84 m.
images of the seabed. For D = Dn50, a considerable difference was found
between the experimental data and Eq. (10) (Fig. 12). The cause of this
difference is unknown, though there was a slight difference in the way
the stones were placed; Kirchner et al. (1990) placed individual test
stones on a horizontal bed by dropping them from above to randomize
not only the location but also the orientation of the individual stones,
whereas in the current work, individual stones were placed on a horizontal bed at random positions but with a specified orientation. In
addition, they pointed out that Eq. (10) could underestimate the friction
angles for a smooth, flat bed (thus ultimately resulting in a high value of
Dn50/B50), because θ50 should approach some non-zero value close to
that of bed CM. Therefore, a modified empirical model was introduced
to predict θ50, with an asymptotic value equal to the median friction
angle of bed CM (θ50 = 32.5°).
θ50 = 32.5 + β exp(−γ Dn50 / B50)
5.2. Observation procedure and results
The field observations began on July 29, 2015, aiming to determine
the critical velocity at which the quarry rocks began to move due to
typhoon waves. An MSR Electronics data logger (MSR145) was attached to each quarry rock to sense its three-axis accelerations at 5 Hz
such that it would record accelerations that were 0.5 G (where
1 G = 9.80665 m/s2) higher or lower than the initial value, along with
the time. At the same time, a wave meter (WH-501, IO Technic Co. Ltd.,
Machida, Tokyo) with a pressure gauge and a two-dimensional electromagnetic velocimeter (EMV) was placed on the seabed near the
quarry rocks to measure the pressure and velocities at 2 Hz for 20 min
every 1 h. The pressure and velocity sensors were approximately 57 and
66 cm above the seabed, respectively. The height at which the velocity
measurement was taken relative to the water depth, 0.66/9.6 ≈ 0.07,
was smaller than that in the wave flume experiment, 7/56 = 0.125, so
that the velocity measurement could be regarded as the near-bed velocity. However, prior to the observation, 2 of the 10 quarry rocks had
already fallen on their sides (thus they were moved outside their initially occupied areas and ‘damaged’; Table 2) because of severe waves
(hereafter referred to as Storm Waves I) associated with Typhoon 1511,
which passed near Shiwagi on July 16, 2015. Severe breaking waves
(hereafter referred to as Storm Waves II) occurred again during August
22–23, 2015, because of two strong typhoons (Typhoons 1515 and
1516) 1100–1700 km away from the study site in the Pacific Ocean. The
significant wave height reached 4.5 m, with a spectral peak period of
17.4 s. The measurement of the distances between the quarry rocks and
the benchmarks set on 10 t armor blocks (2.6 m × 2.6 m × 1.5 m)
placed approximately 2–4 m away from the rocks, and the top-view
images of the quarry rocks (Fig. 14) show no shift or only a slight shift
in the rocks after Storm Waves II, with the maximum displacement of
the rocks being 8 cm (Table 2). Although the quarry rocks should have
become more stable because of the sideways toppling of the two rocks
during Storm Waves I, the rocks were displaced a little further because
of Storm Waves II. It can be inferred from the above facts that if the
quarry rocks had first encountered Storm Waves II, the damage to the
rocks would have been similar to that inflicted by Storm Waves I, or in
other words, the damage ratio would have been 0.2.
As wave oscillatory velocities during Storm Waves II exceeded the
maximum measurable velocity of the EMV (3 m/s), the velocity time
series was estimated from the pressure data using linear theory spectral
transformation (Guza and Thornton, 1980). Fig. 15 shows the time
variation in the wave oscillatory velocity estimates when maximum
peak-to-peak velocity amplitude occurred during Storm Waves II. The
estimated wave oscillatory velocity largely agrees with the measured
velocity component in the principle wave direction, which is defined as
the direction in which the variance in the velocity component is maximum. In addition, the EMV may output a lower value because of
biofouling on the probe, whereas the pressure measurement is hardly
affected by biofouling. Thus, the wave oscillatory velocity time series
during Storm Waves II can be expected to be reasonably estimated from
the pressure. Fig. 15 shows that umax = 3.53 m/s, umin = −1.47 m/s,
and Tzp = 2.0 s, therefore, Ua = 2.50 m/s.
