On Upwind Galerkin Finite Element Method for Incompressible Flow of Contamination, from Nuclear Waste, in Porous Media

(, , 0 (2 Ω ∞ L T L. ABSTRACT Anon-linear parabolic system is derived to describe incompressible nuclear waste disposal contamination in porous media. Galerkin method is applied for the pressure equation. For the concentration, a kind of partial upwind finite element scheme is constructed. The finite element solution satisfies discrete maximum principle and converges to the solution in norm)) (, , 0 (2 Ω ∞ L T L .

. ABSTRACT Anon-linear parabolic system is derived to describe incompressible nuclear waste disposal contamination in porous media.Galerkin method is applied for the pressure equation.For the concentration, a kind of partial upwind finite element scheme is constructed.The finite element solution satisfies discrete maximum principle and converges to the solution in norm

. I In nt tr ro od du uc ct ti io on n
The proposed disposal of high-level nuclear waste in underground repositories is an important environmental topic for many countries.Decisions on the feasibility and safety of the various sites and disposal methods will be based, in part, on numerical models for describing the flow of contaminated brines and groundwater through porous or fractured media under severe thermal regimes caused by the radioactive contaminants.A fully discrete formulation is given in some detail to present key ideas that are essential in code development.The non-linear couplings between the unknowns are important in modeling the correct physics properties of flow.
In this model, we obtain a convection-diffusion equations which represent a mathematical model for a case of diffusion phenomena in which underlying flow is present; w b and w ∇ ∆ correspond to the transport of w through the diffusion process and the convection effects, respectively, where ∆ ∇ and denoted respectively the gradient operater and the Laplacian operator in the spatial coordinates.
In this paper we will consider the fluid flow in porous media using a Galerkin method for the pressure equation and a kind of partial upwind finite element scheme is constructed for the convection dominated saturation (or concentration).For more details of this subject see Douglas, 2002;Douglas, 2001 andHuang, 2000.

2. .M Mo od de el l E
Eq qu ua at ti io on ns s The model for incompressible flow and transport of contaminated brine in porous media can be described by a differential system that can be put into the following form, (see Douglas, 2002).
H He ea at t: :
And the initial conditions , u is the Darcy velocity, P the pressure, is the trace concentration of the i-th radionuclide , and .
The reservoir Ω will be taken to be of unit thickness and will be identified with a bounded domain in 2  R .We shall omit gravitational terms for simplicity of exposition, no significant mathematical questions arises the lower order terms are included.We define . and introduce the corresponding scalar products and norms dx ) , ( ,where Г is the boundary of Ω and we equip ) ( 10 Ω H with the same scalar product and norm as with the usual modification for s = ∞.When s=2, let (see Johnson ,1987).We assume that (A1) The solutions of the problem (1-6) are regular: , the boundary values problem: Manaa, 2000).h to be the maximum side length of triangles and k to be minimum perpendicular length of triangles for all h T e ∈ .

Definition 3.1: A family h
T of triangulations is of weakly acute type, if there exists a constant 0 0 > θ independent of h such that, the internal angle θ of any triangle for any nodal point p j.
We denote M h as the linear span of ), 1 ( , We associate the index set } to adjacent is : { .Let P i P j ,P k , be three vertices of triangular element e and Using interpolation theory in Sobolev space (see Ciarlet,1978) and inverse inequality, with step length c h we have the relation between w ˆand w from the following lemma Lemma 3.1: There exists a constant C such that: , which is piecewise polynomial space of degree less or equal to r-1 with the similar property as M h and .
We also assume that the families } { h M and } { h N satisfy inverse inequalities: (see Manaa,2000).

4.Error Estimates
Let 0 > τ is a time step and τ τ T N = . We use a Galerkin finite element method for the pressure and velocity and partial upwind finite element scheme for brine, radionuclides, and heat equation.Let , with three steps .Let (.,.) denote the inner product in ,and ,here n ij is the unit outer normal to ij Γ .The partial upwind coefficients should be required that (see hu & Tian(1992)) Step2 -Find P m+1 such that: Step3 -Find U m+1 as : then there exists a constant 1 k such that Quarteroni, 1997).(Manaa,2000).3. We will make the inductive assumption that if

Lemma (4.4):
There exists a positive constant k 2 such that:

F
Fl lu ui id d: : is the permeability of the rock, and ) (c µ is the viscosity of the fluid, is dependent upon c , the concentration of the brine in the fluid , T is the temperature of the fluid, .
C satisfies discrete mass conservation law We will prove first the inductive assumption (17) .We have Which is complete the proof.