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Modelling of groundwater flow and solute transport
PGI: Aug. 22-24, 2002
Dr.ir. Gualbert Oude Essink1. Netherlands Institute of Applied Geosciences
2. Faculty of Earth Sciences, Free University of Amsterdam
Mathematical models in hydrogeology
Summer School
Polish Geological Institute
• Delft University of Technology, Civil Engineering: till 1997Ph.D.-thesis: Impact of sea level rise on groundwater flow
regimes
• Utrecht University, Earth Sciences: till 2002Lectures in groundwater modelling and transport processes
Variable-density groundwater flowSalt water intrusion and heat transport
• Netherlands Institute of Applied Geosciences (NITG): till 2004Salinisation processes in Dutch coastal aquifers
• Free University of Amsterdam, Earth Sciences: till 2004EC-project CRYSTECHSALIN (o.a. with J. Bear)Crystallisation processes in porous media
NITG: g.oudeessink@nitg.tno.nlVU: oudg@geo.vu.nl
Curriculum Vitae
• Modelling protocol• Discretisation Partial Differential Equation
(PDE)• Groundwater flow: MODFLOW• Solute transport: MOC3D
Modelling of groundwater flow and solute transport
PGI: Aug. 22-24, 2002
Programme (I)
Programme (II)
PGI: Aug. 22-24, 2002
Thursday Augustus 22, 2002: modelling protocol
Darcy’s Law, steady state continuity equationLaplace-equationTaylor series developmentboundary conditions: Dirichlet, Neumann, Cauchynon-steady state: explicit, implicit, Crank-Nicolson
Friday Augustus 23, 2002:groundwater flow with MODFLOW: mathematical description
short introduction Graphical User Interface PMWIN
Saturday Augustus 24, 2002:solute transport with MOC3D: mathematical description
numerical dispersion
The Hydrological Circle
Ten steps of the Modelling Protocol
1. Problem definition2. Purpose definition3. Conceptualisation4. Selection computer code5. Model design6. Calibration7. Verification8. Simulation9. Presentation10. Postaudit
Why numerical modelling?
-:• simplification of the reality• only a tool, no purpose on itself• garbage in=garbage out: (field)data important• perfect fit measurement and simulation is
suspicious
Modelling protocol
+:• cheaper than scale models• analysis of very complex systems is possible• a model can be used as a database• simulation of future scenarios
3. Conceptualisation (I)
Model is only a schematisation of the reality
Which hydro(geo)logical processes are relevant?
Which processes can be neglected?
Boundary conditions
Variables and parameters:• subsoil parameters• fluxes in and out• initial conditions• geochemical data
Mathematical equations
Modelling protocol
3. Concept (II): example of salt water intrusion
polder areadunes
Salt Water Intrusion
saline
seepage
naturalgroundwater recharge
brackish
fresh
infiltration
sea
extractioninfiltration
freshwater lens
Modelling protocol
Boundary conditions• no flow at bottom• flux in dune area• constant head in polder
area
Relevant processes:• groundwater flow in a heterogeneous porous medium• solute transport• variable density flow• natural recharge• extraction of groundwater
Negligible processes:• heat flow• swelling of clayey aquitard• non-steady groundwater flow
The Netherlands
4. Selection computer code (II)
Groundwatercomputercodes
M at hemat ical models Phys ical models
A nalyt ical models S cale modelsN umer ical models A nalogue models
D et er minist ic S t ochast ic
D ir ect comput at ion I nver se modelling
S at ur at ed fl ow U nsat ur at ed fl ow Coupled models
O ne fl uid T wo or mor e fl uids D ef or mat ion models
O nly gr oundwat er fl ow D issolved solut es H eat t r anspor t
Por ous media F r act ur ed r ocks
S t eady- st at e T r ansient
1D 2 D 3 D
one aquif er mult ilayer ed syst em
D iscr et e f r act ur e modelsDual por osit y models
- M ont e Car lo simulat ion- Random W alk model
- O pt imalisat ion t echnique- Pumping t est s
- f ault s- j oint s
- salt wat er : densit y- dependent- chemical r eact ions- cont aminant t r anspor t
Hemker (1994)
4. Selection computer code
There are numerous good groundwater computercodes available, so don’t make your own code!
See the internet, e.g.:
• U.S. Geological Survey: water.usgs.gov/nrp/gwsoftware/• Scientific Software Group: www.scisoftware.com/• Waterloo Hydrogeologic: www.flowpath.com/• www.groundwatersoftware.com/newsletter.htm• Groundwater Digest: groundwater@groundwater.com
5. Model design (I)
Conditions:• initial conditions• boundary conditions:
1. Dirichlet: head 2. Neumann: flux, e.g., no flow3. Cauchy/Robin: mixed boundary
condition
Modelling protocol
Choice grid x:
•depends on natural variation in the groundwater system•concept model•data collection
Choice time step t
5. Model design (II): example
Geometry, subsoil parameters, boundary conditions
-20
-70 -90
-170
0 1500 5500 9500 125001000
10
Holocene aquitard
Length of the geohydrologic system, [km]
Sand-dunesAmsterdam Waterworks Haarlemmermeer polderRijnland polders
Middle aquifer
Deep aquifer
No flow boundary
Constant piezometriclevel
Extraction points
Sea
No flow boundary
k=30 m/day
c=800 d
k=20 m/day
c=4400 d
-0.15m -0.60m -5.50mnatural groundwater recharge 420 mm/year
rate varies
Phreatic aquifer
Loam layer
k=10 m/day
Co
ast
Dep
th w
ith
res
pec
t to
N.A
.P.,
[m]
Modelling protocol
5. Model design (III): choice time step t
Modelling protocol
m onth
(km )
log tim e (years)
log space
WATER
S O IL
AIR
s e as + o c e an s
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
6
5
4
3
2
1
0
-1
rive rs
la k es
b ank filtra tio ng rou n dw a te r
g lac ie rp o le c aps
g eo lo g ic a l
fo rm a tion s
p res su re z o n e s
fron ts
torn
a do s
clo ud s
hour day w eek century
6. Calibration (I)
Fitting the groundwater model: is your model okay?
