Markov Chain Monte Carlo for Reservoir Uncertainty Assessment

Abstract Accurate representation of reservoir heterogeneity using stochastic modeling techniques requires careful synthesis of the multivariate probability distribution characterizing the spatial variations of petrophysical properties. A well designed scheme for sampling from that distribution is necessary in order to accurately portray the uncertainty in estimating reservoir properties arising from the incomplete knowledge of the reservoir under study. That uncertainty is data-dependent and most importantly model-dependent. The paper presents a Markov Chain Monte Carlo methodology for sampling from the invariant or stationary probability distribution characterizing the reservoir permeability field. An initial random field is perturbed successively following a Gibbs sampling procedure. The updated values at the perturbed node are obtained by sampling from the corresponding kriged distribution. This iterative updating procedure is continued until a prescribed large number of iterations are performed. Hard data, histogram and variogram model are honored as expected. The multiple point histogram (MPH) and entropy of MPH are used to assess the joint spatial uncertainty exhibited by this MCMC model and the impact of data quantity and configuration on that uncertainty are explored. Convergence characteristics of the proposed methodology are investigated and detailed comparisons of the proposed methodology to other established algorithms are presented. Introduction Assessment of uncertainty is a crucial step in the reservoir development decision-making process. Uncertainty in reservoir performance predictions is usually reflected by a number of parameters, such as OIP, production profiles, ultimate recovery and fluid breakthrough characteristics etc. Reservoir complexity, complexity of the fluid flow mechanisms and the lack of exhaustive reservoir specific information are the main reasons for uncertainty in reservoir modeling1. Some papers2,3,4,5 have focused on understanding the source of uncertainties. For example, Massonnat2 proposes a scheme for hierarchical classification of uncertainties into different levels and for quantification of uncertainty in each level separately. Charles et al. 1 propose an approach for translating uncertainties of input parameters into uncertainties of parameters of economic interest. However, any quantification of uncertainty is datadependent and most importantly model-dependent5. Geology plays a major role in reservoir modeling because it drives the understanding of internal reservoir architecture and spatial distribution of reservoir characteristics2. Usually, stochastic simulation is used to assess geological uncertainty by way of multiple equiprobable realizations of a given stochastic model. Petrophysical parameters, such as permeability, porosity and fluid saturation, are described as continuous or discrete random variables in these stochastic models. Reservoir permeability field that reflects heterogeneity (spatial variation in properties) is considered as one of the most important factors governing fluid flow6. The true reservoir permeability field is unknown and is consequently modeled as a manifestation (realization) of a Random Function (RF) characterized by a multivariate probability distribution. The objective of stochastic simulation is to model an isomorphic and conditional random variable function and sample realizations of the permeability field from the underlying multivariate distribution. So our task is to sample values from a posterior probability distribution such that the patterns of variability exhibited by the "true" permeability a

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