Page 229 - Soil and water contamination, 2nd edition

216 Soil and Water Contamination
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C (x , y ,t ) (C (x , y ,t ) 1 V (x , y ,t ) 1
V (x , y ,t )
C (x , 1 y ,t ) 1 u (x , 1 y ,t ) 1 H (x , 1 y ,t ) 1 y t
x
C (x , y ,t ) 1 u (x , y ,t ) 1 H (x , y ,t ) 1 y ) t
x
(11.42)
3
-3
-1
where C = concentration [M L ], V = volume [L ], u = flow velocity [L T ], H = water
x
depth [L], Δy = flow width, i.e. grid cell size [L], and Δt = time-step size [T]. The first term
between brackets represents the mass present in the grid cell in the previous time step and
the second and third terms represent respectively the mass input from the upstream cell and
the mass outflow to the downstream cell. Because the flow field (i.e. V, H, and u ) is constant
x
over space and time and V = H⋅Δx⋅Δy, we may simplify Equation (11.42) to:
u x t
C (x , y ,t ) C (x , y ,t ) 1 (C (x , 1 y ,t ) 1 C (x , y ,t 1 )) (11.43)
x
-1
If we assume a block front of a concentration of 1 mg l entering the system with an initial
-1
-1
concentration of 0 mg l on the left-hand side (x = 0) at t = 0 with a velocity of 0.1 m s
in direction x, it should take 10 seconds before the front of the concentration reaches 1 m.
(Note that we are considering advective transport only.) However, if we discretise the system
with a grid cell size of 1 m and a time step of 1 s, Equation (11.43) shows that after 1 s (one
-1
time-step), the concentration at x = 1 m has already increased to 0.1 mg l . After 150 s, the
front should have moved 15 m, but the results of the simple GIS -based model shows that
the block front is considerably blurred (Figures 11.11a and 11.11b). In two dimensions, a
similar effect can be observed if we consider an instantaneous release of a substance from a
local point source in the upper left corner of the system, which is transported in east/south-
eastern direction. In this case, the concentration spreads in two directions (Figures 11.11c
and 11.11d).
To minimise the undesired effects of numerical dispersion , it is often better to define
a volume of space containing a fixed mass of fluid and to let the boundaries of these cells
move in response to the dynamics of the fluid. The differential equation is transformed into
a form in which the variables are the positions of the boundaries of the cells rather than
the quantity or concentration of a substance in each cell. These methods are known as
Lagrangian methods or particle tracking methods. A well-known example of such a method
is the method of characteristics (MOC) for solute transport in groundwater (Konikow and
Bredehoeft, 1978). Thonon et al. (2007) have implemented this method in a GIS -based
model that simulates sediment transport and deposition over a floodplain .
The first step in the method of characteristics involves placing a swarm of traceable
particles or points in each grid cell, resulting in a uniformly distributed set of points over
the area of interest. Usually, four or nine particles are used per grid cell. An initial mass is
assigned to each particle; it depends on the initial concentration and the water volume in
the grid cell. For each time-step, each individual particle is moved over a distance that is
determined by the product of the time-step size and the flow velocity at the location
of the point, which is derived by interpolating the flow velocities in the grid cell under
consideration and its adjacent cells (Figure 11.12). After all particles have been moved, the
concentration in each grid cell is computed as the total mass of all particles located within
the grid cell, divided by the water volume. This concentration represents the average
concentration as a result of advective transport. Subsequently, using a Eulerian method, the
method of characteristics solves the transport due to dispersion . Alternatively, dispersion
can be simulated using a Lagrangian method. The random walk method, for example,
approximates advective transport in the same manner as the method of characteristics,
after which the particles are moved both parallel and perpendicularly to the flow direction

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