NEW TECHNOLOGY FOR ESTIMATING PLUNGE POOL OR
R. J. Wittler1, G. W. Annandale2, S. R. Abt3, J. F. Ruff4
The Dam Foundation Erosion Study (DFE) is an ongoing cooperative dam safety
research study between the US Bureau of Reclamation, Pacific Gas & Electric (PG&E),
Electric Power Research Institute, Western Area Power Administration, Niagara Mohawk,
the US Federal Energy Regulatory Commission, Colorado State University, and Golder
Associates. The DFE study includes tests with two objectives. The current objectives are 1)
to develop hydraulic equations to calculate the erosive power of water in plunge pools, and
2) to develop a technique that for estimating scour depths in earth materials, including rock.
The basis of the latter is the Erodibility Index Method . There are three facets to the study:
a scale model, a prototype model, and a numerical model. This paper summarizes some of
the tests in the prototype model. The studies conclude that power is a good predictor of
depth of scour, and that the erosion threshold is a function of the erodibility index. The
primary objective of the study is to create a numerical model that simulates the erosion in the
foundation areas of dams under overtopping conditions.
This paper first presents current technology for estimating scour depth. Second, this
paper presents a description of the proposed, new technology. The paper describes the
experimental facility, accompanied by a summary of the experimental results. Scour depths
estimated with the current and proposed technology are subsequently compare to the
CURRENT TECHNOLOGY FOR EROSION PREDICTION
The current equations most often used to calculate plunge pool scour are the
Veronese, Mason, and Yildiz equations. Equation 1 is the Veronese  equation. The
equation yields an estimate of erosion measured from the tailwater surface to the bottom of
the scour hole.
Ys = 1.90H 0.225q 0.54 (1)
Ys = depth of erosion below tailwater (meters)
H = elevation difference between reservoir and tailwater (meters)
q = unit discharge (m3/s/m)
Yildiz  presents a modified version of the Veronese equation, including the angle, α, of
incidence from the vertical, of the jet.
1 Research Hydraulic Engineer, US Bureau of Reclamation, Denver, Colorado, USA.
2 Director, Water Resources Engineering, Golder Associates, Lakewood, Colorado, USA
3 Associate Dean for Research, College of Engineering, Colorado State University, Fort Collins, Colorado, USA
4 Professor of Civil Engineering, Colorado State University, Fort Collins, Colorado, USA
Ys = 1.90H 0.225 q 0.54 cos α (2)
Equation 3 is the Mason  prototype equation.
q x H y hw
Ys = K (3)
h = tailwater depth above original ground surface (meters)
d = median grain size of foundation material, d50 (meters)
g = acceleration of gravity (m/s2)
K = 6.42 − 3.10H 0.10 , x = 0.6 − , y = 0.15 +
v = 0.30,w = 0.15,z = 0.10,d = 0.25m
Unlike the Veronese equation, the Mason equation includes a material factor, d. It is unlikely
that this factor adequately represents the variety of material properties found in foundation
materials. The Mason equation is based upon thorough research including a comprehensive
collection of scale model studies and prototype case studies.
NEW TECHNOLOGY FOR EROSION PREDICTION
The basis of the new technology is Annandale’s Erodibility Index Method . The
Erodibility Index Method relates a geo-mechanical index (known as the Erodibility Index) to
the erosive power of water to define an erosion threshold for any earth material. The
Erodibility Index quantifies the relative ability of earth material to resist erosion, and is
identical to Kirsten’s ripability index . The method estimates the erosion threshold for earth
materials ranging from silt, through sand and gravel, to clays and rock.
Erodibility Index (KH)
One of the principal features of the new technology is the characterization of the
relative ability of earth materials to resist erosion. The Erodibility Index, Kh, is the product of
the Mass Strength Number, Ms, the Block Size Number, Kb, the Shear Strength Number, Kd,
and the Relative Ground Structure Number, Js .
K h = M s K bK d Js (4)
Stream power quantifies the relative magnitude of the erosive power of water. A
generic expression for calculation of stream power is the product of the unit weight of water,
unit discharge and energy loss.
