Scouring around bridge pier: A comprehensive analysis of scour depth predictive equations for clear-water and live-bed scouring conditions

Bridge pier scour is characterized as lowering the riverbed elevation around a bridge pier. It occurs due to the erosive action of flowing water, which excavates and transports materials, potentially leading to bridge failure (Melville & Coleman 2000; Khalid et al. 2021). Local scouring around bridge piers is a common phenomenon that occurs when water flows around a bridge pier and erodes the bed material in its vicinity. The scouring can cause the bridge pier to lose support and stability, potentially leading to bridge failure. There are several factors that can contribute to the development of local scour, including the velocity and depth of the water, the type and size of the bed material, and the shape and size of the bridge pier. Local scour can be minimized through proper design and construction of the bridge pier and its foundation and ongoing maintenance and monitoring of the bridge and its surroundings. Research shows that scour caused by floods and other hydraulic conditions is responsible for over half of all bridge failures (Shirole & Holt 1991; Barbhuiya & Dey 2004). For example, Aggarwal (2001) studied scouring around a bridge over the Ambala–Kalka segment of the Jhajjar River in India, recording a depth of 3.0, 4.5, and 4.5 m below the riverbed during floods in 1976, 1982, and 1988, respectively. Basu & Gupta (2003) reported the tilting of a bridge pier occurs when scouring levels fall below the lowest bed level over the Lohit River in India. Aminoroayaie Yamini et al. (2018) investigated sediment scour phenomena for offshore piles due to sea waves and current interaction and suggested a numerical model to predict the shape and depth of the scour pit. Qasim et al. (2022) studied the impact of bed flume discordance on weir-gate structure hydraulics and found that bed flume configuration significantly alters the water surface path. Widyastuti et al. (2022) identified a structural solution for mitigating scouring around bridge abutments by implementing energy absorbers. Their findings indicate that the placement of these absorbers results in damping forces. Abdulkathum et al. (2023) verified different machine learning (ML) approaches, such as multiple nonlinear regression analysis (MNLR), gene expression programming (GEP), and artificial neural network (ANN) models to predict the local scouring around a bridge pier and found that the ANN model has better predicted the SDR (d_s/y) values than other models followed by the GEP model.

Numerous research studies on scour depth lack a common equation to predict the scour depth under clear water scouring (CWS) conditions and live bed scouring (LBS) conditions across a wide range of data points. Many researchers, as available in the literature, commonly used scour depth predictive equations without distinguishing their applicable/suitable scouring conditions and their limitation of data point range. Also, many equations very poorly predicted scour depth as they are developed only by considering one parameter (b/y) (Laursen 1958; Neill 1964; Breusers 1965) and two parameters (F_r and b/y) (Coleman 1971). So, there is a need to review different research papers and a proper classification of all available scour depth predictive equations between CWS and LBS conditions.

The novelty of the paper is as follows: (1) this paper discusses separately about CWS and LBS around the bridge pier, (2) different influencing parameters which affect the equilibrium scour depth have been summarized, (3) the laboratory data set and field data set of scour depth conducted by different researchers are presented in the Supplementary Material (4) the effect of flow parameters and roughness parameters are discussed separately for CWS and LBS. The present study focuses only on time-independent and equilibrium scour depth conditions. All the equations mentioned in the present manuscripts are equilibrium-based scour depth predictive empirical equations.

Tison (1940) laid the foundation for further research in the field of bridge pier scour and helped to better understand the factors that contribute to equilibrium scour depth around different shapes of bridge piers. Since then, numerous studies have been conducted to improve our understanding of bridge pier scour and to develop effective methods to mitigate its impacts. For example, research has shown that the shape of the bridge pier, the size of the sediment particles, and the flow velocity are all important factors that influence scour depth (Laursen & Toch 1956; Raudkivi & Ettema 1983; Melville & Sutherland 1988; Raikar & Dey 2005; Dey & Sarkar 2006; Vijayasree et al. 2019). Different shapes of bridge piers, such as rectangular, round-nosed, triangular, flared, and lenticular, can have varying effects on equilibrium scour depth and must be carefully considered when designing bridges. In addition, researchers have developed various remedial measures (Chiew & Lim 2000; Garg et al. 2005; Clopper et al. 2007; Akhlaghi et al. 2020; Pandey et al. 2022) to reduce the impact of scouring on bridge piers, such as installing scour protection structures, controlling sediment transport, and designing bridge piers with shapes that are less susceptible to scouring.

The design of the shape of a bridge pier is a crucial factor in determining the amount of local scour it will undergo. Chabert & Engeldinger (1956) classified the pier shape into blunt-nosed and sharp-nosed. The formation of a horseshoe vortex system around the upstream nose of the pier, where most scour occurs, defines a blunt-nosed pier. On the other hand, a sharp-nosed pier splits the flow and experiments have shown that a horseshoe vortex system does not form around its upstream face. As a result, no scour occurs at the sharp-nosed shape of the bridge pier when it is properly aligned with the flow. However, if tested at an angle to the flow, the bridge pier transforms into a blunt-nosed pier, leading to deeper scour.

Many researchers have studied different bridge pier shapes such as chamfered, cylindrical, diamond, elliptic, flared, hexagonal, joukowsky, lenticular, oblong, octagonal, sharp nose, square, rectangular, round-nosed, and triangular (Tison 1940; Inglis 1949; Chabert & Engeldinger 1956; Shen et al. 1966; Ettema 1980; Chiew & Melville 1987; Kumar et al. 1999; Ettema et al. 2006; Hassanzadeh et al. 2019; Garg et al. 2022; Baranwal et al. 2023a).

Different researchers performed experiments and provided empirical equations to predict equilibrium scour depth around the bridge pier (Laursen 1962; Jain & Fischer 1979; Melville & Sutherland 1988; Sheppard et al. 2004; Ismael et al. 2015; Pandey et al. 2018; Rathod & Manekar 2022). The existing equations on equilibrium scour depth show that the scour depth is the function of different geometry, flow, and roughness parameters. The input parameters used to model scour depth are as follows:

• Pier width (b): The width of the pier influences the scour depth, as a wider pier may cause more turbulence and a stronger scour hole (W/b ≥ 8; Shen et al. 1966; Chiew 1984) where W is the width of the flume.

