Page 209 - Soil and water contamination, 2nd edition

196 Soil and Water Contamination

Box 11.II Numerical solution to differential equation (11.7): Simple Euler’s method

If we discretise time in time steps of Δt, then:
t t i 1 t (11.IIa)
i
The reservoir lake water level at time step t is:
i
dH
H( t ) H( t ) t (11.IIb)
i
i 1
dt
Combining Equation (IIb) with Equation 11.7 gives:
Q in
H t ( ) H t ( ) - kH(t ) t (11.IIc)
i i 1 1 - i
A
By means of this simple numerical integration method the lake level at time-step t can
i
be explicitly calculated from the lake level at time step t using a linear estimate of the
i-1
rate at which the lake level is falling at time step t . Note that this Euler’s method is
i-1
only a simple numerical method with a considerable numerical error that can be reduced
by using small time-step of size Δt. Too large values for Δt may result in numerical
instability: for example, because the fall in the lake level during one time-step becomes
larger than possible rise due to water inflow plus the lake water level itself.
For other, more sophisticated numerical methods, see textbooks on numerical methods,
such as Press et al. (1992).

the contrasting chemical behaviour in soil, groundwater, and surface water dealt with in the
subsequent chapters.

11.2 ADVECTION

11.2.1 Advection equation
Advection is the process of transporting substances in solution or suspension with the
movement of the medium (usually water). Water flows due to gravity forces and is retarded
by internal friction (viscosity ) and friction at the contact between the moving water and the
sediment over or through which it flows. The physics of water movement are dealt with in
many hydrological textbooks (e.g. Chow et al., 1988; Domenico and Schwarz, 1998; Ward
and Robinson, 2000; Brutsaert, 2005) and are summarised in Boxes 11.III and 11.IV. If a
substance is brought into flowing water, it is transported in the same direction and at the
same velocity as the water. The change in the mass of a substance in a control volume over
time due to advection equals the difference between mass inflow and mass outflow (see
Figure 11.1) and may be written as the following difference equation:

( VC ( QC) t (11.11)
)
-3
3
where V = the volume of water in the control volume [L ], C = concentration [M L ], and
-1
3
Q = discharge [L T ]. The negative sign on the right-hand side of the equation depicts an
increasing concentration within the control volume if the mass inflow is greater than the

10/1/2013 6:44:47 PM
Soil and Water.indd 208 10/1/2013 6:44:47 PM
Soil and Water.indd 208