Urban Flood modelling and Green Infrastructure

When Hurricane Harvey stalled over Houston in August 2017 and delivered 1,220 mm of rain over four days, the flooding that followed was catastrophic — but not entirely surprising to hydrologists who had been tracking what Houston had done to its landscape. Between 1992 and 2017, the Brays Bayou watershed had gone from roughly 15% impervious surface to more than 53%. Every new parking lot, highway ramp, rooftop, and cleared lot had taken land that once absorbed and delayed rainfall and turned it into a surface that shed water almost instantly. Researchers estimated that the development alone — independent of the rainfall — had increased peak flood flows by 20–30%. Harvey overwhelmed engineered defences designed for a different city.

The relationship between urbanisation and flooding is not mysterious; it follows directly from the physics of runoff generation. Rainfall that lands on soil infiltrates, evaporates, and moves slowly through the unsaturated zone. Rainfall that lands on concrete flows immediately to the nearest drain. The peak discharge from a post-development watershed can be three to five times higher than from the same area in its natural state, and it arrives sooner, leaving less time to evacuate or respond.

Urban flood modelling quantifies this relationship. The rational method translates impervious fraction, catchment area, and design rainfall intensity into a peak discharge number. The SCS Curve Number method tracks runoff volume across mixed land covers. Hydrograph routing follows that water pulse as it moves through pipe networks, detention basins, and channels. Green infrastructure — bioswales, permeable pavement, rain gardens — works by recovering some of the infiltration and storage that development destroyed. This model builds all of that machinery from first principles, starting with a single parking lot and working outward to a full urban watershed.

1. The Question

How much will this parking lot increase downstream flooding?

Urban development problem:

Natural landscapes → impervious surfaces

Consequences: - Increased runoff volume: 50-90% more water reaches streams - Higher peak flows: 2-5× pre-development magnitudes - Faster response: Time to peak reduced from hours to minutes - Reduced infiltration: Groundwater recharge eliminated - Water quality degradation: Pollutant loading (oil, nutrients, sediment) - Stream erosion: Channel incision from increased energy

Before development (forest/grassland): - Rainfall → Infiltration (60-80%) - Interception by vegetation (10-20%) - Runoff (10-30%)

After development (urban): - Rainfall → Infiltration (5-15%) - Runoff (70-90%) - Direct conveyance to streams

Regulatory requirement:

Zero-increase standard: Post-development peak ≤ pre-development peak

Achieved through: - Detention basins (temporary storage) - Retention basins (permanent storage + infiltration) - Green infrastructure (distributed controls)

Applications: - Site development permits - Stormwater management plans - Municipal separate storm sewer systems (MS4) - Total Maximum Daily Load (TMDL) compliance - Green building certification (LEED)

2. The Conceptual Model

Rational Method

Peak discharge estimation:

Q = C \times I \times A

Where: - Q = peak discharge (m³/s or cfs) - C = dimensionless runoff coefficient (0-1) - I = rainfall intensity (mm/hr or in/hr) - A = drainage area (ha or acres)

Unit conversion (SI):

Q \text{ (m}^3\text{/s)} = \frac{C \times I \text{ (mm/hr)} \times A \text{ (ha)}}{360}

Runoff coefficient C:

Represents fraction of rainfall that becomes runoff.

Physical meaning:

C = 0.1: 10% runoff, 90% infiltrates (forest)
C = 0.9: 90% runoff, 10% infiltrates (pavement)

Typical values:

Composite C:

For mixed land use:

C_{composite} = \frac{\sum C_i A_i}{\sum A_i}

Intensity I:

From IDF curves (Model 68) at duration = time of concentration.

Time of concentration (t_c):

Time for runoff from most distant point to reach outlet.

Kirpich equation:

t_c = 0.0078 L^{0.77} S^{-0.385}

Where: - t_c = time (min) - L = longest flow path (m) - S = slope (m/m)

Limitations: - Assumes uniform rainfall - Peak-only (no hydrograph) - Small watersheds (<80 ha typical) - Steady-state flow

SCS Curve Number Method

Runoff depth from rainfall depth:

Q = \frac{(P - I_a)^2}{P - I_a + S}

Where: - Q = runoff depth (mm) - P = rainfall depth (mm) - I_a = initial abstraction (interception, depression storage) - S = potential maximum retention (mm)

Standard assumption:

I_a = 0.2S

Simplified equation:

Q = \frac{(P - 0.2S)^2}{P + 0.8S}

Curve Number (CN):

S = \frac{25400}{CN} - 254 (mm)

Or: S = \frac{1000}{CN} - 10 (inches)

CN ranges: 0 to 100

CN = 100: Impervious (S = 0, all rainfall becomes runoff)
CN = 0: Infinite retention (Q = 0, no runoff)

