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Predictive model for salt intrusion in estuaries
Journal o f Hydrology, 148 (1993) 203-218
203
0022-1694/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
[4]
Predictive model for salt intrusion in estuaries Hubert H.G. Savenije International Institute for Infrastructural, Hydraulic and Environmental Engineering (IHE), P.O. Box 3015, 2601 DA, Delft, Netherlands (Received 7 November 1992; revision accepted 21 January 1993)
Abstract A model which requires prior calibration on prototype measurements can not be used as a predictive model. In a predictive model, the model parameters are determined on the basis of measurable or quantifiable variables, mostly through empirical relations. This paper describes such relations for a predictive steady-state salt intrusion model in alluvial estuaries. The method has been successfully applied to 45 salt intrusion measurements in 15 estuaries worldwide.
Introduction For the m a n a g e m e n t of estuarine water resources, the water manager needs to have an instrument to determine the salt intrusion length and the longitudinal distribution o f the salinity as a function of directly measurable parameters such as geometry, fresh water flow and tide. Such an instrument is called a predictive model; it can be used to simulate a 'what-if' situation in case one or more o f these parameters are changed. Existing predictive models all have an empirical c o m p o n e n t and are either inaccurate or limited to special conditions. TheSe limitation can refer to estuary shape (most commonly that the estuary should have a constant cross-section) or to the mixing mechanism (e.g. predominantly density or tide driven). In general, existing predictive models are either unsuitable for use in real estuaries - - being based on laboratory experiments in channels of constant cross-section - - (e.g. Rigter, 1973) or can only be used after one or more calibration coefficients have been determined for the particular estuary under study. The m e t h o d of S a n m u g a n a t h a n and Abernethy (1975, 1979) belongs to the latter category. This m e t h o d involves two calibration coefficients which are to be determined on the basis of an extensive salinity intru-
204
H.H.G. Savenije / Journal of Hydrology 148 (1993) 203--218
sion survey. Although the method is useful and accurate, it is expensive, timeconsuming and incomplete since it has no universally applicable coefficients. The method presented in this paper is also empirical, but it has wider applicability and does not require prior calibration (although calibration can be used to refine the estimates made). It applies equally well to densityor tide- driven mixing and it can be used for estuaries with a wide range of shapes, as long as the estuary is alluvial and the longitudinal variation of the cross-sectional area can be described by an exponential function. The fact that the method works well in a wide range of estuaries suggests that it can be applied to other estuaries as well, particularly during the reconnaissance stage of an estuarine management study. In the following, first a short summary is given of earlier work on the salt intrusion length after which the predictive model is derived and subsequently tested in 15 estuaries world-wide with quite different characteristics. Finally, the results obtained by using different methods are compared.
Intrusion length The most important output of a predictive model is the salt intrusion length L, the distance from the estuary mouth to the point where the salinity reaches the river salinity. Some researchers, such as Rigter (1973), Fischer (1974) and Van Os and Abraham (1990), used the intrusion length at low-water slack L Lws, others, e.g. Van der Burgh (1972) and Savenije (1986), used the tidal average intrusion length L TA, but in this paper the intrusion length at highwater slack L uws is used. This is the maximum intrusion length occurring during a tidal period. The difference between the intrusion length at HWS and LWS is the tidal excursion E. The tidal average intrusion length may be considered to be the average of the intrusion lengths at HWS and LWS (Savenije, 1989). The following review deals with empirical work by Rigter (1973), Fischer (1974), Van der Burgh (1972) and Van Os and Abraham (1990). With the exception of Van der Burgh, all these investigators based their analysis on laboratory tests and prototype measurements in estuaries with constant crosssection. Rigter (1973), on the basis of flume data of Delft Hydraulics Laboratory and of the Waterways Experiment Station (WES), arrived at the following empirical relation: L Lws = 1 . 5 7 r ~ ( F ~ ' N - ' - 1.7)~ 4 . 7 ~ F ~ l N -'
(1)
where h0 is the tidal average depth at the estuary mouth, f is Darcy-Weis-
H.H.G. Savenije / Journal of Hydrology 148 (1993) 203-218
205
bach's roughness, N is Canter Cremers' estuary number defined as the ratio of the fresh water entering the estuary during a tidal cycle to the flood volume of salt water entering the estuary over a tidal cycle, Pt. Hence:
N _ Q f T _ - Q f T - 7r -Q____£f 7r - u o Pt AoEo 1.08 Aovo 1.08 v0
(2)
where Qf is the fresh water discharge and u0 = Q f / h o is the fresh water velocity at the estuary mouth, both of which are negative since the positive x direction is taken upstream; A 0 is the cross-sectional area at the estuary mouth and Tis the tidal period. Savenije (1992) showed that the flood volume can be very well approximated by the product of A0 and the tidal excursion E 0 at the estuary mouth and that E0 = 1.08voT/Tr where v0 is the tidal velocity amplitude at the estuary mouth. In addition, the densimetric Froude number F d in Eq. (1) is defined as: Fd--
pv°2 -- p F Apgho Ap
(3)
where F = v2/(gho) is the Froude number, p is the density of the water and Ap is the density difference over the intrusion length. It is observed that, since in alluvial estuaries both N and F d are much smaller than unity, the number 1.7 in Eq. (1) can be disregarded and that Rigter's intrusion length, hence, is inversely proportional to N and Fd. Fischer (1974), in a discussion of Rigter's results, proposed the following formula based on the same data: 1.7.7 h0 F_0.75~T_0.25 " " f 0--67~ d ~"
LLWS
(4)
Van der Burgh (1972) made use of limited prototype information from the Rotterdam Waterway, the Schelde, the Haringvliet (a tidal branch of the Rhine-Meuse delta) and the Eems. He obtained a similar relation as the earlier researchers. Elaboration of his expression for the mean tidal dispersion yields (see Savenije, 1992): LTA
=
-
26ho gv/~oVONO.5 ,~, ho ,-,-o.s,~-o.5 -K v0 u0
(5)
where K is Van der Burgh's coefficient defined by Savenije (1993). It is clear that, although Van der Burgh used F instead of F d, there is similarity between the three methods presented. In all estuaries debouching in open sea, the density difference Ap over the intrusion length is about the same (Ap ~ 25); consequently F is proportional to F d (F a ~40F). Also the roughness in alluvial estuaries does not appear to vary much from estuary to
H.H.G. Savemje : Journal ol tIydrology 148 I1993J 203 218
206
estuary (Savenije, 1992). Hence there is a great similarity between the different formulae, most importantly, they are linear in h0. The major difference between the methods is that they use different exponents for the two bulk parameters, the Froude number F and the Canter Cremers' number N. This was also observed by Kranenburg (1983) in a personal communication to the author (in Dutch) and by Van Os and Abraham (1990) who developed a formula similar to Rigter's for use at Delft Hydraulics: LLWS4•4- ~ ho F~IN t
(6)
It can be observed that the exponent of N in Eq. (6) is negative since the salt intrusion length reduces if the fresh water discharge increases. Similarly, the intrusion length decreases linearly with the tidal velocity (Fd ~ decreases with the second power of v 0 and N --1 increases linearly with v0), which is understandable since the salt intrusion length at LWS is short if E is large. For a method based on HWS, however, one expects the intrusion length L Hws to be long if E is large. In the method presented here, use is made of the expression for the longitudinal tidal average dispersion presented by Savenije (1993) and the exponential variation of the cross-sectional area, dD d.v
(7)
KQ f A
A = A0 e x p ( - x ] \
(8)
a /
where a is the convergence length, a length scale for the rate of convergence, and A0 is the cross-sectional area at the estuary mouth• In some estuaries the geometry of the estuary can only be described well by a branched exponential function where upstream and downstream of an inflection point at a distance .vl from the mouth the convergence length has different values: a~ and a2, respectively. The Maputo estuary is a good example of such an estuary (see Fig. 1). In all cases encountered, the convergence length al in the downstream reach is shorter than a2, resulting in a trumpet shape• In these estuaries the following equations are used instead of Eq. (8): A .... 4oexp
-
if0
A -- A1 exp
i f x > _vl a2
where A1 =- A(xj).
