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Please help me in solving these

  1. Dec 3, 2007 #1
    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)
    [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]


    3. 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...
     

    Attached Files:

  2. jcsd
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