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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
An Introduction to Models and Modeling in the Earth and Environmental Sciences
offers students and professionals the opportunity to learn about groundwater modeling, startingfrom the basics. Using clear, physically-intuitive examples, the author systematically takes
us on a tour that begins with the simplest representations of fluid flow and builds through
the most important equations of groundwater hydrology. Along the way, we learn how
to develop a conceptual understanding of a system, how to choose boundary and initial
conditions, and how to exploit model symmetry. Other important topics covered include
non-dimensionalization, sensitivity, and finite differences. Written in an eclectic and readable
style that will win over even math-phobic students, this text lays the foundation for a
successful career in modeling and is accessible to anyone that has completed two semesters
of Calculus.
Although the popular image of a geologist or environmental scientist may be the rugged
adventurer, heading off into the wilderness with a compass and a hand level, the disciplines
of geology, hydrogeology, and environmental sciences have become increasingly quantitative.
Today’s earth science professionals routinely work with mathematical and computer models,
and career success often demands a broad range of analytical and computational skills.
An Introduction to Models and Modeling in the Earth and Environmental Sciencesis written forstudents and professionals who want to learn the craft of modeling, and do more than just
run “black box” computer simulations.
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Table of Contents
Cover
Title Page
About the companion website
Introduction
CHAPTER 1: Modeling basics
Chapter summary
1.1 Learning to model
1.2 Three cardinal rules of modeling
1.3 How can I evaluate my model?
1.4 Conclusions
CHAPTER 2: A model of exponential decay
Chapter summary
2.1 Exponential decay
2.2 The Bandurraga Basin, Idaho
2.3 Getting organized
2.4 Nondimensionalization
2.5 Solving for
θ
2.6 Calibrating the model to the data
2.7 Extending the model
2.8 A numerical solution for exponential decay
2.9 Conclusions
2.10 Problems
References
CHAPTER 3: A model of water quality
Chapter summary
3.1 Oases in the desert
3.2 Understanding the problem
3.3 Model development
3.4 Evaluating the model
3.5 Applying the model
3.6 Conclusions
3.7 Problems
CHAPTER 4: The Laplace equation
Chapter summary
4.1 Laplace’s equation
4.2 The Elysian Fields
4.3 Model development
4.4 Quantifying the conceptual model
4.5 Nondimensionalization
4.6 Solving the governing equation
4.7 What does it mean?
4.8 Numerical approximation of the second derivative
4.9 Conclusions
4.10 Problems
References
CHAPTER 5: The Poisson equation
Chapter summary
5.1 Poisson’s equation
5.2 Alcatraz island
5.3 Understanding the problem
5.4 Quantifying the conceptual model
5.5 Nondimensionalization
5.6 Seeking a solution
5.7 An alternative nondimensionalization
5.8 Conclusions
5.9 Problems
References
CHAPTER 6: The transient diffusion equation
Chapter summary
6.1 The diffusion equation
6.2 The Twelve Labors of Hercules
6.3 The Augean Stables
6.4 Carrying out the plan
6.5 An analytical solution
6.6 Evaluating the solution
6.7 Transient finite differences
6.8 Conclusions
6.9 Problems
References
CHAPTER 7: The Theis equation
Chapter summary
7.1 The Knight of the Sorrowful Figure
7.2 Statement of the problem
7.3 The governing equation
7.4 Boundary conditions
7.5 Nondimensionalization
7.6 Solving the governing equation
7.7 Theis and the “well function”
7.8 Back to the beginning
7.9 Violating the model assumptions
7.10 Conclusions
7.11 Problems
References
CHAPTER 8: The transport equation
Chapter summary
8.1 The advection–dispersion equation
8.2 The problem child
8.3 The Augean Stables, revisited
8.4 Defining the problem
8.5 The governing equation
8.6 Nondimensionalization
8.7 Analytical solutions
8.8 Cauchy conditions
8.9 Retardation and dispersion
8.10 Numerical solution of the ADE
8.11 Conclusions
8.12 Problems
References
CHAPTER 9: Heterogeneity and anisotropy
Chapter summary
9.1 Understanding the problem
9.2 Heterogeneity and the representative elemental volume
9.3 Heterogeneity and effective properties
9.4 Anisotropy in porous media
9.5 Layered media
9.6 Numerical simulation
9.7 Some additional considerations
9.8 Conclusions
9.9 Problems
References
CHAPTER 10: Approximation, error, and sensitivity
Chapter summary
10.1 Things we almost know
10.2 Approximation using derivatives
10.3 Improving our estimates
10.4 Bounding errors
10.5 Model sensitivity
10.6 Conclusions
10.7 Problems
References
CHAPTER 11: A case study
Chapter summary
11.1 The Borax Lake Hot Springs
11.2 Study motivation and conceptual model
11.3 Defining the conceptual model
11.4 Model development
11.5 Evaluating the solution
11.6 Conclusions
11.7 Problems
References
CHAPTER 12: Closing remarks
12.1 Some final thoughts
References
APPENDIX A: A heuristic approach to nondimensionalization
APPENDIX B: Evaluating implicit equations
B.1 Trial and error
B.2 The graphical method
B.3 Iteration
B.4 Newton’s method
References
APPENDIX C: Matrix solution for implicit algorithms
C.1 Solution of 1D equations
C.2 Solution for higher dimensional problems
C.3 The tridiagonal matrix routine TDMA
References
Index
End User License Agreement
List of Illustrations
Chapter 02
Figure 2.1 Conceptual representation of a lumped capacitance (baseflow recession) model of the Bandurraga Basin.