• trial and error• automatic parameter estimation/inverse
modelling (PEST, UCODE)
Modelling protocol
6. Calibration (II): example
Measured and computed freshwater heads
-6
-5
-4
-3
-2
-1
0
1
2
-6 -5 -4 -3 -2 -1 0 1 2
measured freshwater head [m]
co
mp
ute
d f
res
hw
ate
r h
ea
d [
m]
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
model layer
Modelling protocol
6. Calibration (III): example
Measured and computed freshwater heads
• 95 useful observation wells• maximum measured head in area: +1.83 m• minimum measured head in area: -4.98 m
• average (measured -computed) = -0.04 m• average absolute|measured -computed| =
0.34 m• standard deviation = 0.46 m
Modelling protocol
6. Calibration: errors during modelling protocol
•Wrong model conceptresistance layer not consideredheat transport on groundwater flow not considered ( not constant)
•Incomplete equationsdecay term of solute transport not considered
•Inaccurate parameters and variables
•Errors in computer code
•Numerical inaccuraciest, t, numerical dispersion
Modelling protocol
7. Verification: testing the calibrated model
‘verification problem’: there is always a lack of data
Modelling protocol
• 1D, 2D -->3D• stationary dataset --> non-steady state simulation
9. Presentation
Simulation of scenarios
Computation time depends on:• computer speed• size model• efficiency compiler• output format
8. Simulation
Modelling protocol
10. Postaudit
Postaudit: analysing model results after a long time
Anderson & Woessner(’92):four postaudits from the 1960’s
Errors in model results are mainly caused by:•wrong concept•wrong scenarios
Modelling protocol
Modelling scheme
Modelling protocol
Anderson & Woessner (1992)
2
A
Q
Lre ference leve l
Q
I. Darcy’s law (1856)
21 QL
Q1
AQ L
AQ 21 gives
LKAQ 21
where K= hydraulic conductivity [L/T ]
Discretisation PDE
II. Darcy’s Law
Discretisation PDE
In pressures: x
pq xx
y
pq yy
gz
pq zz
Definition piezometric head: zgpzg
p
g ro u n d surfa ce (gro n do p p e rv la k)
fi lte r= p ie zo m e tric h e ad
(s tijgh o o g te )
w ate r ta b le (gro n dw a te rsp ie g e l)
h = = p re ssu re h ea d (d rukh oo g te )
pg
z= e le vatio n h e ad (p la a tsho o g te )
q=Darcian specific discharge [L/T]=piezometric head [L]p=pressure [M/LT2]=intrinsic permeability [L2]=dynamic viscosity [M/LT]=density [M/L3]g=gravitaty [L/T2]ki=hydraulic conductivity [L/T]
III. Darcy’s Law
Discretisation PDE
if then:ii kg
givesx
kq xx
ykq yy
zkq zz
x
gq xx
y
gq yy
z
gq zz
Gives:
Analogy physical processes
xkq
x
Vi
x
Th
Discretisation PDE
Continuity equation: steady state
z
y
y
x
x
z q y z
q y z
q x y
q x z
q x y
z
y
y
x2
x
z
q x z
x1
z1
z2
y2
y1
0
z
q
y
q
x
q zyx
Mass balance [M T-1]
Discretisation PDE
=density [M L-3]
Steady state groundwater flow equation (PDE=Partial Differential Equation)
0
z
q
y
q
x
q zyx +Continuity equation
If ki=constant and =constant then:
02
2
2
2
2
2
zyx
Laplace equation
02
Discretisation PDE
xkq xx
ykq yy
zkq zz
Flow equation (Darcy’s Law)
0
zz
k
y
yk
xx
k zyx
=Groundwater flow equation
3
3211
6
1
2 xx
xxiiii
Taylor series development (I)
4
44
3
33
2
22
124
1
6
1
2
1
xx
xx
xx
xx
i i i ii i
4
44
3
33
2
22
124
1
6
1
2
1
xx
xx
xx
xx
i i i ii i
3
33
11 3
12
xx
xx ii
ii
-
Discretisation PDE
Best estimate of i+1 is based on i
4
44
3
33
2
22
124
1
6
1
2
1
xx
xx
xx
xx
i i i ii i
4
44
3
33
2
22
124
1
6
1
2
1
xx
xx
xx
xx
i i i ii i
4
42
21 1
2
2
12
1 2
xx
x xi i i i i
Taylor series development (II)
4
44
2
22
1 112
12
xx
xx
i ii i i
+
Discretisation PDE
Laplace equation in 2D
002
2
2
22
yx
2
,1,,1
2
,2 2
xxjijijiji
Discretisation in x-direction (i ):
2
1,,1,
2
,2 2
yyjijijiji
Discretisation in y-direction (j ):
022
2
1,,1,
2
,1,,1
yxjijijijijiji
If x = y then: 04 1,1,,,1,1 jijijijiji
41,1,,1,1
,
jijijijiji
‘Fivepoint
operator’
Discretisation PDE
Fivepoint operator (I): constant head example
Const ant head boundar y
1 2 3 4 5
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
i
j6 7 8 9 10
41,1,,1,1
,
jijijijiji
48181214
13
Discretisation PDE
Fivepoint operator (II): no-flow example
421057
6
0
y
kq
210210
2
yy
4
2 10576
1
3 42
5
7 86
9
11 1210
hor izont al no- fl ow boundar y
undefi ned
x
y
x
y
Nodes on the edges of an element
Discretisation PDE
Fivepoint operator (III): example
41,1,,1,1
,
jijijijiji
Hydraulic conductivity k=10 m/dag
Thickness aquifer D=50m
Characteristics:
Groundwater recharge N=0 mm/dag
BOUNDARY 1: no-flow near the mountainsBoundary conditions
BOUNDARY 2: linear from 100 m (near mountains) to 0 m (near sea)BOUNDARY 3: constant seawater level of 0 m
BOUNDARY 4: no-flow
Discretisation PDE
Fivepoint operator (IV): example
41,1,,1,1
,
jijijijiji
On BOUNDARY 1:
4
2 1,,1,1,
jijiji
ji
On BOUNDARY 4:
4
2 1,1,,1,
jijiji
ji
Discretisation PDE
EXCEL example!
