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Accuracy of stream gauging by dilution methods
Journal of Hydrology, 137 (1992) 231-243
231
Elsevier Science Publishers B.V., Amsterdam
[1]
Accuracy of stream gauging by dilution methods Kazuo Okunishi a, Takashi Saito a and Toshio Yoshida b aDisaster Prevention Research lnstilute, Kyo;o U'iiversity, Gokanosho, Uji 611, Japan bOyo Corporation, Ichigaya-Building, Kudan-kita, Chiyoda-Ku, Tokyo 102, Japan (Received 29 October 1991; accepted 3 December 199 !)
ABSTRACT Okunishi, K., Saito, T. and Yoshida, T., 1992. Accuracy of stream gauging by dilution methods. J. Hydrol., i 37:231-243. The constant flux diluUon ,,,,::5,~d and the finite mass dilution method are examined for accurate measurement of stream discharge. The measut,~om procedures are reviewed on a mathematical basis. Their precision is quantitatively evaluated following the assumption that the requirement of homogeneous dispersion is satisfied. The total reliability is discussed on the basis of field experience with the finite mass dilution method. Finally, a summary of the procedures necessary for better field measurements is given.
INTRODUCTION
Among a variety of stream gauging methods, gauging weirs provide the most accurate measurements. However, the construction of the weir is difficult in many streams. To measure the stream water by any type of bucket also requires another construction. Measurement with a current meter or by buoys is possible where the stream is so deep that the mean velocity can be accurately determined from "the measured velocity. Estimation of discharge frem the river stage is possible where the friction factor does not change with time. The dilution methods can be applied to torrential streams where other methods are difficult to adopt. Although it takes a long time to measure stream discharge at a point, measurements at different points can be carried out easily, because no fixed structure is needed. One of the dilution methods consists of pouring a solution of the tracer at a constant discharge into a stream and measuring the tracer concentration in the stream when it reaches the equilibrium level. This method (constant flux dilution method, or C F D method) has been described in many textbooks of hydrology and river Correspondence to: K. Okunishi, Disaster Prevention Research Institute, Kyoto University, Gokanosho, Uji 611, Japan.
232
K. OKUNISHI ET AL.
hydraulics. Another method (finite mass dilution method, or FMD method) involves injection of a finite quantity of the tracer and continuous observation of the concentration at a downstream point until it returns to the background level. This method was first proposed by Hanya et al. (1957), and was described in more detail by Hanya (1959). Although the measurement procedure seems complicated, recent innovations in data logging techniques have made continuous recording of the tracer concentration very easy. An advantage of this method is that it needs a smaller quantity of the tracer and almost no apparatus for the tracer injection. Both of the dilution methods assume that the tracer quickly disperses in the transverse direction, presenting a uniform distribution of its concentration in the stream cross-section. This assumption is satisfied, e.g. in turbulent streams with narrow widths and shallow depths. The precision and reliability of the dilution methods have not been discussed in detail, perhaps because they have been regarded only as auxiliary methods for stream gauging. The recent experience of the authors (Okunishi and Saito, 1988; Okunishi et al., 1990) has suggested that the precision of the finite mass dilution method is much higher than commonly expected, but that particular caution is needed in fieldwork for reliable gauging. PRINCIPLE OF THE DILUTION METHODS
The fundaraental principle is that the dilution of the tracer is proportional to the discharge, when the mass of the injected tracer is conserved between the injection point and the point of concentration measurement. In the CFD method, the equation of mass conservation is written as, cr(o + Or) =
ctat + c b o
(1)
or,
a
= at(ct - c f ) / ( c r -
Cb)
(la)
where Ct is the concentration of the tracer in the solution injected into the channel at a discharge of Qt, Cb is the background concentration in the flow of discharge Q to be measured, and Cf is the tracer concentration an infinite time after the injection. The tracer solution should be injected at a constant discharge until the measured concentration C presents no time change, or is approaching a certain value. In the FMD method (Hanya et al., 1957), the time change in the tracer concentration C(t) in the stream is continuously measured after a certain mass of the tracer, M, is injected into the stream until the measured concentration returns to the background value C b. The equation of mass conservation is
STREAM GAUGING BY DILUTION METHODS
A Z
233
/ - d°
./-
2 I-
u
o.s!
Z O
B .
,,.
-.
06"--
2
4
6
8
lo
TIME
Fig. 1. Theoretical curves of the tracer concentration according to eqn. (4) for the FMD method (Curve A) and according to eqn. (5) for the CFD method (Curve B) on an arbitrary scale and with arbitrary parameters.