Figure 2.2 Schematic representation of the function
θ
(
τ
).
Figure 2.3 A plot of the function
θ
(
τ
) as defined by Equation 2.27.
Figure 2.4 Illustration of the determination of characteristic time by the semilog method. The characteristic time of 55 days is identified as the point at which the dimensionless discharge (
) has decreased to
of its initial value.
Figure 2.5 Determination of characteristic time by trial and error. The curve labeled
a
has a characteristic time of 38 days; the curve labeled
b
has a characteristic time of 55 days (the best fit curve); and the
c
curve has a characteristic time of 60 days. The data, plotted once per 7 days, are shown as circles.
Figure 2.6 Attempting to determine the characteristic time by trial and error for a synthetic data set representing the Bandurraga Basin, Idaho. The curve labeled
a
has a characteristic time of 20 days; the curve labled
b
has a characteristic time of 40 days. Neither curve fits the data, which are shown as circles (data shown once per 7 days).
Figure 2.7 Conceptual model for the two‐domain representation of the Bandurraga Basin.
Figure 2.8 A plot of the same data shown in Figure 2.6, fit with a two‐domain exponential decay model. The model as fit uses the parameters:
,
days, and
days.
Chapter 03
Figure 3.1 Conceptual sketch for a salt tank model of water quality in an artificial lake.
Figure 3.2 Plot of the solution (Equation 3.29) for
.
Figure 3.3 Plot of the solutions for the two extreme values of
ε
, where
or
, as labeled. The shaded area indicates the possible solution space, assuming an initial and input concentration of
.
Chapter 04
Figure 4.1 Schematic diagram of the Elysian Fields. The representation shown here is a composite of accounts from Greek mythology and
Il Inferno
.
Figure 4.2 Diagram showing the relationship between the 1D model domain and the conceptual representation of the Elysian Fields. The model domain is shown above the ground level of the Elysian Fields for clarity of illustration; in actuality, the model domain is under the ground level.
Figure 4.3 The one‐dimensional model domain for the Elysian Fields problem. The
x
‐axis of the model domain runs from
at the World Encircling Sea to
at the River Acheron. Head is equal to
H
0
at the World Encircling Sea, and
H
L
at the River Acheron. The potentiometric surface is drawn at arbitrary locations–this is the quantity we are trying to find.
Figure 4.4 Detail of the control volume (CV) for the Elysian Fields problem. The CV is
w
wide,
b
tall, and Δ
x
thick in the
x
‐direction. The flux into the CV is labeled
q
x
, while the flux out of the CV is labeled
. Both fluxes are drawn positive in the positive
x
‐direction.
Figure 4.5 Plot of the solution of the Elysian Fields model.
Figure 4.6 Schematic representation of a finite difference grid for the Elysian Fields problem. The 1D grid is discretized into nodes with a Δ
ξ
spacing of 0.2. The left‐hand boundary node (
θ
0
) has a constant head of 1, while the right‐hand side node (
θ
5
) has a constant head of 0.
Chapter 05
Figure 5.1 Schematic diagram of a radial domain for the Alcatraz model. The model domain is shown as a circle (dotted line) of radius
R
. As discussed in the text, this model is appealing, but it does not fit well with the actual site geometry.
Figure 5.2 Scale map of Alcatraz island, showing selected buildings. Some buildings are omitted for clarity of illustration.
Figure 5.3 Drawing of a rectangular model boundary (dotted line), superposed over an outline of Alcatraz island (map outline drawn to scale). Along with the proposed model boundary, two possible sets of coordinate axes are shown; the first set (
x
1
,
y
1
) is centered on the lower left‐hand corner of the rectangular boundary. The origin of the second possible set of axes (
x
2
,
y
2
) is at the center of the model domain.
Figure 5.4 Schematic representation of the Alcatraz model domain. The boundaries at
and
are constant head boundaries (fixed at sea level), while the boundaries at
and
are no‐flow (symmetry) boundaries.
Figure 5.5 Control volume (CV) for the Alcatraz model. Recharge is depicted as coming into the CV from above (
R
), although mathematically it is generated within the CV. The height of the CV,
b
[L], is the same as the thickness of the aquifer.
Figure 5.6 Illustration of positive and negative curvature as defined by the second derivative. The upper curve is concave downward, which is negative curvature (as labeled). The lower, concave upward surface shows positive curvature.
Figure 5.7 Plot of the approximate analytical solution (i.e., 1D flow approximation) for the Alcatraz island problem.
is at the center of the island, while
is the boundary at sea level.
Chapter 06
Figure 6.1 Schematic diagram of the Augean Stables problem. The upper and lower layers of low‐permeability materials confine a high‐permeability aquifer connecting the two rivers.
Figure 6.2 Schematic diagram of the Augean Stables problem, showing the
x
‐axis and conditions on the boundaries.
Figure 6.3 Control volume (CV) for the Augean Stables problem. The width of the CV,
w
, is arbitrary, while the height of the CV,
b
, is associated with the thickness of the aquifer.
