Jan 5, 2008

Rainy Day Math Fun

Well, wasn't that refreshing?! Nothing like two inches of solid rain to wake up the winter, and more is coming down today. It's the perfect test of my vernal pool, which began filling around 8 a.m. yesterday and ultimately filled to about 6" below capacity within 12 hours. The water comes primarily from our roof, with about 2/3 of its downspouts tied into a central outlet at the mouth of the pool. (The other downspouts let out onto the permeable driveway, so there's some percolation, although I intend to tie those into the pool as well.) There also was some water siphoned in from drainage problem areas in the back yard, but probably less than 20 gallons.

A couple of things were really amazing to me. The first is that my math was more or less right! Assuming a maximum daily rainfall of about 1 inch (not quite the record, but also above the norm), I wanted the pool to hold all the runoff from my 1200 square foot roof. Ready for some algebra? Follow along:
    1200 square feet
    * (1 inch of runoff ÷ (12 inches/foot))
    100 cubic feet of runoff capacity needed

So I built the pool's surface area to about 110 square feet, at an average depth of about 12":
    110 square feet
    * (12 inches ÷ (12 inches/foot))
    110 cubic feet capacity

Sure enough, this storm dumped 2 inches of rain; and again, the pool collects about 66% of the roof area:
    (1200 sq. ft. * 66%) = 800 sq. ft.
    * (2 inches ÷ (12 in./ft.))
    133 cubic feet of runoff

But wait! Why didn't the pool overflow? Ah, let's not forget about the soil's absorption capacity. My soil isn't completely compacted, so it should still percolate at a rate of at least .75 inch per hour (not that I've actually tested it yet). We don't know the rainfall rates per hour; we do know the day's average was 2" over 24 hours, so let's assume it fluctuated between 1/16" and 1/2" per hour. The pool was receiving those rates (depths) of rain across its 110 sq. ft. area, as well as the 800 sq. ft. of roof.
    910 sq. ft. total surface area

To determine what percentage of "its own" area the pool was receiving:
    910 sq. ft. total area
    ÷ 110 sq. ft. pool area

The pool was receiving 8.27 times, or 827%, of its own area. Multiply that by the rate (depth) of rain to find out what volume it was taking on:
    8.27 * 1/16" per hour = .52" per hour (assumed minimum flow into pool)
    8.27 * 1/2" per hour = 4.14" per hour (assumed maximum flow into pool)

This means at the peak periods, precipitation was outpacing percolation by about 4 inches per hour. That's why the pool filled as much as it did. But that was a small minority of the time: more often, percolation outpaced precipitation by .25 per hour... and that's why the pool never overflowed.

(Knowing that the pool takes on 8.27x its area, we can also determine the "equilibrium" rainfall rate:
    .75"/hour pool percolation rate
    ÷ 8.27
    0.91"/hour precipitation rate)

(By the way, the second amazing thing to me was that, after about 12 hours of no rain, the pool had no water this morning. Let's see what the fill/percolation rates are today, now that the soil is saturated.)

This kind of math is what landscape designers and architects have to do all the time in designing drainage systems, water retention basins (like vernal pools), even landscape irrigation systems. It's a lot more complex than just this algebra, but I hope it shows a little bit of the "behind the scenes" work we do — it's not all just pretty pictures.

If you've hung on this long, you're probably doing the math yourself. Please correct me if I've missed something, or feel free to ask questions. It's been a fun rainy-day diversion.

PS: This whole episode has alerted me that I might consider renaming my little water feature. Technically, a vernal pool has the geographic feature of a semi-imperpeable "pan" beneath the grade which slows drainage throughout the winter and spring. Mine, obviously, drains fairly readily; so I suspect rain garden — although becoming nauseatingly trite — may be a better description. Stay tuned.

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