written as, oO
Q f [C(t)-
Cb]dt =
M
(2)
0
If the dispersion of the tracer in the stream of a mean velocity, v, is assumed to have a constant dispersion coefficient, D, the basic equation, 3C t~C = & + v ~x
D
2dC t~x2
(3)
holds. By solving this equation with appropriate boundary conditions (e.g. Bear, 1972), the concentration C~ (t) in the case of the CFD method is written as,
C,(t) = CtQtv Q
1
i 1
(4reD)l/2 ~ e x p
[ (x-- vz)2]dz -
(4)
4Dr
The solution for the FMD method is written as, C2(t) =
Q (4n/)t)l/2 exp
[
-
(x z _ 4Dt
J
(5)
The time changes in the tracer concentration as represented by eqns. (4) and (5) are shown in Fig. I. The relationship between Cl(t) arid C2(t) is shown as,
C,(t) = CtQt i C2(t M o MEASUREMENT
-
-
z) dr,
AND CALCULATION
(6) PROCEDURES
Equations (4) and (5) show that it takes infinite time for the tracer concentration in the gauged stream to reach the equilibrium level and the
234
K. O K U N I S H ! E T AL.
background level, respectively. To finish the measurement in finite time, therefore, the concentration at the infinite time Cr in eqn. (1) or the integral in eqn. (2) should be estimated from the finite time series of the measured values. As is usually done, the concentration as described in eqns. (4) and (5) can be approximated as, C(t)
Co exp ( - t / T )
=
Cf -
=
Co exp ( - t / T )
(7)
and C(t)
(8)
4- Cb
respectively, when t is large enough (t > tl), where Co and T are constants. The integral in eqn. (2) is thus rewritten as, oo
I =
|[C(t)-
Ii
Cb]dt = | [ C ( t ) -
L/
t/
0
0
Cb]dt 4- C(h) T
(9)
It is obvious from eqns. (7) and (9) that a continuous recording of the tracer concentration is needed in both methods. In order to satisfy the prerequisites for the validity of eqns. (1) or (2), the reach of the stream between the injection point and the measurement point should be large enough to ensure the complete dispersion of the tracer in the cross-section of the stream. On the other hand, longer distances between these points may involve a larger inflow to and/or outflow from the stream which cause an ambiguity of the obtained stream discharge, and the time needed for the measurement lengthens. In the case of finite mass injection, moreover, the longer' distance lowers the level of the measured concentration, lessening its precision. Therefore, the selection of an optimum distance is essential. Sea salt (NaCl) is thought to be the best tracer in many cases, because it is cheap, exerts minimum environmental impact, and its concentration is easily and precisely measured by a conductivity meter. The background conductivity of the stream water reflects the concentrations of different substances. However, it can be hypothetically replaced with the NaC1 solution of the same conductivity. ANALYSIS OF PRECISION
Under field conditions, the measured values are subject to non-systematic fluctuations and systematic deviations. The observed tracer concentration fluctuates when a vortex prevails in the stream and obstructs the homogeneous dispersion required by the theory. Sometimes the background tracer concentration or the background conductivity fluctuates. This is caused by the confluence of tributaries of different water chemistry as well as by human
STREAM GAUGING BY DILUTION METHODS
235
activities. These fluctuations have a wide range of frequency. A short-period fluctuation makes it difficult to determine the parameters in eqns. (7) and (8) exactly. A long-period fluctuation makes the value of Cb used in eqns. (I) or (2) uncertain. These factors are quantitatively evaluated below. It is assumed here that the errors in the preparation and injection of the tracer solution are negligible, that the measurement of the tracer concentration is exact enough, and that the tracer is homogeneously dispersed in the stream cross-section. Precision of the constant flux dilution (CFD) method The precision of the stream discharge as estimated by eqn. (1) is practically determined by Cf - Cb. Introducing the quantity C~ which is defined as Cf = C(t,) -P C~ C~ =
(10)
Co exp ( - t , / T )
(10a)
derived from eqn. (7). The precision of C~ is uncertain because, in the field it is difficult to ensure that the approximation of eqn. (7) holds at t > t ! . Therefore, the error in C~ denoted by ACe might be as large as C~/2. The error in Cb, or ACb, depends on the field conditions. Therefore, if C(h ) is assumed to be error free, AQ/Q = c~/cf/2 + ACb/C f
(11)
It is obvious from eqn. (10a) that the first term of the right-hand side of eqn. (11) becomes smaller as t~ increases. Precision of the finite mass dilution (FMD) method The first term of the right-hand side of eqn. (9) ordinarily prevails in the integral I. Introducing a mean concentration (7,, which will be much larger than Cb, the integral I is roughly approximated by, I
--
(C m --
Cb)t !
~---
Cmt I
The error in the first term of the right-hand side of eqn. (9) is caused by the error in Cb because the fluctuation in C(t) is smoothed out by integration. On the other hand, the estimation of the second term of the right-hand side of eqn. (9) is not reliable, just as mentioned above. Thus, it is assumed again that this term involves an error of 50%. Therefore, the estimated error in I is, AI = ACbt, + C(t,)T/2 and the relative error in Q is approximately written as, AQ/Q
= AI/I = C(tl)T/(2Cmt~) h-ACb/C m
(12)
236
K. OKUNISH! ET AL
.,I
I
I
--° o
[--
°~
,~
0
~
~,..
vII/I/1/1/1//I//l ,~
o
,~
~.
STREAM G A U G I N G BY DILUTION METHODS
237
300Is "1 /
'~' 200
//~-
z 150
/Z/'-----,.
a lOO
./~.,
• ....
/
50
10
15 WATER
20
30cm
LEVEL
Fig. 3. A logarithmic regression of the discharge of a pond as gauged by the CFD method at site I to the water level of the pond.
