Hydraulic geometry relationships and their relevance at near bank-full flow O. Navratil, M.B. Albert & J.M. Grésillon CEMAGREF, Lyon, France

ABSTRACT: Hydraulic Geometry Relationships (HGR) describe the variations of hydraulic parameters with discharge. In this paper, we examined how to assess HGR (1) at a relevant reach length and (2) for in bank flows, up to bank-full flows. Indeed, bank-full discharge is essential according to aquatic habitat and main-channel morphology structuring. With the help of a 1D hydraulic model, we defined a simple survey protocol applied on one river reach, in order to estimate an accurate HGR. This methodology would provide an interesting tool to develop synthetic approach of physical functioning of a river reach and therefore to link it with biological requirements. reaches, would provide a robust protocol to 1 INTRODUCTION estimate accurate HGR. Leopold and Maddock (1953) introduced the concept of hydraulic geometry as a quantative 2 METHOD description of the way in which river width, depth, 2.1 A four steps methodological framework velocity vary with changing discharge. HGR are At a first step, we build a 1D hydraulic model at used by hydrologists, geomorphologists and river reach scale. The hydraulic model allows to ecologists (e.g., Jowet, 1998, Lamouroux et al., test the relevancy of HGR at reach scale and to 1998) to express hydraulic parameters as simple estimate the reference HGR coefficients and power function relationships of flow discharge. exponents (step 2). A test of robustness is led to Different survey protocols exist to estimate study the conditions, length of the reach, number coefficients and exponents of HGR. However, as of cross-sections, choice of calibrating discharges, no generalized survey protocol exists, results show at which HGR well describe mean hydraulic a great variability. The purpose of this paper is to parameter’s values and their variability at river present a methodological framework to define a reach scale (step 3). At step 4, we estimate errors simple survey protocol (1) in accordance with the induced by this survey protocol. hydraulic processes at reach scale and (2) relevant at near bank-full flows. Indeed, most of the time 2.2 Study site and data survey HGR are focused on low flows and don’t take into account near bank-full discharges although they We aimed at selecting an homogeneous river reach are very important from a geomorphic point of to be in good agreement with HGR requirements. view. Bank-full discharge is considered as a The Olivet gauged site is located in the Loire river formative discharge for natural alluvial rivers (e.g., basin, France. It is an alluvial river with perennial Emmett, 2001) and is sensitive to human actions flow. Its associated river basin is a 75.5 km² (e.g., Gregory, 1974). Then, hydraulic parameter’s limestone drainage area. Bank-full discharge is values for discharges varying from low to bankabout 3.9 m3/s (estimated with the 1D hydraulic full flows are highly dependent on bank-full model and Leopold’s definition (Leopold, 1994)). discharge. HGR that consider this range of Median and mean annual low flows are discharge would integrated morphological dynamic respectively 0.13 m3/s and 0.4 m3/s. of the river reach. Topographic measurements were practiced in In this paper, an application at the Olivet river the stream channel and adjacent floodplain, and at Beaumont Village, France, illustrates that linked to the staff gauge elevation using an electronic, digital, total-station theodolite. Fifty-six protocol. This method, leaded on different river

cross-sections were surveyed to describe the main morphological variations over a 380 meters long reach (50 times the mean bank-full width (Wb=7.5 m)). Morphological features were systematically surveyed at each cross-section to describe the main breaks in the cross-section geometry. Two longitudinal water profiles were surveyed (at 0.182 m3/s and 1.99 m3/s). A rating curve with its range of validity (from 0.047 to 19 m3/s) was provided by the French Regional Environmental Department. 2.3 The hydraulic model (step 1 ) A one dimensional code of simulation for steady and sub-critical flow was used (Baume & Poirson 1984). The topographical model was built with all surveyed cross-sections. As we surveyed longitudinal variations of the morphology, crosssections were linearly interpolated at a one meter step. We used water surface profiles surveyed and the Single Channel Method to calibrate our model (Lyness et al. 1997). We fitted the model for other discharges with the rating curve. The hydraulic model with a single Manning’s n value simulated with good accuracy the hydraulic behavior at reach scale. Mean absolute errors were about 3 ± 3 cm. It allowed to interpolate results between measured cross-sections and for discharges within the range of validity of the rating curve. 2.4 Relevance of HGR at reach scale (step 2) 2.4.1 Spatial organization of hydraulic parameters along the river reach Hydraulic parameters considered are the maximum depth, the mean depth, the mean velocity and the channel width averaged at each cross-section. In a first time, we wanted to know if mean parameter’s values were relevant at river reach scale. We studied their evolutions all along the river reach. Linear regressions were applied on each hydraulic parameters evolution over the reach. This method allowed us to verify the hypothesis that no spatial trend occurred and that errors were normally distributed. Then the spatial autocorrelation was studied to show the organization of hydraulic parameters along the reach (Cessie, 1991). The experimental variogram was computed into the downstream direction with a lag distance varying from one meter to the half of the reach length (with a step of one meter). 2.4.2 Reference HGR and its range of validity The evolution of mean hydraulic parameters with discharge was estimated with the hydraulic model