where β and γ are experimental constants. To determine B50, the topview images of the test beds except for bed CM were considered. Subsequently, the horizontal short-axis diameters (assumed to be identical
to the intermediate-axis because sediment grains are normally settled
such that the shortest-axis is directed vertically) and surface areas of
100–160 randomly sampled grains were analyzed for each bed using
the AreaQ image processing software (Kawamata, 2011). The nonlinear
least squares regression of Eq. (11) performed on the current data gave
β = 10.8 and γ = 0.230 with R2 = 0.916, SE = 0.809, and n = 5. The
regression, shown in Fig. 12, is statistically significant (F1,3 = 33.2,
P = 0.010).
5. Comparison with a field experiment
5.1. Test rocks and site
A field experiment was conducted to determine the rock stability of
10 samples of granite quarry rocks at an exposed site off the Pacific
coast of Shiwagi, Minami, Tokushima Prefecture, Japan (33°46.91′ N,
134°36.18′ E). The rocks were visually classified as “2–3 t class” rubbles, which have been generally used to create seaweed beds in
Tokushima Prefecture. Their masses, measured using a 40-t load cell,
ranged from 1.16 to 3.01 t with M50 = 2.04 t (Table 2). As in the laboratory experiments, each quarry rock was placed on the ground with
its maximum surface area oriented downward. An inverted U-shaped
reinforcing steel bar with a diameter of 19 mm was embedded and fixed
using chemical anchors onto the top surface of the rock. The rocks were
then placed by hanging then from the inverted U-shaped bar at intervals
of approximately 2.5 m on the seabed off the Shiwagi coast on December 25, 2014. The deployment site was approximately 9.6 m below
mean sea level and was a relatively flat rocky bottom covered with a
thin layer of sand (Fig. 13). The mean spring tidal range at the nearest
Table 2
Observations of the quarry rocks in the field test.
Rock No.
Mass (t)
Beginning situationa
Displacementb (cm)
no change
fell on its side
no change
fell on its side
no change
no change
no change
no change
no change
5.3. Comparison between measurement and prediction
The mass density of the quarry rocks was determined to be
2592 ± 4.8 kg/m3 (mean ± 1 SE, n = 4) using a water immersion
method with the help of debris samples (dry mass in the range of
1.513–2.836 kg) collected from the same stockpile as the rocks. Hence,
Dn50 = (M50/ρs)1/3 = 0.923 m. The short surface grain diameters of the
seabed in the experimental site off the Shiwagi coast was measured by
processing the photographic images of the seabed. The results show
that B50 = 52 cm (n = 89). The substitution of these values into Eq.
Posture at the onset of the observation period (July 29, 2015 after Storm
Waves I).
Change in the measurements from benchmarks on September 4, 2015 (after
Storm Waves II).
Coastal Engineering 140 (2018) 257–271
S. Kawamata et al.
Fig. 13. Underwater images taken one month after deploying the 10 quarry rocks on the test site off the Shiwagi coast. The term “Nos.” indicates the identification
numbers of the rocks.
Fig. 14. Top-view images of the 10 quarry rocks taken before (July 29, 2015; upper image) and after (September 4, 2015; lower image) Storm Waves II. The two
arrows in the pair of images indicate the same direction and position of the rocks.
(11) gives θ50 = 39.2°, and thus μ50 = tanθ50 = 0.81. Assuming
ρ = 1025 kg/m3, the observed maximum velocity oscillation with
umax = 3.53 m/s, Tzp = 2.0 s, and Ua = 2.5 m/s leads to KC = 4umaxTzp/
Dn50 = 30.6 and ψ2 = Ua2/(μ50ΔgDn50) = 0.559, and thus ϕ = 2.93.
