Surface-water quality and flow Modeling Interest Group
USGS, Water Resources Division
415 National Center
Reston, VA 20192
Internet: hejobson@usgs.gov
Phone: (703) 648-5224
FAX: (703) 648-5295
Citation:
Jobson, H.E., 1997, Predicting travel time and dispersion in rivers and
streams: Journal of Hydraulic Engineering, 123(11), p.
971-979.
The document displayed below is based on the final draft provided to the journal. Minor discrepancies between this document and the published version, therefore, may exist.
Measured tracer-response curves produced from the injection of a known quantity of soluble tracer provide an efficient method of obtaining the necessary data. The purpose of this paper is to use previously presented concepts along with extensive data collected by the U.S. Geological Survey on time-of-travel and dispersion to develop prediction methods that will be useful for responding to spills. This is done by providing methods to estimate:
Results from rivers of all sizes can be combined by defining the unit concentration as that concentration of a conservative pollutant that would result from injecting a unit of mass into a unit of flow. Unit-peak concentrations are compiled for more than 60 different rivers representing a wide range of sizes, slopes, and geomorphic types. Analyses of these data indicate that the unit-peak concentration is well correlated with the time required for a pollutant cloud to reach a point in the river.
The prediction of the traveltime is more difficult than the prediction of unit-peak concentration; but the logarithm of stream velocity can be assumed to be linearly correlated with the logarithm of discharge. Prediction equations are developed for mean velocity based on the drainage area, the mean annual discharge, and the discharge at the time of the measurement, using data from more than 980 subreaches in 90 different rivers. The highest probable velocity, which will result in the highest concentration, is usually of concern after an accidental spill. An envelope curve for which more than 99 percent of the velocities were smaller is developed to address this concern.
The time of arrival of the leading edge of the pollutant indicates when a problem will first exist and defines the overall shape of the tracer-response function. The traveltime of the leading edge is generally about 89 percent of the traveltime to the peak concentration.
The area under a tracer-response function (a known value when unit concentrations are used) can be closely approximated as the area under a triangle with a height of the peak concentration and a base extending from the leading edge to a point where the concentration has reduced to 10 percent of the peak. Knowing the time of the leading edge and the peak, the peak concentration, and the time when the response function has reduced to 10 percent of its peak value allows the complete response function to be sketched with fair accuracy.
Two example applications are included to illustrate how the prediction equations developed in this paper can be used either to calibrate a mathematical model or to make predictions directly.
With the widespread availability of computers today, it is natural to think of numerical models as a means of answering these questions. Although many excellent models are available to make the types of calculations needed, none can be used with confidence before calibration and verification to the particular river reach in question. That is to say, all models must be provided with information from which flow velocities and mixing rates can be computed. In general there are no reliable methods of predicting dispersion coefficients (mixing rates) from commonly available hydraulic information. Stream velocities, typically predicted by use of a flow model, generally require very detailed channel geometry and flow resistance coefficients, which are seldom available. The availability of reliable input information is, therefore, almost always the weakest link in the chain of events needed to predict the rate of movement, dilution, and mixing of pollutants in rivers and streams.
Soluble tracers can be used to simulate the transport and dispersion of solutes in surface waters because they have virtually the same physical characteristics as water (Feurstein and Selleck, 1963; Smart and Laidlaw, 1977). This is the case in either a steady flowing river or in the unsteady oscillatory stage and flow of a tidal estuary. Measured tracer-response curves produced from the injection of a known quantity of soluble tracer provides an efficient method of obtaining the data necessary to calibrate and verify pollutant transport models. These data can also be used, in conjunction with the superposition principle, to simulate potential pollution buildup in streams, lakes, and estuaries without the need to use numerical models.
Extensive use of fluorescent dye tracers to quantify the transport and dispersion in streams and rivers began in the United States in the early to mid-1960's. Kilpatrick (1993), using the concept of unit-peak concentration and the superposition principle, illustrated how these data, obtained in the time-of-travel studies, could be generalized to a wide range of flow conditions and even to other sites.
