### Sammendrag

The Monte Carlo (MC) method is an appealing candidate for uncertainty quantification in reservoir simulation for three reasons: (i) It is the preferred approach for systematic reduction in variance for cases with high-dimensional uncertainty with a strongly nonlinear effect (robustness); (ii) it is seamlessly compatible with any reservoir simulator (non-intrusive); and (iii) it is straight-forward to parallelize (embarrassingly parallel). The method does, however, suffer from a painstakingly slow convergence rate of one half. This means that in many cases, an unacceptably large number of samples are required to achieve a sufficiently low variance.
Multilevel Monte Carlo (MLMC) methods were proposed as a work-around for this slow convergence rate. The premise of these methods is that we can approximate the solution at different levels of accuracy, where less accurate approximations are also less expensive to compute. By computing more samples on lower levels, and less samples on higher levels, we can then obtain a low estimation error at a significantly reduced cost compared to a regular MC method. In reservoir simulation, different levels are typically interpreted as different degrees of spatial upscaling. However, complex geomodels (e.g., channelized reservoirs with high-permeability contrast, different rock types and multiphase behavior) may be very challenging to upscale, and the best methods are generally expensive.
Different levels may also be interpreted as different solver accuracy, where a more accurate solver/discretization is used at higher levels - a view taken by a number of authors. In fact, the only requirement for MLMC to work is that the accuracy and cost of computing a sample increases with level, whereas the variance between two consecutive levels decreases with level. Herein, we follow this line of thought, and outline an MLMC method that uses the same fine spatial resolution for all levels. At the finest level, we use a standard fully-implicit finite volume solver, whereas coarser levels use simple approximate solutions obtained using flow diagnostics and multiscale methods. This gives a flexible and efficient framework for uncertainty quantification that is well suited for geomodels with industry-grade complexity. We demonstrate the applicability of the method on complex geomodels for which efficient upscaling techniques are not readily available, and compare it to standard MC, and MLMC with upscaling.

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