This ϕ-value was plotted against the previously mentioned rd-value
(0.2) in Fig. 11 as a red circle. The small sample size (n = 10) led to a
large 95% confidence interval for the true rd (indicated by an error bar),
which was calculated based on the binomial distribution. Equation (6)
and the above ϕ-value predicts an rd = 0.12, which is close to the observed value. The critical stable masses for rd,cr = 0.1 (0.2) can be
predicted under the same external force conditions as above. Equation
(8), when solved using the Newton's method, leads to Dcr = 0.974
(0.769) m and thus Mcr = 2.40 (1.18) t. The predictions agree well with
a wide range of test rock masses, compared to the minimum stable mass
of the isolated rocks determined using the conventional method, i.e.,
Eq. (1) with C = 25, at 46.0 t.
6. Discussion
Most of the previous methods for predicting the minimum stable
stone mass are based on the assumption that the maximum hydrodynamic force acting on the stones is proportional to the square of the
Coastal Engineering 140 (2018) 257–271
S. Kawamata et al.
Fig. 15. Time variation in the pressure-estimated orbital velocity (solid line) and measured velocity component in the principal wave direction (circles) for a
maximum velocity at the test site off the Shiwagi coast. The time series data were obtained from 01:00 am, August 22, 2015.
“regular” waves or oscillatory flows with or without currents, wherein
the damage (if any) to individual stones is determined using only the
largest peak-to-peak velocity amplitude in the experienced cycles of
oscillatory flow. The wave trains generated in wave flumes sometimes
included far larger waves, and Ua sometimes occurred in a wave different from that with the maximum velocity Um (thus umax < Um).
Nevertheless, the damage ratio was closely related to Ua or half the
maximum peak-to-peak velocity amplitude, supporting the above assumption.
The experiments in the present study employed the largest stones
possible to minimize the scale effect and therefore did not use stones of
different size groups. The experiments were conducted instead with
different larger scales of oscillatory flows. In general, the maximum
force coefficient for an object in an oscillatory flow decreases with the
increase in KC (Chakrabarti and Armbrust, 1987; Grace and Zee, 1981;
Keulegan and Carpenter, 1958). This was consistent with the result
indicating that the predictor ϕ for the damage ratio of stones was a
decreasing function of KC. Additionally, KC can be defined, from a
physical perspective, as the ratio of the moving distance of a water
particle in a half cycle of the oscillatory flow to the object diameter. The
reasonable goodness-of-fit of the resulting model, i.e., Eq. (6), may
support the assertion that the model is applicable over a wide range of
stone sizes.
With respect to the scale effect on the rock stability, no apparent
dependence of the damage ratio on the Reynolds number was found in
the laboratory tests. The scale effect due to the viscous force may become negligible at a high Reynolds number. Although no studies regarding the critical Reynolds number for isolated rocks have been
conducted, the Reynolds number in this study (8 × 103–7 × 104) was
higher than the lower limit of the critical values for the armor layer of
rubble-mound structures, which range from 6 × 103–4 × 105 (Hughes,
1993). The stability of the armor layer depends on the scale effect due
to the viscous force of flow through the porous structure (Hughes,
1993). This leads to the expectation that no scale effects would occur in
experimental tests on isolated rocks. The Reynolds number, defined as
umaxDn50/ν, was 3.3 × 106 for the maximum wave-induced velocity
estimate from the field test. This value was higher than the Reynolds
number determined in the laboratory test by approximately two orders
of magnitude. However, a reasonable agreement was found between
the predicted stability and the observed stability in the field. Therefore,
crest velocity, but independent of the preceding trough velocity (Van
Gent and Wallast, 2001; Tørum et al., 2010; Wallast and van Gent,
2002; Lorang, 2000; Goto et al., 2009). However, with the increase in
velocity asymmetry, the semi-velocity amplitude Ua exhibited a more
explanatory power for the stability of the stones than the maximum
crest velocity Um, suggesting that the hydrodynamic forces increase not
only with the crest velocity but also with the preceding trough velocity.