Fivepoint operator (V): effect convergence criterion
Discretisation PDE
Solution I:•Initial estimate head=50m•Convergence-criterion=1m•Number of iterations=13•Head 1=29,17m•Head 2=25,11m
Solution II:•Initial estimate head=50m•Convergence-criterion=0.01m•Number of iterations=74•Head 1=20,67m•Head 2=23.07m
Discretisation PDE
Numerical schemes: iterative methods
1. Jacobi iteration: only old values
nji
nji
nji
nji
nji 1,1,,1,11
, 4
1
11,1,
1,1,1
1, 4
1
nji
nji
nji
nji
nji
• Gauss-Siedel iteration: also two new values
11,1,
1,1,1,
1, 4
1
nji
nji
nji
nji
nji
nji
n jinjir ,1
, rnji
nji
,1
,
• Overrelaxation:
Non-steady state groundwater flow equation
Multiply with constant thickness D of the aquifer gives:
Wt
Sz
Ty
Tx
T
2
2
2
2
2
2S=elastic storage coefficient [-]T=kD= transmissivity [L2/T]
xkqx
ykqy
zkqz
Flow equation (Darcy’s Law)
'Wt
Sz
q
y
q
x
qs
zyx
+Non-steady state continuity equation =
Ss=specific storativity [1/L]W’=source-term
'Wt
Sz
zk
y
yk
x
xk
s
Groundwater flow equation
non-steady state
Explicit numerical 1D solution (I)
One-dimensional non-steady state groundwater flow equation:
Nx
Tt
S
2
2
Properties:• direct solution• can be numerical instable
tt
ti
ttii
Explicit (‘forwards in space’):
211
2
2 2
xx
ti
ti
tii
ti
ti
ti
ti
tti xS
tT
S
tN112
2
non-steady state
Explicit numerical 1D solution (II): example
Subsoil parameters:• storage=1/3• T=kD=10 m2/day• N=0.001 m/day
Numerical parameters: two sets• timestep t=1 day & t=10
days• length step x=10m• Nt/=0.003 m & 0.03 m• Tt/x2=0.3 & 3
ti
ti
ti
ti
tti x
tTtN112
2
ti
ti
ti
ti
tti 11 23.0003.0
ti
ti
ti
ti
tti 11 2303.0
I.
II.
Groundwater flow in aquifer between two ditches:
10 0 m
10m
N eer s lag N =1mm per dag
x =10 m
sloot sloot
F r eat ische ber ging =1/ 3
1=1
0m
2 3 11
init ieelditch ditch
recharge=1mm/day
phreatic storage=1mm/dayx=10
non-steady state
EXCEL example!
Explicit numerical 1D solution (III): stability analysis
ti
ti
ti
ti
tti x
tTtN112
2
non-steady state
10ti 1011
ti
ti
22
4110410x
tT
x
tTtti
1412
x
tT
141 14122
x
tT
x
tT
5.0 02
x
tTt
Suppose:
Implicit numerical solution
Implicit (‘backwards in space’):
One-dimensional non-steady groundwater flow equation(now no source-term W):
2
2
xT
tS
tt
ti
ttii
211
2
2 2
xx
tti
tti
ttii
Characteristics:• not a direct solution• solving with matrix A=R• numerical always stable• needs more memory
ti
tti
tti
tti tT
xS
tT
xS
2
1
2
1 2
non-steady state
Crank-Nicolson numerical solution
Crank-Nicolson (‘central in space’): =0.5
One-dimensional non-steady state groundwater flow equation (now no source term W):
2
2
xT
tS
tt
ti
ttii
Characteristics:• solving as implicit• stable with temporarily oscillations
211
211
2
2 21
2
xxx
ti
ti
ti
tti
tti
ttii
ti
ti
ti
tti
tti
tti tT
xS
tT
xS1
2
11
2
1 2222
non-steady state
Numerical schemes
t
t + t
unknownknown
explicit : f or war d in t ime
t
unknownknown
implicit : backwar d in t ime
i- 1
i- 1 i+1
i+1i
i
t
unknownknown
Cr ank- N icholson: cent r al in t ime
i- 1 i+1i
t + t
t + t
non-steady state
MODFLOW
MODFLOW
a modular 3D finite-difference ground-water flow model
(M.G. McDonald & A.W. Harbaugh, from 1983 on)
• USGS, ‘public domain’• non steady state• heterogeneous porous medium• anisotropy• coupled to reactive solute transport
MOC3 (Konikow et al, 1996)MT3D, MT3DMS (Zheng, 1990)RT3D
• easy to use due to numerous Graphical User Interfaces (GUI’s)PMWIN, GMS, Visual Modflow, Argus One, Groundwater Vistas, etc.
