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Thread: boundary problem

  1. #1
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    Dec 2007
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    boundary problem

    The 3 equations are:
    Mass conservation laws for water and for CO2:
    (1)
    $\displaystyle
    \phi\frac{\partial s} {\partial t} + \frac{\partial u}{\partial x} \( F(s) \) =0 $
    (2)
    $\displaystyle
    \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)
    $\displaystyle
    u = -k (\frac{s(k_(rw))}{\mu_w}+\frac{s(k_(rg))}{\mu_g})
    (\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:
    $\displaystyle s(x,t); 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).
    $\displaystyle {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
    $\displaystyle \frac{\partial s}{\partial t}= \frac{({s_i}^(k+1)-{s_i}^(k))}{\Delta t}$
    and
    $\displaystyle
    \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).

    So , please help me as its urgent...thank you so much...[/quote]
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  2. #2
    Forum Admin topsquark's Avatar
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    I have translated the tex from the previous post to the "brand" that is used on this forum.

    -Dan

    The 3 equations are:
    Mass conservation laws for water and for CO2:
    (1)
    $\displaystyle
    \phi\frac{\partial s} {\partial t} + \frac{\partial u}{\partial x} ( F(s) ) =0 $
    (2)
    $\displaystyle
    \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)
    $\displaystyle
    u = -k (\frac{s(k_{rw})}{\mu_w}+\frac{s(k_{rg})}{\mu_g})
    (\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:
    $\displaystyle s(x,t); 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...
    $\displaystyle mu_w$ and $\displaystyle mu_g$ are viscosities for water and gas.
    Numerical solution adopted by me:
    Consider finite steps,
    $\displaystyle (\Delta x)$ and $\displaystyle (\Delta t)$.
    $\displaystyle {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
    $\displaystyle \frac{\partial s}{\partial t}= \frac{({s_i}^{k+1}-{s_i}^k)}{\Delta t}$
    and
    $\displaystyle
    \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 $\displaystyle s_i^{k+1}$, for each i, from Eq. (s);
    2) Find the values of $\displaystyle \rho_i^{k+1}= \rho(P_i^{k+1})$ from Eq. (r);
    3) Re-calculate $\displaystyle P_i^{k+1}$ based on the known values of $\displaystyle \rho_i^{k+1}$;
    4) Find the values of $\displaystyle u_i^{k+1} $ from Eq. (u).

    So , please help me as its urgent...thank you so much...
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  3. #3
    Newbie
    Joined
    Dec 2007
    Posts
    2
    Thanks, Dan...
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