An improved multilevel Monte Carlo method for estimating probability distribution functions in stochastic oil reservoir simulations

Abstract

In this work, we develop an improved multilevel Monte Carlo (MLMC) method for estimating cumulative distribution functions (CDFs) of a quantity of interest, coming from numerical approximation of large‐scale stochastic subsurface simulations. Compared with Monte Carlo (MC) methods, that require a significantly large number of high‐fidelity model executions to achieve a prescribed accuracy when computing statistical expectations, MLMC methods were originally proposed to significantly reduce the computational cost with the use of multifidelity approximations. The improved performance of the MLMC methods depends strongly on the decay of the variance of the integrand as the level increases. However, the main challenge in estimating CDFs is that the integrand is a discontinuous indicator function whose variance decays slowly. To address this difficult task, we approximate the integrand using a smoothing function that accelerates the decay of the variance. In addition, we design a novel a posteriori optimization strategy to calibrate the smoothing function, so as to balance the computational gain and the approximation error. The combined proposed techniques are integrated into a very general and practical algorithm that can be applied to a wide range of subsurface problems for high‐dimensional uncertainty quantification, such as a fine‐grid oil reservoir model considered in this effort. The numerical results reveal that with the use of the calibrated smoothing function, the improved MLMC technique significantly reduces the computational complexity compared to the standard MC approach. Finally, we discuss several factors that affect the performance of the MLMC method and provide guidance for effective and efficient usage in practice.

A Machine Learning Based Hybrid Multi-Fidelity Multi-Level Monte Carlo Method for Uncertainty Quantification

“…Fagerlund et al (2016) combined selective refinement technique with the MLMC for estimating the sweep efficiency in a two-phase flow scenario where an absolute accuracy of failure probability in a magnitude 5 to 10 percent is required. Lu et al (2016) applied MLMC method for estimating cumulative distribution functions of QoI obtained from the numerical approximation of largescale stochastic subsurface simulations. For a complete review of MLMC method, we refer the readers to the following papers by Giles (2013) and Giles (2015).…”

Inverse Modeling of Hydrologic Systems with Adaptive Multifidelity Markov Chain Monte Carlo Simulations

“…It is noted that, multi-fidelity simulation methods, that is, using paired simple and complex models to achieve a balance between accuracy and efficiency, have become increasingly popular in hydrologic science (Doherty & Christensen, 2011;Linde et al, 2017;Lu et al, 2016;Moslehi et al, 2015;Watson et al, 2013), while the application of these methods in Bayesian inference is very limited. Here we adopt the method originally developed by Kennedy and O'Hagan (2000) to build the multifidelity GP system.…”

Uncertainty Quantification in Discrete Fracture Network Models: Stochastic Geometry

“…We limit ourself to the description of this case, which is the one adopted here. This approach has been successfully applied in the framework of PDEs with random coefficients in several recent papers, see, e.g., Cliffe et al () and Teckentrup et al (); see also Icardi et al () for an application in the framework of pore‐scale simulations and Lu et al () for an application to stochastic oil reservoir simulations. We remark that, in the framework of DFN simulations, the application of this approach strongly hinges upon the availability of an underlying solver capable to work with very coarse meshes.…”