Oslo

Gardermoen

Oslo

N12

N10N18

N20

N34

N32

N30

N40

N38

N36

N46

N44

N42

N12

N10N18

N20

N34

N32

N30

N40

N38

N36

N46

N44

N42

GPR(44) GPR(46)

GPR(47)

GPR(45)

5 m

Oslo airport Gardermoen

500m

Moreppenresearch site

railwayrunways

utm-E

utm-N

614 000 616 000 618 000

6672500

6674500

6676500

6678500

Ground Penetrating Radar profiles

Figure 1. The Moreppen research site

INVERSE MODELLING OF UNSATURATED FLOW COMBINED WITH STOCHASTIC SIMULATION USING INVERSE MODELLING OF UNSATURATED FLOW COMBINED WITH STOCHASTIC SIMULATION USING EMPIRICAL ORTHOGONAL FUNCTIONS (EOF)EMPIRICAL ORTHOGONAL FUNCTIONS (EOF)

Nils-Otto KitterødUniversity of Oslo, Department of Geophysics, Norwaye-mail: nilsotto@geofysikk.uio.no

Background

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.

Stefan FinsterleUniversity of California, Lawrence Berkeley National Laboratory

Berkeley, California, USA

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):

2m

m1

e21

er S11Sk

n1

m1

e 1S1

p

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.

Conditional simulation of the parameter set p can be done according to the Proper Orthogonal Decomposition Theorem (Loève, 1977):

and:

Uncertainty propagation analysis by Empirical Orthogonal Eigenfunctions

Given the covariance matrix Cpp of the best-estimate parameter set p (Carrera and Neuman, 1986):

11

ZZ

T2

0 JJs CCpp

n

1kkkβp

where is the eigenvector derived from:

n,...,1kkk

T

k ββCpp

n,...,1j,k}{E kkjjk

where is the eigenvalue, andij = 0 if i j, else 1.

Conclusions:

• EOF simulation reproduces Cpp, and

thereby automatically avoides unlikely parameter combinations

• EOF simulation does not rely on second order stationarity

• Truncation of p reduces the quality of Cpp

reproduction

• Geological architecture is critical

• Liquid saturation data can be used to estimate optimal parameters for flow simulation, but a priori information is necessary

• Non-steady infiltration improves parameter estimation

Figure 5. Local Sedimentological Architecture

1050

Dep

th <

m>

0

1

2

3

Top 1

Top 2

Dip 1

Dip 2

lp4

West-East <m>1050

Dep

th <

m>

0

1

2

3

sp4

c11 c16

Figure 6. Liquid saturation, May 11, 1995

2D flow test

Inverse modeling:

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:

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)

)(y*y jjj pr

Figure 2. Constitutive relations between pressure p, permeability kr and saturation S

• Use liquid saturation as primary data for Bayesian Maximum Likelihood Inversion of unsaturated flow parameters.

• Simulate parameter uncertainties by Karhunen-Loève expansion.

The purpose of this study is to:• Estimate sedimentological architecture and liquid saturation by

Ground Penetrating Radar.

Input data

• 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)

-1.16E-05

-6.60E-06

-1.60E-06

3.40E-06

8.40E-06

1.34E-05

1.84E-05

2.34E-05

2.84E-05

25-Apr-95

26-Apr-95

27-Apr-95

28-Apr-95

29-Apr-95

30-Apr-95

1-May-95

2-May-95

3-May-95

4-May-95

5-May-95

6-May-95

7-May-95

8-May-95

9-May-95

10-May-95

11-May-95

time

kg/s

-4

-2

0

2

4

6

8

10

12

mm

/d

Figure 7. Effective Infiltration rate

Results

• 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)

Figure 8. Reproduction of observed liquid saturation in location c11 and c16 (cf.fig 6)

Figure 9. Difference between observed and calculated liquid saturation

Figure 10. Observed and estimated hydraulic conductivities

Figure 11. Cpp reproduced by EOF-simulation

Figure 12. Improved simulation by EOF

p47

p45

Figure 4. Ground Penetrating Radar

foresets

p43

Ground-water-

table

Delta -

topsets

Flow model

p41

Figure 3. Inverse modeling procedure

trueunknownsystem

response

measuredsystem

response

TOUGH2model

calculatedsystem

response

stoppingcriteria

minimizationalgorithm

objectivefunction

best estimateof model

parameters

maximum-likelihood

theory

a posteriorierror

analysis

uncertaintypropagation

analysis

priorinformation

correctedparameterestimate