PUBLS. INST. GEOPHYS. POL. ACA. SC., E-0 (406), 008 Sediment Transport in the Middle Odra River. Verification of Ackers-White s Formulae with Reference to Big Flows Zygmunt MEYER and Adam KRUPIŃSKI Technical University of Szczecin, epartment of Geotechnical Engineering Al. Piastów 50, 7-30 Szczecin, Poland e-mails: meyer@ps.pl, krupina@ps.pl Abstract The paper presents a method of sediment calculation for a compound river cross-section. An essential point for each calculation of sediment is to keep it in relation to the water flow condition. In the paper, a method is shown in which the verification of sediment calculation can be based upon well-defined Chezy constant.. Introduction Sediment transport in rivers has been a research subject for many years. In spite of the fact that the literature concerning this problem is very rich, mechanism of this phenomenon is still subject to study. In the literature there are many formulae which make the sediment transport rate dependent on both the composition of transported material and elements of water motion. These are mainly empirical formulae to be verified by experimental tests and the range of their application is restricted. Pluta (003) has presented a comparison of the results of calculations by different methods for Odra River. Ackers-White s formulae (Ackers and White 975) have found a wide application among many formulae which allow to calculate sediment transport rate in the rivers with movable alluvial bottom. This results from the fact that they include elements of water motion, which are well described in the literature, as, e.g., the flow pattern. This allows to introduce an additional element (Coufal 993) to the sediment transport rate, such as wind action on the water surface or influence of river junction upon the sediment sorting (Krupiński 00). At the epartment of Geotechnical Engineering of the Technical University in Szczecin, many tests have been made to analyse application possibilities of Ackers- White s formulae for the Lower and Middle Odra River (Meyer et al. 998, 997, Meyer and Coufal 007).
8 The tests results confirm the possibility of this method s application for calculating the sediment transport rate, however the accuracy of these calculations varies. Analysis of the Ackers-White s procedure allows us to find which calculation elements have the biggest impact on the results distribution. The calculations show that the parameter which defines the beginning of grain movement in the formula of sediment transport rate has a very big influence on the calculation results. The estimation of this parameter s correctness in the Ackers-White s method is carried out in the present paper. Calculation of sediment transport rate in the river during flood, when inundation areas are filled by flowing water, is still a problem. These issues make the subject of the present elaboration.. Phenomenon analysis. The Ackers-White s method The Ackers-White s method (Ackers and White 975) assumes that the content of sediment which was taken from the bottom is known as the sieve curve { pi; i}, and also the water depth H and river slope I are known. On such a basis it is possible to calculate the average velocity Q q υ o = = () B H H and the shear velocity u* = ghi. () The dimensionless diameter is calculated for each fraction i () i gr gs ( ) = i ν /3, (3) where ν is the kinematic coefficient of water viscosity, while S is the density ratio (sediment grains material to water). The following parameters are calculated on the grounds of gr n= 0.56 loggr 0.3 A = + 0,4 gr 9.66 m = +,34 gr = logc.86log gr (log gr ) 3.53 Further, the following parameters are calculated: grain mobility F gr (4)
9 η υo u αh Fgr = 3 log g (S-) αh υ, (5) 3 log o function of sediment transport G gr according to the formula m Fgr Ggr = C A, (6) value of dimensionless parameter X of sediment movement: X n S υo = Ggr H u, (7) and sediment transport rate ω: ω = ρ gq X. (8) Calculation of sediment transport rate, for which bottom samples are taken and a sieve curve { i, p i} is prepared, is carried out separately for each fraction, and a weighted average is calculated afterwards. It is necessary to underline that this method takes into consideration both the bed load and the suspended sediment. We get N ω = pi w ( i). (9) Ackers and White (975) emphasise clearly that this method should not be applied to the cohesive sediment (with