(1)
Giampaolo
Di Silvio
Università di Padova, Dipartimento di Ingegneria Idraulica, Marittima e Geotecnica
via Loredan 20 - I35131 Padova (Italy)
tel.
+39 049 827 5424 - fax +39 049 827 5446 - E-mail: disilvio@idra.unipd.it
Abstract: Although extremely simplified, a linear one-dimensional model of a prismatic reservoir taking into account the non-uniform composition of sediments, gives a much better description of the sedimentation process than a zero-dimensional (i.e. concentrated volume) reservoir based on the conventional ratios between the storage volume and the volumes of the annual runoff and sediment transport. A detailed classification of a variety of reservoirs, ranging form high dams in steep torrents to low weir in plain rivers and conveying coarse and fine particles in different proportions, can be made on the basis of a small number of simple parameters indicated by the model. These parameters are related to the morphology of the dam, the requirements of the volume regulation and the sorting of sediments. The applicability of "sluicing" can also be assessed with this model.
Keywords: sediment, reservoir, dam, sorting, grainsize composition, transport, trapping, flushing, sluicing
For comparing the behaviour of different reservoirs as far as sedimentation is concerned, is useful a classification based on a limited number of overall parameters. A simple but significant classification (see, for example, Zhou-Zhide, 1996; Di Silvio, 1996; Basson and Rooseboom, 1997) is based on the ratios of the reservoir volume VD (m3) to, respectively, the sediment annual transport of the river PR (m3/year, including porosity) and the annual runoff QR (m3/year). In Fig. 1 the position of a number of reservoirs in Italy and in the world is represented on the graph having these ratios as coordinate axis. Note that on the same graph are also drawn the straight lines (PR /QR) = const, namely the iso-lines of the average sediment concentration of the input. This concentration usually varies between 10-2 (largely erodible watersheds) to 10-4 (less erodible watersheds). Both ratios (VD/QR) and (VD/PR) have the unit of a time (years) and represent the time required by the reservoir to be filled up, respectively, by the water input and by the sediment input if the output from the reservoir were zero. It is obvious, however, that the output of water from the reservoir is, on the long term, equal to the input, and even the output of sediments is never equal to zero.
The ratio between the amount of sediment intercepted by the reservoir and PR is called trapping coefficient, Ctrapping. The ratio between the useful volume and the total volume is called utilization coefficient Cutilization. The time required by the reservoir to be filled-up by sediment is therefore:
(1)
The utilization coefficient depends on the volume regulation requirements. The trapping coefficient depends above all on the morphology of the reservoir, on the granulometry of the transported material and on the regulation of the water level. Trapping coefficient ranges from almost one (for large reservoirs with little or no regulation, created by high dams on relatively small rivers) to almost zero (for low barrages provided with ample openings for letting big floods pass through). In general large values of the trapping coefficient is associated to relatively large values of the ratio (VD/QR) and viceversa. Therefore the filling time of a reservoir tends to increase with (VD/PR) and to decrease with (VD/QR).
The relationship of the trapping coefficient with (VD/QR), however, is not unique as Ctrapping depends very much on the possibility of performing the so-called "sluicing". This technique of sedimentation control is not applicable for very large values of the ratio (VD/QR) while is somehow automatically performed when the ratio (VD/QR) is very small (see Fig. 1). By contrast, additional conditions are required by sluicing for intermediate values of (VD/QR), which is the case of most important reservoirs (Brandt, 2000).
Fig. 1 Classification of reservoirs (from Di Silvio, 1996)
Thus, besides the ratios (VD/QR) and (VD/PR), other parameters are significant for a proper classification. In the next sections an attempt will be made to single out a number of overall parameters capable to predict, at least approximately, the filling time of a reservoir and to assess the mitigation measures to be taken for reducing its sedimentation rate. With this purpose a simple mathematical model will be proposed.
