Cylindrical Permeability Problem: Help on how to set up

[Bcavender]

TL;DR Summary

I wish to learn how to model the radial pressure gradient across a cylindrical, permeable strata and possibly develop tighter ranges for estimated porosity and permeability.

I was asked to give an estimate for a three dimensional permeate problem and I need an assist in how to setup a model equation.

Picture a 6" underground cylindrical bore. The depth of the cylindrical bore is 150 feet. The bore is sealed at the bottom and the permeability below the bore bottom in a horizontal plane is zero. Likewise there is a similar impermeable, horizontal layer at the top of the 150' bore, except there is a pipe leaving the bore where the permeate leaves and its size only offers negligible resistance to flow by comparison to the much lower permeability of the strata composition surrounding the bore.

• Viscosity of the permeate is 0.0111 cP.
• The porosity of the surrounding strata is homogeneous estimated between 3-7%.
• Permeability of the strata over the 150' cylinder depth is estimated to be between 100 and 1000 micro-Darcy.
• Strata pressure at a very long, but unknown, distance from the bore is uniformly 45 PSIG in all directions surrounding the bore.
• Free permeate pressure inside the cylinder is 30 PSIG for a differential pressure of 15 psi.
• The measured flow rate is 4 cubic feet per hour in actual volume at 30 psig.

Is is possible to develop a better estimate for porosity and permeability numbers than the estimated ranges?

How do I set up an equation to model the gradient pressure radiating out from the cylindrical bore?

All comments and suggestions are most welcome!B

 

[Chestermiller]

Are you familiar with the formulation and expression of the Theis equation, and approximations to the Theis equation?

 

[Bcavender]

Chester,
Thank you for your reply.

I had not heard of this, but searched it down.

It looks quite analogous in that Theis storativity and transmissivity parallel porosity and permeability that appear more commonly in Oil&Gas analyses, but the work appears to only target incompressible aquifer flow vs a compressible gas.

It clearly captures the nature of non-steady (non-linear) flow, at different radii, but I could not find an example of this approach utilized for a compressible gas.

Best detail I could find:

http://www.math.clemson.edu/~warner/Projects/GroundWater/NoName30

Your suggestion raised other search ideas that brought numerous research papers on compressible gas permeation, but all were very specific to specialized situations like horizontal wells that didn’t match my less complicated geometry.

I probably need a more basic, elementary introductory teaching model to get traction towards a model.

Any other suggestions?Bruce

 

[Chestermiller]

So you have two phase flow in the formation, with gas displacing water, and both phases precent within the pores within the part of the formation near the well? Are you familiar with using "relative permeabilities" in modeling such flows?

 

[Chestermiller]

See Flow of Fluids Through Porous Materials by R. E. Collins. I don't know whether this book is still in print, but it is very good. Also see the literature on CO2 sequestration by Chin Fu Tsang, formerly of Lawrence Berkeley Lab.

Google 'Two Phase Flow in Porous Media'

 

[Bcavender]

The R.E. Collins book carries a specific double cylinder permeability derivation that looks spot on. (Hat tip to Baluncore for the connection!)

After a first read through, I will have to back up two sections to get the context and variable definitions to try to gel the equations down to a practical unit calculations.

This read brings back great memories of Dr. Raymond Bell, Ace Calc III prof from 1972 e-school, cleaning our 20 year old clocks. (Jeez, has it been 50 years?)

Collins is also two+ pay grades above my mathematics ability and is fascinated at the elegant orthogonality between concentric, constant flow contours and radial, equal pressure gradients … more than calculating the real world, down hole parameters LOL … but now I think I’ve got a shot to get there.

Thank you Gentlemen!

 

[Chestermiller]

The R.E. Collins book carries a specific double cylinder permeability derivation that looks spot on. (Hat tip to Baluncore for the connection!)

After a first read through, I will have to back up two sections to get the context and variable definitions to try to gel the equations down to a practical unit calculations.

This read brings back great memories of Dr. Raymond Bell, Ace Calc III prof from 1972 e-school, cleaning our 20 year old clocks. (Jeez, has it been 50 years?)

Collins is also two+ pay grades above my mathematics ability and is fascinated at the elegant orthogonality between concentric, constant flow contours and radial, equal pressure gradients … more than calculating the real world, down hole parameters LOL … but now I think I’ve got a shot to get there.

Thank you Gentlemen!

Back in the day, Gene Collins and I worked together in my groundwater projects (deep well injection of aqueous wastes) at DuPont. Gene was a consultant, and a great guy to work with.

 

[Bcavender]

Back in the day, Gene Collins and I worked together in my groundwater projects (deep well injection of aqueous wastes) at DuPont. Gene was a consultant, and a great guy to work with.

Ah, Rock Stars! Very cool!

Seems that I read somewhere about a deep disposal well in KY that DuPont was working on and eventually completed the work. Might that have been the same?

————-

Ultimately, I need to move expeditiously to develop an equation that represents the dependent radial pressure profile variable as a function of the independent radius variable from the wellbore boundary condition of 30 psig. I have been fortunate to have come in possession of the well logs so there may be the possibility of closely estimating porosity and permeability.

The other boundary condition is taken from the reasonable assumption that at the r sub e ‘drainage’ radius, the pressure equals 45 psig, as that was the wellbore pressure before flow began from the well (as it had 15-25 years of zero flow and it was highly probable that the wellbore pressure equalized to the overall strata pressure.)

The pivotal value of the data here is to get a real handle on the r sub e distance to see if it is ~60 feet (very good) or ~5000 feet (very bad).

My difficulty is seeing a process to map detailed gas parameters like supercompressibility, etc into a form I can have some internal confidence that it is reasonably accurate.

Thanks!
Bruce

 

[Chestermiller]

Ah, Rock Stars! Very cool!

Seems that I read somewhere about a deep disposal well in KY that DuPont was working on and eventually completed the work. Might that have been the same?

The KY site involved in the work I did was at Louisville, and produced TiO2.

————-

Ultimately, I need to move expeditiously to develop an equation that represents the dependent radial pressure profile variable as a function of the independent radius variable from the wellbore boundary condition of 30 psig. I have been fortunate to have come in possession of the well logs so there may be the possibility of closely estimating porosity and permeability.

The other boundary condition is taken from the reasonable assumption that at the r sub e ‘drainage’ radius, the pressure equals 45 psig, as that was the wellbore pressure before flow began from the well (as it had 15-25 years of zero flow and it was highly probable that the wellbore pressure equalized to the overall strata pressure.)

The pivotal value of the data here is to get a real handle on the r sub e distance to see if it is ~60 feet (very good) or ~5000 feet (very bad).

My difficulty is seeing a process to map detailed gas parameters like supercompressibility, etc into a form I can have some internal confidence that it is reasonably accurate.

Thanks!
Bruce

I strongly suggest you enlist the help of a consultant on this. Regarding to $r_e$, in a transient problem like this, it is going to be increasing with time (depending on the hydraulic diffusivity). The problem is probably going to require numerical solution.