1D Single Phase Flow in Porous Media

[phil ess]

Homework Statement

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).

Homework Equations

The Attempt at a Solution

Combining the continuity equation,

$$\epsilon\frac{\partial\rho}{\partial t} + \nabla\cdot\left(\rho k \nabla v\right) = 0$$

Ignore above line, not showing up correctly,

$$\epsilon\frac{\partial\rho}{\partial t} + \nabla\cdot\left(\rho k \nabla v\right) = 0$$

with Darcy's law,

$$v = \frac{-k}{\mu}\nabla P$$

we get,

$$\epsilon\mu\frac{\partial\rho}{\partial t} = \nabla\cdot\left(\rho k \nabla P\right)$$

Then assuming a liquid with compressibility c,

$$\rho = \rho_{o}e^{c(P-P_{o}}$$

which combines with the above to give the governing equation:

$$\epsilon\mu c \frac{\partial P}{\partial t} = \nabla\cdot\left(k \nabla P\right)$$

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

$$\frac{\partial P}{\partial t} = \alpha\frac{\partial^{2}P}{\partial x^{2}}$$

*** 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!

 

[phil ess]

Bump!

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!