Experimental Study on Scour at a Sharp-Nose Bridge Pier with Debris Blockage

Experiments were carried out in a horizontal, 605-mm-wide, 10-m-long sediment recirculating flume at the University of Exeter, United Kingdom. The flume is equipped with a traverse system (with a movement precision of 1 μm) for positioning instruments at specified $x$, $y$, and $z$ coordinates, where $x$, $y$, and $z$ refer to streamwise, spanwise, and vertical directions as shown in Figs. 1(a and b). Experiments were started with a flat bed with $z=0$ representing bed level at $t=0$. The discharge was controlled with the aid of a variable-speed drive on the centrifugal pump driving the flow. Discharge was measured using an electromagnetic flowmeter (resolution $\pm 0.1\mathrm{L}/\mathrm{s}$) installed in the suction pipe of the water recirculating system. A digital point gauge mounted 3.5 m upstream of the pier was used to read the undisturbed elevation of free surface. The precision of this measurement was 0.5 mm due to free surface fluctuations.

Sediment was coarse, uniform silica sand with particle size $d_{50}$, uniformity coefficient $d_{60}/d_{10}$, and geometric standard deviation of sediment particle size distribution $d_{84}/d_{50}$ equal to 1.37 mm, 1.37, and 1.21, respectively. This type of sand translates to medium gravel in prototype scale for the scaling ratio $1:10$ chosen in this work. This choice of sediment was made to ensure

The authors, however, acknowledge that the assumption of uniform bed material is rarely fulfilled in the field (Breusers et al. 1977; Melville and Sutherland 1988) because there is often a gradation of sediment. Consequently, the scour depths observed in field conditions are smaller than the depths predicted on the assumption of uniform bed material.

Geometry and dimensions of pier were defined according to available data on typical sharp-nose masonry bridge piers in the United Kingdom. As shown in schematic in Fig. 1(a), the pier featured wedge-shaped (triangular) cutwaters at its two noses. Each cutwater was an isosceles triangle with a base angle of 45°. Pier width was 50 mm, which is less than one-sixth of the width of the flume. This was chosen to minimize adverse effects of side walls, as recommended by Frostick et al. (2011). It also ensured that there will be no contraction scour according to the equation proposed by Richardson and Davies (2001). Streamwise length of the straight part of the pier was 156 mm, resulting in a width-to-length ratio of 0.32 that matches the average aspect ratio estimated for several masonry bridge piers in Devon, United Kingdom (Devon County Council, private communication, 2015–2017). The pier was fabricated out of PVC and fixed to the flume bed.

Various terms are used in the literature to refer to debris, e.g., log jam, debris raft, drift, and debris mass. While these terms can imply various debris geometries, they generally refer to accumulation of woody debris. In the present work, the process of debris accumulation, studied by Lyn et al. (2003) and Panici and de Almeida (2017), was not investigated. The latter investigated how debris dimensions vary with the Froude number of the characteristic length of individual debris elements. They found that the width to thickness of debris blockage is generally between 3 and 27. In this study, simplified debris shapes, with width-to-thickness ratios within the range found by Panici and de Almeida (2017), were considered. The shapes represented broadly the debris geometries observed to result from woody debris accumulation in full-scale scenarios. Specifically, four simplified debris shapes were used [Fig. 1(c)]. These included: (1) two cylindrical (log) shapes, (2) an inverted half-pyramid (triangular base), and (3) a plate. The cylindrical shapes represent tree trunks, which are the most common debris and can be the initiators for much larger debris mass (Diehl 1997). The inverted half-pyramid and plate shapes represent solid debris mass of small thickness, typical of debris pile-up. The inverted half-pyramid shape was chosen according to Wellwood and Fenwick (1990) and Diehl (1997), who identified a triangular-in-depth shape as one of the main geometries of debris accumulation. It is also similar to the debris shape used by Lagasse et al. (2010) and is a simplified version of the perfect inverted half-cone used by Panici and de Almeida (2017). The plate shape was chosen to represent a mat of small woody elements.

