Groundwater Modeling - 1
Groundwater Hydraulics
Daene C. McKinney
Models …?
Input
(Explanatory
Variable)
Precipitation
Model
(Represents the
Phenomena)
Soil
Characteristics
ET
Evaporation
Infiltration
Output
(Results – Response
variable)
Run off
Models and more models …
Input
(Explanatory
Variable)
Model
(Phenomena)
Output
(Results)
Hydrologic
Simulation
Simulation
Model
Optimization
Model
Precip. & Soil
Charact.
Inflow Data
Inflow Data
Mimic Physics of
the Basin
Basin Water
Allocation
Policy
Basin
Objectives and
Runoff
Response to
the Policy
Optimum
Policy
Constraints
Source for Input
Predict Response Identify optimal
data of other models
to given
design/policy
design/policy
Modeling Process
•
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
•
1
Model
conceptualization
2
Model
development
3
Problem identification (1)
– Important elements to be modeled
– Relations and interactions between them
– Degree of accuracy
•
Problem identification
and description
Verification (4)
– Independent set of input data used
– Results compared with measured outputs
Data
Model calibration &
parameter estimation
4
Model verification &
sensitivity analysis
Model Documentation
5
Model application
6
Present results
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
W
¶ æ ¶h ö ¶ æ ¶h ö K'
¶h
çTx ÷ + çTy ÷ + (h0 - h) ± å Qwd(x - x w ) = S
¶x è ¶x ø ¶y è ¶y ø b'
¶t
w=1
Flux
Leakage
Source/Sink Storage
Ground surface
• Governing Equations
• Boundary Conditions
• Initial conditions
Head in confined aquifer
Confining Layer
Qx
z
y
x
Bedrock
Confined aquiferb
K
h
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
x
close to x
x
x  Dx
• 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
h(x)
derivative
h(x)
Dx
x  Dx
Dx
x
x  Dx
x
First Derivative - Backward
• Consider the backward Taylor series expansion of a function
f(x) near a point x
• Solve for 1st derivative
h(x)
h(x)
Dx
x  Dx
Dx
x
x  Dx
x
First Derivative Approximations
h(x)
hi-1
hi+1
hi
Dx
ii-1
1st x Derivative
(Backward)
1st x Derivative
(Forward)
Dx
i
i +1
x
First Derivative Approximations
t, l
i, l  1
1st
t Derivative
(Backward)
Dt
i  1, l
i  1, l
i, l
Dx
i, l  1
1st t Derivative
(Forward)
x, i
Second Derivative Approximation
h(x)
hi-1
hi+1
hi
Dx
i -1
i -1/ 2
i
i +1/ 2
i +1
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
Notation
y, j
Mesh
Domain
i,j+1
Dy
i-1,j
i,j
i+1,j
i,j-1
Dx
h(x, y, z,t) = hi,l j,k
Grid cell
Node point
x, i
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
j, columns
– Confined
– Unconfined
– Convertible
i, rows
k, layers
Layers can be
different materials
1-D Confined Aquifer Flow
• Homogeneous, isotropic,
1-D, confined flow
• Governing equation
¶ æ ¶h ö
¶h
çT ÷ = S
¶x è ¶x ø
¶t
Ground surface
Confining Layer
hA
Dx
Aquifer
Node
hB
• Initial Condition
h(x, t = 0) = h0
b
z
y
x
i= 0
• Boundary Conditions
h(x = 0, t) = hA
h(x = L, 0) = hB
1
2
3
4
5
Grid Cell
6
7
8
9
10
Derivative Approximations
• Governing Equation
¶2 h
S ¶h
=
¶x 2 T ¶t
t, l
i, l  1
• LHS - 2nd derivative WRT x
Dt
i  1, l
i  1, l
i, l
Dx
i, l  1
• RHS - 1st derivative WRT t
Which one to use?
Forward
Backward
x, i
Time Derivative
t - axis (index l)
• 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.
i, l + 1
Dt
i - 1, l
Dx
x - axis (index i)
i, l - 1
• 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.
i + 1, l
i, l
t, l
i - 1, l + 1
i + 1, l + 1
i, l + 1
Dt
i - 1, l
i + 1, l
i, l
Dx
i - 1, l - 1
i, l - 1
x, i
i + 1, l - 1
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.
t - axis (index l)
i, l  1
Dt
i  1, l
i  1, l
i, l
Dx
i, l  1
¶2 h
S ¶h
=
¶x 2 T ¶t
PDE
Finite Difference Approx.