Nomenclature MODFLOW element
row (i)colum n (j)
layer (k) NCO L
NRO W
NLAY
r
c
v
y
x
z
layer d irection
colum n d irection
i
k
j
row d irection
MODFLOW
MODFLOW: start with water balance of one element [i,j,k]
z
y
i,j ,k - 1
x
i-1,j ,k
i,j +1,ki,j -1,k
i,j ,k
i,j ,k+1i+1,j ,k
MODFLOW
Continuity equation (I)
Vt
SQ si
In - Out = Storage
MODFLOW
tSW
zk
zyk
yxk
x szzyyxx
Continuity equation (II)
Vt
SQ si
Vt
SS
QQQQQQQttkji
tkji
kji
kjiextkjikjikjikjikjikji
,,,,,,
,,,2/1,,2/1,,,,2/1,,2/1,2/1,,2/1,
MODFLOW
y
j x
z
k y
x
z
Q i-1/2,j,ki
Q i+1/2,j,k
Q i,j+1/2,k
Q i,j-1/2,k
Q i,j,k-1/2
Q i,j,k+1/2
Q ext,i,j,k
In = positive
Flow equation (Darcy’s Law)
xzykQ kjikji
kjikji
,,,1,,2/1,,2/1,
MODFLOW
kjikjikjikji CRQ ,,,1,,2/1,,2/1,
where is the conductance [L2/T] x
zykCR kji
kji
,2/1,,2/1,
xksurfaceqsurfaceQ
**
z
element i,j - 1,k element i,j ,k
x
y
i,j - 1,k
i,j ,k
i,j - 1/ 2 ,kQ
Groundwater flow equation
MODFLOW
kjikjikjikji CRQ ,,,1,,2/1,,2/1,
kjikjikjikji CRQ ,,,1,,2/1,,2/1,
kjikjikjikji CCQ ,,,,1,,2/1,,2/1
kjikjikjikji CCQ ,,,,1,,2/1,,2/1
kjikjikjikji CVQ ,,1,,2/1,,2/1,,
kjikjikjikji CVQ ,,1,,2/1,,2/1,,
y
j x
z
k y
x
z
Q i-1/2,j,ki
Q i+1/2,j,k
Q i,j+1/2,k
Q i,j-1/2,k
Q i,j,k-1/2
Q i,j,k+1/2
Q extern
Vt
SS
QQQQQQQttkji
tkji
kji
kjiextkjikjikjikjikjikji
,,,,,,
,,,2/1,,2/1,,,,2/1,,2/1,2/1,,2/1,
Takes into account all external sources
MODFLOW
kjittkjikjikjiext QPQ ,,,,,,,,, '
The term kjiextQ ,,,
Rewriting the term:
Thé MODFLOW Groundwater flow equation
MODFLOW
VSSSC
tSCQRHS
tSCPHCOF
with
RHSCVCCCR
HCOFCVCCCRCRCCCV
CRCCCV
kjikji
tkjikjikjikji
kjikjikji
kjittkjikji
ttkjikji
ttkjikji
ttkjikjikjikjikjikjikjikji
ttkjikji
ttkjikji
ttkjikji
,,,,
,,,,,,,,
,,,,,,
,,1,,2/1,,,,1,,2/1,1,,2/1,
,,,,2/1,,,,2/1,2/1,,2/1,,,2/12/1,,
,1,,2/1,,,1,,2/11,,2/1,,
1
/1'
/1
:
Vt
SS
QQQQQQQttkji
tkji
kji
kjiextkjikjikjikjikjikji
,,,,,,
,,,2/1,,2/1,,,,2/1,,2/1,2/1,,2/1,
kjittkjikjikjiext QPQ ,,,,,,,,, '
and:
gives:
Boundary conditions in MODFLOW (I)
Numeric model0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 00 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 00 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 00 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 00 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 00 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0
aquif er boundar y
ar ea of const ant head
ar ea wher e heads var y wit h t ime
MODFLOW
Example of a system with three types of boundary conditions
Boundary conditions in MODFLOW (II)
For a constant head condition: IBOUND<0For a no flow condition: IBOUND=0 For a variable head: IBOUND>0
MODFLOW
Packages in MODFLOW
MODFLOW
1. Well package2. River package3. Recharge package4. Drain package5. Evaporation package6. General head package
1. Well package
MODFLOW
kjiwell QQ ,,
Example: an extraction of 10 m3 per day should be
inserted in an element as:
10,,, kjiextQ (in = positive)
kjittkjikjikjiext QPQ ,,,,,,,,, '
10' ,, kjiQ
2. River package (I)MODFLOW
MKLWQ kjiriv
riv,,
M
riv i,j,k
element i,j,k
M
element i,j,k
river gains water
RBOT
riv
kjirivriv M
KLWQ ,, kjirivrivriv CQ ,,
river loses water
river bottom
2. River package (II)MODFLOW
Example: the river conductance Criv is 20 m2/day and
the rivel level=3 m, than this package should be
inserted in an element as:
kjikjiextQ ,,,,, 320