for discharges varying within the range of validity of our model. Reference HGR was estimated by fitting a power law on these relations for discharges varying from the flow that just fills the channel to the base of the banks to bank-full flow (from 0.7 to 3.9 m3/s). The reference HGR’s coefficients and exponents (EXREF and COREF) were estimated considering (1) the whole river reach, (2) one cross-section each meter and (3) a discharge step of 0.3 m3/s. The validity of the fit was evaluated with the correlation coefficient. Difference at each discharge between mean hydraulic values when using the 1D hydraulic model and the HGR provided the range of validity of this simple function. 2.5 Robustness of HGR (step 3) 2.5.1 Reach length (step 3a) To estimate an accurate survey protocol, we had first to define the minimum reach length from which mean hydraulic behavior was well described. We tested the robustness of mean hydraulic parameter’s values and standard errors, considering different length: from one meter to the whole reach length. The minimum length, from which mean values and standard errors don’t vary will be defined (with a tolerance of ± 5%). For this test, we used interpolated cross-sections. 2.5.2 Number of cross-sections (step 3b) A test of robustness was led on the number of cross-sections surveyed. We supposed that for the survey protocol, cross-sections would be measured with a constant step along the reach. We used the hydraulic model with interpolated cross-sections at each meter. Then, we estimated the mean values and standard errors when considering (1) all the cross-sections, (2) one out of every two crosssections, and up to considering only the upstream and downstream cross-sections. The minimum number of cross-sections from which mean parameter’s values and standard errors don’t vary (with a tolerance of ± 10%) was estimated. We used the minimum reach length (step 3a) and simulations at three different discharges (step 1). 2.5.3 Choice of calibration discharges (step 3c) A classical way to calibrate HGR is to survey hydraulic parameters at two discharges (Jowett, 1997). Exponents and coefficients were computed by using different couples of discharges (called Q1 and Q2). These results were compared with COREF and EXREF. The couple of discharges that minimize the difference will be retained. Errors at bank-full flow were also considered. For this study, all interpolated cross-sections and the whole

2.6 Application of the protocol and errors (step 4) We built a survey protocol based on previous criteria: (1) the minimum reach length estimated at step 3a, (2) the minimum number of cross-sections (step 3b) and (3) two discharges chosen with the test led at step 3d. With the use of the hydraulic model, we estimated exponents and coefficients of HGR. Comparison with EXREF/ COREF and errors at near bank-full flow provided the accuracy and limits of this survey protocol. 3 RESULTS 3.1 Relevance of HGR at reach scale (step 2)

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Figure 2. Experimental variogram for the maximum depth parameter (Q=1.9 m3/s).

The variogram (Figure 2) shows the periodic organization of maximum depth along the reach. This parameter has a return period of about 50 meters long. When considering other parameters, periods are about 40 to 50 meters long. Then, to survey all the hydraulic variability, we have to study a river reach longer than this length and a distance between surveyed cross-section smaller than this period. 3.1.2 Reference HGR and its range of validity A simple power law fitted on the evolution of hydraulic parameters with discharge, for in-bank flows is adapted (see Table 2). EXREF and COREF are resumed in table 2. Table 2. EXREF and COREF values. Mean depth Maximum Velocity Width depth Dm=0.68Q0.40 D=0.96Q0.46 V=0.40Q0.24 W=4.48Q0.29 R²=0.98 R²=0.98 R²=0.87 R²=0.90

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Figure 1. Maximum depth at each cross-section along the reach for three different discharges: low flow (0.1 m3/s), 1.9 m3/s and near bank-full flow (4 m3/s). Linear regression are drawn with plain lines.

