INVERSE MODELLING OF UNSATURATED FLOW COMBINED WITH STOCHASTIC SIMULATION USING
EMPIRICAL ORTHOGONAL FUNCTIONS (EOF)
Stefan Finsterle
University of California, Lawrence Berkeley National Laboratory
Berkeley, California, USA
Nils-Otto Kitterød
University of Oslo, Department of Geophysics, Norway
e-mail: nilsotto@geofysikk.uio.no
Background
Input data
Oslo Airport is a potential hazard to the unconfined groundwater aquifer at Gardermoen (fig. 1).
Biological remediation may prevent serious pollution of the groundwater, but this protection
requires that the transport to the groundwater is not too fast. Spatial and temporal variation in
unsaturated flow properties however, make short cuircuiting and preferential flow very likely under
extreme conditions.
• Sedimentological architecture from Ground Penetrating Radar (fig.4 and 5)
• Liquid saturation measured by Neutron Scattering and interpolated by kriging (fig.6)
• A priori statistical data on flow parameters
• Effective infiltration (fig. 7)
Top 1
Figure 4. Ground Penetrating Radar
0
lp4
1
utm-N
• Use liquid saturation as
primary data for
Bayesian Maximum
6678500
Likelihood Inversion of
unsaturated flow
parameters.
6676500
• Simulate parameter
uncertainties
by Karhunen6674500
Loève
expansion.
Gardermoen
2
p45
Dip 2
5
0
10m
W
Ground Penetrating Radar profiles
E
N
615,700
6,677,740
p42
10
p43
c11
0
c16
foresets
sp4
Groundwatertable
p41
k20
p43
k26
k10
p45
k28
k2 k4
2
3
k30
k8
k24
k6
p47
6,677,820
2D flow
test
tr
5
0
10
West-East <m>
p35
Figure 6. Liquid saturation, May 11, 1995
6,677,840
Moreppen research site
1.84E-05
N18
Results
N32
N12
N34
railway
runways
kg/s
N30
6672500
N36
N38
N40
N42
616 000
N44
N46
N20
8.40E-06
4
• Main character of observed liquid saturation
is simulated (fig. 8 and 9) and
absolute permeabilities are estimated
according to independent
observations (fig. 10)
• Simulation by EOF reproduce Cpp (fig. 11)
• Neglecting Cpp imply unphysical parameter combinations and overestimation of parameter
uncertainty (fig. 12)
2
0
25- 26- 27- 28- 29- 3012345678910- 11Apr- Apr- Apr- Apr- Apr- Apr- May- May- May- May- May- May- May- May- May- May- May95
95
95
95
95
95
95
95
95
95
95
95
95
95
95
95
95
-2
-1.16E-05
-4
tim e
618 000
GPR(46)
The Forward Flow Model:
The numerical code TOUGH2 (Preuss, 1991) is used to solve Richards equation with
constitutive relations between pressure p, permeability kr and saturation S according to the van
Genuchten model (fig.2):
 1

6
-6.60E-06
GPR(44)
1
p   S

8
1.34E-05
-1.60E-06
utm-E
1

m
e
10
3.40E-06
5m
614 000
Figure 7. Effective
Infiltration rate
2.34E-05
N10
500m
12
2.84E-05
Oslo
Oslo
Figure 1.
The Moreppen research site
Delta topsets
1
p41
6,677,800
Figure 5.
Local Sedimentological Architecture
p48
p44 p46
p33
6,677,780
0
615,800
615,750
6,677,760
Moreppen
research
site
Dip 1
3
S
Oslo airport
Gardermoen
Top 2
Flow model
1
n


k r  S 1  1  S  
 
 
1
2
e
1
m
e
m
2
where Se is effective
saturation,
• Se = (S- Sr)/(1- Sr),
• Sr is called residual liquid
saturation,
• 1/ is called air entry value,
and
• m=1-1/n where
• n is called the pore size
distribution index.
Figure 8. Reproduction of observed liquid
saturation in location c11 and c16 (cf.fig 6)
A priori grainsize distribution data courtesy Anne Kristine Søvik
Figure 2. Constitutive relations between
pressure p, permeability kr and saturation S
predef
no_lim
Inverse modeling:
lim
where y*j is observation of liquid
saturation in space and yi(p) is
the forward model response,
p={ki, Sri 1/i,ni}, i=1,2,…,number
of sedimentological units, in this
case equal to 4 (top1, top2, dip1
and dip2)
true
unknown
system
response
TOUGH2
model
8.00E-01
1.00E-01
6.00E-01
1.00E-02
1/a-vg_n
1/a-K
vg_n-vg_n
vg_n-K
Slr-vg_n
Slr-K
K-vg_n
stopping
criteria
Hazens m/s
Gustafson m/s
Top 1
Top 2
Dip 1
Dip 2
1.00E-03
Ks [m/s]
1.00E-04
0.00E+00
1.00E-05
-2.00E-01
1.00E-06
-4.00E-01
1.00E-07
-6.00E-01
1.00E-08
0
0.5
1
parameters
calculated
system
response
measured
system
response
minimization
algorithm
best estimate
of model
parameters
objective
function
maximumlikelihood
theory
Uncertainty propagation analysis by
Empirical Orthogonal Eigenfunctions
1.00E+00
2.00E-01
K-K
corrected
parameter
estimate
prior
information
1.00E+00
4.00E-01
Cpp
The code iTOUGH2 (Finsterle, 1999) is used. The general inverse modeling procedure is
illustrated in fig.3 . In this case the inverse problem is to estimate the parameter p in such a
way that the residual vector r is minimized:
rj  y * j y j (p)
Figure 9. Difference between observed
and calculated liquid saturation
Figure 3. Inverse
modeling procedure
Figure 11. Cpp reproduced by EOFsimulation
1.5
2
2.5
3
3.5
4
depth [m]
Figure 10. Observed and estimated
hydraulic conductivities
a posteriori
error
analysis
Conclusions:
uncertainty
propagation
analysis
• EOF simulation reproduces Cpp, and
thereby automatically avoides unlikely
parameter combinations
Given the covariance matrix Cpp of the best-estimate parameter set p (Carrera and Neuman, 1986):
Cpp  s02 JT C ZZ1 J
• EOF simulation does not rely on second
order stationarity
Conditional simulation of the parameter set p can be done according to the Proper Orthogonal
Decomposition Theorem (Loève, 1977):
• Truncation of p reduces the quality of Cpp
reproduction
1
p
n

k 1
k
• Geological architecture is critical
βk
• Liquid saturation data can be used to
estimate optimal parameters for flow
simulation, but a priori information is
necessary
where b is the eigenvector derived from:
CppβkT  kβk
k  1,..., n
and:
E{  k  j }  kj k
k, j  1,...,n
where  is the eigenvalue, and ij = 0 if i j, else 1.
Figure 12. Improved simulation by EOF
• Non-steady infiltration improves
parameter estimation
mm/d
The purpose of this study is to:
• Estimate sedimentological architecture and liquid saturation by
Ground Penetrating Radar.
p47