Typical values:

Composite CN:

CN_{composite} = \frac{\sum CN_i A_i}{\sum A_i}

Advantages over Rational: - Runoff volume (not just peak) - Accounts for soil type - Antecedent moisture conditions

Unit Hydrograph Theory

Discharge over time (not just peak):

Unit hydrograph: Response to 1 unit (1 cm, 1 inch) of effective rainfall

SCS Dimensionless Unit Hydrograph:

Peak occurs at:

t_p = \frac{\Delta t}{2} + t_{lag}

Where: - \Delta t = rainfall duration - t_{lag} = 0.6 \times t_c

Peak discharge:

q_p = \frac{C \times A}{t_p}

Where C = 2.08 (SI units)

Hydrograph shape: Triangular or curvilinear (SCS dimensionless)

3. Building the Mathematical Model

Detention Basin Sizing

Objective:

Reduce post-development peak to ≤ pre-development peak.

Continuity equation:

\text{Inflow} - \text{Outflow} = \frac{dS}{dt}

Where S = storage volume

Simplified design:

V_{required} = (Q_{in} - Q_{out}) \times t_d

Where: - Q_{in} = post-development peak - Q_{out} = allowable release rate (= Q_{pre}) - t_d = detention time (typically $2-3 t_c$)

More accurate (routing):

Solve modified Puls method:

\frac{2S_2}{\Delta t} + O_2 = \frac{2S_1}{\Delta t} - O_1 + I_1 + I_2

Where: - S = storage - O = outflow - I = inflow - Subscripts 1, 2 = time steps

Stage-storage relationship:

V = A_{surface} \times d + \frac{1}{3}(A_{surface} - A_{bottom}) \times d

(Approximate basin as truncated pyramid)

Stage-discharge (orifice):

Q = C_d A_{orifice} \sqrt{2gh}

Where: - C_d \approx 0.6 (discharge coefficient) - A_{orifice} = outlet area (m²) - h = head (m)

Green Infrastructure Performance

Bioretention (rain garden):

Design equation:

A_{bio} = \frac{A_{imperv} \times P \times (1 - C)}{n \times d}

Where: - A_{imperv} = contributing impervious area - P = design storm depth - C = runoff coefficient of impervious area - n = media porosity (0.25-0.40) - d = media depth (typically 0.6-1.2 m)

Drawdown time:

t = \frac{n \times d}{K}

Where K = infiltration rate (typically 25-150 mm/hr)

Design criteria: t < 48 hours (mosquito breeding)

Permeable pavement:

Infiltration capacity:

Must exceed design storm intensity:

K_{pavement} > I_{design}

Typical: K = 200-1000 mm/hr (vs I_{10yr} = 50-100 mm/hr)

Void ratio: 15-40% (aggregate base)

Green roofs:

Retention capacity:

R = d_{media} \times n \times \rho_w

Where: - d_{media} = 50-200 mm - n = 0.30-0.45 - Results: 15-75 mm retention

Peak reduction: 50-90% for small storms (<25 mm)

4. Worked Example by Hand

Problem: Shopping center development and stormwater management

Site characteristics:

Pre-development: - Area: 5 hectares - Land use: Grass/pasture - Soil: Hydrologic group B (silty loam) - Slope: 2% - Longest flow path: 350 m

Post-development: - Building: 1.5 ha (30%) - Parking: 2.5 ha (50%) - Landscaping: 1.0 ha (20%)

Design storm: 10-year, IDF equation: I = \frac{1200 T^{0.2}}{(D+10)^{0.8}} (mm/hr)

Calculate: 1. Pre-development peak discharge 2. Post-development peak discharge 3. Required detention volume 4. Bioretention area needed

Solution

Step 1: Time of concentration (pre-development)

Kirpich equation:

t_c = 0.0078 \times 350^{0.77} \times 0.02^{-0.385}

= 0.0078 \times 54.1 \times 6.54 = 2.76 \text{ min}

Round up: t_c = 5 min (minimum practical)

Step 2: Design intensity (10-year, 5-min)

I = \frac{1200 \times 10^{0.2}}{(5+10)^{0.8}} = \frac{1200 \times 1.585}{15^{0.8}} = \frac{1902}{10.1} = 188 \text{ mm/hr}

Step 3: Pre-development peak (Rational Method)

Grass/pasture, soil B: C = 0.20

Q_{pre} = \frac{0.20 \times 188 \times 5}{360} = \frac{188}{360} = 0.522 \text{ m}^3\text{/s}

