Felipe Pereira

Original article: Rapid quantification of uncertainty in permeability and porosity of oil reservoirs for enabling predictive simulation

One of the most difficult tasks in subsurface flow simulations is the reliable characterization of properties of the subsurface. A typical situation employs dynamic data integration such as sparse (in space and time) measurements to be matched with simulated responses associated with a set of permeability and porosity fields. Among the challenges found in practice are proper mathematical modeling of the flow, persisting heterogeneity in the porosity and permeability, and the uncertainties inherent in them. In this paper we propose a Bayesian framework Monte Carlo Markov Chain (MCMC) simulation to sample a set of characteristics of the subsurface from the posterior distribution that are conditioned to the production data. This process requires obtaining the simulated responses over many realizations. In reality, this can be a prohibitively expensive endeavor with possibly many proposals rejection, and thus wasting the computational resources. To alleviate it, we employ a two-stage MCMC that includes a screening step of a proposal whose simulated response is obtained via an inexpensive coarse-scale model. A set of numerical examples using a two-phase flow problem in an oil reservoir as a benchmark application is given to illustrate the procedure and its use in predictive simulation.

Original article: Design and implementation of a multiscale mixed method based on a nonoverlapping domain decomposition procedure

We use a nonoverlapping iterative domain decomposition procedure based on the Robin interface condition to develop a new multiscale mixed method to compute the velocity field in heterogeneous porous media. Hybridized mixed finite elements are used for the spatial discretization of the equations. We define local, multiscale mixed basis functions to represent the discrete solutions in subdomains. Appropriate subspaces of the vector space spanned by these basis functions can be considered in the numerical approximations of heterogeneous porous media flow problems. The balance between numerical accuracy and numerical efficiency is determined by the choice of these subspaces. A detailed description of the numerical method is presented. Following that, numerical experiments are discussed to illustrate the important features of the new procedure and its comparison to the traditional fine grid simulations.

A prefetching technique for prediction of porous media flows

In many applications in flows through porous media, one needs to determine the properties of subsurface to detect, monitor, or predict the actions of natural or induced forces. Here, we focus on two important subsurface properties: rock permeability and porosity. A Bayesian approach using a Markov Chain Monte Carlo (MCMC) algorithm is well suited for reconstructing the spatial distribution of permeability and porosity, and quantifying associated uncertainty in these properties. A crucial step in this approach is the computation of a likelihood function, which involves solving a possibly nonlinear system of partial differential equations. The computation time for the likelihood function limits the number of MCMC iterations that can be performed in a practical period of time. This affects the consistency of the posterior distribution of permeability and porosity obtained by MCMC exploration. To speed-up the posterior exploration, we can use a prefetching technique, which relies on the fact that multiple likelihoods of possible states into the future in an MCMC chain can be computed ahead of time. In this paper, we show that the prefetching technique implemented on multiple processors can make the Bayesian approach computationally tractable for subsurface characterization and prediction of porous media flows.

Multiple Markov chains Monte Carlo approach for flow forecasting in porous media

Abstract Predictions in subsurface formations consists of two steps: characterization and prediction using the characterization. In the characterization, we reconstruct the subsurface properties, such as distributions of permeability and porosity, with a set of limited data. A Bayesian approach using Markov Chain Monte Carlo (MCMC) methods is well suited for reconstructing permeability and porosity fields. This statistical approach aims at generating a Markov chain from which a stationary, posterior distribution of the characteristics of the subsurface may be constructed. A crucial step in this framework is the calculation of the likelihood information which can be computationally very demanding. This limitation hinders the application of the Bayesian framework for the flow predictions in porous media in a practical period of time. The parallel computation of multiple MCMCs can substantially reduce computation time and can make the framework more suitable to subsurface flows. In this paper, we consider multi–MCMC and compare the multi–MCMC with the MCMCs for the predictions of subsurface flows.

Forecasting Production in an Oil Reservoir Simulation and Its Challenges

A Bayesian approach for uncertainty quantification of oil reservoir parameters in forecasting the production is straightforward in principle. However, the complexity of flow simulators and the nature of the inverse problem at hand present an ongoing practical challenges to addressing uncertainty in all subsurface parameters. In this paper, we focus on two important subsurface parameters, permeability and porosity, and discuss quantifying uncertainty in those parameters.

Mixing regimes and the scale up problem for multiphase flow in porous media

We examine fluid mixing dynamics for two-phase, immiscible flow in heterogeneous porous media. To assess the combined effect of nonlinear flow instability and heterogeneity, we solve the model problem numerically, using large suites of realizations of stochastically generated permeability fields. Different mixing regimes are identified, depending on the relative importance of nonlinearity in the flow equations and heterogeneity.