Most reservoirs have a basically one-dimensional configuration. This means that their lenght is orders of magnitude larger than their depth and width. The relative longitudinal dimension of a reservoir is a fundamental parameter which should be taken into account for describing the filling process.
Another aspect to be considered is the distinction between the general longitudinal movement of sediments along the reservoir, controlled by the cross-section averaged velocity, and the localized movement near the dam controlled by the flow in the outlets. In fact, the sediments are selectively transported towards the dam through the entire lenght of the reservoir, more or less slowly according to their grain-size distribution, and eventually flushed through the outlets. If the outlets near the dam are closed, the sediments accumulate against the gates which will get more and more covered by the deposited material. For removing the accumulated material the gates should be opened when the water level in the reservoir is high enough to exert a sufficient pressure on the sediments deposited in the outlet and to prevent further clogging of the outlet itself. If the pressure is sufficient, the opening of the gates will immediately remove all the deposited material by creating a zone of concentrated erosion ("flushing cone") all around the outlet.
After the almost instantaneous creation of the flushing cone, however, the sediment discharge through the outlet will suddenly decrease. In order to increase the sediment discharge the water level of the reservoir must be lowered. In other words, once the local equilibrium configuration around the flushing cone has been established, the outlets will let pass any amount of sediment coming from upstream as dictated by the waterflow and water level (i.e. by the cross-section averaged velocity) in the reservoir (Di Silvio, 1990). When the bottom outlets are closed again, the flushing cone will be slowly filled up by the incoming material and the accumulation near the dam will recommence. If the outlets are kept closed, the accumulation zone near the dam will progressively expand.
In principle, flushing operations through the outlets are performed more or less regularly, so that the accumulation zone is more or less extended. For sake of simplicity, here, the hypothesis will be made that all the sediments approaching the outlets will continuously be discharged downstream, so that the extension of the accumulation zone is zero. In other words it will be assumed that the sediment transport through the dam is solely controlled by the waterflow and water level in the reservoir. This hypothesis is acceptable if both water intake and bottom outlet are close to the dam and flushing operations are performed efficiently and rather frequently.
The sedimentation control technique called "sluicing" consists in flushing the sediment through the outlets during the floods characterized by high sediment concentrations. As said before, in order to increase the transport in these conditions, the waterlevel in the reservoir should be kept substantially low. In this case, since a large amount of water must be released before and during the floods, only a relatively small part of the annual runoff can be stored and destined to the utilization. This is what is called in China "impound the clear and discharge the turbid (I&D)", which not only requires large outlet works but inevitably implys a sufficiently small ratio (VD/QR).
The simulation of the sediment transport and the consequent morphological evolution of a reservoir is made by the simplest possible model, still retaining the most significant features of the filling-up process. The model is based on the linearized one-dimensional waterflow equations along a rectangular prismatic channel and on the sediment balance of two distinct grainsize fractions d1 and d2 accounting for the markedly distinct behaviour of fine and coarse material (Di Silvio and Marin, 1996). The dependent variables of the linear system are the non-dimensional deviations (z, h, p and b) of the following quantities (Z, H, P and b1) from the respective constant values of the base-flow (Z, H, P and b1), as reported in Fig. 2:
·bottom elevation, z = (Z - Z)/H;
·water depth, h = (H - H)/H;
·sediment transport, p = (P - P)/P;
·percentage of the finer grainsize (d1), b = (b1 - b1)/b1 in the bottom, being b1 + b2 = 1.
The non-dimensional independent variables are the space x and the time t, respectively scaled with L and (L/U), where L and U are the lenght of the reservoir and the cross-section averaged velocity of the base flow.
Due to the extremely low values of the Froude number in
reservoirs, the so-called "rigid-lid approximation", ¶(H+Z)/
¶x
= 0, is introduced in the waterflow equation, allowing a substantial
simplification of the system (Bagagiolo et al., 1999). The solution of the
simplified partial differential equations consists in the superposition of two
waves propagating downstream along the reservoir with markedly different speeds.
