GROUNDWATER - Water Quality Engineering in Natural Systems - page 159

5.5 FATE PROCESSES

and taking α * as 0 1 .

α L

= 0 1 0 003 . ( . ) = 0 0003 . m , Equa-

tion (5.42) gives

Fate processes include all mechanisms that remove
tracer mass from the aqueous phase. These processes
include chemical reactions, decay, and sorption onto the
solid matrix. Sorption processes include adsorption ,
chemisorption , and absorption , where adsorption is the
process by which a solute attaches itself to a solid
surface, chemisorption occurs when the solute is incor-
porated onto a solid surface by ion exchange, and
absorption occurs when the solute diffuses into the solid
matrix and is sorbed onto the interior surfaces. Fate
processes in groundwater are complex and difficult to
study at the field scale, and are usually studied under
idealized laboratory conditions, with the results fitted to
idealized models. The most commonly considered fate
processes in groundwater are sorption and decay.

D
V

1 73 10
1 5 0 5

.
( . )( . )

×

− 4

m

*

α = α + = 0 0003 . + = 0 00053 . m

T T

τ

The principal components of the dispersion coeffi-
cient are then given by

2

D = ( α + α L ) V = ( . 1 9 0 0032 0 5 + . )( . ) = 0 95 . m /d

11 11

2

D = ( α + α T ) V = ( . 0 19 0 00053 0 5 + . )( . ) = 0 095 . m /d

22 22

2

D = ( α + α T ) V = ( . 0 019 0 00053 0 5 + . )( . ) = 0 0097 . m /d

33 33

It must be emphasized that these estimates of the
dispersion coefficient are order-of-magnitude estimates
only, and it would be entirely appropriate to take
D 11 = 1 m 2 /d, D 22 = 0.1 m 2 /d, and D 33 = 0.01 m 2 /d. The
dominance of macrodispersion over pore scale mechan-
ical dispersion and molecular diffusion in the longitudi-
nal and horizontal-transverse directions are evident.

5.5.1 Sorption

models that describe the partitioning of dissolved mass
onto solid surfaces are called sorption isotherms , since
they describe sorption at a constant temperature. The
most widely used sorption isotherm is the Freundlich
isotherm , given by

Field experiments to estimate local values of disper-
sivity at a particular site typically consist of releasing
dye from an injection well, measuring the breakthrough
dye concentrations at a downstream well, and estimat-
ing the dispersivity based on the rate of growth of
variance of the dye cloud or matching the measured
concentrations to a solution to the advection-diffusion
equation. These tests can be either natural gradient tests
or forced gradient tests . natural gradient tests are con-
ducted under natural flow conditions, and forced gradi-
ent tests are conducted under artificial (pumping) stress
conditions. Examples of forced gradient conditions
include converging radial flow, and flows induced by
placing an injection well upstream of a pumping well. It
is important to keep in mind that the dispersivities esti-
mated under different stress conditions tend to be dif-
ferent. Results reported by Tiedeman and Hsieh (2004)
show that among forced gradient tests, a converging
radial flow test tends to yield the smallest longitudinal
dispersivity ( α 11 ), an equal strength two-well test tends
to yield the largest α 11 , and an unequal strength two-well
test tends to yield an intermediate value of α 11 . Tiede-
man and Hsieh (2004) also showed that values of α 11
estimated under forced gradient conditions can signifi-
cantly underestimate α 11 under natural gradient flow
conditions. In support of this result, Chao et al. (2000)
reported that based on numerical simulations using
point sources, α 11 from radial flow tests were 5-10 times
smaller than α 11 for natural gradient tests.

F K c n

=

F aq

(5.43)

where F is the mass of tracer sorbed per unit mass of
solid phase (mm −1 ), c aq is the concentration of the tracer
dissolved in the water (aqueous concentration) (mL −3 ),
and K F and n are empirical constants. The constant, n , is
typically in the range of 0.7-1.2, and for many contami-
nants at low concentrations, n is approximately equal to
unity. In this case, the Freundlich isotherm, Equation
(5.43), is linear and is written in the form

F K c =

d aq

(5.44)

where K d is called the distribution coefficient (L 3 m −1 )
and is defined as the ratio of the sorbed tracer mass per
unit mass of solid matrix to the aqueous concentration.
Although the Freundlich isotherm is widely used in
practice, it has an important limitation in that it places
no limit on the sorption capacity of the solid matrix. The
Langmuir isotherm allows for a maximum sorption
capacity on the solid matrix and is defined by

K Sc
K c

l aq

F = (5.45)

1 +

l aq

where K l is the Langmuir constant (L 3 m −1 ) and S is
the maximum sorption capacity of the solid matrix
(mm −1 ). At low solute concentrations, when K 1 c aq << 1,