Is a sediment rating curve really curved?, Victor M. Ponce

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Redrawn from ASCE Manual 75, p. 476 (1975)

IS A SEDIMENT RATING CURVE REALLY CURVED?

Victor M. Ponce

San Diego State University, San Diego, California

27 September 2023

ABSTRACT. The nature of a sediment rating curve has been reviewed, explained, and clarified. The "ultimate sediment concentration" is the maximum concentration that may be attained in the absence of form friction. Its value may be readily calculated based on flow and sediment properties. The shape of the actual sediment rating is curved, wherein the slope of the curve is high for the lower flows and decreases asymptotically for the higher flows to the characteristically low value of 3. This behavior is a direct consequence of the increase in total boundary friction that may be attributed to the forms of bed roughness in lower regime, that is, ripples and dunes. The latter interfere with the flow's potential to transport the ultimate sediment concentration, thereby reducing sediment discharge. The above rationale clarifies the reason for the demonstrated drop in sediment transport for low flows and, therefore, explains the shape of a typical sediment rating curve. A verification with field data confirms the findings of this study: The higher the boundary friction, the lower the amount of sediment transport, and vice versa.

1. INTRODUCTION

A sediment rating curve is a relation between water discharge Q in the abscissas and sediment discharge Q_s in the ordinates. It is used in the field of river mechanics and sedimentation engineering to readily convert from one discharge to the other, normally from Q to Q_s (ASCE, 1975; 2007). In engineering practice, the curve is readily grasped for what it accomplishes; however, the understanding of its true nature often goes unnoticed.

In this article we endeavor to explain the intricacies of the sediment rating curve. We explain how ecology took some liberties with alluvial streams to create an environment where fish and other stream biota would continue to flourish in spite of the fact that the local climate did not appear to be cooperating. In this process, we tie the related fields of climatology, geology, geomorphology, hydrology, and ecology into a seamless fabric, where the avowed purpose is the preservation of stream biota. To set the proper stage to explain the concepts that will follow in later sections, we begin with three basic truths of sediment transport mechanics.

2. THREE BASIC TRUTHS

Truth No. 1. The ratio between sediment discharge Q_s and water discharge Q is referred to as the sediment concentration C_s (Eq. 1). Under steady equilibrium flow conditions in an alluvial stream, the sediment concentration is the highest sediment load that the stream is able to carry for the prevailing flow. This is because a lower value of C_s would lead to bed degradation; conversely, a higher value would lead to bed aggradation; both of these situations would defy equilibrium. In sedimentation engineering, the equilibrium sediment concentration is referred to as the "sediment transport capacity," expressed in F/T units, i.e., lbs. per second, or tons per day. Steady equilibrium flow in an alluvial stream must always transport the maximum sediment load that it can carry.

Q_s
C_s = ^_____
Q

(1)

Truth No. 2. An alluvial stream will transport sediments originating in the streambed, i.e., the bed-material load, in two distinct ways: (1) by rolling and sliding along the bed, that is, the bed load, and (2) in suspension throughout the depth, by action of the flow's turbulence on the entrained sediment particles, i.e., the suspended bed-material load. In addition, an alluvial stream carries a third type of load, referred to as washload, consisting of the finer-sized sediments which ostensibly did not originate in the streambed, but rather, in the catchment's uplands (ASCE, 1975; Ponce, 2014a). The collection of the three types of sediment load constitutes the total sediment load (Fig. 1).

Fig. 1 Types of sediment load in an alluvial stream.

Truth No. 3. Flow in an alluvial channel may be of two types: (1) lower regime, and (2) upper regime. Under lower regime, which takes place for the lower Froude numbers, say, F < 0.5, the bed load itself generates forms of bed roughness, referred to as ripples and dunes [Fig. 2 (a)]. These forms act to increase the total boundary friction, now consisting of grain and form friction (Einstein, 1950). Conversely, under upper regime, which takes place for higher Froude numbers, say, F ≥ 0.5, the swiftness of the flow acts to obliterate the ripples and dunes, decreasing the total boundary friction to only grain friction, leading to a plane bed configuration [Fig. 2 (b)] (Simons and Richardson, 1966).