In this paper, the concepts presented by Kilpatrick (1993), along with extensive data collected by the U.S. Geological Survey on time of travel and dispersion, are used to develop methods that should be useful in responding to spills. This will be done by providing methods to estimate (1) the rate of movement of a solute through a river reach, (2) the rate of attenuation of the peak concentration of a conservative solute with time, and (3) the length of time required for the solute plume to pass a point in the river. It will be shown how these estimates can be used alone to make the required predictions. In addition, they are precisely the data required to calibrate or verify pollutant transport models. The accuracy of these predictions will be greatly increased by performing time-of-travel studies on the river reach in question; but the emphasis of this paper is on providing methods for making estimates in rivers where few data are available. Traveltime and concentration attenuation of pollutants not dissolved in the water are beyond the scope of this paper.
The paper begins with a short discussion of the theory of movement and dispersion of dissolved pollutants and introduces the unit-peak concentration concept. Methods are recommended for estimating the rate of movement and attenuation of conservative pollutants based on an analysis of the data compiled by Jobson (1996). The paper concludes by illustrating the application of these results by use of two examples.
The dispersion and mixing of a tracer in a receiving stream take place in all three dimensions of the channel (fig. 1). In this paper, vertical and lateral diffusion will be referred to in a general way as mixing. The elongation of the tracer-response cloud longitudinally will be referred to as longitudinal dispersion. Vertical mixing is normally completed rather rapidly, within a distance of a few river depths. Lateral mixing is much slower but is usually complete within a few kilometers downstream. Longitudinal dispersion, having no boundaries, continues indefinitely. In other words, vertical mixing is likely to be complete at section I in figure 1, which is a very short distance downstream of the injection. At section II lateral mixing is still taking place rapidly, so mixing and dispersion are both significant processes between the injection and section III on figure 1. Downstream of section III the dominant mixing process is longitudinal dispersion, so the tracer concentration can generally be assumed to be uniform in the cross section.
For a midpoint injection, the tracer cloud moves faster than the mean stream velocity upstream of section III because the bulk of the tracer is in the high velocity part of the cross section. Preferably, all measurement cross sections for a time-of-travel study are at least as far downstream as the optimum distance (section III in fig. 1) so that longitudinal dispersion is the dominant process acting between measurement cross sections and so the tracer moves downstream at the mean stream velocity.
The conventional manner of displaying the response of a stream to a slug injection of tracer is to plot the variation of concentration with time (the tracer-response curve) as observed at two or more cross sections downstream of the injection, as illustrated on figure 2. The tracer-response curve, defined by the analysis of water samples taken at selected time intervals during the tracer-cloud passage is the basis for determining time-of-travel and dispersion characteristics of streams. A detailed explanation of the analysis and presentation of time-of-travel data are covered in the report by Kilpatrick and Wilson (1989).
The characteristics of the tracer-response curves shown in figure 2 are described in terms of elapsed time after an instantaneous tracer injection:
Cp, peak concentration of the tracer cloud;
Tl, elapsed time to the arrival of the leading edge of a tracer cloud at a sampling location;
Tp, elapsed time to the peak concentration of the tracer cloud;
Tt, elapsed time to the trailing edge of the tracer cloud;
Td, duration of the tracer cloud (Tt-Tl);
T10d, duration from leading edge until tracer concentration has reduced to within 10 percent of the peak concentration; and
n, number of sampling site downstream of injection.
The mass of tracer to pass a cross section, Mr, is computed as:
(eqn 1)where W is the total width of the river, Cv is the vertically averaged tracer concentration, and q is the unit discharge (discharge per unit width). Both Cv and q are given at time t and distance w from one bank. After mixing is complete in the cross section, the equation simplifies to:
(eqn 2)where C is assumed to be uniform in the cross section and Q is the total discharge in the cross section at time t. If mixing is not complete, equation 2 can still be used as long as the concentration C is the discharge-weighted, cross-sectional-average concentration. If discharge is constant during the passage of a tracer cloud, it can also be factored out of the integral.