Linear regressions applied at each hydraulic parameter’s evolution show that no downstream trend occurs at reach scale (mean slope of about 0.0001). Moreover, variability is normally distributed all along the reach (with a 95% confidence interval). Mean hydraulic values are then relevant at river reach scale.

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Maximum depth (in meter

3.1.1 Spatial organization of hydraulic parameters along the river reach Figure 1 provides a good example on how hydraulic parameter’s values vary from upstream to downstream. They are organized around a mean value. The periodic variations are due to the sequences of pools and riffles.

Gamma (in mete

reach was considered. The discharge step was 0.3 m3/s.

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Figure 3. Evolution of mean hydraulic parameters for in-bank and over-bank flows (from Q=0.1 to 10 m3/s): with the hydraulic model (circle plot) and with HGR (plain line).

However, Figure 3 shows that for discharges (1) greater than bank-full discharge and (2) smaller than 0.7 m3/s, this simple power law is inaccurate to describe mean hydraulic parameters.

3.2 Robustness of HGR (step 3)

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Mean Hmean (m)

3.2.1 Reach length (step 3a) Maximum depth averaged at reach scale is relevant for reach length of about 150 meters (Figure 4a). Standard error is relevant, when considering a reach of about 300 meters (Figure 4b). For longer reach, mean parameter and standard error have stable value (with a tolerance of ±5%). For other parameters, the minimum reach lengths are about 150 meters for mean parameters with a tolerance of ±5%. For standard error, reach lengths must be about 300 meters with a tolerance of ±5% or about 200 meters with ±10% tolerance.

10% and 30 % in gray color and greater than 30 % are in white color.

3.3 Application of the protocol and errors (step 4) Coefficients and exponents are estimated with the following protocol: 1/ a reach length of 200 meters; 2/ 25 cross-sections; 3/ Q1=1 m3/s and Q2=2.8 m3/s as calibrating discharges. The relative differences between these values and EXREF/COREF are less than 10 %. This protocol lead to about 5% of error to estimate mean hydraulic parameters at bank-full flow.

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4 DISCUSSION AND CONCLUSION

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Distance (m) Figure 4 a), b): Mean values and standard errors of mean depth parameter, considering different reach lengths (Q=1.9 m3/s).

3.2.2 Number of cross-sections (step 3b) A test of robustness on the number of crosssections shows that we have to survey at least 20 cross-sections to have a good estimation of mean parameters and 30 cross-sections to have a good estimation of standard error (20 cross-sections with a tolerance of ±20%). The result of this test is in accordance with the periodic distribution of hydraulic parameters. 3.2.3 Choice of calibration discharge (step 3c) Figure 5 a) and b) show the results of the test of robustness on the choice of two calibrating discharges (Q1, Q2) applied with mean depth parameter. Different couples of discharges are interesting. When considering Q1=1 m3/s and Q2=[2-4 m3/s], absolute errors are less than 10% on exponent and coefficient. The analysis leaded with other parameters gives equivalent results.

Q2

For this study site, different conditions are required to estimate accurately HGR and hydraulic variability: 1/ a minimum reach length of 25 times bank-full width; 2/ one cross-sections at each bank-full width; 3/ Q1= 2.5*Qmean and Q2=[5-10* Qmean] as calibrating discharges. The first two conditions validate literature’s suggestions (Leopold, 1994). Errors associated to this protocol are less than 10% when estimating exponents, coefficients and bank-full conditions. Then, this protocol is accurate to estimate hydraulic parameters for flows that fills the channel to the base of the banks (0.7 m3/s), up to bank-full discharge. However, at lower flows, this HGR do not model accurately hydraulic parameters (mean errors of about 50% at median flows). Further studies have to be leaded on other river reaches (1) to test the relevance of a single power function for the whole range of flow within the main channel and (2) to generalize the survey protocol. This survey protocol would allow (1) to estimate the aquatic habitat for discharges varying between low flow to bank-full flow with good accuracy and, (2) to link the variability between the HGR estimated at different river reaches to the characteristics of associated river basins. REFERENCES

Q1 Exponent Coefficient Figure 5 a), b) : Relative difference between coefficient and exponant of mean depth parameter considering different couples of calibrating discharges (Q1 and Q2) and EXREF, COREF. Errors less than 10% are in black color, between

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