small diameter), such as dusts and loams, because the calculations result in unrealistic, large quantities of sediment, and that is why the sieving curve should be cut off from the lower side (Pluta 003), limiting it to sands and gravels.. Verifying procedure by the Ackers-White s method in case of big flows It is assumed that during field measurement the samples were taken from three different places in the river cross-section: from the river midstream and from the flood areas. The obtained hydraulic and geometrical values are shown in Fig.. On the grounds of Fig. the basic relationships can be written as: and where Q= H B υ + H B υ + H B υ (0) o o 3 3 o3 ω = ω + ω + ω3, () ( i ) () () pi ω, () ω =
0 ( i ) () () pi ω, (3) ω = 3 ( i ) (3) (3) pi ω. (4) ω = Fig.. River cross-section and sampling sites. Moreover, the sediment continuity equation is fulfilled in the range of the successive fractions in the river. We have ω( ) p = ω ( ) p + ω ( ) p + ω ( ) p. (5) (0) () () 3 i i i i i i 3 i i Relationship (5) allows to obtain the so-called resultant sieving curve in the crosssection where the samples were taken from three different places. It is p (0) i = 3 j= ω ( ) p ω ( j) j i i (0) 0( i ). (6) As far as the sieve curve is concerned it is known that the total sum of the successive fractions content must make 00%. So we have: N (0) pi =. (7) It is an additional equation, which enables calculating one additional unknown value. The shear velocity to average velocity ratio is presumed as both the additional unknown value and the parameter for optimization u = ξ. (8) υ o
Then the formula of grain mobility (5) is obtained in the following form F gr n υo αh = Fgr ( ξ) = ξ 3 log g (S-) αh. (9) 3 log Afterwards, the parameter ξ appears in other terms of this procedure and we get S Ggr = f( ξ); X = X( ξ) = Ggr ( ξ) H ξ and ω = ωξ ( ) (0) ω = ωξ ( ) and ω( ξ); ω( ξ). 3 The optimisation still requires ξ to be made dependent on the velocity. On the grounds of the Chezy formula it is possible to write down (for I = I 3 ) and /3 3 n3 H υ30 = υ0 H n /3 n n () H υ0 = υ0 H. () n In order to simplify calculations, n = const is assumed in the first approximation. After substituting it to formula (0), we get υ = 0 /3 /3 H H3 BH BH B3H3 H + + H Q The following relationships result from the shear-stress velocity formula and u u / H = u* H / H3 3 = u* H. (3) (4). (5) If the parameter ξ from formula (8) is substituted to the above equations, the searched shear-stress velocity to average velocity ratio will be obtained for each of three parts of riverbed cross-section. We have u* ξ υ = (6) 0
and u υ H = ξ H * 0 /6, (7) u3* H = ξ υ 30 H. (8) 3 Finally, we obtain relationships that each component of the sediment transport rate, ω ; ω ; ω 3 = f(ξ), and the entire sediment transport rate ω = f(ξ) are also functions of ξ..3 Calculation results of sediment transport rate s parameters On the grounds of the presented procedure it is possible (assuming the value ξ in the beginning) to calculate a joined sediment transport rate for three parts of the river bed cross-section, and then the principle of one (Eq. 7) is checked. There is one such a value ξ, that verifies this formula. It is possible to check how the joint sediment transport rate changes downstream, because the sediment samples were taken in six different cross-sections on the ca. 0-km-long section. For the steady motion there is another equation verifying the calculation d ω = 0. (9) dx α H This equation was used to optimize the second parameter. There is a term in formula (0) for F gr ; the parameter α =.3 is assumed. Verification was made to check which parameter α makes the sum of standard deviation of sediment transport rate the smallest. The calculation results are presented in Table. Table /6 Optimisation of parameter α calculations for the assumed values α : km 5.0.3 0.0 7.0 5.0 4.0 3.0.0.0 0.5 ξ ω 0 ξ ω 0 ξ ω 0 ξ ω 0 ξ ω 0 ξ ω 0 ξ ω 0 ξ ω 0 ξ ω ξ ω 594 0.0 0.00 0.0 0.00 0.0 0.00 0.0 0.00 0.05 0.399 0.05 0.394 0.030.907 0.07.67 0.03 5.50 0.07 5.76 598 0.033.70 0.03.345 0.09.08 0.07.568 0.0 0.630 0.0 0.53 0.06.549 0.08 4.780 0.03 4.376 0.09 4.999 604 0.075.84 0.073.57 0.07.3 0.068.087 0.065.047 0.063.04 0.06 0.995 0.057 0.958 0.05 0.903 0.087 8.649 608 0.04 0.6 0.040 0.47 0.039 0.33 0.038 0. 0.050.306 0.035 0.08 0.033 0.069 0.03 0.054 0.038 0.778 0.05 0.05 609 0.039 0.6 0.038 0.588 0.037 0.567 0.036 0.53 0.034 0.50 0.033 0.486 0.03 0.47 0.03 0.458 0.038 3. 0.07 74.8 64 0.054 0.68 0.05 0.54 0.05 0.40 0.049 0.8 0.047 0.099 0.045 0.087 0.043 0.073 0.04 0.055 0.037 0.03 0.046 0.80 ω śr 0.9 0.7 0.7 0.7 0.6 0.5..6 3.0 8.7 δ 4.7 4.0 3..0.0 0.64 5.4 6.5 6. 44.9 δ / ω śr 5.4 7.4 6.9 4.7.8.5 3.8 6.7 3.0.0 In column one km denotes location along the river. As a result, α opt = 4.3 is obtained and this value is different from α =.3. The parameter ξ was the second element that was verified. On the grounds of the Chezy equation it is known that