The two celerities depend in principle on the following quantities pertaining to
the base-flow: ratio between the representative grainsizes (d = d1/d2
< 1); average sediment concentration (Y);
ratio between grainsize percentages on the bottom (b1/b2);
and thickness of the bottom active layer scaled with the water depth (D=
d/H).
By choosing d1 and d2 as, respectively, d16 and d84, the value of b1/b2 results to be usually about 1(b1 = b2 = 0.5). In this case, the two non-dimensional wave celerities, lf,s = Cf,s/U, in which U represents the cross-section averaged velocity of the base-flow in the reservoir, are provided by the following expressions:
Eq. (2) is obtained by assuming a monomial transport
formula (like Engelund-Hansen) and a hiding
and exposure coefficient ri equal to one ("intrinsic
mobility" of the grains). A
more complicated expression of ff,s = (lf,s/Y)
is obtained if the transport is multiplied by a hiding-exposure coefficient ri
= (di/Sbidi)s,
which reduces the "intrinsic mobility" of the finer particles and
increases the"intrinsic mobility" of the coarser ones. Fig. 3 reports
the values of (lf,s/Y),
by assuming a hiding exposure
coefficient either with s = 0 (ri = 1, intrinsic mobility) or with s
= 0.8 (more realistic assumption, closer to "equal mobility"). The
relative thickness of the active layer has been reasonably estimated as D
= 0.05.
It appears from the graphs of Fig. 3 that, in any case, the
celerities of the two waves are extremely different and that this difference
increases when the sorting of the sediment composition is very high (d = d16/d84
<< 1). This reproduces pretty well the behaviour of most reservoirs, where
two basic types of deposition can be observed: a first one, mostly composed by
fines, which rapidly reaches the dam and a second one, mostly composed by the
coarser particles, which propagates slowly from the river through the lenght of
the reservoir.
It will be seen later that the
relative importance of the two types of sedimentation also depends on the ratio
d = d16/d84. In terms of bed elevation, most of the
perturbation is associated with the slow wave. By contrast, in terms of sediment
transport it is associated with the fast wave. It has to be noted, however, that
although the fast waves carries finer material than the slow one, it also
conveys coarse sediments depending upon the value of d.
Perturbation to the base flow are determined by
the boundary conditions. By assuming the rigid-lid hypothesis, Z + H = H
+ Y, boundary conditions can be prescribed in terms of non-dimensional water
elevation y(t) = Y/H and/or water discharge q(t) all over the reservoir,
as well as in terms of non-dimensional sediment input p(0,t) and/or sediment
composition b(0,t)
at its upstream end. In any section x, however, the following relations should
be satisfied (Bagagiolo et al., 1999):
h(t) +
z(t) = y(t)
(3)
p(t) =
6q(t) – 6h(t) + b(t)
(1– d)/(1 + d)
(4)
Before applying the simplified linear solution to a real reservoir, however, the geometric and hydrological schematization should be properly discussed. An acceptable geometric schematization of most reservoirs is represented (Fig. 2) by a pyramid characterized by the angles a and b (respectively, longitudinal slope of the dammed river and transversal slope of the valley) and by the height HD of the dam. In relation to the hydrological regime and to the required water utilization, the stored volume in the reservoir will be subject to oscillations around the averaged value V, with a maximum amplitude VW. In Fig. 2 are represented the dimensions corresponding to the volumes VD and V.
With due consideration to the space- and time-scale of the model, the base-flow in the equivalent prismatic rectangular reservoir should conserve, with respect to the real one, the average water velocity in the cross section U = Q/A and the average travel time of the water through the reservoir L/U. For the similitude of sediment transport, moreover, also the velocity in the river should be conserved. From the geometric point of view, it suffices that the equivalent rectangular reservoir and the real one have the same ratio (V/VD) between average and total volume and the same ratio (A/AR) between the average cross section of the reservoir and the average cross section of the river. For a rectangular prism these conditions correspond to prescribe, respectively, the following constants: kD = HD/H = VD/V and kS = HS/H = (H – HR)/H = 1 – (AR/A).
