Two-dimensional numerical algorithm for water qualitymodeling in the topographically complicated riv

Joumnal of Environmental Sciences Vol. 12 , No.4,pp 469- -473,2000CN1-2629/XArticle ID: 1001-0742(2000)04-0469-05 CLC number; X131 Docment code; ATwo-dimensionalnumericalalgorithmforwater qualitymodeling in the topographically complicated riverZENG Guang ming' , JIANG Yi-min' , GUO Huai-cheng2 , Gordon Guo He Huang31. Environmental Protection Institute, Hunan University, Hunan 410082, China; 2. Environmental Science Center, PekingUniersity. Bejing 100871, China; 3. Environmental Systems Engineering. Universty of Regina, Sask S4S 0A2, Canada)Abstract:In this paper, a two-dimensional numerical calculation algorithm for the water quality rmodeling in the Hengyang Gity sectionof the Xiangjiang River is researched considering the efete of the Deyuandu navigational key project. The research river is windingof applying topographic map of the river course and the finite elenent method. The calculation result for the water quality modelingincludes the concentrtion felds for various pollutants. The numerical calculation algorithm for the water quality modeling stl up inthis paper can be applied to shallww river with similar topographically complicated river c∞urse.Key words: two dimensional river; water quality; numerical calculation algorithm1 IntroductionAccording to the Hunan component of the proposed inland waterways multipurpose project ,the Dayuandu navigational key project, located 62 km downstream from the Hengyang City, willbe built with the help of the world bank loan. After the project being constructed, the XiangjiangRiver section within the Hengyang City will be changed into a reservoir resulted from the projectand the water quality will be surely influenced as the river hydrological condition upstream from theproject will be changed. To make it clear how much negative influence the project will have on thewater quality in the Xiangjiang River section within the Hengyang City, the research river section(about 13 km) from the Tongqiaogang upstream from the Hengyang City to the Chengbei drinkingwater plant downstream from the Hengyang City was selected with the consideration of thepollutants' discharge distribution and the water quality management of the Hengyang City. Thelocation of the research river section is shown in Fig.1.The research river section is topographically complicated as it is winding, relatively shallowwith two branches resulted from an isle and its cross section changes irregularly along the river. Forthese reasons, the traditional one- dimensional water quality model and the traditional two-dimensional water quality model fit for straight river with uniform rectangle cross section can not beused for this research, and the two dimensional numerical calculation algorithm based on the finiteelement method is proposed. To finish the water quality modeling, totally about 2921 unit cellshave been divided in the researched 13 km river section and a lot of application problems have beenmet and solved. These problems include how to build the modeling algorithm, how to decide thewidth and length of the modeled unit cell， how to estimate the hydraulic and hydrologicalparameters required in the model and so on. As the water quality modeling problem based on thefinite element method for the topographically complected river is always difficult and not well solved(Cheng，1996; Heidtke， 1986; White， 1977 )，the two-dimensional numerical calculationalgorithm proposed in this paper and some details中国煤化工deling are quitepossible to be used as a reference for future waterMHC N MH Grparaphiallcomplicated river.Foundation item; The National Natural Science Foundation of China (No. 49201015) and the Science Foundation of EducationMinistry of Chine for Outstanding Young Teachers470ZENG Guang-ming et al.Vor. 122Water qualityFlow zoneRiver crOSS soctionFlow linemodelThe modelbuildingYapproech for the two-Deyuandu projoctxdimensionalnumerial气十Xiangiang Rivercalculation algorithm mainlyincludes two steps. The firstZengshui Riverstep is to divide the river flowFig.2 The rectangular curved coordinateinto m flow zones followingsystem for the river weter qualitythe flow direction as well asmodelingintoriversections入-' Rexcceractonon perpendicularto the flow direction and then a series of(sce Fi23)Leishui Rivermathematical models for all the unit cells to model the waterquality based on the mass conservation law are established.To make the research easier, the flow discharge in each flowHengyang Ciyzone has to be kept constant. In this way there will be onlypollutant exchange resulting from the dispersion and not any flowFig.1 The loeation of the research riverdischarge exchange existing between the neighboring filow zones.Correspondingly, the width for each flow zone must change alongsectionthe river as the shape of the river cross section changes along theriver (Fig.2).Take one unit cell (given ij unit cell), assuming pollutant ( given BOC5) concentration isuniform in each unit cell, the pure added amount for each unit cell resulted from the river waterflow is:q;(L-1.s- Lq),(1)the pure added amount for BOD3 in each unit cell resulted from the longitudinal dispersion is:Di-.,.6(L-.- Ly) - D.i1.(Lj - Li+1.;)，(2)the pure added amount for BODs in each unit cell resulted from the river cross dispersion is:Di,j-l).6(L;,s-1L:)- D,(i,+1)(Lj- L,j+1).As the decreased BODs amount in each ij unit cell is V;KawLj and the input BODs amountfrom the outside of each ij unit cell is Wj, therefore, the following equation can be obtainedacording to the mass conservation principle:VydL。/di =q(L-1;- Lg)+ Di-,(L1,- Lg)- Di,i+1.,)(L,- ..1,) .+ Di.1-n,(,.,1- L;) - Di,r.,+t)(L, - .,.1)- V;KuJLe+ W略,(4)in whichD(i-1,i),j = D(i-1,j).6i ●A(i1,i),g/5i-.j)j，(5)Dj.<(i+1,j) = Dij(i+1.J) ●Ang,(i+1,ji)/xi,(i+1,i)，(6)D'(i,j-1). = Di,j-1),. A-//1i.(7)Di(5i,j+1) = Dij(i,+1中国煤化工(8)q,is the flow in the j flow zone, m'/s; L:CNMHGU unit ell, mg/;V; the volume for the ij unit cell, m3 ; Kai the'DuLs deoxygenauing Tale wueificient in the ij unitcell, 1/d; Dy.从the dispersion cofficient between the ij unit cell and the h unit cell, m2/s;Aj,h the contact surface area between the ij unit cell and the h unit cell, m'; xj.从the averagelongitudinal distance for the two neighboring cells i and kh, m; yi,a the average crosswise万鬥数狱for two neighboring clls j and外h，m.No.4Two-dimensional numerical algorithm for water quality modeling in the topograplail.....When the state is steady, dL;/dt =0, the Equation (4) can be changed into:W沿=- q,(L;-1,- Lg)- D(i-1.j).(L;-1.j- L,) + D.(+1.)(Lj - L,+1.) .D(,j-1).,(Lj-1 - Lj) + Dj.(i.j+1)(L;,;- L,j+1) + V;KajLj.(9)Letting the BODs concentrations in all unit cells be written in the formof (m.n) X I matrix: L=(L11L12...L1mL21L... Lj... Lm)T, and the outside inputs for all unit cells be witten inthe formof (m.n) x l matrix: WL= (Wh Wl2... Wtm W... W... wtm)T, the matrixequation for the two-dimensional water quality modeling in the topographically complicated shallowriver can be obtained as :GL = Wh,(10)where G isthe m X n BODs mdtrix. According to Equation (9), each element gn(k = 1, 2,....,m; h = 1, 2,...,n) in G can be calculated by the following equations:gk = qj+ D(ij-1),j+ D,(i,j+1) + Di-1,j),i + D.,(i+1,s) + V;Ka，(11)8k.k+1 =- Dj,(i,J+1)，(12)8t,k-1 =- Dir,j-1).的，(13)8&,e+m =- Di;,