Comparing eqn. (12) with eqn. (11), it can be said that the precision is not the same but comparable between the two dilution methods. E S T I M A T I O N O F T H E R E L I A B I L I T Y O F ]'HE D I L U T I O N M E T H O D S F R O M F I E L D EXPERIENCE
Stream gauging by the F M D method was carried out ala~"5 :~ r~ach of the Otani River in Shiga Prefecture, Japan. This reach is located at the apex o~ tiic alluvial fan, and the river water flows into a pond from the downstream end of this reach. The location of the injection points and the measurement points are shown in Fig. 2. Although the measurement points were fixed, injection points were chosen at an appropriate distance from the measurement points, according to the stream discharge. The water supply from the pond for domestic use was gauged in the channel downstream from the outlet of the pond (site 1 of Fig. 2) and correlated with the water level in the pond (Fig. 3). The outlet of the pond had a fixed shape similar to a weir. The artificial channel between the injection points and the measurement point has a homogeneous turbulence owing to the regular form of the channel, an appropriate channel gradient and relatively great roughness of the bottom. The existence of the pond stabilizes the background conductivity of the water flowing out of the pond. These factors are thought to have enabled a high reliability (8% of the calculated value with 95% confidence) of the stream gauging at this point. Table 1 shows that the precision of this method at this point is around 1% except the first measurement. When the observed curve of the tracer concentration is much different from the theoretical one, it is doubtful whether the assumption of complete
238
K. OKUNISHI ET AL.
TABLE 1 Evaluation of the precision of the finite mass dilution method for the measurements along the reach of the Otani River Site
Date
1 1 1 1 1 1 1 1 1 1 3 7 8
21Apr 15 Sep 26 Sep 10Nov 2Dec 20Dec 17 Jan 20 Jul I Aug 29 Jul 21Apr 21Apr 17 Jan
1989 1989 1989 1989 1989 1989 1990 1990 1990 1991 1989 1989 1990
M
I
Q
Cb/Cm
C(tl)T 2Cmtl
AQ/Q
1017 5002 5002 5002 5002 5002 5002 5002 5030 3026 1018 2036 5002
4847 23184 29223 44566 66369 58268 43620 19027 29602 17487 148346 7763 150145
210 216 171 112 75 86 115 263 170 173 6.9 262 33.3
0.039 0.009 0.005 0.004 0.003 0.004 0.004 0.001 0,001 0.002 0.0003 0.016 0.0004
0.019 0.002 0.020 0.006 0.016 0.003 0.003 0.004 0.002 0.001 0.005 0.028 0.007
0.058 0.011 0.025 0.010 0.019 0.007 0.007 0.005 0,003 0,003 0,095 0.044 0.007
dispersion in the stream cross-section is satisfied, and the estimated value of the discharge is not reliable. The time changes in the tracer concentration at different measurement points on different occasions are shown in Fig. 4. The curves of site 1 are similar to the theoretical curve (B) in Fig. 1, reflecting the homogeneous turbulence in the channel. The measurements in artificial channels of smooth bottom frequently present a flat peak as exemplified by the one at site 3. It suggests a piston flow component in the channel, presumably due to the smooth bottom. The curve at site 7 presents a low peak concentration. Coupled with heterogeneous dispersion and with a large fluctuation in the background concentration, it caused a large error in the stream discharge as mentioned below. At site 8, the tracer concentration involved marked periodical fluctuation. Because of a succession of sh~ot and pool between the injection and measurement points, vortices of similar sizes prevailed in the mixing processes of the tracer. Determination of the time constant T in eqn. (8) was carried out by trial-and-error method, assuming ~he background level Cb and plotting log [C(t) - Cb] VS. t, until the plots constituted a straight line. In many cases, the optimum value of Cb was larger than the real b~ckground level Cb0 as measured prior to the injection of the tracer as exemplified by Fig. 5 (B). If the value of Cb0 is used, the plots often constitute a concave curve as shown in Fig. 5(A). This might be caused by the dispersion coefficient decreasing with
STREAM
GAUGING
1"~1
'~
BY DILUTION
Site
1
17 Jan.
1.01
1990
1
2
i
[-'\
site 3
t
/
21
\
|
!
]
i
•
°'~t t '
• - "
2
-
'
\.
"
/
4
6rain.
1989
: ,.
~ !._~
~
Apr.
',
I
~
0
239
METHODS
"
0
5
10rain.
3
d
Site
an.
1 990
2
Site 0.31
7
21 Apr.
1989
l
0,2
1
0
[\ ..
~
f" 2
0
F
I
'l 4
~
- -I 6min,
0
--"--" 0
5
10
15min,
Fig. 4. The time change in the observed tracer concentration (in g l- ~) at different sites. The time interval for which the exponential approximation was applied is shown by the double arrows.
100[
'
I
0 % .... ;
2
3
4
5
f
~
6
7
TIME
"'
0
1
2
t
i
3
4
5
~
7
(rain)
Fig. 5. Diagrams of ( C - Cb)/Cb vs. time plotted on sc;nilogarithmic paper. The background concentration Cb is assumed to b~ identical to the conceutration before the tracer injection (26.8 mg l -t ) in the left-hand diagram, and 28.4 mg 1-' in the right-hand diagram.
