BRISENT: An Entropy-Based Model for Bridge-Pier Scour Estimation under Complex Hydraulic Scenarios

: The goal of this paper is to introduce the ﬁrst clear-water scour model based on both the informational entropy concept and the principle of maximum entropy, showing that a variational approach is ideal for describing erosional processes under complex situations. The proposed bridge–pier scour entropic (BRISENT) model is capable of reproducing the main dynamics of scour depth evolution under steady hydraulic conditions, step-wise hydrographs, and ﬂood waves. For the calibration process, 266 clear-water scour experiments from 20 precedent studies were considered, where the dimensionless parameters varied widely. Simple formulations are proposed to estimate BRISENT’s ﬁtting coefﬁcients, in which the ratio between pier-diameter and sediment-size was the most critical physical characteristic controlling scour model parametrization. A validation process considering highly unsteady and multi-peaked hydrographs was carried out, showing that the proposed BRISENT model reproduces scour evolution with high accuracy.


Introduction
Bridges are important for society because they allow social, cultural, and economic connectivity; as such, they are key pieces in the development process and progress. At the same time, bridges continually suffer the action of natural hazards which exposes the road network to risk. The main cause of bridge collapse is related to hydraulic conditions [1][2][3][4][5][6], where flood events can compromise the safety of the whole structure, including its failure. Therefore, the pier-scour phenomenon has become a subject of research for civil and environmental engineers who have proposed many formulas in this field (see e.g., [7][8][9][10]).
Estimating the maximum scour depth around bridge piers and its temporal evolution is a critical step in the design of foundations. Previous researchers have proposed formulas to reproduce the time-dependent scour depth and its maximum value. Among them, Zanke [10] proposed a semi-empirical formulation based on the principle of mass conservation of the bedload. Richardson and Davis [8] used literature data generated by several authors to develop the well-known HEC-18 scour equation, later extended for wide piers. Manes and Brocchini [11] proposed a novel approach combining theoretical arguments with considerations taken from empirical evidence in order to provide new avenues for the development of general predictive models founded more on physical than empirical grounds. Dey [12] elaborated a theoretical model, considering that the key agent in the scour process is the horseshoe vortex. Melville and Chiew [13] and Oliveto and Hager [14] The goal of this paper is to introduce the first scour model based on both the informational entropy concept and the principle of maximum entropy (POME), showing that a variational approach is ideal for describing erosional processes under complex hydraulic situations.
The paper is organized as follows: first, the informational entropy concept and POME are introduced; the BRISENT model is then formulated and presented in detail; literature data for the model test are described; the calibration/estimation of model parameters is presented and discussed; and the BRISENT model is validated with multi-peaked hydrographs containing sequences of flood events. Conclusions are drawn at the end.

The Informational Entropy Concept
The informational entropy concept and its theory have been applied before in statistical mechanics, information theory, hydrology, and water resources (e.g., [46][47][48][49][50][51][52][53]), providing good results and an easy way to introduce probabilities into hydraulics. Informational entropy, Shannon entropy or simple entropy is defined in terms of probability. Taking into account the continuous random variable X, entropy is defined as: in which Φ(X) is the entropy, f (X) is the probability density function of X, and E[ ] denotes expectation. Note that for random variable notation, the Dutch convention is adopted [54] according to which a random variable is underlined. Informational entropy may be considered a measure of the uncertainties as well as a quantity of information contained in the data [43]. Theoretically, a uniform probability distribution maximizes the entropy defined in Equation (1) [47]; however, due to the constraints that define the physics of different phenomena, the probability density function is not always uniform. According to Chiu [47], the entropy concept can be thought of as a measure of how close the real probability density function is to the uniform function. Therefore, a maximization of entropy will make the probability density function as uniformly as possible, considering the constraints. In this context, and from a physical perspective, the tendency toward maximal entropy is the driving force of natural change, where POME is the logic counterpart of the second law of thermodynamics [43,46].
The maximization of entropy to determine f (X) may be resolved using the method of variational calculus. Maximizing Equation (1) and satisfying the "n" constraints defined in Equation (2) is equivalent to maximizing the Lagrange function presented in Equation (3): in which λ i are the Lagrange multipliers and i is the number of constraints. Thus, the mathematical function of f (X) can be obtained by solving Equation (4) and therefore, a simple function of X can be deduced,

