SDSU GEOL 651 - Numerical
Modeling of Ground-Water Flow
SDSU Coastal Waters Laboratory
USGS San Diego Project Office
1st Floor conference room
4165 Spruance Road
San Diego CA 92101-0812
Tuesdays 4 -7 PM
Introductions
• Claudia C. Faunt
• Ph.D. in Geological Engineering from
Colorado School of Mines
• Hydrologist with U.S. Geological Survey
• (619) 225-6142
• ccfaunt@usgs.gov
• Office 2nd floor NE corner
Introductions
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Please introduce yourself
explain who you are
where you are from
what your current endeavor is (for example,
MS student; state government hydrologist;
or consulting hydrologist)
• explain why you would like to learn more
about ground-water modeling (knowing your
motives helps me improve the class)
Course Organization
• Organizational Meeting
– Part of the first class meeting will be dedicated
to an organizational meeting, at which time a
general outline of the class topics, and any
desired changes in schedule will be discussed.
• Grading (details next week)
– 25% miscellaneous assignments
– 25% paper critique assignment
– 50% final project (paper and presentation)
• Syllabus
Course Organization
• Classes
– First few mostly lectures
– Majority
• First half lectures
• Second half
– Problem set related to lecture
– Model project work
Course Topics
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Introduction, Fundamentals, and Review of Basics
Conceptual Models
Boundary Conditions
Analytical Modeling
Numerical Methods (Finite Difference and Finite Element)
Grid Design and Sources/Sinks
Introduction to MODFLOW
Transient Modeling
Model Calibration
Sensitivity Analyses
Parameter Estimation
Predictions
Transport Modeling
Advanced Topics including new MODFLOW packages
Others?
Tentative Syllabus
(subject to change to adjust our pace)
• Handout
Introduction to
Ground-Water
Modeling
OUTLINE:
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What is a ground-water model?
Objectives
Why Model?
Types of problems that we model
Types of ground-water models
Steps in a geohydrologic project
Steps in the modeling process
What is a ground-water model?
• A replica of a “real-world” ground-water
system
OBJECTIVE:
• UNDERSTAND why we model ground-water systems
and problems
• KNOW the TYPES of problems we typically model
• UNDERSTAND what a ground-water model is
• KNOW the STEPS in the MODELING PROCESS
• KNOW the STEPS in a GEOHYDROLOGIC PROJECT
and how the MODELING PROCESS fits in
• KNOW HOW to FORMULATE & SOLVE very SIMPLE
ground-water MODELS
• COMPREHEND the VALUE of SIMPLE ground water
MODELS
Why model?
• SOLVE a PROBLEM or
make a PREDICTION
• THINKING TOOL
– Understand the system and
its responses to stresses
Types of problems that we model
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WATER SUPPLY
WATER INFLOW
WATER OUTFLOW
RATE AND DIRECTION
CONCENTRATION OF CHEMICAL
CONSTITUENTS
• EFFECT OF ENGINEERED FEATURES
• TEST ANALYSIS
Types of ground-water models
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CONCEPTUAL MODEL
GRAPHICAL MODEL
PHYSICAL MODEL
ANALOG MODEL
MATHEMATICAL MODEL
• We will focus on numerical models in this
class
Conceptual Model
• Qualitative description of the system
– Think of a cartoon
Graphical Model
• FLOW NETS
– limited to steady state, homogeneous
systems, with simple boundary conditions
Physical Model
• SAND TANK
– which poses scaling problems, for example
the grains of a scaled down sand tank model
are on the order of the size of a house in the
system being simulated
Sand Tank Model
Analog Model
• ELECTRICAL CURRENT FLOW
– circuit board with resistors to represent
hydraulic conductivity and capacitors to
represent storage coefficient
– difficult to calibrate because each change of
material properties involves removing and
resoldering the resistors and capacitors
Electrical Analog Model
Hele Shaw Model
(viscous liquid)
Mathematical Model
• MATHEMATICAL DESCRIPTION OF
SYSTEM
– SIMPLE – ANALYTICAL
• provides a continuous solution over the model
domain
– COMPLEX - NUMERICAL
• provides a discrete solution - i.e. values are
calculated at only a few points
• we are going to focus on numerical models
Numerical Model
Numerical Modeling
• Formation of conceptual models
• Manipulation of modeling software
• Represent a site-specific ground-water
system
• The results are referred to as:
– A model or
– A model application
Steps in a geohydrologic project
1. Define the problem
2. Conceptualize the system
3. Envision how the problem will affect your system
4. Try to find an analytical solution that will provide some
insight to the problem
5. Evaluate if steady state conditions will be indicative of
your problem
(conservative/non-conservative)
6. Evaluate transients if necessary but always consider
conditions at steady
state
Steps in a geohydrologic project
7. SIT BACK AND ASK - DOES THIS RESULT MAKE
SENSE?