240
K. O K U N I S H I ET AL.
390-
/ /
- 200-
0
|
8+9
|
i
|
i
i
|
7
8
5
4
2
1
LOCATION NUMBER Fig. 6. Changes in the stream discharge along the reach of the Olani River c i different occasions.
time as the smaller scale turbulence prevails in the dispersion process. Therefore, the exponential approximation of eqn. (8) cannot be considered reliable. Changes in the stream discharge along the reach shown in Fig. 2 are shown in Fig. 6. The notation of 8 + 9 means that the total discharge of these tributaries is plotted on the diagram. The discharge at point 7 on 21 April 1989 was unnaturally large. At this time, the tracer was injected into one of the two tributaries. It is not certain, but possible, that the concentration of the tracer presented a deviated distribution in the stream cross-section at the measure,, ment point. Moreover, a large fluctuation in the background conductiv! ies and the low peak level of the measured concentration caused a low precision of the calculated discb ~;ge. On the other occasions, the tracer was injected at the confluence of the two channels although the distance to the measurement point is not enough for complete dispersion. Except the measurement on 21 April 1989, the general trend of the spatial distribution of the stream discharge is similar through all measurements. Increase and decrease in the stream discharge along the reach is inferred as reflecting the inflow of ground water from both sides of the valley and percolation into the debris flow deposit which constitutes the channel bed (Okunishi et al., 1990). The river water is
STREAM GAUGING BY DILUTION METHODS
241
introduced to the pond through a channel at a point near site 2. However, not all of the river water is collected. Moreover, there are several overflow structures for the adjustment of the discharge along the channel between site 2 and the pond. This is the reason for a marked decrease between site 2 and site I. The evaluation of the precision of the gauging is shown in Table 1 for several cases of gauging in this reach. In most cases, the evaluated precision is better than 3%. However, the factors which are not quantitatively evaluated in the previous section should have affected the reliability of this method. The gauging error at site 7 on 21 April 1989 appears to be as large as 20% according to Fig. 6. A comparison of the discharge obtained at site 8 and the water level of the Parshal flume above this site suggests that the average error is about 10% (Okunishi et al., 1990). In generv~, it can be said that the FMD method has a reliability better than 10% of the tream discharge, unless the field conditions are extremely bad. The CFD method was not tested in the field by the authors. However, it is expected that similar precision would have been obtained in the Otani River, if this method had been applied with the same injection and measurement points, because of the similarities of the principles underlying both methods and their calculation procedures. As mentioned above, the precision of the CFD method can be improved by measuring the tracer concentration over a longer period. Unlike the FMD method, longer measurement does not lead to lower reliability, e:~c~-~ptthat the background concentration might drift during the measuremeut. Uneven tracer concent~-ation in the stream cross-section at the measurement point can essentially affect the reliability of the CFD method. In applying this method, the tracer is usually injected to the centre of the stream. If the dispersion in the transverse direction is incomplete, the concentration at the bankside parts of the stream at the measurement point is lower than at the central part where the measurement is usually carried out. The average concentration, which leads to the correct discharge is, therefore, lower than the measured concentrati~'.~ (,\;,~ 7). In applying the FMD method, on the other hand, the tracer is easily spread across the stream when it is injected. In this case, the measured concentration may be different between the central and bankside parts of the measurement point due to the difference in the dispersion coefficient as shown in Fig. 7. However, the integration ! in eqn. (9) is expected to be essentially the same. C O N C L U D I N G REMARKS
When improving the precision of the FMD method according to eqn. (12),
242
K. OKUNISHI ET AL.
TIME
TIME
Fig. 7. Schematic illustrations of the effects of uneven distribution of the tracer concentration across the stream in the CFD method (left) and the FMD method (right). (A) Central part; (B) bankside part; (C) cross-sectional average.
the value of t, should be chosen so that the relative error be at a minimum. This choice can be made at the time of calculation if the time series of C(t) has b~en obtained in the field for a time long enough for the change in C(t) to b~come as small as the fluctuation of the background concentration. Homogeneous dispersion is essential to obtain reliable values of stream discharge using the FMD method. Although this factor is difficult to quantify, heterogeneity in the dispersion process involved can be detected by examining the observed curve of C(t). A domain of piston flow, large space of dead water, or dominapt vortex which occurs in a pool in the stream channel are usually the major causes of heterogeneous dispersion. The precision of the CFD method can be improved by measuring the tracer concentration over a long period, until the increase in C(t) as expected in eqn. (7) becomes less than the fluctuation of the background concentration. Among the factors which make the reliability of tile CFD method uncertain, the inhomogeneity of the tracer concentration in the stream cross-section at the measurement point can be checked in the field, when the time change in C(t) has become small. A long-period fluctuation of the background concentration cannot be checked in the field during the measurement, it should be examined prior to the ~aeasurement. The quantity or the flux of the tracer is also an important factor determining the reliability of the dilution methods because the measured concentration should be much greater than the background level or the resolution of the measuring equipment. However, using too much of the tracer leads to a lower precision of the measurement, as well as damage to the environment. In conclusion, it is pointed out that the FMD method is suitable. Ior experimental measurements in the reaches where the discharge is not frequently garaged because the tracer injection is simple and easy. On the other hand, the CFD method is suitable for more or less fixed stations, where the characteristics of the dispersion and the background fluctuation are known, and a high reliability of stream gauging is needed.