Debris was considered static for simplicity and fixed perpendicular to the pier at the desired elevation [Fig. 1(b)]. This was achieved by a 3 or 5-mm-deep 90° triangular notch at the midlength of the debris. The dynamics of fluid–structure interaction between debris and flow was not studied due to the significant challenge in managing scaling issues, such as creating a debris model of suitable density. Also, understanding the influence of static debris on scour was an important first step to design the experiments that would be able to characterize the scour effect of the dynamics of debris.

Due to the large width and the elongated geometry of masonry bridge piers, debris seldom extends downstream in real-life bridges and was hence considered accordingly in the experiments. Also, all debris models used were solid and smooth as illustrated in Fig. 1(c). This is sufficient for the purposes of this study because, as found by Pagliara and Carnacina (2010) and Lagasse et al. (2010), porosity and roughness of debris do not affect depth and pattern of scour in any appreciable way and can only be considered secondary factors when compared with dimensions of debris and flow intensity.

The debris dimensions were chosen carefully considering real-life debris sizes and the experimental constraints. The spanwise length of debris, $L_{d,y}$, was expected to affect mainly the lateral (transverse) extent of scour and have minimal influence on the maximum scour depth (Lagasse et al. 2010). The value of $L_{d,y}$ was hence set to 300 mm for all debris shapes to maximize the lateral extent of scour while minimizing the effect of flume sidewalls. Diameter/length ratio of the smaller cylindrical debris was defined according to diameter/length ratio of tree debris observed in field conditions. Accordingly, this debris shape with diameter $T_d=16\mathrm{mm}$ had a diameter/length ratio $T_d/L_{d,y}=0.059$, which is the average diameter/length ratio of trees suggested in several field surveys of woody debris in the United States and Europe (Beechie and Sibley 1997; Diehl 1997; Kail 2003; Comiti et al. 2006; Magilligan et al. 2008). The second cylindrical debris with diameter $T_d=32\mathrm{mm}$ was used to quantify the relationship between diameter of the debris and scour. The inverted half-pyramid debris had the same thickness and streamwise length as the large cylindrical debris. This was done to enable comparison between scour maps generated by the two geometries. The plate debris had a thickness of 16 mm and a streamwise length of 150 mm.

The debris models were either 3D-printed or made from steel. The cylindrical shapes and the inverted half-pyramid shape were represented using 3D-printed nylon objects. Steel was used to fabricate the plate geometry as well as a second model of the smaller cylindrical debris. This was aimed specifically at investigating the influence of the streamwise length of resting-on-bed debris on scour.

Hydraulic conditions of experiments are presented in Table 1. Here, $D_0$ = pier width; $B$ = width of approach flow (= flume width); $h$ = depth of approach flow; $Q$ = flow discharge; $\mathsf{R}$ = flow Reynolds number {$=Uh/\nu$, with $U$ and $\nu$ being mean velocity of approach flow [$=Q/(Bh)$] and fluid kinematic viscosity, respectively}; $\mathsf{F}$ = Froude number {$=U/(gh{)}^{1/2}$, where $g$ stands for acceleration due to gravity}; ${\mathsf{R}}_{*}$ = roughness Reynolds number [$={u_{*}k}_s/\nu$ with $u_{*}$ = shear velocity calculated as proposed by Sheppard et al. (2014), and $k_s$ = granular roughness = $2d_{50}$ (Kamphuis 1974), where $d_{50}$ is median particle size of sand]. The term $d_s$ is the maximum measured scour depth.

