Advection Dispersion equation

The solution of the advection dispersion equation according to Leibundgut et al. (2009) is:

$$ c_(t) = \frac{M}{Q} \cdot \frac{x}{\sqrt{4 \cdot pi \cdot D \cdot t^3}} \cdot exp\left[-\frac{(x-v \cdot t)^2}{4 \cdot D \cdot v \cdot t}\right] $$

with the dispersion coefficient:

$$ D = v \cdot \alpha = v \cdot f_D * x $$

and with the variables $Q(t)$ discharge in $m^3/s$, distance $x$ in $m$, time $t$ in seconds, velocity $v$ in m/s. $f_D$ is the scale factor.

The advection dispersion equation written and plotted in Python reads:

import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt
 
M = 1 # mass injected kg
Q = 1 # discharge m³/s
v = 0.1 # flow velocity
D = 1000 # dispersion coefficient m²/s
x = 10000 # distance
# time 
t = np.linspace(0.1,86400,60)
c = M/Q*x/(4*np.pi*D*t**3)**(.5)*np.exp(-(x-v*t)**2/(4*D*t))
plt.plot(t,c)