Three graphs define the erosion threshold relating the Erodibility Index and Stream
Power . The U.S. Department of Agriculture determined the threshold by analysis of a
large prototype database. These graphs are not repeated here, due to space limitations.
Calculation of Scour Depth
The essence of the method that was developed to calculate scour depth by using the
Erodibility Index Method entails a comparison between available stream power and stream
power that is required to initiate scour. If the available stream power exceeds the stream
power that is required to initiate scour, erosion will occur. When the available stream power
is less than the required power, erosion ceases. The Dam Foundation Erosion study team
developed equations for estimating stream power in plunge pools.
The power (kW/m2) available to erode material is a function of the jet hydraulics. From
Bohrer , the velocity along the centerline of a jet in a plunge pool is a function of the
velocity at impact, the angle of impact, the air concentration of the jet at impact, given by the
ratio of air and water densities, and gravitational acceleration. Equation 5 describes this
functional relationship, followed by the limits of application. Equation 5 yields the distance
along the centerline.
v ρ Vi
V = −0.5812 ln i
gL + 2.107 (5)
i ρ w
ρ Vi 2 z j − z j +1
-0.29< ln i
v = Velocity “L” distance along jet centerline beneath water surface (m/s)
Vi = Velocity at jet impact with water surface (m/s)
ρ i = mass density of aerated jet at impact with water surface (kg/m3)
ρ w = mass density of water (kg/m3)
The rate of energy dissipation, or available power, is a discretized function of the total
head at various elevations along the centerline of the submerged jet. Equation 6 shows a
discrete calculation for the change in energy, ∆Ej, between points j and j+1. As the velocity
decays, with decreasing elevation, or increasing displacement along the jet centerline, the
total head decreases. Equation 7 yields the corresponding available power, pAj.
v 2 − v 2+1
j j Pj − Pj +1
∆E j = + + z j − z j +1 (6)
γv j ∆E j
pA j = (7)
1000(z j − z j +1 )
From Annandale , the power (kW/m2) required, pR, to erode material is a function of
Kh. The required power for granular material is a function of the erodibility index, Kh.
pR = Kh (8)
Companion papers  describe the prototype experimental facility located at
Colorado State University in Fort Collins, Colorado, USA. The facility includes a basin, 10 m
(30.5 ft) wide by 16.75 m (55 ft) long and 4.5 m (15 ft) deep, and an 8.7 cm (3.4375 in) by
3.05 m (10 ft) wide nozzle discharging up to 3.4 m3/s (120 ft3/s) at angles ranging from zero
to forty-five degrees from vertical. Figure 1 shows a profile of the facility containing road
base after scour occurred. Figure 2 shows a profile of the facility with simulated rock.
10' 42" Diffuser
24" Delivery Pipe/Manifold
18' 15.4' 11
15° Road Base 9
Figure 1. Profile of experimental facility at Colorado State University.
Road base and concrete blocks simulated earth materials in these experiments.
Table 1 shows the results of the gradation analysis of sample material collected at a
dam site in northern California. The sample material consists of basaltic boulders and
cobble-sized fragments set in a silty and sandy matrix, characteristic of the material below
the buttress section of the dam, adjacent to the left abutment. The material in the sample
has an aggregate specific gravity of roughly 2.65. A locally available road-base material, ¾”-
minus, matches this scaled sample gradation, except sizes greater than D90.
This material was placed in the basin and compacted to roughly 95% of optimum
density. The material was placed horizontally in 8” lifts prior to compaction. The elevation of
the material varied between 2.09m and 2.87m (7 and 9 feet) above the floor of the basin for
the various experiments.
Table 1. Material size fractions.
Prototype Prototype Fall
D0 0.03 0.07
D20 0.19 2.10
D40 0.54 6.00
D60 1.30 12.54
D80 5.00 24.60
D100 100.00 110.00
Cu = = 15.000 Cc = = 0.798
D10 D10 D60
Light weight concrete blocks, placed in two layers and dipped 45 degrees in the
downstream direction, simulated a fractured rock mass. Figure 2 is a profile of the prototype
facility showing the concrete blocks simulating a fractured rock mass. The nominal
dimensions of each fluted lightweight concrete block is 3x8x16 inches. Figure 3 details the
commercially available concrete blocks. The flutes are roughly one-half inch wide by one
inch deep transverse grooves in the face of the block, as Figure 3 illustrates.
