ABSTRACT GOVERNING EQUATIONS
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1 A three dimensional finite element model for fluid flow and transport in confined or unconfined aquifer F. Jacob*, J.M. Crolef, P. Lesaint*, J. Mania* "Laboratoire de calcul scientifique, ^Laboratoire de geologic structurale et appliquee, Universite de Franche-Comte, Besangon, France ABSTRACT A three dimensional finite element model was developed for the simulation of ground water flow and solute or energy transport in confined or unconfined aquifer. Density and viscosity was considered variable in space and in time as functions of concentration or temperature. Numerical implementation has been realized, in order to minimize computational time in respect with the most efficient methods. INTRODUCTION Actually it is more and more important to simulate or to predict pollution of the ground water by toxic contaminant or industrial settling. Development of computer allows to make some three dimensional models to simulate flow and transport in porous media. In the last previous years, we can mention MOSE [6], GENTILE [10], HUYAKORN et al. [12), which one have proposed some three dimensional models using the finite element method. GOVERNING EQUATIONS The present model has been written as a system of two equations with two mains unknowns: the pressure P of the fluid and either the concentration C of the solute or the temperature T of the fluid.
2 90 Water Pollution The first one allows to determine the pressure P, and the second one gives the concentration C of the solute or the temperature T in each point of saturated media. In order to simplify the description only the case of solute transport is presented. A full explanation of the physical problem is made by JACOB [3], according to BEAR [1,2] or VOSS [4]. The flow in a confined or unconfined aquifer and the solute transport are respectively governed by the equations (1) and (2). ap ep + (l-e)pacs ] + cp(q.vc) D).VC= 1= f(c,cg) (2) where c P k U Ps D dm : storage coefficient porosity fluid density tensor of intrinsic permeability viscosity of the fluid bulk density tensor of dispersivity coefficient of molecular diffusion In the equation (2) it is important to note the term containing "C^,", which comes from the chemical reaction on the bulk matrix. Some parameters have to be defined by other physical relationships : 1- the average speed in saturated porous media is derived from Darcy's law BEAR [1) : = - (9P - P9) (3) 2- the relationships between the density p or the viscosity jit and the concentration C, were given by WEAST et al. [7], or VOSS [4] :
3 for the density variation : Water Pollution 91 p = Po + (C - Co) where C, = C(tJ (4) for the viscosity variation : U = Ho + g(c - Co) (5) where ILI^ and p<> are respectively the viscosity and the density of the fluid at initial time to- It can be noted that the previously written equations are quite different from those given in the literature. In many previously published papers, the second member takes into account the source of fluid, or the source of energy or polluant. In the present study, the source term is not taking into account in the second member of the system but it is considered as a boundary conditions on the flux. For the flow, three kinds of boundary can be used : condition - the Dirichlet condition P=P* on H - the Neumann condition -FvP - pgl.n = d on T^ - the mixed condition ~VP - pgl.n +? P = A on For the solute transport, two kinds of boundary condition can be used : - the Dirichlet condition U = U* on H - the Neumann condition D.VU. n = d on F^ The Dirichlet condition allows to impose a pressure, a concentration or a temperature on a surface, whereas, the Neumann condition is used to simulate all type of flux, for example infiltration, seepage...
4 92 Water Pollution COMPUTATIONAL METHOD In this section the mains algorithms and numerical schemes used are presented. The physical problem to solved is described by relations (1) and (2). Each equation was solved by a finite element method. Resolution of the problem was realized with the same mesh and the same finite element for the two equations. The finite element which was chosen is a hexahedron, with an interpolation of degree two/ on the boundary a C -continuity was assumed. The two degrees interpolation was choosen for its advantages into approximation of average speed at the center of the element, since the first derivates of shape functions are continuous excepted in a finite number of points. The finite element method has needed the resolution for flow equation of a linear symmetric system, and for the transport equation of a non linear and non symmetric system. One of the most efficient method to solve linear system is preconditioned conjugate gradient method as it was proved by GAMBOLATI [9] and JOLY [11]. Thus, to solve linear symmetric system, an over relaxation method was used in order to get a better preconditioning of the matrix. Several tests have been made to obtain the best over relaxation coefficient. When the system was non symmetric, the matrix was preconditioned by an incomplete LU decomposition. 1) Schemes for resolution In the case where the density and the viscosity are functions of the temperature or concentration, then the flow and transport equations are coupled. We had to solve a system with two differential equations and two unknowns. M(c)g + A(c)p = f (c) (6) N(c)gg + B(c,p)c = g(c) (7) P(to) = Po c(to) = CQ An iterative method based on the pressure P of the fluid was used in order to solve this system.