kjittkjikjikjiext QPQ ,,,,,,,,, '
20 and 60' ,,,, kjikji PQ
kjirivrivriv CQ ,,
2. River package (III)
MODFLOW
where is the
conductance [L2/T] of the river
M
KLWCriv
L =lengt h of r each
M =t hickness of r iver bed
K=hydr aulic conduct ivit y of r iver bed
r iver
W =widt h of t he r iver bed
Determine the conductance of the river in one element:
2. River package (IV)
MODFLOW
Leakage to the groundwater system
M
riv
i,j,k
element i,j,k
RBOT
Special case:
if i,j,k<RBOT, then RBOTCQ rivrivriv
2. River package (V)
MODFLOW
i,j
,ki,j
,kri
v
i,j,k
riv
<RB
OT
R
BO
T<
<>
e levat ion o f t he bot t omof t he r iver be d R B O T
R ive r s t age
Losi
ng r
iver
G
aini
ng r
iver
head
in
ele
men
t i,j
,k (
L)
Leakage t hr ough r iver bed
I nt o t he aquif er O ut of t he aqui f er
River
3. Recharge package
MODFLOW
yxIQrec
4. Drain package
MODFLOW
d r ain e levat ion
head
in
ele
men
t i,j
,k (
L)
Leakage int o a dr ain
N o leak age int o d r ain L eak age int o d r ain
leak age r at e
leak age r at e =0
Dr ain
dCQ kjidrndrn ,,
0drnQ
Special case:
if i,j,k < d than
i,j ,k
element i,j ,k
di,j ,k
dr ain
Cd
5. Evapotranspiration package
MODFLOW
M ax imum E T r at e
E T r at e=0
E X E L =ex t inc t ion e levat ion
S U R F =E T sur f ac e e le vat ion
E
XD
P=ex
tinc
tion
dep
th
head
in
ele
men
t i,j
,k (
L)
E vapot r anspir at ion (E T )
E vapot r anspir at ion
6. General head boundary package
MODFLOW
kjighbghbghb CQ ,, h
ead
in
elem
ent
i,j,k
(L)
F low t o a gener al head boundar y
F low int o t he e le me nt F low out o f t h e e lement
s lope =c onduc t anc e bet ween e le me nt and b ound ar y
Gener al head boundar y
i,j ,k
element i,j ,k
ghb
Cghb
Time indication MODFLOW
ITMUNI=1: seconde ITMUNI=2: minuteITMUNI=3: hour ITMUNI=4: dayITMUNI=5: year
MODFLOW
MODFLOW: implementation of geology
Vcont = vertical leakage [T-1]:
vertical hydraulic conductivity / thickness element
MODFLOW
kD = transmissivity [L2 T-1]:
horizontal hydraulic conductivity * thickness element
Layertypes in MODFLOW (LAYCON): 4*
I. LAYCON=0
Confined: T=kD=constant
Storage in element:t
SAQtkji
ttkji
,,,,
MODFLOW
II. LAYCON=1
tAQ
tkji
ttkji
,,,,
TOPwhenBOTTOPkT kji ,,)(
TOPBOTwhenBOTkT kjikji ,,,, )(
BOTwhenT kji ,,0 B O T
sit uat ion b .
T O P
sit uat ion a.
s it uat ion c .
i,j,k
MODFLOW
Unconfined: T=variable
Storage in element:
II. LAYCON=2
t
TOPAS
t
TOPASQ
tkji
ttkji
,,2
,,1
SusethenTOPif ttkji
,,
B O T
T O P
i,j,k usethenTOPif tt
kji ,,
MODFLOW
Confined or unconfined: T=kD=constant
Storage in element:
II. LAYCON=3
MODFLOW
Confined of unconfined: T=variable as with LAYCON=1
Storage in element as with LAYCON=2
Solving of the MODFLOW matrix
MODFLOW
{ 1 NCOLNCOL*NROW
{1NCOLNCOL*NROW
LU-decomposition
RA RLU rewriting as
Files in MODFLOW: infile.nam fileMODFLOW
{ 1 NCOLNCOL*NROW
{1NCOLNCOL*NROW
INFILE.NAMlist 16 ext.lstbas 95 ext.basbcf 11 ext.bcfsip 19 ext.sipwel 12 ext.welconc 33 ext_moc.nam
Additional information (good documentation):MODFLOW: http://water.usgs.gov/nrp/gwsoftware/modflow.html
MOC3D: http://water.usgs.gov/nrp/gwsoftware/moc3d/moc3d.html
Files in MODFLOW: *.bas fileMODFLOW
{ 1 NCOLNCOL*NROW
{1NCOLNCOL*NROW
EXT.BASGroundwater extraction NLAY NROW NCOL NPER ITMUNI 1 19 19 1 1FREE 0 1 ; IAPART,ISTRT 95 1(19I3) 3 ; IBOUND1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.00 ; HNOFLO 95 1.0(19F5.0) 1 ; HEAD1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 86400.0 20 1.0 ; PERLEN,NSTP,TSMULTend
