SDSU GEOL 651 - Numerical Modeling of Ground-Water Flow

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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

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Introductions

• Claudia C. Faunt
• Ph.D. in Geological Engineering from Colorado
  School of Mines
• Hydrologist with U.S. Geological Survey
• [phone]
• ccfaunt_at_usgs.gov
• Office 2nd floor NE corner

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Introductions

• 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)

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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

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Course Organization

• Classes
• First few mostly lectures
• Majority
• First half lectures
• Second half
• Problem set related to lecture
• Model project work

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Course Topics

• 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?

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Tentative Syllabus(subject to change to adjust
our pace)

• Handout

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Introduction to Ground-Water Modeling
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OUTLINE

• 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

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What is a ground-water model?

• A replica of a real-world ground-water system

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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

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Why model?

• SOLVE a PROBLEM or make a PREDICTION
• THINKING TOOL
• Understand the system and its responses to
  stresses

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Types of problems that we model

• WATER SUPPLY
• WATER INFLOW
• WATER OUTFLOW
• RATE AND DIRECTION
• CONCENTRATION OF CHEMICAL CONSTITUENTS
• EFFECT OF ENGINEERED FEATURES
• TEST ANALYSIS

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Types of ground-water models

• CONCEPTUAL MODEL
• GRAPHICAL MODEL
• PHYSICAL MODEL
• ANALOG MODEL
• MATHEMATICAL MODEL
• We will focus on numerical models in this class

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Conceptual Model

• Qualitative description of the system
• Think of a cartoon

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Graphical Model

• FLOW NETS
• limited to steady state, homogeneous systems,
  with simple boundary conditions

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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

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Sand Tank Model
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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

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Electrical Analog Model
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Hele Shaw Model(viscous liquid)
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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

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Numerical Model
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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

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Steps in a geohydrologic project

• 1. Define the problem2. Conceptualize the
  system3. Envision how the problem will affect
  your system4. Try to find an analytical
  solution that will provide some insight to the
  problem5. Evaluate if steady state conditions
  will be indicative of your problem(conservative/n
  on-conservative)6. Evaluate transients if
  necessary but always consider conditions at
  steadystate

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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
  RESULT9. Decide if you have solved the problem
  or if you need
• a. more field data
• b. a numerical model (time, cost, accuracy)
• c. both

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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?

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Steps in a geohydrologic project

• 9b. If a numerical model is needed, select
  appropriate code and when setting up the model
• 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

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Steps in a geohydrologic project

• 10.Keep the question in focus and the objective
  in mind11.Evaluate Sensitivity12.Evaluate
  Uncertainty

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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.

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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

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Steps in the Modeling Process

• 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

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(K.J. Halford, 1991)
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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

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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?

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Modeling Code Administration

• Is there support for the code?
• Is there a users manual?
• What does it cost?
• Is the code proprietary?
• Are user references available?
• Is the code widely used?

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Types of Modeling Codes

• Objective based
• Ground-water supply
• Well field design
• Process Based
• Saturated or unsaturated flow
• Contaminate transport
• Physical System Based
• Mathematical

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Components of a Mathematical Model

• Governing Equation (Darcys law water balance
  eqn) with head (h) as the dependent variable
  
• Boundary Conditions
• Initial conditions (for transient problems)

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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

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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

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Theis Equation
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Toth Problem
Water Table
Groundwater divide
Groundwater divide
AQUIFER
Impermeable Rock
Steady state system inflow equals outflow
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Toth Problem
Water Table
Groundwater divide
Groundwater divide
Laplace Equation
Impermeable Rock
2D, steady state
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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

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• Finite difference models
• may be solved using
• a computer program or code (e.g., a FORTRAN
  program)
• a spreadsheet (e.g., EXCEL)

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Finite Difference Grid -- Simple
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Finite Difference Grid -- Complex
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• 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

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Finite Element Methods

• 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

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Structural features create anisotropy in this
karst system
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Finite-Element Mesh for system
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Class Focus

• Will use USGS finite-difference model, MODFLOW,
  for class presentations and exercises
• More details on mathematics and simplifications
  used in MODFLOW later

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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
  Darcys 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

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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.

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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 steady-state. 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) --- Darcys Law

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Governing Equations for Ground Water Flow

• Darcys Law
• 1856 experiment measured flow through sand pack
• generalized relationship for flow in porous media

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Darcys 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), andh hydraulic head

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Darcys Law
If the soil did not have uniform properties, then
we would have to use the continuous form of the
derivative
Notice the minus sign on the right hand side of
Darcys Law. We do this because in standard
notation Q is positive in the same direction as
increasing x, and we take x1 gt x0. Notice that
since H0 gt H1, the slope of H(x), DH/Dx, is
negative. If it had been the other way around,
with H1 gt 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.aquarie
n.com/sptutor/index.htm
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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 andP/pg pressure head
  (water table 0).

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Darcys 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
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"Darcy tube" to flow in simple aquifers

• LaPlaces 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/dx2d2h/dy2d2h/dz20
• 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

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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

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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
• GMS (Ground Water Modeling System)
• Visual MODFLOW
• Groundwater Vistas
• MFI (USGS for setting up smaller models)