The shape and magnitude of the observed tracer-response curves shown in figures 1 and 2 are determined by four factors:
1. the quantity of tracer injected;
2. the degree to which the tracer is conservative;
3. the magnitude of the stream discharge; and
4. longitudinal dispersion.
All of these factors must be taken into consideration to predict the concentration of solutes from tracer-concentration data.
It is obvious that the magnitude of the tracer concentration in a stream is in direct proportion to the mass of tracer injected, Mi. Doubling the amount of injected tracer will double the observed concentrations, but the shape and duration of the tracer-response curve will remain constant. Thus, most investigators have normalized their data by dividing all observed tracer concentrations by the mass of tracer injected, Mi (Bailey and others, 1966; Martens and others, 1974).
It has also been found that various tracers are lost in transit due to adhesion on sediments and photochemical decay. Scott and others (1969) found fluorescent dyes to be absorbed on fine sediments such as clay. Rhodamine WT dye has been shown both in the field and laboratory to decay photochemically about 2 to 4 percent per day (Hetling and O'Connell, 1966; Tai and Rathbun, 1988). Kilpatrick (1993) noted decay rates tended to be higher in rivers, about 5 percent per day, compared to about 3 percent per day in estuaries.
To compare data and to have it simulate a conservative substance, it is desirable to eliminate the effects of tracer loss. If the stream discharge, Q, is measured at the same time and location as the tracer concentration, it is possible to evaluate the mass of tracer recovered, Mr, from equations 1 or 2. When the mass of the tracer injected, Mi, is known, the tracer recovery ratio Rr can be expressed as:
(eqn 3)A factor that inversely affects the magnitude of the tracer-response curves is the stream discharge. The diluting effect of tributary inflows, as well as that of natural ground-water accretion, differs from stream to stream and with location. To counter the variable diluting effects of differing discharges, it is desirable to adjust observed concentration data by multiplying by the stream discharge.
Observed concentrations can be adjusted for (1) the amount of tracer injected, (2) tracer loss, and (3) stream discharge (three of the four factors affecting the concentration) by use of what is called a "unit concentration." The unit concentration is defined as 1,000,000 times the concentration produced in a unit discharge due to the injection of a unit mass of conservative soluble substance. The unit concentration, Cu (units of inverse time), can be computed by the equation:
(eqn 4)The unit concentration can be visualized as the mass flux of solute (milligrams per liter times liters per second = milligrams per second) per unit of mass injected (milligrams). The 1,000,000 simply makes the numbers closer to unity. The discharge must be expressed in units that are consistent with the denominator of the concentration, and the injected mass must be in the same units as the numerator of the concentration. For example, if the concentration is expressed in milligrams per liter, the injected mass must be expressed in milligrams and the discharge must be expressed in liters per unit time. If the entire tracer cloud is sampled, the value of Mr can be computed and the mass of injected tracer need not be known.
Equation 4 can be used to convert any measured tracer-response curve to a unit-response (UR) curve. This UR curve can be used as the building block for simulating the concentrations to be expected from various pollutant loadings at different stream discharges. Normalizing the tracer-response curves, in effect, fits one unit of mass of tracer into one unit of flow. As such, when the flow is constant and mixing is complete, the area under UR curves is constant (1x106) for any cross section on a stream.
(eqn 5)in which Cup is the unit-peak concentration, t is time since injection, and b is a coefficient. The value of b should be approximately 1.5 for very short dispersion times (section I on fig. 1) and decrease to 0.5 for very long dispersion times (section V on fig. 1). Nordin and Sabol (1974) argue that a Fickian type equation cannot adequately describe longitudinal dispersion in rivers because the value of b never decreases to a value of 0.5. They conclude that a typical value of b is 0.7.
