1. Dec 3, 2007

### Naaani

Also ,see the attachments for clarity...

1. The problem statement, all variables and given/known data

I would like to find a numerical solution for the 3 equations using the conditions.

‘w’ refers to water, g to gas (CO2) and ‘o’ to oil
P=Pressure; u=Velocity; s=Saturation (wetting phase);
L=distance between CO2 injection and oil production;
t=time; x=horizontal distance in X-direction; µ=viscosity; ρ=density; ø=porosity
F=fractional flow of water=relative water mobility/sum of relative mobilities

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

All the variables are known except s, u and P.

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

2. Relevant equations

Mass conservation laws for water and for CO2:
(1)
$$\phi\frac{\partial s} {\partial t} + \frac{\partial u}{\partial x} $$F(s)$$ =0$$

(2)
$$\phi \frac{\partial \rho} {\partial t}(P)(1-s)+\frac{\partial \rho}{\partial x}P(u) (1-F(s))=0$$

The Darcy Law for both phases, water and gas is

(3)
$$u = -k (\frac{s(k_(rw))}{\mu_w}+\frac{s(k_(rg))}{\mu_g}) (\frac{\partial P}{\partial x})$$

3. The attempt at a solution

Consider finite steps,
(\Delta x) and (\Delta t).

$${s_i} ^ k =s (i \Delta x,k \Delta t)$$ and the same for P and u.

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

$$\frac{\partial s}{\partial t}= \frac{({s_i}^(k+1)-{s_i}^(k))}{\Delta t}$$

and

$$\frac{\partial u}{\partial x}F(s)=\frac{uF_(i+1)^k-uF_(i)^k}{\Delta x}$$

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).

Thank you...

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