EXT.SIP 500 5 ; MXITER,NPARM SIP Input 1. 0.0001 0 0.001 0 ; ACCL,HCLOSE,IPCALC,WSEED,IPRSIP
Files in MODFLOW: *.sip file
Files in MODFLOW: *.bcf fileMODFLOW
{ 1 NCOLNCOL*NROW
{1NCOLNCOL*NROW
EXT.BCF 0 0 0.0 0 0.0 0 0 ISS,IBCFBD BCF Input 0 LAYCON 0 1.0 TRPY 0 50.0 DELR 0 50.0 DELC 0 0.00075 Sf1 0 0.0023 TRAN1
EXT.WEL 1 0 AUXILIARY CONC CBCALLOCATE MXWELL,IWELBD 1 ITMP (NWELLS) 1 10 10 -0.004 2.5E3 ; k i j q c -1 1 ITMP (NWELLS) 1 8 7 -0.004 2.5E3 ; k i j q c
Files in MODFLOW: *.wel file
Variable density models:
1. Hypothetical cases:interface between fresh and saltevolution of a freshwater lensHenry caseHydrocoinElder (heat transport)Rayleigh
2. Real cases in the Netherlands:Island of TexelWater board of RijnlandProvince of North-HollandWieringermeerpolderBoard 14 basin
MODFLOW: examples of casesMODFLOW
{ 1 NCOLNCOL*NROW
{1NCOLNCOL*NROW
Formula of Theis (1935)
r
Q
Q0
D
t= 0
t= t2
t= t1
h e a d s o f th e c o n f in e d a q u if e r
confined
Solution:
t
sS
r
s
rr
skD
12
2
),(4
2uWkD
Qs o
t
r
kD
Su
22
4
2
2
22
2 )(u
u
duu
euW
Non-steady state groundwater flow towards an extraction well
PDE:
If u is small, than W(u2)=2ln(0.75/u)
Formula of Theis: dataNon-steady state groundwater flow towards an extraction well
one layer: NLAY=1in the horizontal plane: 19*19 elements of 50*50 m2
initial head is 0 mhydraulic conductivity k is 4 m/dporosity ne=0.25T=kD=0.0023 m2/s (or 200 m2/d), so D=50 mnon-steady state: specific storativity Ss=0.000015 m-1 elastic storagecoefficient S=0.00075extraction in element [i,j,k]=[10,10,1] with Q=345.6 m3/dtimestep: 1 day divided in 20 stepsthree observation points on x=175, 375 and 475 m (y=475 m)top systeem is 0 m M.S.L, bottom is -50 m M.S.L.
Groundwater movement near salt domes:The Hydrocoin case (I)
Distance ( )x mconstant concentration boundary
p Pa=0
0 300 600 900
no-flow
Dep
th
()
zm
0
300 no-flowsalt dom e ( =1, =1200 )C kg/m
initial condition: C
C k g / m’=0, =1000
3
specified pressure boundary
no-flowno-flow
no-flow
p Pa=10 53
s
o
Groundwater movement near salt domes:The Hydrocoin case (II)
Salt fraction:
Distance ( )x mconstant concentration boundary
p Pa=0
0 300 600 900
no-flow
Dep
th
()
zm
0
300 no-flowsalt dom e ( =1, =1200 )C kg/m
initial condition: C
C k g / m’=0, =1000
3
specified pressure boundary
no-flowno-flow
no-flow
p Pa=10 53
s
o
Schematisation Dutch coastal aquifer system
Present ground surface in the Netherlands and position polder
Beemster 1608-1612Wormer 1625-1626Schermer 1633-1635Purmer 1618-1622
Haarlemmermeer polder 1850-1852
Wieringermeer polder ~1930
Flevo polders 1950-60s
Depth saline-fresh interface (150 mg Cl-/l)
Noordzee
0 25 50 75 100 km
<100m
>200m
100-200m
Belgium
Germany
Den H aag
Rotterdam
Am sterdam
Waddenzee
Development of the Dutch polder area
stream s in the low lands sw am py lands
prim itively drained
polderboezem
1-2 m below m ean sea level
pum ping station
2.5-4.0 m below m ean sea level
polder polderboezem boezem
deep polder
polder
A
B
C
D
E
F
Past and future sea level rise in the Netherlands
sea
leve
l wit
h r
esp
ect
to M
.S.L
. [m
]
Historic lowering of the ground surface in Holland
1000 AD 1200 1400 1600 1800 2000
+0.9 m
-2
-3
M.S.L.