(8a) (8b)
H.H.G. Savenije / Journalof Hydrology 148 (1993) 203-218
207
100000 Mac~t°
Bay of
~Maputo
31623 " ~ 10000
~
.
~
.
•
Area(m2)
*
Width (m)
•Depth(m)
3162 1000 316
,o$..
•
100
$
•
•
32 10 3 1 ~, el
Mozambique
•
& • A •
i
10 r~
•
•
• • • •
•
~
20
• &
i
i
i
30
40
50
D i s t a n c e from the m o u t h
(km)
/
,)/,J~
Indian Ocean
Fig. 1. The Maputo estuary and the branched exponential function describing the cross-sectional area variation.
Similarly to Eq. (8), an exponential function can be fitted to the estuary width B: B=B0exp
-
(9)
where b is the width convergence length and B 0 is the width at the estuary mouth. In some cases, such as in the Maputo estuary (see Fig. 1), the width convergence length is shorter downstream (bl) than~ upstream (b2). In the case where the estuary depth is constant, a -- b. In alluvial estuaries the variation of the depth is generally small and a ~ b (see Table 1). In this respect, the Lalang estuary, which becomes deeper upstream, is an exception. Integration of the steady-state salt balance equation for H W S in combination with Eq. (7) yields (see Savenije, 1989, 1992):
, ,f (o)1 Voo where SO and D Oare boundary conditions at x = 0 for H W S and
(lo) Sf
is the river
208
H.H.G. Saven~e ,, Journal o f Hydrology 148 (1993) 203-218
Table I
Characteristics of the estuaries studied Estuaries
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Mae Klong Solo Lalang Limpopo Tha Chin
ChaoPhya Incomati Pungue Maputo Thames Eems Corantijn
Country
Thailand Indonesia Indonesia Mozambique
Thailand Thailand Mozambique Mozambique Mozambique UK
Netherlands Surinam
Schelde Delaware
Netherlands USA
Tejo
Portugal
Bo
ho
al
a~~)
xl
(m)
(m)
(km)
(km)
(kin) (km)
250 220 350 370 3600 600 4500 6500 9000 7500 31600 30000 15200 37700 5820
5.2 9.2 10.6 7.0 5.3 7.2 3.0 4.3 3.6 7.1 3.9 6.5 10.0 6.6 7.0
22 7 4
19
102 226 2t7 50 87 109 42 20 16 23 19 64 26 41 II
22 14 8
18
~, ~ ,
155 226 96 50 87 109 42 20 16 23 19 ,48 28 42 il
H~
K
(m) 2.2 0.8 3.0 1. l 2.6 2.5 1.4 6.0 3.4 6.0 3.6 2.3 3.7 1.8 3.4
0.30 0.60 0.65 0.60 0.35 0.75 0.15 0.30 0.38 0.20 0.30 0.21 0.25 0.22 0.90
water salinity. Integration of Eq. (7) yields (see Savenije, 1989, 1992): D
1- 3(exp ( x
where:
3-
KaQf Ka DoAo c~oAo
(12)
where ;4 is a positive coefficient (since Qf is negative) which determines the longitudinal variation of D. The two parameters Qf and Do, which in this approach appear conveniently together, are combined into a single-model variable s0 = -Do/Qr. From Eq. (10) it can be seen that S = Sf when D = 0. Since S = Sf at x = L Hws, Eq. (11) can be elaborated to yield an expression for the intrusion length: L Hws = a In
(;) + 1
(13)
Since/3 is positive, the argument of the natural logarithm is always larger than unity. In case two branches are required to fit the exponential function, such as in the Tha Chin, the Incomati, the Maputo and the Corantijn (see Table 1), the
209
H.H.G. SavenOe / Journal of Hydrology 148 (1993) 203-218
following equation is used instead of Eq. (13): L nws = xl + a2 In
+ 1
(13a)
where 32 is defined by:
32 = Ka2 OtlA1
(13b)
where ~l is computed by substitution of Xl in Eq. (11) with a = -D/Qf: o~_£1= 1 - 3 1 o~0
((x:)) exp
- 1
where 31 is determined from Eq. (12) by substituting al for a. To make the model predictive, what remains to be determined are empirical equations relating K and oL0 to quantifiable estuary numbers.