3 u g ξ = =. (30) υo C Hence g C =. (3) ξ This means that in each cross section it is possible to control the constant value C for the Chezy equation. This constant value is recognized for the Middle and Lower Odra River. It turns out that if the value A is assumed according to formula (4), the obtained values of C will not be realistic. Therefore, different values of A are presumed for calculations according to the relationship A The calculation results are shown in Table. = A ε. (3) Table Optimisation of parameter C for α=.3.a.a.3a.4a.5a.6a.7a km ω 0 ξ ω ξ ω ξ ω ξ ω ξ ω 3 ξ ω 3 ξ 594 0.488 0.035 89 0.564 0.043 73 0.65 0.05 6 0.76 0.06 5 0.86 0.07 44 0.93 0.084 37.007 0.097 3 598 3.06 0.04 76 5.337 0.058 54 8.4 0.079 40.44 0.05 30 7.99 0.38 3.7 0.70 8 5. 0. 5 604. 0.09 34.30 0.3 8.39 0.36 3.5 0.6 9.66 0.89 7.75 0.0 4.85 0.54 608 0. 0.05 6 0.9 0.06 5 0.37 0.075 4 0.45 0.089 35 0.55 0.05 30 0.6 0. 6 0.68 0.40 609 0.67 0.048 66 0.78 0.058 54 0.90 0.070 45.05 0.083 38.8 0.097 3.8 0.3 8.39 0.30 4 64 0. 0.066 48 0.9 0.08 39 0.37 0.097 3 0.45 0.6 7 0.55 0.36 3 0.6 0.58 0 0.68 0.83 7 average 0.978 6.47.46 49.73.06 40.4.778 33.37 3.797 8.0 0.45 3.86 0.69 0.5 δ 5.9 99 9. 70 50.0 796.7 576 4.6 445 0.0 337 0.0 66 In column one, km denotes location along the river. It appears that in order to obtain the constant value C in the Chezy formula as it results from the flow conditions in the successive six cross-sections, parameter A must change. In every cross-section these changes were made depending on characteristic features of the sieve curve for the current material, assuming the statistic parameters: o = ( pi i) (33) and average values average standard deviation and skewness of sieve curve p ( ) i i o, (34) δ = p ( ) 3 i i o. (35) = ϑ 3
4 The empirical formula to calculate the value of ε was obtained on this way in the following form ε = 0.3 o.875 δ +.05 ϑ. (36) The optimisation results for formula (36) are presented in Table 3. Table 3 Optimisation of parameter ε for α=,3 flow parameters sieve curve parameter km ω 0 H 0 ξ C Chezy 0 δ ϑ calculated n s 594 0.996.70 0.087 36.00..80.50 0.0304 598 9.7.730 0.087 35.84.3.55.4 0.0306 604.97.00 0.086 36.4 0.98 0.98.5 0.0308 608 0.435.90 0.086 36.40 0.8.9.89 0.0306 609.083.870 0.087 36.00 0.7 0.93.47 0.0308 64 0.3.050 0.087 36.0 0.64 0.75.3 0.033 In column one, km denotes location along the river. While making Tables and 3 it was assumed that α =.3 as Ackers suggests. It results not only from the fact that the formulae were verified on the grounds of the field measurements for these relationships by many authors, but also that there is preserved a possibility to compare presented calculation results with other parameters. Moreover, the calculations show that such a variable value of coefficient α has a very little influence upon the sediment transport rate w that is calculated for ξ opt. 3. Conclusions The paper presents the verification of Ackers-White s formulae for calculation of sediment transport rate under the conditions of the Middle Odra River and big flows, when the inundation areas are flooded. The analyses were carried out with the help of optimisation of parameter ξ, which describes the ratio of the obtained velocity to the average velocity in the selected river cross-section. The verification consisted in comparing the constant value C in the Chezy formula resulting from the river water flow to the constant value that was calculated on the grounds of the optimal parameter ξ from the Ackers-White s method. The verification indicates that the formula needs to be replaced by the constant value A, which defines the beginning of grain motion in the river. This constant value was made dependent on the parameters of sieving curve, reaching a good accuracy. Meyer and Coufal (007) did similarly determining a representative grain diameter in dependence on the statistic parameters of sieve curve.