Figure 6.4 Plot of dimensionless head (
θ
) as a function of dimensionless position (
ξ
) for selected values of dimensionless time (
τ
) as indicated in the figure.
Chapter 07
Figure 7.1 Conceptual model for a 1D radial coordinate system. The pumping well is located in the center of the coordinate system (centered on
). The thickness of the aquifer is given as
b
.
Figure 7.2 Control volume (CV) for 1D radial coordinates. Note the fluxes are drawn positive in the
direction. (a) Perspective view of the CV. (b) Looking down from above on the CV (i.e., map view).
Chapter 08
Figure 8.1 Schematic diagram of the Augean Stables problem, revisited. In this revised problem, King Augeus’s well is located on the hill between the two model boundaries at
.
Figure 8.2 Control volume for the Augean Stables redux problem. In this case, the
q
x
and
fluxes refer to the diffusive flux of nitrates, rather than the water flux, as they did in Figure 6.3. In the figure (and in the text),
U
is the water velocity.
Figure 8.3 Steady‐state solution space for the transport of nitrate in the Augean Stables aquifer. The line
shows the steady solution for the diffusion‐only problem, while the line
is the steady solution for the case in which advection dominates over diffusion. All possible steady‐state solutions must fall between these limits; therefore, the gray area denotes the space of possible steady‐state solutions for this problem.
Figure 8.4 Transient solution of the diffusion‐dominant (advection negligible) equation for nitrate transport in the Augean Stables aquifer. The concentration of nitrate can be seen sweeping through the aquifer from left to right. The approximate final (steady‐state) concentration of nitrate in the aquifer differs from the line
by an amount that is not visible at the scale of this figure (compare with the diffusion limit in Figure 8.3). The curves in the figure are the same as those in Figure 6.4, with dimensionless time scaled by the factor
α
.
Figure 8.5 Plot of characteristics for the advection equation, with
(as an example). The value of
θ
N
at
ξ
0
is preserved along the entire length of the characteristic, but the
ξ
at which that characteristic occurs translates with time
τ
, as shown schematically for some sample characteristics. Some characteristics are drawn with a dashed extension to the left of the
τ
‐axis to illustrate that characteristics that come into positive values of
ξ
at times
originate on the negative half of the
ξ
‐axis. The characteristics are all straight lines in this example, because the velocity (in this case
) is constant.
Figure 8.6 Perspective diagram showing the method of characteristics solution to the advection equation problem (Eq. 8.65). The solution, given by Equation 8.90, is shown in
θ
N
,
ξ
,
τ
‐space, where
θ
N
is the dimensionless concentration of transported quantity (in this problem, nitrate),
ξ
is the normalized distance, and
τ
is the nondimensional time. The gray panels, which mark the location of the advecting front, can be seen to track toward progressively greater values of
ξ
as time increases.
Figure 8.7 Schematic diagram showing the translation and diffusion of an advecting and diffusing contaminant front with increasing time. The curves labeled
,
, and
indicate three sample times, with
.
Chapter 09
Figure 9.1 A conceptual layered porous medium for layer‐parallel flow. The goal is to find a single effective conductivity,
K
eff
, that will give the same flow across the model domain (from
to
) as the total flow along all four layers.
Figure 9.2 Conceptual layered porous medium for layer‐perpendicular flow. We are seeking an effective conductivity,
K
eff
, that represents the conductivity of all the layers shown in this figure for flow moving perpendicular to the layers (i.e., in the
z
‐direction).
Figure 9.3 Refraction of streamlines at a material boundary. The two materials pictured, material 1 and material 2, have contrasting properties such that
(for the refraction pictured).
Chapter 11
Figure 11.1 Study area location and site map, showing the distribution of springs (indicated by black dots) in the vicinity of Borax Lake, with elevations given in meters above sea level.
Figure 11.2 Schematic diagram showing the conceptual model for the Borax Lake hot springs problem.
Figure 11.3 Diagram showing the control volume for a flowpath in the Borax Lake problem.
Figure 11.4 Diagram showing the control volume for heat transport (i.e., the energy balance) of the Borax Lake problem.
Figure 11.5 Plot of the solution for the dimensionless temperature,
θ
T
, of a hydrothermal fluid, migrating from a reservoir (at
) to the land surface (at
) for a selection of values of the dimensionless heat transfer coefficient
β
.
Guide
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Models and Modeling
An Introduction for Earth and Environmental Scientists
Jerry P. Fairley
This edition first published 2017 © 2017 by John Wiley & Sons, Ltd
Registered OfficeJohn Wiley & Sons, Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK
Editorial Offices9600 Garsington Road, Oxford, OX4 2DQ, UKThe Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK111 River Street, Hoboken, NJ 07030‐5774, USA
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All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.
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Library of Congress Cataloging‐in‐Publication Data
Names: Fairley, Jerry P., author.Title: Models and modeling : an introduction for earth and environmental scientists / Jerry P. Fairley.Description: Chichester, UK ; Hoboken, NJ : John Wiley & Sons, 2016. | Includes bibliographical references and index.Identifiers: LCCN 2016024807 (print) | LCCN 2016033087 (ebook) | ISBN 9781119130369 (pbk.) | ISBN 9781119130383 (pdf) | ISBN 9781119130376 (epub)Subjects: LCSH: Earth sciences–Mathematical models. | Environmental sciences–Mathematical models.Classification: LCC QE33.2.M3 F35 2016 (print) | LCC QE33.2.M3 (ebook) | DDC 550.1/5118–dc23LC record available at https://lccn.loc.gov/2016024807
A catalogue record for this book is available from the British Library.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.
