CE5504 – Surface Water Quality Modeling
General Lecture 2. Model Development
1. Modeling Approaches
We recognize two general approaches in modeling –

empirical models, based on an inductive or data-based approach. Induction
being the process of discovering explanations based on observational evidence.

mechanistic models, based on a deductive or theoretical approach. Deduction
being the process of drawing conclusions based on something known or assumed.
2. Empirical Models
Data-based or empirical models utilize paired observations to estimate coefficients (e.g.
a, the assimilation factor) and establish relationships between state variables or between
loads and in-lake conditions.
[.ppt] Chapra (1997, Fig. 29.6) Secchi disk depth and chlorophyll
[.ppt] Chapra (1997, Fig. 29.1) Vollenweider plot
Empirical models offer the advantage of being straightforward and cost-effective in their
development. Shortfalls include their non-site-specific nature, their level of uncertainty
and the fact that they do not contribute to our understanding of system behavior.
Empirical models are sometimes used in conjunction with mechanistic models as we will
discuss in a future lecture.
[.ppt] Chapra (1997, Fig. 29.4) linked empirical/mechanistic models
3. Mechanistic Models
Mechanistic models are often termed mechanistic mass balance models as they are based
on the principal of conservation of mass, i.e. within a finite control volume, mass is
neither created nor destroyed.
This type of model is based upon a mass balance equation that accounts for all transfers
of mass across the system boundaries (loads, mass transport) and all transformations
occurring within the system (kinetics).
[.ppt] Chapra (1997, Fig. 1.5) Mass balance schematic and equations
Our efforts to develop mechanistic models will be organized around an exploration of
ways to best understand, and represent mathematically, the three components of the
generalized mass balance:

loads (point source, surface runoff, atmospheric, sediment)

mass transport (advection and diffusion)

kinetics (reactions of various orders, microbial dynamics)
The term mechanistic reminds us that we are called to understand and describe
mathematically the mechanisms describing system response.
4. Model Resolution
Model resolution is the degree to which time, space, and matter are segmented.
In the generalized mass balance equation, we identify sources (loads, mass transport in,
growth kinetics) and sinks (mass transport out, loss kinetics) of the parameter being
modeled. The concentration of that material varies with the relative magnitudes of the
source-sink terms. If the sources are greater than the sinks, concentrations increase. If
the sinks are greater than the sources, concentrations decrease. If the sources and sinks
are in balance, the system is said to be at steady state (a poised or dynamic equilibrium)
and concentration remains unchanged.
Steady state solutions are straightforward in their application, both mathematically and
with respect to required model inputs. However, certain substances and certain systems
may fail to reach a steady state due to variability in source-sink terms and related
environmental forcing conditions. In this case, a time-variable approach is necessary.
The calculation of time to steady state (t95%) may offer some guidance in this respect,
t95% 
3
1

k
Also consider the pseudo-steady state condition shown by algal biomass in some lakes.
[.ppt] Chapra (1997, Fig. 18.8) Chlorophyll in lakes
Our selection of a spatial scale for a model is influenced by the physical characteristics of
the system (e.g. presence of embayments, influence of thermal stratification, Chapra, Fig.
16.1). Models may be zero dimensional (completely-mixed), 1-D (e.g. varying over
depth), 2-D (e.g. varying over depth and length), or 3-D (e.g. varying over depth, length
and width). The ‘work-load’ associated with model development increases dramatically
as one moves from zero- and one-D to two- and three-D.
[.ppt] Chapra (1997, Fig. 16.1) Spatial segmentation
There are some generalizations which may be made about the time and space scales
which are appropriate for certain materials.
[.ppt] Chapra (1997, Fig. 18.9) Time and space scale of water quality problems
Another important feature of model development is kinetic resolution. By this, we mean
the degree to which we compartmentalize the substance under study.
[.ppt] Kinetic resolution
For example, we may consider:

total phosphorus or soluble reactive, dissolved organic and particulate P

total nitrogen or ammonia, nitrite, nitrate and organic N

chlorophyll or any number of algal species or species groups
Again, the choice carries a great burden in terms of required inputs (both loads and
kinetics) and is generally dictated by the nature of the problem under consideration as
well as data availability. In the end, decisions regarding model resolution are made based
on experience and a clear understanding of the resource availability and needs of the
problem at hand.
5. Complexity and Reliability
Finally, one must consider trade-offs in model complexity and model reliability. This is
related to the ‘garbage in – garbage out’ paradigm of many computer applications. In the
simplest sense, the reliability of a model increases with its complexity because a more
complex model can best simulate the intricacies of natural systems. However, if the field
and laboratory efforts providing model inputs fail to keep pace with the complexity of
model constructs, the model becomes less reliable. The solution is to develop a model
which is sufficiently complex to capture features of system behavior critical to the
problem at hand with the available level of input information.
[.ppt] Chapra (1997, Fig. 18.2) Complexity and reliability
5. Historical Development of WQ Modeling





1925 – 1960 (Streeter-Phelps)
1960 – 1970 (Computerization)
1970 – 1977 (Biology)
1977 – 2000 (Toxics)
2000 – present (Ecosystems)
[.ppt] Chapra (1997; Fig. 1.6)