The BRISENT Model
The dimensionless, effective flow work by the flow on the streambed around a pier, W * , was deduced from dimensional analysis and physical considerations by Pizarro et al. [21]: where t con is the considered time, t R is a reference time (= z R /u R ), z R = D 2 /2d 50 is a reference length (D is the pier-diameter and d 50 the sediment-size), u is the section averaged flow velocity, u R is a reference velocity (= ρ gd 50 , where ρ = (ρ S − ρ W )/ρ W is the relative density between sediment "s" and water "w", and g is the gravitational acceleration). u c is the critical flow velocity for incipient motion of sediment particles. Based on experimental evidence (e.g., [10,[55][56][57]), it is considered that scour occurs when u(t) ≥ 0.5u c , thus: W * is an energetic parameter applied to local scour phenomena. Consequently, POME applied to W * results in a variational/thermodynamic approach for treating the local scour process. Pizarro et al. [21] evidenced that under constant sediment properties, geometrical scale, and clear-water conditions the mathematical relation between the relative scour depth Z * and W * is unique. Z * = z/z R is defined as a normalized parameter, where z is the dimensional scour depth and z R the reference length established previously. In agreement with Link et al. [22], Z * increases monotonously over W * , taking values from zero at non-eroded conditions to a maximum value at the equilibrium state. Based on the Laplace principle of insufficient reason and therefore, assuming that all values of W * between zero and a maximum considered value, W * max are equally likely to be attached (not necessarily true), then the probability of the relative scour depth to be equal to or less than Z * , Pr(Z * ≤ Z * ) = F(Z * ), can be specified by W * /W * max . Thus, and the probability density function of Z * is the derivative of Equation (7): Unfortunately, Equation (8) cannot be used to determine f (Z * ) because the derivate of W * over Z * is unknown. However, POME can be exploited for this purpose. Equation (9) reveals the informational entropy applied to the scour phenomenon: Resolving the issue to find the probability density function of Z * implies using Equation (9) with a few constraints that completely define the boundary conditions and physics. The first constraint must satisfy the axiom of probability: and the second constraint, related to the mass conservation law for bridge-pier scour phenomenon at equilibrium condition, can be written as: in which Z * is the (W * -averaged) mean relative scour depth, i.e., the average value of Z * during the scour process. The scour entropy given by Equation (9) needs to be maximized subjected to the constraints in Equations (10) and (11). For this purpose, the method of Lagrange multipliers is adopted. The probability density function f (Z * ) is: where λ 1 and λ 2 are Lagrange multipliers. After substitution of Equation (12) into the constraint equations, and considering the boundary condition Z * = 0 at W * = 0, the final equations after some algebra are: in which Z * max = max( Z * ) is the maximum relative scour depth associated to the maximum dimensionless, effective flow work W * max = max( W * ), λ = λ 2 is a Lagrange multiplier that needs to be estimated, and S is defined as the entropic-scour parameter. Note that due to the physical relationship between W * max and Z * max , the BRISENT model contains two fitting parameters: λ and Z * max . These parameters depend on scour influencing parameters and can be estimated as described in the following sections.
Note that for run names a standard notation was selected in order to provide a simple and characteristic code that contains the year of publication, author names, number of experiment, and its hydraulic conditions. For example, "2017PIZARRO05U" is the unsteady experiment number 5 carried out by Pizarro et al. [21] and "2004SHEPPARD04S" is the steady experiment number 4 performed by Sheppard et al. [60].
The scour data in Table 1 can be classified as steady or unsteady, depending on the hydraulic conditions in which the experiments were performed. For time-dependent runs, the flow-discharge, flow-velocity, and flow-depth vary in time, whereas the values presented in Table 1 refer to peak flow conditions. Figure 1 shows the hydrographs corresponding to the 14 unsteady runs considered herein.
Stepwise hydrographs are presented in Figure 1j-l. The dataset is composed of natural and artificial sediments as well, with relative densities ranging from 0.04 to 1.65. The ratio between pier-diameter and sediment-size D/d 50 ranges from 11.11 to 4159.09, and the ratio between flow-depth and pier-diameter takes values from 0.12 to 13.20. D * ranges from 5.06 to 121.42 and thus covered flows with hydraulically smooth, transitional smooth-rough, and rough walls. The considered experimental data refer to clear-water conditions (u/u c ≤ 1), allowing the identification of local scour without effects of overlapping processes, such as bedforms migration and scour hole refilling. Only four experiments are catalogued as exceptions to such decisions, in which their flow intensity values are slightly greater than one (2002OLIVETO18S, 2002OLIVETO33S, 1956CHABERT07S, and 1956CHABERT08S). Such exceptions were considered in order to have at least three experiments with the same D/d 50 and different D * , with the aim of analyzing the sediment and geometrical dependency on λ (Section 5.2.1).
The scour data in Table 1 can be classified as steady or unsteady, depending on the hydraulic conditions in which the experiments were performed. For time-dependent runs, the flow-discharge, flow-velocity, and flow-depth vary in time, whereas the values presented in Table 1 refer to peak flow conditions. Figure 1 shows the hydrographs corresponding to the 14 unsteady runs considered herein. Stepwise hydrographs are presented in Figure 1j-l. The dataset is composed of natural and artificial sediments as well, with relative densities ranging from 0.04 to 1.65. The ratio between pier-diameter and sediment-size ⁄ ranges from 11.11 to 4159.09, and the ratio between flow-depth and pier-diameter takes values from 0.12 to 13.20.
in which f 1 represents the water depth effect on bridge pier scour, f 2 the velocity influence, and f 3 denotes scale effects between pier-diameter and sediment-size. Accordingly, f 1 and f 2 contain the hydraulic impacts on local scour, while f 3 denotes scale ratio effects.
Note that the BRISENT model is able to reproduce the scour dynamic until the maximum selected value of W * . Thus, it is convenient to choose its maximum value, i.e., relating W * max with W * at equilibrium scour conditions. The laboratory data employed by Sheppard et al. [9] was used in correspondence with their formulation for estimating W * max , S * max , and S. Such a dataset consists of 441 laboratory experiments and 791 field data. The original dataset was filtered by three conditions to ensure the calculation of W * max = W * equilibrium : (1) having complete information about the experiments; (2) clear-water scour; and (3) W * max ≥ 1 × 10 5 in order to ensure the scour equilibrium state.

