Finite Volume Method & Evaluating Integral at Borders for Two-Phase Flow

[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 :)

Additional info(NOT REALLY RELEVANT):

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.

 

[jvicens]

Hi there! It's great to hear that you are working on a one-dimensional simulator for two-phase flow using the finite volume method. This approach is indeed a good choice for keeping track of the oil/water ratio in the area.

To answer your question about evaluating the integral at the border between volumes, there are a few things to consider. Firstly, the integral should be evaluated at the interface between the two volumes, as this is where the flux and pressure gradients will be changing. This means that you will need to use the values of k(x) and dP/dx at the interface.

In terms of evaluating k(x), this can be done using the concept of a transmissibility factor. This factor takes into account the permeability, porosity, and viscosity of each phase, as well as the geometry of the grid cells at the interface. It is used to calculate the effective permeability at the interface, which is then used in the integral.

I would also like to mention that when updating the saturation and permeability for each timestep, it is important to consider the impact of capillary pressure and relative permeability on these properties. These factors can significantly affect the flow behavior and should be taken into account in your simulation.

I hope this helps and good luck with your simulator! Let me know if you have any further questions.