-1
+1
land subsidence
digging d ike w indm ill m echanicalchannels building drainage drainage
tidalfluctuation
rise of m ean sea level
digging d ike w indm ill m echanical
tidehigh
-0.7 m
tidelow
channels building drainage drainage
Salt water intrusion in the Netherlands
Salt water intrusion (3D) in Texel: geometry
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 km
0.0
-2.5
-5.0
-7.5
-10
.0-1
2.5
-15
.0-1
7.5
-20
.0-2
2.5
-25
.0-2
7.5
km
-1 .25-
-1 .00-
-0 .75
-0 .75-
-0 .50
-0 .50-
0 .00
0 ..00-
0 .50
0 .50-
1 .00
1 .00-
1 .50
1 .50-
2 .00
2 .00-
3 .00
3 .00-
4 .00
4 .00-
-1 .00
Phreatic water level [m ]:
6.50
-2 .50-
-1 .50-
-1 .25
-1 .50
Groot Geohydrologisch Onderzoek Texel
Eijerland (-2)
Waal en Burg/Het Noorden (-3)
Dijkm anshuizen/De Schans (-4)
Prins Hendrikpolder (-5)
x=250 m y=250 m z=1.5-20 mn =80 n =116 n =25
t=1 year conv. crit.=10 m8 partic les per elem ent
f=1 m m /d=1000 kg/m=1025 kg/mn =0.35
k =1-30 m /danisotropy=0.25
=2 m= =0.2 m
h
zyx-5
f
e
s3
3
TV
L
TH
Subsoil parameters:
Numerical parameters:
Boundary conditions:constant head in poldersconstant flux in dune-areano-flow boundary a t sea-side
sea
sea
Geometry of the groundwater system in Noord-Holland
L =51.25 km
S ubsoil param eters : Num erica l param eters:
x=1250 m y=1250 m z= 10 mn =41 n =52 n =29
t=1 yr conv. crit.=10 m27 partic les per elem ent
f=1 m m /d=1000 kg /m=1025 kg /m
n =0.35
layer4
L =290 m
constantfreshwaterhead=-0.15m
Noordzee
z
sand-dune 1 m m /d
z
xy
k=10-70 m /dc =500-10000 dc =83 d
=2 m= =0.2 m
layer2
IJsselm eer
zyx
T V
L
T H
inactive elements
salinegroundwater
W ieringermeerpolder (-3.8m,-6.2 m)
IJsselm eer
Scherm er -4.2m
Beem ster
-4.4mWormer Purmer -4.3m
-4.5m
Boundary conditions:
constant head in all poldersconstant recharge in duneshydrostatic pressure at sea side
L =65.0 km
x
y
3f 3se
-5
Solute transport equation
MOC3D
Partial differential equation (PDE):
CR
n
WCCCV
xx
CD
xt
CR d
ei
ijij
id
'
Dij=hydrodynamic dispersion [L2T-
1] Rd=retardation factor [-]=decay-term [T-1]
dispersiondiffusion
advection source/sink decaychange in concentration
Solute transport equation: column test (I):
MOC3D
Solute transport equation: column test (II):
MOC3D
Solute transport equation: column test (III):
MOC3D
Hydrodynamic dispersion
MOC3D
hydrodynamic dispersion=
mechanical dispersion+ diffusion
mechanical dispersion:tensor velocity dependant
diffusion: molecular processsolutes spread due to concentration
differences
Mechanical dispersion
Differences in velocity due to variation in pore-dimension
Differences in velocity in the pore
Differences in velocity due to variation invelocity direction
MOC3D
Solute transport equation: diffusionMOC3D
diffusion is a slow process: diffusion equation
2
2
z
CD
t
C
only 1D-diffusion means: Rd=1, Vi=0, =0 and W=0
Nx
Tt
S
2
2
ti
ti
ti
ti
tti xS
tT
S
tN112
2
similarity with non-steady state groundwater flow equation
5.02
xS
tT
ti
ti
ti
ti
tti CCC
z
tDCC 112
2
5.02
z
tD
Solute transport equation: diffusion
MOC3D
diffusion is a slow process: diffusion equation2
2
z
CD
t
C
0
4000
8000
12000
16000
0 50 100 150 200
Depth (m)
Ch
lori
de
co
nte
nt
(mg
Cl/l
)
0
500
5000
20000
80000
175000
220000
350000
Time (year)
Method of Characteristics (MOC)
MOC3D
e
iij
iji n
WCCCV
xx
CD
xt
C '
Solve the advection-dispersion equation (ADE)with the Method of Characteristics
Splitting up the advection part and the dispersion/source part:
•advection by means of a particle tracking technique
•dispersion/source by means of the finite difference method
Lagrangian
approach:
Advantage of the approach of MOC?
MOC3D
It is difficult to solve the whole advection-dispersionequation in one step, because the so-called Peclet-numberis high in most groundwater flow/solute transport problems.
(hyperbolic form of the equation is dominant)
The Peclet number stands for the ratio between advection and dispersion
Procedure of MOC: advective transport by particle tracking
MOC3D
•Average values of all particles in an element to one node value
•Place a number of particles in each element
•Move particles during one solute time step tsolute
•Determine the effective velocity of each particle by (bi)linear interpolation of the velocity field which is derived from MODFLOW
•Calculate the change in concentration in all nodes due to advective transport
•Add this result to dispersive/source changes of solute transport
Steps in MOC-procedure
MOC3D
5. Determine concentration in element node after advective, dispersive/source transport on timestep k
4. Determine concentration gradients on new timestep k*
3. Concentration of particles to concentration in element node
1. Determine concentration gradients at old timestep k-1
2. Move particles to model advective transport
Causes of errors in MOC-procedure
MOC3D
4. Empty elements
3. Concentration of sources/sinks to entire element
1. Concentration gradients
2. Average from particles to node element, and visa versa
5. No-flow boundary: reflection in boundary
Reflection in boundary
MOC3D
z
x
j
j - 1
j +1
i-1 i i+1
par t icle on pr evious locat ioncomput ed locat ion of par t ic le
cor r ect ed locat ion of par t ic le
no- fl ow boundar y
fl ow line comput ed pat hof fl ow
cent er of gr id cell
cor r ect ion
Stability criteria (I)MOC3D
Stability criteria are necessary because the ADE issolved explicitly
222
5.0
zD
y
D
xD
tzzyyxx
s
1. Neumann criterion:
5.0222
z
tD
y
tD
x
tD szzsyysxx
Stability criteria (II)MOC3D
',,
,,
kji
kkjie
s Q
bnt
Change in concentration in element is not allowed to be larger than the difference between the present concentration in the element and the concentration in the source
2. Mixing criterion
** *
** *
** *
Stability criteria (III)MOC3D
3. Courant criterium
max,xs V
xt
max,ys V
yt
max,zs V
zt
** *
** *
** *** *
** *
** *
** *
** *
** *
** *
** *
** *
* N ode element Par t ic le
Velocit y d ir ect ion
M ovement par t ic les
Files in MOC3D: *.moc fileMODFLOW
{ 1 NCOLNCOL*NROW
{1NCOLNCOL*NROW
EXT.MOCGroundwater extraction ISLAY1 ISLAY2 ISROW1 ISROW2 ISCOL1 ISCOL2 1 1 1 19 1 19 1 0.0 0.0 ; NODISP, DECAY, DIFFUS 20000 9 ; NPMAX, NPTPND 0.5 0.05 2 ; CELDIS, FZERO, INTRPL -2 0 0 -1 0 0 0 ; NPNTCL, ICONFM, NPNTVL, IVELFM, NPNTDL, IDSPFM, NPRTPL 35000.0 ; CNOFLO 96 1(19F5.0) 3 ; initial concentration1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0 ; NZONES to follow 0 0 ; IGENPT1 0 1.0 ; retardation factor 0 50.0 ; thickness1 0 0.25 ; porosity1end 0 35000.0 ; C' inflow subgrid 0 0.001 ; longitudinal disp. 0 0.0001 ; transverse disp. horiz. 0 0.0001 ; transverse disp. vert.
