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

Abstract

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 Morkov 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.

A multiscale direct solver for the approximation of flows in high contrast porous media

“…However, it is important to note that it is not our intention to construct solutions on larger subdomains at all. We have computed local solutions on the smallest subdomains and want to know what would be their coefficients in the global solution, i.e., P h in (5), that is a linear combination of all local solutions. Thus, we now focus on how to determine P h |Ω a .…”

“…The remaining portion of the argument for combining subdomains remains unaltered for m > 1 and m = 1. We obtain {P cl , u cl }, an approximation for (5), that is discontinuous at the boundaries of the primal subdomains, and has to be post-processed to become a continuous approximation.…”

“…Therefore, the goal of MS methods is to have data exchange between fine and coarse quantities in order to improve the accuracy of upscaled coarse grid calculations. This strategy is useful in applications such as uncertainty quantification, where thousands of simulations have to be performed [5]. For a wide range of applications these methodologies have been proven to be effective in terms of accuracy and improved efficiency.…”

A comparative PEST study of diesel engine research in India, China, USA and Turkey

“…With this, it is possible to develop highly accurate chemical kinetic models to estimate reactions at low temperatures [4], self-ignition, heat release [5], such as exhaust emissions and soot produced [6], [7]. By analyzing gasoline engines, diesel engines have the advantage of producing the required torque with optimum fuel and exhaust gas consumption according to regulations [8], [9]. Studies have also been conducted on the combustion of a single-cylinder dual-cylinder diesel engine with gasoline injection to show the rate of heat expelled, the fuel consumption in a controlled space [10].…”