(adsbygoogle = window.adsbygoogle || []).push({}); 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!

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# Homework Help: 1D Single Phase Flow in Porous Media

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