The 3 equations are:

Mass conservation laws for water and for CO2:

(1)

(2)

The Darcy Law for both phases, water and gas is

(3)

Initial and boundary conditions

(x is distance and t is time,P is pressure,s is saturation):

At t=0, s=1

At x=0. s=s_wi

At x=0, P=P_1

At x=L, P=P_2

Three variables to be found:

F is a function of s and F is ratio of relative premeability and viscosity.

All others are known....rw and rg are relative premeabilities for water and gas...

mu_w and mu_g are viscosities for water and gas.

Numerical solution adopted by me:

Consider finite steps,

(\Delta x) and (\Delta t).

and the same for P and u.

Then (for the explicit method), we can write approximately using discretization as

and

On substitution in (1), we get an equation for s at (i,k+1) .

Now , i tried to do the same for the other 2 equations but could not separate the

variables u and p.Also did not know how to use the initial and boundary conditions.

But i think the procedure could be like:

The solution at the layer k=0 (t=0) is known from initial conditions.

Assume that the solution at layer k has been calculated. In order to find the solution at the layer k+1,

1) Find the values of saturation s_i^k+1, for each i, from Eq. (s);

2) Find the values of rho_i^k+1= rho(P_i^k+1) from Eq. (r);

3) Re-calculate P_i^k+1 based on the known values of rho_i^k+1;

4) Find the values of u_i^k+1 from Eq. (u).

So , please help me as its urgent...thank you so much...[/quote]