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

A parametric acceleration of multilevel Monte Carlo convergence for nonlinear variably saturated flow

We present a multilevel Monte Carlo (MLMC) method for the uncertainty quantification of variably saturated porous media flow that is modeled using the Richards equation. We propose a stochastic extension for the empirical models that are typically employed to close the Richards equations. This is achieved by treating the soil parameters in these models as spatially correlated random fields with appropriately defined marginal distributions. As some of these parameters can only take values in a specific range, non-Gaussian models are utilized. The randomness in these parameters may result in path-wise highly nonlinear systems, so that a robust solver with respect to the random input is required. For this purpose, a solution method based on a combination of the modified Picard iteration and a cell-centered multigrid method for heterogeneous diffusion coefficients is utilized. Moreover, we propose a non-standard MLMC estimator to solve the resulting high-dimensional stochastic Richards equation. The improved efficiency of this multilevel estimator is achieved by parametric continuation that allows us to incorporate simpler nonlinear problems on coarser levels for variance reduction while the target strongly nonlinear problem is solved only on the finest level. Several numerical experiments are presented showing computational savings obtained by the new estimator compared with the original MC estimator

A parametric acceleration of multilevel Monte Carlo convergence for nonlinear variably saturated flow

We present a multilevel Monte Carlo (MLMC) method for the uncertainty quantification of variably saturated porous media flow that is modeled using the Richards equation. We propose a stochastic extension for the empirical models that are typically employed to close the Richards equations. This is achieved by treating the soil parameters in these models as spatially correlated random fields with appropriately defined marginal distributions. As some of these parameters can only take values in a specific range, non-Gaussian models are utilized. The randomness in these parameters may result in path-wise highly nonlinear systems, so that a robust solver with respect to the random input is required. For this purpose, a solution method based on a combination of the modified Picard iteration and a cell-centered multigrid method for heterogeneous diffusion coefficients is utilized. Moreover, we propose a non-standard MLMC estimator to solve the resulting high-dimensional stochastic Richards equation. The improved efficiency of this multilevel estimator is achieved by parametric continuation that allows us to incorporate simpler nonlinear problems on coarser levels for variance reduction while the target strongly nonlinear problem is solved only on the finest level. Several numerical experiments are presented showing computational savings obtained by the new estimator compared with the original MC estimator