• Flow depth (y): The flow depth of the approaching water affects scour depth, as a deeper flow will generally create a deeper scour hole (y/b < 3.0; Ettema 1980).

• Flow velocity (V): The flow velocity of the approaching water also affects scour depth, as a higher velocity will cause a stronger scour hole and velocity ratio (V/V_c) (if V/V_c ≤ 1.0, CWS and if V/V_c > 1.0, LBS) (Raudkivi & Ettema 1983).

• Bed sediment size (d₅₀): The size of the sediment in the riverbed affects scour depth, as the scour depth at the gravel bed (4.10 mm ≤ d₅₀ ≤ 14.25 mm) is found to be more compared to the sand bed (d₅₀ ≤ 4.0 mm) (Raikar & Dey 2005). The size of river bed particles is a vital bed roughness parameter to model the scour depth. The critical velocity (Equation (2)) is also calculated using mean particle size, which is necessary to classify CWS and LBS conditions. Also, during the modeling using ML approaches, the parameter of bed sediment size is non-dimensionalized in terms of b/d₅₀ and σ_g (Bateni et al. 2007; Khan et al. 2012; Shamshirband et al. 2020; Baranwal et al. 2023a; Kumar et al. 2023; Nil et al. 2023).

• Standard deviation of the bed material particle size (σ): The standard deviation of the bed sediment size distribution affects scour depth as a higher standard deviation indicates a more heterogeneous sediment size distribution, which can impact scour (σ_g < 1.4; Dey & Sarkar 2006).

• Froude number (F_r): The Froude number, a dimensionless quantity that describes the relative importance of inertial forces to gravitational forces in a fluid flow, affects scour depth. A higher Froude number indicates a stronger scour hole (If F_r ≤ 1, then subcritical flow and F_r > 1 supercritical flow) (Ettema 1980).

• Pier correction factors (K): Various pier correction factors were used to account for the impact of specific pier geometry and flow conditions on scour depth. These correction factors can vary depending on the specific empirical equation being used (Silvia et al. 2021). The pier correction factor for different shapes of bridge pier is considered for cylinder and round nose shape (K = 1.0) Tison (1940), rectangular and square nose shape (K = 1.1) (Melville & Sutherland 1988), oblong (K = 0.86) (Laursen & Toch 1956), and sharp nose shape (K = 0.9) (Vijayasree et al. 2019).

It is important to note that the scour depth around a bridge pier is a complex and dynamic phenomenon influenced by various factors. The parameters listed earlier are just some key factors that can impact scour depth. Additionally, the specific values of these parameters can vary widely depending on the specific site and flow conditions, making it difficult to predict scour depth with certainty.

Bridge pier scouring experimental models have been conducted for CWS conditions by many investigators in the last eight decades (Laursen 1958; Neill 1964; Jain & Fischer 1979; Melville & Sutherland 1988; Kothyari et al. 1992a; Mia & Nago 2003; Ebrahimi et al. 2018; Vaghefi et al. 2021; Kayadelen et al. 2022; Sharp & McAlpin 2022; Baranwal et al. 2023a, 2023b; Choudhary et al. 2023; Nil et al. 2023).

A list of local scour studies under CWS conditions by various researchers is presented in Supplementary Material, Table S1. The flow and roughness parameters effect on CWS are as follows:

• Ettema (1980) reported that the suspended fine silt has a major effect on scouring depths.

• Raudkivi & Ettema (1983) found that the equilibrium scour depth decreases faster as flow depth declines with smaller values of relative flow depth (y/b). The equilibrium local scour depth decreases for a b/d₅₀ value less than 20–25.

• Yanmaz & Altinbilek (1991) observed an inverted cone-shaped circular scour hole around a cylindrical pier; its shape does not change over time.

• Chiew (1992) studied the influence of sediment size on scour depth around circular-shaped bridge piers for uniform sediments.

• Kothyari et al. (1992a) defined the effective size of non-uniform sediment for scouring purposes.

• Ahmed & Rajaratnam (1998) discovered that bed roughness caused a steeper pressure gradient and raised the amount of bed shear stress.

• Chang et al. (2004) developed a method for the computation of equilibrium scour depth in non-uniform sediment based on the mixing layer concept.

• Raikar & Dey (2005) reported that equilibrium scour depth increases as the size of gravel, i.e., 4.10 mm ≤ d₅₀ ≤ 14.25 mm, decreases in CWS condition for circular and square shape of bridge pier. When compared to sand beds, the effect of gravel size on scour depth is noticeably different. It is also found that generally, the scour depth at the gravel bed (4.10 mm ≤ d₅₀ ≤ 14.25 mm) is more than that of the sand bed (d₅₀ ≤ 4.0 mm).

• Kothyari et al. (2007) proposed a scour depth model considering actual and entrainment densiometric particle Froude numbers and gave a criterion for the ‘end scour’ condition.

• Aksoy et al. (2017) reported that scour depth increases with pier diameter and flow velocity.

• Ebrahimi et al. (2018) investigated the influence of debris collection on the upstream face of a sharp nose bridge pier and found that scour depth decreases when the debris is near the bed and increases when the debris is just under the free flow surface.

• Akhlaghi et al. (2019) found that maximum local scour depth in uniform sediments occurs at the threshold condition.

• Pandey et al. (2019) reported that the maximum scour depth around the bridge pier increases with pier diameter, approach velocity and critical velocity ratio. However, it decreases with bed particle size in CWS and decreases as armor layer particle size increases.

A comprehensive list of scour depth predictive Equations (3)–(25) is presented in Table 2, along with the data sets used to develop each model. Yanmaz (1989) developed a semi-theoretical equation (Equation (12)) for calculating scour depth around circular and square piers under clear water conditions. This model is based on the solid sediment continuity equation and can compute the maximum scour depth under critical discharge. Lee & Sturm (2009) proposed two scour depth models using least square regression analysis (Equation (20)). Guo (2012) reported that the maximum probable scour depth is generally equal to the square root of the product of the pier diameter and approach flow depth (Equation (24)). Pandey et al. (2018) proposed a model (Equation (25)) to predict scour depth and two empirical equations to calculate the maximum scour length and maximum affected scour width for cohesionless bed sediment.

