Stochastic Imaging for Reservoir Characterization
Authors: Thadani S. G. 
Published: December 2007

Abstract
One of the key problems in Reservoir Characterization involves the description and visualization of reservoir heterogeneities (as represented by the spatial variability of properties such as porosity, permeability, thickness, lithofacies types, fracture / fault orientations, sand body geometry, etc). The inherent nonuniqueness associated with this problem has prompted considerable interest in the development and application of stochastic (ie. probabilistic) imaging techniques. Such techniques are designed to generate a family of equiprobable descriptions or stochastic images of these parameters - each image being consistent with all the available quantitative (well-logs, cores, seismic derived constraint intervals, well-test information) and qualitative (geological interpretation) information, and its spatial correlation characteristics. These stochastic images may be viewed as samples from approximations to the "optimal’’ Bayesian posterior distribution on the reservoir parameters. Statistical analysis of the "spread" of this distribution allows for the quantification of risk/uncertainty associated with the spatial variability of these parameters. Also, selected subsets of stochastic images may be "passed" through dynamic flow simulators to assess the distribution of significant production response variables. Such distributions can be incorporated with statistical decision theoretic techniques in order to aid in the optimal forecasting and management of the reservoir.
A number of stochastic imaging techniques have been developed in the past few years - indeed this number is rapidly growing. The object of this paper is to assess the current state of the art in stochastic imaging techniques for reservoir characterization, along with associated statistical methodologies for integrating seismic data, and for reservoir performance forecasting and management. The following paragraphs provide an overview of the body of the paper.
Section II of the paper provides a comparitive review of stochastic imaging techniques. The reviewed set covers both discrete and continuous single/multivariable methods - these include Boolean algorithms and Marked Point Processes, Indicator methods, (truncated) Gaussian Random Functions. Fractal fields, Simulated Annealing, Markov Random Fields and direct Bayesian Imaging algorithms. Particular attention is given to underlying assumptions* data integration, internal consistency, performance (eg. exactitude, reproduction of spatial correlation structure, quality of approximation to the Bayesian posterior), computational and inferential complexity, and practical limitations. Also, the techniques are compared with respect to their capabilities for incorporating "soft" information (such as inequality constraints), handling anisotropy and trends (ie.lst order non-stationarities), and for imaging vector variables.
The potential of seismic data for adding detail to reservoir descriptions "between the wells", is now generally acknowledged. Section III reviews known techniques for integrating seismic data in reservoir descriptions. This includes recent developments in techniques such as External drift, Cokriging, Markov Random Fields, M.A.P. algorithms, Bayesian (Hard/Soft) Inversion, Markov_Bayes algorithms, and ID Stochastic Inversion. Brief descriptions are also provided of methods for conditioning the stochastic images to physics-based "forward models", and to qualitative geological information.
Section IV summarizes current techniques for utilizing the stochastic images in performance forecasting and reservoir management. This review emphasizes the use of statistical decision theoretic approaches. Also, current progress in conditioning stochastic reservoir models to production / well-test information is summarized. In section V, illustrative test results are presented of the application of these techniques to both synthetic and where available, "real" reservoir data sets. Reservoir description applications of hybrid multistep approaches are also summarized - here multiple stochastic imaging algorithms are applied in sequence to compute progressively more detailed descriptions.
Section VI presents general guidelines for the use of stochastic imaging techniques on specific reservoir characterization problems. The paper concludes with an overview of open problems and current research directions in this field. These include (computationally feasible) multivariable stochastic imaging, incorporation of seismic information, visualization, utilization of stochastic images in dynamic flow simulations, and decision theoretic techniques for reservoir performance forecasting and management.