Step 4: Post-development composite C

Buildings (roof): C = 0.95
Parking (asphalt): C = 0.95
Landscaping: C = 0.25

C_{comp} = \frac{0.95 \times 1.5 + 0.95 \times 2.5 + 0.25 \times 1.0}{5}

= \frac{1.425 + 2.375 + 0.25}{5} = \frac{4.05}{5} = 0.81

Step 5: Post-development time of concentration

Paved surfaces reduce t_c:

Assume t_c = 8 min (sheet flow on pavement + pipe flow)

New intensity:

I = \frac{1200 \times 1.585}{(8+10)^{0.8}} = \frac{1902}{12.7} = 150 \text{ mm/hr}

Step 6: Post-development peak

Q_{post} = \frac{0.81 \times 150 \times 5}{360} = \frac{608}{360} = 1.69 \text{ m}^3\text{/s}

Step 7: Peak increase

\Delta Q = 1.69 - 0.52 = 1.17 \text{ m}^3\text{/s}

224% increase!

Step 8: Detention volume required

Detention time: t_d = 30 min (approximate, routing required for exactness)

V = (Q_{post} - Q_{pre}) \times t_d

V = (1.69 - 0.52) \times 30 \times 60 = 1.17 \times 1800 = 2106 \text{ m}^3

Required storage: 2100 m³

Basin dimensions (example):

Depth: 1.5 m
Bottom area: \frac{2100}{1.5 \times 1.5} = 933 m² (30 × 31 m)
Top area (with 3:1 side slopes): ~1200 m² (35 × 35 m)

Step 9: Bioretention alternative

Design to capture first 25 mm (water quality storm):

Impervious area: $1.5 + 2.5 = 4$ ha

A_{bio} = \frac{40000 \times 0.025 \times 0.9}{0.35 \times 0.9}

= \frac{900}{0.315} = 2857 \text{ m}^2

Bioretention area: 2860 m² (6% of impervious area)

Typically distributed as: - Parking lot islands: 1200 m² - Perimeter gardens: 1000 m² - Roof runoff planters: 660 m²

Step 10: Combined approach

Bioretention handles frequent storms (<25 mm).

Detention handles rare storms (10-year design).

Reduced detention: With bioretention pre-treatment, detention can be 20-30% smaller.

5. Computational Implementation

Below is an interactive urban hydrology calculator.

<label>
  Site area (ha):
  <input type="range" id="site-area" min="1" max="20" step="1" value="5">
  <span id="area-val">5</span>
</label>
<label>
  Imperviousness (%):
  <input type="range" id="imperv" min="10" max="95" step="5" value="70">
  <span id="imperv-val">70</span>
</label>
<label>
  Design storm (years):
  <input type="range" id="return-period" min="2" max="100" step="1" value="10">
  <span id="rp-val">10</span>
</label>
<label>
  Green infrastructure:
  <select id="gi-type">
    <option value="none">None</option>
    <option value="bioretention">Bioretention</option>
    <option value="permeable">Permeable pavement</option>
    <option value="greenroof">Green roof</option>
  </select>
</label>
<div class="hydro-info">
  <p><strong>Pre-dev Q:</strong> <span id="q-pre">--</span> m³/s</p>
  <p><strong>Post-dev Q:</strong> <span id="q-post">--</span> m³/s</p>
  <p><strong>Increase:</strong> <span id="increase">--</span>%</p>
  <p><strong>Detention volume:</strong> <span id="detention">--</span> m³</p>
</div>

Observations: - Post-development hydrograph peaks higher and faster - Green area (pre-dev) shows gradual response - Red area (post-dev) shows flashy response - Time to peak reduced by 30-50% - Total runoff volume increased significantly - Green infrastructure reduces both peak and volume - Detention storage needs proportional to shaded area between curves

Key insights: - Imperviousness directly controls runoff coefficient - Higher return periods amplify differences - Green infrastructure provides meaningful mitigation - Combined strategies most effective

6. Interpretation

Stream Channel Impacts

Geomorphic consequences:

Effective discharge (2-year flood) increased 2-5×.

Channel response: - Incision (downcutting) - Bank erosion - Bed coarsening (fines washed out) - Loss of pool-riffle structure

Timeline: Years to decades for equilibrium

Mitigation insufficient:

Traditional detention controls peaks but not duration.

Stream still receives elevated flows for longer periods.

Solution: Volume control + peak control

Water Quality Trading

Total Maximum Daily Load (TMDL):

Regulatory limit on pollutant loading.