Cover image: iStockphoto/Alexander Shirokov
To my father, for teaching me to do math one step at a time.
About the companion website
This book is accompanied by a companion website:
www.wiley.com/go/Fairley/Models
This website includes:
Powerpoints of all figures from the book for downloading
Solutions to many of the problems given in the chapters
Introduction
Before serious discussion on any topic can take place, it is usually necessary to define one’s terms. In the present instance, we need to ask the following question: “what, exactly, do we mean when we speak of a ‘model’?” A reasonable definition of the term might be a simplified or idealized representation of reality. Although this is a pretty broad definition, it is perhaps the only one that really embraces everything we refer to as “a model.”
In order to better understand what is, and what is not, a model, we can examine some possible examples. Which of the following would be considered models?
A map
A model train
A model United Nations
A fashion model
An atom
Of course, by our definition all of these things would qualify as models. Students will sometimes argue that “an atom” is a physical object and, therefore, not a model. However, everything we know about atoms comes from observing their effects, and our understanding and depiction of atoms has changed and evolved over time. In fact, our current understanding of atoms and subatomic particles is so far outside our everyday experience that our descriptions are entirely probabilistic. As a result, I maintain that any discussion of “atoms” is necessarily a discussion of models—simplifications and idealizations of things that are beyond our direct knowledge.
In this book, we will primarily concentrate on a subset of models known as mathematical models. A mathematical model is a kind of model that is formulated in terms of mathematical concepts such as constants, variables, functions, derivatives, and so on. The nominal goal of a mathematical model is to give quantitative indications of the behavior of some aspect of a system.
Note that I said we would primarily be concentrating on mathematical models. In fact, the word “primarily” glosses over an important fact about the development of mathematical models: every mathematical model is the quantification of an underlying conceptual model; that is, a mathematical model is the quantification of our idea of how a system works. This is a very important, but often overlooked, aspect of model development. The fact is that model development requires many decisions to be made about which aspects of the system are important, and which aspects may be neglected. If the decisions are poor, a poor model will result. If the decisions are good, a good model may result (no guarantee). In any case, before a mathematical model can be developed, it is first necessary to define a conceptual model, or conceptual understanding of the system. The usefulness of the quantitative model, and the reliability of its representation of the system, will be wholly dependent on this conceptual understanding.
Another aspect of models that is deserving of discussion is the uses to which models may be put. A sample of these might be the following:
To teach
To “predict” system behavior or the consequences of actions
To develop or test theories
To plan or design field/laboratory tests
As propaganda to support a partisan viewpoint
Of these, the use of models as propaganda is probably the most common application for models (e.g., think about the usefulness of fashion models and model cars in selling products). It is tempting to think that “scientific” (i.e., mathematical) models are exempt from issues of partisanship, but this is far from the truth. Because every model is a quantification of an individual’s conceptual understanding, all models are developed from a partisan point of view; that is, every model contains bias! Therefore, the first step in evaluating any model is to understand the motivations and biases of the developers.
Another topic it would be appropriate to touch on is model fallibility. It is not uncommon to hear statements similar to “it must be true, because the model says so,” and models are commonly depicted in popular culture as unfailing guides to the future. In reality, the knowledgeable modeler understands that a model embodies our conceptual understanding of some system, and is therefore incomplete and prone to error. Furthermore, there are an infinite number of alternative models that could be devised to represent any given system. Since it is impossible to test all the alternatives for a superior model, a model can never be shown to be the “true” representation of a system.
It is also easy to make the mistake of thinking that a model is reliable because it has been “validated” or “verified.” In my opinion, the idea of “model validation” is misleading at best; it implies that, because a model has produced good results in the past, future predictions may be taken uncritically. In fact, every new prediction or new situation to which a model is applied will take a model into scenarios outside of the conditions for which it was calibrated and tested. Under these circumstances, the modeler must always be alert to the possibility that new processes, changes in parameters, or insufficiencies of the underlying conceptual model will cause the model to fail.
Instead of using the terms “validated” or “verified,” I prefer to think in terms of confidence building. If a model performs well when tested against observations, we gain confidence in the model’s output, and each additional test increases our confidence. We retain our confidence as long as the model continues to perform reliably; however, we never lose sight of the fact that, at some time in the future, the model parameterization or underlying conceptualization may be insufficient for the task at hand, and the model may fail. Keep in mind that, although a model can never be proven to be “right,” it only takes one counterexample to prove it wrong.
Usually, models are created for some useful purpose—to understand some phenomenon, plan some test, or make some prediction—and I am a believer in the idea that models should be useful. However, I would be less than honest if I didn’t say that one of the reasons for developing models was the pleasure to be had in their development. I believe this is as good a reason to develop a model as any, and perhaps better than many others. As you work through the models presented in the following chapters, I hope that a little of the enjoyment and gratification that comes from working with models will shine through.
