Assessment of Rock Scour Depth in Plunge Pools

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  • ICSE6 Paris - August 27-31, 2012 Sara Rashidian and Dr. Pooyan Asadollahi

    ICSE6-053

    Assessment of Rock Scour Depth in Plunge Pools

    Sara RASHIDIAN1, Pooyan ASADOLLAHI2

    1 The Catholic University of America Postal address: 620 Michigan Ave., N.E. Washington, DC 20064, USA - e-mail: sarahrshdn@gmail.com

    2 PARSONS Corporation

    Postal address1: 100 M St. SE, Washington, DC 20003, USA - e-mail: pooyanas@gmail.com Significant erosion of earth materials, known as scour, occurs when the erosive capacity of water exceeds the ability of the earth materials to resist it.. Assessment of the extent of scour is necessary to ensure the safety of a dam and to guarantee the stability of its abutments. There are some empirical approaches to predict dynamic water pressure fluctuation in rock fractures. However, there exists no perfect formulation for evaluating scour caused by general failure modes (static failure modes such as translation and rotation as well as dynamic failure modes such as flutter and divergence) of rock blocks. In this paper, statistical pattern recognition techniques are employed to estimate the depth of scour in plunge pools caused by high-velocity water jet impact. The results of an experimental study are investigated considering two classes: (1) scour to a specific depth, and (2) no scour. Appropriate dimensionless features are selected by using fluid mechanics concepts. Then, performing a Fishers Linear Discriminate Analysis reduces the number of features. Finally, the minimum distance classifier is simplified to predict the scour depth.

    Key words Scour, plunge pool, pattern recognition, rock, dam

    I INTRODUCTION

    High-velocity plunging jets, issuing from hydraulic artificial or natural structures, can result in scouring of the rock riverbed or the dam toe foundation. Assessment of the extent of scour is necessary to ensure the safety of the dam and to guarantee the stability of its abutments. Figure 1 schematically depicts a jet discharging over a dam.

    Figure 1: Nomenclature for a jet discharging over an ogee spillway and plunging into a pool [Bollaert, 2002].

    Currently, there is no perfect formulation for evaluating scour caused by general failure modes of rock

    blocks (such as translation, roto-translation, flutter, and divergence). Existing approaches to scour evaluation are mostly combinations of empirical equations and either an analytical or a numerical method. These 1 Corresponding author

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  • ICSE6 Paris - August 27-31, 2012 Sara Rashidian and Dr. Pooyan Asadollahi

    approaches, however, have some limitations. In this paper, we propose a method which utilizes the pattern recognition techniques to estimate the depth of the scour in plunge pool.

    Section II reviews the existing approaches and discusses the limitation of these approaches. Section III classifies the results of an experimental study performed by Martins [1973] into two classes: scour to a specific depth and no scour. Section IV explains our feature selection, feature extraction, and the classifier. A formulation is proposed to estimate the scour depth by using a Minimum Distance (MINDIS) classifier obtained in this Section. The ability of the suggested approach in estimating scour is checked and validated in Section V followed by summary and conclusion in Section VI.

    II LIMITATIONS OF AVAILABLE APPROACHES

    In this section, the abilities of two existing approaches (Dynamic Impulsion and Block Stability Three- Dimensional) in predicting scour depth are reviewed. The limitations of each approach are also discussed.

    II.1 Dynamic Impulsion (DI) Method

    Dynamic Impulsion (DI) method [Bollaert, 2002; Bollaert and Schleiss, 2005] is limited to vertical translational failure of parallelepiped rock blocks with one face at the plunge pool bottom. Indeed, roto-translational failures are common even for parallelepiped blocks subjected to pressure fluctuations [Fiorotto and Rinaldo, 1992].

    DI method estimates the water pressure fluctuation in plunge pools using empirical equations or curves (such as those developed by Ervine et al. [1997]). Asadollahi [2009] demonstrated that these empirical approaches have lots of estimation errors.

    Moreover, DI method requires a scour threshold, which itself was calibrated from experimental studies. Different researchers [Bollaert, 2002; Bollaert and Schleiss, 2005; Asadollahi, 2009] suggested different scour thresholds, which in turn decreases the precision of the scour depth predicted using DI method.

    II.2 Block Stability 3-Dimensional (BS3D)

    Block Stability 3-Dimensional (BS3D) is a code developed by Asadollahi [2009] to analyse the stability of single rock blocks subjected to general forces, including non-conservative forces such as water pressure. The code is based on an algorithm introduced by Tonon [2007] and validated by Asadollahi and Tonon [2008].

    BS3D can capture all failure modes of rock blocks. Therefore, using it in predicting the scour depth will resolve one of the limitations of DI method. Dynamic water pressure fluctuations in plunge pools can be estimated using empirical methods (the same as those that are being used in DI method). The scour threshold again originates from the calibration of experimental results.

    BS3D has several advantages over DI method [Asadollahi and Tonon, 2010], one of which is the capability of capturing all failure modes. However, it still suffers from some numerical inaccuracies (which can be eliminated in Civil Engineering problems) as well as errors in estimating dynamic water pressure and scour threshold.