Empirical analysis In the empirical analysis, use is made of an estuary with a simple geometry where a = ax = a2 and b = bl = b2. In the case where an estuary requires a branched exponential function to describe the geometry, the values of a2 and b2 are used instead of a and b in the empirical equations for K and a0. First the relation for Van der Burgh's K is addressed. Each different estuary appears to have its own characteristic value of K (Savenije, 1986, 1989). Table 1 shows the values obtained for 15 estuaries. Since K varies from estuary to estuary, but not significantly over time, it is logical to look for a relation with bulk parameters which are not time-dependent. The following dimensionless ratios appeared to correlate well with K: h0 II 1 = -ff
(14)
II 2 = - ~
(15)
h0 1-I3 = B--o
(16)
II4
(17)
T gv/~°
/-/0 where H0 is the tidal range at the estuary m o u t h at spring tide (a constant for a given estuary). The first parameter is the estuary shape number, a geometrical
210
H.H.G. Savenije i Journal o/Hydrology 148 (1993,) 203 218
parameter which Savenije (1993) showed to be a good indicator tor the shape of the intrusion curve. If ho/b (or ho/a) is large, the salt intrusion has a dome shape; if ho/b is small, the intrusion curve has a recession shape. The second parameter is the tidal range to depth ratio, which characterises the tidal wave. The third parameter is the depth to width ratio at the estuary mouth, characterising the shape of the estuary mouth. The fourth parameter is a ratio between the celerity of an undamped progressive wave ( x / ~ ) and Ho/T, the tidal range per tidal period. Since 11o/Tis related to the tidal flow velocity, II 4 is a sort of Froude number. Regression analysis (see Fig.2) on the estuaries presented in Table 1 yielded the following relation: K = 0.16 x 10-6Hil°H~661IT13II224(0 < K < 1)
(18)
with an explained variance of 84%, and standard errors of the exponents of 0.21, 0.44, 0.06 and 0.28, respectively. Substitution of the ratios Hi yields: K ---- 0.16 x 10
6
h°69gli2T224 HOo.59bl.lOBO.13(0
< K<
1)
(19)
1 /.<. 0,9
i, ; i
/
!
/r~
0.8
3 0.7
i i
//.
0.6
0.5
0.4
/"
i / /// 5 ,/ .-/<
~ 13
1
//" /
/ 6 4
//" /~9"--
0.3
7
0.2
~
8
0.1
/ / 0 0
0,2
0.4
0.6 K
Fig. 2. Empirical relation for K
(calibrated)
08
1
H.H.G. Savenije / Journal of Hydrology 148 (1993) 203-218
211
The dispersion at the mouth
For the prediction of the boundary condition for the dispersion D~ ws and hence the calibration parameter a0, an analysis was made of the measurements presented in Table 2. As with the earlier researchers, a relation was sought between a non-dimensional dispersion parameter on the one hand, and Canter Cremers' number N and the Froude number F on the other. An expression for the dimensionless dispersion can be derived from scaling the steady-state one-dimensional salt balance equation: dS
OfS - AD~x = 0
(20)
through the introduction of ~({), the dimensionless salinity as a function of the dimensionless distance {: S ~(~) = ~
(21)
x
= E
(22)
where L is a longitudinal length scale and S is a salinity scale. Consequently, the salinity gradient is defined by: dS dg d~ S dg dx - S d~dx - Ld~
(23)
Substitution of Eqs. (21)--(23) in (20) yields: uf~
Dd~ Ld~-
0
(24)
where uf = Of~A, the velocity of the fresh water discharge. It can be seen from Eq. (24) that the advective transport in the first term is counterbalanced by a dispersive transport with a velocity D/L. Hence the dimension of the dispersion is the product of a longitudinal length scale and a longitudinal velocity. A good choice for the longitudinal length scale is the tidal excursion E0: the horizontal length scale of tidal mixing, while a good choice for the longitudinal velocity is the tidal velocity amplitude v 0. Hence the dimensionless dispersion is defined as D~WS/(voEo). The following empirical relation was obtained between the dimensionless dispersion and estuary numbers: D~Iws voEo -- 220h°a F-°'SN°'5 = 220v/~ boaV ~ R (25)
212
.==o
0
0
H.H.G. Savenije / Journal of Hydrology 148 (1993) 203--218
H.H.G. Saven(je / Journal of Hydrology 148 (1993) 203-218
0
0
0
0
0
0
0
~
0
0
0
0
0
0
0
0
0
213
0
•
~
.
"~,
~ ~ ~ oo o oo o o o o ..=az
H.H.G. Savenije Journal q/Itydrology 14~ (1993) 203 218
214
If the relative density difference Ap/p, present in Fd, is considered constant for different estuaries (Fd = 40F), then it can be observed that the dimensionless dispersion varies with the root of the Estuarine Richardson number NR = N/Fd. The Estuarine Richardson number is generally used as an indicator for the degree of stratification. Equation (25) is a simple power relation between the dimensionless dispersion, the estuary shape number ho/a and the Estuarine Richardson number. It is interesting to note that Fischer (1972) also observed that the dimensionless dispersion is a function of the Estuarine Richardson number. The main difference between Eq. (25) and Fischer's relation is the presence of the estuary shape number (Fischer disregarded estuary shape) and that Fischer used u*hoinstead of v0E0 to make the dispersion dimensionless (u* is the root mean square bottom shear velocity). The result of this empirical analysis is presented in Fig. 3, where the numbers indicate the estuaries. The middle line is the line of perfect agreement and the lines above and below represent deviations by a factor 2. Owing to the large uncertainty in the determination of the fresh water discharge, a considerable scatter in the data points on which the relation of Eq. (25) is based is unavoidable. An error of a factor 2 in the determination of D0 implies an error of a factor 4 in Qr- Of course, this is an unacceptably large error, but then, all points, except the points of the Tha Chin, lie well between these lines. In the Tha Chin, an estuary in the Chao Phya delta in Thailand, an overestimation of the fresh water discharge by a factor 2 is well possible in view of the considerable water consumption in the area which has been disregarded in the determination of Qf. The observed values in the Tha Chin would be quite in agreement with Eq. (25) if a withdrawal of fresh water of about 20 m 3 swere taken into account. Such an abstraction is quite possible in view of the water demands of the area (ILACO, 1987). It is often very difficult to estimate the fresh water discharge in the downstream part of an estuary. Moreover, there can be substantial uncertainty about withdrawals in the area. Therefore, in most estuaries, an error of a factor 2 in the estimation of Qr is quite possible. In light of this, the results obtained seem satisfactory. The only way to improve the reliability of Eq. (25) is by extending the empirical analysis to a larger set of data points. In Fig. 4, the same empirical relation is presented with ~0 as the variable. This relation is obtained after division of Do by -Qr. Hence Eq. (25) can be elaborated into:
ao= 220 h° E 0T °
V--gZrAo
(26)
H.H.G. Savenije / Journal of Hydrology 148 (1993) 203-218
215
2500
1000
400 10
250
Q.
E 0 0
100
40 25
10
i
10
i
i
25
40
t
~
i
100
i
250
I
400
I
I
1000
i
2500
calibrated
Fig. 3. Empirical relation for Do
This is a rather simple formula that relates the second model parameter to quantifiable hydrological, hydraulic and geometrical variables.