After mixing in the cross section is complete, the decrease of the unit-peak concentration with time (as measured by b) is a measure of the longitudinal mixing efficiency. Larger values of b indicate more rapid longitudinal mixing. The presence of pools and riffles, bends, and other channel and reach characteristics will increase the rate of longitudinal mixing and almost always yield a value of b greater than the Fickian value of 0.5.
Unit-peak concentrations were compiled for 422 cross sections obtained from more than 60 different rivers in the United States (Jobson 1996). These data represent mixing conditions in rivers with a wide range of size, slope, and geomorphic type. For example, the slope in the study reach of the Mississippi River is 0.01 m/km and the mean annual discharge is about 11,000 m3/s, whereas the study reach of Bear Creek has a slope of 36.0 m/km and a mean annual discharge of only about 1.3 m3/s.
Figure 3 is a plot of the unit-peak concentrations (Cup) as a function of traveltime (Tp) of the peak concentration of all the data for which the mean annual flow was available. A tight correlation is shown by the data, indicating that a reasonable estimate of the unit-peak concentration can be determined from an expression of the form of equation 5. The regression equation based only on traveltime that best fit all of the data was:
(eqn 6)This equation predicted the 422 available data points with a root mean square (RMS) error of 0.502 natural log units. The coefficient of variation was 0.112 and the coefficient of determination (R2) value was 0.893. The standard error of estimate of the coefficient is 4.9 percent and the standard error of estimate for the exponent is 1.7 percent.
Other river characteristics that were available to help define the relation included the drainage area (Da), the reach slope (S), the mean annual river discharge (Qa), and the discharge at the time of the measurement (Q). The most significant other variable in the correlation was the ratio of the river discharge to mean annual discharge giving a prediction equation:
(eqn 7)in which Q is the river flow at the section at the time of the measurement and Qa is the mean annual flow at the section. This equation predicted the 410 available data points with an RMS error of 0.426 natural log units. The coefficient of variation was 0.100 and the R2 value was 0.910. The standard error of estimate of the coefficient is 4.3 percent, and the standard error of estimate for the exponent (0.760) is 1.6 percent.
The data in figure 3 are separated into two groups--one with values of relative discharge (Q/Qa) greater than 0.5 (high flow) and one with a relative discharge less than 0.5 (low flow). The solid lines for high flow and low flow are plotted assuming constant values of relative discharge of 1.0 and 0.2, the approximate median value for each group of data.
Slope was not significant as an explanatory variable. Various regression models based on different combinations of discharge, mean annual discharge, and drainage area were tried. None of the equations produced a smaller RMS error or a larger R2 value than equation 7.
Results for individual rivers generally define a much closer relation. For example, figure 4 presents measured concentrations of dye for the Shenandoah River as published by Taylor and others (1986). The points labeled as Q/Qa=0.65 were actually taken at relative discharges ranging from 0.57 to 0.79 and the points labeled as Q/Qa=0.27 actually ranged from 0.21 to 0.32. Notice that the data for the Shenandoah River show almost no correlation with relative discharge. Equations 6 and 7 are also plotted on the figure for reference. In this case the equations fit the data very closely.
The Sangamon River shows strong correlations with relative discharge (fig. 5). It should be noted, however, that one set of measurements was made at extremely low flow. At any rate, the scatter among points for a single river is typically much less than the scatter among all rivers (fig. 3) so there is significant value in collecting data for individual rivers to improve the ability to predict the variation of unit-peak concentration.
The more efficient the mixing in a river, the steeper will be the relation between unit-peak concentration and traveltime. At high flow, river channels generally tend to be relatively uniform in shape, and they tend to increasingly exhibit a pool and riffle structure as the flow decreases. A pool and riffle structure offers great opportunities for tracer trapping; therefore, a pool and riffle structure tends to be efficient in mixing and attenuating the peak concentration. Equation 7 accounts for this process by decreasing the slope of UR curve for lower relative discharges.
