NEW TECHNOLOGY FOR ESTIMATING PLUNGE POOL OR SPILLWAY SCOUR R
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NEW TECHNOLOGY FOR ESTIMATING PLUNGE POOL OR SPILLWAY SCOUR R. J. Wittler1, G. W. Annandale2, S. R. Abt3, J. F. Ruff4 INTRODUCTION 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 experimental results. 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) gvd z h = tailwater depth above original ground surface (meters) d = median grain size of foundation material, d50 (meters) g = acceleration of gravity (m/s2) H H K = 6.42 − 3.10H 0.10 , x = 0.6 − , y = 0.15 + 300 200 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) Erosive Power 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. Erosion Threshold 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. Available Power 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 2 − ln V = −0.5812 ln i gL + 2.107 (5) i ρ w ρ Vi 2 z j − z j +1 -0.29< ln i <2.6 gL L= ρ w cos α 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) 2g γ γv j ∆E j pA j = (7) 1000(z j − z j +1 ) Required Power 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. 480 0.44 pR = Kh (8) 1000 EXPERIMENT FACILITY 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 3.4375" Nozzle Jet 15 14 13 12 18' 15.4' 11 Flow 10 15° Road Base 9 8 7 6 5 4 7' 3 2 1 0 50' Figure 1. Profile of experimental facility at Colorado State University. MATERIAL SIMULATION Road base and concrete blocks simulated earth materials in these experiments. Granular Material 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 Size Velocity (mm) (cm/s) 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 2 D60 D30 Cu = = 15.000 Cc = = 0.798 D10 D10 D60 Simulated Rock 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 3.4375" Nozzle Jet 15 14 Block Field 13 12 18' 15.4' 11 10 15° Flow Road Base 9 5' 20' 0.91' 1.83' 8 7 6 5 14.85' 4 7' 3 2 1 0 50' Figure 2. Profile of prototype facility. (flow is left to right) Head Wall 15' Piezometer Taps 8' 8' 18 places 45° 4' 4' TYP 15.5'' 1.75'' 7.5'' 1.5'' 3.0'' Figure 3. Concrete block details including locations of piezometer taps. Granular Material 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 0 RESULTS 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 2 Yildiz Mason Prototype Identity Line Linear (Mason Prototype) Linear (Yildiz) 1.5 Predicted Erosion Elevation (m) y = 0.6856x 2 R = 0.4358 y = 1.0983x 2 R = 0.5345 1 0.5 0 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. 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. 1000 pA 100 pR Rate of Energy Dissipation (kW/m ) 2 10 1 0.1 0.01 0.001 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 Elevation (m) Figure 5. Power available and power required. Simulated Rock 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 3 3 m /s m /s m3/s Depth of Scour - 0.67 m 0.70 m 0.73 m Experiment Depth of Scour – 0.67 m 0.67 m 0.67 m Erodibility Index Depth of Scour - Mason 0.95 m 1.01 m 1.08 m 2 y = 0.9996x 1.75 2 R = 0.6784 1997 USBR 1.5 Identity Calculated Scour Elevation (m) Linear (1997 USBR) 1.25 1 0.75 0.5 0.25 0 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. CONCLUSIONS 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 submerged jet. 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 observed threshold. ACKNOWLEDGEMENTS 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. REFERENCES  Wittler, R.J., et. al., “Pit 4 Dam: Slab and Buttress Foundation Scale Model Simulation.” PAP 681, US Bureau of Reclamation Water Resources Research Laboratory, April 1995.  Kuroiwa, J., Ruff, J.F., Wittler, R.J., Annandale, G.W., “Prototype Scour Experiment in Fractured Rock Media.” Proceedings of 1998 International Water Resources Engineering Conference and Mini-Symposia, ASCE, Memphis, TN, August, 1998.  Annandale, G.W., Wittler, R.J., Ruff, J.F., Kuroiwa, J., “Prototype Validation of Erodibility Index for Scour in Fractured Rock Media.” Proceedings of 1998 International Water Resources Engineering Conference and Mini-Symposia, ASCE, Memphis, TN, August, 1998.  Wittler, R.J., Annandale, G.W., Ruff, J.F., Abt, S.R., “Prototype Validation of Erodibility Index for Scour in Granular Media.” American Society of Civil Engineers, Proceedings of the 1998 International Water Resources Engineering Conference, Memphis, Tennessee, August, 1998.  Veronese, A., “Erosioni de Fondo a Valle di uno Scarico.” Annali dei Lavori Publicci, Vol. 75, No. 9, pp. 717-726, Italy, September 1937.  Yildiz, D., Üzücek, E., “Prediction of Scour Depth From Free Falling Flip Bucket Jets.” Intl. Water Power and Dam Construction, November, 1994.  Mason, P.J., Arumugam, K., “Free Jet Scour Below Dams and Flip Buckets.” Journal of Hydraulic Engineering, Vol. 111, No. 2, ASCE, February 1985.  Kirsten, H. A. D., 1982, A classification system for excavation in natural materials, The Civil Engineer in South Africa, pp. 292 – 308, July 1982.  Annandale, G.W., “Erodibility.” Journal of Hydraulic Research, Vol. 33, No. 4, pp. 471- 494. 1995.  Barton, N., Lien, R., Lunde, J., “Engineering Classification of Rock Masses for the Design of Tunnel Support.” Rock Mechanics, Vol. 6, No. 4, pp. 189-236. 1974.  Bohrer, J.G., Abt, S.R., “Plunge Pool Velocity Prediction of Rectangular, Free Falling Jets.” US Bureau of Reclamation, Dam Foundation Erosion Study, Phase II report. R.J. Wittler Study Team Leader. November, 1996.  Juergenson, J.P., Abt, S.R., “Flow Pattern and Circulation Velocity Prediction in Plunge Pools Formed by Overtopping Steep Dams.” US Bureau of Reclamation, Dam Foundation Erosion Study, Phase IV report. R.J. Wittler Study Team Leader. May, 1998. KEY WORDS Foundation, Erosion, Erodibility, Plunge Pool, Hydraulics, Abutment. PUBLISHED REFERENCE 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.