# Finite volume method

1. Feb 11, 2014

### maka89

Hi! I am trying to make a one-dimentional simulator for two-phase flow. I am going to use the finite volume method, because it is conservative and thus it's easier to keep track of the oil/water ratio in the area.

Say you have a conservation equation on the form $\nabla \cdot (k(x) \nabla P(x))$. Then you integrate, apply the divergence theorem and get:
$\int \nabla \cdot (k(x) \nabla P(x)) dV = \int k(x)\nabla P(x) d\overline{A} = A\cdot ([k(x)\frac{dP}{dx}]_{i+1/2} - [k(x)\frac{dP}{dx}]_{i-1/2}).$

So finally, my question: How do you evaluate the integral at the border between the volumes? In particular: How do you evaluate k(x) ?

Help appriciated :)

The equation used is Darcy's law:
$\overline{q}= \frac{k_{i}(x)}{\mu}\cdot\frac{dP}{dx} \widehat{x}$.

q is flux, P is pressure, $\mu$ is viscosity(constant) $k_{i}$ is the permeability for each phase. The permeability will change with the saturation(the ratio between volume of the phase divided by total fluid volume).

Mass conservation:
$\nabla \cdot \overline{q} = s_{i}c_{i}\phi\frac{dP}{dt}$.
$\phi$ is porosity(ratio between volume available for fluids and total rock volume), $c_{i}$ is the compressibility for each phase and is taken to be constant. $s_{i}$ is the saturation mentioned earlier. The equations for each phase can be combined to one equation for the pressure, knowing $s_{oil} + s_{water} = 1$. Then updating $s_{i}$ and $k_{i}$ for each timestep.

Last edited: Feb 11, 2014