A kind of Upwind Finite Element Approximations for Compressible Flow of Contamination from Nuclear Waste in Porous Media

ABSTRACT A non-linear parabolic system is derived to describe compressible nuclear waste disposal contamination in porous media . Galerkin method is applied for the pressure equation . For the concentration of the brine of the fluid, a kind of partial upwind finite element scheme is constructed. A numerical application is included to demonstrate certain aspects of the theory and illustrate the capabilities of the kind of partial upwind finite element approach.


Introduction
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 of flow .
In this model one 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 operator 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) equation. For more details of this subject see [7, 6, and 4].

Model Equations
The model for compressible flow and transport of contaminated brine in porous media can be described by a differential system that can be put into the following form [2]. is a production term, .. (5) Shifting and the initial conditions The reservoir  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 arise the lower order terms are included. We assume that The solution of the problem (from eq.(1) to eq.(6)) is regular: , the boundary value problem: And a subspace of We associate the index set , which is piecewise polynomial space of degree less or equal to r with step length hP and the following property: for , which is piecewise polynomial space of degree less or equal to r-1 with the similar property as Mh and We also assume the Lemma(3.1): [5] There exists a constant C such that:

Error Estimates
Let 0   is time step and We use a Galerkin finite element method for the pressure and velocity and partial upwind finite element scheme for brine. Let ,here nij is the unit outer normal to ij  . The partial upwind coefficients should be required that [5]. (11) step2. Find P m+1 such that: step3. Find U m+1 as : for J t  , where the constant  is chosen to be large enough to insure the coercivity of the bilinear form over ...(16) A standard result in the theory of Galerkin methods give [2].

Proof:
We shall begin by deriving an evolution inequality for the difference  between the projection p and the approximation solution P.
The weak form of eq.(1) is so there can be difference between equations (12) and (15) to show that.
which complete the proof.

Numerical Application
In this example, we solve a purely convective problem in one dimension [12]  we apply two methods: Galerkin and a kind of partial upwind finite element for this example. We discrete the region -2 ≤ y ≤ 0 into at first 100 quadrilateral elements and second 200 triangular elements with 202 nodes, also the distance between any two nodes is 0.02 and take uy =1.0, theta =0.5 , τ =0.04 , and the number of steps (N) = 25 on a 1*100 mesh for quadrilateral element and 1*200 mesh for triangular element.
The correct solution to the problem is described by a rectangular pulse moving with unit velocity in the y direction. Table (1) contains numerical results where θ =1/2 at all nodes lie on the right-hand side of the finite element mesh in this example. From the boundary conditions we can see that the solution at nodes 1 and 2 is held at the value 1.0 for the first 0.2 seconds of convection. Figure (1) shows the computed solution after one second and it draws between the concentration and coordinate y, we can see that while y convergence to zero the value of concentration convergence oscillation to solution.   The numerical solutions at all nodes of the right -hand side of mesh, where theta = 0.5, τ = 0.04, N = 25

Conclusions
We used the system with large coupled of strongly non-linear partial differential equations which arise from the contamination of nuclear waste in porous media .We used a Galerkin method for the pressure equation and a kind of partial upwind finite element method for the concentration.For the compressible case, we obtained the error estimates for approximate Darcy velocity U, concentrations C in )) ( , , 0 ( 2   L T L .From the numerical results presented in this application, we have got a kind of partial upwind finite element method for triangular element convergent to the exact solution and in comparison with Galerkin method , we found that a kind of Exact solution at t=1 second