CHAPTER 1Modeling basics
Chapter summary
Regardless of whether the subject to be modeled is a groundwater flow system, the growth of a bacterial colony, the flight of a projectile shot from a cannon, or any other system, the construction of a model is not a simple step‐by‐step procedure. Because the modeler must make decisions about which processes should be included in the model and which will be neglected, what domain will be used, and so on, the development of a model is as much art as science. Notwithstanding the creative aspects of model development, there are some simple but important rules that should be followed. In this section, we will examine three basic rules for model development, briefly discuss some important aspects of model formulation, and make some suggestions for evaluating a model’s performance. These rules and suggestions will form the basis for all of the examples of model development in the following chapters.
1.1 Learning to model
How to develop a model of a physical system, when to believe and when not to believe the model output, and how to determine whether the model predictions have any relevance to real life are common and central questions that must be answered by the would‐be modeler every time a new situation is encountered. Most commonly, modelers are shown how to use a software package (or asked to read the documentation for a software package), and then assumed to be sufficiently competent to produce reliable predictions of system behavior. Even a moment’s reflection will show that this is a nonsensical way to go about learning the craft of modeling, and this attitude has been largely responsible for the proliferation of bad models and the subsequent lack of confidence in modeling (and modelers).
Ideally, a modeler would gain experience and a deep understanding of the process of model development while working as an “apprentice” under a skilled modeler. This desirable state of affairs is seldom met with in the real world, however. The purpose of this book is to provide some guidance for those aspiring modelers who do not have the advantage of serving such an apprenticeship. Although not a substitute for the teaching and advice of an experienced modeler, it is to be hoped that the rules and examples in this and the following sections will at least keep the novice modeler from falling into some of the more obvious pitfalls associated with the mathematical modeling of physical systems.
1.2 Three cardinal rules of modeling
It is probable that, over time, many hundreds of “rules” have been made up regarding the construction, evaluation, and application of mathematical models. Most of these purported rules would be better classified as “suggestions,” “considerations,” or even, in some cases, as “superstitions.” Over many years of making and using models, however, I have become convinced that the following three cardinal rules should be followed at all times in the development of a mathematical model:
Always know exactly the objective of model development.
The model you develop should be appropriate for the available data.
Start with the simplest possible model of the system, even if it is completely unrealistic. Once you thoroughly understand this preliminary model, add complexities to the model
one at a time
until you arrive at a satisfactory representation of the system.
In my experience, all three of these rules are routinely overlooked by modelers, and many poor and inappropriate models have resulted. We will briefly consider each of these rules here, but, more importantly, they are bound into the fiber of every model developed in the following chapters.
1.2.1 Rule 1: Know your model objective
It is common for a modeler to start a modeling investigation with the objective of “making a model of the aquifer” or with a similarly vague idea of what is to be accomplished. I cannot state strongly enough that a modeler must know exactly what s/he is trying to accomplish before ever putting pen to paper (or typing an input parameter). The more precisely the objectives of the model are known, the more likely the investigation will be successful. Make a habit of writing down the objective of your model, and be ready and willing to reduce your objective to a single sentence. The objective “to model the wells in the Grande Ronde Aquifer” is a very poor statement of purpose; a better (although still insufficient) objective is “to determine the influence of pumping well MW‐4 on nearby wells.” Better yet (and possibly sufficient to begin an investigation) is “to estimate the change in head in wells MW‐1 and MW‐3 that results from a 24‐hour constant rate pump test in well MW‐4.” The examples in the following chapters always include a statement of that which is to be found; hopefully, after working through the examples, the reader will have a clear idea of how to formulate model objectives. Although the temptation to “just get modeling” and “show some results” may be strong, you will always be better off if you first make certain you understand exactly what it is you want to achieve, and formulate a plan to reach that goal.
1.2.2 Rule 2: Make your model appropriate for your data
Hydrogeologists and environmental scientists are often working in data‐poor environments. There is usually little to be gained from building a three‐dimensional (3D), coupled saturated–unsaturated zone model with heterogeneous property sets when the only data to constrain the model come from a single aquifer test. Often in these situations a simple analytical model will give results that are as reliable as (or more reliable than) a complex numerical simulation. Furthermore, complex numerical simulations are often misleading, since it is tempting to think that, because they are complicated, they are realistic. What non‐modelers (and many modelers) are unaware of is the fact that any computer model is solving the same equations that the modeler can write down with a pencil and paper. If there are few data to support the added complexity in terms of spatially varying properties, time‐dependent recharge or boundary conditions, and so on, then the complicated numerical simulation may in fact be a worse representation of the system than a greatly simplified analytical model.
It should be said that many clients, regulatory agencies, and other downstream users of model output will push for complex numerical simulations in spite of the paucity of data to support such simulations. Although economic, political, or regulatory pressures may force a modeler to undertake the development of 3D simulations when only 1D simulations are justified, or a transient model when a steady‐state model would do, the modeler should at all times be aware of the limitations of the models s/he is working with. By following Rule 3 (Section 1.2.3), the savvy modeler will be able to develop the more complex model demanded by the client while still maintaining her or his integrity and a high standard of modeling ethics.
