SDSU GEOL 651 - Numerical Modeling of Ground-Water Flow SDSU Coastal Waters Laboratory USGS San Diego Project Office 1st Floor conference room 4165 Spruance.

SDSU GEOL 651 - Numerical SDSU GEOL 651 - Numerical Modeling of Ground-Water FlowModeling 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

IntroductionsIntroductions

• Claudia C. Faunt

• Ph.D. in Geological Engineering from Colorado School of Mines

• Hydrologist with U.S. Geological Survey

• (619) 225-6142

• [email protected]

• Office 2nd floor NE corner

IntroductionsIntroductions

• 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 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 Course Organization • Classes

– First few mostly lectures– Majority

• First half lectures• Second half

– Problem set related to lecture– Model project work

Course TopicsCourse 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?

Tentative SyllabusTentative Syllabus(subject to change to adjust our pace)(subject to change to adjust our pace)

• Handout

Introduction to Introduction to Ground-Water Ground-Water

ModelingModeling

OUTLINE: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

What is a ground-water model?What is a ground-water model?

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

OBJECTIVE: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?Why model?• SOLVE a PROBLEM or

make a PREDICTION• THINKING TOOL

– Understand the system and its responses to stresses

Types of problems that we modelTypes of problems that we model• WATER SUPPLY• WATER INFLOW• WATER OUTFLOW• RATE AND DIRECTION• CONCENTRATION OF CHEMICAL

CONSTITUENTS• EFFECT OF ENGINEERED FEATURES• TEST ANALYSIS

Types of ground-water modelsTypes of ground-water models

• CONCEPTUAL MODEL• GRAPHICAL MODEL• PHYSICAL MODEL• ANALOG MODEL• MATHEMATICAL MODEL

• We will focus on numerical models in this class

Conceptual ModelConceptual Model

• Qualitative description of the system– Think of a cartoon

Graphical ModelGraphical Model

• FLOW NETS– limited to steady state, homogeneous

systems, with simple boundary conditions

Physical ModelPhysical 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 ModelSand Tank Model

Analog ModelAnalog 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 ModelElectrical Analog Model

Hele Shaw ModelHele Shaw Model(viscous liquid)(viscous liquid)

Mathematical ModelMathematical 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 ModelNumerical Model

Numerical ModelingNumerical 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 projectSteps in a geohydrologic project1. 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 steadystate

Steps in a geohydrologic projectSteps in a geohydrologic project7. 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 datab. a numerical model (time, cost, accuracy)c. both

Steps in a geohydrologic projectSteps 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 projectSteps in a geohydrologic project9b. 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

Steps in a geohydrologic projectSteps 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 projectSteps in a geohydrologic projectKEEP 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 ProjectComponents 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 ProcessSteps 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

(K.J. Halford, 1991)

Model AccuracyModel 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 Determination of Modeling NeedsNeeds

• 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 AdministrationModeling 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?

Types of Modeling CodesTypes 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 ModelComponents 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 MethodsSolution 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 MethodsAnalytical 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 EquationTheis Equation

Toth Problem

Impermeable Rock

Groundwater divide

Groundwater divide

Water Table

AQUIFER

Steady state system: inflow equals outflow

Toth Problem

Impermeable Rock

Groundwater divide

Groundwater divideLaplace Equation

2D, steady state

Water Table

Finite Difference MethodsFinite 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 modelsmay be solved using:

• a computer program or code (e.g., a FORTRAN program)

• a spreadsheet (e.g., EXCEL)

Finite Difference Grid -- SimpleFinite Difference Grid -- Simple

Finite Difference Grid -- Finite Difference Grid -- ComplexComplex

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 MethodsFinite 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

Structural features create Structural features create anisotropy in this karst systemanisotropy in this karst system

Finite-Element Mesh for systemFinite-Element Mesh for system

Class FocusClass 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 Governing Equations for Ground Water Flowfor 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 Governing Equations for Ground Water Flowfor 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 Governing Equations for Ground Water Flowfor Ground Water Flow

• Conservation of MassStarting 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

Governing Equations Governing Equations for Ground Water Flowfor Ground Water Flow

• Darcy’s Law– 1856 experiment measured flow through sand pack– generalized relationship for flow in porous media

Darcy’s LawDarcy’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 LawDarcy’s LawIf 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 AdH

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), H/x, 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 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).

HeadHead

Darcy’s LawDarcy’s LawDupuit 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

• 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

"Darcy tube" to "Darcy tube" to flow in simple aquifers flow in simple aquifers

Assignment: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- ProcessorsPre- 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)