# Math Help - boundary problem

1. ## boundary problem

The 3 equations are:
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})$

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:
$s(x,t); u(x,t); P(x,t);$
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.
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).

2. 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)
$
\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})$

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:
$s(x,t); u(x,t); P(x,t);$
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.
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).