An efficient Bayesian inversion of a geothermal prospect using a multivariate adaptive regression spline method

In this study, an efficient Bayesian framework equipped with a multivariate adaptive regression spline (MARS) technique is developed to alleviate computational burdens encountered in a conventional Bayesian inversion of a geothermal prospect. Fast MARS models are developed from training dataset generated by CPU-intensive hydrothermal models and used as surrogate of high-fidelity physical models in Markov Chain Monte Carlo (MCMC) sampling. This Bayesian inference with MARS-enabled MCMC method is used to reduce prior estimates of uncertainty in structural or characteristic hydrothermal flow parameters of the model to posterior distributions. A geothermal prospect near Superstition Mountain in Imperial County of California in USA is used to illustrate the proposed framework and demonstrate the computational efficiency of MARS-based Bayesian inversion. The developed MARS models are also used to efficiently drive calculation of Sobol’ total sensitivity indices. Only top sensitive parameters are included in Bayesian inference to further improve the computational efficiency of inversion. Sensitivity analysis also confirms that water circulation through high permeable structures, rather than heat conduction through impermeable granite, is the primary heat transfer method. The presented framework is demonstrated an efficient tool to update knowledge of geothermal prospects by inversing field data. Although only thermal data is used in this study, other type of data, such as flow and transport observations, can be jointly used in this method for underground hydrocarbon reservoirs.

[1]  Ilya M. Sobol,et al.  Sensitivity Estimates for Nonlinear Mathematical Models , 1993 .

[2]  Jun S. Liu,et al.  The Multiple-Try Method and Local Optimization in Metropolis Sampling , 2000 .

[3]  Farid Melgani,et al.  Kernel ridge regression with active learning for wind speed prediction , 2013 .

[4]  R. Dennis Cook,et al.  Cross-Validation of Regression Models , 1984 .

[5]  Yunwei Sun,et al.  Surrogate-based optimization of hydraulic fracturing in pre-existing fracture networks , 2013, Comput. Geosci..

[6]  J. Friedman Multivariate adaptive regression splines , 1990 .

[7]  Farrokh Mistree,et al.  Kriging Models for Global Approximation in Simulation-Based Multidisciplinary Design Optimization , 2001 .

[8]  Kourosh Behzadian,et al.  Stochastic sampling design using a multi-objective genetic algorithm and adaptive neural networks , 2009, Environ. Model. Softw..

[9]  D. Higdon,et al.  Accelerating Markov Chain Monte Carlo Simulation by Differential Evolution with Self-Adaptive Randomized Subspace Sampling , 2009 .

[10]  Christine A. Shoemaker,et al.  A Stochastic Radial Basis Function Method for the Global Optimization of Expensive Functions , 2007, INFORMS J. Comput..

[11]  M. D. McKay,et al.  A comparison of three methods for selecting values of input variables in the analysis of output from a computer code , 2000 .

[12]  Jesús Carrera,et al.  State of the Art of the Inverse Problem Applied to the Flow and Solute Transport Equations , 1988 .

[13]  Raghavan Srinivasan,et al.  Approximating SWAT Model Using Artificial Neural Network and Support Vector Machine 1 , 2009 .

[14]  Ilya M. Sobol,et al.  Theorems and examples on high dimensional model representation , 2003, Reliab. Eng. Syst. Saf..

[15]  Leah L. Rogers,et al.  Solving Problems in Environmental Engineering and Geosciences with Artificial Neural Networks , 1996 .

[16]  G. Mariéthoz,et al.  Bayesian inverse problem and optimization with iterative spatial resampling , 2010 .

[17]  Agus Sudjianto,et al.  Computer Aided Reliability and Robustness Assessment , 1998 .

[18]  Tyler Smith,et al.  Bayesian methods in hydrologic modeling: A study of recent advancements in Markov chain Monte Carlo techniques , 2008 .

[19]  Yalchin Efendiev,et al.  An efficient two‐stage Markov chain Monte Carlo method for dynamic data integration , 2005 .

[20]  Albert Tarantola,et al.  Inverse problem theory - and methods for model parameter estimation , 2004 .

[21]  Tiangang Cui,et al.  Bayesian calibration of a large‐scale geothermal reservoir model by a new adaptive delayed acceptance Metropolis Hastings algorithm , 2011 .

[22]  C. Tiedeman,et al.  Effective Groundwater Model Calibration: With Analysis of Data, Sensitivities, Predictions, and Uncertainty , 2007 .

[23]  K. Dyer,et al.  STOCHASTIC JOINT INVERSION OF A GEOTHERMAL PROSPECT , 2013 .

[24]  Qian Fan,et al.  Multi-source information fusion based fault diagnosis of ground-source heat pump using Bayesian network , 2014 .

[25]  J. Doherty,et al.  Calibration‐constrained Monte Carlo analysis of highly parameterized models using subspace techniques , 2009 .

[26]  J. Gómez-Hernández,et al.  Uncertainty assessment and data worth in groundwater flow and mass transport modeling using a blocking Markov chain Monte Carlo method. , 2009 .

[27]  Robert J. Mellors,et al.  EVALUATION OF A GEOTHERMAL PROSPECT USING A STOCHASTIC JOINT INVERSION MODELING PROCEDURE , 2013 .

[28]  S. Kollet,et al.  Groundwater Availability Within the Salton Sea Basin Final Report , 2008 .

[29]  Bryan A. Tolson,et al.  Review of surrogate modeling in water resources , 2012 .

[30]  G. Marsily,et al.  Regards sur 40 ans de problèmes inverses en hydrogéologie , 1999 .

[31]  Dongxiao Zhang,et al.  A sparse grid based Bayesian method for contaminant source identification , 2012 .

[32]  D. Oliver,et al.  Markov chain Monte Carlo methods for conditioning a permeability field to pressure data , 1997 .

[33]  Andres Alcolea,et al.  Inverse problem in hydrogeology , 2005 .

[34]  Sylvia Richardson,et al.  Markov Chain Monte Carlo in Practice , 1997 .

[35]  Hsien-Chie Cheng,et al.  Assessing a Response Surface-Based Optimization Approach for Soil Vapor Extraction System Design , 2009 .

[36]  Ning Liu,et al.  Inverse Theory for Petroleum Reservoir Characterization and History Matching , 2008 .

[37]  T. Simpson,et al.  Comparative studies of metamodelling techniques under multiple modelling criteria , 2001 .