Stream velocity and, consequently, traveltime commonly vary with discharge. The relation of mean stream velocity, V, to discharge is generally assumed to take the form:
(eqn 8)which is a straight line when the logarithm of discharge, Q, is plotted against the logarithm of velocity. For accurate estimates the constant, K, and exponent, a, must be defined for each river reach of interest, and two or more time-of-travel measurements are required to define the transport characteristics of the river reach. Geomorphic analyses by many investigators, however, suggest that the exponent in equation 8 typically has a value of about 0.34 (Jobson, 1989).
The velocity of the peak concentration and associated hydraulic data are compiled in Jobson (1996) for more than 980 subreaches of about 90 different rivers in the United States representing a wide range of river sizes, slopes, and geomorphic types. Four variables were available in sufficient quantities for regression analysis. These included the drainage area (Da), the reach slope (S), the mean annual river discharge (Qa), and the discharge at the section at time of the measurement (Q). It was reasoned that these variables should be combined into the following dimensionless groups. The dimensionless peak velocity is defined as:
(eqn 9)The dimensionless drainage area is defined as:
(eqn 10)in which g is the acceleration of gravity. The dimensionless relative discharge is defined as:
(eqn 11)These equations are homogeneous, so any consistent system of units can be used in the dimensionless groups. The regression equations that follow, however, have a constant term that has specific units, meters per second. The most convenient set of units for use with the equations is, therefore, velocity in meters per second, discharge in cubic meters per second, drainage area in square meters, acceleration of gravity in m/s2, and slope in meters per meter.
The most accurate prediction equation, based on 939 data points, for the peak velocity in meters per second was:
(eqn 12)The standard error of estimates of the constant and slope are 0.026 m/s and 0.0003, respectively. This prediction equation has an R2 of 0.70 and an RMS error of 0.157 m/s. Figure 6 contains a plot of the observed velocities as a function of the variables on the right side of equation 12.
For responses to accidental spills, the highest probable velocity, which will result in the highest concentration, is usually a concern. On figure 6 an envelope line for which more than 99 percent of the observed velocities are smaller is also shown. The equation for this line, the maximum probable velocity, in meters per second (Vmp) is:
(eqn 13)The best equation for the velocity of the peak concentration, in meters per second, that did not include slope as a variable was:
(eqn 14)The standard error of estimates of the constant and slope are 0.009 m/s and 0.0013, respectively. The root-mean-square error of the prediction equation, based on 986 points, is 0.17 m/s with an R2 of 0.62. Figure 7 presents a plot of the observed velocities as a function of the variables on the right side of equation 14.
Also shown on the figure is a line for which 99 percent of the data points indicate a smaller velocity. The equation for this line, for the probable maximum velocity, in meters per second, is:
(eqn 15)
Fewer data are available for the time-of-arrival of the leading edge (520 sites) than are available for the velocity of the peak concentration. Eight variables were available in sufficient quantities for regression analysis. These included the drainage area (Da), the reach slope (S), the mean annual river discharge (Qa), the discharge at the section at time of the measurement (Q), the velocity of the peak concentration (Vp), the width of the river, the depth of the river, and the time from the injection to the passage of the peak concentration (traveltime of the peak concentration, Tp). No significant correlation could be found between any of the variables and the time from injection to the arrival of the leading edge (Tl) except for the traveltime to the peak concentration. Figure 8 contains a plot of the traveltime of the leading edge as a function of the traveltime of the peak concentration. As can be seen from the figure, the correlation between these two variables is very good with an R2 of 0.989, a coefficient of variation of 0.13, and a RMS error of 3.78 hours. These data indicate that the traveltime of the leading edge can be estimated from:
(eqn 16)
(eqn 17)Furthermore, the area under the tail of the tracer-response curve should approximately balance the area between the falling limb portion of the tracer-response curve and the falling limb of the scalene triangle (fig. 2). This allows a complete tracer-response curve to be sketched in with reasonable accuracy based on the peak concentration and the times to the leading edge and peak.