If the ratio kW = HW/HS between the regulation volume and the useful volume is prescribed, it should also be kD = kS(1 + kW). Note that the ratio (HS - HW)/HD = kS(1 + kW)/kD represents the percentage of the total volume of the dam destined to accumulate sediments (dead storage).
From the hydrological point of view, the reservoir is characterized by the following principal quantities: annual runoff QR (m3/year) corresponding to the average flow discharge Q = QR/365×86400 (m3/s); annual sediment input from the river PR (m3/year, including porosity); peak flow Qo, number N and duration To of the major floods in the year (possibly requiring sluicing, if feasible) which convey altogether most of the annual sediment transport PR. This implys prescribing the constants: kQ = Qo/Q and kT = To/(L/U).
If no sediment management by sluicing is applied, the sedimentation of the reservoir takes place in following way. A non-dimensional perturbation p(0,0) = pR is created, at the upstream boundary of the reservoir immediately after the construction of the dam, by the surplus of sediment input from the river PR with respect to the transport capacity P of the base-flow in the reservoir void of sediment. Let us suppose first that the water surface elevation of the reservoir is kept constant (y(t) = 0) and that the water and sediment input is also stationary and equal, respectively, to the annual average discharge of the base-flow (q = 0) and to the average total transport of the river (p = pR and b = 0). In this case, the water-depth perturbation h(0,0) = hR corresponding to pR is the one which restores an equilibrium condition in the reservoir, namely the same water velocity (i.e. the same water depth HR) as in the river. By putting
hR = (HR – H)/H = –HS/H = –kS (5)
from eqs. (3) and (4) one obtains
zR = –hR = kS and pR = –6hR = 6kS (6), (7)
where: kS = (HS/H) = 1 – (HR/H) = (HD/H) – HW/H) (8)
is a fundamental morphological parameter of the reservoir under consideration.
The dimensional bottom perturbation ZR can be split int wo parts, ZRf and ZRs, propagating in the downstream direction respectively with the fast and with the slow dimensional celerity Cf and Cs. The bottom perturbations must satisfy the compatibility and continuity conditions:
ZR = ZRf + ZRs and PR = CfZRf + CsZRs (9), (10)
After substitution of (6) and (7), eqs. (9) and (10) provide the non-dimensional perturbations:
(11),(12)
in which the relative celerities ff and fs are given by eqs. (2) or by the graphs in Fig. 3.
Following the construction of the dam, also the grainsize composition of the original bottom (reservoir void of sediments) will change. The perturbations bf,s carried by the fast and the slow waves can be computed from eq. (4) in which the dimensional transport P = pP is put respectively: Pf = PR – CsZs and Ps = PR. This provides:
(13),(14)
where zf and zs are given by eqs. (11) and (12). The travel time Tf,s = L/Cf,s of the two waves along the reservoir, expressed in years, results from the definition of Cf,s = lf,sU = ff,sYU, where U = P/YA = PRL/(1 + pR) YV. One finds:
(15)
The quantity Tf,s is the time respectively spent by the finer and the coarser sedimentation to reach the dam for the prescribed stationary boundary conditions, i.e., in particular, for a stationary water elevation in the reservoir. Under these conditions the travel time of the slower perturbation, Ts, represents the filling time of the useful volume (BLHs), in the equivalent prismatic rectangular reservoir of width B.
