We study improvements of standard and multilevel Monte Carlo methods for point evaluation of the cumulative distribution function (failure probability) applied to porous media two-phase flow simulations with uncertain permeability. In an injection scenario with sweep efficiency of the injected phase as quantity of interest, we seek the probability that this quantity of interest is smaller than a critical value. In the sampling procedure, we use computable error bounds on the sweep efficiency functional to solve only a subset of all realizations to highest accuracy by means of what we call selective refinement. We quantify the performance gains possible by using selective refinement in combination with both the standard and multilevel Monte Carlo method. We also identify issues in the process of practical implementation of the methods. We conclude that significant savings (one order of magnitude) in computational cost are possible for failure probability estimation in a realistic setting using the selective refinement technique, both in combination with standard and multilevel Monte Carlo.
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