Instantaneous Release into Groundwater

This section of Fate allows you to map the passage of a pollutant in groundwater. It models the migration of the pollutant through an aquifer over time or can determine how long it will take a pollutant to reach a given point, such as a river. In this situation, the pollutant may come from a spill or other one-time release such as a tanker truck accident.

Step 1 Manually convert input data to metric units: meters, kilograms, or curies.

Step 2: Calculate the retardation factor from the distribution coefficient Bulk Density
ρ_b = g/cm³ Distribution Coefficient
K_d = cm³/g Porosity
n = (unitless) Retardation factor
R = 14.333 (unitless)

$R = 1 + \frac{ρ_bK_b}{n}$

Step 3: Correct for void volume of spill site Cross-sectional area
A = m² Porosity
n = 0.3 (unitless) Void volume
VV = 15 m²

$VV= A * n$

Step 4: Calculate the total mass of the pollutant spilled Pollutant is a

Volume of solution spilled
V = m³ Concentration of solution
C = mg/m³ Total mass of pollutant
M_T = 100,000 mg

$M_T = V * C$

Step 5: Calculate first order rate constant

Half-Life

$ln(\frac{C}{C_o}) = -kt_\frac{1}{2}$

k = 0.693/year

Step 6: Enter velocity and dispersion coefficient Groundwater velocity
v = m/year Dispersion Coefficient
d = m²/year

Step 7: Verify data

Data Point	Value	Unit
Void volume (VV)	15	m²
Retardation factor (R)	14.333
Mass of contaminent (M_T)	100,000	mg
First Order Rate Constant (k)	0.693	/year

Point calculation Distance
m Time
Years

Result: 0.15 mg/L

Graph varying time Distance (fixed)
meters

$C_{(x,t)} = \frac{M}{A\sqrt{4\pi \frac{D}{R}t}}e^{-\frac{(x-\frac{v}{R}t)^2}{4\frac{D}{R}t}-kt}$

0,0

Graph varying distance Time
Years

$C_{(x,t)} = \frac{M}{A\sqrt{4\pi \frac{D}{R}t}}e^{-\frac{(x-\frac{v}{R}t)^2}{4\frac{D}{R}t}-kt}$