In reality, when the
reservoir is subject to regulation, the water elevation is not stationary but
oscillates around its average value. Oscillations also occur for the water and
sediment input from the river, so that other negative and positive perturbations
propagate along the reservoir superposed to zf and zs
given by eqs. (11) and (12). In any case, if one assumes a linear response for
all the relevant quantities, the filling time of the useful volume is still
given by Ts in eq. (15). This can also be written in terms of the
entire volume of the dam VD:
(16)
where HS = (kS/kD)/HD and kS/kD = Cutilization in eq: (1). By comparing (eq. (16) to eq. (1) one finds:
(17)
Eqs. (16) and (17) show the dependence of the filling time and the trapping coefficient on the specific characteristics of the reservoir: dam morphology (kS = HS/H); regulation requirements (kw = Hw/H); and sediment grainsize distribution (fs, basically function of the non-uniformity ratio).
Eq. (17) is valid when the waterlevel oscillations are exclusively dictated by regulation requirements of the stored volumes, without any correlation with the input of sediment entering the reservoir. The trapping coefficient can be strongly reduced if, by contrast, the waterlevel is purposedly lowered in concomitance with the major floods which convey large amount of sediments. In this way the output of sediments from the reservoir is augmented and the sedimentation is reduced. As already mentioned, however, this technique is applicable only when the runoff is abundant and when the stored water can be released just before the arrival of the flood. The linear model can be applied for evaluating the benefits of sluicing compared to the disadvantages of forgoing a large part of the incoming runoff and available storage. Due to the limitation of the present text lenght, this analysis is not reported here but will published elsewhere. It sufficies to say that, to this purpose, other non-dimensional parameters of the reservoir (like the relative peak flow and duration of the major floods during the year), should be compared to the regulation requirements of the reservoir.
A one-dimensional linear model, accounting for the non-uniform grainsize distribution of the transported material, indicates that the sedimentation of a reservoir is produced by successive couples of waves propagating with two extremely different celerities. The faster wave mainly conveys fine material by relatively thin layers. The slower wave, by contrast, is usually larger in terms of amplitude and conveys coarser particles. The relative size and speed of the two waves control the filling-up modalities of the reservoir and, in particular, the trapping coefficient. The one-dimensional linear model indicates that besides the basic ratios (VD/QR) and (VD/PR) between the reservoir capacity and the annual volumes of, respectively, runoff and sediment transport, other non-dimensional parameters should be properly introduced for a realistic description of the sedimentation process.
These parameters are simple functions of the reservoir morphology (kS = HS/H), volume regulation requirements (kW = HW/H) and granisize non-uniformity (d16/d84). Other simple parameters related to the hydrological regime of the river should be combined with the parameters mentioned above for assessing the possibility of reducing sedimentation by the technique of "sluicing" (the so called I&D, "impound the clear and discharge the turbid").
References
[1] BAGAGIOLO, F., DI SILVIO, G. , MARION, A., River profile evolution with graded sediments: a linear analysis. Atti del XXVIII Congresso IAHR, memoria in CD-ROM, Graz (Austria), 1999.
[2] BASSON, G.R. and ROOSEBOOM, A., Dealing with Reservoir Sedimentation. Water Resources Commission Report No. TT 91/97, Pretoria, xxxiii+395 pp.
[3] BRANDT, S.A. A review of reservoir desiltation. International Journal of Sediment Research, Vol. 15, No. 3, 2000, pp. 321-342.
[4] DI SILVIO, G., Modelling desiltation of reservoirs by bottom-outlet flushing. NATO Workshop on Movable Bed Physical Models, De Voorst (The Netherlands), August 18-21, 1987. Also in NATO Asi-Series C n. 313, Kluwer Academic Publisher.
[5] DI SILVIO, G. and MARIN, A.: Analytical approach to river morphodynamics: effects of space-and time-irregularities and grain-size non-uniformity. Commission of the European Communities, Directorate General XII for Science, Research and Development, Research and Technical Development Programme in the Field of Environment, FRIMAR Project, Technical Report n. 2, 1996, pp. 48.
[6] ZHOU ZHIDE: Overview of preventive remedial measures. Proceed. Int. Conf. on Reservoir Sedimentation, Vol. 2, Colorado State University, Ft. Collins, Sept. 9-13, 1996, p. 904.