# Must flow be supersonic for disturbances not to affect upstream?

I'm studying oil production and found a fact that puzzled me. It states that fluid flow downstream of the wellhead must be supercritical in order not to disturb the flow upstream of it. From PetroWiki:

A wellhead choke controls the surface pressure and production rate from a well. Chokes usually are selected so that fluctuations in the line pressure downstream of the choke have no effect on the production rate. This requires that flow through the choke be at critical flow conditions.

So I learned that (super)critical flow is when the Froude number is $\ge1$, and, according to multiple sources, the Froude number is the speed of the flow divided by some characteristic speed, which varies from case to case:

• Thermopedia seems to suggest this is the speed of sound:

A choked plane forms at this location, and further reductions in downstream pressure have no effect on conditions upstream as the rarefaction waves travel at the local sound speed and are stalled at the choked plane.

• Wikipedia seems to suggest this is the group speed of some disturbance (like ripple):

Information travels at the wave velocity. This is the velocity at which waves travel outwards from a pebble thrown into a lake.

I tend to believe more in the latter, since I find it highly unlikely that oil travels at over 1500 m/s in the piping, but I'm confused.

So, what must be the speed of the oil in the piping? The speed of sound (~1500m/s), the speed of some kind of ripple (1-10 m/s -- much more reasonable) or something else entirely?

• According to Wikipedia, oil pipelines flow between 1 and 6 m/s. Note that you can define the characteristic velocity as characteristic length divided by the characteristic time. – Kyle Kanos Dec 1 '14 at 20:26
• @KyleKanos thanks. Like you said in that answer, I also find it strange that $c=sqrt(gH)$ and I think it doesn't make sense for pipe flow. Also, I wouldn't know what would be a characteristic time for this scenario. – André Chalella Dec 2 '14 at 1:51
• Slight correction: using $c=L/\tau$ instead of $c=\sqrt{gH}$ for shallow waves confounds me. I am perfectly comfortable with the latter--note that this does not apply to your problem here because you don't have gravity waves. – Kyle Kanos Dec 2 '14 at 1:57

The sound velocity depends on the sound frequency (dispersion). The flow must be locally faster than the frequency of the downstream disturbances. If the latter are such that their sound velocity is small, the local flow velocity may be chosen small too.

Note, that the sound velocity depends also on the void fraction. If there are bubbles (even locally - due to local depressurization (cavitation)), the critical or sound velocity will be smaller.

• Is it the phase or group velocity of the disturbances that must be exceeded by the local flow speed? Also, are all the individual phase velocities approximately equal to the "official" speed of sound in the medium? – André Chalella Dec 2 '14 at 1:59
• I do not know about oil properties, so I cannot answer. And I would compare propagation of a wave front with the local velocity. – Vladimir Kalitvianski Dec 2 '14 at 9:51
• Forgive me, but I don't have a solid background in wave theory. A quick search led me to an article (goo.gl/sJ1u6V) that led me to believe that wavefront speeds for a valve closing in voidless water flow would be close to the speed of sound (he finds ~1432m/s). With voids, however, this could be much less, I suppose. – André Chalella Dec 2 '14 at 14:56

Seventeen months later, I found the answer. What I described in the question is called choked flow.

It means that, in the choke, fluid speed must be supersonic in order to "cut" disturbances coming from downstream. Ergo, fluid speed doesn't need to be supersonic in the entire pipeline.

Also, it is achieved only for pure gas or multiphase (oil + gas) flow, never for pure oil flow, which is what baffled me initially.