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Finite volume method

  1. Feb 11, 2014 #1
    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 [itex]\nabla \cdot (k(x) \nabla P(x))[/itex]. Then you integrate, apply the divergence theorem and get:
    [itex]\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}).[/itex]

    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:
    [itex]\overline{q}= \frac{k_{i}(x)}{\mu}\cdot\frac{dP}{dx} \widehat{x}[/itex].

    q is flux, P is pressure, [itex]\mu[/itex] is viscosity(constant) [itex]k_{i}[/itex] 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:
    [itex] \nabla \cdot \overline{q} = s_{i}c_{i}\phi\frac{dP}{dt} [/itex].
    [itex]\phi[/itex] is porosity(ratio between volume available for fluids and total rock volume), [itex]c_{i}[/itex] is the compressibility for each phase and is taken to be constant. [itex]s_{i}[/itex] is the saturation mentioned earlier. The equations for each phase can be combined to one equation for the pressure, knowing [itex]s_{oil} + s_{water} = 1[/itex]. Then updating [itex] s_{i} [/itex] and [itex]k_{i}[/itex] for each timestep.
    Last edited: Feb 11, 2014
  2. jcsd
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