\[ \newcommand\si[1]{\mathrm{#1}} \newcommand\SI[2]{#1\,\si{#2}} \newcommand\matr[1]{\mathbf{#1}} \DeclareMathOperator*{\argmax}{arg\,max} \DeclareMathOperator*{\argmin}{arg\,min} \]

# The rational method

## 1 The rational method

\[Q=CIA\]

- $Q$: Runoff discharge (volume per time)
- $C$: Runoff coefficient (unitless)
- $I$: Rainfall intensity (depth per time)
- $A$: Watershed area (area)

See also hydrologic cycle and development.

### 1.1 Watershed with a single land cover

There is a small grassy watershed whose area is $10{,}000\,\si{m}^2$. It has lawn on the average slope above heavy soils. Last night, there was a storm event with a rainfall intensity of 20 mm/hr. What is the estimated runoff discharge from this storm event in $\si{m^3/hr}$?

- Find the range of C for “lawn on the average slope above heavy soils.” In other words, that’s “lawns; heavy soils; average, 2%–7% slopes.” Its range of C is 0.18–0.22.
- Take the average of the minimum 0.18 and maximum 0.22: $C=\frac{0.18+0.22}{2}=0.2$.
- Convert the rainfall intensity to m/hr: $I=20\,\si{mm/hr}\times 0.001\,\si{m/mm}=0.02\,\si{m/hr}$.
- Its area is given: $A=10{,}000\,\si{m^2}$.
- Calculate the runoff discharge: $Q=C\times I\times A=0.2\times 0.02\,\si{m/hr}\times 10{,}000\,\si{m^2}=40\,\si{m^3/hr}$.

### 1.2 Watershed with multiple land covers

You are an environmental consultant who is responsible for estimating the total runoff discharge from a planned shopping mall. The total area of the shopping mall is 10 acres. It consists of 60% building footprint with concrete roofs, 20% asphalt roads and parking lots, 10% lawns over flat sandy soils, 5% walkways, and 5% playground area. Your design storm intensity is 20 in/hr. What is your estimated runoff discharge in $\si{ft^3/hr}$?

- Calculate five areas: 60%, 20%, 10%, 5%, and 5% of $435,600\,\si{ft^2}$ (10 acres)
- Find the average runoff coefficients for the five land covers.
- Convert the rainfall intensity to ft/hr.
- Calculate five individual runoff discharges in $\si{ft^3/hr}$.
- Add the five runoff discharges to obtain the total runoff discharge.
- The total runoff discharge is $Q=0.751\times 1.667\times 435{,}600=545{,}517\,\si{ft^3/hr}$, which is huge.
- Convert the total runoff discharge to cfs ($\si{ft^3/s}$) by dividing it by 3,600 because there are 3,600 seconds in an hour: $Q=152\,\si{cfs}$.

### 1.3 Composite runoff coefficient

\[C_w=\frac{\sum_{i=1}^n C_iA_i}{\sum_{i=1}^n A_i}\]

- $n$: Number of land covers inside the watershed
- $A_i$: Area of land cover $i$
- $C_i$: Runoff coefficient of land cover $i$
- $C_w$: Composite runoff coefficient

## 2 Homework: Composite runoff coefficient

We are designing a mini city whose watershed area is 50 mi^{2}. We want to build the following components inside the city:

- downtown areas (5 mi
^{2}) - one industrial park (10 mi
^{2}, mostly concrete) - suburban residential areas (15 mi
^{2}) - playgrounds (2 mi
^{2}) - parks (10 mi
^{2}) - drives and walks (5 mi
^{2}) - unimproved areas (3 mi
^{2})

The rainfall intensity is 1.3 inches per hour. Calculate the composite runoff coefficient and runoff discharge in cubic feet per second (cfs). Use the average runoff coefficient for each land cover. Show your work for full credits.