groundwater modeling - 1 groundwater hydraulics daene c. mckinney
TRANSCRIPT
Models …?
Input(Explanatory
Variable)
Model(Represents the
Phenomena)
Output(Results – Response
variable) Run off
InfiltrationEvaporation
ET
Precipitation Soil Characteristics
Models and more models …
Input(Explanatory
Variable)
Model(Phenomena)
Output(Results)
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
SimulationModel
OptimizationModel
HydrologicSimulation
Predict Response to given
design/policy
Identify optimal design/policy
Source for Input data of other models
Modeling Process
• 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
Problem identificationand description
Model verification & sensitivity analysis
Model Documentation
Model application
Model calibration & parameter estimation
Model conceptualization
Model development
Data
Present results
1
2
3
4
5
6
7
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
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?
What Do We Really Want To Solve?
• Horizontal flow in a leaky confined aquifer
• Governing Equations• Boundary Conditions• Initial conditions
Ground surface
Bedrock
Confined aquiferQx
K
xyz
h
Head in confined aquifer
Confining Layer
b
Flux Leakage Source/Sink Storage
Finite Difference Method
• Finite-difference method– Replace derivatives in governing equations with
Taylor series approximations– Generates set of algebraic equations to solve
1st derivatives
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
xxx x
First Derivative - Forward • Consider the forward Taylor series expansion of a function
h(x) near a point x
• Solve for 1st derivative
xxx
x
xxx
x
First Derivative - Backward • Consider the backward Taylor series expansion of a function
f(x) near a point x
• Solve for 1st derivative
xxx
x
xxx
x
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
D y
Grid cell
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
Layers can be different materials
i, rows
j, columns
k, layers
1-D Confined Aquifer Flow
• Homogeneous, isotropic, 1-D, confined flow
• Governing equation
• Initial Condition
• Boundary Conditions
Ground surface
Aquifer
x
yz
hB
Confining Layer
b
hA
Dx
i = 0 1 2 3 4 5 6 7 8 9 10
Node
Grid Cell
Derivative Approximations• Governing Equation
• 2nd derivative WRT x
• Need 1st derivative WRT t
Forward Backward
li ,1
ix,
lt,
1, li
li ,1
1, li
x
t
li,
Which one to use?
Time Derivative• Explicit
– 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.
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.
li ,1
1, li
li ,1
1, li
x
tli,
FD Approx.Forward
1-D Confined Aquifer Flow
• Initial Condition
• Boundary Conditions
Ground surface
Aquifer
x
yz
hB
Confining Layer
b
hA
Dx
i = 0 1 2 3 4 5 6 7 8 9 10
Node
Grid CellL
Dx = 1 m
L = 10 m
T=bK = 0.75 m2/d
S = 0.02
Explicit MethodGround surface
Aquifer
hB
Confining Layer
b
hA
Dx
i = 0 1 2 3 4 5 6 7 8 9 10
Node
Grid CellConsider: r = 0.48
r = 0.52 Dx = 1 mL = 10 mT = 0.75 m2/dS = 0.02
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
Aquifer
hB
Confining Layer
b
hA
Dx
i = 0 1 2 … i-1 i i+1 … 8 9 10
Dx
Grid Cell i
r > 0.5Tme interval is too large Cell doesn’t contain enough water Causes instability
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
li ,1
ix,
lt,
1, li
li ,1
1, li
x
tli,
1,1 li1,1 li
1,1 li 1,1 li
FD Approx. Backward
2-D Steady-State Flow
• General Equation
• Homogeneous, isotropic aquifer, no well
• Equal spacing (average of surrounding cells)
jy,
ix,x
y
)4,1( )4,2( )4,3( )4,4(
)3,1( )3,2( )3,3( )3,4(
)2,1( )2,2( )2,3( )2,4(
)1,1( )1,2( )1,3( )1,4(
)0,1( )0,2( )0,3( )0,4(
)5,1( )5,2( )5,3( )5,4(
)4,0(
)3,0(
)2,0(
)1,0(
)4,5(
)3,5(
)2,5(
)1,5(
)4,5(
)5,1(Node No. Unknown heads
Known heads
2-D Heterogeneous Anisotropic Flow
j+ 1
j-1
j
i-1
i i+ 1
i+ 1 /2
j+ 1 /2
j-1 /2
x
y
Q x ,i+ 1 /2 Q x ,i-1 /2
Q y ,j+ 1 /2
Q y ,j-1 /2
x
y
n o d e ( i ,j) i-1 /2
ce ll ( i ,j)
Tx and Ty are transmissivities in the x and y directions
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 (www.groundwater-vistas.com)
– GMS (www.ems-i.com)
– PMWIN (www.ifu.ethz.ch/publications/software/pmwin/index_EN)
– Each includes MODFLOW code
What Can MODFLOW Simulate?
1. Unconfined and confined aquifers2. Faults and other barriers3. Fine-grained confining units and
interbeds 4. Confining unit - Ground-water flow
and storage changes 5. River – aquifer water exchange6. Discharge of water from drains
and springs7. Ephemeral stream - aquifer water
exchange8. Reservoir - aquifer water exchange9. Recharge from precipitation and
irrigation 10. Evapotranspiration 11. Withdrawal or recharge wells12. Seawater intrusion