Issue 013 - Hydrology - Runoff rate
How big is a Texas Hill Country flood wave?
AP reported that a Guadalupe River location rose above the height of a two-story house in about five hours, while Reuters reported that parts of the Texas Hill Country had already received 10 to 15 inches of rain, with another 10 to 15 inches possible. Convert intense rainfall into river flow.
The problem
Estimate the peak river flow rate produced when 10 to 20 inches of rain falls over a steep Texas Hill Country watershed.
Is the resulting flood-wave flow closer to a large creek, the Mississippi River, or Niagara Falls?
Because Fermi problems target an order of magnitude, I normally use no more than two significant digits and write most calculations in scientific notation; the Fermi reference explains both conventions.
Before checking sources
Matt's first pass
The biggest weakness in my estimate is that I do not have a good concept of the watershed area. I worked with about 5 square kilometers, but any order-of-magnitude change in that area changes the answer by the same amount.
I also expected runoff percentage to change over time and with total rainfall. The first couple centimeters of rain might be almost entirely absorbed or otherwise not become runoff, while later rainfall on saturated ground might become mostly runoff. I used 20% uniformly across the rainfall volume.
Starting with a 5 km2 area, about 15 inches of rain, and a 5-hour runoff window:
watershed area ~= 5 km2
~= 5 x 10^6 m2
rainfall depth ~= 15 inches
~= 0.42 m
runoff time ~= 5 hours
~= 1.8 x 10^4 seconds
rain volume ~= (5 x 10^6 m2) x (0.42 m)
~= 2.1 x 10^6 m3
runoff volume ~= 20% x 2.1 x 10^6 m3
~= 4.2 x 10^5 m3
flow rate ~= (4.2 x 10^5 m3) / (1.8 x 10^4 s)
~= 23 m3/sThat is about the rate of a creek.
If the affected area is 10 times larger, runoff is about 230 m3/s, which is closer to a serious flood. If the runoff percentage is double that, the runoff rate doubles to about 460 m3/s.
Calibration Score
Matt's Calibration Score: 40 / 100
Higher is better: earn points for accurate pegs, sound models, correct math, and a result close to the sourced answer. The image shows percent full of it: 100 minus the Calibration Score.
Pegs: 0/30. Watershed area was off by orders of magnitude, which dominated everything.
Model: 30/30. Runoff volume divided by concentration time was the right hydrology model.
Math: 10/10. The arithmetic was clean.
Result: 0/30. The final flow estimate was far below the corrected flood-wave scale.
Grounding facts
Niagara Falls' high-season tourist flow is about 2,800 m3/s. The Mississippi River near New Orleans averages about 17,000 m3/s. A flood wave of a few thousand cubic meters per second is therefore not just a swollen creek; it is a regional-river torrent carrying Niagara-like flow through channels, crossings, trees, bridges, debris, and floodplains.
A simple area lesson: 1 km2 receiving 0.4 m of rain gets 4 x 10^5 m3 of water. Scale that to 100 km2, and the rain volume is already 4 x 10^7 m3 before runoff fraction or timing are considered.
After checking sources
Check and recalibrate
Matt correctly identified the dominant uncertainty: watershed area. A 5 km2 drainage area is more like a small local creek basin. A dangerous Hill Country river flood can collect water from tens to hundreds of square kilometers, and downstream reaches can reflect still larger drainage areas.
Keep the same structure, but use a broader watershed range:
rainfall depth ~= 10 to 20 inches
~= 0.25 to 0.50 m
rapid runoff fraction ~= 30% to 70%
runoff time ~= 3 to 8 hours
~= about 1 x 10^4 to 3 x 10^4 secondsFor a 100 km2 steep watershed:
middle case:
runoff volume ~= (1 x 10^8 m2) x (0.4 m) x (50%)
~= 2 x 10^7 m3
flow ~= (2 x 10^7 m3) / (1.8 x 10^4 s)
~= 1.1 x 10^3 m3/sFor a 500 km2 watershed under similar conditions:
runoff volume ~= (5 x 10^8 m2) x (0.4 m) x (50%)
~= 1 x 10^8 m3
flow ~= (1 x 10^8 m3) / (1.8 x 10^4 s)
~= 5.6 x 10^3 m3/sA reasonable Fermi bracket is therefore about 1,000 to 5,000 m3/s for a major flash-flood wave, with smaller tributaries below that and larger/wetter/faster-concentrating basins above it. Peak flow can be higher than the event-average estimate if the runoff arrives in a sharp pulse.
That puts the event far beyond creek-scale. It is plausibly Niagara-scale in the core flood wave, but generally below the lower Mississippi River's average flow unless an exceptionally large drainage area contributes quickly.
Post-check reflection
Matt's reflection
My biggest error was the watershed area. I really did not have a mental model for how big it should be, and it looks like I was off by several orders of magnitude, which skewed my answer badly.
I am not super surprised by the actual answer. Approaching Niagara-scale is what I might have expected just from videos on the news. I have been near and in large fast-moving bodies of water, so I already have a lot of respect for their scale and for how incredibly powerful they are as a consequence of that scale.
Digging in on this calculation, particularly on the size of the affected areas, reinforces that respect.
Recommended memory peg
Remember that 1 inch of rain over 1 square mile is about 65,000 m3 of water. For flood Fermi problems, use flow = watershed area x rain depth x runoff fraction / runoff time.
Reader results
Bars show how submitted estimates sort into the answer choices from the gut-check prompt.