STREAM GAUGING BY DILUTION METHODS
243
REFERENCES Bear, J., 1972. Dynamics of Fluid in Porous Media. Elsevier, New York, pp. 627-629: Hanya, T., 1959. On the measurement of stream velocity and discharge by a chemical method. Water Temp. Res., 2(5): 1-19 (in Japanese). Hanya, T., Oya, M. and Yokoyama, T., 1957. Discharge measurement based on the dilution ratio (Preliminary Report), Case study of R. Tanigawa. Water Temp. Res., 1(5): 1-5 (in Japanese). Okunishi, K. and Saito, T., 1988. Hydrological characteristics of the mountains draining to the northern part of Lake Biwa Part 1. Catchment area of River rshida. Rep. of Water Resources Center, Kyoto University, No. 8, pp. 25-39 (in Japanese). Okuaishi, K., Saito, T. and Yokoyama, K., 1990. Report of Stream Discharge Investigations of the Otani River. 24 pp (Unpubl.), (in Japanese).
231
Elsevier Science Publishers B.V., Amsterdam
[1]
Accuracy of stream gauging by dilution methods Kazuo Okunishi a, Takashi Saito a and Toshio Yoshida b aDisaster Prevention Research lnstilute, Kyo;o U'iiversity, Gokanosho, Uji 611, Japan bOyo Corporation, Ichigaya-Building, Kudan-kita, Chiyoda-Ku, Tokyo 102, Japan (Received 29 October 1991; accepted 3 December 199 !)
ABSTRACT Okunishi, K., Saito, T. and Yoshida, T., 1992. Accuracy of stream gauging by dilution methods. J. Hydrol., i 37:231-243. The constant flux diluUon ,,,,::5,~d and the finite mass dilution method are examined for accurate measurement of stream discharge. The measut,~om procedures are reviewed on a mathematical basis. Their precision is quantitatively evaluated following the assumption that the requirement of homogeneous dispersion is satisfied. The total reliability is discussed on the basis of field experience with the finite mass dilution method. Finally, a summary of the procedures necessary for better field measurements is given.
INTRODUCTION
Among a variety of stream gauging methods, gauging weirs provide the most accurate measurements. However, the construction of the weir is difficult in many streams. To measure the stream water by any type of bucket also requires another construction. Measurement with a current meter or by buoys is possible where the stream is so deep that the mean velocity can be accurately determined from "the measured velocity. Estimation of discharge frem the river stage is possible where the friction factor does not change with time. The dilution methods can be applied to torrential streams where other methods are difficult to adopt. Although it takes a long time to measure stream discharge at a point, measurements at different points can be carried out easily, because no fixed structure is needed. One of the dilution methods consists of pouring a solution of the tracer at a constant discharge into a stream and measuring the tracer concentration in the stream when it reaches the equilibrium level. This method (constant flux dilution method, or C F D method) has been described in many textbooks of hydrology and river Correspondence to: K. Okunishi, Disaster Prevention Research Institute, Kyoto University, Gokanosho, Uji 611, Japan.
232
K. OKUNISHI ET AL.
hydraulics. Another method (finite mass dilution method, or FMD method) involves injection of a finite quantity of the tracer and continuous observation of the concentration at a downstream point until it returns to the background level. This method was first proposed by Hanya et al. (1957), and was described in more detail by Hanya (1959). Although the measurement procedure seems complicated, recent innovations in data logging techniques have made continuous recording of the tracer concentration very easy. An advantage of this method is that it needs a smaller quantity of the tracer and almost no apparatus for the tracer injection. Both of the dilution methods assume that the tracer quickly disperses in the transverse direction, presenting a uniform distribution of its concentration in the stream cross-section. This assumption is satisfied, e.g. in turbulent streams with narrow widths and shallow depths. The precision and reliability of the dilution methods have not been discussed in detail, perhaps because they have been regarded only as auxiliary methods for stream gauging. The recent experience of the authors (Okunishi and Saito, 1988; Okunishi et al., 1990) has suggested that the precision of the finite mass dilution method is much higher than commonly expected, but that particular caution is needed in fieldwork for reliable gauging. PRINCIPLE OF THE DILUTION METHODS
The fundaraental principle is that the dilution of the tracer is proportional to the discharge, when the mass of the injected tracer is conserved between the injection point and the point of concentration measurement. In the CFD method, the equation of mass conservation is written as, cr(o + Or) =
ctat + c b o
(1)
or,
a
= at(ct - c f ) / ( c r -
Cb)
(la)
where Ct is the concentration of the tracer in the solution injected into the channel at a discharge of Qt, Cb is the background concentration in the flow of discharge Q to be measured, and Cf is the tracer concentration an infinite time after the injection. The tracer solution should be injected at a constant discharge until the measured concentration C presents no time change, or is approaching a certain value. In the FMD method (Hanya et al., 1957), the time change in the tracer concentration C(t) in the stream is continuously measured after a certain mass of the tracer, M, is injected into the stream until the measured concentration returns to the background value C b. The equation of mass conservation is
STREAM GAUGING BY DILUTION METHODS
A Z
233
/ - d°
./-
2 I-
u
o.s!
Z O
B .
,,.
-.