Calibration and Validation Procedures
All λ fitting coefficients were determined by MATLAB(MathWorks, Natick, MA, USA) nonlinear curve-fit function, considering Z * max and W * max as the last measured point. Therefore, W * max and Z * max are not fitting coefficients when the BRISENT model is calibrated with time-dependent scour depth experiments; i.e., experiments that contain the whole scour depth evolution over time. The experiments are presented in Table 1.
Based on the unique relationship between W * and Z * (under clear-water conditions, constant sediment properties, and geometrical scales) [21], unsteady experiments were calibrated in two ways: (a) Calibration employing unsteady data: Each unsteady run was calibrated with the aim of finding the best performance of the model. This kind of calibration will be called "unsteady calibration" in the rest of the paper. and Link et al. [22]. This kind of calibration will be called "steady calibration" in the rest of the paper.  The BRISENT model is completely determined when λ, Z * max , and W * max are known. Z * max and W * max are estimated employing the S/M formulation and the dataset of Sheppard et al. [9] in order to guarantee equilibrium conditions. This is a critical step for the proper use of the model due to Equation (11).
For validation procedures, the most complex experiments were used among those available to the authors: two highly unsteady and multi-peaked runs pertained to Link et al. [22] (2017LINK01U and 2017LINK02U).

BRISENT Performance under Steady Hydraulic Conditions
Ninety percent of the experimental data presented in Section 4 were performed under steady hydraulic conditions. Figure 2 illustrates the time-dependent scour depth for different experiments that consider the extreme values of the dimensionless parameters. In all cases, the BRISENT model is able to reproduce the scour evolution with root mean square error (RMSE) values between 0.04 and 1.60 cm.

