Direct Pattern-Based Simulation of Non-stationary Geostatistical Models

Non-stationary models often capture better spatial variation of real world spatial phenomena than stationary ones. However, the construction of such models can be tedious as it requires modeling both statistical trend and stationary stochastic component. Non-stationary models are an important issue in the recent development of multiple-point geostatistical models. This new modeling paradigm, with its reliance on the training image as the source for spatial statistics or patterns, has had considerable practical appeal. However, the role and construction of the training image in the non-stationary case remains a problematic issue from both a modeling and practical point of view. In this paper, we provide an easy to use, computationally efficient methodology for creating non-stationary multiple-point geostatistical models, for both discrete and continuous variables, based on a distance-based modeling and simulation of patterns. In that regard, the paper builds on pattern-based modeling previously published by the authors, whereby a geostatistical realization is created by laying down patterns as puzzle pieces on the simulation grid, such that the simulated patterns are consistent (in terms of a similarity definition) with any previously simulated ones. In this paper we add the spatial coordinate to the pattern similarity calculation, thereby only borrowing patterns locally from the training image instead of globally. The latter would entail a stationary assumption. Two ways of adding the geographical coordinate are presented, (1) based on a functional that decreases gradually away from the location where the pattern is simulated and (2) based on an automatic segmentation of the training image into stationary regions. Using ample two-dimensional and three-dimensional case studies we study the behavior in terms of spatial and ensemble uncertainty of the generated realizations.     

Quantifying geological uncertainty for flow and transport modeling in multi-modal heterogeneous formations

In this work, we address the problem of characterizing the heterogeneity and uncertainty of hydraulic properties for complex geological settings. Hereby, we distinguish between two scales of heterogeneity, namely the hydrofacies structure and the intrafacies variability of the hydraulic properties. We employ multiple-point geostatistics to characterize the hydrofacies architecture. The multiple-point statistics are borrowed from a training image that is designed to reflect the prior geological conceptualization. The intrafacies variability of the hydraulic properties is represented using conventional two-point correlation methods, more precisely, spatial covariance models under a multi-Gaussian spatial law. We address the different levels and sources of uncertainty in characterizing the subsurface heterogeneity, and explore their effect on groundwater flow and transport predictions. Typically, uncertainty is assessed by way of many images, termed realizations, of a fixed statistical model. However, in many cases, sampling from a fixed stochastic model does not adequately represent the space of uncertainty. It neglects the uncertainty related to the selection of the stochastic model and the estimation of its input parameters. We acknowledge the uncertainty inherent in the definition of the prior conceptual model of aquifer architecture and in the estimation of global statistics, anisotropy, and correlation scales. Spatial bootstrap is used to assess the uncertainty of the unknown statistical parameters. As an illustrative example, we employ a synthetic field that represents a fluvial setting consisting of an interconnected network of channel sands embedded within finer-grained floodplain material. For this highly non-stationary setting we quantify the groundwater flow and transport model prediction uncertainty for various levels of hydrogeological uncertainty. Results indicate the importance of accurately describing the facies geometry, especially for transport predictions.     

Comparing Training-Image Based Algorithms Using an Analysis of Distance

As additional multiple-point statistical (MPS) algorithms are developed, there is an increased need for scientific ways for comparison beyond the usual visual comparison or simple metrics, such as connectivity measures. In this paper, we start from the general observation that any (not just MPS) geostatistical simulation algorithm represents two types of variability: (1) the within-realization variability, namely, that realizations reproduce a spatial continuity model (variogram, Boolean, or training-image based), (2) the between-realization variability representing a model of spatial uncertainty. In this paper, it is argued that any comparison of algorithms needs, at a minimum, to be based on these two randomizations. In fact, for certain MPS algorithms, it is illustrated with different examples that there is often a trade-off: Increased pattern reproduction entails reduced spatial uncertainty. In this paper, the subjective choice that the best algorithm maximizes pattern reproduction is made while at the same time maximizes spatial uncertainty. The discussion is also limited to fairly standard multiple-point algorithms and that our method does not necessarily apply to more recent or possibly future developments. In order to render these fundamental principles quantitative, this paper relies on a distance-based measure for both within-realization variability (pattern reproduction) and between-realization variability (spatial uncertainty). It is illustrated in this paper that this method is efficient and effective for two-dimensional, three-dimensional, continuous, and discrete training images.