06"--
2
4
6
8
lo
TIME
Fig. 1. Theoretical curves of the tracer concentration according to eqn. (4) for the FMD method (Curve A) and according to eqn. (5) for the CFD method (Curve B) on an arbitrary scale and with arbitrary parameters.
written as, oO
Q f [C(t)-
Cb]dt =
M
(2)
0
If the dispersion of the tracer in the stream of a mean velocity, v, is assumed to have a constant dispersion coefficient, D, the basic equation, 3C t~C = & + v ~x
D
2dC t~x2
(3)
holds. By solving this equation with appropriate boundary conditions (e.g. Bear, 1972), the concentration C~ (t) in the case of the CFD method is written as,
C,(t) = CtQtv Q
1
i 1
(4reD)l/2 ~ e x p
[ (x-- vz)2]dz -
(4)
4Dr
The solution for the FMD method is written as, C2(t) =
Q (4n/)t)l/2 exp
[
-
(x z _ 4Dt
J
(5)
The time changes in the tracer concentration as represented by eqns. (4) and (5) are shown in Fig. I. The relationship between Cl(t) arid C2(t) is shown as,
C,(t) = CtQt i C2(t M o MEASUREMENT
-
-
z) dr,
AND CALCULATION
(6) PROCEDURES
Equations (4) and (5) show that it takes infinite time for the tracer concentration in the gauged stream to reach the equilibrium level and the
234
K. O K U N I S H ! E T AL.
background level, respectively. To finish the measurement in finite time, therefore, the concentration at the infinite time Cr in eqn. (1) or the integral in eqn. (2) should be estimated from the finite time series of the measured values. As is usually done, the concentration as described in eqns. (4) and (5) can be approximated as, C(t)
Co exp ( - t / T )
=
Cf -
=
Co exp ( - t / T )
(7)
and C(t)
(8)
4- Cb
respectively, when t is large enough (t > tl), where Co and T are constants. The integral in eqn. (2) is thus rewritten as, oo
I =
|[C(t)-
Ii
Cb]dt = | [ C ( t ) -
L/
t/
0
0
Cb]dt 4- C(h) T
(9)
It is obvious from eqns. (7) and (9) that a continuous recording of the tracer concentration is needed in both methods. In order to satisfy the prerequisites for the validity of eqns. (1) or (2), the reach of the stream between the injection point and the measurement point should be large enough to ensure the complete dispersion of the tracer in the cross-section of the stream. On the other hand, longer distances between these points may involve a larger inflow to and/or outflow from the stream which cause an ambiguity of the obtained stream discharge, and the time needed for the measurement lengthens. In the case of finite mass injection, moreover, the longer' distance lowers the level of the measured concentration, lessening its precision. Therefore, the selection of an optimum distance is essential. Sea salt (NaCl) is thought to be the best tracer in many cases, because it is cheap, exerts minimum environmental impact, and its concentration is easily and precisely measured by a conductivity meter. The background conductivity of the stream water reflects the concentrations of different substances. However, it can be hypothetically replaced with the NaC1 solution of the same conductivity. ANALYSIS OF PRECISION
Under field conditions, the measured values are subject to non-systematic fluctuations and systematic deviations. The observed tracer concentration fluctuates when a vortex prevails in the stream and obstructs the homogeneous dispersion required by the theory. Sometimes the background tracer concentration or the background conductivity fluctuates. This is caused by the confluence of tributaries of different water chemistry as well as by human
STREAM GAUGING BY DILUTION METHODS
235
activities. These fluctuations have a wide range of frequency. A short-period fluctuation makes it difficult to determine the parameters in eqns. (7) and (8) exactly. A long-period fluctuation makes the value of Cb used in eqns. (I) or (2) uncertain. These factors are quantitatively evaluated below. It is assumed here that the errors in the preparation and injection of the tracer solution are negligible, that the measurement of the tracer concentration is exact enough, and that the tracer is homogeneously dispersed in the stream cross-section. Precision of the constant flux dilution (CFD) method The precision of the stream discharge as estimated by eqn. (1) is practically determined by Cf - Cb. Introducing the quantity C~ which is defined as Cf = C(t,) -P C~ C~ =
(10)
Co exp ( - t , / T )
(10a)
derived from eqn. (7). The precision of C~ is uncertain because, in the field it is difficult to ensure that the approximation of eqn. (7) holds at t > t ! . Therefore, the error in C~ denoted by ACe might be as large as C~/2. The error in Cb, or ACb, depends on the field conditions. Therefore, if C(h ) is assumed to be error free, AQ/Q = c~/cf/2 + ACb/C f
(11)
It is obvious from eqn. (10a) that the first term of the right-hand side of eqn. (11) becomes smaller as t~ increases. Precision of the finite mass dilution (FMD) method The first term of the right-hand side of eqn. (9) ordinarily prevails in the integral I. Introducing a mean concentration (7,, which will be much larger than Cb, the integral I is roughly approximated by, I
--
(C m --
Cb)t !
~---
Cmt I
The error in the first term of the right-hand side of eqn. (9) is caused by the error in Cb because the fluctuation in C(t) is smoothed out by integration. On the other hand, the estimation of the second term of the right-hand side of eqn. (9) is not reliable, just as mentioned above. Thus, it is assumed again that this term involves an error of 50%. Therefore, the estimated error in I is, AI = ACbt, + C(t,)T/2 and the relative error in Q is approximately written as, AQ/Q
= AI/I = C(tl)T/(2Cmt~) h-ACb/C m
(12)
236
K. OKUNISH! ET AL
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STREAM G A U G I N G BY DILUTION METHODS
237
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20
30cm
LEVEL
Fig. 3. A logarithmic regression of the discharge of a pond as gauged by the CFD method at site I to the water level of the pond.
