ESTIMATING STAGE-DISCHARGE RATING SHIFTS DUE TO RIVER BED
SCOUR-FILL DURING FLOODS
J. Le Coz,1,2 G.M. Smart,2 B. Camenen,1
1
Irstea, Hydrology-Hydraulics, Lyon-Villeurbanne, France
2
NIWA, Hydrodynamics, Christchurch, New Zealand
Aims
When a flood wave propagates along a mobile-bed channel, bed scour and fill may be measured
or indicated during the rising and falling limbs of the hydrograph, respectively (cf. e.g. Laronne et
al. 1994, Capapé & Martín-Vide 2015). While many processes can make each local situation
complex, this general trend can be explained by the bed response to the propagation of the flood
wave and of the associated bedload wave. One of the practical issues related to transient scour-fill
of river beds during floods is that the stage-discharge relation is changed in an unknown way, since
only the stage, not the bed elevation, is usually monitored at hydrometric stations. As a
consequence, large uncertainties in the flood hydrograph may arise and the flood peak discharge
is potentially underestimated. In order to correct the rating curves of hydrometric stations, we need
to derive simple equations based on readily identifiable parameters and time derivatives (instead
of spatial derivatives) of quantities that are measured (stage h) or computed at a station.
Method
Based on the Exner equation for sediment continuity, the bedload wave celerity can be related to
the bedload rate derivative with respect to bed level. The rate of change of the active bed elevation
is found to be negatively proportional to the rate of change of bedload. Most common transport
capacity formulas imply that the bedload wave celerity varies much less with time than the bedload
rate. This allows establishing the following simple at-a-station bed response model:
𝜕𝑞
𝑧(𝑡) = 𝑧(0) + [ 𝜕𝑧𝑏 |𝑧(𝑡) ]
−1
𝑞𝑏 (𝑡)
(1)
where z is the bed elevation, t is the time since initiation of sediment transport, and qb is the volume
bedload rate per unit width.
Exact solutions of Eq. 1 were derived for five bedload capacity equations: Meyer-Peter & Mueller
(1948), Smart (1984), Nielsen (1992), Ribberink (1998), and Camenen & Larson (2005) formulas.
Assuming that the flow is fairly uniform and the channel wide enough to approximate the hydraulic
radius by the flow depth, z(t) can be expressed as a function of the water level, h(t), and of the
critical flow depth for incipient motion ycr = (s-1) d τ*,cr / S0, where S0 is the bed longitudinal slope, s
is the relative sediment density and d is the grain diameter. The critical bed shear stress for incipient
motion, τ*,cr, was computed using the formula proposed by Soulsby & Whitehouse (1997).
Results
The exact solutions for the five bedload formulas were applied to experimental data from a physical
model of the Waimakariri at Gorge, in the South Island of New Zealand (Davies & Griffiths 1996).
At that site, the braided river is contracted by narrow gorges, which enhances bed changes in
response to flood events with intense bedload transport. In such a situation, the active width of the
river (where bedload and bed changes occur) is equal to the total width of the cross-section. Results
using the experiment reported by Davies & Griffiths (1996) are shown in Fig. 1. The paper provides
partial information on the main parameter values, some of which remain relatively uncertain.
The results appear to be quite sensitive to the bedload equation selected and to its parameters.
With the set of parameter values presented in Fig. 1a, the Camenen & Larson formula yields the
best results while other formulas overestimate the scour magnitude, up to more than twice using
the Nielsen formula. However, the temporal dynamics of the bed evolution is accurately predicted
by all bedload formulas, and the maximum scour depth could be acceptably reproduced if
parameters such as d, τ*,cr, etc. were calibrated within the range of realistic values. Both simulated
and observed data suggest that significant scour is still present during flood recessions, with bed
adjustments lasting for long times after flood peaks. The stage-discharge relation is also acceptably
predicted, except for moderate hysteresis visible in the data at low flow (cf. Fig. 1b).
(a)
(b)
Figure 1 Simulation of run 27 of experiments by Davies and Griffiths (1996). (a): observed stage and bed level
(blue and red dashed lines), simulated bed level evolution using different bedload capacity equations:
Camenen & Larson (gray), Ribberink (green), Smart (blue), Meyer-Peter & Mueller (orange), Nielsen
(magenta); (b): stage-discharge and bed-discharge relations observed and simulated using Camenen &
Larson formula; the predicted stage-discharge relation ignoring bed scour is shown in black dashed.
Application to field data from similar sites like the Rakaia river at Gorge also yielded encouraging
results as regards the capability of the model to predict the timing and magnitude of scour-fill cycles
induced by floods. Nevertheless, the predictive uncertainties can be expected to be as huge as
those of bedload capacity equations, even when calibrated using some direct observations
(gaugings, scour chains, etc.) or experiments (physical models).
The bottom shear stress, hence the bedload rate, are often found to be affected by the impact of
unsteady flow conditions and changes in the turbulence intensity. This may be accounted for by
correcting the friction slope, Sf, with Jones (1916) formula and use it instead of S0 in the
computations. However, this would make exact solutions of Eq. 1 much more difficult to derive.
Alternatively, Davies & Griffiths (1996) proposed a model based on a comprehensive development
of the 1-D hydraulics theory to account for the stage-discharge hysteresis due to transient scourfill. The related loop effects would be interesting to be checked using field observations, though
experimental evidence might be challenged by the more direct impact of cross-sectional changes.
References
Camenen, B. & Larson, M. (2005) A general formula for non-cohesive bed load sediment transport. Estuarine,
Coastal and Shelf Science, 63, 249-260.
Capapé, S. & Martín-Vide, J.P. (2015) Evidence of transient scour and fill. 36th IAHR World Congress, The
Hague, the Netherlands.
Jones, B. (1916) A method of correcting river discharge for a changing stage. Water Supply Paper 375, 117130, U.S. Geological Survey.
Laronne, J.B., Outhet, D.N., Carling, P.A., & McCabe, T.J. (1994) Scour chain employment in gravel bed
rivers. Catena, 22, 299-306.
Davies, T.R. & Griffiths, G.A. (1996) Physical model study of stage-discharge relationships in a gorge of a
braided river. Journal of Hydrology (New Zealand), 35, 239-258.
Smart, G. (1984) Sediment transport formula for steep channels. J. of Hydraulic Eng., 110(3), 267-276.
Soulsby, R. & Whitehouse, R. (1997) Threshold of sediment motion in coastal environment. Proc. Pacific
Coasts and Ports '97 Conf', University of Canterbury, Christchurch, New Zealand, 149-154.