RMSE is defined as RMSE
where z j is the measured scour depth j,ẑ j is the computed scour depth j, and n is the number of measured points. Therefore, the proposed BRISENT model can reproduce the scour evolution with high accuracy under steady hydraulic conditions.   Figure 3 presents the time-dependent evolution of the measured scour depth in contrast to the BRISENT model calibrated according to the procedure presented in Section 4.3. Table 3 summarizes the model benchmarking in terms of RMSE for each experiment. The results show that, independently of the kind of calibration, BRISENT correctly reproduces the time-dependent scour depth with RMSE values less than 1.57 and 0.52 cm for steady and unsteady calibrations, respectively.  Note that the scour evolution presents similar behavior in two groups of experiments (triangle markers). The first group consists of the experiments "2012MEYERING06S" and "2012MEYERING07S", while the second group consists of the experiments "2012MEYERING01S", "2002OLIVETO31S", and "1956CHABERT08S". Such a similar scour evolution is due to the reduced variability of the dimensionless D/d 50 that takes values around 10 and 120 for the first and second groups, respectively. Figure 3 presents the time-dependent evolution of the measured scour depth in contrast to the BRISENT model calibrated according to the procedure presented in Section 4.3. Table 3 summarizes the model benchmarking in terms of RMSE for each experiment. The results show that, independently of the kind of calibration, BRISENT correctly reproduces the time-dependent scour depth with RMSE values less than 1.57 and 0.52 cm for steady and unsteady calibrations, respectively.

Effects of ⁄ and * on
A selected group of 15 experimental runs was considered to analyze the sediment and geometrical dependency on . This number was established in order to have at least three experiments with the same ⁄ and different * . The data were categorized in four classes depending on the geometrical scale between pier-diameter and sediment-size ( ⁄ 35, 90, 190, and 350). The range for the dimensionless particle diameter * takes values from 6.58 to 78.42. Figure  4a shows the evolution of * over * according to the four ⁄ classes. All data collapse into a Steady calibration tends to overestimate the maximum scour depths after flood waves, while unstedy calibration presents better performances in all analyzed experiments. Furthermore, the average RMSE values are less than 1 cm, independetly of the kind of calibration. Therefore, the proposed BRISENT model can reproduce the scour evolution with high accuracy under stepwise hydrographs and 100% unsteady hydraulic conditions.

Effects of D/d 50 and D * on λ
A selected group of 15 experimental runs was considered to analyze the sediment and geometrical dependency on λ. This number was established in order to have at least three experiments with the same D/d 50 and different D * . The data were categorized in four classes depending on the geometrical scale between pier-diameter and sediment-size (D/d 50 ≈ 35, 90, 190, and 350). The range for the dimensionless particle diameter D * takes values from 6.58 to 78.42. Figure 4a shows the evolution of Z * over W * according to the four D/d 50 classes. All data collapse into a single and geometrically-dependent curve, independently of the hydraulic conditions. Z* decreases with D/d 50 for equal W * .
Water 2017, 9,889 13 of 21 single and geometrically-dependent curve, independently of the hydraulic conditions. Z* decreases with ⁄ for equal * . Figure 4b shows calibrated on * . No correlation between these two variables is detected and it can be observed that is strongly controlled by ⁄ . A power function is adopted in order to describe this dependency: where and are fitting coefficients that were determined by the procedure presented in Section 4.3, obtaining = 4.237 and = 0.957 with a determination coefficient = 0.99 and a RMSE equal to 49. Equation (22) is plotted in Figure 5a in comparison to the fitted values and Figure 5b presents the residuals between calibrated and estimated .  The fitting coefficient is always affected by the ratio between pier-diameter and sediment size, even for high values. Thus, a stabilizer or equilibrium threshold for it was not found.  Figure 4b shows calibrated λ on D * . No correlation between these two variables is detected and it can be observed that λ is strongly controlled by D/d 50 . A power function is adopted in order to describe this dependency: where a 1 and a 2 are fitting coefficients that were determined by the procedure presented in Section 4.3, obtaining a 1 = 4.237 and a 2 = 0.957 with a determination coefficient R 2 = 0.99 and a RMSE equal to 49. Equation (22) is plotted in Figure 5a in comparison to the fitted values and Figure 5b presents the residuals between calibrated and estimated λ. The fitting coefficient λ is always affected by the ratio between pier-diameter and sediment size, even for high values. Thus, a stabilizer or equilibrium threshold for it was not found.
The hydraulic effects in the original S/M formulation are represented by and . On the other hand, * has the capacity to integrate these hydraulic effects on the local scour process. Therefore, and can take a convenient value with the aim of writing it in a function of * . Considering = 1 = makes it possible to be on the safe side from a design perspective and, consequently, this value was selected: * = 2.5 2 = 5 0.4( ⁄ ) . + 10.6( ⁄ ) .
From the experimental data of Sheppard et al. [9], only 129 of the total experiments were considered after applying the filters described in Section 4.3. Table 4 summarizes the source for laboratory data, the number of experiments from the source, and the range of selected dimensionless parameters according to the filtered dataset.
The hydraulic effects in the original S/M formulation are represented by f 1 and f 2 . On the other hand, W * has the capacity to integrate these hydraulic effects on the local scour process. Therefore, f 1 and f 2 can take a convenient value with the aim of writing it in a function of W * max . Considering f 1 = 1 = f 2 makes it possible to be on the safe side from a design perspective and, consequently, this value was selected: From the experimental data of Sheppard et al. [9], only 129 of the total experiments were considered after applying the filters described in Section 4.3. Table 4 summarizes the source for laboratory data, the number of experiments from the source, and the range of selected dimensionless parameters according to the filtered dataset.  Figure 6, in which the solid line represents Equation (25): where a and b are fitting coefficients that were calibrated by the procedure described in Section 4.3, obtaining a = 21.84 and b = −0.295 with a determination coefficient R 2 = 0.65 and a RMSE = 1.19. versus ⁄ values are plotted in Figure 6, in which the solid line represents Equation (25): (25) where and are fitting coefficients that were calibrated by the procedure described in