    III EXPERIMENTAL RESULTS

    Martins [1973] built a river-bed test facility of equal, cubic, comparatively large blocks, systematically arranged, without cohesion. The number of tests carried out was 90, which resulted from the combination of three angles of impact, with three openings of the gate closing the orifice discharging the jet, with two values of the sides of the blocks, with five depth of the water cushion (parameter z in Figure 1). The opening of the gates was chosen so as to ensure an approximately square form in the initial cross section of the jet. The blocks were made of cement/sand mortar with a unit weight of about 2.2 g/cm3.

    In this paper, we considered all 90 cases reported by Martins [1973]. The effective parameters (features) are as follows:

    Size of the rock blocks, a. Height of falling jet, H. Angle of impact, . Jet diameter, Dj.

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  • ICSE6 Paris - August 27-31, 2012 Sara Rashidian and Dr. Pooyan Asadollahi

    Depth of scour, h. Depth of water cushion, z. Unit weight of blocks, b.

    In order to analyse these experimental results, we introduced two classes as follows:

    a) Class 1: plunge pool is at its initial condition. Failure will be extended to scour depth of h. The water cushion is z.

    b) Class 2: plunge pool is at its final condition. No failure will occur and no block can be removed. Thus, the water cushion is Y and the first block with depth of a is fixed in its place.

    Consequently, we produced a database for our pattern recognition analysis. Our database contains 180 data points (two classes for each of the 90 cases tested by Martins [1973]).

    IV PATTERN RECOGNITION ANALYSIS

    IV.1 Feature Selection

    Feature selection was performed using fluid mechanics concepts in an attempt to have dimensionless parameters as appropriate features.

    First, the falling jet height, H, and the angle of impact, , were combined to create a new feature: the jet velocity, Vj, using the following equation:

    ,)sin(

    2Hg

    V j

    (1)

    where, g is the gravity acceleration. Second, the jet velocity was expressed in terms of hydraulic head as follows:

    ,2

    2

    wj

    hyd gV

    h

    (2)

    in which, w is water unit weight. Finally, we introduced the first dimensionless feature by dividing the hydraulic head by the scour depth multiplied by the block unit weight as follows:

    ,1sb

    hyd

    dh

    x (3) where, ds is depth of scour depth which is equal to h, for Class 1, and is equal to a, for Class 2.

    The only remaining effective parameters are jet diameter and water cushion which both have the same dimension. Thus, the second dimensionless parameter was defined as follows:

    ,2jD

    cx (4) where, c is the water cushion which is equal to z and Y, for Classes 1 and 2, respectively.

    The magnitudes of x1 and x2 were calculated for all 180 data points of our database. Figure 2 depicts the scatter distribution of the two classes on x1-x2 coordinate system.

    IV.2 Feature Extraction

    The two features selected in the previous sections were reduced to one by performing a Fishers Linear Discriminant Analysis [Fisher, 1936; McLachlan, 2004]. The objective is to perform dimensionality reduction while preserving as much of the class discriminatory information as possible. The mean (1 and 2) and covariance matrices (S1 and S2) of both classes were obtained. Then, the direction of the appropriate projection vector was obtained by normalizing the vector calculated using the following formula:

    w S1 S2 1(1 2) (5)

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  • ICSE6 Paris - August 27-31, 2012 Sara Rashidian and Dr. Pooyan Asadollahi

    Our analysis resulted in a projection vector of (0.49, 0.87). Therefore, the final one-dimensional feature for classification would be:

    21 87.049.0 xxy (6)The magnitudes of y were calculated for all 180 data points of our database. Figures 3 presents the scatter

    distribution of the two classes along y-axis. Also, Figure 4 depicts the column distribution (the number of cases at each interval of the magnitude of y) versus y-axis.

    Figure 2: The distribution of the two classes on x1-x2 coordinate system.

    Figure 3: The scatter distribution of the two classes on y-axis.

    Figure 4: The column distribution (the number of cases at each interval of y value) of the two classes versus y-

    axis.

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  • ICSE6 Paris - August 27-31, 2012 Sara Rashidian and Dr. Pooyan Asadollahi

    IV.3 Classifier

    In this research, Minimum Distance (MINDIS) classifier was used to recognize the pattern govern between the two classes. MINDIS is based on the shortest distance to class means. We obtained the normalized or Mahalanobis [Mahalanobis, 1936; De Maesschalck et al., 2000] distance from mean using the following equation:

    dk (y k ) / k , (7)where k and k is the average and standard deviation of magnitudes of y in each classes. Our calculations revealed that the average of y in Classes 1 and 2 are 7.95 and 21.69 with standard deviations of 2.98 and 5.62. Consequently, the normalized distance of y value form means of Classes 1 and 2 should be calculated. Depending on which one is smaller, the experiment can be categorized as Class 1 or 2. Thus, the classifier is the point at which the normalized distance from the mean of Class 1 is equal to that of class 2.

