1. Groundwater Modeling - 1
Groundwater Hydraulics
Daene C. McKinney

2. Models …? Input (Explanatory Variable) Precipitation
Soil Characteristics
Model
(Represents the Phenomena) ET
Evaporation
Infiltration
Output
(Results – Response variable)
Run off

3. Models and more models …
Hydrologic
Simulation
Simulation
Model
Optimization
Model
Input
(Explanatory Variable)
Inflow Data
Basin Water Allocation Policy
Response to the Policy
Inflow Data
Basin Objectives and Constraints
Optimum Policy
Precip. & Soil Charact.
Mimic Physics of the Basin
Runoff
Model
(Phenomena)
Output
(Results)
Source for Input data of other models
Predict Response to given design/policy
Identify optimal design/policy

4. Problem identification
Modeling Process
Problem identification
and description
Model verification &
sensitivity analysis
Model Documentation
Model application
Model calibration &
parameter estimation
Model
conceptualization
development
Data
Present results 1
Problem identification (1)
Important elements to be modeled
Relations and interactions between them
Degree of accuracy
Conceptualization and development (2 – 3)
Mathematical description
Type of model
Numerical method - computer code
Grid, boundary & initial conditions
Calibration (4)
Estimate model parameters
Model outputs compared with actual outputs
Parameters adjusted until the values agree
Verification (4)
Independent set of input data used
Results compared with measured outputs 2 3 4 5 6 7

5. Tools to Solve Groundwater Problems
Physical and analog methods
Some of the first methods used.
Analytical methods
What we have been discussing so far
Difficult for irregular boundaries, different boundary conditions, heterogeneous and anisotropic properties, multiple phases, nonlinearities
Numerical methods
Transform PDEs governing flow of groundwater into a system of ODEs or algebraic equations for solution

6. Conceptual Model
Descriptive representation of groundwater system incorporating interpretation of geological & hydrological conditions
What processes are important to model?
What are the boundaries?
What parameter values are available?
What parameter values must be collected?

7. What Do We Really Want To Solve?
Horizontal flow in a leaky confined aquifer
Governing Equations
Boundary Conditions
Initial conditions
Flux
Leakage
Source/Sink
Storage
Ground surface
Bedrock
Confined aquifer Qx K x y z h
Head in confined aquifer
Confining Layer b

8. Finite Difference Method
Replace derivatives in governing equations with Taylor series approximations
Generates set of algebraic equations to solve
1st derivatives

9. Taylor Series
Taylor series expansion of h(x) at a point x+Dx close to x
If we truncate the series after the nth term, the error will be

10. First Derivative - Forward
Consider the forward Taylor series expansion of a function h(x) near a point x
Solve for 1st derivative

11. First Derivative - Backward
Consider the backward Taylor series expansion of a function f(x) near a point x
Solve for 1st derivative

12. Finite Difference Approximations
1st Derivative
(Backward)
1st Derivative
(Forward)

13. Grids and Discretrization
Discretization process
Grid defined to cover domain
Goal - predict values of head at node points of mesh
Determine effects of pumping
Flow from a river, etc
Finite Difference method
Popular due to simplicity
Attractive for simple geometry
i,j
i,j+1
i+1,j
i-1,j
i,j-1
x, i
y, j
Domain
Mesh
Node point D x y
Grid cell

14. Three-Dimensional Grids
An aquifer system is divided into rectangular blocks by a grid.
The grid is organized by rows (i), columns (j), and layers (k), and each block is called a "cell"
Types of Layers
Confined
Unconfined
Convertible
j, columns
i, rows
k, layers
Layers can be
different materials

15. 1-D Confined Aquifer Flow
Homogeneous, isotropic, 1-D, confined flow
Governing equation
Initial Condition
Boundary Conditions
Ground surface
Aquifer x y z hB
Confining Layer b hA Dx
i = 1 2 3 4 5 6 7 8 9 10
Node
Grid Cell

16. Derivative Approximations
Need 2nd derivative WRT x

17. Derivative Approximations
Governing Equation
2nd derivative WRT x
Need 1st derivative WRT t
Which one to use?
Forward
Backward

18. Time Derivative Explicit Implicit
Use all the information at the previous time step to compute the value at this time step.
Proceed point by point through the domain.
Implicit
Use information from one point at the previous time step to compute the value at all points of this time step.
Solve for all points in domain simultaneously.

19. Explicit Method
Use all the information at the previous time step to compute the value at this time step.
Proceed point by point through the domain.
Can be unstable for large time steps.
FD Approx.
Forward

20. Explicit Method
l+1 time level
unknown
l time level
known

21. 1-D Confined Aquifer Flow
Initial Condition
Boundary Conditions
Ground surface
Aquifer x y z hB
Confining Layer b hA Dx
i = 1 2 3 4 5 6 7 8 9 10
Node
Grid Cell L
Dx = 1 m
L = 10 m
T=bK = 0.75 m2/d
S = 0.02

22. Explicit Method Consider: r = 0.48 r = 0.52 Dx = 1 m L = 10 m
Ground surface
Aquifer hB
Confining Layer b hA Dx
i = 1 2 3 4 5 6 7 8 9 10
Node
Grid Cell
Consider: r = 0.48
r = 0.52
Dx = 1 m
L = 10 m
T = 0.75 m2/d
S = 0.02

23. Explicit Results (Dt = 18.5 min; r = 0.48 < 0.5)

24. Explicit Results (Dt = 20 min; r = 0.52 > 0.5)

25. What’s Going On Here? At time t = 0  no flow At time t > 0  flow
Water released from storage in a cell over time Dt
Water flowing out of cell over interval Dt
Ground surface
Confining Layer hA
Aquifer Dx hB b Dx
i = 1 2 …
i-1 i
i+1 … 8 9 10
Grid Cell i
r > 0.5
Tme interval is too large
Cell doesn’t contain enough water
Causes instability

26. Implicit Method
Use information from one point at the previous time step to compute the value at all points of this time step.
Solve for all points in domain simultaneously.
Inherently stable
FD Approx.
Backward

27. Implicit Method
l+1 time level
unknown
l time level
known

28. 2-D Steady-State Flow General Equation
Homogeneous, isotropic aquifer, no well
Equal spacing (average of surrounding cells)
Node No.
Unknown heads
Known heads

29. 2-D Heterogeneous Anisotropic Flow
Tx and Ty are transmissivities in the x and y directions

30. 2-D Heterogeneous Anisotropic Flow
Harmonic average transmissivity

31. Transient Problems

32. MODFLOW USGS supported mathematical model
Uses finite-difference method
Several versions available
MODFLOW 88, 96, 2000, 2005
(water.usgs.gov/nrp/gwsoftware/modflow.html)
Graphical user interfaces for MODFLOW:
GWV (
GMS (
PMWIN (
Each includes MODFLOW code

33. What Can MODFLOW Simulate?
Unconfined and confined aquifers
Faults and other barriers
Fine-grained confining units and interbeds
Confining unit - Ground-water flow and storage changes
River – aquifer water exchange
Discharge of water from drains and springs
Ephemeral stream - aquifer water exchange
Reservoir - aquifer water exchange
Recharge from precipitation and irrigation
Evapotranspiration
Withdrawal or recharge wells
Seawater intrusion