The experiments were performed for two flow depths, $h=80$ and 131 mm, each with corresponding discharges $Q=19.1$ and $33.4\mathrm{L}/\mathrm{s}$, respectively. For each experiment, the approach flow depth, discharge, and debris distance from the initial bed were adjusted carefully to within 0.5 mm, $0.1\mathrm{L}/\mathrm{s}$, and 0.5 mm, respectively, of the desired values. Reynolds number scaling was not strictly adhered to but $\mathsf{R}$ was kept between 31,000 and 55,000 to ensure the flow was turbulent (Chanson 2004) in all the scenarios. In addition, approach flow was also kept subcritical ($\mathrm{F}<1$). Flow shallowness, $h/D_0$, was 1.6 and 2.6 for the two flow depths, corresponding to shallow flow conditions (Chiew 1992; Melville and Chiew 1999). The value $h/D_0=1.6$ was the smallest feasible value that could be attained under the experimental constraints; $h/D_0=2.6$ was smaller than the flow shallowness used in most of the previous research (Lagasse et al. 2010; Pagliara and Carnacina 2011b).

Contour maps of scour for the measured half of the flume for all scenarios are illustrated in Fig. 3. The location of maximum scour depth, $d_s$, is shown by an enhanced dot on the pier cutwater. In most experiments, this was located at the corner of cutwater with $x=25\mathrm{mm}$ and $y=-25\mathrm{mm}$. This finding is in agreement with results obtained for similar sharp-nose piers without debris in 2-h experiments by Müller et al. (2001) and 8-h experiments by Vijayasree et al. (2017). Khosronejad et al. (2012) also obtained similar results for diamond piers through flume experiments and numerical simulations that were run until equilibrium scour was reached. This finding differs markedly from observations for rectangular/circular piers for which the maximum scour depth $d_s$ is seen to occur at the upstream face of the pier (e.g., Melville 1975; Breusers et al. 1977; Lagasse et al. 2010). The different locations of maximum scour depth for rectangular/circular piers and sharp-nose piers may be due to the following two factors related to the flow around the pier: First, the turbulent horseshoe vortex (THV) system, which is known to be the primary factor causing scour at the upstream face of rectangular/circular piers (Kirkil et al. 2008; Escauriaza and Sotiropoulos 2011; Link et al. 2012; Apsilidis et al. 2015; Bouratsis et al. 2017; Chen et al. 2017), has been observed by previous researchers (Shen et al. 1966; Breusers et al. 1977) to be not as well pronounced at the leading edge of sharp-nose piers. Second, the flow separation and the shedding of the two shear-layers at the two side corners of a sharp-nose pier increases bed shear stresses at these corners (Khosronejad et al. 2012; Vijayasree et al. 2017) and may thereby lead to maximum scour occurring at these locations. In Scenarios 5 and 6, the maximum scour depth occurred between the upstream nose of the pier and lateral corner of cutwater at $x=12.5\mathrm{mm}$. However, the maximum scour depths in these two scenarios were only 0.5 and 1.4 mm larger than the scour depths observed at the corner of the cutwater.

The contour plots and the $d_s$ values for Scenarios 5, 6, 8, 10, and 13 show that debris had a protective effect when on the bed which reduced maximum scour depth. Due to this, the shapes (in plan) of the scour holes for these scenarios differ from the overall shape observed for other scenarios in Fig. 3.

