MATHEMATICAL APPROACH TO RESERVOIR SEDIMENTATION

Abstract

Progressive loSs in the capacity of reservoirs built on alluvial streams due to sedimentation is very important from various considerations. Sedimentation in reservoirs is undesirable because a huge amount is spent on the construction of a dam, hoping that it will serve the community over a long period; but progressive reduction in capacity of reservoir reduces the water that can be stored for irrigation, power generation etc., and also reduces the life of reservoir. It also creates several problems in the upstream region from the hydraulic point of view, such as raising of water level in the reach as compared to that prior to construction of dam. There fore, knowing the manner in which the sediment is deposited on the upstream of dam and the way it changes with time will help the design engineers to estimate correctly the reservoir space requires to fulfil the purposes of building a dam. This will also help to assess accurately the available storage at any time and to predict the performance of the reservoir in future. In this thesis a numerical algorithms has been developed for nonprismptic channel which takes into account the variation li in width of stream channel with respect to distance and dis charge with respect to time' This scheme will enable the field engineers to estimate the successive sediment profiles in up stream of dams'_ The method developed is briefly described belovf BASIC EQUATIONS (a) Irregularity of Cross-Section River cross-sections are invariably irregular and the channel is non-prismatic < Therefore, it is desirable to express the cross-sections in term of Fourier Series, i'e A(y) =1| aQ+E°° (an cos nwy+ bn sin nwy) (l) Where A(y) is the water area with depth y, and e^i a^ \ are the Fourier coefficients given by YL xo "~YYTL"" i A(y) dy (2) o YL an 1 " YL J A(y) cos nwy dy YL (3) bn . __1, • j A(y) sin nwy dy (4) i.X YL i in which w •• ic/YL, YL being the maximum po ssible depth at any sec tion # The Fourier coefficients were <obtained by numerical integration using Weddle' s rule in which the length YL is divided in intervals equal to some multiple of six'. Obviously lii the coefficients are different for different sections along the river reach' Equations similar to _tf.(l) can be written for the wetted perimeter P(y) and the channel width T(yV Prelimi nary analysis indicated that for irregular sections, n = Nil/2 where Nil is the number of intervals in integration by Weddle* h oo rule gives best results' The summation I can be terminated at n=l n - Nil/2 without loss of accuracy, (b) Equation for Water Surface Profile The Water surface profile equation can simply be written by applying Bernoulli's equation between any two sections (say 1 and 2) of the stream channel' This is as follows : 2g A2 ^S Ai A2 1 where ? 2 6^nc. (6) (A1+A2ff VR2? y , y2 _ two successive water depths, S - bed slope, ax = length of reach, 0. _ discharge, g = acceleration due to gravity, A, kp » area of cross-section at the two ends of the reach, K » energy loss coefficient, to account for energy loss "due to expansion or contraction. The standard step method was used to solve the above equation numerically to get the values of y at various sections IV (c) Sediment Transport Law s There are many sediment transport laws available in literature' However, these equations are developed for speci fic morphological conditions'. Therefore an attempt was made to develop an equation based on Indian reservoir data"*. This has been done through analysis of yearly sediment yield data of 20 Indian reservoirs'. The equation obtained is as follows i QT =00001477 ^s^p^Fs1!35 (?) Here 6- is the total sediment load in absolute volume in m3/s> and P , P « are fractions of clay and silt in deposited sediment' It may be mentioned that Sf is used in place of S0 when the flow is non-uniform', (d> Formulation for Deposition Depth Denoting the elevation of stream bed z, upwards from the original bed surface, the following sediment continuity equation in respect of sediment transport in a natural channel can be written : 3t + B "IS - ° Where Bis the bottom width of the channel at any time t' Anumerical algorithm based on finite difference scheme has been developed to solve this non-linear equation; This is as follows ; ^ -t(V--> -**•*>> (9) ax (Bi+ Bi+1)/2 Since Az value computed is in absolute depth of deposition, the apparent deposition depth will be az # -*-— where to is the initial unit weight of deposited material'. It has been observed that, the sediment profile has a major peak followed by minor peaks" These undulations in sediment profile would create instability in almost all numerical solution'. To avoid this problem the constraint has been put in selection of time interval for the computational purpose in all numeri cal solution' To avoid this problem the constraint has been put in selection of time interval for the computational purpose in all numerical schemes". Small time intervals make the numerical solution scheme uneconomical for estimation of sediment deposition hi reservoir for long duration'. In this study this problem has been tackled by introducing a smootiiening process using Fourier Sine Series approach. The elemen tary deposition at time (t + At) is first converted into apparent depth and then smoothened and added to the previous z valves" The profile obtained at the end of each year is modified taking into account the effect of consolidation using Miller's equation (52)'. The following equation is used for this purpose hUi) •Z(i^-D *"f^f ^apparent <~> rav(j) Here i and j refers to cross-section index and time respectively y _, is average unit weight of deposited SLl material which is estimated at the end of each year of deposi tion''. SCHEME OF COMPUTATIONS The step-wise computation recommended, using earlier mentioned equations is as follows j (i) Using Eqs';(2) to (4) generate the Fourier Series co efficients for area, wetted perimeter and channel width (ii) Using Fourier coefficients and Eqs'. (5), (6), and Manning's equation, the water surface profile upstream of fhe dam is computed for the case when dam is not present". (iii) The water surface profile is computed using Eqs' (5) and (6) with the increased depth obtained after the construction of the dam'. Comparison of the two water surface profiles enables determination of the distance unto which the backwater extends^ (iv) By comparison of the shear stress yf R Sf with the critical shear stress, the distance from the dam of the section downstream of which sediment is not in motion is determined". Calculation of rate of deposi tion is carried out only upstream of this section. ( v) Using the 0'»5 year time interval as At, the new bed elevation at different sections is obtained using s vii Eqs" (7), (9) and (10) alongwith the equations for estimation of channel widths from Fourier coefficients (vi) The deposited bed is smoothened using Fourier series appro aoh'. (vii) The new bed slope is computed with respect to new bed elevation' (viii) Steps (i) to (vii) are repeated for the next time interval and the successive bed profile is obtained' The scheme has been tested for hypothetical data with irregular, non-primatic ch.annel using constant and varying discharge' The 20 years sediment profiles has been obtained for which the scheme was found to be fully stable'. Similarly the scheme is used for Go'vindsagar reservoir data'