The change from high friction in low flows to low friction in high flows has the net effect of reducing the variation in stage for a given variation in discharge. Therefore, the stage is higher than normal for low flows, and lower than normal for high flows. This is a decided advantage for the survival of stream biota (Kennedy, 1983). Pointedly, we note that in the Upper Paraguay river, in Mato Grosso do Sul, Brazil, this somewhat unusual phenomenon is referred to as auto-dregding (Ponce, 1995).

In summary, the three basic truths of sediment transport mechanics are:

The equilibrium sediment discharge is at peak capacity: Under equilibrium flow, a stream will always transport the maximum sediment load that it can carry.
The stream transports three types of sediment: (1) bed load, (2) suspended bed-material load, and (3) washload.
The boundary friction varies with the flow regime: In low flows, the flow itself will create forms of bed roughness, i.e., ripples and dunes, which will be obliterated under high flows into plane bed and other smoother configurations.

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Simons and Richardson (1966)

Fig. 2 (a) Forms of bed roughness: Lower regime.

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Simons and Richardson (1966)

Fig. 2 (b) Forms of bed roughness: Upper regime.

3. ULTIMATE SEDIMENT CONCENTRATION

In engineering practice, the following power function is used to provide a fit between measured water and sediment discharge data:

Q_s = c Q^m

(2)

in which c and m are coefficient and exponent of the sediment rating curve, respectively.

In Equation 2, for the special case of m = 1, the sediment concentration (Eq. 1) is independent of Q and equal to c. This sediment concentration is referred to as ultimate sediment concentration (Ponce, 1988). The value m = 1 appears to be a minimum value approached by typical streams at very high flood discharges (ASCE, 1975: p. 476).

A widely used sediment transport formula is the following (Colby, 1964):

q_s = k ρ vⁿ

(3)

in which q_s = unit-width sediment discharge; k = bed-material transport parameter, a function of sediment properties, including mean particle size, gradation, and specific weight; ρ = density of water; v = mean flow velocity; and n = exponent (Ponce, 2014a).

Colby (1964) showed that n ≅ 7 is typical of low-water discharges, while n ≅ 3 is typical of high-water discharges. In fact, n = 3 is a characteristic asymptotic lower limit for n, rendering dimensionless the bed-material transport parameter k in Eq. 4:

q_s = k ρ v³

(4)

Since n = 3 is typically associated with high-water discharges, Eq. 4 may be used to calculate the ultimate sediment concentration.

The water discharge, per unit width, is:

q = v d

(5)

in which d = flow depth.

Equation 1 is recast in terms of unit-width variables as follows:

q_s
C_s = ^_____
q

(6)

Substituting Eqs. 4 and 5 into Eq. 6 leads to the equation for ultimate sediment concentration C_s' :

C_s' = k F² γ

(7)

in which F = Froude number, defined as F = v /(gd )^1/2; g = gravitational acceleration; and γ is the specific weight of water (γ = ρ/g). For example, with k = 0.1, F = 0.4, and γ = 1,000 mg/L, Eq. 7 leads to: C_s' = 0.1 × 0.16 x 1,000 g/L = 16 g/L = 16,000 ppm.

Figure 3 shows a sediment rating curve for Watershed 34, of surface area equal to 87 square miles, at Pigeon Roost Creek, near Holly Springs, Mississippi, measured during the storm of February 18, 1961. [This figure has been extracted from p. 476, Chapter IV-C of the ASCE Sedimentation Engineering Manual 75]. The isolines of sediment concentration are shown for reference. The graph depicts the asymptotic approach of the sediment rating curve to a line of equal concentration (45° orientation). In this example the value of ultimate sediment concentration exceeds 12,500 ppm. The graph shows that there is a limit to the concentration of suspended sediment at very high discharges, which is likely to be the case during an infrequent flood.

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Redrawn from ASCE Manual 75, p. 476 (1975)

Fig. 3 Measured sediment rating curve.