In experiments with $h=80\mathrm{mm}$, $d_s$ increased from 78.3 mm in Scenario 1 without debris to 88.9 and 104 mm in Scenarios 2 and 9 that used 16 and 32-mm cylindrical debris respectively. This corresponds to 14% and 33% increase in $d_s$. For $h=131\mathrm{mm}$, $d_s$ increased from 85.4 mm in Scenario 11 without debris to 110 mm in Scenario 12 with 32-mm cylindrical debris, i.e., 29% increase. Also, $d_s$ for Scenarios 2 and 9 is greater than that for Scenario 11 for which flow depth is comparatively much larger. This shows that the presence of debris in shallow flow ($h/D_0=1.6$) can lead to scour depths exceeding those due to deep flow ($h/D_0=2.6$) without debris, as can be also inferred from Melville and Dongol (1992) for a circular pier. By keeping debris dimensions constant, flow area reduction will be higher in shallow flow compared to deep flow. Therefore, under-free-surface debris will increase velocities (both streamwise and vertical) and, therefore, bed shear stress and consequently scour depth. Similarly, by keeping flow depth constant, thicker and under-free-surface debris increases velocities, bed shear stress, and scour depth more than a thin debris. This result has important implications for scour management of bridges as it highlights the importance of monitoring debris accumulation even when flow is shallow, such as when there may be no flooding. A real-life example of a bridge failure under shallow flow conditions may be Bridge RDG1 48 near Feltham in England, which failed due to scour-induced subsidence in November 2009. The United Kingdom’s Rail Accident Investigation Branch, in its postevent investigation report, had noted that substantial scour had occurred despite flow depth being below the level for a flood alert, and that this was most likely due to debris accumulation that was observed upstream of the subsided abutment (RAIB 2010).

Another finding from the contour maps is that the maximum scour depth $d_s$ for Scenario 7 was 16% smaller than that for Scenario 9. The overall size of scour hole for Scenario 9 was also larger than that for Scenario 7. This can be attributed to the fact that the inverted-pyramid debris has a smaller blockage area in the plane normal to the flow (i.e., in $y\text{–}z$ plane) than the cylindrical debris. Consequently, the inverted-pyramid debris may have caused a smaller increase in flow velocities and bed shear stress than the cylindrical debris of identical thickness.

As can be inferred from Fig. 3, the presence of debris just under the flow free surface (Scenarios 2, 7, 9, and 12) caused maximum scour. For these scenarios, the effect of debris on flow can be qualitatively explained as follows. Debris at the pier reduced flow cross-sectional area. Reduced flow area led to acceleration of the flow and increased bed shear stress. Also, when debris was located just under the flow free surface, it diverted flow streamlines towards the bed, thereby increasing downward flow which loosens and lifts sediment particles (e.g., Hjulstrom 1939; Shen et al. 1966; Yalin 1972; Melville 1975; van Rijn 1993). Together these two phenomena increased sediment transport and hence enhanced scour around the base of the pier. At the beginning of the experiments, the onset of scour was observed at the upstream nose of the pier, possibly due to the presence of a horseshoe vortex. However, as the scour progressed, downflow at the pier base was forced into the scour hole by the three-dimensional vortex generated by flow separation from the bed at the upstream edge of scour hole (Dey et al. 1995). This caused scour to quickly propagate downstream where vortex shedding from the two upstream lateral corners of the pier would further increase scour depth.

A simple but approximate way of determining the scour due to debris in front of a pier is to superimpose the scour depths produced individually by the pier and the debris. Debris can be considered to create a pressurized flow as during inundation of bridge decks proposed by Umbrell et al. (1998). The continuity of flow discharge necessitates that total flow discharge, $Q$, at location of debris can be split as follows:where $Q_{\mathrm{above}}$ and $Q_{\mathrm{below}}$ = discharges passing above and below debris, respectively. For ease of computation, the mean flow velocity above debris can be assumed to be equal to the mean approach velocity, and the mean flow velocity below debris, at equilibrium stage of scour, can be assumed to be equal to the critical velocity. Thenwhere $h_a$ and $h_d$ = height of approach flow above and below debris; and $d_{s,p}$ = scour depth due to pressurized flow. Rearranging Eq. (2) gives

When debris is located just under flow free surface, i.e., for Scenarios 2, 9, and 12, with $h_a=0$ and $h_d=64$, 48, and 99 mm, Eq. (3) yields $d_{s,p}=11.2$, 27.2, 24.1 mm, respectively. Therefore, the total scour depths, i.e., ($d_{s,p}+d_{s,0}$) where $d_{s,0}$ is the scour depth for the scenario without debris, are 89.5, 105.5, and 109.5 mm, respectively. These values are within $\pm 1.5\%$ of the measured values. It is worthwhile to mention that this approach did not predict the measured scour depths for scenarios with debris being at elevations other than just under the flow free surface. This can be due to the complexity that debris at lower elevation introduces into the flow and sediment transport.