10' 42" Diffuser
24" Delivery Pipe/Manifold
Block Field 13
18' 15.4' 11
15° Flow Road Base 9
5' 20' 0.91' 1.83' 8
Figure 2. Profile of prototype facility. (flow is left to right)
15' Piezometer Taps
8' 8' 18 places
Figure 3. Concrete block details including locations of piezometer taps.
Table 2 tabulates the values of the experimental parameters. The experimental matrix
includes three angles of issuance and four tailwater elevations, resulting in twelve
combinations. The datum is the floor of the test basin. Equation 9 yields the angle of
impingement, α , as a function of the initial velocity, vo, the angle of issuance, φ ,
gravitational acceleration, g, and the change in elevation, ∆z .
Table 2. Experimental parameters.
No. φ ∆z α Q vo vi Ai Bed El. TW El. Noz El. H h D50
(m) (m) (m) (m) (m) (m)
1 15º 3.49 11.9º 2.74 10.82 5.75 64.7% 2.78 3.35 6.84 8.26 0.57 1.00E-02
2 15º 3.18 12.1º 2.74 10.82 5.80 63.1% 2.87 3.66 6.84 7.96 0.79 1.00E-02
3 15º 3.76 11.7º 2.74 10.82 5.72 66.0% 2.78 3.08 6.84 8.53 0.30 1.00E-02
4 15º 2.91 12.3º 2.74 10.82 5.86 61.5% 2.12 3.93 6.84 7.68 1.82 1.00E-02
5 25º 2.95 20.2º 2.74 10.82 5.85 61.7% 2.15 3.94 6.89 7.72 1.79 1.00E-02
6 25º 3.55 19.5º 2.74 10.82 5.75 65.0% 2.16 3.34 6.89 8.32 1.18 1.00E-02
7 25º 4.33 18.8º 2.74 10.82 5.65 68.3% 2.15 2.56 6.89 9.10 0.41 1.00E-02
8 25º 3.99 19.1º 2.74 10.82 5.69 67.0% 2.12 2.90 6.89 8.77 0.77 1.00E-02
9 35º 3.61 26.9º 2.74 10.82 5.74 65.3% 2.19 3.35 6.96 8.38 1.16 1.00E-02
10 35º 2.98 27.9º 2.74 10.82 5.84 61.9% 2.17 3.97 6.96 7.76 1.81 1.00E-02
11 35º 3.93 26.5º 2.74 10.82 5.70 66.7% 2.19 3.03 6.96 8.70 0.84 1.00E-02
12 35º 4.37 25.8º 2.74 10.82 5.65 68.4% 2.09 2.59 6.96 9.14 0.50 1.00E-02
v 0 sin φ
α = arctan (9)
(v cos φ )2 + 2g∆z
Comparison with Current Technology
Table 3 lists the equated values, using experimental data, of scour elevation and
scour depth. Figure 4 shows these values compared to the experimental results. The Mason
Prototype equation overestimates the experimental results of the Prototype Facility by an
average of roughly 10% with a coefficient of determination R2=0.54. Figure 4 also shows that
the Yildiz equation under predicts the scour depth by approximately 30% on average, with a
coefficient of determination R2=0.44. The Yildiz equation appears to be significantly different
from the identity line.
Table 3. Predicted and experimental scour elevations for granular material.
Experiment Mason Yildiz Prototype
No. Prototype Experimental
Scour El. (m) Scour El. (m) Scour El. (m)
1 1.41 0.57 1.23
2 1.63 0.90 1.37
3 1.31 0.27 0.95
4 1.64 1.19 1.12
5 1.65 1.37 1.21
6 1.17 0.72 1.13
7 0.69 -0.11 0.66
8 0.85 0.25 0.82
9 1.19 0.98 1.31
10 1.69 1.65 1.54
11 0.96 0.64 1.29
12 0.66 0.17 0.89
Linear (Mason Prototype)
Predicted Erosion Elevation (m)
y = 0.6856x
R = 0.4358
y = 1.0983x
R = 0.5345
0 0.5 1 1.5 2
Experimental Erosion Elevation (m)
Figure 4. Experimental data, Mason Prototype and Yildiz equations.
