A Multi-Stage Bayesian Prediction Framework for Subsurface Flows

Abstract

We are concerned with the development of computationally efficient procedures for subsurface flow prediction that relies on the characterization of subsurface formations given static (measured permeability and porosity at well locations) and dynamic (measured produced fluid properties at well locations) data. We describe a predictive procedure in a Bayesian framework, which uses a single-phase flow model for characterization aiming at making prediction for a two-phase flow model. The quality of the characterization of the underlying formations is accessed through the prediction of future fluid flow production.in comparison to the nonlinear case. Specifically, solving the Darcy velocity, which considerably is the most expensive component, occurs only one time, after which it is proceeded by solving the transport equation.Within the statistical framework, the above subsurface characterization can be thought of as sampling the parameters from a probability distribution that is conditioned to the available output data. The spatially dependent nature of the parameters makes them belong to an infinite dimensional space, which up to some numerical discretization can be approximated by some finite-dimensional space. However, this dimension by far exceeds the dimension of the space of the guiding measurement data. It is not surprising that a successful match between simulated and measured data does not necessarily guarantee obtaining the exact parameters of the subsurface, thus making the characterization an ill-posed mathematical problem in the sense that we attempt to invert a higher dimensional map whose range has a much lower dimension.As mentioned, even at the discretized level, the dimension of the uncertainty space may be exceptionally large, so reduction-of-order techniques are needed to bring down the whole calculation to the extent that it becomes amenable for computational simulations. This can be achieved by the Karhunen-Loève expansion, which allows for parametrization of the uncertainty space. The characterization is undertaken by sampling the parametric variables in the expansion. Still, a practical way of inverting the system in the characterization needs to be elaborated properly. An alternative to direct inversion of the system is to use implicit inverse models in a Bayesian approach to estimate permeability and porosity of the subsurface. These parameter estimates are updated automatically in a Markov chain Monte Carlo Method (MCMC) in response to computed sensitivity of simulated data to changes in these parameters.Many authors studied Bayesian methods for inverse problems in reservoir modeling. We refer the reader to [15,16] for recent works in the study of inverse problems in applications of flow through porous media. In [15], the authors used an intrinsically stationary Markov random field, which compares favorably to Gaussian process models and offers some additional flexibility and computational advantages for the choice of prior for the unknown permeability field. Through a Bayesian approach, usi...

Multiscale Sampling for the Inverse Modeling of Partial Differential Equations

“…Thus, we need to reduce the dimension of the uncertainty parameter space describing the permeability field. We use KLE [20,28] to achieve the desired dimensional reduction of the parameter space. Next, we briefly discuss the KLE.…”

Conditioning by Projection for the Sampling from Prior Gaussian Distributions

“…We use the KLE to reduce the large dimensional uncertainty space describing the permeability field k(x). In this subsection, we present a brief description of the KLE [29,20]. Let x ∈ Ω, and suppose log…”

Multi-physics Markov chain Monte Carlo methods for subsurface flows

“…In this subsection we discuss Karhunen-Loève (KL) expansion [28,33] to reduce dimension of uncertainty space for permeability and porosity. This reduction technique has been applied within a Bayesian statistical framework in [11,17,18,20,19,21]. Here we reproduce the technique for the sake of completeness of the discussion.…”

“…Furthermore, in the same spirit of two-stage procedures [6,11,12,17], the authors introduced a multi-stage Bayesian prediction framework for subsurface flows in [20]. The authors described a predictive procedure in a Bayesian framework, which uses a single-phase flow model for characterization aiming at making prediction for a two-phase flow model.…”