Table 2

Different existing equations used in CWS for the prediction of scour depth ratio

Reference .	Equation .	Citation of the equation in literature (CEL)/Data used to develop the equation (DDE)/developed equation is validated with the model of other researchers (DEV) .	Eq No. .
Laursen (1958)		CEL: Gülbahar (2009) and Yeleğen & Uyumaz (2016)	(3)
Neill (1964)		CEL: Abd El-Hady Rady (2020)	(4)
Breusers (1965)		CEL: Abdelmonem et al. (2009) and Vonkeman & Basson (2019)	(5)
Shen et al. (1966)		CEL: Breusers et al. (1977) and Sheppard et al. (2014)	(6)
Shen et al. (1969)	where	CEL: Breusers et al. (1977), Ting et al. (2001) and Vonkeman & Basson (2019)	(7)
Hancu (1971)		CEL: Melville & Sutherland (1988)	(8)
Breusers et al. (1977)	(a) if then (b) if ⁠, then ⁠, and (c) if ⁠, then	CEL: Ghorbani (2008), Gülbahar (2009) and Vonkeman & Basson (2019)	(9)
Jain & Fischer (1979)	for Max forfor	DDE: Own experimental data DEV: Chabert & Engeldinger (1956) and Hancu (1971)	(10)
Melville & Sutherland (1988)		DDE: Shen et al. (1966), Ettema (1980), Chee (1982), and Chiew (1984), DEV: Laursen & Toch (1956), Hancu (1971), Shen (1971), and Breusers et al. (1977)	(11)
Yanmaz (1989)		CEL: Gülbahar (2009) and Kilinç (2019)	(12)
Kothyari et al. (1992a)	where	DDE: Own experimental data DEV: Laursen & Toch (1956), Shen et al. (1969), Jain & Fischer (1979) and Ettema (1980)	(13)
Simplified Chinese (Gao et al. 1993)		CEL: Shin & Park (2010)	(14)
El-Saiad (1998)		CEL: Shamshirband et al. (2020)	(15)
Ettema et al. (1998)		DDE: Raudkivi & Ettema (1983), Melville & Sutherland (1988), and Breusers & Raudkivi (1991)	(16)
Kumar et al. (1999)		DDE: Schneible (1951), Chabert & Engeldinger (1956), Tanaka & Yano (1967), Ettema (1980), Chiew (1992), and Kumar (1996)	(17)
Melville & Chiew (1999)		DDE: Chabert & Engeldinger (1956), Shen et al. (1966), Hancu (1971), Ettema (1980), Chee (1982), and Chiew (1984)  DDV: By assuming hypothetical data	(18)
Sheppard et al. (2004)		DDE: Own experimental data	(19)
Lee & Sturm (2009)	where where	DDE: Own experimental data,; Ettema (1980), Melville & Sutherland (1988), Ting et al. (2001), Sheppard (2003), Sheppard et al. (2004), and Sheppard & Miller (2006)	(20)
Revised Shen II (1971) equation revised by AbGhani et al. (2010)		CEL: Khan et al. (2012)	(21)
Revised Hancu (1971) equation revised by AbGhani et al. (2010)		CEL: Khan et al. (2012)	(22)
Ettema et al. (2011) or modified Sheppard et al. (2011) equation	(a) (b)	(a) For shallow water or wide piers where (b) For deep water or narrow piers where	(23)
Guo (2012)	where	DDE: CSU equation-based Molinas (2004) data DEV: Laursen (1963) and Sheppard et al. (2011)	(24)
Pandey et al. (2018)		DDE: Own experimental data; Kothyari (1989), Yanmaz & Altinbilek (1991), Dey et al. (1995), Sheppard et al. (2004), Raikar & Dey (2005), Das et al. (2014), Lanca et al. (2013), and Lodhi et al. (2014), DEV: Richardson & Davis (2001), Khan et al. (2012), and Sheppard et al. (2014)	(25)

Reference .	Equation .	Citation of the equation in literature (CEL)/Data used to develop the equation (DDE)/developed equation is validated with the model of other researchers (DEV) .	Eq No. .
Laursen (1958)		CEL: Gülbahar (2009) and Yeleğen & Uyumaz (2016)	(3)
Neill (1964)		CEL: Abd El-Hady Rady (2020)	(4)
Breusers (1965)		CEL: Abdelmonem et al. (2009) and Vonkeman & Basson (2019)	(5)
Shen et al. (1966)		CEL: Breusers et al. (1977) and Sheppard et al. (2014)	(6)
Shen et al. (1969)	where	CEL: Breusers et al. (1977), Ting et al. (2001) and Vonkeman & Basson (2019)	(7)
Hancu (1971)		CEL: Melville & Sutherland (1988)	(8)
Breusers et al. (1977)	(a) if then (b) if ⁠, then ⁠, and (c) if ⁠, then	CEL: Ghorbani (2008), Gülbahar (2009) and Vonkeman & Basson (2019)	(9)
Jain & Fischer (1979)	for Max forfor	DDE: Own experimental data DEV: Chabert & Engeldinger (1956) and Hancu (1971)	(10)
Melville & Sutherland (1988)		DDE: Shen et al. (1966), Ettema (1980), Chee (1982), and Chiew (1984), DEV: Laursen & Toch (1956), Hancu (1971), Shen (1971), and Breusers et al. (1977)	(11)
Yanmaz (1989)		CEL: Gülbahar (2009) and Kilinç (2019)	(12)
Kothyari et al. (1992a)	where	DDE: Own experimental data DEV: Laursen & Toch (1956), Shen et al. (1969), Jain & Fischer (1979) and Ettema (1980)	(13)
Simplified Chinese (Gao et al. 1993)		CEL: Shin & Park (2010)	(14)
El-Saiad (1998)		CEL: Shamshirband et al. (2020)	(15)
Ettema et al. (1998)		DDE: Raudkivi & Ettema (1983), Melville & Sutherland (1988), and Breusers & Raudkivi (1991)	(16)
Kumar et al. (1999)		DDE: Schneible (1951), Chabert & Engeldinger (1956), Tanaka & Yano (1967), Ettema (1980), Chiew (1992), and Kumar (1996)	(17)
Melville & Chiew (1999)		DDE: Chabert & Engeldinger (1956), Shen et al. (1966), Hancu (1971), Ettema (1980), Chee (1982), and Chiew (1984)  DDV: By assuming hypothetical data	(18)
Sheppard et al. (2004)		DDE: Own experimental data	(19)
Lee & Sturm (2009)	where where	DDE: Own experimental data,; Ettema (1980), Melville & Sutherland (1988), Ting et al. (2001), Sheppard (2003), Sheppard et al. (2004), and Sheppard & Miller (2006)	(20)
Revised Shen II (1971) equation revised by AbGhani et al. (2010)		CEL: Khan et al. (2012)	(21)
Revised Hancu (1971) equation revised by AbGhani et al. (2010)		CEL: Khan et al. (2012)	(22)
Ettema et al. (2011) or modified Sheppard et al. (2011) equation	(a) (b)	(a) For shallow water or wide piers where (b) For deep water or narrow piers where	(23)
Guo (2012)	where	DDE: CSU equation-based Molinas (2004) data DEV: Laursen (1963) and Sheppard et al. (2011)	(24)
Pandey et al. (2018)		DDE: Own experimental data; Kothyari (1989), Yanmaz & Altinbilek (1991), Dey et al. (1995), Sheppard et al. (2004), Raikar & Dey (2005), Das et al. (2014), Lanca et al. (2013), and Lodhi et al. (2014), DEV: Richardson & Davis (2001), Khan et al. (2012), and Sheppard et al. (2014)	(25)

