Screening wells by multi-scale grids for multi-stage Markov Chain Monte Carlo simulation

For improved prediction of subsurface flows and representation of the uncertainties of geostatistical properties, we use the framework of Bayesian statistical interface in combination with the Markov Chain Monte Carlo (MCMC) method which needs many fine-scale simulations. Hence it is essential to apply cheap screening stages, such as coarse-scale simulation to remove irrelevant proposals of the generated Markov chain, reduce fine-scale computational cost and increase the acceptance rate of MCMC. We propose a screening step, that is examination of subsurface characteristics around injection/production wells, aiming at accurate breakthrough capturing as well as above mentioned efficiency goals. However this short time simulation needs fine-scale structure of the geological model around wells and running a fine-scale model is not as cheap as necessary for screening steps. On the other hand applying it on a coarse-scale model declines important data around wells and causes inaccurate results, particularly accurate breakthrough capturing which is important for prediction applications. Therefore we propose a multi-scale grid which preserves the fine-scale model around wells (as well as high permeable regions and fractures) and coarsens rest of the field and keeps efficiency and accuracy for the screening well stage and coarse-scale simulation, as well. A discrete wavelet transform is used as a powerful tool to generate the desired unstructured multi-scale grid efficiently. Finally an accepted proposal on coarse-scale models (screening well stage and coarse-scale simulation) will be assessed by fine-scale simulation. Accepted proposals are saved for prediction. Numerical results admit increment in acceptance rate, improvement in breakthrough capturing and significant reduction in computational cost by avoiding many forward simulations.