Figs. 4(a and b) show longitudinal and cross-stream profiles of scour at $x\text{–}z$ and $y\text{–}z$ planes passing through $y=-26.1$ and $x=25\mathrm{mm}$ respectively. These $x$ and $y$ coordinates correspond to the scour measurement point nearest to the lateral corner of the pier cutwater, where the maximum scour depth was observed for most of experiments (Fig. 3). The upstream slope of the scour hole along $x$ in all experiments was $\sim 32°$ [Fig. 4(a)], which is 23% higher than the angle of repose for the chosen sand. This demonstrates that, while equilibrium scour depth may not have been reached, the 5-h duration of the experiments was sufficient to attain quasi-equilibrium scour as defined by Dey et al. (1995). Also, while in some scenarios (e.g., 9 and 10) the scour hole was seen to extend toward the flume side walls [Fig. 4(b)], there was no actual scour observed visually at the walls. This can also be approximately verified using the measured scour data. For example, for Scenario 9 in Fig. 4(b), the bed elevation at $y=-190\mathrm{mm}$ was $-16.9\mathrm{mm}$. As the scour profile has an approximately constant gradient towards the flume wall ($y=-302.5\mathrm{mm}$), then the point where there is no scour ($z=0$) is at $y\simeq -226\mathrm{mm}$, which is 76 mm away from the flume wall. This implies that there was no contraction scour in the experiment, which is also verified by the equation proposed by Richardson and Davies (2001).

As can be seen in Fig. 4, both streamwise and cross-stream extent of scour generally increased with $d_s$. The exceptions are Scenarios 6 and 8 (red plots) where debris was resting on the initial bed and could move downward as scour progressed. This caused the shape of the scour hole to deviate from those for the rest of scenarios. In Scenarios 2–5, $d_s$ and streamwise extent of scour decreased with debris elevation. Spanwise extent of scour however changed negligibly. This is due to the use of the same spanwise length $L_{d,y}$ for all debris models.

The longitudinal extent of scour in Scenario 6 was significantly greater than that for Scenario 5 even though both scenarios used identical debris dimensions. This is due to the debris being fixed at its elevation in Scenario 5, which diverted the flow under debris and toward the bed. This meant flow energy was dissipated mainly via increasing $d_s$ and not extending scour downstream. In Scenario 6, because debris was resting on the bed and moved downward as scour progressed, it protected the bed to a degree and diverted the flow over which caused downstream extension of scour. In Scenario 8, the plate debris protected the bed in all three directions by diverting the flow away from the bed and caused very little longitudinal and spanwise extent of scour. Some small secondary scour was however observed adjacent to the plate edges parallel to the flow direction.

The volume of the scour hole $V_s$, which is indicative of the extent of loss of soil support to the pier foundation, was calculated for each scenario using the scour map for the hatched region around the pier shown in the schematic insert of Fig. 8. The value of $V_s$ is observed to increase from 1.45 L for Scenario 1 without debris to 1.8 and 2.7 L for Scenarios 2 and 9, respectively, with cylindrical debris. These values indicate 24% and 86% increase in $V_s$ for an increase of 14% and 33% respectively in maximum scour depth $d_s$. Also, a 100% increase in debris thickness from Scenario 2 ($T_d/h=0.2$) to Scenario 9 ($T_d/h=0.4$) caused a 50% increase in scour volume.