Urban runoff major contributor: - Nutrients (N, P) - Sediment - Bacteria - Metals (Zn, Cu from brake pads) - Hydrocarbons

Green infrastructure provides treatment:

Bioretention removal: - Suspended solids: 80-90% - Phosphorus: 50-80% - Nitrogen: 30-60% - Metals: 60-90%

Permeable pavement: - Suspended solids: 70-95% - Metals: 50-80%

Cost comparison:

Traditional detention: $50-150/m³ storage
Bioretention: $200-400/m²
Permeable pavement: $25-75/m² premium over conventional

Benefits beyond storage: - Water quality treatment - Groundwater recharge - Urban heat island reduction - Aesthetics

Regulatory Evolution

Traditional approach (1980s-2000s):

Peak rate control only (Q_post ≤ Q_pre for design storm).

Modern approach (2010s+):

Multi-objective: 1. Peak rate control (flooding) 2. Volume reduction (stream protection) 3. Water quality treatment (pollution) 4. Groundwater recharge (base flow)

Performance standards:

Retain first 25 mm on-site (runoff reduction).

Treat 90% of average annual rainfall.

Green infrastructure mandates:

Some jurisdictions require GI for all new development.

Philadelphia: Green City, Clean Waters ($2 billion GI program)

7. What Could Go Wrong?

Detention Basin Failures

Outlet clogging:

Debris, sediment, ice block orifice.

Consequence: Overtopping, dam failure

Solution: - Trash racks - Multiple outlets (redundancy) - Regular maintenance

Improper sizing:

Used wrong storm (5-year vs 10-year).

Used wrong IDF curve (outdated).

Consequence: Insufficient storage, downstream flooding

Solution: Third-party peer review, as-built verification

Green Infrastructure Challenges

Soil clogging:

Bioretention media clogs over time (5-15 years).

Infiltration rate degrades: 50-90% reduction

Consequence: Ponding, bypassing, failure

Solution: - Pretreatment (grass filter, forebay) - Mulch replacement (annual) - Media replacement (every 10-20 years)

Winter performance:

Frozen soil reduces infiltration.

Snowmelt + frozen ground = maximum runoff.

Underdrains freeze.

Solution: - Deep frost-free outlets - Salt management (reduces freezing but damages plants)

Maintenance Neglect

Sediment accumulation:

Detention basins fill with sediment.

Volume reduced: 20-50% after 10 years (if unmaintained)

Solution: - Annual inspection - Sediment removal (5-10 year cycle) - Erosion control in watershed

Vegetation overgrowth:

Bioretention becomes overgrown, channelized.

Solution: - Annual weeding - Mulch refreshing - Plant replacement

8. Extension: Low Impact Development

Paradigm shift:

From “manage runoff” to “prevent runoff generation”

Principles: 1. Minimize impervious area 2. Maximize infiltration 3. Distribute controls throughout site 4. Mimic pre-development hydrology

Techniques:

Narrower streets: 6 m vs 9 m (40% less impervious)

Bioretention in parking islands: Treat runoff at source

Rain gardens: Residential-scale bioretention

Cisterns: Harvest roof runoff for irrigation (volume reduction)

Disconnected impervious: Route roof runoff to lawn (not street)

Performance:

Well-designed LID can achieve: - 80-95% runoff volume reduction (small storms) - 50-70% peak flow reduction - Near pre-development hydrology

Case study - Village Homes (Davis, CA):

Built 1975 with integrated LID.

40 years later: No downstream flooding, despite upstream urbanization.

9. Math Refresher: Mass Balance

Continuity Equation

Conservation of mass:

\text{Inflow} - \text{Outflow} = \frac{d(\text{Storage})}{dt}

Detention basin application:

I(t) - O(t) = \frac{dS}{dt}

Integrated over time step \Delta t:

\frac{I_1 + I_2}{2} \Delta t - \frac{O_1 + O_2}{2} \Delta t = S_2 - S_1

Rearranged (Modified Puls):

\frac{2S_2}{\Delta t} + O_2 = \left(\frac{2S_1}{\Delta t} - O_1\right) + (I_1 + I_2)

Solution procedure:

1. Known: S_1, O_1, I_1, I_2
2. Compute right side
3. Use stage-storage-discharge curves
4. Solve for S_2, O_2
5. Advance time step, repeat

Summary

Urban development increases runoff 2-5× through impervious surfaces requiring stormwater management via detention and green infrastructure. Rational method estimates peak discharge from runoff coefficient intensity and area with coefficients ranging 0.1 (forest) to 0.95 (pavement). SCS Curve Number method calculates runoff volume from rainfall via CN values 25-98 depending on land use and soil type. Detention basins sized to reduce post-development peaks to pre-development levels typically requiring 1000-5000 m³ for commercial sites. Green infrastructure including bioretention permeable pavement and green roofs provides distributed volume reduction and water quality treatment with 50-90% pollutant removal. Low-impact development principles minimize impervious area and maximize infiltration achieving near pre-development hydrology. Critical for flood mitigation stream protection and water quality compliance in urbanizing watersheds requiring integrated multi-objective design.