Multiscale Data Integration Using Markov Random Fields

Summary We propose a hierarchical approach to spatial modeling based on Markov Random Fields (MRF) and multiresolution algorithms in image analysis. Unlike their geostatistical counterparts, which simultaneously specify distributions across the entire field, MRFs are based on a collection of full conditional distributions that rely on the local neighborhoods of each element. This critical focus on local specification provides several advantages:MRFs are computationally tractable and are ideally suited to simulation based computation, such as Markov Chain Monte Carlo (MCMC) methods, andmodel extensions to account for nonstationarity, discontinuity, and varying spatial properties at various scales of resolution are easily accessible in the MRF framework. Our proposed method is computationally efficient and well suited to reconstruct fine-scale spatial fields from coarser, multiscale samples (based on seismic and production data) and sparse fine-scale conditioning data (e.g., well data). It is easy to implement, and it can account for the complex, nonlinear interactions between different scales, as well as the precision of the data at various scales, in a consistent fashion. We illustrate our method with a variety of examples that demonstrate the power and versatility of the proposed approach. Finally, a comparison with Sequential Gaussian Simulation with Block Kriging (SGSBK) indicates similar performance with less restrictive assumptions. Introduction A persistent problem in petroleum reservoir characterization is to build a model for flow simulations based on incomplete information. Because of the limited spatial information, any conceptual reservoir model used to describe heterogeneities will, necessarily, have large uncertainty. Such uncertainties can be significantly reduced by integrating multiple data sources into the reservoir model.1 In general, we have hard data, such as well logs and cores, and soft data, such as seismic traces, production history, conceptual depositional models, and regional geological analyses. Integrating information from this wide variety of sources into the reservoir model is not a trivial task. This is because different data sources scan different length scales of heterogeneity and can have different degrees of precision.2 Reconciling multiscale data for spatial modeling of reservoir properties is important because different data types provide different information about the reservoir architecture and heterogeneity. It is essential that reservoir models preserve small-scale property variations observed in well logs and core measurements and capture the large-scale structure and continuity observed in global measures such as seismic and production data. A hierarchical model is particularly well suited to address the multiscaled nature of spatial fields, match available data at various levels of resolution, and account for uncertainties inherent in the information.1–3 Several methods to combine multiscale data have been introduced in the literature, with a primary focus on integrating seismic and well data.3–9 These include conventional techniques such as cokriging and its variations,3–6 SGSBK,7 and Bayesian updating of point kriging.8,9 Most kriging-based methods are restricted to multi-Gaussian and stationary random fields.3–9 Therefore, they require data transformation and variogram construction. In practice, variogram modeling with a limited data set can be difficult and strongly user-dependent. Improper variograms can lead to errors and inaccuracies in the estimation. Thus, one might also need to consider the uncertainty in variogram models during estimation. 10 However, conventional geostatistical methods do not provide an effective framework to account for the uncertainty of the variogram. Furthermore, most of the multiscale integration algorithms assume a linear relationship between the scales. The objective of this paper is to introduce a novel multiscale data-integration technique that provides a flexible and sound mathematical framework to overcome some of the limitations of conventional geostatistical techniques. Our approach is based on multiscale MRFs11–14 that can effectively integrate multiple data sources into high-resolution reservoir models for reliable reservoir forecasting. This proposed approach is also ideally suited to simulation- based computations, such as MCMC.15,16 Methodology Our problem of interest is to generate fine-scale random fields based on sparse fine-scale samples and coarse-scale data. Such situations arise when we have limited point measurements, such as well data, and coarse-scale information based on seismic and/or production data. Our proposed method is a Bayesian approach to spatial modeling based on MRF and multiresolution algorithms in image analysis. Broadly, the method consists of two major parts:construction of a posterior distribution for multiscale data integration using a hierarchical model andimplementing MCMC to explore the posterior distribution. Construction of a Posterior Distribution for Multiscale Data Integration. A multiresolution MRF provides an efficient framework to integrate different scales of data hierarchically, provided that the coarse-scale resolution is dependent on the next finescale resolution.11 In general, a hierarchical conditional model over scales 1,. . ., N (from fine to coarse) can be expressed in terms of the product of conditional distributions,Equation 1 where p(xn), n=1, . . ., N, are MRF models at each scale, and the terms p(xn|xn-1) express the statistical interactions between different scales. This approach links the various scales stochastically in a direct Bayesian hierarchical modeling framework (Fig. 1). Knowing the fine-scale field xn does not completely determine the field at a coarser scale xn+1, but depending on the extent of the dependence structure modeled and estimated, it influences the distribution at the coarser scales to a greater or lesser extent. This enables us to address multiscale problems accounting for the scale and precision of the data at various levels. For clarity of exposition, a hierarchical model for reconciling two different scales of data will be considered below.Equation 2 From this equation, the posterior distribution of the fine-scale random field indexed by 1 given a coarse-scale random field indexed by 2 can be derived as follows.