As the streamwise and spanwise extents of scour increase with $d_s$, the total volume of scour hole, $V_s$, is also expected to increase with $d_s$. The exact nature of the relationship between $d_s$ and $V_s$ is found by fitting an equation to the data as shown in Fig. 5. The volume of scour is seen to increase quadratically with $d_s$ following the equation $V_s=0.00022d_s^2$, where $V_s$ and $d_s$ are in liters and millimeters, respectively. In prototype scale, the corresponding equation will be $V_s=2.2d_s^2$, where $V_s$ and $d_s$ are in cubic meters and meters, respectively. This equation has a similar form to equations proposed by Link et al. (2008) and Diab et al. (2010) for scour at circular and square piers in the absence of debris. This information can be useful in planning post flood remedial works (e.g., Solaimani et al. 2017) where often only maximum scour depth is initially measured.

The effect of debris elevation on maximum scour depth was investigated using the cylindrical shapes (Table 1). Fig. 6 shows the variation of normalized scour depth ($d_s/d_{s,0}$) versus normalized debris elevation ($h_d/h$) as derived from measured data. The term $d_{s,0}$ is the scour depth observed for the related baseline scenario, i.e., when there is no debris. The term $h_d$ is debris elevation as measured from the initial bed level to the bottom of the debris; $h_d/h=0$ corresponds to scenarios where the bottom of debris was at the initial bed level; and $h_d/h=1$ corresponds to scenarios where the debris is just above the free surface of approach flow (i.e., equivalent to no debris). Data for Scenarios 1–5, with $h/D_0=1.6$ and $T_d/h=0.2$, are shown using squares. As can be inferred from the figure, scour was maximum when debris was located just below the free surface, i.e., for Scenario 2 with $h_d/h=0.8$. The scour depth reduced as debris was moved downwards. This relationship between debris elevation and scour depth can be explained as follows. When debris is just below the free surface, it diverts the streamlines below it downwards toward the bed and a large portion of flow impacts the bed. However, as debris is moved closer to the bed, it splits the flow and only those streamlines obstructed by the debris or below it are directed downward. The rest of the streamlines are above the debris and cannot affect the bed significantly. The streamlines below debris impact the bed but because their velocities are smaller than those close to the free surface, the scour depths they produce also reduce with debris elevation. Lastly, when debris rests on the bed, it protects the bed to a large extent as most of the flow has to pass over the debris. In addition, and compared to the case with debris just under the flow free surface, scour depth reduced if the debris was removed from the flow. These results confirm the overall pattern observed by Lagasse et al. (2010).

Because the characteristics of the physical process of scour were not expected to change with cylindrical debris diameter and flow depth, fewer scenarios were explored for cylindrical debris of 32-mm diameter at $h=80$ and $h=131\mathrm{mm}$. The purpose was to understand how the maximum scour depth varied assuming the overall nature of the relationship between $d_s/d_{s,0}$ and $h_d/h$ remains unchanged. Results for Scenarios 1, 9, and 10 with $h/D_0=1.6$ and $T_d/h=0.40$ and for Scenarios 11, 12, and 13 with $h/D_0=2.6$ and $T_d/h=0.24$ are also shown in Fig. 6. These two latter curves further confirm an earlier observation that debris accumulation in shallower flow can cause scour depths that exceed those during deep flow without debris.

Scour results for Scenarios 1, 6, and 8, in which debris was resting on the bed and only streamwise length of debris $L_{d,x}$ was changed, can be used to characterize the influence of streamwise length of resting-on-bed debris on scour depth. Similar to this analysis, scour depth $d_s$ and streamwise length of debris $L_{d,x}$ are normalized with respect to $d_{s,0}$ and $h$, respectively. Here $d_{s,0}$ corresponds to the scour depth for the related no-debris scenario, i.e., Scenario 1 with $L_{d,x}=0$. The normalized quantities are plotted in Fig. 7. The protective effect of debris is seen to increase with increasing $L_{d,x}$. However, after $L_{d,x}/\mathrm{h}\simeq 1.8$, further increment of $L_{d,x}$ does not affect scour significantly because there is always some scour just downstream of debris.

