Drainage through vertical porous cylinder with open sides.

[xnh11]

Good day.

I'm trying to understand the flow through a vertical porous cylinder with open sides. Please see attached sketch. I'm assuming flow enters the top of the cylinder at some specified flow rate and pressure. The flow may exit through the bottom of the cylinder and *through the sides*. The situation is something like that in

http://www.itg.cam.ac.uk/people/grae/48.pdf,

except that the flow domain is finite.

Now, I expect that capillary forces will prevent flow through the sides until some condition is satisfied. For example, if the cylinder is saturated and taller than the capillary rise, then flow will exit through the sides and bottom.

Also, and of more interest to me, it seems that, even for a short cylinder, if the mass flow rate at the inlet exceeds some limiting value, some flow will be forced through the sides and the rest will exit through the bottom. If the inlet flow rate is less than that limiting value, the flow will stay inside the cylinder until it exits through the bottom. I'm trying to determine that limiting value.

One thought that I had is that there will be flow through the sides if the frictional pressure gradient exceeds the hydrostatic pressure gradient. However, this approach doesn't seem to give reasonable answers.

The seepage and drainage studies I've read don't seems to deal with this situation. I see lots of analyses of flow through porous cylinders, but the cylinders are bounded on the sides by an impermeable boundary.

Thanks for the assistance.

G

 

[bimr]

If the fluid is full flow, then the flow through the porous sidewalls will be dependent on the pressure in the column as well as the porosity of the sidewall.

 

[Compositepro]

If the pressure of the fluid at any point in the porous media is greater than outside the media, then fluid will flow toward the lower outside pressure. The converse is also true, that if the fluid pressure is lower than outside, then air will be drawn into the porous media.

The fluid pressure will be a function of hydrostatic forces due to gravity, frictional forces due to viscosity, and and surface tension forces. Inside a full saturated porous media the surface tension forces cancel out and cause no net effect. At the outside surfaces of the porous cylinder surface tension forces keeping fluid inside the cylinder are very small but the forces that prevent air from entering are much larger because the pores are small. There will be a threshold pressure before the surface tension is broken and, once broken, air can penetrate relatively freely. Air has a low viscosity so fluid draining may be much more rapid once air enters the cylinder.

If the pore size is small enough and the surface tension great enough, then ambient air pressure will never be great enough for air to penetrate the surface of the cylinder, so it will always stay saturated and never drain completely due to gravity.

So, as you can see, the modeling of what you describe is not simple, but it can be done.

 

[xnh11]

Answer found. For steady flow in a homogeneous isotropic porous body (i.e., permeability is constant) that satisfies Darcy's equation, the flow satisfies the equations for potential flow with appropriate boundary conditions (see, for example, Kovks (1981))). The potential flow solution will specify the location of the capillary exposed face and the free exit face, if any.