3.5 Diffusion
A concentration difference between two points in a stagnant solution will be leveled out in the
time by the random Brownian movement of molecules. The process is called molecular diffusion
and the movement of molecules by diffusion is described by Fick’s laws. Fick’s first law relates
the flux of a chemical to the concentration gradient in Equation (3.34). Diffusion changes the
concentraton at a given x and the flux is therefore time dependent. The Fick’s second law of
diffusion was also shown in Equaion (3.38).
F  D
c
x
(3.34)
c
 2c
D 2
t
x
(3.38)
(Source: http://epswww.unm.edu)
3.5.1 Diffusion coefficient
For solute ions the diffusion coefficient, D, will be related to their mobility, u, as derived from
the molar electrical conductivity in m/s per V/m (m2/s/V) given in table 1.7 and determined by
using Equation (3.41).
D  u  kT (z  q e )
(3.41)
Temperature dependance of diffusion coefficient was obtained using Equation (3.42)
D f ,T  D f , 298  T  η298 (298  ηT )
(3.42)
For the ionic mobility, it was decreased toward higher concentration. Moreover, the diffusion
coefficients decreased slightly with increasing concentration for simple salt solutions.
In case of neutral organic molecules, the diffusion coefficient can be estimated with Equation
(3.43)
D f , 298  2.8  10 9 Vl
0.71
(3.43)
Compared to diffusion in free water, solutes diffusing through a sediment-water system must
travel an extra distance because they have to circumnavigate the sediment grains. The effective
diffusion coefficient (De) corrected for the additional pathway, which related to the tortuosity
(defined as the length of the actual travel path taken by a solute in a porous medium devided by
the straight line distance).
De  Df θ 2
(3.44)
Or D e  D f  ε w
(3.47)
When applying diffusion coefficient for free water to sediment-water systems, one should
remember that only the water filled porosity, ε w , participates in the diffusive flux.
3.5.2 Diffusion as a random process
Fick’s second law in Equation (3.38) gives the change of c in space (x) and time (t). By
differentiation with respect to t and x, we can verify the integraton of Equation (3.38) yields a
solution such as:
c  At
1
2
  x2 

exp 
 4Dt 
(3.48)
Mass conservation of solute at any time t requires:
N  At
1

2
  x2 
exp
  4Dt dx
(3.49)
Where A  N /( 4πD)1 / 2 , finally n (x) of any solute can be obtained as follow
n(x) 
  (x  x 0 ) 2 

exp 
2
2
σ
2πσ 2


N
(3.54)
Where σ  2Dt for one dimension
3.5.3 Diffusive transport
Table 3.4 shows results for different time periods, and for comparison also the traveled distance,
L, during the same time, by an advective flow of 10 m/year. It was found that over short time
periods diffusive transport equals advective transport, but over longer time periods (and hence
also over larger distance) advective flow becomes more and more important.
Another integrated form of Fick’s second law can be written as:
 x 
 (3.58)
c( x, t )  c i  (c 0  c i )erfc 
 4D t 
e 

3.5.4 Isotope diffusion
(Considering page 96 – 99)
3.6 Dispersion
Groundwater flowing through a sand layer is forced to move around the sediment grains. The
resulting spreadig of a concenration front is called dispersion. There are two types of dispersion;
differences in travel time along flowlines around grains cause longitudinal dispersion (D L),
whereas transverse dispersion (DT) is due to stepover onto adjacent flowlines by diffusion.
We can still use Fick’s laws to quantify the spreading of concentration fronts. It is also called
ARD equation.
c
c q
 2c
 ν

 DL 2
t
x t
x
(3.62)
The dispersion can be calculated using Equation (3.53) with the difference that x0, the initial
location of the point source, is not zero, as for diffusion, but increases with the distance covered
bythe moving fluid, i.e. x0 = vt.
3.6.1 Column breakthrough curves
When a conservative tracer is injected into a column at concentration c = c0, the front will move
with the average water flow velocity throgh the column. At the same time the front disperses.
The concentration are found by integrating Equation (3.62) for dq = 0 which becomes:
  2c 
 c 
 c 
   ν   D L  2 
 t  x
 x  t
 x  t
(3.64)
Finally, we will get Equation (3.65), which for an infinitely long column,    x   , and for
resident concentration, cr, within the column.
c( x , t )  c i 
 x  vt 
(c 0  c i )
 (3.65)
erfc 
 4D t 
2
L 

If we instead consider a semi-infinite column from x = 0 o x =  , and define a constant
concentration boundary at x = 0, the concentration in the column are:
c r ( x, t )  c i 
 x  vt 
(c 0  c i ) 
  exp  vx
erfc 
D
 4D t 
2

 L
L 

3.6.2 Dispersion coefficients and disperivity
The Peclet number was defined as Pe  vd / Df
 x  vt 


erfc 


4
D
t

L 


(3.66)
3.6.3 Macrodispersivity
The larger spreading in aquifers, compared to laboratory columns, is due to the heterogenous
structure of natural sediments with alternating sands of different hydraulic conductivity, and with
intercalated layers of clay, silt or gravel. It is no longer the individual grain size, but rather the
variation in flow length around low conductivity bodies that determines the dispersivity. This
effect is known as macrodispersivity.