Hamiltonian Monte Carlo algorithm for the characterization of hydraulic conductivity from the heat tracing data

Abstract Estimating spatial distributions of the hydraulic conductivity in heterogeneous aquifers has always been an important and challenging task in hydrology. Generally, the hydraulic conductivity field is determined from hydraulic head or pressure measurements. In the present study, we propose to use temperature data as source of information for characterizing the spatial distributions of the hydraulic conductivity field. In this way, we performed a laboratory sandbox experiment with the aim of imaging the heterogeneities of the hydraulic conductivity field from thermal monitoring. During the laboratory experiment, we injected a hot water pulse, which induces a heat plume motion into the sandbox. The induced plume was followed by a set of thermocouples placed in the sandbox. After the temperature data acquisition, we performed a hydraulic tomography using the stochastic Hybrid Monte Carlo approach, also called the Hamiltonian Monte Carlo (HMC) algorithm to invert the temperature data. This algorithm is based on a combination of the Metropolis Monte Carlo method and the Hamiltonian dynamics approach. The parameterization of the inverse problem was done with the Karhunen-Loeve (KL) expansion to reduce the dimensionality of the unknown parameters. Our approach has provided successful reconstruction of the hydraulic conductivity field with low computational effort.

[1]  Frédéric Nguyen,et al.  A salt tracer test monitored with surface ERT to detect preferential flow and transport paths in fractured/karstified limestones , 2012 .

[2]  Martin O. Saar,et al.  Review: Geothermal heat as a tracer of large-scale groundwater flow and as a means to determine permeability fields , 2011 .

[3]  Massimo Civita,et al.  Open-loop groundwater heat pumps development for large buildings: A case study , 2009 .

[4]  Mary F. Wheeler,et al.  Hybrid nested sampling algorithm for Bayesian model selection applied to inverse subsurface flow problems , 2014, J. Comput. Phys..

[5]  Frédéric Nguyen,et al.  Quantitative temperature monitoring of a heat tracing experiment using cross-borehole ERT , 2015 .

[6]  P. Ackerer,et al.  Laboratory tracer tests on three-dimensional reconstructed heterogeneous porous media , 2004 .

[7]  A. Tarantola,et al.  Inverse problems = Quest for information , 1982 .

[8]  M. Girolami,et al.  Riemann manifold Langevin and Hamiltonian Monte Carlo methods , 2011, Journal of the Royal Statistical Society: Series B (Statistical Methodology).

[9]  A. Revil,et al.  Stochastic inversion of permeability and dispersivities from time lapse self‐potential measurements: A controlled sandbox study , 2010 .

[10]  H. Jourde,et al.  Influence of the observation scale on permeability estimation at local and regional scales through well tests in a fractured and karstic aquifer (Lez aquifer, Southern France) , 2011 .

[11]  C. V. Theis.  The relation between the lowering of the Piezometric surface and the rate and duration of discharge of a well using ground‐water storage , 1935 .

[12]  M. Mastrocicco,et al.  Limitation of using heat as a groundwater tracer to define aquifer properties: experiment in a large tank model , 2013, Environmental Earth Sciences.

[13]  M. Mastrocicco,et al.  Combined use of heat and saline tracer to estimate aquifer properties in a forced gradient test , 2015 .

[14]  André Revil,et al.  Self-potential monitoring of a thermal pulse advecting through a preferential flow path , 2014 .

[15]  R. Fanelli,et al.  Patterns of Water, Heat, and Solute Flux through Streambeds around Small Dams , 2008, Ground water.

[16]  Olaf Kolditz,et al.  Geothermal energy systems: research perspective for domestic energy provision , 2013, Environmental Earth Sciences.

[17]  Habib N. Najm,et al.  Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems , 2008, J. Comput. Phys..

[18]  A. Revil,et al.  Stochastic joint inversion of temperature and self-potential data to determine the permeability of geological units in sedimentary basins , 2009 .

[19]  S Wildemeersch,et al.  Coupling heat and chemical tracer experiments for estimating heat transfer parameters in shallow alluvial aquifers. , 2014, Journal of contaminant hydrology.

[20]  Liangping Li,et al.  Inverse methods in hydrogeology: Evolution and recent trends , 2014 .

[21]  James Martin,et al.  A Stochastic Newton MCMC Method for Large-Scale Statistical Inverse Problems with Application to Seismic Inversion , 2012, SIAM J. Sci. Comput..

[22]  Radford M. Neal.  MCMC Using Hamiltonian Dynamics , 2011, 1206.1901.

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

[24]  Andrew Gelman,et al.  The No-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo , 2011, J. Mach. Learn. Res..

[25]  Double forced gradient tracer test : Performance and interpretation of a field test using a new solute transport model , 2006 .

[26]  Vittorio Verda,et al.  Development of the thermally affected zone (TAZ) around a groundwater heat pump (GWHP) system: A sensitivity analysis , 2012 .

[27]  Narendra N. Das,et al.  Characterization of effective saturated hydraulic conductivity in an agricultural field using Karhunen‐Loève expansion with the Markov chain Monte Carlo technique , 2010 .

[28]  Omar M. Knio,et al.  Coordinate Transformation and Polynomial Chaos for the Bayesian Inference of a Gaussian Process with Parametrized Prior Covariance Function , 2015, 1501.03323.