How Do You Find Pressure Distribution In A Porous Material?

[Pchang38]

Homework Statement

Using Darcy’s law, plus another appropriate relationship, derive a
single equation for the pressure distribution in a porous material. Your equation should be stated in
terms of generic operators (i.e. without assuming any specific coordinate system) and allow for the
fact that the permeability K may vary with position, although you can assume that μ is constant.

Homework Equations

Darcy's Law: μ = -K/μ ∇ρ
Where the fluid velocity vector and the pressure are denoted by μ and p, respectively, K is known as the
permeability (which is a property only of the porous material), and μ is the flowing fluid’s dynamic
viscosity.

The Attempt at a Solution

I attempted to create a modified equation using darcy's and navier-stokes but can't seem to put all the pieces together. Can anyone point me in the right direction?

 

[Chestermiller]

Do a differential mass balance on a control volume.

Chet

 

[Pchang38]

Do a differential mass balance on a control volume.

Chet

Hey, thanks for the reply. This is what I have done so far:

We know, Darcy's law: u' = -k/u*grad(P)
So, my thought process is:
1.) We assume steady flow and flow in one direction (linear).

2.) Using the relationship that fluid velocity (u') = Q/A

3.) we get, Q = -kA/u * grad(P)

4.) We know that the shape is circular, so A (cross sectional area) = 2piRH, where R is radius, where H is the height

5.) We know that grad(P) = dP/dR, (only in the radial direction)

5.) so, Q = -k2piR/u * (dP/dR)

6.) now, Separate the variables and integrate from R = Ro to some generic location R:

-> integral(dR/R) from Ro to R = integral(-k2piH/uQ * dP) from Po to P
-> ln(Ro/R) = -2kpiH/uQ * (P-Po)
-> P(R) = Po-uQ/2piKH*(ln(R/Ro)) Final eqn

This equation shows that Pressure varies logarithmically and can be used to find pressure differences.

Does this make sense and seem reasonable? Does it also show that the permeability K may vary with position?

 

[Chestermiller]

Hey, thanks for the reply. This is what I have done so far:

We know, Darcy's law: u' = -k/u*grad(P)
So, my thought process is:
1.) We assume steady flow and flow in one direction (linear).

2.) Using the relationship that fluid velocity (u') = Q/A

3.) we get, Q = -kA/u * grad(P)

4.) We know that the shape is circular, so A (cross sectional area) = 2piRH, where R is radius, where H is the height

5.) We know that grad(P) = dP/dR, (only in the radial direction)

5.) so, Q = -k2piR/u * (dP/dR)

6.) now, Separate the variables and integrate from R = Ro to some generic location R:

-> integral(dR/R) from Ro to R = integral(-k2piH/uQ * dP) from Po to P
-> ln(Ro/R) = -2kpiH/uQ * (P-Po)
-> P(R) = Po-uQ/2piKH*(ln(R/Ro)) Final eqn

This equation shows that Pressure varies logarithmically and can be used to find pressure differences.

Does this make sense and seem reasonable?

This makes sense specifically for the case of radial flow. But, I'm pretty sure they are looking for something more general than this. Think about the problem in 2D flow.

Does it also show that the permeability K may vary with position?

No. The problem says that K is a function of R, so the equation is not integrated correctly.

What they are looking for is a second order partial differential equation in 2 or more dimensions, involving spatial gradients of pressure, and also involving K. Take a differential control volume between x and x + Δx, and y and y + Δy, with flow through the 4 boundaries of the control volume.

Chet

 

[Pchang38]

This makes sense specifically for the case of radial flow. But, I'm pretty sure they are looking for something more general than this. Think about the problem in 2D flow.

No. The problem says that K is a function of R, so the equation is not integrated correctly.

What they are looking for is a second order partial differential equation in 2 or more dimensions, involving spatial gradients of pressure, and also involving K. Take a differential control volume between x and x + Δx, and y and y + Δy, with flow through the 4 boundaries of the control volume.

Chet

Hey, Chet I got some feed back and this is what my TA said:

I would not suggest that approach. Your assumptions are more applicable for a cylinder rather than a sphere, and also ignore the 'no coordinate system' requirement. A different relationship to couple with Darcy's law will probably yield a more appropriate answer. Also, since there is no coordinate system, integration is unnecessary in this problem.

Maybe we were both over-thinking? What do you make of this Chet? I am pretty lost now haha

 

[Chestermiller]

Hey, Chet I got some feed back and this is what my TA said:

I would not suggest that approach. Your assumptions are more applicable for a cylinder rather than a sphere, and also ignore the 'no coordinate system' requirement. A different relationship to couple with Darcy's law will probably yield a more appropriate answer. Also, since there is no coordinate system, integration is unnecessary in this problem.

Maybe we were both over-thinking? What do you make of this Chet? I am pretty lost now haha

Who says that the geometry is circular? I never said that. You were the one who assumed circular geometry. All I said was that what you had done applies exclusively to that limited type of geometry, and that what they are looking for is something more general.

Have you learned about the divergence theorem in your courses yet? Do you know how to apply that to a control volume of arbitrary geometry?

Chet

 

[Pchang38]

Who says that the geometry is circular? I never said that. You were the one who assumed circular geometry. All I said was that what you had done applies exclusively to that limited type of geometry, and that what they are looking for is something more general.

Have you learned about the divergence theorem in your courses yet? Do you know how to apply that to a control volume of arbitrary geometry?

Chet

I do not believe we have covered the divergence theorem yet. But I am somewhat familiar with it. But i believe the theorem requires integration, but the problem I am trying to solve does not

 

[Chestermiller]

Suppose you have a closed fixed control volume, such that the total amount of fluid within the control volume remains constant. If dS represents a differential element of surface area on the surface of the control volume and ##\vec{n}## represents a unit outwardly directed normal to the surface at the location of dS, what is the rate of fluid flow out of the control volume through the surface element dS (if the seepage velocity is ##\vec{u}##)?

Chet

 

[Chestermiller]

Are you familiar with the fact that, if the flow is incompressible, the divergence of the velocity vector is equal to zero?

Chet