Page 207 - Soil and water contamination, 2nd edition

194 Soil and Water Contamination

The actual lake water depth is greater than this equilibrium depth, which implies that to
achieve equilibrium the lake water needs to drop.

b.
The response time (= 1/k = 50 000 s = 13.9 h) gives the time needed to reduce the
difference between the actual and equilibrium lake water depths by 63 percent (see
Section 10.1). Thus, after 13.9 hours the lake water depth is

H ( 13 9 . h ) 5 . 0 63 5 ( . 4 75 ) . 4 84 m

This can also be calculated by using Equation (11.8):
3.8 3.8 2 10 5 50 000
H ( 50 000 s ) 5 - 4 5 5 - 4 e
2 10 4 10 2 10 4 10

. 4 75 . 0 25 . 0 37 . 4 84 m

This is indeed the same result as in the above calculation.

c.
Use Equation (11.10) to calculate what the discharge Q would be if H = 5 m:
in eq
Q in
H 5 m
eq 5 - 4
2 10 4 10
3
4
Q in 5 2 10 -5 4 10 4 m s -1
In more complex systems, for example when we simulate a cascade of reservoir lakes with
different values for parameter k, which vary in time, and an upstream boundary condition
Q that varies in time, finding an analytical solution may be very complex or even
in
impossible. In such cases, a numerical solution of the differential equation must be found
by numerical integration, which involves discretisation in space and time (i.e. dividing space
and time into discrete units) and local linearisation of the problem at the level of the discrete
units. Box 11.II illustrates a simple numerical solution technique for the Equation (11.7). In
cases in which the parameter k is not constant in time, different values of k may be applied
for the different time-steps; this is an important advantage of numerical modelling over
analytical solutions.
The transport of substances in soil and water is principally governed by two kinds of
physical processes: advection , i.e. bulk movement with water from one location to another
(also referred to as convection), and dispersion , i.e. random or seemingly random mixing
with the water. Non-aqueous phase liquids (NAPLs ) can move independently from water
flow, due to density flow. Chemicals may also be transported in the gas phase or transported
biologically by moving organisms, such as the swimming of contaminated fish or the
burrowing of contaminated soil fauna. Furthermore, substances can undergo chemical or
physical transformation . A chemical transport model should take account of the whole
body of these processes and their complex dependencies on different environmental factors.
Because the model parameters vary spatially and temporally, the differential equations
describing chemical transport can rarely be solved analytically; instead, the equations are
usually solved using numerical techniques.

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