4. NATURE OF A SEDIMENT RATING

We have established that a sediment rating curve characteristically has a curved shape, wherein at low Q the slope (Q_s /Q) is quite high, typically around 7, and that it has the tendency to progressively decrease with an increase in Q, eventually attaining a lower limit of 3 at a sufficiently high discharge Q (Fig 3). There is a reason for this typical behavior of a sediment rating curve. We explain it by stating the following facts, borne out by theory and experience.

Fact No. 1. At sufficiently low flows, alluvial streams transport their bed material at a concentration constantly growing with increasing Q. At these low flows, both grain and form friction are present, since the low flow velocities are conducive to the development of form friction, manifested as ripples and dunes. In effect, the bed configuration turns out to be ripples and dunes [Fig. 2 (a): A, B, and C].

Fact No. 2. At sufficiently high flows, alluvial streams transport their bed-material load at a constant maximum concentration, referred to as the "ultimate sediment concentration" (Eq. 7). This concentration is a function of flow properties (Froude number F and specific weight γ) and sediment properties (the dimensionless parameter k). At these high flows, grain friction is the only type of boundary friction present, since form friction has already been obliterated by the swiftness of the current. The resulting bed configuration is plane bed [Fig. 2 (b): E].

There is a reason for this interesting and somewhat fascinating physical behavior of a sediment rating. The local bed-material sediment concentration should reflect the local transport capacity of the stream. If the channel boundary were rigid, the sediment concentration ought to be unique or close to unique, that is, not to vary appreciably with the flow. But in an alluvial channel, the facts bear otherwise: At low flows, the sediment concentration has a tendency to increase with the flow. This must be because the flow is busy dealing with the additional friction created by the lower-regime bedforms, i.e., the form friction originating in the ripples and dunes [Fig. 4 (a)].

We affirm that at low flows, the presence of lower-regime bedforms does not permit the flow to carry its maximum sediment load, in effect reducing it to a fraction of its equilibrium value. Conversely, at high flows, the swiftness of the current acts to obliterate the lower-regime bedforms, reducing the boundary friction to grain friction only [Fig. 4 (b)]. This enables the high flow to transport the maximum sediment load that it possibly can.

A splendid corollary of the present analysis may be stated as follows: If it were not for the lower-regime bedforms, the entire sediment transport problem could be encapsulated into Eq. 7, i.e., the calculation of sediment transport by simply resorting to a sole value of sediment concentration, the "ultimate sediment concentration."

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Extracted from Simons and Richardson (1966)

Fig. 4 (a) Dunes and superposed ripples configuration.

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Extracted from Simons and Richardson (1966)

Fig. 4 (b) Plane bed configuration.

We close this section with a relevant quote from Prof. Hans A. Einstein (1950), p. 9:

"The part of the energy which corresponds to the shape resistance is transformed to turbulence at the intercept between wake and free stream flow, or at a considerable distance away from the grains. This energy does not contribute to the bedload motion of the particles, therefore, and may be largely neglected in the entire sediment picture." (Italics are the author's).

It is seen that Einstein saw fit not to include the energy spent in overcoming the lower-regime bedforms in his bedload function. This fact underscores the concept that the ultimate sediment concentration is indeed a ceiling for the actual sediment concentration, confirming the true nature of the sediment rating.

5. FIELD VERIFICATION

Ponce and others (2012) used a comprehensive data set originally compiled by Williams (1995) to compare six (6) measured sediment rating curves with their corresponding frictional values as measured by Manning's n. The objective was to ascertain the inverse relation between sediment transport and boundary friction: The greater the latter, the smaller the corresponding water and sediment discharge, and vice versa.

The following six sets of Brownlie's data (Brownlie 1981a, 1981b) were used by Ponce and others (2012):

63 data points from the Atchafalaya River.

40 data points from the Niobrara River.

38 data points from the Middle Loup River.

51 data points from the Rio Grande River.

156 data points from the Mississippi River.

29 data points from the Red River.