DDE, data used to develop the equation; DEV, developed equation is validated with the model of other researchers; CEL, citation of the equation in literature; ⁠, center-to-center spacing between two piers; ⁠, local scour depth in case of the pier with collar plate; ⁠, local scour depth at equilibrium; ⁠, local scour depth in case of the pier without appurtenances; ⁠, critical Froude number; ⁠, Densiometric particle Froude number; F_d50, Particle Froude number; F_r, Froude number; H, depth of collar below the free water surface; ⁠, flow intensity factor; ⁠, flow depth pier size factor; ⁠, sediment size factor; K₄, sediment gradation factor or bed armoring factor; K₅, pier nose shape factor; ⁠, pier alignment factor; K₇, flow depth – pier width factor; K₈, time factor; N, shape number; T, dimensionless time; V, flow velocity of the upstream from the pier; V_ic, approach velocity corresponding to critical velocity at the pier; V_c, flow critical velocity; W, channel width; σ_g, standard deviation of grain size distribution ⁠; ⁠, dimensionless coefficient about the shape of the pier nose; α*, opening ratio; ν, kinematic viscosity of water.

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Most of the bridge failures occur in live bed conditions caused by high flow intensities during flood scenarios. LBS around bridge piers has been investigated by many researchers (Chabert & Engeldinger 1956; Carstens 1966; Jain & Fischer 1979; Melville 1984; Froehlich 1988; Kothyari et al. 1992b; Link 2006; Zhao et al. 2010; Ettmer et al. 2015; Bordbar et al. 2021; Okhravi et al. 2022; Rathod & Manekar 2022; Choudhary et al. 2023; Nil et al. 2023).

The effect of flow and roughness parameters on LBS is summarized as follows:

• Chiew (1984) studied the effect of sediment size on scour depth at circular piers in live bed scour conditions.

• Scour depth decreases above the critical velocity and then increases to a maximum value at the transition to flatbed conditions.

• At higher velocities, the equilibrium scour depth decreases due to the formation of antidunes on the bed surface.

• Karim et al. (1986) and Chin et al. (1994) defined the bed armoring is a process in which the prolonged degradation of a riverbed occurs when the flow entrains finer sediment and leaves larger particles on the bed surface. This results in a gradual coarsening of the riverbed. Local scour depths are likely to be lower if bed armoring occurs. The effect of particle size depends on whether the bed sediment forms ripples or not. For ripple-developing sands, the maximum scour depth occurs at the transition to flatbed conditions, while for non-ripple-developing sediments, the maximum scour depth occurs at the threshold condition (Chiew & Melville 1987).

• Zhao et al. (2010) suggested that the scour depth decreases if the height of the cylindrical bridge pier is reduced, with this change occurring exponentially. The scour depth is almost independent of the pier height if the height-to-diameter ratio of the cylindrical bridge pier exceeds 2.0.

• With an increase in the V/V_c value, there is a decreasing tendency in the maximum scour depth (d_s/y) from 2.0 to 1.2 (Shen et al. 1966). When the value of V/V_c is 4.0, the maximum non-dimensional scour depth (d_s/y) increases to 3.1 (Jain & Fischer 1979), and when the value of V/V_c increases to 9.0, the d_s/y values vary from 2.0 to 2.5 (Zanke 1982).

• Sheppard & Miller (2006) reported that bedforms were transported periodically from the scour hole, leading to the attainment of maximum scour depth. Additionally, it is observed that for y/b = 2.7 and b/d₅₀ = 563, scour depth increased from 0.13 to 0.30 m as V/V_c increased from 0.63 to 6.0. Furthermore, for y/b = 2.6 and b/d₅₀ = 181, scour depth remains nearly constant as V/V_c increases from 0.90 to 4.0.

• According to Ettmer et al. (2015), the scour depth under live bed conditions is generally significantly higher than under clear water conditions and further increases with flow intensity. Ettmer et al. (2015) proposed two conditions: (1) If 1 < V/V_c < 4, bed load with dunes is the main transport mode; and (2) If V/V_c ≥ 4, bedforms with entrainment into suspension without development has been dominated.

• The horseshoe primary vortex formed in front of the pier is responsible for developing the scour hole around the pier (Kothyari et al. 1992b).

A detailed list of scour depth predictive Equations (26)–(39), along with the data point used to develop the LBS model and validation of the developed model under LBS conditions, is presented in Table 3.