No data exist for the stream receiving the spill, but topographic maps show that the drainage area is 350 km2 at the spill site and 430 km2 at the intake for the town. A review of available data also indicates that a gaging station exists for a nearby stream with a drainage area of 452 km2 and a mean-annual flow of 5.22 m3/s. At the time of the spill the flow at the gaging station was 3.88 m3/s. The hydrology and weather are assumed to be fairly uniform within the area so it will be assumed that the stream carrying the spill is flowing at about 3.88 (390/452) = 3.35 m3/s, assuming the average drainage area for the reach is (350+430)/2 = 390 km2. Likewise, the mean-annual flow of the ungaged stream is estimated to be about 5.22 (390/452) = 4.50 m3/s.
The first step is to estimate traveltime of the peak concentration. Because the river slope is not available, equations 14 and 15 will be used to estimate the expected and fastest probable traveltimes in the stream. The dimensionless drainage area and discharge are computed first from equations 10 and 11:
Applying equation 14:
while the maximum probable velocity from equation 15 is:
The most probable traveltime of the peak to the water intake is:
and the probable minimum traveltime of the peak is:
With the traveltimes known, the most probable unit-peak concentration at the town intake can be estimated from equation 7 as:
Rearranging equation 4, to give the peak concentration:
and using the injected mass, Mi, of 6x109 mg, the flow rate at the intake, Q, of (3.88x(430/452)x1000)=3,690 L/s, and assuming the recovery ratio, Rr, to be 1.0, the most probable conservative-peak concentration can be computed as:
occurring 15.8 hours after the injection.
At the highest probable velocity, the unit-peak concentration is 202/s giving an estimated conservative-peak concentration of 328 mg/L occurring 6.4 hours after the spill.
When will the pollutant first arrive at the intake? As can be seen from equation 16, the time of arrival of the leading edge of the pollutant cloud should occur 0.89x15.8 = 14 hours after the accident. It is highly unlikely that the pollutant will arrive at the intake sooner than 0.89x6.4 = 5.7 hours after the spill.
How long will the intake be affected? As can be seen from equation 17, the most probable time required for the bulk of the dye cloud to pass the site (the concentration to be reduced to 10 percent of the peak value, 16 mg/L) is:
hours after the time of arrival, or 14+5.6 = 19.6 hours after the spill.
It is highly unlikely that the pollutant concentration will have reduced to less than 20 mg/L before;
hours after the spill.
All of the above computations were carried out assuming no loss of pollutant between the spill and the intake. Losses could occur by chemical reactions, volatilization, absorption on the streambed, or other processes.
Two time-of-travel studies have been completed on this reach of the Apple River and the data are presented in Jobson (1996, table A-1 of Appendix A). One of these studies was conducted at relatively low flow, when the river discharge was about 0.7 times the mean annual flow, and one was conducted at relatively high flow, when the flow rate was about 3.5 times the mean annual flow. The first step is to estimate the times of travel of the leading edge and peak of the pollutant cloud. The traveltimes of the peak concentrations are plotted in figure 9.
From figure 9 it is seen that the traveltime of the peak concentration to Elizabeth is 49.4 hours at a relative discharge of 0.68, while the traveltime to Whitton is 105.8 hours at a relative discharge of 0.62. Also it is seen that the distance from Elizabeth to Hanover is 16.1 km while the distance from Elizabeth to Whitton is 22.5 km, so Hanover is 72 percent of the way between Elizabeth and Whitton. By linear interpolation, it is easily seen that the traveltime from the injection site to Hanover would be about 49.4+(105.8-49.4)x0.72 = 89.8 hours and that the relative discharge at this point would have been about 0.68+(0.62-0.68)x0.72 = 0.64. Likewise, the traveltime from the town of Apple River to the spill site would be 1.30+(20.80-1.30)x(10-1.9)/(35.9-1.9) = 5.95 hours at a relative discharge of 3.7+(3.3-3.7)x(10-1.9)/(35.9-1.9) = 3.6. In a similar manner, the traveltime from Apple River to the spill site would be 14.6 hours at a relative discharge of 0.82.
Assuming a mean annual flow at the spill site of 1.4 m3/s, the relative discharge at the time of the spill is 2.4/1.4 = 1.7. Then by linear interpolation between the relative discharges, it is seen that the traveltime from Apple River to the spill site would be 5.95+(14.6-5.95)x(1.7-3.6)/(0.82-3.6) = 11.9 hours. Likewise the traveltime from Apple River to Hanover would be 67.1 hours. The traveltime from the spill site to Hanover should, therefore, be 67.1-11.9 = 55.2 hours.
With the relatively small amount of data shown on figure 9 for the Apple River, it is possible to estimate the timing of a spill on the river with much better accuracy than would have been possible by use of equations 12 to 13.
Figure 10 is a plot of the unit-peak concentrations measured on the Apple River during the two tests. As can be seen from the figure, the unit-peak concentration should be about 40 s-1 for a traveltime of 55 hours. Converting the spilled mass into milligrams (5x107 mg), the flow rate at Hanover (Qave = 5 m3/s from Jobson, 1996) to liters per second (1.7x5.0x1000 = 8500), and assuming a recovery ratio of 1.0, the peak concentration at the intake can be estimated from equation 4 as:
The time required for the pollution cloud to pass the intake and the river concentration to be reduced to 10 percent of the peak value (0.024 mg/L) can be estimated by use of equation 17 as:
The times for the arrival of the leading edge of the tracer cloud, from Jobson (1996), can also be plotted as in figure 10. The traveltime of the leading edge of the tracer cloud from the spill site to Hanover can then be estimated using the same procedure as for the peak concentration, as 51.1 hours. After 51.1+13.9 = 65 hours the pollution cloud should have passed the intake and the concentration reduced to 0.024 mg/L.
In conclusion, the pollutant should first arrive at Hanover 51 hours after the spill. The peak concentration should pass the site 55 hours after the spill; and if there are no losses, it should arrive with a concentration of 0.24 mg/L. By 65 hours after the spill, the concentration should have fallen back to 0.024 mg/L. If there are losses or chemical reactions between the spill and the intake, the concentrations will be smaller and a numerical model could be used for predictions.
This paper uses information compiled from a large number of time-of-travel and dispersion studies and presents empirical relations that appear to have general applicability. These relations are not recommended as a substitute for field studies but are believed to provide reasonable estimates in situations where adequate field data are not available. Empirical relations are given for the unit-peak concentrations, velocity of the peak concentration, velocity of the leading edge of a solute cloud, and the duration of the time of passage as measured from the leading edge to the point where the solute concentration has fallen to 10 percent of its peak value. It is shown how this information can be used to estimate the complete response function. The recommended methods are demonstrated by presenting two examples.
If the solute transport in the river is to be modeled, the model must be calibrated to provide the correct traveltimes and rates of attenuation of the peak concentration. The relations presented in this paper can be used to calibrate a solute-transport model for use on a river that has little field data.
The relation for unit-peak concentration is the best defined of all the relations needed to predict the transport and dispersion of pollutants. Field data show that the peak concentration tends to decrease more rapidly with time than predicted by Fickian dispersion. Because almost all numerical models are based on the Fickian relation, model dispersion coefficients must be assumed to increase with time for the model results to duplicate observed data.
The relation for predicting mean stream velocity (traveltime) is the least accurately defined of all relations presented in this paper. Traveltime information is, therefore, the most valuable information that can be collected to improve the ability to predict the transport and dispersion in a river. These data should be collected at two or more flows, preferably a low flow and a high flow.
Back to the SMIG Features Page
Home | Mailing List | Features | Conferences | Classes | Reading | Model Archives | Feedback