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The 3 equations are:
Mass conservation laws for water and for CO2:
(1)
 \) =0 )
(2)
(1-s)+\frac{\partial \rho}{\partial x}P(u)<br />
(1-F(s))=0<br />
)
The Darcy Law for both phases, water and gas is
(3)
)}{\mu_w}+\frac{s(k_(rg))}{\mu_g})<br />
(\frac{\partial P}{\partial x}) )
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:
; u(x,t); P(x,t); )
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]