Table 3

Different existing equations used in LBS for the prediction of scour depth ratio

Reference .	Equation .	Citation of the equation in literature (CEL)/data used to develop the equation (DDE)/developed equation is validated with the model of other researchers (DEV) .	Eq No. .
Larras (1963)		CEL: Gülbahar (2009)	(26)
Laursen (1963)		CEL: Gülbahar (2009)	(27)
Carstens (1966)		CEL: Gülbahar (2009)	(28)
Veiga (1970)		CEL: Ghorbani (2008)	(29)
Hancu (1971)		CEL: Melville & Sutherland (1988)	(30)
Froehlich (1988)	or	CEL: Pal et al. (2013)	(31)
Kothyari et al. (1992b)	where	DDE: Own experimental data DEV: Chabert & Engeldinger (1956), Laursen & Toch (1956), Liu et al. (1961), Chitale (1962), Shen et al. (1969), Hancu (1971), Neil (1973), Melville (1975), Jain & Fischer (1979), Chee (1982), and R.D.S.O. (1987)	(32)
Simplified Chinese (Gao et al. 1993)	where	CEL: Mueller (1996) and Shin & Park (2010)	(33)
Johnson (1995)		DDE: Zhuravlyov (1978), Jain & Modi (1986), Froehlich (1988), and Dongguang et al. (1993), DEV: Laursen & Toch (1956), Larras (1963), Shen et al. (1969), Hancu (1971), Breusers et al. (1977), Melville & Sutherland (1988), and Hydraulic engineering circular (HEC-I8) (1993)	(34)
Fischenich & Landers (1999) or Modified Froelich formula		CEL: Abd El-Hady Rady (2020) and Kayadelen et al. (2022)	(35)
Lim & Chiew (2001)	where ∼ Sediment size factor, and ∼ Flow depth adjustment factor	DDE: Own experimental data	(36)
Yanmaz (2001)		DDE: Froehlich (1988), DEV: Chabert & Engeldinger (1956), Laursen & Toch (1956), Tarapore (1962), Laursen (1963), Shen et al. (1966, 1969), Hancu (1971), Basak et al. (1975, 1977), Jain & Fischer (1979), and Melville (1984)	(37)
Sheppard & Miller (2006)	(a) (b)	(a) When live bed scour ranges up to live bed peak i. e., (b) when the live bed scour ranges above the live bed peak where and Vlp is the live bed peak scour velocity or velocity where the bed planes out	(38)
Ismael et al. (2015)		DDE: Own experimental data	(39)

Reference .	Equation .	Citation of the equation in literature (CEL)/data used to develop the equation (DDE)/developed equation is validated with the model of other researchers (DEV) .	Eq No. .
Larras (1963)		CEL: Gülbahar (2009)	(26)
Laursen (1963)		CEL: Gülbahar (2009)	(27)
Carstens (1966)		CEL: Gülbahar (2009)	(28)
Veiga (1970)		CEL: Ghorbani (2008)	(29)
Hancu (1971)		CEL: Melville & Sutherland (1988)	(30)
Froehlich (1988)	or	CEL: Pal et al. (2013)	(31)
Kothyari et al. (1992b)	where	DDE: Own experimental data DEV: Chabert & Engeldinger (1956), Laursen & Toch (1956), Liu et al. (1961), Chitale (1962), Shen et al. (1969), Hancu (1971), Neil (1973), Melville (1975), Jain & Fischer (1979), Chee (1982), and R.D.S.O. (1987)	(32)
Simplified Chinese (Gao et al. 1993)	where	CEL: Mueller (1996) and Shin & Park (2010)	(33)
Johnson (1995)		DDE: Zhuravlyov (1978), Jain & Modi (1986), Froehlich (1988), and Dongguang et al. (1993), DEV: Laursen & Toch (1956), Larras (1963), Shen et al. (1969), Hancu (1971), Breusers et al. (1977), Melville & Sutherland (1988), and Hydraulic engineering circular (HEC-I8) (1993)	(34)
Fischenich & Landers (1999) or Modified Froelich formula		CEL: Abd El-Hady Rady (2020) and Kayadelen et al. (2022)	(35)
Lim & Chiew (2001)	where ∼ Sediment size factor, and ∼ Flow depth adjustment factor	DDE: Own experimental data	(36)
Yanmaz (2001)		DDE: Froehlich (1988), DEV: Chabert & Engeldinger (1956), Laursen & Toch (1956), Tarapore (1962), Laursen (1963), Shen et al. (1966, 1969), Hancu (1971), Basak et al. (1975, 1977), Jain & Fischer (1979), and Melville (1984)	(37)
Sheppard & Miller (2006)	(a) (b)	(a) When live bed scour ranges up to live bed peak i. e., (b) when the live bed scour ranges above the live bed peak where and Vlp is the live bed peak scour velocity or velocity where the bed planes out	(38)
Ismael et al. (2015)		DDE: Own experimental data	(39)

b_e, the width of the bridge pier projected normal to the approach flow; b*, effective structure width (or diameter); d, size of uniform sediment; ⁠, grain number; ϕ, dimensionless coefficient based on the shape of the pier nose; ⁠, angle of attack; EP, expression programming.

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Johnson (1995) studied the performance of various scour depth predictive equations by collecting field data (Equation (34)). Hancu (1971) (Equation (8)) and Breusers et al. (1977) (Equation (9)) were found to have zero biases for the collected data when V/V_c < 0.5, as they assumed that no local scour takes place at low velocities. However, the Shen et al. (1969) (Equation (7)) and Hancu (1971) equations showed biases of less than one for the selected data, making them undesirable for safety purposes. Therefore, it is recommended not to use either of these equations to calculate scour depth around piers. In contrast, it is found that the Melville & Sutherland (1988) Equation (11) equation tends to overpredict scour depth to a greater extent than other equations, particularly if sediment gradation is considered.

The next section, titled ‘Bridge Pier Scour Depth Modeling (Scour Type Not Mentioned or Undistinguishable): Available Data Sets and Existing Equations,’ has been created for the following reasons.

• (a)

  The literature shows that some researchers have not distinguished between CWS and LBS and developed the empirical equation to predict scour depth around the bridge pier.

• (b)

  In some data sets, the range of V/V_c values varies from less than 1.0 to greater than 1.0. So, they cannot be classified either in clear water or LBS types.

