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 = (unitless) \[ R = 1 + \frac{ρ_bK_b}{n}\]

Step 3: Correct for void volume of spill site Cross-sectional area
A = m² Porosity
n = (unitless) Void volume
VV = 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 = mg \[ M_T = V * C\]

Total mass of pollutant
M_T = mCi

Step 5: Calculate first order rate constant

Half-Life

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

k = /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)		m²
Retardation factor (R)
Mass of contaminent (M_T)
First Order Rate Constant (k)		/year

Point calculation Distance
m Time
Years

Result:

Graph varying time Distance (fixed)
meters

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

Graph varying distance Time
Years

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