MODFLOW: A Finite-Difference Groundwater Flow Modelor an Integrated Finite-Difference Groundwater Flow Model?

BALLEAU GROUNDWATER, INC.

Presented by

DAVE M. ROMERO

• Discretization ProcessesFinite DifferenceIntegrated Finite DifferenceMODFLOW

• MODFLOW Modifications

• Test Problem

DISCUSSION TOPICS

Governing PDE with BC’s and IC Discretize System of Linear

Algebraic Equations

Equation SolverApproximate Solution

FINITE-DIFFERENCE DISCRETIZATION

dtdhSq

zhK

zyhK

yxhK

x szyx =+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

∂∂+⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

Governing Flow Equation

Homogenous Form

dtdhSq

zhK

yhK

xhK szyx =+

∂∂+

∂∂+

∂∂

2

2

2

2

2

2

⎥⎦⎤

⎢⎣⎡T1

⎥⎦⎤

⎢⎣⎡T1

x∆ x∆i i+1i-1

xhh

xh ii

∆−≅

∂∂ −1

( )211

11

2

2 2x

hhhx

xhh

xhh

xh

xxh iii

iiii

∆+−=

∆∆−−

∆−

≅⎟⎠⎞

⎜⎝⎛

∂∂

∂∂=

∂∂ +−

+−

First Derivative

Second Derivative

Continuous Governing Equation

( ) ( ) +∆

+−+

∆+− +−+−

2,,1,,,,1

2,1,,,,1, 22

yhhh

Kx

hhhK kjikjikji

ykjikjikji

x

( ) 1

1,,,,

,,,,21,,,,1,, 2

−

−+−

−−

=+∆

+−

mm

mkji

mkji

kjikjikjikjikji

z tthh

Ssqz

hhhK

dtdhSq

zhK

yhK

xhK szyx =+

∂∂+

∂∂+

∂∂

2

2

2

2

2

2

Discrete Finite-Difference Formulation

⎥⎦⎤

⎢⎣⎡T1

⎥⎦⎤

⎢⎣⎡T1

Governing PDE with BC’s and IC

Discretize

System of LinearAlgebraic Equations Equation Solver

Approximate Solution

INTEGRATED FINITE-DIFFERENCE DISCRETIZATION

Spatially Integrate PDE(small finite subregion)

Apply Divergence Theorem(volume integral of net outflux

converted into surface integral)

∑=

−

∆−=+−c

n

km

km

mmmmmn

mnmnmn t

hhVSsVqL

hhAK1

1

dtdhSq

zhK

zyhK

yxhK

x szyx =+⎟⎠⎞

⎜⎝⎛

∂∂

∂∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

∂∂+⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

Governing Flow Equation

Spatially integrate, apply Divergence Theorem& discretize to get

( )∑=

−

∆−=+−

c

n

km

km

mmmmmnmn thhVSsVqhhC

1

1

LKAC =If , then

⎥⎦

⎤⎢⎣

⎡TL3

⎥⎦⎤

⎢⎣⎡T1

Discretize Darcy’s Lawfor flow through the six facesof a 3-D Block-Centered Cell

Discretize Time

System of LinearAlgebraic Equations Equation Solver

Approximate Solution

MODFLOW’S DISCRETIZATION

Derive a Source/Sink Termto Account for External

Flow Rates

Apply Continuity: Sum of All FlowsIn & Out of Cell is Equal to Time

Rate of Change of Storage in Cell

m

( )∑= −

−

−−=++−

6

1 1

1

n kk

kk

mmmkmm

km

knmn tt

hhVSsQhPhhC

Discretize Darcy’s Law, derive source/sink term& apply continuity to get

⎥⎦

⎤⎢⎣

⎡TL3

( )∑= −

−

−−=++−

6

1 1

1

n kk

kk

mmmkmm

km

knmn tt

hhVSsQhPhhC

( )∑=

−

∆−=+−

c

n

km

km

mmmmmnmn thhVSsVqhhC

1

1

( ) ( ) +∆

+−+

∆+− +−+−

2,,1,,,,1

2,1,,,,1, 22

yhhh

Kx

hhhK kjikjikji

ykjikjikji

x

( ) 1

1,,,,

,,,,21,,,,1,, 2

−

−+−

−−

=+∆

+−

mm

mkji

mkji

kjikjikjikjikji

z tthh

Ssqz

hhhK

Finite-Difference

Integrated Finite-Difference

MODFLOW

SUMMARY OF DISCRETIZATION SCHEMES

⎥⎦

⎤⎢⎣

⎡TL3

⎥⎦

⎤⎢⎣

⎡TL3

⎥⎦⎤

⎢⎣⎡T1

Non-Generalized Integrated Finite-Difference Grid

Finite-Difference Grid

IMPLICATION OF IFD NUMERICAL SCHEME

ENABLING MODFLOW TO UTILIZE ITS IFD METHOD

• Calculate Cell Areas

• Two-Dimensional DELR and DELC arrays

• Adjust Conductance Calculation

ADJUSTMENT TO CONDUCTANCE CALCULATION

ijji

jiij LTLT

wTTC

+=

2

Reduced Form of Conductance Calculation

Unreduced Form of Conductance Calculation

ijjjii

jjiiij LwTLwT

wTwTC

+=

2

STEADY-STATE RADIAL FLOW PROBLEM

h1=5 ft h2=15 ft

DIMENSIONLESS HEAD vs DIMENSIONLESS RADIUS

1

1.5

2

2.5

3

3.5

1 1.5 2 2.5 3 3.5 4 4.5

r/r1

h/h 1

ANALYTICAL SOLUTION

ADAPTED MODFLOW

i

j

i

j

MODFLOW Grid withIFD Adaptations

CONCLUSIONS

• MODFLOW implements a non-generalized IFDnumerical scheme within the confines of a finite difference grid.

• Minor modifications can be made to MODFLOW’s source code to enable flow simulations through a curvilinear grid constructed from trapezoidal cells.

• The modifications maintain compatibility with other MODFLOW packages and increase the versatility of grid construction.