Many researchers have conducted laboratory and field experiments on both CWS and LBS without classification of scouring types (Inglis 1949; Shen et al. 1969; Melville 1975; Chiew 1984; Johnson 1992; Kandasamy & Melville 1998; Sheppard & Miller 2006; Ettmer et al. 2015; Hassanzadeh et al. 2019; Sharp & McAlpin 2022). A detailed list of published research in the literature and different data sets for experimental study and field study has been compiled and presented in Supplementary Material, Table S3. Some researchers have conducted experiments in both CWS and LBS and found the range of V/V_c to be between 1 and greater than 1 (as shown in Supplementary Material, Table S3), so it is difficult to distinguish the type of scouring that occurs during the measurement of data sets. According to Hamill (2014), during an actual flood, scour may initially form in clear water, transition to live bed and/or suspended sediment conditions, and eventually return to the initial CWS condition. From Table S3, it is found that the V/V_c value is not reported by Shen et al. (1969), Melville (1975), Chiew (1984), and Mohamed et al. (2007) and additionally, the V/V_c value is reported to be in the range of 0.61–6.07 by Sheppard & Miller (2006) and 0.8–8.5 by Ettmer et al. (2015).

A comprehensive list of scour depth predictive Equations (40)–(63), including the data set used to develop the model and model validation, is presented in Table 4. Note that the previous authors did not mention the scour type in their literature, or it is not distinguishable in some literature.

Table 4

Different scour depth predictive equations where scour type is not mentioned by authors/undistinguishable

Reference .	Equation .	Citation of the equation in literature (CEL)/data used to develop the equation (DDE)/developed equation is validated with the model of other researchers (DEV) .	Eq No. .
Inglis (1949)		CEL: Abdelmonem et al. (2009) and Ghorbani (2008)	(40)
Inglis (1949) or Inglis – Poona I		CEL: Mueller (1996) and Abdelmonem et al. (2009)	(41)
Inglis (1949) or Inglis – Poona II		CEL: Mueller (1996) and Abdelmonem et al. (2009)	(42)
Laursen & Toch (1956)		CEL: Vonkeman & Basson (2019) and Shamshirband et al. (2020)	(43)
Blench (1960) or Blench – Inglis I		CEL: Mueller (1996) and Boehmler & Olimpio (2000)	(44)
Blench (1960) or Blench – Inglis II		CEL: Mueller (1996) and Boehmler & Olimpio (2000)	(45)
Chitale (1962)		CEL: Mueller (1996), Lee et al. (2019), and Vonkeman & Basson (2019)	(46)
Laursen (1962)		CEL: Shin & Park (2010)	(47)
Coleman (1971)		CEL: Lee et al. (2019)	(48)
Neil (1973)		CEL: Vonkeman & Basson (2019)	(49)
Basak et al. (1975)		CEL: Breusers et al. (1977)	(50)
CSU Equation (1975)		CEL: Abd El-Hady Rady (2020), Kayadelen et al. (2022)	(51)
Chitale (1988)		CEL: Abdelmonem et al. (2009)	(52)
Breusers & Raudkivi (1991)		CEL: Zanke et al. (2011) and Hoffmans & Verheij (2021)	(53)
HEC-18 (Richardson et al. 1993)/(Arneson et al. 2012)		CEL: Wilson (1995)	(54)
CSU formula (Richardson & Davis 1995)		CEL: Choi & Chong (2006)  DDE: Jones (1984)	(55)
Wilson (1995)		DDE: Data collected from bridge sites in the Mississippi river DEV: HEC-18 (Richardson et al. 1993)	(56)
HEC-18/Mueller (1996)	(a) K4 = 1.0 for (d50 ≤ 2.0 mm or d95 ≤ 20 mm) (b) (d50 > 2.0 mm or d95 > 20 mm)	CEL: Nadal (2007)  where and are the approach velocity required to initiate scour for grain sizes d50 and d95, respectively, and is the critical velocity for the incipient motion for grain size d50	(57)
Melville (1997)		DDE: Chabert & Engeldinger (1956), Shen et al. (1966), Hancu (1971), Ettema (1980), Chee (1982), and Chiew (1984)  DEV: By assuming hypothetical data	(58)
Kandasamy & Melville (1998)	where for (K = 5, n = 1), for (K = 1, n = 0.5), and for (K = 1, n = 0)	DDE: Own experimental data; Mueller (1996)	(59)
Melville & Coleman (2000)		CEL: Vonkeman & Basson (2019)	(60)
Ali & Karim (2002)	for steady flow case, where and	DDE: Yanmaz & Altinbilek (1991), DEV: Laursen & Toch (1956), Neill (1964), Breusers (1971) and Melville (1975)	(61)
CSU formula Mohamed et al. (2005)		CEL: Azamathulla et al. (2010), DDE: Own experimental data DEV: Laursen & Toch (1956), Jain & Fischer (1979), Melville & Sutherland (1988), HEC-18 (Richardson et al. 1993)	(62)
Sharafi et al. (2016)		CEL: Shamshirband et al. (2020)	(63)