Comparison of methods Combination of the empirical Eq. (26) with Eqs. (12) and (13) yields: L Hws = a In \(-220
h°E°V°KaZuo F-°SN °'5 + 1)
(27)
Since on the interval x 6 (0, 1): ln(x + 1) ~0 x, as a first-order approximation, Eq. (27) can be simplified to facilitate comparison with the results of earlier researchers: LHWS ~
-200h°E°V°Kauo F-°SN 0.5 if
LHWS
a
< 1
(28)
Earlier researchers based their theories on estuaries with constant crosssection, or estuaries where a ~ cx~. In such cases, << 1. Hence Eq. (28) may be compared with the work of these researchers. Substitution of Eq.
LHWS/a
H.H.G..Savenije /' Journal o[ Hydrology 148 (1993) 203-218
216
40 25 , , z -,-Z ~ 6 ~ 6 " 10 i
Jl0
_
s
~ ~"
q259/
10 CX
4.0
0
2.5
E 0
,
j
11 2 1. ~ _ 14 j , ~ 12
1.0
0.4 0.25 !
0.4
1.0
2.5
4.0
10.0
25
40
calibrated Fig. 4. Empirical relation for ~0
(2) in Eq. (28) yields: LHWS
L Hws - 2207rh°E° F-°'SN -°5 if - 1.08 K a a
< 1
(29)
The correspondence with the relations obtained by Rigter (1973), Fischer (1974) and Van Os and Abraham (1990) presented in Eqs. (1), (4) and (6) is high, but it is highest with Van der Burgh's relation, Eq. (5). The difference with Van der Burgh's relation, besides the logarithmic function of Eq. (27), lies in the use of Eo/a. E o is explainable since Eq. (27) has been derived for HWS. As a consequence, unlike in Van der Burgh's equation which applies to mean tide, L Hws should increase with an increase in v0 (or E0 for that matter). The inclusion of the convergence length a is the result of considering an exponentially varying cross-section. If the estuary has a more pronounced funnel shape (a small), the salt intrusion length is large; apparently the salt intrusion through mixing is easier in a funnel-shaped estuary than in a prismatic estuary. This corresponds with what one would think intuitively. The major advantage of this method over earlier methods is that it takes account of estuary shape and that it does not limit itself to estuaries of constant cross-section.
217
H.H.G. Saveniie / Journal of Hydrology 148 (1993) 203-218
6310
+
3981
Rigter 11973)
;~
Fischer (1974)
2512
Savenije
1585
Van der Burgh (1972)
:~
1000
line of perfect agreement
~k
631
×
i
~
x
x x
261 x
>,e~xy X"
. ~,
X &
4O
X X
~
x
~k z
~
,,
~
X
± x ~
i L~IXI~'~N
n
x
X
•
x
X
x
. .
25 16
%
10
I
20
I
32
I
I
50
I
I
79
I
I
126
I
200
measured Intrusion length (kin) Fig. 5. Comparison of different methods to compute the intrusion length at HWS
Model results
In Fig. 5, the four methods for the determination of the intrusion length are compared. In the methods of Rigter (1973), Fischer (1974) and Van Os and Abraham (1990) the tidal excursion is added to L Lws to obtain LHWS; in the method of Van der Burgh El2. It can be seen that the methods of Rigter (1973) and Van Os and Abraham (1990) do not differ significantly. The value of K used for this graph is the one obtained from calibration. This is the approach one would follow once K had been obtained from calibration on a salt intrusion measurement. If the predictive model is used for the first time, in a situation where no salt intrusion measurement is available, then the predicted value of K, Eq. (19), should be used. The use of the predicted value of K then results in slightly more scatter. Conclusion
It can be clearly seen from Fig. 5 that the method presented in this paper applies very well to prototype estuaries; much better than methods based on laboratory experiments. The only d a t a points diverging somewhat from the line of perfect agreement are those belonging to the Tha Chin discussed earlier.
218
H.H.G. Saventye / Journal t4/Hydrolocy 148 11993; 203 218
The earlier methods are clearly hampered by the assumption of constant cross-section. Apparently the influence of estuary shape on salt intrusion is large, In fact, one may conclude that since only man-made estuaries with fixed banks (such as the Rotterdam Waterway) have a constant cross-section, the laboratory experiments have little value for real estuaries. Although this paper has concentrated on the salt intrusion length, it has been shown in previous studies (Savenije 1986, 1989, 1992) that the model can be adequately applied to describe the longitudinal salinity variation in estuaries also. The values of K and c~0 derived from Eqs. (19) and (26) on the basis of quantifiable hydraulic, hydrologic and geometric parameters can be substituted in Eqs. (10)-(12) to compute the longitudinal variation of the salinity in a predictive model.
References Fischer. H.B., 1972. Mass transport mechanisms in partially stratified estuaries. J. Fluid Mech., 53:671-687. Fischer, H.B., 1974. Discussion of "Minimum length of salt intrusion in estuaries' by B.P. Rigter, 1973. J. Hydraul. Div. Proc., ASCE, 100:708 712. ILACO, 1987. Agricultural development and crop diversification project covering the Chao Phya west bank area, Thailand, Annex B (Hydrology). Euroconsult, Arnhem, Netherlands. Rigter, B.P., 1973. Minimum length of salt intrusion in estuaries. J. Hydraul. Div. Proc., ASCE, 99:1475 1496. Sanmuganathan, K. and Abernethy, C.L., 1975. A mathematical model to predict long term salinity intrusion in estuaries. Proc. Second World Congress, International Water Resources Association, New Delhi, Vol. 3, pp. 313-324. Sanmuganathan, K. and Abernethy C.L., 1979. Long term prediction of salinity intrusion in branched estuaries. Proc. 18th IAHR Congress. International Association for Hydraulic Research, Cagliari Savenije, H.H.G., 1986. A one-dimensional model tbr salinity intrusion in alluvial estuaries. J. Hydrol., 85: 87--109. Savenije, H.H.G., 1989. Salt intrusion model for high-water slack, low-water slack and mean tide on spreadsheet. J.Hydrol., 107: 9-18. Savenije, H.H.G., 1992. Rapid assessment technique for salt intrusion in alluvial estuaries. 1HE Report Series, Report No. 27, International Institute for Infrastructural, Hydraulic and Environmental Engineering, Delft, Netherlands Savenije, H.H.G., 1993. Composition and driving mechanisms of longitudinal tidal average salinity dispersion in estuaries. J. Hydrol., 144: 127--141. Van der Burgh, P., 1972. Ontwikkeling van een methode voor her voorspellen van zoutverdelingen in estuaria, kanalen en zeeen. Rijkswaterstaat Rapport, pp. 10 72. Van Os, A.G. and Abraham, G., 1990. Density Currents and Salt Intrusion. Lecture Note for the Hydraulic Engineering Course at IHE-Delft, Delft Hydraulics, Delft, Netherlands.