Comparing eqn. (12) with eqn. (11), it can be said that the precision is not the same but comparable between the two dilution methods. E S T I M A T I O N O F T H E R E L I A B I L I T Y O F ]'HE D I L U T I O N M E T H O D S F R O M F I E L D EXPERIENCE
Stream gauging by the F M D method was carried out ala~"5 :~ r~ach of the Otani River in Shiga Prefecture, Japan. This reach is located at the apex o~ tiic alluvial fan, and the river water flows into a pond from the downstream end of this reach. The location of the injection points and the measurement points are shown in Fig. 2. Although the measurement points were fixed, injection points were chosen at an appropriate distance from the measurement points, according to the stream discharge. The water supply from the pond for domestic use was gauged in the channel downstream from the outlet of the pond (site 1 of Fig. 2) and correlated with the water level in the pond (Fig. 3). The outlet of the pond had a fixed shape similar to a weir. The artificial channel between the injection points and the measurement point has a homogeneous turbulence owing to the regular form of the channel, an appropriate channel gradient and relatively great roughness of the bottom. The existence of the pond stabilizes the background conductivity of the water flowing out of the pond. These factors are thought to have enabled a high reliability (8% of the calculated value with 95% confidence) of the stream gauging at this point. Table 1 shows that the precision of this method at this point is around 1% except the first measurement. When the observed curve of the tracer concentration is much different from the theoretical one, it is doubtful whether the assumption of complete
238
K. OKUNISHI ET AL.
TABLE 1 Evaluation of the precision of the finite mass dilution method for the measurements along the reach of the Otani River Site
Date
1 1 1 1 1 1 1 1 1 1 3 7 8
21Apr 15 Sep 26 Sep 10Nov 2Dec 20Dec 17 Jan 20 Jul I Aug 29 Jul 21Apr 21Apr 17 Jan
1989 1989 1989 1989 1989 1989 1990 1990 1990 1991 1989 1989 1990
M
I
Q
Cb/Cm
C(tl)T 2Cmtl
AQ/Q
1017 5002 5002 5002 5002 5002 5002 5002 5030 3026 1018 2036 5002
4847 23184 29223 44566 66369 58268 43620 19027 29602 17487 148346 7763 150145
210 216 171 112 75 86 115 263 170 173 6.9 262 33.3
0.039 0.009 0.005 0.004 0.003 0.004 0.004 0.001 0,001 0.002 0.0003 0.016 0.0004
0.019 0.002 0.020 0.006 0.016 0.003 0.003 0.004 0.002 0.001 0.005 0.028 0.007
0.058 0.011 0.025 0.010 0.019 0.007 0.007 0.005 0,003 0,003 0,095 0.044 0.007
dispersion in the stream cross-section is satisfied, and the estimated value of the discharge is not reliable. The time changes in the tracer concentration at different measurement points on different occasions are shown in Fig. 4. The curves of site 1 are similar to the theoretical curve (B) in Fig. 1, reflecting the homogeneous turbulence in the channel. The measurements in artificial channels of smooth bottom frequently present a flat peak as exemplified by the one at site 3. It suggests a piston flow component in the channel, presumably due to the smooth bottom. The curve at site 7 presents a low peak concentration. Coupled with heterogeneous dispersion and with a large fluctuation in the background concentration, it caused a large error in the stream discharge as mentioned below. At site 8, the tracer concentration involved marked periodical fluctuation. Because of a succession of sh~ot and pool between the injection and measurement points, vortices of similar sizes prevailed in the mixing processes of the tracer. Determination of the time constant T in eqn. (8) was carried out by trial-and-error method, assuming ~he background level Cb and plotting log [C(t) - Cb] VS. t, until the plots constituted a straight line. In many cases, the optimum value of Cb was larger than the real b~ckground level Cb0 as measured prior to the injection of the tracer as exemplified by Fig. 5 (B). If the value of Cb0 is used, the plots often constitute a concave curve as shown in Fig. 5(A). This might be caused by the dispersion coefficient decreasing with
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Fig. 5. Diagrams of ( C - Cb)/Cb vs. time plotted on sc;nilogarithmic paper. The background concentration Cb is assumed to b~ identical to the conceutration before the tracer injection (26.8 mg l -t ) in the left-hand diagram, and 28.4 mg 1-' in the right-hand diagram.
240
K. O K U N I S H I ET AL.
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LOCATION NUMBER Fig. 6. Changes in the stream discharge along the reach of the Olani River c i different occasions.