5 The verification allowed also to optimize the parameter α, which appears informula (0) for the grain mobility. Using the constant stream of the sediment transport rate along the river, it is possible to define that α opt = 4.6 and this is a different value from what was assumed both by Ackers and in the literature, i.e. α =.3. ue to the fact that this change does not affect the verification results, concerning the constant value C in the Chezy formula it is suggested to preserve α =.3 in calculations. The program of further researches includes: field measurements in other places of the Middle and Lower Odra River in order to gain materials for verification, especially for big flows; verification of Ackers-White s method for different sections and for changeable flows in the Odra River; searching for more universal relationships defining parameters A, α and bottom roughness for longer river sections and big flows. 4. Notations A parameter of grain movement in the Ackers-White s method, B riverbed width, C constant value in the Chezy formula, o average grain diameter of bottom sediment, gr dimensionless parameter defining sediment size, F gr mobility parameter, g acceleration due to gravity, G gr dimensionless parameter of sediment transport rate, H river depth, I river slope, n roughness coefficient by Manning, {p i, i } set of values of sieve curve, u * shear velocity, υ o average flow velocities, ω sediment stream, X dimensionless sediment transport rate, x horizontal co-ordinate of the assumed system of coordinates, α constant value in the average velocity formulae, ε function of change of parameter A, ξ shear velocity ratio, to average velocity. Note: lower indices denote sediment fractions; upper indices (Eqs. to 6) denote the number of riverbed cross-section part.
6 References Ackers, P., and W.R. White (975), Sediment transport, new approach and analysis, J. Hydraul. iv. ASCE, 99, HY. Meyer, Z., R. Coufal, and R. Kotiasz (998), Influence of the sediment transport on the steady water flow in the river, 3 rd International Conference Hydro-Science and Engineering, ICH' 98 Cottbus. Sawicki, I.M. (998), Flows with Free Water Surface, Wydawnictwo Naukowe PWN, Warszawa. Pluta, M. (003), Applicability of Sediment Transport Formulae for Odra River. XI seminarium naukowe z cyklu Regionalne Problemy Ochrony Środowiska w Ujściu Odry. Szczecin-Ystad-Świnoujście. Meyer, Z., R. Coufal, and A. Roszak (997), ivision of Sediment Stream in River Bifurcation during Flood at Widuchowa Weir in lower Odra River. XVII Ogólnopolska Szkoła Hydrauliki, wrzesień 997. Coufal, R. (993), Wind Shear Stress Affected Sediment. Proceedings of the Advances of Hydro-Science and Engineering. Part B, pp.749-754, S. Wang (ed.), The University of Mississippi. Krupiński, A. (00), Sediment Sortation Influence on Bottom Changes in River Junction at Kostrzyn, Praca doktorska. Szczecin 00. Meyer, Z., and R. Coufal (007), etermination of Sediment Transport Characteristic iameter for the Odra River Section. XXVII International School of Hydraulics Hydraulic and Environmental Hydraulics, Hucisko 8- Sept. 007. Accepted ecember 0, 008