203
0022-1694/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
[4]
Predictive model for salt intrusion in estuaries Hubert H.G. Savenije International Institute for Infrastructural, Hydraulic and Environmental Engineering (IHE), P.O. Box 3015, 2601 DA, Delft, Netherlands (Received 7 November 1992; revision accepted 21 January 1993)
Abstract A model which requires prior calibration on prototype measurements can not be used as a predictive model. In a predictive model, the model parameters are determined on the basis of measurable or quantifiable variables, mostly through empirical relations. This paper describes such relations for a predictive steady-state salt intrusion model in alluvial estuaries. The method has been successfully applied to 45 salt intrusion measurements in 15 estuaries worldwide.
Introduction For the m a n a g e m e n t of estuarine water resources, the water manager needs to have an instrument to determine the salt intrusion length and the longitudinal distribution o f the salinity as a function of directly measurable parameters such as geometry, fresh water flow and tide. Such an instrument is called a predictive model; it can be used to simulate a 'what-if' situation in case one or more o f these parameters are changed. Existing predictive models all have an empirical c o m p o n e n t and are either inaccurate or limited to special conditions. TheSe limitation can refer to estuary shape (most commonly that the estuary should have a constant cross-section) or to the mixing mechanism (e.g. predominantly density or tide driven). In general, existing predictive models are either unsuitable for use in real estuaries - - being based on laboratory experiments in channels of constant cross-section - - (e.g. Rigter, 1973) or can only be used after one or more calibration coefficients have been determined for the particular estuary under study. The m e t h o d of S a n m u g a n a t h a n and Abernethy (1975, 1979) belongs to the latter category. This m e t h o d involves two calibration coefficients which are to be determined on the basis of an extensive salinity intru-
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H.H.G. Savenije / Journal of Hydrology 148 (1993) 203--218
sion survey. Although the method is useful and accurate, it is expensive, timeconsuming and incomplete since it has no universally applicable coefficients. The method presented in this paper is also empirical, but it has wider applicability and does not require prior calibration (although calibration can be used to refine the estimates made). It applies equally well to densityor tide- driven mixing and it can be used for estuaries with a wide range of shapes, as long as the estuary is alluvial and the longitudinal variation of the cross-sectional area can be described by an exponential function. The fact that the method works well in a wide range of estuaries suggests that it can be applied to other estuaries as well, particularly during the reconnaissance stage of an estuarine management study. In the following, first a short summary is given of earlier work on the salt intrusion length after which the predictive model is derived and subsequently tested in 15 estuaries world-wide with quite different characteristics. Finally, the results obtained by using different methods are compared.
Intrusion length The most important output of a predictive model is the salt intrusion length L, the distance from the estuary mouth to the point where the salinity reaches the river salinity. Some researchers, such as Rigter (1973), Fischer (1974) and Van Os and Abraham (1990), used the intrusion length at low-water slack L Lws, others, e.g. Van der Burgh (1972) and Savenije (1986), used the tidal average intrusion length L TA, but in this paper the intrusion length at highwater slack L uws is used. This is the maximum intrusion length occurring during a tidal period. The difference between the intrusion length at HWS and LWS is the tidal excursion E. The tidal average intrusion length may be considered to be the average of the intrusion lengths at HWS and LWS (Savenije, 1989). The following review deals with empirical work by Rigter (1973), Fischer (1974), Van der Burgh (1972) and Van Os and Abraham (1990). With the exception of Van der Burgh, all these investigators based their analysis on laboratory tests and prototype measurements in estuaries with constant crosssection. Rigter (1973), on the basis of flume data of Delft Hydraulics Laboratory and of the Waterways Experiment Station (WES), arrived at the following empirical relation: L Lws = 1 . 5 7 r ~ ( F ~ ' N - ' - 1.7)~ 4 . 7 ~ F ~ l N -'
(1)
where h0 is the tidal average depth at the estuary mouth, f is Darcy-Weis-
H.H.G. Savenije / Journal of Hydrology 148 (1993) 203-218
205
bach's roughness, N is Canter Cremers' estuary number defined as the ratio of the fresh water entering the estuary during a tidal cycle to the flood volume of salt water entering the estuary over a tidal cycle, Pt. Hence:
N _ Q f T _ - Q f T - 7r -Q____£f 7r - u o Pt AoEo 1.08 Aovo 1.08 v0
(2)
where Qf is the fresh water discharge and u0 = Q f / h o is the fresh water velocity at the estuary mouth, both of which are negative since the positive x direction is taken upstream; A 0 is the cross-sectional area at the estuary mouth and Tis the tidal period. Savenije (1992) showed that the flood volume can be very well approximated by the product of A0 and the tidal excursion E 0 at the estuary mouth and that E0 = 1.08voT/Tr where v0 is the tidal velocity amplitude at the estuary mouth. In addition, the densimetric Froude number F d in Eq. (1) is defined as: Fd--
pv°2 -- p F Apgho Ap
(3)
where F = v2/(gho) is the Froude number, p is the density of the water and Ap is the density difference over the intrusion length. It is observed that, since in alluvial estuaries both N and F d are much smaller than unity, the number 1.7 in Eq. (1) can be disregarded and that Rigter's intrusion length, hence, is inversely proportional to N and Fd. Fischer (1974), in a discussion of Rigter's results, proposed the following formula based on the same data: 1.7.7 h0 F_0.75~T_0.25 " " f 0--67~ d ~"
LLWS
(4)
Van der Burgh (1972) made use of limited prototype information from the Rotterdam Waterway, the Schelde, the Haringvliet (a tidal branch of the Rhine-Meuse delta) and the Eems. He obtained a similar relation as the earlier researchers. Elaboration of his expression for the mean tidal dispersion yields (see Savenije, 1992): LTA
=
-
26ho gv/~oVONO.5 ,~, ho ,-,-o.s,~-o.5 -K v0 u0
(5)
where K is Van der Burgh's coefficient defined by Savenije (1993). It is clear that, although Van der Burgh used F instead of F d, there is similarity between the three methods presented. In all estuaries debouching in open sea, the density difference Ap over the intrusion length is about the same (Ap ~ 25); consequently F is proportional to F d (F a ~40F). Also the roughness in alluvial estuaries does not appear to vary much from estuary to
H.H.G. Savemje : Journal ol tIydrology 148 I1993J 203 218
206
estuary (Savenije, 1992). Hence there is a great similarity between the different formulae, most importantly, they are linear in h0. The major difference between the methods is that they use different exponents for the two bulk parameters, the Froude number F and the Canter Cremers' number N. This was also observed by Kranenburg (1983) in a personal communication to the author (in Dutch) and by Van Os and Abraham (1990) who developed a formula similar to Rigter's for use at Delft Hydraulics: LLWS4•4- ~ ho F~IN t
(6)
It can be observed that the exponent of N in Eq. (6) is negative since the salt intrusion length reduces if the fresh water discharge increases. Similarly, the intrusion length decreases linearly with the tidal velocity (Fd ~ decreases with the second power of v 0 and N --1 increases linearly with v0), which is understandable since the salt intrusion length at LWS is short if E is large. For a method based on HWS, however, one expects the intrusion length L Hws to be long if E is large. In the method presented here, use is made of the expression for the longitudinal tidal average dispersion presented by Savenije (1993) and the exponential variation of the cross-sectional area, dD d.v
(7)
KQ f A
A = A0 e x p ( - x ] \
(8)
a /
where a is the convergence length, a length scale for the rate of convergence, and A0 is the cross-sectional area at the estuary mouth• In some estuaries the geometry of the estuary can only be described well by a branched exponential function where upstream and downstream of an inflection point at a distance .vl from the mouth the convergence length has different values: a~ and a2, respectively. The Maputo estuary is a good example of such an estuary (see Fig. 1). In all cases encountered, the convergence length al in the downstream reach is shorter than a2, resulting in a trumpet shape• In these estuaries the following equations are used instead of Eq. (8): A .... 4oexp
-
if0
A -- A1 exp
i f x > _vl a2
where A1 =- A(xj).