Comparison with New Technology
The stream power that is required to initiate erosion was calculated with equation (9)
after the granular material was indexed with equation (4), using tables to assess the values
of Ms, Kb, Kd and Js from . The available stream power, as a function of scour hole
elevation, was calculated with equations (5) to (8). Figure 5 shows an example of the
comparison between calculated available and required stream power, as a function of
elevation. The elevation where the available and required stream powers cross is the
estimated elevation of the ultimate scour depth for the particular experiment shown in this
Figure 6 compares calculated and observed scour elevations for all twelve
experiments. On average, the predicted scour depths are approximately equal to the
observed scour depths, with a coefficient of determination of approximately 68%. This new
procedure for estimating the depth of scour in a plunge pool accounts for angle of impact,
aeration of the jet, hydraulic cushion, and material properties. Empiricism is limited to the
relationship between the erodibility index and rate of energy dissipation, and velocity decay
in a plunge pool. Otherwise, the procedure directly calculates the scour in a plunge pool.
Rate of Energy Dissipation (kW/m )
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50
Figure 5. Power available and power required.
In a companion paper  Annandale et al., confirm the hypothesis that the erodibility index is
a valid means for expressing the relative ability of earth material to resist erosion. Annandale
et al.’s paper  favorably compares observed and calculated stream power required to
initiate scour. Table 4 shows the Mason calculations compared to the experimental values
measured following each of three unit discharges. The Mason equation overestimates the
depth of scour of the simulated rock.
Table 4. Comparison of experimental data with Mason equation.
1.133 1.274 1.416
m /s m /s m3/s
Depth of Scour - 0.67 m 0.70 m 0.73 m
Depth of Scour – 0.67 m 0.67 m 0.67 m
Depth of Scour - Mason 0.95 m 1.01 m 1.08 m
y = 0.9996x
R = 0.6784
Calculated Scour Elevation (m)
Linear (1997 USBR)
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
Observed Scour Elevation (m)
Figure 6. Comparison of calculated and observed scour elevations.
This paper summarizes thirteen experiments simulating an overtopping jet plunging
into a forming plunge pool. In twelve of the experiments, the eroding material is a locally
available road-base. In the thirteenth experiment, concrete blocks simulate a fractured rock
mass. The Erodibility Index and the corresponding power required for erosion are functions
of the geotechnical properties of the material. The power available to erode the material is a
function of the velocity and air concentration at impact, and the rate of velocity decay of the
Calculated and observed scour depths are compared for current and new technology.
The comparisons find that the Mason equation slightly over-predicts scour depth and that the
Erodibility Index method improves the scour depth estimate in granular material. The Yildiz
equation, modified from the Veronese equation, under-predicts scour depth in granular
material. The Mason and Yildiz equations do not account for the relevant rock properties.
The Erodibility Index Method does. Annandale et al.  found that the erosion threshold for
simulated rock, as predicted with the Erodibility Index Method, compares favorably with an
The Dam Foundation Erosion Study Team received financial and technical support from the
following agencies and persons: Electric Power Research Institute, Federal Energy
Regulatory Commission. Pacific Gas & Electric Company. Western Area Power
Administration. Niagara-Mohawk. US Bureau of Reclamation Research & Technology
Function Applied Science & Technology Development program. US Department of Interior
Dam Safety Program. Colorado State University. Golder Associates. Kerrin Spurr.
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PAP 681, US Bureau of Reclamation Water Resources Research Laboratory, April 1995.
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Fractured Rock Media.” Proceedings of 1998 International Water Resources Engineering
Conference and Mini-Symposia, ASCE, Memphis, TN, August, 1998.
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Index for Scour in Fractured Rock Media.” Proceedings of 1998 International Water
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Foundation, Erosion, Erodibility, Plunge Pool, Hydraulics, Abutment.
Wittler, R.J., Annandale, G.W., Abt, S.R., Ruff, J.F., “New Technology for Estimating Plunge
Pool or Spillway Scour.” Proceedings of the 1998 Annual Conference of the Association of
State Dam Safety Officials. October 11-14, Las Vegas, NV.