Streamline-Based History Matching and Uncertainty--Markov-Chain Monte Carlo Study of an Offshore Turbidite Oil Field
Authors: Maucec Marko , Douma Sippe G. , Hohl Detlef , Leguijt Jaap , Jimenez Eduardo , Datta-Gupta Akhil 
Published: December 2007

Abstract
Streamline models have shown great potential for the inversion of dynamic reservoir models using well production data. The solution of "history matching" inverse problem is greatly simplified when sensitivities are known. This applies to both single-model optimization and multi-realization stochastic sampling of the objective function with uncertainty evaluation.
Bayesian modeling with sequential Markov chain Monte Carlo (MCMC) algorithms provides a rigorous framework for stochastic sampling. However, a naive approach of calculating the likelihood term in Bayes' formula using a dynamic reservoir simulator incurs prohibitively high computational cost. Each MCMC proposal requires a time-consuming reservoir simulation, and when no Jacobian information is available acceptance ratios are very low.
To reduce these high computational costs, approximate models have been introduced. In this contribution we present a Bayesian multi-stage MCMC approach, based on an approximation with a linear expansion around the current model state using the semi-analytically computed streamline sensitivities. The algorithm is benchmarked to a simple synthetic permeability model with one injecting well and four producing wells to study efficiency.
Another crucial ingredient in our inversion and sampling scheme is a fast method to generate new realizations of reservoir models (here: permeability fields) from the prior probability density function that obey known geostatistics (variograms) and well constraints. This is accomplished using a novel Fourier-space filter method that can be used with very large numbers of variables (~106) without the computational and memory cost of traditional algorithms like the Cholesky decomposition.
We ultimately demonstrate the application of the multi-stage MCMC algorithm for an efficient dynamic inversion of water-cut data in structurally complex and faulted offshore turbidite oil reservoir. Timing studies, validation and implementation of MCMC algorithm convergence criteria as well as full parallelization of the computer code, rendering substantial reduction of computing time, are described in detail.

Introduction
Uncertainty represents an inherent component in dynamic reservoir modeling, primarily because of data noise, the systematic modeling error, and non-uniqueness of the inverse problem solution. Reducing this uncertainty can be achieved by integrating additional static data (e.g. well logs, cores, seismic traces) as well as dynamic data (e.g. production history, pressure transient tests, etc.) in subsurface modeling. Integration of the data from different sources is a non-trivial assignment, because deployed data span variety in length scales of heterogeneity and usually have a different degree of precision.
Streamline models have shown great potential for the inversion of dynamic reservoir models using well production data1–3. Streamline-based models offer the calculation of sensitivities (derivatives of a suitable objective function with respect to reservoir model parameters such as permeability) at little extra computational expense. When the sensitivities are known, the solution of the "history matching" inverse problem is greatly simplified. This applies to both single-model optimization and multi-realization sampling of the objective function with uncertainty evaluation. Streamline-based sensitivities have been successfully applied for the inversion of well-production data of a large-scale, carbonate reservoirs in the Middle East4 and in South America5.