Bayesian Uncertainty Quantification for Channelized Reservoirs via Reduced Dimensional Parameterization

Anirban Mondal,Jia Wei

doi:10.3390/math9091067

Abstract

In this article, we study uncertainty quantification for flows in heterogeneous porous media. We use a Bayesian approach where the solution to the inverse problem is given by the posterior distribution of the permeability field given the flow and transport data. Permeability fields within facies are assumed to be described by two-point correlation functions, while interfaces that separate facies are represented via smooth pseudo-velocity fields in a level set formulation to get reduced dimensional parameterization. The permeability fields within facies and pseudo-velocity fields representing interfaces can be described using Karhunen–Loève (K-L) expansion, where one can select dominant modes. We study the error of posterior distributions introduced in such truncations by estimating the difference in the expectation of a function with respect to full and truncated posteriors. The theoretical result shows that this error can be bounded by the tail of K-L eigenvalues with constants independent of the dimension of discretization. This result guarantees the feasibility of such truncations with respect to posterior distributions. To speed up Bayesian computations, we use an efficient two-stage Markov chain Monte Carlo (MCMC) method that utilizes mixed multiscale finite element method (MsFEM) to screen the proposals. The numerical results show the validity of the proposed parameterization to channel geometry and error estimations.

Highlights

The distribution of subsurface properties are mainly controlled by the location of distinct geologic facies with sharp contrasts in properties, such as permeability and porosity, across facies boundaries [1]
Traditional geostatistical techniques for subsurface characterization have typically relied on two-point correlation functions to describe the spatial variability
We describe the parameterization of the permeability field within the facies

Summary

Introduction

The distribution of subsurface properties are mainly controlled by the location of distinct geologic facies with sharp contrasts in properties, such as permeability and porosity, across facies boundaries [1]. In a fluvial setting, high permeability channel sands are often embedded in a nearly impermeable background causing the dominant fluid movement to be restricted within these channels. Under such conditions, the channel geometry plays an important role in determining the flow behavior in the subsurface. The success of object-based models, such as discrete Boolean or object-based models [5], is heavily dependent on the parameters to specify the object size, shapes, proportion, and orientation These parameters are highly uncertain, in the early stages of subsurface characterization [2,6]. All the parameters have considerable uncertainty associated with them and will impact fluid flow in the subsurface

Objectives

Results

Conclusion