BRISENT Validation: Highly Unsteady and Multi-Peaked Hydrographs
The BRISENT model was validated using highly unsteady and multi-peaked hydrographs pertaining to Link et al. [22]. Validation runs are plotted on Figure 7a-d, presenting the time-dependent evolution of the measured scour depth in contrast to the BRISENT model calibrated and estimated according to the proposed equations (Equations (22), (26) and (27)).

BRISENT Validation: Highly Unsteady and Multi-Peaked Hydrographs
The BRISENT model was validated using highly unsteady and multi-peaked hydrographs pertaining to Link et al. [22]. Validation runs are plotted on Figure 7a-d, presenting the timedependent evolution of the measured scour depth in contrast to the BRISENT model calibrated and estimated according to the proposed equations (Equations (22), (26) and (27)). Table 5 summarizes the model benchmarking in terms of RMSE for experiments "2017LINK01U" and "2017LINK02U", and the results show that the proposed equations correctly reproduce the main dynamic of the time-dependent scour depth. Note that the unsteady calibration provides the most accurate results with RMSE values less than 0.32 cm, while the observed higher RMSE values for the estimated BRISENT model rely on the computed values of * . Steady calibration shows good performance as well, with RMSE values less than 0.81 cm.   Images (a,b) are experiments 2017LINK01U and 2017LINK02U, respectively. Images (c,d) present measured scour depths over time, BRISENT model calibrated using a steady run, BRISENT model calibrated using unsteady runs, and BRISENT model considering Equation (13) to Equation (14) and Equation (19) to Equation (22) for experiments 2017LINK01U and 2017LINK02U, respectively. Table 5 summarizes the model benchmarking in terms of RMSE for experiments "2017LINK01U" and "2017LINK02U", and the results show that the proposed equations correctly reproduce the main dynamic of the time-dependent scour depth. Note that the unsteady calibration provides the most accurate results with RMSE values less than 0.32 cm, while the observed higher RMSE values for the estimated BRISENT model rely on the computed values of Z * max . Steady calibration shows good performance as well, with RMSE values less than 0.81 cm. From a practical perspective, traditional bridge scour formulas are used in order to estimate a potential scour depth that could occur during bridge life. However, neither a vectorial nor a empirical approach can estimate the scour process by considering long duration events, discharge regime, and the interactions of different erosional processes. Therefore, scour depth estimations are uncertain.
The informational entropy concept has long been recognized as a measure of uncertainties, and is the ideal concept for analyzing the bridge scour phenomenon. Moreover, BRISENT is based on the effective flow work parameter (W * ) and thus, the combination of W * and the principle of maximum entropy (POME) is a variational/thermodynamic approach for treating the scour process. The fact that the core of the model is based on the informational entropy concept (uncertainty) presents a clear advantage in comparison to other formulations by taking into consideration bridge design, bridge life, and hydrologic regime.

