Stream Reaeration Coefficient: Calculation Methods in QUAL2K

Reaeration is the primary source of dissolved oxygen in rivers, and a critical term in the dissolved oxygen balance. It describes the rate at which atmospheric oxygen transfers across the water surface into the stream, as originally characterized by O'Connor and Dobbins (1958). The reaeration coefficient $k_a$ (day⁻¹) governs the first-order rate of oxygen exchange:

\text{Reaeration flux} = k_a \cdot (DO_{sat} - DO)

where $DO_{sat}$ is the dissolved oxygen saturation concentration and $DO$ is the current DO in the stream.

O'Connor-Dobbins Formula (Default)

QUAL2K uses the O'Connor-Dobbins formula as the default reaeration model. This is the most widely used empirical formula for natural streams:

k_a = 3.93 \cdot \frac{U^{0.5}}{H^{1.5}}

Where:

$U$ — stream velocity (m/s)
$H$ — stream depth (m)
$k_a$ — reaeration coefficient at 20°C (day⁻¹)

Alternative Reaeration Formulas

Empirical reaeration formulas

Formula	Equation	Best For
O'Connor-Dobbins (1958)	ka = 3.93 · U⁰·⁵ / H¹·⁵	Deep, slow rivers (H > 0.6 m)
Churchill et al. (1962)	ka = 5.026 · U⁰·⁹⁶⁹ / H¹·⁶⁷³	Large rivers with dams
Owens-Gibbs (1964)	ka = 5.32 · U⁰·⁶⁷ / H¹·⁸⁵	Shallow streams (H < 0.6 m)
Tsivoglou-Neal (1976)	ka = 31,183 · U · S	Streams with known slope S
User-specified	ka = value	Custom / laboratory-measured rates

Temperature Correction

All reaeration rates are corrected for temperature using the Arrhenius equation:

k_a(T) = k_a(20) \cdot \theta^{(T - 20)}

where $\theta = 1.024$ for reaeration (Elmore and West, 1961). At 15°C, the correction factor is $1.024^{-5} \approx 0.888$ , reducing $k_a$ by about 11%.

Dissolved Oxygen Saturation

The DO saturation concentration decreases with temperature and altitude. QUAL2K uses the APHA formula (Standard Methods, 23rd Edition):

\ln(DO_{sat}) = -139.3441 + \frac{1.5757 \times 10^5}{T_K} - \frac{6.6423 \times 10^7}{T_K^2} + \frac{1.2438 \times 10^{10}}{T_K^3} - \frac{8.6219 \times 10^{11}}{T_K^4}

where $T_K = T + 273.15$ is the absolute temperature in Kelvin. An altitude correction factor is applied:

DO_{sat,elev} = DO_{sat} \cdot \left(1 - 0.0001148 \cdot elevation \right)

Physical Mechanism

Figure 1: Oxygen flux at the air-water interface

Python Implementation

pythonhydraulics.py — Reaeration calculation

# Reaeration ka (1/d) - O'Connor-Dobbins
if h > 0:
    ka = 3.93 * (u ** 0.5) / (h ** 1.5)
else:
    ka = 0

pythonkinetics.py — DO saturation with altitude correction

def calculate_dosat(temp, elev):
    """Calculate DO saturation (mg/L) using APHA formula."""
    tk = temp + 273.15
    ln_dos = (-139.34411 + 1.575701e5/tk
              - 6.642308e7/tk**2
              + 1.243800e10/tk**3
              - 8.621949e11/tk**4)
    dos = math.exp(ln_dos)
    # Altitude correction
    dos *= (1 - 0.0001148 * elev)
    return max(dos, 0.0)