8. CONSIDER WHAT YOU MIGHT HAVE LEFT OUT
ENTIRELY
AND HOW THAT MIGHT AFFECT YOUR RESULT
9. Decide if you have solved the problem or if you need
a. more field data
b. a numerical model (time, cost, accuracy)
c. both
Steps in a geohydrologic project
9a. If field data are needed, use your analysis to guide data
collection
what data are needed?
what location should they be collected from?
Steps in a geohydrologic project
9b. If a numerical model is needed, select appropriate
code and when setting up the model
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keep the question to be addressed in mind
keep the capabilities and limitations of the code in mind
plan at least three times as much time as you think it will take
draw the problem and overlay a grid on it
note input values for
• material properties,
• boundary conditions, and
• initial conditions
– run steady-state first!
– plan and conduct transient runs
– always monitor results in detail
Steps in a geohydrologic project
10.Keep the question in focus and the objective in mind
11.Evaluate Sensitivity
12.Evaluate Uncertainty
Steps in a geohydrologic project
KEEP THESE THOUGHTS IN MIND:
1. Numerical models are valuable thinking tools to help you understand the
system. They are not solely for calculating an "answer". They are also
useful in illustrating concepts to others.
2. A numerical modeling project is likely a major undertaking.
3. Capabilities of state-of-the-art models are often primitive compared to the
analytical needs of current ground-water problems.
4. Data for model input is sparse therefore there is a lot of uncertainty in
your results. Report reasonable ranges of answers rather than single
values.
5. DO NOT get discouraged! 99% of modeling is getting the model set up
and working. The predictive phase comprises only a small percentage of
the total modeling effort.
Components of Modeling Project
• Statement of objectives
• Data describing the physical system
• Simplified conceptual representation of the
system
• Data processing and modeling software
• Report with written and graphical
presentations
Steps in the Modeling Process
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Modeling objectives
Data gathering and organization
Development of a conceptual model
Numerical code selection
Assignment of properties and boundary
conditions
• Calibration and sensitivity analysis
• Model execution and interpretation of results
• Reporting
(K.J. Halford, 1991)
Model Accuracy
• Dependant of the level of understanding of
the flow system
• Requirements:
– Some level of site investigation
– Accurate conceptualization
• Old quote:
– “All models are wrong but some are useful”
• Accuracy is always a trade-off between
– resources and
– goals
Determination of Modeling
Needs
• What is the general type of problem to be
solved?
• What features must be simulated to answer the
questions about the system?—study objective
• Can the code simulate the hydrologic features of
the site?
• What dimensional capabilities are needed?
• What is the best solution method?
• What grid discretization is required for simulating
hydrologic features?
Modeling Code Administration
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Is there support for the code?
Is there a user’s manual?
What does it cost?
Is the code proprietary?
Are user references available?
Is the code widely used?
