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Inversion of coupled groundwater flow and heat transfer
M. Bücker 1, V.Rath 2 & A. Wolf 1
1 Scientific Computing, 2 Applied Geophysics
Bommerholz 14.8-18.8, Sommerschule 2006: Automatisches Differenzieren
2
x (km)
020
4060
80100y
(km)
0
20
40
60
z(km
)
0
1
2
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z(k
m)
Y
X
Z
Contents
• Geothermal modeling in SHEMAT
• Bayesian inversion method
• Validation of the inversion code
• Analytical solution
• Numerical experiments
• Covariance and Resolution matrices
• Quality Indicator
• Summary
3
FD solution of 3-D steady-state coupled fluid flow and heat transfer:
SHEMAT equations
Data h : hydraulic heads T : temperatures
Parameter k : hydraulic permeability λ : thermal conductivity
Others v : filtration velocity depending on hydraulic head Q, A : sources …
4SHEMAT intern
• Dirichlet and Neumann boundary conditions
• fluid and rock properties dependent on temperature
and pressure
• nonlinear solution by simple alternating fixed point
iteration
• linear solvers
•direct (from LAPACK) and iterative solvers
(BiCGstab, parallelized with OpenMP )
6
Data from Boreholes: temperatures and hydraulic heads
Parameters: e.g. permeability, thermal conductivity. Underground structure sometimes well known, but measurements of parameters values often inadequate
Questions:
Is it possible to distinguish between advective and conductive effects?
Which uncertainties will be present in the estimated parameters?
Which data are necessary to constrain the estimate?
Inverse geothermal modeling
d g(p)
1d g (p)
nPmD
forward modeling,“SHEMAT”
Inverse modeling,“SHEM_AD”
8
General assumptions:
• A-priori error bounds of
data and parameters
• Arbitrary integration of
boundary conditions
• No ad-hoc regularisation
parameters
Bayesian Inversion
Thomas Bayes, 1702-1761
9
with: covariance matrices a priori
residual
Differentiate with respect to p, and apply Gauss-Newton method
1 1( ( )) ( ( )) ( ) ( ) min!T TB a a
d pd g p C d g p p p C p p
1 1 1 1 1
1 1
( ) [ ( )] (parameter space)
( ) [ ( )] (data space)
k a T T kd p d
k a T T kp p d
g
g
p p J C J C J C d p
p p C J JC J C d p
,
,
( )( )
( )( )
d ij i i j j
p ij i i j j
C E d d d d
C E p p p p
( ) r d g p
11 1 1( )apo T T Tp d p p p p d p
C J C J C C C J JC J C JC
with Jacobian
parameter covariance a posteriori
Bayesian Estimation
10
Analytical solution for coupled flow and heat transport
00 1 0
exp ( ) 1
exp 1( )
z z
Pe
Pe zT z T T T
fc v zPe
Péclet Number
Validation by analytical solution
12
Free Convection(driven by density differences)
Forced Convection(driven by surface topography )
Synthetic Models
T
h
15
Test setup
• 8 data sets / runs
• 8 boreholes chosen at random (position and depth)
• consisting of temperatures, heads, or both
• Data errors: ΔT = 0.5 K, Δh = 0.5 m
• Error bars: ,apo
i p iip C
Numerical experiment: type 1
Reference model
• Generate original data from initialisation parameters
Goal
• Estimate parameters
Boreholes
19
Test setup
• 8 data sets / runs
• consisting of 4, 8, or 12 boreholes
• each chosen at random (position and depth)
• Data errors: ΔT = 0.5 K, Δh = 0.5 m
• Error bars: ,apo
i p iip C
Numerical experiment: type 2
Reference model
• Same original data from initialisation parameters
Goal
• Estimate parameters
Boreholes
21Data Fit: Forced Convection
• One of the runs with 8 randomly chosen boreholes• Temperature and head data• Inversion converged
• Adequate parameters estimated
22Information Discussion
Questions:
• Which uncertainties will be present in the estimated
parameters?
• Which data are necessary to constrain the estimate?
23
Covariance matrix a posteriori near the minimum of θB:
11 1 1( )apo T T Tp d p p p p d p
C J C J C C C J JC J C JC
Covariance a posteriori
Disadvantage: full Jacobian matrix
units units
units
Thermal Conductivity Permeability
24
,
( )
est truep
apo apr Tp p p
p R p
R I -C C
Parameter resolution matrix(solution inverse problem)Free Convection
Thermal Conductivity
Resolution matrices
units
units
25
x (km)
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80100y
(km)
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z(km
)
0
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z(k
m)
Y
X
Z
kz
2E-141.8E-141.6E-141.4E-141.2E-141E-148E-156E-154E-152E-15
x (km)
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80100y
(km)
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z(km
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m)
Y
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Z
Permeability
Model (14 zones)
x (km)
020
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(km)
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z(km
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Y
X
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temp
1201101009080706050403020
Temperature
Experimental Design: 3-D Model
26
x (m)
y (
m)
TEMP method: t2
0 2 4 6 8
x 104
0
1
2
3
4
5
x 104
5
10
15
20
25
30
Design Quality Indicators
• Top view• Sensitivities for 0 to 2000m
† 1 1
†
( )T Td p
G J C J C J
UΛV G
“Generalized Inverse”
†
21 max max
†3
1
trace( )
det( )
...
Ni
i
N
ii
t
t
G
G
M
M
Advantage: row compressed Jacobian matrix
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• Successful validation without data from real experiments, which can be expensive
• Covariance and Resolution matrices can help to decide which parameters needs to be determent more exactly– But their computation may be expensive
Future work:• Reverse-Mode AD version make it possible to use
algorithms with:– improved convergency (matrix free Newton-Krylow)– smaller memory requirements for larger models
• Validation with real experimental data
Summary and Conclusions