Reference .	Equation .	Citation of the equation in literature (CEL)/data used to develop the equation (DDE)/developed equation is validated with the model of other researchers (DEV) .	Eq No. .
Inglis (1949)		CEL: Abdelmonem et al. (2009) and Ghorbani (2008)	(40)
Inglis (1949) or Inglis – Poona I		CEL: Mueller (1996) and Abdelmonem et al. (2009)	(41)
Inglis (1949) or Inglis – Poona II		CEL: Mueller (1996) and Abdelmonem et al. (2009)	(42)
Laursen & Toch (1956)		CEL: Vonkeman & Basson (2019) and Shamshirband et al. (2020)	(43)
Blench (1960) or Blench – Inglis I		CEL: Mueller (1996) and Boehmler & Olimpio (2000)	(44)
Blench (1960) or Blench – Inglis II		CEL: Mueller (1996) and Boehmler & Olimpio (2000)	(45)
Chitale (1962)		CEL: Mueller (1996), Lee et al. (2019), and Vonkeman & Basson (2019)	(46)
Laursen (1962)		CEL: Shin & Park (2010)	(47)
Coleman (1971)		CEL: Lee et al. (2019)	(48)
Neil (1973)		CEL: Vonkeman & Basson (2019)	(49)
Basak et al. (1975)		CEL: Breusers et al. (1977)	(50)
CSU Equation (1975)		CEL: Abd El-Hady Rady (2020), Kayadelen et al. (2022)	(51)
Chitale (1988)		CEL: Abdelmonem et al. (2009)	(52)
Breusers & Raudkivi (1991)		CEL: Zanke et al. (2011) and Hoffmans & Verheij (2021)	(53)
HEC-18 (Richardson et al. 1993)/(Arneson et al. 2012)		CEL: Wilson (1995)	(54)
CSU formula (Richardson & Davis 1995)		CEL: Choi & Chong (2006)  DDE: Jones (1984)	(55)
Wilson (1995)		DDE: Data collected from bridge sites in the Mississippi river DEV: HEC-18 (Richardson et al. 1993)	(56)
HEC-18/Mueller (1996)	(a) K4 = 1.0 for (d50 ≤ 2.0 mm or d95 ≤ 20 mm) (b) (d50 > 2.0 mm or d95 > 20 mm)	CEL: Nadal (2007)  where and are the approach velocity required to initiate scour for grain sizes d50 and d95, respectively, and is the critical velocity for the incipient motion for grain size d50	(57)
Melville (1997)		DDE: Chabert & Engeldinger (1956), Shen et al. (1966), Hancu (1971), Ettema (1980), Chee (1982), and Chiew (1984)  DEV: By assuming hypothetical data	(58)
Kandasamy & Melville (1998)	where for (K = 5, n = 1), for (K = 1, n = 0.5), and for (K = 1, n = 0)	DDE: Own experimental data; Mueller (1996)	(59)
Melville & Coleman (2000)		CEL: Vonkeman & Basson (2019)	(60)
Ali & Karim (2002)	for steady flow case, where and	DDE: Yanmaz & Altinbilek (1991), DEV: Laursen & Toch (1956), Neill (1964), Breusers (1971) and Melville (1975)	(61)
CSU formula Mohamed et al. (2005)		CEL: Azamathulla et al. (2010), DDE: Own experimental data DEV: Laursen & Toch (1956), Jain & Fischer (1979), Melville & Sutherland (1988), HEC-18 (Richardson et al. 1993)	(62)
Sharafi et al. (2016)		CEL: Shamshirband et al. (2020)	(63)

K, coefficient; n_e, exponent; ⁠, channel geometry factor; ⁠, angle of attack of flow factor; ⁠, bed condition factor; s, specific gravity of the bed material.

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Inglis (1949) used the flow depth (y), pier width (b), and Froude no. (F_r) to predict the scour depth and proposed Equation (40). Neil (1973) suggested Equation (49) in which a correction factor (K_s) is multiplied by the width of the pier to calculate the scour depth. The correction factor (K_s) value equals 2.0 for the rectangular shape of piers and 1.5 for circular and rounded-nosed piers, respectively. From the literature, it is noted that Johnson (1992) used all the key scour parameters (b, y, F_r, and σ)) to formulate the scour depth predictive equation, i.e., Equation (34). Equation (51) – the CSU (1975) equation is also known as Richardson et al. (1993) equation which is only applicable for the circular shape of the pier. Melville (1997) investigated scour depth as a function of sediment size, gradation factor, flow depth, flow intensity, channel geometry, and pier shape and proposed a common equation that contained various parameter correction variables to allow for more accurate scour prediction for all types of piers which is given in Equation (58). Richardson & Davis (2001) proposed an empirical equation (Equation (55)) to calculate the scour depth which is also known as modifying Colorado State University (CSU) eq or HEC-18 equation. Inglis-Poona II (1949) (Equation (42)) and Blench-Inglis I (1962) (Equation (44)) provide a fair approximation of scour depth value in the case of round and sharp nose shape piers with a large width. Sheppard et al. (2014) reviewed different scour depth predictive equations and suggested that all the predictive equations are better in experimental data sets but overpredict scour depth for field data sets. It is found that the value of the densiometric Froude number (F_rd) has a substantial effect on the maximum equilibrium scour depth.

As shown in Tables 2–4, scour depth predictive equations were developed by using only the width (b) of the bridge pier as the input parameter. Larras (1963) developed a scour depth predictive equation, as shown in Equation (26), Breusers (1965) developed Equation (5), Neil (1973) developed Equation (49), Basak et al. (1975) developed Equation (50), and Chitale (1988) developed Equation (52). Many researchers, such as Johnson (1995), Mohamed et al. (2005), Choi & Chong (2006), Briaud (2015), Abd El-Hady Rady (2020), Kayadelen et al. (2022), and Baranwal et al. (2023b) generally used CSU equation (Equation (51)) for predicting the local scour depth in both clear water scour (CWS) and live bed scour (LBS) conditions.

In Table 6, all the selected models predict poor results based on statical indices, yet among all, the CSU (1975) equation provides MAE, RMSE, and R² values of 1.04, 1.47, and 0.04, respectively, whereas Shen et al. (1966) equation predicted high value of MAE and RMSE and very low R² value. The Pandey et al. (2018) model utilizes the particle Froude number parameter to predict the scour depth, which provides an error in terms of MAE = 2.31, RMSE = 25.55, and R² = 0.02. These predictive models are providing poor results, and this may be due to the input parameter range used in the present research being in a wider range of data sets, as mentioned in Table 5 for the CWS condition and Table 7 for the LBS condition. The various author models are derived for a limited range of data sets. In the present study, the critical velocity of flow is calculated by the Neill (1968) formula. Still, it may differ when V_c is calculated from other models, so it may also be a possible cause of poor prediction of the model. A scour depth model is needed to handle a wide range of data for both CWS and LBS conditions.