(8a) (8b)
H.H.G. Savenije / Journalof Hydrology 148 (1993) 203-218
207
100000 Mac~t°
Bay of
~Maputo
31623 " ~ 10000
~
.
~
.
•
Area(m2)
*
Width (m)
•Depth(m)
3162 1000 316
,o$..
•
100
$
•
•
32 10 3 1 ~, el
Mozambique
•
& • A •
i
10 r~
•
•
• • • •
•
~
20
• &
i
i
i
30
40
50
D i s t a n c e from the m o u t h
(km)
/
,)/,J~
Indian Ocean
Fig. 1. The Maputo estuary and the branched exponential function describing the cross-sectional area variation.
Similarly to Eq. (8), an exponential function can be fitted to the estuary width B: B=B0exp
-
(9)
where b is the width convergence length and B 0 is the width at the estuary mouth. In some cases, such as in the Maputo estuary (see Fig. 1), the width convergence length is shorter downstream (bl) than~ upstream (b2). In the case where the estuary depth is constant, a -- b. In alluvial estuaries the variation of the depth is generally small and a ~ b (see Table 1). In this respect, the Lalang estuary, which becomes deeper upstream, is an exception. Integration of the steady-state salt balance equation for H W S in combination with Eq. (7) yields (see Savenije, 1989, 1992):
, ,f (o)1 Voo where SO and D Oare boundary conditions at x = 0 for H W S and
(lo) Sf
is the river
208
H.H.G. Saven~e ,, Journal o f Hydrology 148 (1993) 203-218
Table I
Characteristics of the estuaries studied Estuaries
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Mae Klong Solo Lalang Limpopo Tha Chin
ChaoPhya Incomati Pungue Maputo Thames Eems Corantijn
Country
Thailand Indonesia Indonesia Mozambique
Thailand Thailand Mozambique Mozambique Mozambique UK
Netherlands Surinam
Schelde Delaware
Netherlands USA
Tejo
Portugal
Bo
ho
al
a~~)
xl
(m)
(m)
(km)
(km)
(kin) (km)
250 220 350 370 3600 600 4500 6500 9000 7500 31600 30000 15200 37700 5820
5.2 9.2 10.6 7.0 5.3 7.2 3.0 4.3 3.6 7.1 3.9 6.5 10.0 6.6 7.0
22 7 4
19
102 226 2t7 50 87 109 42 20 16 23 19 64 26 41 II
22 14 8
18
~, ~ ,
155 226 96 50 87 109 42 20 16 23 19 ,48 28 42 il
H~
K
(m) 2.2 0.8 3.0 1. l 2.6 2.5 1.4 6.0 3.4 6.0 3.6 2.3 3.7 1.8 3.4
0.30 0.60 0.65 0.60 0.35 0.75 0.15 0.30 0.38 0.20 0.30 0.21 0.25 0.22 0.90
water salinity. Integration of Eq. (7) yields (see Savenije, 1989, 1992): D
1- 3(exp ( x
where:
3-
KaQf Ka DoAo c~oAo
(12)
where ;4 is a positive coefficient (since Qf is negative) which determines the longitudinal variation of D. The two parameters Qf and Do, which in this approach appear conveniently together, are combined into a single-model variable s0 = -Do/Qr. From Eq. (10) it can be seen that S = Sf when D = 0. Since S = Sf at x = L Hws, Eq. (11) can be elaborated to yield an expression for the intrusion length: L Hws = a In
(;) + 1
(13)
Since/3 is positive, the argument of the natural logarithm is always larger than unity. In case two branches are required to fit the exponential function, such as in the Tha Chin, the Incomati, the Maputo and the Corantijn (see Table 1), the
209
H.H.G. SavenOe / Journal of Hydrology 148 (1993) 203-218
following equation is used instead of Eq. (13): L nws = xl + a2 In
+ 1
(13a)
where 32 is defined by:
32 = Ka2 OtlA1
(13b)
where ~l is computed by substitution of Xl in Eq. (11) with a = -D/Qf: o~_£1= 1 - 3 1 o~0
((x:)) exp
- 1
where 31 is determined from Eq. (12) by substituting al for a. To make the model predictive, what remains to be determined are empirical equations relating K and oL0 to quantifiable estuary numbers.
