The 3 equations are:
Mass conservation laws for water and for CO2:
The Darcy Law for both phases, water and gas is
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
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]