The results indicate that debris resting on bed can reduce scour depth by around 50% relative to the scour depth for no-debris scenario ($d_s/d_{s,0}\sim 0.5$). This is in contrast to the scour worsening effects of debris, when it is just under the flow free surface shown previously. In fact, for this scenario (i.e., $h_d=h$), Lagasse et al. (2010) and Pagliara and Carnacina (2011a) found that maximum scour occurs when streamwise length of debris is equal to flow depth and $3D_0$, respectively.

Based on the scour depths in Scenarios 1, 2, and 9, and data from Melville and Dongol (1992), Lagasse et al. (2010), and Pagliara and Carnacina (2011b) an attempt is made to characterize the influence of debris thickness $T_d$ on scour depth. Data from each of the four sources that have identical hydraulic conditions and debris characteristics excluding $T_d$ are grouped together. Groups that have at least three data points are considered in this analysis. Only debris with a rectangular cross-section section in the plane normal to the flow (i.e., $y\text{–}z$ plane) is investigated here. Scour depth and debris thickness are presented in normalized form, i.e., $d_s/d_{s,0}$ and $T_d/h$ in Fig. 8. Two exponential curves were fitted to data; $d_s/d_{s,0}=1+3.2(T_d/h{)}^{1.237}$ for data from Pagliara and Carnacina (2011b) and $d_s/d_{s,0}=1+0.32(T_d/h{)}^{1.237}$ for the rest of data. The general form of the equation is adopted from an extensive study by Pagliara and Carnacina (2011b), with a new coefficient and exponent derived to minimize differences with measured values. The value of $d_s/d_{s,0}$ increases from 1 for $T_d/h=0$ (Scenario 1) to 1.04 for $T_d/h=0.2$ (Scenario 2) and then to 1.1 for $T_d/h=0.4$ (Scenario 9).

The performance of the following four widely used equations for estimation of maximum scour depth is compared using the collected data:

All these equations were developed solely for a pier-only case and were not intended for estimating scour due to debris blockage. However, due to the lack of an approach that explicitly accounts for debris effects, these equations are still used by practitioners for estimating debris-induced scour by using the equivalent pier width concept proposed by Melville and Dongol (1992) and Lagasse et al. (2010). In this section, the experimental results are used to compare the performance of the equations when using the equivalent pier width approach.

To account for the sharp-nose geometry of the pier, a shape factor of 0.9 was applied to the original width of pier $D_0$ as recommended by Melville (1997) and Richardson and Davies (2001). Also, chosen equations, except that of Oliveto and Hager (2002), calculate equilibrium scour depth while the laboratory data were collected after 5 h. To enable comparisons with the measured scour depth, the equilibrium scour depth predicted by each equation was converted to scour depth at $t=300\min$ using the equation for temporal development of scour proposed by Melville and Chiew (1999).

Results from the equations were only analyzed for scenarios with debris located just under the free surface, i.e., Scenarios 1, 2, 7, 9, 11, and 12. For cases with debris, an equivalent pier width $D_e$ was evaluated to account for the effect of debris blockage. The value of $D_e$ was estimated using the equations by Melville and Dongol (1992), and its modified version proposed by Lagasse et al. (2010) (Table 2). For each scenario, measured scour depth $d_s$ was compared to the estimated scour depth from each equation $d_{s,EST}$ [Figs. 9(a and b)]. Also, the mean percentage error relative to the measured value was calculated for each equation (Table 3). From Figs. 9(a and b), and Table 3, it is clear that the equation by Richardson and Davies (2001) consistently resulted in estimations closest to the measured values compared to the other three equations. The equation by Oliveto and Hager (2002) was observed to underestimate scour. The equations by Melville and Chiew (1999) and Sheppard et al. (2014) were seen to overestimate scour and may hence be preferred for design purposes in cases in which a degree of conservatism is sought.