Application of multilevel concepts for uncertainty quantification in reservoir simulation

Uncertainty quantification is an important task in reservoir simulation and is an active area of research. The main idea of uncertainty quantification is to compute the distribution of a quantity of interest, for example oil rate. That uncertainty, then feeds into the decision making process. A statistically valid way of quantifying the uncertainty is a Markov Chain Monte Carlo (MCMC) method, such as Random Walk Metropolis (RWM). MCMC is a robust technique for estimating the distribution of the quantity of interest. RWM is can be prohibitively expensive, due to the need to run a huge number of realizations, 45% - 70% of these may be rejected and, even for a simple reservoir model it may take 15 minutes for each realization. Hamiltonian Monte Carlo accelerates the convergence for RWM but may lead to a large increase computational cost because it requires the gradient. In this thesis, we present how to use the multilevel concept to accelerate convergence for RWM. The thesis discusses how to apply Multilevel Markov Chain Monte Carlo (MLMCMC) to uncertainty quantification. It proposes two new techniques, one for improving the proxy based on multilevel idea called Multilevel proxy (MLproxy) and the second one for accelerating the convergence of Hamiltonian Monte Carlo is called Multilevel Hamiltonian Monte Carlo (MLHMC). The idea behind the multilevel concept is a simple telescoping sum: which represents the expensive solution (e.g., estimating the distribution for oil rate on finest grid) in terms of a cheap solution (e.g., estimating the distribution for oil rate on coarse grid) and `correction terms', which are the difference between the high resolution solution and a low resolution solution. A small fraction of realizations is then run on the finer grids to compute correction terms. This reduces the computational cost and simulation errors significantly. MLMCMC is a combination between RWM and multilevel concept, it greatly reduces the computational cost compared to the RWM for uncertainty quantification. It makes Monte Carlo estimation a feasible technique for uncertainty quantification in reservoir simulation applications. In this thesis, MLMCMC has been implemented on two reservoir models based on real fields in the central Gulf of Mexico and in North Sea. MLproxy is another way for decreasing the computational cost based on constructing an emulator and then improving it by adding the correction term between the proxy and simulated results. MLHMC is a combination of Multilevel Monte Carlo method with a Hamiltonian Monte Carlo algorithm. It accelerates Hamiltonian Monte Carlo (HMC) and is faster than HMC. In the thesis, it has been implemented on a real field called Teal South to assess the uncertainty

On the predictivity of pore-scale simulations : estimating uncertainties with multilevel Monte Carlo

A fast method with tunable accuracy is proposed to estimate errors and uncertainties in pore-scale and Digital Rock Physics (DRP) problems. The overall predictivity of these studies can be, in fact, hindered by many factors including sample heterogeneity, computational and imaging limitations, model inadequacy and not perfectly known physical parameters. The typical objective of pore-scale studies is the estimation of macroscopic effective parameters such as permeability, effective diffusivity and hydrodynamic dispersion. However, these are often non-deterministic quantities (i.e., results obtained for specific pore-scale sample and setup are not totally reproducible by another “equivalent” sample and setup). The stochastic nature can arise due to the multi-scale heterogeneity, the computational and experimental limitations in considering large samples, and the complexity of the physical models. These approximations, in fact, introduce an error that, being dependent on a large number of complex factors, can be modeled as random. We propose a general simulation tool, based on multilevel Monte Carlo, that can reduce drastically the computational cost needed for computing accurate statistics of effective parameters and other quantities of interest, under any of these random errors. This is, to our knowledge, the first attempt to include Uncertainty Quantification (UQ) in pore-scale physics and simulation. The method can also provide estimates of the discretization error and it is tested on three-dimensional transport problems in heterogeneous materials, where the sampling procedure is done by generation algorithms able to reproduce realistic consolidated and unconsolidated random sphere and ellipsoid packings and arrangements. A totally automatic workflow is developed in an open-source code [1], that include rigid body physics and random packing algorithms, unstructured mesh discretization, finite volume solvers, extrapolation and post-processing techniques. The proposed method can be efficiently used in many porous media applications for problems such as stochastic homogenization/upscaling, propagation of uncertainty from microscopic fluid and rock properties to macro-scale parameters, robust estimation of Representative Elementary Volume size for arbitrary physics

Multilevel and quasi-Monte Carlo methods for uncertainty quantification in particle travel times through random heterogeneous porous media

In this study, we apply four Monte Carlo simulation methods, namely, Monte Carlo, quasi-Monte Carlo, multilevel Monte Carlo and multilevel quasi-Monte Carlo to the problem of uncertainty quantification in the estimation of the average travel time during the transport of particles through random heterogeneous porous media. We apply the four methodologies to a model problem where the only input parameter, the hydraulic conductivity, is modelled as a log-Gaussian random field by using direct Karhunen–Loéve decompositions. The random terms in such expansions represent the coefficients in the equations. Numerical calculations demonstrating the effectiveness of each of the methods are presented. A comparison of the computational cost incurred by each of the methods for three different tolerances is provided. The accuracy of the approaches is quantified via the mean square error

Multilevel Markov Chain Monte Carlo Method for High-Contrast Single-Phase Flow Problems

In this paper we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow problems. It is based on the generalized multiscale finite element method (GMsFEM) and multilevel Monte Carlo (MLMC) methods. The former provides a hierarchy of approximations of different resolution, whereas the latter gives an efficient way to estimate quantities of interest using samples on different levels. The number of basis functions in the online GMsFEM stage can be varied to determine the solution resolution and the computational cost, and to efficiently generate samples at different levels. In particular, it is cheap to generate samples on coarse grids but with low resolution, and it is expensive to generate samples on fine grids with high accuracy. By suitably choosing the number of samples at different levels, one can leverage the expensive computation in larger fine-grid spaces toward smaller coarse-grid spaces, while retaining the accuracy of the final Monte Carlo estimate. Further, we describe a multilevel Markov chain Monte Carlo method, which sequentially screens the proposal with different levels of approximations and reduces the number of evaluations required on fine grids, while combining the samples at different levels to arrive at an accurate estimate. The framework seamlessly integrates the multiscale features of the GMsFEM with the multilevel feature of the MLMC methods following the work in \cite{ketelson2013}, and our numerical experiments illustrate its efficiency and accuracy in comparison with standard Monte Carlo estimates.Comment: 29 pages, 6 figure