1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

1D Single Phase Flow in Porous Media

  1. May 15, 2010 #1
    1. The problem statement, all variables and given/known data

    I have been asked to model the flow of a slightly compressible liquid through a rigid, incompressible porous medium in 1D, assuming the flow obeys Darcy's law. I am given a rectangular prism of cross sectional area A with length L, a source of liquid (say water) on one side with total discharge Q, along with variables concerning the properties of the medium (porosity, permeability, etc). I am ignoring dispersion and diffusion effects.

    Basically I am looking to find the pressure in the tube as a function of position and time, P(x,t).

    2. Relevant equations

    3. The attempt at a solution

    Combining the continuity equation,

    [tex]\epsilon\frac{\partial\rho}{\partial t} + \nabla\cdot\left(\rho k \nabla v\right) = 0[/tex]

    Ignore above line, not showing up correctly,

    [tex]\epsilon\frac{\partial\rho}{\partial t} + \nabla\cdot\left(\rho k \nabla v\right) = 0[/tex]

    with Darcy's law,

    [tex]v = \frac{-k}{\mu}\nabla P[/tex]

    we get,

    [tex]\epsilon\mu\frac{\partial\rho}{\partial t} = \nabla\cdot\left(\rho k \nabla P\right)[/tex]

    Then assuming a liquid with compressibility c,

    [tex]\rho = \rho_{o}e^{c(P-P_{o}}[/tex]

    which combines with the above to give the governing equation:

    [tex]\epsilon\mu c \frac{\partial P}{\partial t} = \nabla\cdot\left(k \nabla P\right)[/tex]

    Combining the constant terms together, and since we are only considering the 1D flow in the x direction:

    [tex]\frac{\partial P}{\partial t} = \alpha\frac{\partial^{2}P}{\partial x^{2}}[/tex]

    *** The first few equations are not showing up correctly. But what is important is the final results which seem ok.

    Now all I need to do is solve this equation to get P(x,t).

    What I am having problems with is the boundary conditions. I haven't taken any PDE courses or anything, so I'm not sure how they are supposed to be formulated, but I'll give some guesses:

    P(x,0) = Pi ie. the pressure initially is some constant throughout the medium
    Q = constant ie. the input flow rate is constant for all t
    P(L,inf) = ?

    I am assuming a finite medium with no discharge, so the liquid, and the pressure, will build up indefinitely, but I don't know what to put for the boundary at the end.

    Like I said I am unfamiliar with PDEs which makes this very difficult. Later this problem will be extended to multi-phase flow, but this was given as a warm up for now.

    Any help with this problem would be greatly appreciated!! Thank you!
  2. jcsd
  3. May 18, 2010 #2

    I think I have the boundary conditions:

    P(0,t) = some constant (non-zero, from Darcy's law)
    P(L,t) = Po ??
    P(x,0) = Pi or do I use dP/dx = 0 since the end is 'insulated'

    Now I know how to solve pde's for both ends at 0 or both ends insulated, but not for one end at constant presure and the other end insulated. I'm thinking in terms of the analogy of heat flow in a wire.

    Again any help is very much appreciated!!!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook