Application of Element-Free Method to water quality analysis of Miyun reservoir

Science in China Ser. D Earth Sciences 2005 Vol.48 Supp. II 303- -310Application of Element-Free Method to water qualityanalysis of Miyun reservoirGU Zhaosheng12, YANG Tianxing2 & XU Qing1. Changchun Institute of Technology, Changchun 130012, China;2. College of Geo-Exploration Science and Technology, Jlin University, Changchun 130026, China;3. National Research Center of Geoanalysis, Beijing 100037, ChinaCorrespondence should be addressed to Gu Zhaosheng (email: guzhaosheng@ 126.com)Received March 15, 2005Abstract This paper presents a 3D mathematical model for transfer and transform of watercontaminants in reservoir, describes the discrete methods and the compute process of using anelement-free collocation method. By these models and methods, the water quality conditions ofMiyun reservoir area, the quantified description of three dimensions concerning the distributionand change of diversified contaminants are obtained within the assigned time. The distributionsof total nitrogen and total phosphorus of the reservoir area in 2003, 2004 and 2005 are simulatedand forecasted by the boundany conditions in 2003, so the main development trend is shown. It isfound through calculation that: The contribution of the bed mud with regard to total phosphorus iscomparatively marked. On the whole, the influence factor of the bed mud in the reservoir areaand other comprehensive factors to total phosphorus are quite great. The influence of the livingthings or other function in the surface water of the Chaohe River valley in the east of the reservoiron the total phosphorus is comparatively conspicuous. Within the reservoir area in spring, sum-mer and autumn, the concentration distribution of total nitrogen is basically in the trend of a slowprogressive decrease, but the concentration of total phosphorus increases progressively. It is justcontrary in autumn, winter and spring, the concentration of total nitrogen increases and theconcentration of total phosphorus decreases. On the whole of the year's development, the totalnitrogen is basically in a steady state, and the total phosphorus is in the increasing trend withyears, but the kind of the trend is not conspicuous.Keywords: Miyun reservoir, water wuality analysis, Element-Free Method.DOI: 10.1360/05yd0019With regard to solving the problem of transfer andMethod in numeral calculation of differential equationtransform of water contaminants, a traditional methodhas been followed with more and more interest by a .is finite element method or finite difference method,lot of fields in engineering calculation' 3. Especiallywhich is very difficult for the three-dimensional areain the problems of three-dimensional space and dis-that differ in depth and shallow, that is complicated inpers中国煤化工Method shows im-geometric shape, so generally it is only applied tomens)ds have been widelytwo-dimensional models or one-dimensional models.appiMYH. CN M H G16-10,， and some apIn recent years, the application of every Element-Freeplied in water science research. In ref. [1] based onCopyright by Science in China Press 2005304Science in China Ser. D Earth Sciencesthe moving least squares method and the finite differ-represents release cofficient of the pore water and bedence scheme, a new element-free method, is proposedmud, 1/d. β represents sedimentation coefficient, 1/d.and used to simulate horizontal two dimension riverI(x, y, z, t) represents the rate of change of concentra-flow with complicated boundary. A collocation methodtion caused by the function of iving things and otherbased on compactly supported radial basis function, afactors (such as the reproduce of alga or other plank-mesh-less method, is presented in ref. [5] to solveton), mgL":d'.Poisson equation problem. But the discussion aboutDiffusion coefficient, release coefficient, and sedi-the direct application of Element-Free Method tomentation coefficient usually are mensurated by ex-solving the 3D models of reservoir or lake water pol-perimentation. Commonly I(x, y, z, [) is not knownlution is not found. So this article expatiates on theeasily, but it can be determined by parameters inverseprocess to directly solve the 3D water quality modelscalculation based on the concentration observationof the reservoir by using the Element-Free Method,values from different periods.and makes the detailed introduction about the applica-Give out the following initial conditions (statetion in analysis of total nitrogen and total phosphorusequation)in Miyun reservoir transfer and transform.C(x, y, z, D)=o=u(x, y, z), (x, y, z)∈Q (2) .1 The mathematic models of water quality analy-sis in reservoirand boundary conditions (control equation)The whole reservoir area is denoted by Q. The res-C(x,y,z,t)r,=v(x,y,z,I),(x,y,z)∈T,t≥0, (3)ervoir boundary is divided into two parts T =Ti∪N2.Fi is all the water inlets and water outlet. T2 is the Wa-。C(x,y,z,t) =0,(x,y,z)∈T2,1≥0, (4)terproof boundary (including the bottom surface andEnr2water surface of the reservoir). Considering the fol-u(x, y, z), v(x, y, z, 1) are known initial conditions andlowing factors: convection-diffusion function of waterboundary conditions separately.body, sedimentation of the contaminants in the waterThe same total nitrogen model can be establishedbody, the releasing anew of the contaminants in thebed mud, storage, hunting for food, delivery process ofyet. Heavy metal model such as Cd, Cr, Hg, etc. canliving things and other function of random sourcebe identical model with total phosphorus by suitablyterms, the equation of transfer and transform of totaladjusting.phosphorus is established as follows:2 The solving process with Element-Free MethodTake the models of total phosphorus (1), (2), (3), (4)axl(D.0)(,0)+影(“D:H)for example. Write P+=(xyz,)i1,2"", l is all theacaCsampling points in three. dimensional area. Qi= (xi Yodydzz), i= l+1, l+2, .. n is the boundary points located inthe outlet and the inlet nearby.+ aCedud - βC + I(x,y,z,t),(x,y,z)∈Q, t>0,(1)Letwhere C represents the concentration of total phos-C(x,y,z,t)= Za()(x,v,), (x,y,z)∈S2ur, (5)phorus in the water body, mg/L. x, y, z are Descartescoordinates variables. Dx, Dy, Dz represent diffusioncoefficient, m'/d. Ux, Uy, uz represent the division quan-as th中国煤化Ilution of the problemtity of velocity of flow, m/d. CedMud represents the(1),YHC N M H Gz) takes Multi Quad-concentration of total phosphorus in bed mud. aric type basic function of Hardy φ(x, y, z) = (c'+305Application of Element-Free Method to water quality analysis of Miyun reservoir(x-x)}+(>- y)2+(z -的一C+-C=q-9)aC+1, aCjt.9(0dyNotice: φ is satisfied lim φ(r)=0，when 0< β<2。aC;+1aCj+1. dCj+10.5，limdQ;(r)d中;(r)=0， limd中(r)2 =0,D2-厂“x- xr→dyaCt+r=√(x-x)2 +(y- y,)2 +(z-z)B. It makes definedu--CH+(x,y,1tt) |+solution C(x, y, z, t) of formula (5) be satisfied foraC)n~C(x,y,),≈0 and (x,y,)|r,≈0 natu-D.xj+rally in boundary T2. This is an advantage which the8CjaC;method of radial basic in being adapted to some kindL-u-βC,+3x'dy“zdof boundary conditions has.Notice that it also has this character if A(x,y, z) isl(x,y,z,1,)]+ aCBeMud,(6)weighted cofficient θ∈[0, 1]. When θ=0, for implicttaken for Gauss distribution function φ(r)=escheme; when θ =1, for explicit scheme. For every jNow in the period of the assigned time [0, T], solvetaken, let each sampling point (xk Yk, zx) and boundarythe problem of the concentration of terminal time C(x,point in Q be substituted in, then we arrange it andy,z, T).obtain n linear equations about a(j+1)Let time interval be divided into m, the divided之a:(j+1)point0=to