Files in MOC3D: *_moc.nam and *.obs filesMODFLOW
{ 1 NCOLNCOL*NROW
{1NCOLNCOL*NROW
EXT.OBS 3 1 ;NUMOBS IOBSFL Observation well data 1 10 10 45 ;layer, row, column, unit number 1 8 10 ;layer, row, column 1 4 10 ;layer, row, column
EXT_MOC.NAMclst 94 ext.outmoc 96 ext.mocobs 44 ext.obsdata 45 ext.oba
Numerical dispersion and oscillation
C
Concent r at ion
E x ac t solut ion
N umer ical d isper sion
Concent r at ionO ver sh oot ing
U nd er sh oot ing
E x ac t solut ion
O scillat ion
N ume r ic al d i spe r s ion
xx
C
Derivation of numerical dispersion: 1D (I)
x
CV
x
CD
t
C
2
2 Discretisation:backwards in spacebackwards in time
x
CCV
x
CCCD
t
CC ki
ki
ki
ki
ki
ki
ki
1
211
1 2
By means of Taylor series development:
Derivation of numerical dispersion: 1D (II)
44
42
2
2
211
12
2xO
x
Cx
x
C
x
CCC ki
ki
ki
33
32
2
21
62xO
x
Cx
x
Cx
x
C
x
CC ki
ki
33
32
2
21
62tO
t
Ct
t
Ct
t
C
t
CC ki
ki
Now Taylor series development with truncation erros!:
Derivation of numerical dispersion: 1D (III)
3
32
2
2
4
42
2
2
2
2
62122 x
Cx
x
Cx
x
CV
x
Cx
x
CD
t
Ct
t
C
x
CCV
x
CCCD
t
CC ki
ki
ki
ki
ki
ki
ki
1
211
1 2
Neglect 3rd and 4th order terms:
2
2
2
2
2
2
22 x
Cx
x
CV
x
CD
t
Ct
t
C
Derivation of numerical dispersion: 1D (IV)
Rewriting term:2
2
t
C
t
C
xV
t
C
xD
x
CV
x
CD
tt
C
tt
C2
2
2
2
2
2
x
CV
x
CD
xV
x
CV
x
CD
xD
t
C2
2
2
2
2
2
2
2
2
22
3
3
3
3
4
42
2
2
x
CV
x
CVD
x
CVD
x
CD
t
C
2
22
2
2
x
CV
t
C
Derivation of numerical dispersion: 1D (V)
Rewrite all extra terms:
2
22
2
2
x
CV
t
C
2
2
2
2
2
2
22 x
Cx
x
CV
x
CD
t
Ct
t
C
2
2
2
2
2
22
22 x
CxV
x
CV
x
CD
x
CVt
t
C
x
CV
x
CVtx
VDt
C
2
22
22
Derivation of numerical dispersion: 1D (VI)
Remedy to reduce numerical dispersion:
1. x en t smaller
2. Use a different numeric scheme (Crank-Nicolson)
3. Use a smaller Dreal
2
22Vtx
VDD real
Numerical dispersion:
Conclusions
• Modelling protocol helps you in making a numerical model• Conceptualisation is important• Stepwise from hydrogeological process to model:
-Concept-Darcy and Continuity gives groundwater flow equation-Taylor series development gives numerical model-Stability criterion
•MODFLOW packages in mathematical formulation•Use General Head Boundary package for a change!•MOC3D: particle tracking technique•Check input and output files in case of errors•See documentation of USGS
Modelling of groundwater flow and solute transport
Additional information:Groundwater modelling: ftp://ftp.geo.uu.nl/pub/people/goe/gwm1/gwm1.pdf
Density dependent groundwater flow: ftp://ftp.geo.uu.nl/pub/people/goe/gwm2/gwm2.pdf
Gualbert Oude EssinkFaculty of Earth, Free University (VU)Netherlands Institute of Applied Geosciences (NITG)
PGI: Aug. 22-24, 2002
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