This is consistent with the hydrodynamic force measurements conducted on marine pipelines, wherein the maximum force in a given
velocity half cycle has been found to strongly depend on the magnitude
of the velocity in the preceding half cycle because of the wake effect
(Lambrakos et al., 1987). The force coefficients normalized with the
square of the crest velocity substantially increase with the increase in
the absolute ratio of the preceding trough velocity to the crest velocity.
The wake effect has been modeled to more accurately predict forces due
to waves, even with superimposed steady current, acting on marine
pipelines (Lambrakos et al., 1987). However, it is difficult to develop a
“wake force” model of irregularly shaped stones because it requires
introducing numerous additional hydrodynamic coefficients and determining the coefficients from direct measurements of the forces acting
on an object. To avoid such difficulty in modeling the hydrodynamic
forces, the developed model given by Eq. (6) simply takes Ua as a representative velocity instead of Um, leading to a substantial improvement over the predictions obtained using the conventional method.
The above finding is of great importance in the establishment of a
reasonable formula for rock stability under asymmetric velocity flow
due to shallow waves because the minimum stable mass is fundamentally proportional to the sixth power of the representative velocity. For
example, in the field test, taking Ua = 2.5 m/s as the representative
velocity instead of the maximum velocity (= 3.53 m/s) leads to the
reduction of the minimum stable mass by a factor of (2.5/
3.53)6 = 0.13. However, this finding is contrary to the well-known
Morison equation, which expresses a time-varying wave force as the
sum of the inertia force proportional to a and the drag force proportional to u2. Thus, for example, if the velocity skew umax/(umax − umin)
increases with an increase in umax and an even more substantial decrease in |umin| (thus amax increases but Ua decreases), then the maximum force prediction from the Morison equation should increase while
that predicted using Ua will decrease.
The proposed model is based on the stability test conducted under
Coastal Engineering 140 (2018) 257–271
S. Kawamata et al.
obtained using the conventional method, Eq. (1), was considerably
For the proper application of Eq. (6), it is necessary to determine
three characteristic quantities of the wave-induced near-bed velocity,
Ua, umax, and Tzp, as well as μ50. The three velocity quantities could be
predicted using scale model experiments or numerical models that can
simulate irregular wave trains, and μ50 could be predicted using photogrammetry of the surface geometry of the seabed using Eq. (11). The
reasonable prediction of rock stability using Eq. (6), however, would
still require a novel, reliable method to predict Ua and its associated
parameters umax and Tzp under the action of random waves.
it might be concluded that the developed equations are valid for predicting the stability of isolated quarry rocks in the field.
The equations for rock stability, Eqs. (6) and (11), assume rocks on
relatively flat beds whose roughness height is apparently lower than the
rocks. If the bed roughness increases relative to the rocks size, the rocks
might be more deeply embedded and thus more stabilized. In such a
case, stabilization might also be enhanced by the accompanying decrease in rock protrusion (thus a decrease in drag). The empirical Eq.
(6) also assumes a single isolated rock or an array of rocks spaced
sparsely enough to eliminate flow interference. By analogy with the
maximum wave force on a slender pile group (Bonakdar et al., 2015),
the wave force on a rock within a more closely spaced rock array can
change from that on a single isolated rock, depending on the spacing
relative to the rock diameter, the Keulegan–Carpenter number, rock
arrangement, etc. In the field test, quarry rocks were arrayed in a 5 × 2
arrangement and directed obliquely offshore. The relative spacing was
approximately 1.7 times the rock size, which was considerably closer
than in the laboratory tests. Nevertheless, the interference effect in the
field test was likely small not only because the change in the maximum
force due to the side-by-side or oblique (referred to as “staggered” in
reference (Bonakdar et al., 2015)) arrangement of slender piles is typically less than 10% for a relative spacing greater than 1.5, but also
because the low height of the rocks may enable over-the-top flow, resulting in less of an effect on the surrounding side flows than long
vertical cylinders like piles that force all flow around the sides.