x - axis (index i)
Explicit Method
t - axis (index l)
i, l  1
Dt
i  1, l
i  1, l
i, l
Dx
i, l  1
(
l
l
hil+1 = hil + r hi-1
- 2hil + hi+1
l+1 time level l time level
unknown
known
)
x - axis (index i)
1-D Confined Aquifer Flow
• Initial Condition
Ground surface
h(x,0) = 6.1 m
• Boundary Conditions
Confining Layer
hA
h(0, t) = 6.1 m
Dx
Aquifer
Node
h(L, t) = 1.5 m
hB
b
z
Dx = 1 m
y
x
i= 0
L = 10 m
T=bK = 0.75 m2/d
S = 0.02
1
2
3
4
5
Grid Cell
6
7
L
8
9
10
Explicit Method
hil+1 = hil
l
+ r(hi-1
- 2hil
Ground surface
l
+ hi+1
)
Confining Layer
hA
Dx
Aquifer
Node
hB
b
i= 0
1
2
3
4
5
6
Grid Cell
Consider:
r = 0.48
r = 0.52
Dt = rDx 2
S
0.02
= (0.48)(1)2
= 0.0128d » 18.5min
T
0.75
7
8
9
10
Explicit Results (Dt = 18.5 min; r = 0.48 < 0.5)
Explicit Results (Dt = 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 Dt
Ground surface
Confining Layer
hA
Dx
Aquifer
• Water flowing out of cell
over interval Dt
hB
b
Dx
i= 0
1
2
…
i-1
i
i+1
…
8
Grid Cell i
T
Dh
Dt £ SDxDh
Dx / 2
r=
T Dt 1
£
2
S Dx
2
r > 0.5
Tme interval is too large
Cell doesn’t contain enough water
Causes instability
9
10
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
t, l
i  1, l  1
i  1, l  1
i, l  1
Dt
i  1, l
i  1, l
i, l
x, i
Dx
i  1, l  1
i, l  1
i  1, l  1
¶2 h
S ¶h
=
¶x 2 T ¶t
FD Approx.
Backward
Implicit Method
l+1
l+1
-rhi-1
+ (1+ 2r)hil+1 - rhi+1
= hil
l+1 time level
unknown
l time level
known
2-D Steady-State Flow
y, j
• General Equation
¶ æ ¶h ö ¶ æ ¶h ö W
¶h
çTx ÷ + çTy ÷ ± å Qw = S
¶x è ¶x ø ¶y è ¶y ø w=1
¶t
• Homogeneous, isotropic aquifer, no well
¶ h
2
¶x
2
+
¶ h
2
¶y
2
Node No.
(1,5) Unknown heads
(5,4) Known heads
(1,5) (2,5) (3,5) (4,5)
(0,4) (1,4) (2,4) (3,4) (4,4) (5,4)
Dy
=0
(0,3) (1,3) (2,3) (3,3) (4,3) (5,3)
(0,2) (1,2) (2,2) (3,2) (4,2) (5,2)
(0,1) (1,1) (2,1) (3,1) (4,1)
(5,1)
(1,0) (2,0) (3,0) (4,0)
Dx
• Equal spacing (average of surrounding cells)
hi,j =
hi-1,j + hi+1,j + hi,j-1 + hi,j+1
4
x, i
2-D Heterogeneous Anisotropic Flow
y
¶ æ x ¶h ö ¶ æ y ¶h ö
çT
÷ + çT
÷= 0
¶x è ¶x ø ¶y è ¶y ø
node (i,j)
i+1/2
i-1/2
Dx
cell (i,j
j+1
Qy,j+1/2
j+1/2
Tx and Ty are transmissivities in the
x and y directions
Qx,i-1/2
Qx,i+1/2
j
Dy
j-1/2
Qy,j-1/2
j-1
i-1
i
i+1
2-D Heterogeneous Anisotropic Flow
• Harmonic average transmissivity
x
Ti+1,j
Ti,jx
x
Ti+1/2,j
=2 x
Ti+1,j+Ti,jx
Ai,j hi+1,j + Bi,j hi-1,j + Ci,j hi,j+1 + Di,j hi,j-1 + Ei,j hi,j = 0
Transient Problems
¶ æ x ¶h ö ¶ æ y ¶h ö
¶h
=
S
çT
÷ + çT
÷
¶x è ¶x ø ¶y è ¶y ø
¶t
l+1
l+1
l+1
l+1
l+1
l
¢ hi,j
Ai,j hi+1,j
+ Bi,j hi-1,j
+ Ci,j hi,j+1
+ Di,j hi,j-1
+ Ei,j
= Fi,j hi,j
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 (http://www.groundwatermodels.com/)
– GMS (http://www.aquaveo.com/software/gms-groundwater-modeling-system-introduction)
– PMWIN (www.ifu.ethz.ch/publications/software/pmwin/index_EN)
– Each includes MODFLOW code
What Can MODFLOW Simulate?
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
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