Empirical analysis In the empirical analysis, use is made of an estuary with a simple geometry where a = ax = a2 and b = bl = b2. In the case where an estuary requires a branched exponential function to describe the geometry, the values of a2 and b2 are used instead of a and b in the empirical equations for K and a0. First the relation for Van der Burgh's K is addressed. Each different estuary appears to have its own characteristic value of K (Savenije, 1986, 1989). Table 1 shows the values obtained for 15 estuaries. Since K varies from estuary to estuary, but not significantly over time, it is logical to look for a relation with bulk parameters which are not time-dependent. The following dimensionless ratios appeared to correlate well with K: h0 II 1 = -ff
(14)
II 2 = - ~
(15)
h0 1-I3 = B--o
(16)
II4
(17)
T gv/~°
/-/0 where H0 is the tidal range at the estuary m o u t h at spring tide (a constant for a given estuary). The first parameter is the estuary shape number, a geometrical
210
H.H.G. Savenije i Journal o/Hydrology 148 (1993,) 203 218
parameter which Savenije (1993) showed to be a good indicator tor the shape of the intrusion curve. If ho/b (or ho/a) is large, the salt intrusion has a dome shape; if ho/b is small, the intrusion curve has a recession shape. The second parameter is the tidal range to depth ratio, which characterises the tidal wave. The third parameter is the depth to width ratio at the estuary mouth, characterising the shape of the estuary mouth. The fourth parameter is a ratio between the celerity of an undamped progressive wave ( x / ~ ) and Ho/T, the tidal range per tidal period. Since 11o/Tis related to the tidal flow velocity, II 4 is a sort of Froude number. Regression analysis (see Fig.2) on the estuaries presented in Table 1 yielded the following relation: K = 0.16 x 10-6Hil°H~661IT13II224(0 < K < 1)
(18)
with an explained variance of 84%, and standard errors of the exponents of 0.21, 0.44, 0.06 and 0.28, respectively. Substitution of the ratios Hi yields: K ---- 0.16 x 10
6
h°69gli2T224 HOo.59bl.lOBO.13(0
< K<
1)
(19)
1 /.<. 0,9
i, ; i
/
!
/r~
0.8
3 0.7
i i
//.
0.6
0.5
0.4
/"
i / /// 5 ,/ .-/<
~ 13
1
//" /
/ 6 4
//" /~9"--
0.3
7
0.2
~
8
0.1
/ / 0 0
0,2
0.4
0.6 K
Fig. 2. Empirical relation for K
(calibrated)
08
1
H.H.G. Savenije / Journal of Hydrology 148 (1993) 203-218
211
The dispersion at the mouth
For the prediction of the boundary condition for the dispersion D~ ws and hence the calibration parameter a0, an analysis was made of the measurements presented in Table 2. As with the earlier researchers, a relation was sought between a non-dimensional dispersion parameter on the one hand, and Canter Cremers' number N and the Froude number F on the other. An expression for the dimensionless dispersion can be derived from scaling the steady-state one-dimensional salt balance equation: dS
OfS - AD~x = 0
(20)
through the introduction of ~({), the dimensionless salinity as a function of the dimensionless distance {: S ~(~) = ~
(21)
x
= E
(22)
where L is a longitudinal length scale and S is a salinity scale. Consequently, the salinity gradient is defined by: dS dg d~ S dg dx - S d~dx - Ld~
(23)
Substitution of Eqs. (21)--(23) in (20) yields: uf~
Dd~ Ld~-
0
(24)
where uf = Of~A, the velocity of the fresh water discharge. It can be seen from Eq. (24) that the advective transport in the first term is counterbalanced by a dispersive transport with a velocity D/L. Hence the dimension of the dispersion is the product of a longitudinal length scale and a longitudinal velocity. A good choice for the longitudinal length scale is the tidal excursion E0: the horizontal length scale of tidal mixing, while a good choice for the longitudinal velocity is the tidal velocity amplitude v 0. Hence the dimensionless dispersion is defined as D~WS/(voEo). The following empirical relation was obtained between the dimensionless dispersion and estuary numbers: D~Iws voEo -- 220h°a F-°'SN°'5 = 220v/~ boaV ~ R (25)
212
.==o
0
0
H.H.G. Savenije / Journal of Hydrology 148 (1993) 203--218
H.H.G. Saven(je / Journal of Hydrology 148 (1993) 203-218
0
0
0
0
0
0
0
~
0
0
0
0
0
0
0
0
0
213
0
•
~
.
"~,
~ ~ ~ oo o oo o o o o ..=az
H.H.G. Savenije Journal q/Itydrology 14~ (1993) 203 218
214
If the relative density difference Ap/p, present in Fd, is considered constant for different estuaries (Fd = 40F), then it can be observed that the dimensionless dispersion varies with the root of the Estuarine Richardson number NR = N/Fd. The Estuarine Richardson number is generally used as an indicator for the degree of stratification. Equation (25) is a simple power relation between the dimensionless dispersion, the estuary shape number ho/a and the Estuarine Richardson number. It is interesting to note that Fischer (1972) also observed that the dimensionless dispersion is a function of the Estuarine Richardson number. The main difference between Eq. (25) and Fischer's relation is the presence of the estuary shape number (Fischer disregarded estuary shape) and that Fischer used u*hoinstead of v0E0 to make the dispersion dimensionless (u* is the root mean square bottom shear velocity). The result of this empirical analysis is presented in Fig. 3, where the numbers indicate the estuaries. The middle line is the line of perfect agreement and the lines above and below represent deviations by a factor 2. Owing to the large uncertainty in the determination of the fresh water discharge, a considerable scatter in the data points on which the relation of Eq. (25) is based is unavoidable. An error of a factor 2 in the determination of D0 implies an error of a factor 4 in Qr- Of course, this is an unacceptably large error, but then, all points, except the points of the Tha Chin, lie well between these lines. In the Tha Chin, an estuary in the Chao Phya delta in Thailand, an overestimation of the fresh water discharge by a factor 2 is well possible in view of the considerable water consumption in the area which has been disregarded in the determination of Qf. The observed values in the Tha Chin would be quite in agreement with Eq. (25) if a withdrawal of fresh water of about 20 m 3 swere taken into account. Such an abstraction is quite possible in view of the water demands of the area (ILACO, 1987). It is often very difficult to estimate the fresh water discharge in the downstream part of an estuary. Moreover, there can be substantial uncertainty about withdrawals in the area. Therefore, in most estuaries, an error of a factor 2 in the estimation of Qr is quite possible. In light of this, the results obtained seem satisfactory. The only way to improve the reliability of Eq. (25) is by extending the empirical analysis to a larger set of data points. In Fig. 4, the same empirical relation is presented with ~0 as the variable. This relation is obtained after division of Do by -Qr. Hence Eq. (25) can be elaborated into:
ao= 220 h° E 0T °
V--gZrAo
(26)
H.H.G. Savenije / Journal of Hydrology 148 (1993) 203-218
215
2500
1000
400 10
250
Q.
E 0 0
100
40 25
10
i
10
i
i
25
40
t
~
i
100
i
250
I
400
I
I
1000
i
2500
calibrated
Fig. 3. Empirical relation for Do
This is a rather simple formula that relates the second model parameter to quantifiable hydrological, hydraulic and geometrical variables.
