# Fluid dynamics: emptying a chasm

Deep underground, there's a huge limestone cavern, roughly 50m x 100m. It's 250m deep, and completely filled with still water. There is a very small inlet, but the system is now effectively closed, so it appears to be a vast underground lake.

Now down at the bottom, there's an ancient stone door, 4m on a side. Using a magical bomb of sorts, the heroes blow this door off its hinges. On the other side is nothing to speak of -- the water can flow out freely and descend well below the bottom of the chasm/lake.

1) How long will it take for the lake to empty?

2) Will the emptying process accelerate over time, building up an increasing vortex, or remain basically steady? The water can tear the door-frames to bits as it races by.

3) What will this all look like, seen from the shore of the lake above?

4) The walls of the cavern are dressed elaborately -- think of the great gate of Petra, all the way around, halfway up the sides. What will happen to these sculptures as the water races out down below?

Note: If this belongs in some sort of fluid-dynamics SE, please direct me there. I figured it's such a broad hypothetical, especially the part about the Petra-like carved facades, that it belongs here.

• Is the door centered at the bottom of the lake, and is the lakebed level? Dec 19 '15 at 14:56
• Roughly level lakebed, yes. I hadn't realized the placement of the door mattered. I was imagining it centered in one long side. Would things change much if it were significantly closer to a short wall? Dec 19 '15 at 14:58
• It would only matter if the lakebed wasn't flat. For example, if the bed was like an inverse cone, and the door wasn't at the center, some of the water would not flow into it because of an elevation difference. Thanks for clarifying. Dec 19 '15 at 14:59
• Oh, I see. That makes sense. Dec 19 '15 at 15:00

I ran a model on a similar tank. I recomputed everything in terms of a circular tank and drain, keeping the areas equivalent. I used a slightly smaller drain area (based on my judgment) to account for the non-circular drain -- the non-circular tank will have little effect. Though I got an A in fluid mechanics, it was a long time ago - this is not a professional estimate.

My software only works for circular drains.

1) I got the result of 1430 minutes to drain the tank - just 10 minutes short of 24 hours.

Due to imperfect model and lack of detail, I would expect this to be accurate within an hour or so. One key assumption is that the access inlet above the pool is sufficiently large that airflow coming in is not restricted. Based on your description, this is very likely true.

There are plenty of variables that could affect the answer in the real world. The temperature of the water, the local gravity, salt or fresh water, condition of the drain hole (you mentioned bombs and ripping door frame - both are messy complications).

I found a place with an online tank drain calculator, but you can only pick standard pipe dimensions for the drain (24 inch max.)

2) The maximum flowrate is when the drain is first opened because the water pressure is the highest. As the basin drains the flow gets slower and slower.

3) There will be very little to see from the surface initially. Due to the depth of the water, there will be little obvious for quite a while - having never drained a pool this deep and large not sure just how long this period will last. Eventually you will see the circular motion and a vortex. The circular motion will not be noticable unless there is some stuff floating on the surface that makes the circulation motion apparent. Initially the vortex appears as no more than a very shallow "cone-shape" in the water. As the vortex becomes more intense, it is more obviously a vortex and you will see the characteristic whirlpool shape at its center. Later the vortex becomes more and more active as the level drops close to the drain point.

4) Anything on the side of the pool will be largely unaffected by the water motion as the water only reaches significant speed when in gets fairly close to the vortex. You can safely swim in the pool for most of the drainage time as long as you remain a reasonable distance away from the vortex.

• I have a similar background but wasn't planning on putting in the work to do the actual calculations. Thank you so much for doing this! Dec 19 '15 at 18:00
• @Jim2B Not too hard when you have an old spreadsheet built for the purpose. All I had to do was convert areas and units to match the spreadsheet. I actually keep a worldbuilder directory with a number of spreadsheets that I build when a problem interests me. Dec 19 '15 at 18:21
• so do I but mine's more around planet formation and orbital dynamics :) Dec 20 '15 at 1:07
• Wouldn't people on the surface see bubbles emerging and popping as the air from the chamber below escapes? Also, if the chamber below the stone door contained a vacuum, would that speed up the emptying process? Dec 20 '15 at 6:36
• @CAgrippa -- At this level of detail, I ignored everything but the area of the drain and the fact that it was square. If the rushing water erodes out a larger drain area, the basin would drain more quickly. So, it could be a real world effect. Given the initial pressure and flow rate, this is a definite issue unless the drain is solidly constructed. Given time and pressure, water can eat through reinforced concrete or even steel. If not strongly constructed the drain could collapse in a cascade failure just like a dam collapse. Dec 20 '15 at 9:57