Effects of Bottom-Up Blockage on Entrance Loss Coefficients and Head-Discharge Relationships for Pipe Culvert Inlets: Comparisons of Theoretical Methods and Experimental Results

Flow through unblocked culverts is commonly classified using the USGS system, which distinguishes between six or seven flow types, based on the location of the control section, and the relative submergence of the inlet and outlet. Descriptions of the flow types and corresponding design equations can be found in publications such as Bodhaine (1968), Schall et al. (2012), and Chin (2013). Flow types 1–3 occur when the inlet is unsubmerged, and types 4–7 occur when the inlet is submerged (Fig. 1). For gated culverts, additional flow types can occur based on the relative inlet blockage caused by the gate (Cantero-Chinchilla et al. 2023).

Under outlet control conditions, energy is lost due to the contraction and expansion of the flow field near the inlet, and the resulting entrance head loss ($\mathrm{\Delta }{H}_e$) can be calculated as follows (Schall et al. 2012):

All variables are defined at the end of the paper. Here, the entrance loss coefficient “$k_e(v)$” is used to indicate that $\mathrm{\Delta }{H}_e$ is calculated using the average culvert barrel velocity ($v$). For the purpose of this study, the hydraulic efficiency of an inlet was defined by $k_e(v)$, with a low value indicating high efficiency. Values for $k_e(v)$ are commonly given as a single, constant value for each inlet configuration (inlet type, cross section shape, and edge geometry) for type 4, 6, and 7 flow (Schall et al. 2012). $k_e(v)$ is also known to vary with the approach-to-culvert contraction ratio (Idelchik 1986; Tullis et al. 2008) and the entrance length (French 1956; White 2016). For type 2 and 3 flow, $k_e(v)$ has been found to depend on the relative headwater elevation (Smith and Oak 1995; Tullis et al. 2005, 2008).

For pressure flow in a circular pipe, the friction slope through the culvert barrel can be estimated using the method of Swamee and Jain (1976) and the Darcy–Weisbach equation (White 2016):

For a culvert inlet with a given blockage ratio ($A_b/A$), the ratio of the average velocity in the blocked section ($v_b$) to the average velocity in the full cross section area of the pipe ($v$) can be found through continuity

To indicate entrance loss coefficients for partially blocked inlets, “$k_{eb}$” is used. It follows from Eqs. (1) and (4) that for the same energy loss, $k_{eb}$ will have different values depending on if $v$ or $v_b$ is used. For reasons of clarity, “$k_{eb}(v)$” and “$k_{eb}(v_b)$” will therefore be used to indicate which velocity is used. The relationship between the two loss coefficients can be found as follows:

Eq. (5) allows for comparison of entrance loss coefficients that use different velocities to determine $\mathrm{\Delta }{H}_e$.

ELM accounts for the effects of the inlet blockage ratio ($A_b/A$) under outlet control conditions (Weeks et al. 2009; Witheridge 2009)

ELM assumes that the inlet is blocked by a thin-walled orifice plate with a circular opening concentric with the inlet and that the flow contraction is constant with increasing blockage ratio (Witheridge 2009). $k_{eb}(v)$ can be substituted for $k_e(v)$ in Eq. (1) to determine the entrance loss under blockage conditions. Idelchik (1986) gives a modified version of ELM where the value of $k_e(v)$ is given as a function of the edge size relative to the hydraulic diameter of the unblocked part of the inlet.

For inlet control conditions, the semidimensionless head-discharge relationship depends on the inlet configuration, inlet cross section rise, and the barrel slope and can be accurately estimated using a continuous 5th-degree polynomial equation, covering both unsubmerged (type 1) and submerged (type 5) inlet control conditions (Schall et al. 2012)

The effective head ($H_w^{’}$) is calculated as $H_w-(S{L}^{’})$, using $L^{’}=-0.5D$ for nonmitered inlets, and $L^{’}=0.7D$ for the mitered inlet, consistent with French (1955, 1961) and Schall et al. (2012). For the purpose of this study, efficiency was defined by the $(H_w^{’}/D)/(K_{u}Q/A{D}^{0.5})$ ratio, with a low ratio indicating high efficiency.