BRISENT: A Multipurpose Model
BRISENT was inferred without imposing the hydraulic conditions, under which the model can run. In particular, the results confirm that the BRISENT model can be used under different and complex hydraulic conditions (steady flows, step-wise hydrographs, and flood waves with multi-peaked hydrographs) to describe bridge scour phenomena, and independently of the type of calibration. In consequence, BRISENT can be considered a multipurpose model which allows calibration under simple hydraulic conditions and application to more complex and realistic hydraulic situations. Since, in the past, most scour experiments were carried out under steady hydraulic conditions, these are clear advantages over other published formulations.

Wide Range of the Considered Dimensionless Parameters
Two hundred and sixty-six clear-water scour experiments were considered for the calibration process, with widely ranging dimensionless parameters. For example, D * ranges from 5.06 to 121.42 and, thus, covered flows with hydraulically smooth, transitional smooth-rough, and rough walls, according to Oliveto and Hager [66]. The ratio between pier-diameter and sediment-size D/d 50 ranges from 11.11 to 4159.09 and, thus, the obtained results can be used in both laboratory and natural scales. The same can be said for the ratio between flow-depth and pier-diameter, which takes values from 0.12 to 13.20.

Applicability of the Model: Advantages and Limitations
Simple equations are proposed to estimate the time-dependent scour depth independently of hydraulic conditions, geometrical scale, and hydraulic flow roughness. Equation (26) and Equation (27) are empiric equations for S and W * max which are based on the dataset published by Sheppard et al. [9]. Such dataset contains a wide value range of dimensionless parameters, making it possible to treat the bridge scour phenomenon under different flow regimes and different geometrical scales. Furthermore, the results show that the proposed equations correctly reproduce the main dynamic of the time-dependent scour depth under clear-water conditions, where no overlapping of other erosional processes occur.
6.5. Key Dimensionless Parameters: W * and D/d 50 The ratio between pier-diameter and sediment-size (geometrical scale) is the most critical physical characteristic controlling the scour model parametrization. From a physical modeling perspective, keeping constant a high value of D/d 50 at laboratory scales is very hard due to the sediment cohesion (sediment size has to be reduced 10 times or more in laboratory flumes). As a consequence, theoretical efforts to develop new and more accurate models are a major challenge in this field, where the BRISENT model represents the first model of its kind. Additionally, our results highlight that the key dimensionless parameters are W * and D/d 50 . Such parameters contain the energy for undermining and the geometrical scale between pier diameter and sediment size, respectively. More energy to reach the same relative scour depth at a real scale was observed, thus positioning laboratory-deduced formulas on the side of safety from a design perspective.

Conclusions
Bridge-pier scour evolution has been analyzed in the present work, using the first mathematical formulation for simulating scour phenomena based on energy concepts and entropy theory. The proposed BRISENT model has been established on the effective flow work parameter (W * ) and on the principle of maximum entropy (POME). The results confirm the idea that the local-scour phenomenon indirectly depends on the time-dependent mean velocity, and thus the model can be calibrated using steady scour data and applied to more complex hydraulic scenarios. Literature data under clear-water conditions were employed, in which the key dimensionless parameters took values over a wide range, allowing analysis of the local scour phenomenon from a holistic point of view. Simple formulations were proposed to estimate the fitting coefficient λ (Equation (22)) and the entropic-scour parameter S (Equation (27)). Moreover, we observed that the ratio D/d 50 is the most critical physical characteristic controlling the scour model parametrization.
Our results support the idea that POME and W * could be used to explain erosional processes and, in particular, to describe clear-water local-scour phenomena with high accuracy and simple formulations. This is an entry point for treating more complex river interaction processes at different spatial scales (e.g., channel instability, channel bed gradation, channel migration, contraction scour, local scour, etc.). The authors are keen to apply these concepts with the aim of answering the following questions: (i) how long does it take to reach the equilibrium state under clear-water, bed-load, suspension conditions and/or a possible combination of scour modes at real piers? (ii) is it achievable during the lifetime of bridges? (iii) how does the scour-hole refilling process affect the whole scour dynamic? (iv) what is the time-evolution of W * and Z * ?