1.2.3 Rule 3: Start simple and build complexity
When faced with a complex and difficult real‐life situation, it is tempting to start out by developing a model that includes the most important processes. For example, if the goal is to understand the impact of a pumping well on other nearby wells, a novice modeler might want to build a model that includes variations in the rate of pumping, recharge from rainfall or snowmelt, the influence of changing water levels in a nearby lake, and other similar items that are clearly needed for a realistic representation of the system. The problem is that there is no way, in such a complex conceptualization, for the modeler to determine if the model output makes sense or not. Your first attempt at modeling a system should always be the simplest possible representation. Rather than modeling a 3D transient system, begin by modeling a 1D or 2D steady‐state system with no source terms or other complexities. Although this may not be a realistic representation of the system, at least the modeler will know if the results are reasonable. Next, the modeler may add a spatially and temporally constant source term; again, evaluate the results. Do they make sense? Can you convince yourself the output is reasonable? If so, add another complexity, reevaluate, and so on, until the final product is one that you both understand and believe in. Never go on to the next step until you have complete confidence in, and an intimate understanding of, the current step—as well as all the steps that lead to the current step.
1.3 How can I evaluate my model?
Particularly with regard to Rule 3 (Section 1.2.3), one may rightly ask the question: “how do I know my model is giving reasonable results?” There is no easy answer to this question; in part, knowing a model is giving reasonable results is the product of experience with models. There are, however, some simple suggestions that can help to uncover problems with a model; these are described in the following text.
1.3.1 Test model behavior in the limits
The quickest way to begin an examination of model behavior is to check the behavior of a model in the limits. What is the model behavior at time ? What about as ? If you set a parameter to 0 or check the limit as it goes to , is the result what you would expect? These kinds of tests are most readily carried out with analytical models, but, with some ingenuity, they can usually be applied even to complex numerical simulations.
1.3.2 Look for behavior congruent with the governing equations
Even very complex numerical simulations are based on a few well‐known equations such as the Laplace equation, Poisson’s equation, and the transient diffusion equation. Each of these equations implies particular behaviors, and you should check to make certain your model is behaving in a fashion that accords with your understanding of the underlying governing equations. (We will examine the characteristic behaviors of the most common equations in the following chapters.) Check for maxima and minima, look at the curvature of the predicted model surface, and examine any discontinuities in the output. Are these features present (if you expect them) or not (if you don’t expect them), and are they in the appropriate places? It is a good sign if your expectations, based on your understanding of the characteristics of the governing equations, are realized. If the behavior you observe in your model output is different than your expectations, you need to understand why these differences arise before moving forward.
1.3.3 Nondimensionalization
Whenever possible, you should nondimensionalize your model (nondimensionalization is described in Appendix A, and illustrated throughout this text). Note that, although you will rarely be able to nondimensionalize a commercial simulation package, the model is based on mathematical equations that can always be nondimensionalized. Nondimensionalization carries with it a number of benefits; in particular the following:
Nondimensionalization reduces the governing equations to their most basic functional form, which helps to clarify the expected behavior.
Examination of the nondimensional form of the governing equation is the easiest and most certain way to identify which parts of an equation are relevant and which parts may be neglected. In this way, simplifications of the original equations may often be made (and, equally important, justified).
Nondimensional plots of model output are the most compact way of presenting the model results. Admittedly, your target audience may not be equipped to understand dimensionless results; in this case, it is up to the modeler to either present dimensional results or educate their audience regarding the nondimensional ones.
The nondimensionalization process results in the identification of the controlling dimensionless parameters. These parameters control the behavior of the equation, allowing the modeler to readily identify parameter ranges over which behavioral changes will take place. Furthermore, the dimensionless parameters show the modeler which dimensional parameters can be uniquely identified and which cannot.
1.4 Conclusions
As was stated in Section 1.1, following the rules and suggestions laid out in this chapter won’t guarantee success, nor will it make you a modeler (only time and experience will do that). Hopefully, however, the ideas presented here will help you avoid some of the most common traps that inexperienced modelers tend to fall into. In the following chapters, I will develop a number of models; as you follow these developments, watch for the application of these basic principles. Ask yourself how you could apply these principles to your own problems. As with any creative endeavor, there are rules and guidelines that can be applied, but the ultimate responsibility for the final product belongs with the artist.
CHAPTER 2A model of exponential decay
Chapter summary
Possibly, the most basic building block for mathematical models of physical systems is the process of exponential growth and decay; models of exponential decay processes are encountered in every branch of science and engineering. Here, we examine the problem of modeling discharge from a high alpine basin, which leads to the representation of basin discharge as an exponential decay process. Although simple, the model is quite general and may apply to many springs and streams, as well as to the electrical charge stored in a capacitor, the temperature of a heated piece of steel quenched in cold water, and the decay of radioactive materials.
2.1 Exponential decay
Exponential decay processes are ubiquitous in nature. The essence of exponential decay is that the rate at which some quantity is lost is determined by the amount of the quantity that remains behind. We are all familiar with this process on an intuitive level; for example, a can of soda placed in the refrigerator contains quite a bit of heat (relative to the temperature of the refrigerator). The temperature gradient between the refrigerator and the soda is large at early times; thus, the soda at first cools rapidly. However, as time goes on, the difference in temperature between the soda and the refrigerator lessens; as a result, the rate of cooling also decreases. Theoretically, the soda will never actually reach the temperature of the refrigerator, but it will come arbitrarily close over a long period of time. We say that the temperature of the soda decays exponentially to the temperature of the refrigerator.
