1. 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. 2 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. 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 …

4. 4 SHEMAT 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 )

5. 5 SHEMAT in action

6. 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 forward modeling, “SHEMAT” Inverse modeling, “SHEM_AD”

7. 7 Parameterisation

8. 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. 9 with: covariance matrices a priori residual Differentiate with respect to p, and apply Gauss-Newton method  with Jacobian  parameter covariance a posteriori Bayesian Estimation

10. 10 Analytical solution for coupled flow and heat transport Péclet Number Validation by analytical solution

11. 11 Synthetic Models

12. 12 Free Convection (driven by density differences) Forced Convection (driven by surface topography ) Synthetic Models T h

13. 13 Free Convection Forced Convection Temperature Sensitivities + boundary conditions

14. 14 Free Convection Forced Convection Hydraulic Head Sensitivities

15. 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: Numerical experiment: type 1 Reference model Generate original data from initialisation parameters Goal Estimate parameters Boreholes

16. 16 Parameter Fit units

17. 17 units Numerical experiment: data types

18. 18 Numerical experiment (II) Forced Convection Free Convection units

19. 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: Numerical experiment: type 2 Reference model Same original data from initialisation parameters Goal Estimate parameters Boreholes

20. 20 units Synthetic Models: how many data?

21. 21 Data Fit: Forced Convection One of the runs with 8 randomly chosen boreholes Temperature and head data Inversion converged Adequate parameters estimated

22. 22 Information Discussion Questions: Which uncertainties will be present in the estimated parameters? Which data are necessary to constrain the estimate?

23. 23 Covariance matrix a posteriori near the minimum of θ B : Covariance a posteriori  Disadvantage: full Jacobian matrix units Thermal ConductivityPermeability

24. 24 Parameter resolution matrix (solution inverse problem) Free Convection Thermal Conductivity Resolution matrices units

25. 25 Permeability Model (14 zones) Temperature Experimental Design: 3-D Model

26. 26 Design Quality Indicators Top view Sensitivities for 0 to 2000m “Generalized Inverse” M M  Advantage: row compressed Jacobian matrix

27. 27 Parameter Sensitivity Matrix

28. 28 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

29. 29 The End THANK YOU FOR YOUR ATTENTION !