5 Water Pollution 93 Let be c = c and p = p ; using a implicit o o scheme for the discretization in time we had to solve the following equations: n+1 n+1 n M(C ) P -P " L J n+1 N(c ) k+1 n+1 n c -c k+1 n+1 k+1 n+1 n+1 n+1 Atn+i.A(c )p = At^i-f(c ) k k+1 k n+1 n+1 n+1 Atn+i-B (C,p ) C k+1 k+1 k+1 n+1 This last equation is non linear on c, also a k+1 simple fixed iterative method was employed. It can be n+1 proved that the sequence (p ) converges in norm k k (N L*. Thus the solution of the system was obtained. FLOW IN UNCONFINED AQUIFER The general problem was presented at the beginning of this paper, and for the case of flow in an unconfined aquifer only comments on the used computational method are given. Here h is defined as the piezometric level. Flow in unconfined aquifer was modeled with an equation which can be written on the following form A.h = f. Sy ^ -V.fKVh] - 0 on Qxfo,Tl (8) where Sy is the effective porosity TO is noted the free boundary and the new boundary condition was : h = z on TO
6 94 Water Pollution The idea of this method was to consider the flow in an unconfined aquifer as a sequence of state flows in confined aquifer. The advantage of this method is that we get a sequence of problems which were easier to solve. The principle was based on the evaluation of the mesh at each iteration, following the piezometric level. Only thickness of the aquifer was evaluated. A such method is valuable only if there are small variations of the piezometric level. This assumption has to be verified for many schemes to solve such problems BODVARSSON [8]. n Let be h = h(t^) the piezometric level at time step tn, b the thickness of aquifer at time step k tn+i, and at iteration k. The variation of thickness is solution to : = h b + k-l h k - b k-l for k>l for k=0 The system A.h =f is solved iteratively until k convergence on h CONCLUSION On the last years, contamination of aquifer became a more and more important problem. In particular, previous studies have suggested that realistic simulations of contaminant in an aquifer should, at least take into account the three dimensional distribution. Some phenomena which are typically three dimensional, and which need to be treated like it, are for example dispersivity or permeability.
7 Water Pollution 95 Knowledge of speed on an accurate way is quite important in order to solve correctly problem of energy or contaminant transport. In this way, the previously described model uses a two degree interpolation and gives a good accuracy of velocity of fluid in each point of the domain because a two degree interpolation gives some continuous first derivates of shape functions. It is true that this method is quite expensive, because it needs 20 nodes by element, in place of only 8 for a one degree interpolation. However the using of a such method allows a homogeneous scheme for the two equations induces with the same finite element and the same mesh. Can we say that time of calculus it is quite important, in front of development of machines. Flow in phreatic aquifer is solved by an iterative method based on knowledge of piezometric level. It is possible to think that this method is not available for porous media with high gradient of pressure for example near a well. Thus, this problem is one of our preoccupation and it will be nearly taking into account changing the first iteration. Numerical schemes have been tested on different cases and have given excellent results, so for the stability or so for the convergence. It seems that discretisation in time does not induce a condition for time step, whenever we have to respect the classical condition on the Peclet number in mesh given by KINZELBACH [5]. It is important to note that our model uses a conjugate gradient method to solve linear system even if it isn't no symmetric. BIBLIOGRAPHY 1. BEAR.J : Hydraulics of ground water. Mc-graw Hill,New-york, BEAR.J : Dynamics of fluid in porous media.american Elsevier, New-york JACOB.F : Modelisation tridimensionnelle par la methode des elements finis de 1'ecoulement et du transport en milieu poreux sature. These de 1'Universite de Franche-Comte, Mai VOSS.C.I : SUTRA a finite element simulation model, Reston Virginia KINZELBACH.W : Groundwater modelling, an introduction with simple programs in Basic, Elsevier New-York 1986.
8 96 Water Pollution 6. MOSE.R : Application de la methode des elements finis mixtes hybrides et de la "marche au hasard" a la modelisation de 1'ecoulement et au transport de masse en milieu poreux. These de 1'Universite de Strabourg WEAST R.C, SELBY S.M : Handbook of chemistry and physics, 48 erne edition, Chemical Rubber, C.O Cleveland, BODVARSSON.G : Linearization techniques and surfaces operators in the theory of unconfined aquifers. Water Ress. Research vol.20 n 9 p Sept GAMBOLATI.G, PUTTI.M : Multigrid vs. Conjugate Gradient in the FE solutions of flow problems. Groundwater Modelling and pressure flow, Vol. 1 Springer Verlag GENTILE.G, MAFFIO.A, MOLINARO.P, RANGOGNI.R : application of F.E.M to 3D problems of transport and diffusion of polluant into aquifer. Goundwater modelling and pressure flow. Oct JOLY.P, : Resolution de systemes lineaires, acceleration de 1'algorithms du double gradient conjugue. Rapport interne de 1'INRIA HUYAKORN.P : Improved three dimensional element techniques for field simulation of variably saturated flow and transport. Journal of Contaminant Hydrology, 12(1993)3.33, Elsevier Sciences Publushers Amsterdam
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