- #1

Naaani

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Also ,see the attachments for clarity...

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

Mass conservation laws for water and for CO2:

(1)

[tex]

\phi\frac{\partial s} {\partial t} + \frac{\partial u}{\partial x} \( F(s) \) =0 [/tex]

(2)

[tex]

\phi \frac{\partial \rho} {\partial t}(P)(1-s)+\frac{\partial \rho}{\partial x}P(u)

(1-F(s))=0

[/tex]

The Darcy Law for both phases, water and gas is

(3)

[tex]

u = -k (\frac{s(k_(rw))}{\mu_w}+\frac{s(k_(rg))}{\mu_g})

(\frac{\partial P}{\partial x}) [/tex]

Consider finite steps,

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

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

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

[tex]\frac{\partial s}{\partial t}= \frac{({s_i}^(k+1)-{s_i}^(k))}{\Delta t}[/tex]

and

[tex]

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

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.

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...

## Homework Statement

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

## Homework Equations

Mass conservation laws for water and for CO2:

(1)

[tex]

\phi\frac{\partial s} {\partial t} + \frac{\partial u}{\partial x} \( F(s) \) =0 [/tex]

(2)

[tex]

\phi \frac{\partial \rho} {\partial t}(P)(1-s)+\frac{\partial \rho}{\partial x}P(u)

(1-F(s))=0

[/tex]

The Darcy Law for both phases, water and gas is

(3)

[tex]

u = -k (\frac{s(k_(rw))}{\mu_w}+\frac{s(k_(rg))}{\mu_g})

(\frac{\partial P}{\partial x}) [/tex]

## The Attempt at a Solution

Consider finite steps,

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

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

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

[tex]\frac{\partial s}{\partial t}= \frac{({s_i}^(k+1)-{s_i}^(k))}{\Delta t}[/tex]

and

[tex]

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

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...