    IV.4 Scour Depth

    The boundary between the two classes (the boundary between having scour and no scour) can be expressed as follows by substituting Equations (2), (3), (4), and (6) in MINDIS classifier as follows:

    69.387.02

    49.046.02

    26.022

    jb

    wj

    jb

    wj

    Dz

    hgV

    Dhz

    agV

    (8)

    As it was mentioned in section 1, we are interested in estimating the scour depth, h. As a fact, the values of all parameters are known and we can solve Equation (9) to obtain the magnitude of h. Equation (8) can be expressed as a traditional quadratic equation of:

    h2 B h C 0 (9) in which:

    B 0.57 V j2wD j

    2g ba 8.02D j 0.89z (10)

    C 1.07 V j2wD j

    2g b (11)While the scour depth is always a positive value, it can be estimated using the following equation:

    h B B2 4C

    2 (12)

    V VALIDATION

    In order to validate the effectiveness of our approach in estimating the scour depth, the proposed method was applied to 90 cases reported by Martins [1973]. The scour depth was then calculated for all cases. It was found that the ratio of the predicted value to the measured scour depth has an average of 1.34 with a standard deviation of 0.43. This implies that Equation (12) overestimates the scour depth by 34% on average.

    While the suggested approach overestimates (not underestimates) the scour depth, employing that in designing a dam is conservative. Moreover, the overestimation rate (34%) is acceptable in a Civil Engineering application since in general they are using a factor of safety of 1.5 for such a problem. Although BS3D [Asadollahi, 2009] can predict the scour depth better than our pattern recognition approach (error of 10% versus 34%), Equation (12) is much simpler. In fact, anyone can predict the scour depth using the approach presented here and a calculator. However, there is no commercial version of BS3D and few persons have access to the current academic version of the code.

    VI CONCLUSION

    In this paper, statistical pattern recognition techniques have been employed to estimate the depth of scour in plunge pools caused by high-velocity water jet impact. The results of an experimental study have been investigated considering two classes: (1) scour to a specific depth and (2) no scour. Appropriate dimensionless features have been selected using fluid mechanics concepts. The features have been reduced

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    to one feature by performing a Fishers Linear Discriminate Analysis. Finally, the Minimum distance classifier has been simplified to predict the scour depth.

    It was demonstrated that the proposed approach works well in predicting the depth of scour. It is simple to apply and its error of estimation is in acceptable range.

    Finally, the current paper introduced a new approach to address solid-fluid interaction problems. While evaluating dynamic water pressures developed by interaction between water and rock is so complicated, the pattern recognition methods can appropriately estimate what we need to design a structure (such as dam foundations or plunge pools).

    VII REFERENCES

    Asadollahi, P. (2009) - Stability Analysis of a Single Three Dimensional Rock Block: Effect of Dilatancy and High-velocity Water Jet Impact, PhD dissertation, University of Texas at Austin.

    Asadollahi, P., Tonon, F. (2008) - Validation of single rock blocks stability analysis. In: 42nd US Rock Mechanics Symposium. San Francisco, CA.

    Asadollahi, P., Tonon, F. (2010) - Stability of rock blocks subjected to high-velocity water jet impact. In: 44th US Rock Mechanics Symposium. Salt Lake City, UT.

    Bollaert, E. (2002) - Transient water pressures in joints and formation of rock scour due to high-velocity jet impact, Communication No. 13 of the Laboratory of Hydraulic Constructions, EPFL, Lausanne, Switzerland.

    Bollaert, E., Schleiss, A. (2005) - Physically based model for evaluation of rock scour due to high-velocity jet impact. ASCE J. of Hydraulic Engineering, March: 153-165.

    De Maesschalck, R., Jouan-Rimbaud, D., Massart, D.L. (2000) - The Mahalanobis distance. Chemometrics and Intelligent Laboratory Systems, 50: 118

    Ervine, D.A., Falvey, H.T., Withers, W. (1997) - Pressure fluctuations in plunge pool floors. J. Hydraulic Research. 35: 257-279.

    Fiorotto, V., Rinaldo, A. (1992) - Turbulent pressure fluctuations under hydraulic jumps. J. Hydraulic Research. 30:499-519.

    Fisher, R. (1936) - The use of multiple measurements in taxonomic problems. In: Annals of Eugenic, 7: 179-188.

    Mahalanobis, P.C. (1936) - On the generalizes distance in statistics. In: Proc. of the National Institute of Sciences of India. 2 (1): 4955.

    Martins, R. (1973) - Contribution to the knowledge on the scour action of free jets on rocky river-beds. In: 11th ICOLD Congress, Madrid, pp. 799-814.

    McLachlan (2004) - Discriminant Analysis and Statistical Pattern Recognition, Willey Interscience. Tonon, F. (2007) - Analysis of single rock blocks for general failure modes under conservative and non-

    conservative forces. Int. J. for Numerical and Analytical Methods in Geomechanics. 31(14): p. 1567-

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