06 groundwater modeling 1
TRANSCRIPT
Groundwater Modeling - 1
Groundwater Hydraulics
Daene C. McKinney
Modeling Process
• Problem identification– Important elements to be modeled – Relations and interactions between them– Degree of accuracy
• Conceptualization and development– Mathematical description– Type of model – Numerical method - computer code– Grid, boundary & initial conditions
• Calibration– Estimate model parameters– Model outputs compared with actual outputs– Parameters adjusted until the values agree
• Verification– Independent set of input data used – Results compared with measured outputs
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
www.epa.state.oh.us
www.isws.illinois.edu
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
2nd derivatives
Taylor Series
• Taylor series expansion of h(x) at a point x+x close to x
• If we truncate the series after the nth term, the error will be
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
Second Derivative - Central
Add and solve for
Finite Difference Approximations
xx x
1st Derivative(Backward)
1st Derivative(Forward)
2nd Derivative(Central)
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
x
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
1-D Confined Aquifer Flow • Homogeneous, isotropic,
1-D, confined flow• Governing equation
• Initial Condition
• Boundary Conditions
Ground surface
Aquifer
xyz
hB
Confining Layer
b
hA
x
i = 0 1 2 3 4 5 6 7 8 9 10
Node
Grid Cell
Derivative Approximations• Governing Equation
• Need 2nd derivative WRT x
• Need 1st derivative WRT t
Forward Backward
li ,1
ix,
lt,
1, li
li ,1
1, li
x
t
li,
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.
Explicit Method
l+1 time levelunknown
l time levelknown
1-D Confined Aquifer Flow • Initial Condition
• Boundary Conditions
Ground surface
Aquifer
xyz
hB
Confining Layer
b
hA
x
i = 0 1 2 3 4 5 6 7 8 9 10
Node
Grid CellL
x = 1 m
L = 10 m
T=bK = 0.75 m2/d
S = 0.02
Explicit MethodGround surface
Aquifer
hB
Confining Layer
b
hA
x
i = 0 1 2 3 4 5 6 7 8 9 10
Node
Grid CellConsider: r = 0.48
r = 0.52 x = 1 mL = 10 mT = 0.75 m2/dS = 0.02
Explicit Results (t = 18.5 min; r = 0.48 < 0.5)
Explicit Results (t = 20 min; r = 0.52 > 0.5)
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 t
• Water flowing out of cell over interval t
Ground surface
Aquifer
hB
Confining Layer
b
hA
x
i = 0 1 2 … i-1 i i+1 … 8 9 10
x
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.
Implicit Method
l+1 time levelunknown
l time levelknown
2-D Steady-State Flow
• General Equation
• Homogeneous, isotropic aquifer, no well
• Equal spacing (average of 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
i+ 1 /2
j+ 1 /2
y
Q x ,i+ 1 /2 Q x ,i-1 /2
Q y ,j+ 1 /2
x
y
n o d e ( i,j ) i-1 /2
c e ll ( i ,j)
Tx and Ty are transmissivities in the x and y directions
2-D Heterogeneous Anisotropic Flow• Harmonic average transmissivity
Transient Problems
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