Much of the behavior observed in the natural world can be expressed in terms of differential equations—equations that show relationships between the rates of change in different quantities—and, because they possess the property of being their own derivatives, the solutions to differential equations very commonly involve exponential functions. Thus, it should be no surprise to find that exponential functions are encountered widely in natural phenomena.
Although exponential decay models are common to many disciplines, each discipline refers to them by a different name. In heat transfer, models of exponential decay are called “lumped capacitance” models; similarly, electrical engineers use the term “RC models” (for “resistance–capacitance”). Hydrogeologists, who use exponential decay models to represent spring flow, basin discharge, and similar processes, often refer to such models as “baseflow recession models.” In the following sections, we will develop a simple exponential decay model for basin discharge, using as an example a hypothetical watershed in central Idaho.
2.2 The Bandurraga Basin, Idaho
Suppose you are approached by the US Forest Service regarding the management of water yield from an alpine basin in central Idaho. The basin receives a variable amount of precipitation in the form of rain and snow each year; in general, there is little or no precipitation after April. As one would expect, discharge in the stream that drains the basin (the “main stem”) peaks about the time precipitation in the basin ceases, and stream flow decreases over time until precipitation resumes in the fall. Water from the basin is captured in a down‐valley reservoir and used to operate a micro‐hydroelectric generator system that provides power for the Forest Service wilderness visitor center. The water manager’s problem is this: some years there is an overabundance of discharge from the basin, and the excessive runoff can damage the reservoir; in other years, the water manager drains (draws down) the reservoir in early spring, making room in the reservoir to store the basin yield, but there is not always enough water to refill the reservoir. When spring and summer flows are too low to refill the reservoir, the visitor center can experience power shortages in late summer. The water manager’s question is, can you help?
2.3 Getting organized
Most modelers are familiar with this kind of request: can you help? Can you make a model of this? Whenever you receive a request of this type, the first thing you should do is work to understand the exact nature of the problem and what is expected of you (i.e., Section 1.2.1). Requests for models of physical systems are rarely well‐thought‐out, at least in part because few non‐modelers really understand the craft of modeling. Regardless of whether or not the client is clear in her/his description of the problem and desired objectives of the model, you can be certain that you, as the modeler, will bear the brunt of the client’s unhappiness if your deliverable does not meet the client’s (often unstated) expectations. As a result, it is up to you to discuss the problem with the client until you clearly understand what is required; furthermore, you must share your understanding with the client, to be certain that the client knows what to expect when s/he receives your final report.
2.3.1 Observable quantities
This is probably a good point to pause and reflect on what I call “observable quantities.” When formulating your model objectives, it is important to keep in mind what can be observed and what cannot. A model that makes predictions that cannot be checked is rarely useful; as a result, the quantity you select as your model output must, at least in principle, be amenable to measurement. In this example, there would be very little point to making predictions of average saturation in the basin, or even of the quantity of water stored in the basin, because there is no way to measure an average basin saturation or independently verify the amount of water in the basin at any given time. Furthermore, it is not clear how the amount of water in the basin (or the saturation) can be related to the quantity of discharge from the basin. In a sort of “worst case,” the Forest Service’s water manager may equate the “total water stored in the basin” with the amount of water available for power generation. This will almost certainly translate into a resource shortfall, and the modeler will bear the responsibility for such a shortfall because s/he provided misleading model predictions.
2.3.2 Stating the model objective
On the basis of the foregoing discussions (Sections 2.3.1 and 1.2.1), it is clear we must formulate our model objective with consideration of our observable quantity (in this case, the discharge from the basin) and with other expectations we have for the solution (i.e., to what use will we put the model output?). In this case, I propose a good expression for that which is to be found may be the following:
FIND
: An expression for discharge from the Bandurraga Basin, Idaho, at any time following the cessation of precipitation in an annual cycle. The expression must allow the calculation of cumulative discharge between any two specified points in time (e.g., from May 1 to August 1).
Note that we have stated exactly what the model output should be (the basin discharge as a function of time) and, additionally, we have stated the expectation that we will be able to integrate the solution between any two points in time (either analytically or numerically). The first of these objectives tells us we don’t necessarily need to know the spatially distributed head in the basin, the temperature of the discharge, the dissolved load in the main stem, the amount of power that can be generated from the water in the basin, or similar extraneous information. The second part of the statement of model objectives starts us thinking about what kinds of models might or might not be appropriate; for example, we probably don’t want to devise a statistical model that gives the mean discharge of all previous years on day X, because such a model can’t give predictions of cumulative discharge specific to the current water year (although it would certainly be possible to devise a probabilistic model for this situation).
2.3.3 What data are available?
Once we have a clear statement of our model objective, the next step is to marshal our resources in terms of how much we know about the situation. What data are available? What facts about the model domain are known to us? In this case, we will assume we have the type of information we could reasonably expect to be available for a wilderness basin under Forest Service management.
KNOWN:
Discharge from the basin at some initial time (say May 1)
Basin area, topography, and geometry
Type of vegetative cover in the basin
Twenty‐year record of precipitation, discharge, and temperatures in the basin
Basin geology (rock types, geological structures, faults, etc.)
Information on soil types and thicknesses
It is likely that not all of this information will be useful to us, but it isn’t always clear in advance what will be and what will not be useful. Furthermore, the process of thinking through what is known about a system is a crucial part of developing an overall understanding of the problem and of conceptualizing an approach to its solution.
