1. 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
Tuesdays 4 -7 PM

2. Introductions Claudia C. Faunt
Ph.D. in Geological Engineering from Colorado School of Mines
Hydrologist with U.S. Geological Survey
(619)
Office 2nd floor NE corner

3. 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)

4. Course Organization Organizational Meeting Grading (details next week)
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

5. Course Organization Classes First few mostly lectures Majority
First half lectures
Second half
Problem set related to lecture
Model project work

6. 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?

7. Tentative Syllabus (subject to change to adjust our pace)
Handout

8. Introduction to Ground-Water Modeling

9. 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

10. What is a ground-water model?
A replica of a “real-world” ground-water system

11. 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

12. Why model? SOLVE a PROBLEM or make a PREDICTION THINKING TOOL
Understand the system and its responses to stresses

13. 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

14. Types of ground-water models
CONCEPTUAL MODEL
GRAPHICAL MODEL
PHYSICAL MODEL
ANALOG MODEL
MATHEMATICAL MODEL
We will focus on numerical models in this class

15. Conceptual Model Qualitative description of the system
Think of a cartoon

16. Graphical Model FLOW NETS
limited to steady state, homogeneous systems, with simple boundary conditions

17. 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

18. Sand Tank Model

19. 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

20. Electrical Analog Model

21. Hele Shaw Model (viscous liquid)

22. 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

23. Numerical Model

24. 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

25. 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

26. 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

27. 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?

28. 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

29. Steps in a geohydrologic project
10.Keep the question in focus and the objective in mind 11.Evaluate Sensitivity 12.Evaluate Uncertainty

30. 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.

31. 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

32. 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

34. (K.J. Halford, 1991)

35. 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

36. 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?

37. Modeling Code Administration
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?

38. Types of Modeling Codes
Objective based:
Ground-water supply
Well field design
Process Based:
Saturated or unsaturated flow
Contaminate transport
Physical System Based
Mathematical

39. 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)

40. 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

41. 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

42. Theis Equation

43. Toth Problem Water Table Groundwater Groundwater AQUIFER divide divide
Impermeable Rock
Steady state system: inflow equals outflow

44. Toth Problem Laplace Equation 2D, steady state Water Table Groundwater
divide
Groundwater
divide
Laplace Equation
Impermeable Rock
2D, steady state

45. 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

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

47. Finite Difference Grid -- Simple

48. Finite Difference Grid -- Complex

49. 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

50. 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

51. Structural features create anisotropy in this karst system

52. Finite-Element Mesh for system

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

54. 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

55. 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.

56. 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) --- Darcy’s Law

57. Governing Equations for Ground Water Flow
Darcy’s Law
1856 experiment measured flow through sand pack
generalized relationship for flow in porous media

58. 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

59. From D.L. Baker online tutorial
Darcy’s 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 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

60. 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).

62. Darcy’s Law Dupuit Simplification Mainly used for unconfined aquifers
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

63. "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

64. 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

65. 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)