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the model descriptions to the regional or supra-regional scale (e.g. De Wit, 1999). A better
insight into the major governing processes and their rates may lead to improved models;
this would improve the prediction of chemical concentrations or loads and therefore benefit
the other purposes too. The development of improved models of contaminant transport
need not result in more complex models, but may even lead to simpler models from which
irrelevant processes have been omitted; this would save computational effort and therefore
time and money.

10.3 CLASSIFICATION OF MATHEMATICAL MODELS

In general, mathematical models can be divided into two broad classes. Physical models
use known science to model the processes on the basis of systems behaviour. Examples of
physical models are groundwater and surface water transport models based on physical
principles from fluid mechanics. Empirical models are used when insufficient is known
about the underlying physics, or when application of the underlying physics would require
an unpractically detailed resolution to characterise the system. These models are based on
experimental observation and then functionally fitting the models to these observations
either by trial and error or by statistical techniques. The fitting procedure is known as
model calibration . Examples of empirical models include models describing the temperature
dependency of the biodegradation rate of organic chemicals. In practice, there is no sharp
boundary between strict physical models and complete empirical models. Complex
environmental models, however, are often a mix of models based on physical laws and
empirical models, so this book introduces both approaches.
Another important classification of mathematical environmental models involves the
distinction between dynamic models , i.e. models that involve change over time and time-
varying interactions, and static models , i.e. models that describe a steady state . If the
temporal variation in the system state is negligible, the system may be considered to be at
steady state and the time dimension may be disregarded and a static model is sufficient
to describe the state of the system. Conversely, if the system state changes significantly in
time, a dynamic model is necessary. When solving a mathematical model, the main model
input besides the model parameters is the so-called boundary conditions that describe the
constant or time-varying values of the state variables at the system boundaries. A dynamic
model also requires initial conditions ; this implies that the state of a system at the beginning
of the model period should be defined. A static model does not require initial conditions,
but because chemical transport processes are dynamic, these models are only applied under
special conditions.
To summarise, mathematical environmental models are comprised of five conceptual
components (Jørgensen, 1988):
• The set of state variables that define the state or condition of the system;
• The set of input variables or forcing functions that influence the state of the system;
• The set of mathematical equations that represent the processes within the system
and the relations between the forcing functions and the state variables, including the
mathematical solution technique;
• The set of universal physical constants that are not subject to variation, such as
Avogadro’s number or the gas constant ;
• The set of other coefficients or parameters in the mathematical equations, which are
either constant within the system or vary in space and time.

Thus, the definition of an environmental model used here comprises more than just an
executable computer program.

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