In Figure 4(a), almost all the predictive equations give good results, but the Neill (1964) data point is overestimated. In Figure 4(b), Neill (1964) and CSU (1975) data are overestimated and some data from Yanmaz (1989) between 0.5 and 0.7 underestimate the SDR. In Figure 4(c), only Yanmaz (1989) gives a good result, and others poorly estimated the result for this data set. Figure 4(d) shows that all the Mueller & Wagner (2005) data come in the 0.05–0.3 range after the normalization, and Pandey et al. (2018) provide good results, and other models are overestimating. In Figure 4(e), Yanmaz (1989) and Ettema et al. (1998) overestimate the scour depth from 0.05 to 0.6 and Pandey et al. (2018) underestimate the scour depth in the range of 0.4–0.8. In Figure 4(f), Yanmaz (1989) data underestimate the scour depth, Neill (1964) data overestimate in the range of 0.15–0.5, and Pandey et al. (2018) are good at predicting the scour depth among all. In Figure 4(g), it is found that Pandey et al. (2018) overestimate the scour depth in the range of 0.2–0.24 and underestimate the scour depth in the range of 0.3–0.6, Yanmaz (1989) underpredicts the scour depth 0.3–0.6 and Neill (1964) estimates the good scour depth in the range of 0.3–0.5. In Figure 4(h), CSU (1975) poorly overestimates the scour depth, Neill (1964), Yanmaz (1989), and Ettema et al. (1998) overestimate the scour depth, and Pandey et al. (2018) predict the good scour depth in the range of 0.3–0.6. In Figure 4(i) and 4(j), all the models are over predicting scour depth. The reason may be the estimation of the selected five equations in the condition in which the existing developed models do not fall in the category of the data set obtained by Yang et al. (2020) and Garg et al. (2022), respectively. A spurious effect is introduced to the regression parameters when regressing predicted vs observed (PO) values and comparing them against the 1:1 line and reported to opt for observed (on the y-axis) versus predicted (on the x-axis) (OP) regressions instead (Pineiro et al. 2008) so in the present study, observed (on the y-axis) versus predicted (on the x-axis) (OP) axis is used.

In Table 8, all the models predict poor results based on statical indices, yet among all, Yanmaz (2001) equation provides MAE, RMSE, and R² values of 0.36, 0.53, and R² = 0.54, respectively, whereas Hancu (1971) equation predicted high value of MAE and RMSE and very low R² value. CSU (1975) model also predicts the scour depth, which provides an error in terms of MAE = 0.77, RMSE = 0.91, and R² = 0.55.

In Figure 6(a), the CSU (1975) model poorly overestimated the scour depth, and Kothyari et al. (1992b) overestimated the scour depth in the range of 0.1–0.4. Figure 6(b) depicts that Veiga (1970), Kothyari et al. (1992b), Johnson (1995), and Yanmaz (2001) models over-predicted the SDR, whereas only the CSU (1975) model underpredicted the SDR in the range of 0.4–1.0. In Figure 6(c), CSU (1975) overestimates the scour depth, and Veiga (1970) underestimates the scour depth in the range of 0.38–0.6. In Figure 6(d), all the models poorly overestimate the scour depth, in which Johnson (1995) overestimated the scour depth maximum among all. In Figure 6(e), CSU (1975) data poorly overestimated the scour depth, and Veiga (1970) data underpredicted the scour depth in the range of 0.35–0.6. In Figure 6(f), all the models poorly predicted the scour depth, so it can be found that for different author data points, the empirical scour depth predictive equation behaves differently.

Different researchers have performed laboratory experiments in CWS and LBS conditions and developed the scour depth predictive model in which different non-dimensional input parameters, such as b/y, V/V_c, F_r, b/d₅₀, and σ_g, have been considered. These non-dimensional parameters have different input data ranges. The model may be used only when the input parameters for the prediction of d_s/y for field conditions are available; otherwise, the predictive results may produce a large error in estimation. Precaution must be taken before applying any empirical models to estimate d_s/y in CWS and LBS to avoid scale effect. The reader may refer to Link et al. (2019) for more details on the scale effect of different scour depth predictive models for CWS and LBS conditions.

The paper presents more than 80 experimental/field data sets and more than 60 scour depth predictive equations developed over the past seven decades for CWS and LBS conditions. Each equation has limitations, including the type of scouring, the number of data used, the combination of input parameters, and the range of input data used in their development. From the present review of different scour depth predictive models and the application of each to different data points, the following conclusions and recommendations are made:

• The factors affecting scour depth under CWS and LBS are found to be pier width, flow depth, velocity of flow, bed sediment size, standard deviation of bed material, Froude number, and pier correction factors.

• In the case of the CWS condition, it is found that the equilibrium scour depth decreases at a faster rate as flow depth declines with smaller values of relative flow depth (y/b). The bed roughness caused a steeper pressure gradient and raised the amount of bed shear stress. The maximum scour depth around the bridge pier increases with pier diameter, approach velocity, and critical velocity ratio and decreases with bed particle size in CWS. It also decreases as armor layer particle size increases.

• From the LBS condition, it is found that scour depth decreases above the critical velocity and then increases to a maximum value at the transition to flatbed conditions. At higher velocities, the equilibrium scour depth decreases due to the formation of antidunes on the bed surface. It is also found that the scour depth under live bed conditions is generally significantly higher than under clear water conditions and further increases with flow intensity.

• For CWS, Neill (1964), CSU (1975), Yanmaz (1989), Ettema et al. (1998), and Pandey et al. (2018) and in the case of LBS around bridge pier, Veiga (1970), CSU (1975), Kothyari et al. (1992b), and Yanmaz (2001) are recommended to use for predicting scour depth. While utilizing these equations, the input data set range of influencing parameters must be considered while selecting the aforementioned scour depth predictive equations.

Application of all available scour depth predictive models to estimate scouring around the bridge pier is inappropriate for field scenarios. If clear water scour or live bed scour conditions are present around the bridge pier, it is recommended to prefer the clear water scour equation or live bed scour equation, respectively.

The primary reason for global bridge collapses is recognized as the scour of bridge piers. Climate change associated with global warming has the capacity to worsen bridge scour, primarily due to increased river flooding. Bridge engineers need to understand evolving climate risks for effective long-term strategies. Insufficient data and omissions in scour risk assessments create uncertainties. Therefore, more research is recommended by considering different flow, roughness, and non-circular shapes, i.e., square, rectangular, elliptical, and chamfered under temporal scouring conditions. Additionally, exploring the impact of climate change linked to global warming is essential for advancing studies on river flooding conditions.

A.B. rendered support in collecting experimental data, analyzing the data, using different scour depth predictive empirical equations and writing the first draft of the manuscript; and B.S.D. rendered support in article supervising, data analysis and discussion, concluding remarks, and manuscript proofreading.

The authors declare that the research described in this work is not supported financially.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.