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Homework Help: 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!!!
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