TECHNICAL .NOTES Estimation of Average Stream ...

TECHNICAL .NOTES

Estimation of Average Stream Velocity Michael G. Waldon, M.ASCE1

Abstract: Stream tracer studies provide information supporting diverse applications in environmental research and management. Average stream velocity through a reach is often estimated from tracer temporal profiles. This Technical Note addresses the calculation of average reach velocity. It is shown here that. under steady ftow, average reach velocity over a fixed distance equal~ the spatial harmonic mean velocity. Similarly. the average reach velocity from the point of tracer injection to a fixed downstream measurement site is equal to downstream distance divided by the harmonic mean tracer time-of-travel, rather than the commonly used temporal centroid .

DOl: 10. 1061/(ASCE)073J-9429(2004)130:ll(ll19) CE Database subject headings: Mass transport: Mathematical models: Dispersion: Spills: Water quality: Streams: Velocity.

Introduction Understanding processes that result in longitudinal transport and dispersion is fundamental in a wide range of applications in various media. An understanding of longitudinal dispersion in freeflowing streams is important for pollution management and modeling. ecological studies. emergency spill management. spill contingency planning, and scientific study of water quality processes. Tracer studies provide a technique to efficiently and economically study stream transpon and dispersion. In water quality modeling. tracer studies are often used to estimate average stream

velocity and cross-sectional area. Soluble dye, chemical. or radioactive tracers have been commonly used in stream tracer investigations !.Buchanan 1964: Huhbard et al. 1982; Jobson 1996, 1997). Since the mid 1960s. numerous dye tracer studies have been performed on streams in the Cnited States. For example, in a study of travel-times. Jobson ( 1997) analyzed tracer data from nearly a thousand subreacbes of united States streams. Typically, the tracer is injected in a nearly instantaneous manner across a stream. or in the fastest flowing region of the cross section. Observed tracer transport is then assumed to correspond to advection and dispersion within the stream. The leading section of fhe tracer cloud extends downstream of tbe spatial peak, and the trailing section extends upstream. As the tracer cloud spreads, tracer velocity in the leading section is faster than the average water velocity because of the forward dispersive flux. Similarly, in the trailing section tracer velocity is less than the average water velocity because dispersive flux transports the 1

Senior Hydrologist. A.R.M. Loxahatchee National Wildlife Refuge. Boynton Beach, FL 33437. E-mail: walden@ members.asce.org Note. Discussion open until April I, 2005. Separah:: discussions must be submitted for indi11idual papers. To extend the closing date by one month, a written requeM must be filed with the ASCE Managing Editor. The manuscript for this technical note was submitted for review and possible publication on April II. 2002: approved on July 29. 2003. This technical note is part of the Journal of Hydraulic Engineering , Vol. 130. No. 11. November I, 2004. ©ASCE. ISSN 0733-9429/2004/11-1 I 1910216 Lee Rd.,

11221$18.00.

tracer in the upstream direction relati ve 10 the spatial tracer peak. Tracer profile evolution may be measured by observing concentrations along a one-dimensional spatial dimension at fixed limes. or over time at ri.xed locations. These observations are termed. respectively, spatial profiles and temporal profiles. Ylodels of tracer dispersion are often analyzed in a more straightforward manner using spatial profiles. L'nfortunately. in most situations it is impractical to directly observe the spatial tracer protiles: instead. temporal tracer profiles are observed. This Technical Note addresses the question of how to correctly calculate average velocity from slug-injection stream tracer observations. Throughout this Technical Note it is assumed that ( I) t he tracer is conservative during passage of the measurement site; (2) ftows are steady. that is, discharge and velocity do not vary with time but may vary spatially with stream location; (3) average water velocity and average tracer velocity are equal; and (4) that concentrations are sufficiently uniform laterally and venically to allow spatial variation to be appropriately represented by a onedimensional spatial axis. Assumptions of specific dispersion models have been excluded from consideration here. This Technical Note uses the concept of the harmonic mean. The harmonic mean is the reciprocal of the expected value of the reciprocal of a random variable. Thai is. for a random variable. ;;, the harmonic mean i~ defined by

~HM = E( 1/z)

(!)

where E=expected value operator. The harmonic mean of a set of positive numbers is always less than or equal to their arithmetic mean. with equality restricted to the special case in which all of the numbers are equal. Harmonic means of variables are often found to be the correct statistic for calculation of parameters when the variable appears in the denominator of the parameter's formula. Often these parameters a re related to rates (e.g.. velocity or d ischarge). Harmonic mean stream discharge. for example. is used to assess the level of exposure re~ulting from constant pollutant loads (Rossman I 990). The following section presents a simple example of the application of the harmonic mean for estimation of average reach velocity.

JOURNAL OF HYDRAULIC

ENGINEERING~

ASCE I NOVEMBER 2004 / 1119

Table 1. Amitl! River Study Sampling Site Descriptions and Statistics

Site

Description Hwy 432 Bridgl! Darlington, La. Grangeville. La. Magnolia, La. Denham Springs. La.

Injection Site I Site 2 Site J Site -l

October mean Q

Dr.tinage areu (km"l 1.502 1,919

2.290 3,315

C.onsider a parcel of water traveling downstream at the cross:.ectional average water velocity. u(x), where x=distance downl>tream from an initial site. x =O. The spatially varying function. u(x), might be estimated through direct measurement using a currenr meter, or from subreach tracer based velocity c::stimates. Average water velocity over the reach (O,x 1) is

,,

Q

Distance

( m 'l