The developed method is more complex than the conventional
method, requiring three characteristic quantities of near-bed oscillatory
velocity (Ua, umax, and Tzp) as well as the median friction coefficient
between the quarry rock and the seabed. With respect to the friction
coefficient, an empirical equation for the friction angle, Eq. (11), was
proposed in the present study. As the bed surface grain size can be
easily obtained from photogrammetry of the surface geometry of the
seabed, the friction angle and therefore the friction coefficient can be
estimated. Previously, flow quantities have been studied as characteristic properties, describing the asymmetry of near-bed orbital velocities
under shoaling and breaking waves (Dibajnia et al., 2001; Elfrink et al.,
2006; Tajima and Madsen, 2002). However, no empirical formulae
have been validated in terms of predicting the waveform from the
maximum peak-to-peak velocity amplitude (thus Ua) under irregular
wave trains. Reliable prediction of Ua is particularly important for
reasonable prediction of rock stability using Eq. (6) because the
minimum stable mass is proportional to Ua6 and because Ua might not
be as large as umax due to the effect of the sea bottom on the waveform
in shallow waters. Further study is required to address this issue. At the
moment, the only available techniques for waveform prediction may be
scale model experiments and numerical models, such as the extended
Boussinesq-type model BOUSS-2D (Nwogu and Demirbilek, 2001) and
the non-hydrostatic model SWASH (Zijlema et al., 2011), which can
efficiently compute the propagation and transformation of waves over
large-scale coastal areas from offshore to shore, and can accurately
reproduce the variations in waves and wave-driven currents.
This study was performed within the framework of the research
project “The sophistication of the methods for designing facilities of
fishing ports and grounds,” funded by the Fisheries Infrastructure
Development Project, Japan Fisheries Agency. We thank Akihiko
Moriguchi for assistance with laboratory work. We also thank Editage
( for English language editing. We are grateful to two
anonymous reviewers for their useful comments that helped improve
the manuscript.
List of symbols
acceleration of fluid
maximum acceleration of fluid in the individual oscillation
cycle with Ua
experimental constant in Eq. (6)
median bed surface grain size
experimental constant in Eq. (6)
coefficient of critical stable mass in Eq. (1)
size of stone on the bed
critical value of Dn50
Dn50=(M50/ρs)1/3 median nominal diameter of stone
acceleration due to gravity
Keulegan–Carpenter number, defined by Eq. (5)
median stone mass
critical stable mass of stone
damage ratio
critical damage ratio
coefficient of determination
Re=umaxDn50/v Reynolds number
time varying velocity
maximum velocity at the crest in the individual oscillation
cycle with Ua
minimum velocity at the trough in the individual oscillation
cycle with Ua
maximum semi-velocity amplitude
maximum peak velocity
period of individual oscillation cycle
zero-to-peak period in the individual oscillation cycle with Ua
experimental constant in Eq. (6b)
experimental constant in Eq. (11)
experimental constant in Eq. (11)
Δ=ρs/ρ − 1 submerged specific density of the stone
friction angle
median friction angle
μ=tanθ friction coefficient
median friction coefficient
kinematic viscosity of the fluid
mass density of the fluid
mass density of the stone
mobility index defined by Eq. (2)
mobility index defined by Eq. (3)
mobility index defined by Eq. (4)
7. Conclusions
The results of the wave flume and CWC experiments show that the
damage ratio of isolated rocks was most closely related to ψ2 based on
Ua, and not to ψ1 based on Um. The finding is particularly important for
the accurate prediction of stable rock mass under the effects of asymmetric waves in shallow waters. There remained significant discrepancies in terms of the relationship of the damage ratio, with respect
to Ua, between the wave flume and CWC experiments; however, the
introduction of the correction factor expressed using a function of KC
into the ψ2 equation helped to obtain the empirical formula, Eq. (6), the
results of which were largely in good agreement with the experimental
results. A field experiment showed that the proposed formula could
provide a reasonable estimate of the stable mass, whereas the mass
Coastal Engineering 140 (2018) 257–271
S. Kawamata et al.
predictor defined by Eq. (6b)
critical ϕ value corresponding to rd,cr
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