time as the smaller scale turbulence prevails in the dispersion process. Therefore, the exponential approximation of eqn. (8) cannot be considered reliable. Changes in the stream discharge along the reach shown in Fig. 2 are shown in Fig. 6. The notation of 8 + 9 means that the total discharge of these tributaries is plotted on the diagram. The discharge at point 7 on 21 April 1989 was unnaturally large. At this time, the tracer was injected into one of the two tributaries. It is not certain, but possible, that the concentration of the tracer presented a deviated distribution in the stream cross-section at the measure,, ment point. Moreover, a large fluctuation in the background conductiv! ies and the low peak level of the measured concentration caused a low precision of the calculated discb ~;ge. On the other occasions, the tracer was injected at the confluence of the two channels although the distance to the measurement point is not enough for complete dispersion. Except the measurement on 21 April 1989, the general trend of the spatial distribution of the stream discharge is similar through all measurements. Increase and decrease in the stream discharge along the reach is inferred as reflecting the inflow of ground water from both sides of the valley and percolation into the debris flow deposit which constitutes the channel bed (Okunishi et al., 1990). The river water is
STREAM GAUGING BY DILUTION METHODS
241
introduced to the pond through a channel at a point near site 2. However, not all of the river water is collected. Moreover, there are several overflow structures for the adjustment of the discharge along the channel between site 2 and the pond. This is the reason for a marked decrease between site 2 and site I. The evaluation of the precision of the gauging is shown in Table 1 for several cases of gauging in this reach. In most cases, the evaluated precision is better than 3%. However, the factors which are not quantitatively evaluated in the previous section should have affected the reliability of this method. The gauging error at site 7 on 21 April 1989 appears to be as large as 20% according to Fig. 6. A comparison of the discharge obtained at site 8 and the water level of the Parshal flume above this site suggests that the average error is about 10% (Okunishi et al., 1990). In generv~, it can be said that the FMD method has a reliability better than 10% of the tream discharge, unless the field conditions are extremely bad. The CFD method was not tested in the field by the authors. However, it is expected that similar precision would have been obtained in the Otani River, if this method had been applied with the same injection and measurement points, because of the similarities of the principles underlying both methods and their calculation procedures. As mentioned above, the precision of the CFD method can be improved by measuring the tracer concentration over a longer period. Unlike the FMD method, longer measurement does not lead to lower reliability, e:~c~-~ptthat the background concentration might drift during the measuremeut. Uneven tracer concent~-ation in the stream cross-section at the measurement point can essentially affect the reliability of the CFD method. In applying this method, the tracer is usually injected to the centre of the stream. If the dispersion in the transverse direction is incomplete, the concentration at the bankside parts of the stream at the measurement point is lower than at the central part where the measurement is usually carried out. The average concentration, which leads to the correct discharge is, therefore, lower than the measured concentrati~'.~ (,\;,~ 7). In applying the FMD method, on the other hand, the tracer is easily spread across the stream when it is injected. In this case, the measured concentration may be different between the central and bankside parts of the measurement point due to the difference in the dispersion coefficient as shown in Fig. 7. However, the integration ! in eqn. (9) is expected to be essentially the same. C O N C L U D I N G REMARKS
When improving the precision of the FMD method according to eqn. (12),
242
K. OKUNISHI ET AL.
TIME
TIME
Fig. 7. Schematic illustrations of the effects of uneven distribution of the tracer concentration across the stream in the CFD method (left) and the FMD method (right). (A) Central part; (B) bankside part; (C) cross-sectional average.
the value of t, should be chosen so that the relative error be at a minimum. This choice can be made at the time of calculation if the time series of C(t) has b~en obtained in the field for a time long enough for the change in C(t) to b~come as small as the fluctuation of the background concentration. Homogeneous dispersion is essential to obtain reliable values of stream discharge using the FMD method. Although this factor is difficult to quantify, heterogeneity in the dispersion process involved can be detected by examining the observed curve of C(t). A domain of piston flow, large space of dead water, or dominapt vortex which occurs in a pool in the stream channel are usually the major causes of heterogeneous dispersion. The precision of the CFD method can be improved by measuring the tracer concentration over a long period, until the increase in C(t) as expected in eqn. (7) becomes less than the fluctuation of the background concentration. Among the factors which make the reliability of tile CFD method uncertain, the inhomogeneity of the tracer concentration in the stream cross-section at the measurement point can be checked in the field, when the time change in C(t) has become small. A long-period fluctuation of the background concentration cannot be checked in the field during the measurement, it should be examined prior to the ~aeasurement. The quantity or the flux of the tracer is also an important factor determining the reliability of the dilution methods because the measured concentration should be much greater than the background level or the resolution of the measuring equipment. However, using too much of the tracer leads to a lower precision of the measurement, as well as damage to the environment. In conclusion, it is pointed out that the FMD method is suitable. Ior experimental measurements in the reaches where the discharge is not frequently garaged because the tracer injection is simple and easy. On the other hand, the CFD method is suitable for more or less fixed stations, where the characteristics of the dispersion and the background fluctuation are known, and a high reliability of stream gauging is needed.
STREAM GAUGING BY DILUTION METHODS
243
REFERENCES Bear, J., 1972. Dynamics of Fluid in Porous Media. Elsevier, New York, pp. 627-629: Hanya, T., 1959. On the measurement of stream velocity and discharge by a chemical method. Water Temp. Res., 2(5): 1-19 (in Japanese). Hanya, T., Oya, M. and Yokoyama, T., 1957. Discharge measurement based on the dilution ratio (Preliminary Report), Case study of R. Tanigawa. Water Temp. Res., 1(5): 1-5 (in Japanese). Okunishi, K. and Saito, T., 1988. Hydrological characteristics of the mountains draining to the northern part of Lake Biwa Part 1. Catchment area of River rshida. Rep. of Water Resources Center, Kyoto University, No. 8, pp. 25-39 (in Japanese). Okuaishi, K., Saito, T. and Yokoyama, K., 1990. Report of Stream Discharge Investigations of the Otani River. 24 pp (Unpubl.), (in Japanese).