Comparison of methods Combination of the empirical Eq. (26) with Eqs. (12) and (13) yields: L Hws = a In \(-220
h°E°V°KaZuo F-°SN °'5 + 1)
(27)
Since on the interval x 6 (0, 1): ln(x + 1) ~0 x, as a first-order approximation, Eq. (27) can be simplified to facilitate comparison with the results of earlier researchers: LHWS ~
-200h°E°V°Kauo F-°SN 0.5 if
LHWS
a
< 1
(28)
Earlier researchers based their theories on estuaries with constant crosssection, or estuaries where a ~ cx~. In such cases, << 1. Hence Eq. (28) may be compared with the work of these researchers. Substitution of Eq.
LHWS/a
H.H.G..Savenije /' Journal o[ Hydrology 148 (1993) 203-218
216
40 25 , , z -,-Z ~ 6 ~ 6 " 10 i
Jl0
_
s
~ ~"
q259/
10 CX
4.0
0
2.5
E 0
,
j
11 2 1. ~ _ 14 j , ~ 12
1.0
0.4 0.25 !
0.4
1.0
2.5
4.0
10.0
25
40
calibrated Fig. 4. Empirical relation for ~0
(2) in Eq. (28) yields: LHWS
L Hws - 2207rh°E° F-°'SN -°5 if - 1.08 K a a
< 1
(29)
The correspondence with the relations obtained by Rigter (1973), Fischer (1974) and Van Os and Abraham (1990) presented in Eqs. (1), (4) and (6) is high, but it is highest with Van der Burgh's relation, Eq. (5). The difference with Van der Burgh's relation, besides the logarithmic function of Eq. (27), lies in the use of Eo/a. E o is explainable since Eq. (27) has been derived for HWS. As a consequence, unlike in Van der Burgh's equation which applies to mean tide, L Hws should increase with an increase in v0 (or E0 for that matter). The inclusion of the convergence length a is the result of considering an exponentially varying cross-section. If the estuary has a more pronounced funnel shape (a small), the salt intrusion length is large; apparently the salt intrusion through mixing is easier in a funnel-shaped estuary than in a prismatic estuary. This corresponds with what one would think intuitively. The major advantage of this method over earlier methods is that it takes account of estuary shape and that it does not limit itself to estuaries of constant cross-section.
217
H.H.G. Saveniie / Journal of Hydrology 148 (1993) 203-218
6310
+
3981
Rigter 11973)
;~
Fischer (1974)
2512
Savenije
1585
Van der Burgh (1972)
:~
1000
line of perfect agreement
~k
631
×
i
~
x
x x
261 x
>,e~xy X"
. ~,
X &
4O
X X
~
x
~k z
~
,,
~
X
± x ~
i L~IXI~'~N
n
x
X
•
x
X
x
. .
25 16
%
10
I
20
I
32
I
I
50
I
I
79
I
I
126
I
200
measured Intrusion length (kin) Fig. 5. Comparison of different methods to compute the intrusion length at HWS
Model results
In Fig. 5, the four methods for the determination of the intrusion length are compared. In the methods of Rigter (1973), Fischer (1974) and Van Os and Abraham (1990) the tidal excursion is added to L Lws to obtain LHWS; in the method of Van der Burgh El2. It can be seen that the methods of Rigter (1973) and Van Os and Abraham (1990) do not differ significantly. The value of K used for this graph is the one obtained from calibration. This is the approach one would follow once K had been obtained from calibration on a salt intrusion measurement. If the predictive model is used for the first time, in a situation where no salt intrusion measurement is available, then the predicted value of K, Eq. (19), should be used. The use of the predicted value of K then results in slightly more scatter. Conclusion
It can be clearly seen from Fig. 5 that the method presented in this paper applies very well to prototype estuaries; much better than methods based on laboratory experiments. The only d a t a points diverging somewhat from the line of perfect agreement are those belonging to the Tha Chin discussed earlier.
218
H.H.G. Saventye / Journal t4/Hydrolocy 148 11993; 203 218
The earlier methods are clearly hampered by the assumption of constant cross-section. Apparently the influence of estuary shape on salt intrusion is large, In fact, one may conclude that since only man-made estuaries with fixed banks (such as the Rotterdam Waterway) have a constant cross-section, the laboratory experiments have little value for real estuaries. Although this paper has concentrated on the salt intrusion length, it has been shown in previous studies (Savenije 1986, 1989, 1992) that the model can be adequately applied to describe the longitudinal salinity variation in estuaries also. The values of K and c~0 derived from Eqs. (19) and (26) on the basis of quantifiable hydraulic, hydrologic and geometric parameters can be substituted in Eqs. (10)-(12) to compute the longitudinal variation of the salinity in a predictive model.
References Fischer. H.B., 1972. Mass transport mechanisms in partially stratified estuaries. J. Fluid Mech., 53:671-687. Fischer, H.B., 1974. Discussion of "Minimum length of salt intrusion in estuaries' by B.P. Rigter, 1973. J. Hydraul. Div. Proc., ASCE, 100:708 712. ILACO, 1987. Agricultural development and crop diversification project covering the Chao Phya west bank area, Thailand, Annex B (Hydrology). Euroconsult, Arnhem, Netherlands. Rigter, B.P., 1973. Minimum length of salt intrusion in estuaries. J. Hydraul. Div. Proc., ASCE, 99:1475 1496. Sanmuganathan, K. and Abernethy, C.L., 1975. A mathematical model to predict long term salinity intrusion in estuaries. Proc. Second World Congress, International Water Resources Association, New Delhi, Vol. 3, pp. 313-324. Sanmuganathan, K. and Abernethy C.L., 1979. Long term prediction of salinity intrusion in branched estuaries. Proc. 18th IAHR Congress. International Association for Hydraulic Research, Cagliari Savenije, H.H.G., 1986. A one-dimensional model tbr salinity intrusion in alluvial estuaries. J. Hydrol., 85: 87--109. Savenije, H.H.G., 1989. Salt intrusion model for high-water slack, low-water slack and mean tide on spreadsheet. J.Hydrol., 107: 9-18. Savenije, H.H.G., 1992. Rapid assessment technique for salt intrusion in alluvial estuaries. 1HE Report Series, Report No. 27, International Institute for Infrastructural, Hydraulic and Environmental Engineering, Delft, Netherlands Savenije, H.H.G., 1993. Composition and driving mechanisms of longitudinal tidal average salinity dispersion in estuaries. J. Hydrol., 144: 127--141. Van der Burgh, P., 1972. Ontwikkeling van een methode voor her voorspellen van zoutverdelingen in estuaria, kanalen en zeeen. Rijkswaterstaat Rapport, pp. 10 72. Van Os, A.G. and Abraham, G., 1990. Density Currents and Salt Intrusion. Lecture Note for the Hydraulic Engineering Course at IHE-Delft, Delft Hydraulics, Delft, Netherlands.