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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
Contents • Geothermal modeling in SHEMAT • Bayesian inversion method • Validation of the inversion code • Analytical solution • Numerical experiments • Covariance and Resolution matrices • Quality Indicator • Summary
SHEMAT equations FD solution of 3-D steady-state coupled fluid flow and heat transfer: Data h : hydraulic heads T : temperatures Parameter k : hydraulic permeability λ : thermal conductivity Others v : filtration velocity depending on hydraulic head Q, A : sources …
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 )
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”
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
Bayesian Estimation with: covariance matrices a priori residual Differentiate with respect to p, and apply Gauss-Newton method with Jacobian • parameter covariance a posteriori
Analytical solution for coupled flow and heat transport Validation by analytical solution Péclet Number
Synthetic Models Free Convection(driven by density differences) Forced Convection(driven by surface topography ) T h
Temperature Sensitivities Free Convection Forced Convection +boundary conditions
Hydraulic Head Sensitivities Free Convection Forced Convection
Numerical experiment: type 1 Boreholes Reference model • Generate original data from initialisation parameters Goal • Estimate parameters 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:
Parameter Fit units
units units Numerical experiment: data types
Numerical experiment (II) units units Forced Convection Free Convection
Numerical experiment: type 2 Boreholes Reference model • Same original data from initialisation parameters Goal • Estimate parameters Test setup • 8 data sets / runs • consisting of4, 8, or12 boreholes • each chosen at random (position and depth) • Data errors: ΔT = 0.5 K, Δh = 0.5 m • Error bars:
Synthetic Models: how many data? units units
Data Fit: Forced Convection • One of the runs with 8 randomly chosen boreholes • Temperature and head data • Inversion converged • Adequate parameters estimated
Information Discussion Questions: • Which uncertainties will be present in the estimated parameters? • Which data are necessary to constrain the estimate?
Covariance a posteriori Thermal Conductivity Permeability units units units Covariance matrix a posteriori near the minimum of θB: • Disadvantage: full Jacobian matrix
Resolution matrices Parameter resolution matrix (solution inverse problem) Free Convection Thermal Conductivity units units
Experimental Design: 3-D Model Model (14 zones) Permeability Temperature
Design Quality Indicators “Generalized Inverse” M M • Top view • Sensitivities for 0 to 2000m • Advantage: row compressed Jacobian matrix
Parameter Sensitivity Matrix
Summary and Conclusions • 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
The End THANK YOU FOR YOUR ATTENTION !