Measured sediment rating curves, including a graphical best fit, are shown in Fig. 5. In addition, for each data unit, the water discharge, channel width, hydraulic depth, and bottom slope were used to calculate Manning's n (Ponce, 2014b). The calculated values of Manning's n were plotted on a secondary arithmetic y-axis, so that the trends in sediment rating and boundary friction could be readily compared. In all cases, it is clearly seen that Manning's n decreases as the sediment discharge increases, confirming the fact that larger boundary friction leads to smaller sediment discharge, and vice versa.

♦ Click on each figure to display ♦

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 5 (a) to 1 (f) Sediment rating curves and corresponding values of Manning's n
(friction data points are shown in magenta) (Ponce and others, 2012).

6. CONCLUSIONS

The nature of a sediment rating curve has been reviewed, explained, and clarified. The "ultimate sediment concentration" is the maximum concentration that may be attained in the absence of form friction. Its value may be readily calculated based on flow and sediment properties using Eq. 7.

The shape of the actual sediment rating is curved, wherein the slope of the curve is high for the lower flows, around 7, and decreases asymptotically for the higher flows to the characteristically low value of 3. This behavior is a direct consequence of the increase in boundary friction that may be attributed to the forms of bed roughness in lower regime, that is, ripples and dunes. The latter interfere with the flow's potential to transport the ultimate sediment concentration, thereby reducing sediment discharge.

The above rationale clarifies the reason for the demonstrated drop in sediment transport for low flows and, therefore, explains the shape of a typical sediment rating curve. A verification with field data confirms the findings of this study: The higher the boundary friction, the lower the amount of sediment transport, and vice versa.

REFERENCES

ASCE, 1975. Sedimentation Engineering. Manuals and Reports on Engineering Practice, Manual 54, Vito A. Vanoni, editor, New York.

ASCE, 1975. Sedimentation Engineering. Manuals and Reports on Engineering Practice, Manual 54: p.476.

ASCE, 2007. Sedimentation Engineering: Processes, Measurements, Modeling, and Practice. Manuals and Reports on Engineering Practice, Manual 110, Marcelo H. Garcia, editor, New York.

Brownlie, W. L. (1981a). Prediction of flow depth and sediment discharge in open channels. Report KH-R-43A, W.M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena, California.

Brownlie, W. L. (1981b). Compilation of alluvial channel data: Laboratory and field. Report KH-R-43B, W.M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena, California.

Colby, B. R. 1964. Discharge of sands and mean velocity relations in sand-bed streams. U.S. Geological Survey Professional Paper 462-A, Washington, DC.

Einstein, H. A. (1950). The bed-load function for sediment transportation in open-channel flows. USDA Soil Conservation Service, Technical Bulletin No. 1026, Washington, DC, September.

Kennedy, J. F. 1983. Reflections on rivers, research, and Rouse. Journal of Hydraulic Engineering ASCE, 109(10), 1257-1260.

Ponce, V. M. 1988. Ultimate sediment concentration. Proceedings, National Conference on Hydraulic Engineering, Colorado Springs, Colorado, August 8-12, 311-315.

Ponce, V. M. 1995. Hydrologic and environmental impact of the Parana-Paraguay waterway on the Pantanal of Mato Grosso, Brazil. https://ponce.sdsu.edu/hydrologic_and_environmental_impact_of_the_parana_paraguay_waterway.html

Ponce, V. M., D. S. Smith, and R. D Aguilar. 2012. Effect of form friction on the sediment rating curve. Online article.
https://ponce.sdsu.edu/effect_of_form_friction.html

Ponce, V. M. 2014a. Engineering Hydrology: Principles and Practices. Online textbook.
https://ponce.sdsu.edu/enghydro/index.html

Ponce, V. M. 2014b. Fundamentals of Open-channel Hydraulics. Online textbook.
https://ponce.sdsu.edu/openchannel/index.html

Simons, D. B., and E. V. Richardson. 1966. Resistance to flow in alluvial channels. Geological Survey Professional Paper 422-J, U.S. Government Printing Office, Washington, D.C.

Williams, D. T. (1995). Selection and predictability of sand transport relations based upon a numerical index. Ph.D. dissertation, Department of Civil Engineering, Colorado State University, Fort Collins, Colorado.

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