2.3.4 What can we assume?
The next step in conceptualizing an approach to model development is to begin a list of our model assumptions. This list of assumptions will change as we formulate the model; it may grow or shrink as we relax some of our initial assumptions. Regardless of the way the list evolves over time, you must keep careful track of your modeling assumptions! You will need to know your model assumptions for your final report (or journal article, thesis, etc.), and it will be helpful to have them all gathered together in one place.
There are many possible assumptions that could be made at this point in the development of our model; for example, we could assume the basin comprises homogeneous and isotropic geological materials (this is a common assumption of groundwater models), or that precipitation in the basin is equal to the average of the precipitation over the past 20 years. In general, however, it is best to make a minimal number of assumptions at the start, and then to impose additional assumptions as needed during model development (while keeping careful track as new assumptions arise). In this case, we can (at least initially) make do with a very short list of assumptions.
ASSUME:
No additional precipitation in the basin after time
.
We can neglect hydraulic gradients and variations on a subbasin scale, and consider the basin as a “lumped system.”
In general, if you state your assumptions clearly when you report the findings of your study, you are on safe ethical ground. If you don’t state your assumptions explicitly, however, you will be in a very poor position when your conclusions don’t pan out. As a result, I strongly recommend that you keep a careful and accurate list of your model assumptions, and keep it up to date.
2.3.5 Finding an approach
At this point, we have reached one of those critical junctures that appears in the formulation of every model: we must settle on a conceptual model of our system. How does the system work? Our mathematical model will be a quantification of our conceptualization of the system, so the success of our endeavor depends on how well we understand the system and on the appropriateness of our abstraction of the system. Our conceptualization must be simple enough to be tractable (the simpler, the better, in terms of time and money). At the same time, it must include sufficient detail to provide a realistic representation of the system (i.e., to represent the system accurately enough to allow us to achieve our objectives).
Given our need to develop a realistic conceptualization of the system, it is tempting to rush off to construct some kind of numerical model of the basin that includes evaporation, transpiration, recharge from rainfall and snowmelt, and so on. The question we must ask, though, is, what model would be appropriate, given the level of information we have about the system and the objectives of our model?
As it turns out, a more complex model is rarely a better model. William of Ockham is generally credited with formulating the guiding principle of model development in the 1300s.1 His statement advocating parsimony and elegance in reasoning, Numquam ponenda est pluralitas sine necessitate (“Plurality is never to be posited without necessity”) (Badius and Trechsel, 1495), has come to be known as “Ockham’s Razor.” These days, this principle is usually stated as follows:
Ockham’s Razor:
When choosing between two explanations for a phenomenon, the simplest explanation is most likely to be the best, all other things being equal.
Of course, if a simple model neglects important processes or fails to produce output that adequately reproduces the observations, a more complex model may be warranted. Following the cardinal rules of modeling, however (Rules 2 and 3, Section 1.2), suggests that you should first seek a simple model that is appropriate to the existing data, and add complexity only if it is needed. In other words, you should remember William of Ockham’s principle and apply it to all your model development problems.
For the problem at hand, a little thought will show that a simple analytical model is likely to serve us better than a complex numerical simulation. In part, this is because of the time that is required to set up a numerical simulation, and in part because we have few data to constrain the input parameters of such a model. Furthermore, a numerical model of the basin—if developed correctly and with sufficient data—may yield a great deal of information about the distribution of hydraulic head in the basin, the evolution of soil moisture content over time, and so on, but these quantities are completely irrelevant to our model objectives. In this case, the time and expense of developing a numerical simulation of the basin are not justified based on our objectives and the existing data. At the very least, if we develop a simple representation of the system, we can get an overall idea of how the basin works, and then reevaluate our need for a more complex model.
On the basis of these considerations, we can propose the following approach to modeling discharge from the Bandurraga Basin:
APPROACH:
Conceptualize the basin as a large tank, discharging water from an outlet in the bottom of the tank to represent streamflow draining from the basin into the main stem. Use mass balance to formulate a relationship for the discharge of water from the tank as a function of time.
This simple formulation is known as a “lumped system” model, because most of the processes affecting discharge are dealt with in aggregate via a group of effective parameters. Drawing a cartoon of the conceptual model and labeling the salient quantities (Figure 2.1) will help to solidify the important points of the model, as well as assisting in developing a notation for the problem. In the following sections, we will see how this lumped system model applies to the problem of modeling discharge from our hypothetical basin.
Figure 2.1 Conceptual representation of a lumped capacitance (baseflow recession) model of the Bandurraga Basin.
2.3.6 Executing the plan
The principle of mass balance (or energy balance for heat transfer models) underlies all models in this book, and the majority of deterministic groundwater flow models in use today. Simply stated, mass balance requires the mass coming into our model domain (Min) to equal the outgoing mass (Mout), with any differences between those two quantities reflected in a change of mass stored in the model domain (ΔM):
In the present instance, we have assumed that no mass is entering the model domain after time ; therefore, and we can rewrite Equation 2.1 as:
We now need to find expressions for each of the terms in Equation 2.2. The Mout term can be written as:
where ρ is the density of water [M/L3], Q [L3/T] is the discharge, and Δt is an arbitrary, but small, unit of time. Similarly, ΔM can be written:
