Solving 3 Equations for Water, Gas, and Oil: A Numerical Solution Approach

[Naaani]

Also ,see the attachments for clarity...

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

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...

 

[KataKoniK]

Thank you for sharing your equations and initial/boundary conditions. I understand that you are trying to find a numerical solution for these equations, specifically for the variables of saturation (s), velocity (u), and pressure (P). It seems like you have made a good start by discretizing the equations and using the explicit method.

To further progress with your solution, I suggest using an iterative approach. This means that you will need to make an initial guess for the values of s, u, and P, and then use the equations to update these values until they converge to a solution. This will involve using the boundary and initial conditions, as well as the equations you have listed.

I also recommend considering using a numerical software or coding language to help with the calculations. There are many available options that can handle complex equations like these. Additionally, you may want to look into different numerical methods, such as the implicit method, which may be more accurate for your problem.

Overall, solving these equations numerically can be a challenging task, but with careful consideration and use of appropriate methods, you can find a solution that accurately represents the physical system you are studying. I wish you the best of luck with your research.

 

[piercas]

I would like to commend the effort and thought put into this numerical solution approach. It is clear that you have a good understanding of the equations and variables involved in this problem. However, I would like to offer some suggestions for improvement and clarification.

Firstly, it would be helpful to define the units for each variable, as well as any relevant physical constants. This will ensure consistency and accuracy in your calculations.

Secondly, I noticed that the equations you have provided are not fully written out, and some symbols are missing. This can make it difficult for others to follow your work. I would recommend fully writing out the equations and including all necessary symbols.

Additionally, it would be helpful to provide some context or background information about the problem you are trying to solve. This will help others understand the significance and relevance of your approach.

Regarding your attempt at a solution, it is a good start. However, I would suggest breaking down the problem into smaller steps and focusing on one equation at a time. This will make it easier to understand and implement your solution.

Finally, I would recommend seeking feedback and assistance from a colleague or mentor who has experience in solving similar problems. They may be able to provide valuable insights and suggestions for improvement. Good luck with your work!