Types of Modeling Codes
• Objective based:
– Ground-water supply
– Well field design
• Process Based:
– Saturated or unsaturated flow
– Contaminate transport
• Physical System Based
• Mathematical
Components of a Mathematical Model
• Governing Equation (Darcy’s law + water
balance eqn) with head (h) as the
dependent variable
• Boundary Conditions
• Initial conditions (for transient problems)
Solution Methods
• In order of increasing complexity:
– Analytical
– Analytical Element
– Numerical
• Finite difference
• Finite element
• Each solves the governing
equation of ground-water flow
and storage
• Different approaches,
assumptions and applicability
Analytical Methods
• Classical mathematical methods
• Resolve differential equations into exact
solutions
• Assume homogeneity
• Limited to 1-D and some 2-D problems
• Can provide rough approximations
• Examples are the Theis or Theim
equations
Theis Equation
Toth Problem
Water Table
Groundwater
divide
AQUIFER
Groundwater
divide
Impermeable Rock
Steady state system: inflow equals outflow
Toth Problem
Water Table
Groundwater
divide
Laplace Equation
Impermeable Rock
2D, steady state
Groundwater
divide
Finite Difference Methods
• Solves the partial differential equation
• Approximates a solution at points in a
square or rectangular grid
• Can be 1-, 2-, or 3-Dimensional
• Relatively easy to construct
• Less flexibility, especially with boundary
conditions
Finite difference models
may be solved using:
• a computer program or code (e.g.,
a FORTRAN program)
• a spreadsheet (e.g., EXCEL)
Finite Difference Grid -- Simple
Finite Difference Grid -Complex
MODFLOW
 a computer code that solves a
groundwater flow model using finite
difference techniques
Several versions available
• MODFLOW 88
• MODFLOW 96
• MODFLOW 2000
• MODFLOW 2005
Finite Element Methods
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Allows more precise calculations
Flexible placement of nodes
Good at defining irregular boundaries
Labor intensive setup
Might be necessary if the direction of
anisotropy varies in the aquifer
Structural features create
anisotropy in this karst system
Finite-Element Mesh for system
Class Focus
• Will use USGS finite-difference model,
MODFLOW, for class presentations and
exercises
• More details on mathematics and
simplifications used in MODFLOW later
Governing Equations
for Ground Water Flow
Conditions and requirements:
• Mass of water must be conserved at every
point in the system
• Rate and direction of flow is related to head
by Darcy’s Law
• Water and porous medium behave as
compressible, elastic materials, so the
volume of water “ stored” in the system can
change as a function of head
Governing Equations
for Ground Water Flow
• Many forms depending on the
assumptions that are valid for the problem
of interest.
• In most cases, it is assumed that the
density of ground water is spatially and
temporally constant.
Governing Equations
for Ground Water Flow
• Conservation of Mass
Starting point for developing 3-D flow equation
Mass In – Mass Out = Change in Mass Stored
(If there is no change in storage, the condition is said to be steadystate. If the storage changes, the condition is said to be transient.)
Small control volume over time in 3 directions
-finite difference and differential forms
-to be useful must be able to express flow rates
and change in storage in terms of head
(measurable variable) --- Darcy’s Law
Governing Equations
for Ground Water Flow
• Darcy’s Law
– 1856 experiment measured flow through sand pack
– generalized relationship for flow in porous media
Darcy’s Law
• Relates direction and rate of ground-water flow to the
distribution of head in the ground-water system
where,
Q = volumetric flow rate (discharge),
A = flow area perpendicular to L (cross sectional area),
K = hydraulic conductivity,
L= flow path length (L = x1 - x0), and
h = hydraulic head
Darcy’s Law
If the soil did not have uniform properties, then we would have
to use the continuous form of the derivative:
Darcy ' s Law: Q( x )   K ( x )  A 
dH
dx
Notice the minus sign on the right hand side of Darcy’s Law. We
do this because in standard notation Q is positive in the same
direction as increasing x, and we take x1 > x0. Notice that since
H0 > H1, the slope of H(x), DH/Dx, is negative. If it had been the
other way around, with H1 > H0, then the negative sign would
ensure that Q would be flowing the other way.
*** hydraulic head always decreases in the direction of flow ***
From D.L. Baker online tutorial
http://www.aquarien.com/sptutor/index.htm
Head
• Head is defined as the elevation to which ground
water will rise in a cased well. Mathematically, head
(h) is expressed by the following equation:
• where
• z = elevation head and
P/pg = pressure head (water table = 0).
Darcy’s Law
Dupuit Simplification
Dupuit's simplification uses the approximate gradient (difference in h over the
distance x rather than the flow path length, l), and uses the average head to
determine the height of the flow area.
Mainly used for unconfined aquifers
"Darcy tube" to
flow in simple aquifers
• LaPlace’s Equation:
– Steady groundwater flow must satisfy not only Darcy's Law but also
the equation of continuity
– 3-Dimensional Steady State flow: Homogeneous, Isotropic Conditions
where there are no changes in storage of fluid
d2h/dx2+d2h/dy2+d2h/dz2=0
– Steady-state version of diffusion equation
– the change of the slope of the head field is zero in the x direction
– hydraulic head is a harmonic function, and has many analogs in other
fields
Assignment:
• If you chose to purchase Applied
Groundwater Modeling:
– read the Preface and Chapters 1 and 2.
• Begin thinking about class project
• Begin looking at journal articles
Pre- and Post- Processors
• Many commercially available programs
• Best allow placement of model grid over a base
map
• Allow numerical output to be viewed as contours,
flow-path maps, etc
• Some popular codes are:
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GMS (Ground Water Modeling System)
Visual MODFLOW
Groundwater Vistas
MFI (USGS for setting up smaller models)