Page 220 - Soil and water contamination, 2nd edition

Substance transport 207

This Equation (11.19) says that the sum of the concentration changes in space and time
equals zero. Combining this equation with Equation (11.18) yields:
C 2 C
D (11.20)
t x 2

This is a second-order partial differential equation and is known as Fick’s second law. To
solve this Equation (11.20), one initial condition and two boundary conditions (one for each
order) are needed and for each set of boundary and initial conditions there is a solution. For
a pulse injection of a tracer or pollutant into a water body at t=0, we may pose the following
initial condition:
C(x) = 0 at t = 0 (11.21)

The boundary conditions are formulated so that the concentration integrated over the semi-
infinite length equals the mass M of the substance injected and the concentration at infinite
distance remains zero:
C(∞) = 0 for all t (11.22)

M A C dx (11.23)

2
where A = the cross-sectional area [L ]. The integration of the dispersion Equation (11.20)
may be achieved by trial and error methods and the analytical solution reads:
M x 2 /( 4 D
C (x , ) t e ) t x (11.24)
2 A D x t

where D = the longitudinal dispersion coefficient This Equation (11.24) describes the
.
x
substance concentration as function of space and time and may be rewritten as:
M x 2 /( 2 2 D
C (x , ) t e ) t x (11.25)
A 2D x t 2
This equation is analogous to the description of the ‘bell-shaped’ probability density curve of
the normal or normalised Gaussian distribution:
1 x 2 2 / 2
(x ) e (11.26)
2

where the arithmetic mean of φ(x) equals zero and σ = the standard deviation of φ(x). Thus,
Equation (11.25) describes the concentration distributions as function of time and distance
from the point at which the substance was injected and the concentration profile in the water
body is described by a Gaussian curve with a mean of x = 0 and a standard deviation of

2 D x t (11.27)

This implies that the centre of mass remains at the point of injection and the standard
deviation increases with time. Accordingly, the Gaussian curve becomes broader with time.

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