RAM calculates the ratio of discharge capacities for blocked ($Q_b$) and unblocked ($Q$) conditions (Weeks et al. 2009)

RAM was derived using dimensional analysis and assumes that the efficiency is constant with increasing blockage ratio (Witheridge 2009). The $Q_b/Q$ ratio is mathematically identical to $(K_u{Q}_b/A{D}^{0.5})/(K_{u}Q/A{D}^{0.5})$ for given values of the diameter $D$ and cross section area $A$. It should be noted that the $Q_b/Q$ ratio is not used in Eq. (7) or in other commonly used design equations for type 1 and 5 flow, but that RAM can be included in the head-discharge relationship using the relationship $Q_b=Q_b/Q\times Q$.

The entrance loss coefficients for the unblocked inlets were found to be in agreement with those of both Schall et al. (2012) and Idelchik (1986) within uncertainty for all inlets, except the $1:1$ bevel edge, which was in better agreement with Idelchik (1986) (Table 2). While $k_e(v)$ varied with $H_w/D$, the average value of $k_e(v)$ could be determined within the uncertainty of all values for each experiment. The results show repeatability within uncertainty for the three experiments using the $1:1.5$ bevel inlet (Table 2).

For the blockage ratio experiments, the relationships between $k_{eb}(v)$, $k_{eb}(v_b)$, and $A_b/A$ for the square edge in headwall and thin-walled projecting inlets were analyzed, and it was found that $k_{eb}(v_b)$ could be expressed as a linear function of $A_b/A$ within uncertainty [Fig. 11(a)]. From Eq. (5), it follows that $k_{eb}(v)$ is a linear function of $(A_b/A)/(1\text{-}{A}_b/A{)}^2$ [Fig. 11(b)]. The following results are in the form of $k_{eb}(v_b)$, as it is more easily expressed as a function of the blockage ratio, but values of $k_{eb}(v)$ used in Eq. (1) can be calculated using Eq. (5).

For the blockage shape and roughness experiments, the results showed a significant variation in $k_{eb}(v_b)$ with blockage shape for all inlets (Table 3). The maximum difference in $k_{eb}(v_b)$ between blockage shapes varied between the different inlets; 14%–37% for $A_b/A=0.374$, and 19%–56% for $A_b/A=0.748$, indicating that blockage shape effects increased with the blockage ratio. The highest values of $k_{eb}(v_b)$ were generally found for the square or plate blockage elements (blunt bodies), and the lowest loss coefficients were found for the triangular or curved blockage elements (streamlined bodies). No significant differences were found between blockage elements of similar shape but different roughness. It should be noted that the roughness was limited to $d_{100}/\mathrm{\Delta }{z}_b\le 0.053$ and that there were significant differences between the triangular and stepped blockage elements. The difference varied between the different inlets; 2%–28% for $A_b/A=0.374$ and 9%–43% for $A_b/A=0.748$, indicating significant effects of larger blockage element surface structures. These findings are in general agreement with the theory of hydrodynamic drag, where form drag is large compared to friction drag (White 2016). To the degree that $k_{eb}(v_b)$ is a function of the drag of the blockage element, the effects can be expected to vary with the approach flow Reynolds number (White 2016). However, the variation in $k_{eb}(v_b)$ with the Reynolds number was found to be within the estimated uncertainty for each experiment. For the mitered inlet, $k_{eb}(v_b)$ varied significantly with both blockage shape and location, indicating a change in the zone of flow separation from the US to the DS blockage position. As noted, the blockage ratio was calculated perpendicular to the culvert barrel axis, giving the same value of $A_b/A$ for the two cases, but this difference indicates a dependence of $k_{eb}(v_b)$ on the unblocked cross section area perpendicular to the mitered section (Fig. 10). However, determination of the cross section area perpendicular to the culvert barrel axis is consistent with existing design methods and allowed for direct comparison of the mitered and nonmitered inlets, and this effect was therefore not investigated further. It was found that the average, minimum, and maximum $k_{eb}(v_b)$ values could be expressed as linear functions of $A_b/A$ for the nonmitered inlets, while the mitered inlet gave a non-linear relationship with $A_b/A$

The blockage regression coefficient $\beta$ accounts for the differences between the inlets and blockage shapes. Values of $\beta$, ${\beta }_1$, and ${\beta }_2$ for the average, maximum, and minimum and values of $k_{eb}(v_b)$ are given in Table 4.

Eqs. (10) and (11) were compared to ELM using the formulations given in Idelchik (1986) and Witheridge (2009) and the embedded culverts of Tullis et al. (2008), using ${\beta }_{\max }$ as the most conservative estimate of blockage effects. For the mitered inlet, the DS position was used for comparison. The results for the different inlets are plotted in Fig. 12, and estimation errors and uncertainty are given in Table 5. The results show that ELM significantly overestimates the experimental $k_{eb}(v_b)$ values for inlets with higher hydraulic efficiency, indicating that flow contraction is significantly reduced with increasing blockage ratio. This is in agreement with Idelchik (1986), where $k_e(v)$ is reported to decrease as the size of the edge or socket increases relative to the hydraulic diameter. However, using values of $k_e(v)$ based on the relative inlet edge size yielded values of $k_{eb}(v_b)$ close to those of ELM, except for the thin-walled projecting inlet, indicating that bottom-up blockage conditions differ fundamentally from the concentric blockage assumptions of ELM (Fig. 12). For embedded culverts, $k_{eb}(v)=k_{eb}(v_b)$, as the barrel and inlet cross sections are identical. For this condition, $k_{eb}(v_b)$ remained largely constant with increasing blockage ratio, indicating that $k_{eb}(v_b)$ is dependent on the abrupt expansion of the flow field downstream of the blocked inlet section for bottom-up blockage conditions (Fig. 12).

The head-discharge relationships for the unblocked inlets were estimated using Eq. (7), resulting in estimation errors no larger than 0.5% (Table 6). The experimental $H_w^{’}/D$ values were compared to those of French (1961), and it was found that the results agreed within uncertainty for $H_w/D>0.7$, but gave larger estimation errors for lower relative headwater elevations (Table 7). This difference was attributed to the differences between the inlet invert edges in the two studies, which is consistent with French (1961) where it is reported that flow separation is suppressed in the region around the inlet invert at higher relative headwater elevations. The repeatability tests showed that the regression curves from all three runs using the $1:1.5$ bevel edge inlet were within the uncertainty of the average regression curve for $H_w^{’}/D\ge 0.5$ (Fig. 13).

For the blockage ratio experiments, it was found that $Q_b/Q$ increased rapidly from $H_w^{’}/D=\mathrm{\Delta }{z}_b/D$ and converged to an approximately constant value for $H_w^{’}/D\gtrsim 1.3$ for all inlets, showing a clear difference between unsubmerged (type 1) and submerged (type 5) inlet conditions. Analysis of the results showed that the average submerged (type 5) $Q_b/Q$ ratio could be expressed as a linear function of ($1–A_b/A$), within uncertainty. The results are shown in Fig. 14 and Table 8

For the blockage shape and roughness experiments, it was found that the more efficient inlets yielded higher $Q_b/Q$ ratios for both the average, minimum, and maximum $Q_b/Q$ ratios, but this trend was only significant for $A_b/A=0.748$. Fig. 15 shows the minimum $Q_b/Q$ ratio for all inlets and blockage conditions. During the experiments, it was observed that for the socket and bevel inlets, the control section moved from the upstream edge for $A_b/A=0.374$ to the downstream inlet edge for $A_b/A=0.748$, consistent with increased efficiency (French 1961). The two conditions are shown in Fig. 16. However, the experimental data are not sufficient to accurately determine the blockage ratio required for control edge change. No significant differences were found between the maximum, minimum, and average $Q_b/Q$ ratios for any of the tested inlets, indicating insignificant shape and roughness effects (Table 9). These findings are in general agreement with the results of French (1961) where the head-discharge relationship is reported to be mainly influenced by the orientation of the control surface relative to the culvert barrel axis, showing limited variation with upstream factors when using vented inlets. For the mitered inlet, it was found that the DS blockage position yielded the lowest efficiency, indicating a dependency on the location of the blockage element and the cross section area perpendicular to the mitered face, similar to the results for outlet control (type 4 flow).

The minimum $Q_b/Q$ ratios from Table 9 were compared to RAM and the embedded culverts of Tullis et al. (2008). For the mitered inlet, the $Q_b/Q$ ratio for the DS position was used. The results are plotted in Figs. 15 and 17, and uncertainty and estimation errors are given in Table 10. The comparison shows that RAM underestimates the experimental $Q_b/Q$ ratios, indicating that the method does not account for the increased efficiency associated with changes in cross section shape and control edge location caused by partial blockage (French 1961). The constant value of $Q_b/Q$ for the square edge in the headwall inlet in Table 9 also supports this finding since this inlet configuration only has one potential control surface oriented 90° to the culvert barrel axis (French 1961). Eq. (12) significantly underestimates $Q_b/Q$ for the $1:1.5$ bevel and projecting socket edge inlet and overestimates $Q_b/Q$ for the mitered inlet for $A_b/A=0.748$. It should also be noted that only the blockage plates allowed for a blockage element control surface perpendicular to the culvert axis, consistent with the higher $Q_b/Q$ ratios found for the shape and roughness experiments, compared to the blockage ratio experiments for $A_b/A=0.748$ (Figs. 14 and 17).

The comparisons to embedded culverts show differences from the relationship of Eq. (12) and are generally closer to RAM (Fig. 17). Tullis et al. (2008) did not test the unblocked case, and so the $Q_b$ values for embedded culverts were compared to the $Q$ values from the present study for corresponding inlet configurations. Differences in the experimental setup between the present study and Tullis et al. (2008), such as small differences in inlet edge geometry and the use of vented inlets in the present study are therefore sources of uncertainty. However, it should be noted that the differences between inlet blockage and embedment are small under inlet control (type 1 and 5 flow) compared to under outlet control (type 4 flow) (Figs. 12 and 17).

Due to the large number of measurements, a full analysis of the relationship between the uncertainties used to determine the uncertainty of the entrance loss coefficients and head-discharge relationships was not included in this study. However, it was generally found that for $k_e(v)$ and $k_{eb}(v_b)$, the absolute uncertainty decreased with increasing discharge, leading to higher uncertainty for combinations of less efficient inlets, and higher blockage ratios. For inlet control, it was found that the absolute uncertainty of $Q_b/Q$ decreased with increasing blockage ratio. Previous studies have generally not included the uncertainty of the results (French 1955, 1961; Idelchik 1986), and further work on the application of the method of Kline and McClintock (1953) or similar methods to the different USGS flow types is therefore warranted.

The relative estimation error and experimental uncertainty were compared for the full sets of entrance loss coefficients and the head-discharge relationships, and it was found that 56 of the 1,272 entrance loss coefficients (4.4%) and 5 of the 1,760 $H_w^{’}/D$ measurements (0.3%) had estimation errors larger than the estimated uncertainty. The larger errors were generally found for single datapoints in each experiment and at different values of $H_w/D$ and $H_w^{’}/D$. It was therefore assumed that these errors were related to transient flow conditions and/or errors with the measurement or data registration equipment. These datapoints were therefore removed from the data sets before the analysis. For the reduced data set, the estimated uncertainty accounted for the observed variations in entrance loss coefficients and head-discharge relationships (Table 11). While the unblocked inlet control experiments showed significant deviation from French (1961) at low headwater elevations (type 1 flow), only data in the range $1.3<H_w^{’}/D<3.0$ were used to determine the $Q_b/Q$ ratio. It is therefore argued that the results of the present study are of similar validity to those that form the basis for the existing minimum performance methods (French 1955, 1961; Schall et al. 2012).

For the outlet control (type 4 flow) experiments, the minimum culvert barrel Reynolds number was found to be ${6.8\times 10}^4$, indicating Reynolds invariance of $k_e(v)$ and $k_{eb}(v_b)$ for all experiments (Idelchik 1986). Other potential scale effects were not evaluated, but since the average $k_e(v)$ and $k_{eb}(v_b)$ could be determined within the uncertainty for each measurement for the reduced data set, it is argued that effects of approach flow conditions, entrance length, and scale effects were not significant. For the inlet control experiments, the use of vented inlets is consistent with previous experimental setups that form the basis of existing minimum performance methods (French 1955, 1961; Schall et al. 2012). While scale effects and effects of the approach-to-culvert contraction ratio and entrance length cannot be ruled out completely, it is argued that the results are of similar validity to those used in existing design frameworks.

The main finding of this study is the general verification of the ELM and RAM; that hydraulically efficient inlets are less affected by blockage, illustrating the importance of the inlet configuration and the evaluation of potential blockage effects in hydraulic culvert design. However, ELM and RAM do not account for observed increases in hydraulic efficiency with increased blockage ratio and are conservative for the range of idealized bottom-up blockage conditions used in this study. Due to the large variation in in situ blockage conditions, it is recommended to use the maximum estimates for $k_{eb}(v_b)$ and the minimum estimates for $Q_b/Q$ as a conservative approach. Application of the results of this study, as well as ELM, RAM, and top-down blockage effects estimated based on gated culverts depend on the ability to estimate the blockage ratio for a given design flood discharge. The results indicate that the blockage ratio is the most important blockage factor, and further research into the relationship between blockage ratio, catchment conditions, and flood magnitude is warranted. Further study into type 2 and 3 flow and flow classification for bottom-up blocked inlets, the uncertainty of the different USGS flow types, and the effects of porous plug blockage is also warranted.