### Abstract

Markov Chain Monte Carlo (MCMC) methods are often used to probe the posterior probability distribution in inverse problems. This allows for computation of estimates of uncertain system responses conditioned on given observational data by means of approximate integration. However, MCMC methods suffer from the computational complexities in the case of expensive models as in the case of subsurface flow models. Hence, it is of great interest to develop alterative efficient methods utilizing emulators, that are cheap to evaluate, in order to replace the full physics simulator. In the current work, we develop a technique based on sparse response surfaces to represent the flow response within a subsurface reservoir and thus enable efficient exploration of the posterior probability density function and the conditional expectations given the data.

Polynomial Chaos Expansion (PCE) is a powerful tool to quantify uncertainty in dynamical systems when there is probabilistic uncertainty in the system parameters. In the context of subsurface flow model, it has been shown to be more accurate and efficient compared with traditional experimental design (ED). PCEs have a significant advantage over other response surfaces as the convergence to the true probability distribution when the order of the PCE is increased can be proved for the random variables with finite variances. However, the major drawback of PCE is related to the curse of dimensionality as the number of terms to be estimated grows drastically with the number of the input random variables. This renders the computational cost of classical PCE schemes unaffordable for reservoir simulation purposes when the deterministic finite element model is expensive to evaluate. To address this issue, we propose the reduced-terms polynomial chaos representation which uses an impact factor to only retain the most relevant terms of the PCE decomposition. Accordingly, the reduced-terms polynomial chaos proxy can be used as the pseudo-simulator for efficient sampling of the probability density function of the uncertain variables.

The reduced-terms PCE is evaluated on a two dimensional subsurface flow model with fluvial channels to demonstrate that with a few hundred trial runs of the actual reservoir simulator, it is feasible to construct a polynomial chaos proxy which accurately approximates the posterior distribution of the high permeability zones, in an analytical form. We show that the proxy precision improves with increasing the order of PCE and corresponding increase of the number of initial runs used to estimate the PCE coefficient.

Polynomial Chaos Expansion (PCE) is a powerful tool to quantify uncertainty in dynamical systems when there is probabilistic uncertainty in the system parameters. In the context of subsurface flow model, it has been shown to be more accurate and efficient compared with traditional experimental design (ED). PCEs have a significant advantage over other response surfaces as the convergence to the true probability distribution when the order of the PCE is increased can be proved for the random variables with finite variances. However, the major drawback of PCE is related to the curse of dimensionality as the number of terms to be estimated grows drastically with the number of the input random variables. This renders the computational cost of classical PCE schemes unaffordable for reservoir simulation purposes when the deterministic finite element model is expensive to evaluate. To address this issue, we propose the reduced-terms polynomial chaos representation which uses an impact factor to only retain the most relevant terms of the PCE decomposition. Accordingly, the reduced-terms polynomial chaos proxy can be used as the pseudo-simulator for efficient sampling of the probability density function of the uncertain variables.

The reduced-terms PCE is evaluated on a two dimensional subsurface flow model with fluvial channels to demonstrate that with a few hundred trial runs of the actual reservoir simulator, it is feasible to construct a polynomial chaos proxy which accurately approximates the posterior distribution of the high permeability zones, in an analytical form. We show that the proxy precision improves with increasing the order of PCE and corresponding increase of the number of initial runs used to estimate the PCE coefficient.

Original language | English |
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Pages (from-to) | 385–399 |

Number of pages | 15 |

Journal | Advances in Water Resources |

Volume | 86 |

Issue number | Part B |

Early online date | 19 Oct 2015 |

DOIs | |

Publication status | Published - Dec 2015 |

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## Cite this

Bazargan, H., Christie, M. A., Elsheikh, A. H., & Ahmadi, M. (2015). Surrogate accelerated sampling of reservoir models with complex structures using sparse polynomial chaos expansion.

*Advances in Water Resources*,*86*(Part B), 385–399. https://doi.org/10.1016/j.advwatres.2015.09.009