06 groundwater modeling 1
TRANSCRIPT
![Page 1: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/1.jpg)
Groundwater Modeling - 1
Groundwater Hydraulics
Daene C. McKinney
![Page 2: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/2.jpg)
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
![Page 3: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/3.jpg)
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
![Page 4: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/4.jpg)
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?
![Page 5: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/5.jpg)
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
![Page 6: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/6.jpg)
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
![Page 7: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/7.jpg)
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
![Page 8: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/8.jpg)
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
![Page 9: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/9.jpg)
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
![Page 10: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/10.jpg)
Second Derivative - Central
Add and solve for
![Page 11: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/11.jpg)
Finite Difference Approximations
xx x
1st Derivative(Backward)
1st Derivative(Forward)
2nd Derivative(Central)
![Page 12: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/12.jpg)
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
![Page 13: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/13.jpg)
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
![Page 14: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/14.jpg)
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
![Page 15: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/15.jpg)
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,
![Page 16: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/16.jpg)
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.
![Page 17: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/17.jpg)
Explicit Method
l+1 time levelunknown
l time levelknown
![Page 18: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/18.jpg)
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
![Page 19: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/19.jpg)
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
![Page 20: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/20.jpg)
Explicit Results (t = 18.5 min; r = 0.48 < 0.5)
![Page 21: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/21.jpg)
Explicit Results (t = 20 min; r = 0.52 > 0.5)
![Page 22: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/22.jpg)
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
![Page 23: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/23.jpg)
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.
![Page 24: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/24.jpg)
Implicit Method
l+1 time levelunknown
l time levelknown
![Page 25: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/25.jpg)
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
![Page 26: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/26.jpg)
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
![Page 27: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/27.jpg)
2-D Heterogeneous Anisotropic Flow• Harmonic average transmissivity
![Page 28: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/28.jpg)
Transient Problems
![Page 29: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/29.jpg)
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
![Page 30: 06 Groundwater Modeling 1](https://reader033.vdocuments.mx/reader033/viewer/2022061105/54409ca4b1af9f650b8b46c3/html5/thumbnails/30.jpg)
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