Jef Caers

Geostatistical reservoir modelling using statistical pattern recognition

The traditional practice of geostatistics for reservoir characterization is limited by the variogram which, as a measure of geological continuity, can capture only two-point statistics. Important curvi-linear geological information, beyond the modelling capabilities of the variogram, can be taken from training images and later used in model construction. Training images can provide multiple-point statistics which describe the statistical relation between multiple spatial locations considered jointly. Stochastic reservoir simulation then consists of anchoring the borrowed geo-structures in the form of multiple-point statistics to the actual subsurface hard and soft data.

Combining geologic-process models and geostatistics for conditional simulation of 3-D subsurface heterogeneity

[1] The goal of simulation of aquifer heterogeneity is to produce a spatial model of the subsurface that represents a system such that it can be used to understand or predict flow and transport processes. Spatial simulation requires incorporation of data and geologic knowledge, as well as representation of uncertainty. Classical geostatistical techniques allow for the conditioning of data and uncertainty assessment, but models often lack geologic realism. Simulation of physical geologic processes of sedimentary deposition and erosion (process-based modeling) produces detailed, geologically realistic models, but conditioning to local data is limited at best. We present an aquifer modeling methodology that combines geologic-process models with object-based, multiple-point, and variogram-based geostatistics to produce geologically realistic realizations that incorporate geostatistical uncertainty and can be conditioned to data. First, the geologic features of grain size, or facies, distributions simulated by a process-based model are analyzed, and the statistics of feature geometry are extracted. Second, the statistics are used to generate multiple realizations of reduced-dimensional features using an object-based technique. Third, these realizations are used as multiple alternative training images in multiple-point geostatistical simulation, a step that can incorporate local data. Last, a variogram-based geostatistical technique is used to produce conditioned maps of depositional thickness and erosion. Successive realizations of individual strata are generated in depositional order, each dependent on previously simulated geometry, and stacked to produce a fully conditioned three-dimensional facies model that mimics the architecture of the process-based model. We demonstrate the approach for a typical subsea depositional complex.

Modeling of a Deepwater Turbidite Reservoir Conditional to Seismic Data Using Principal Component Analysis and Multiple-Point Geostatistics

Geological interpretation and seismic data analysis provide two complementary sources of information to model reservoir architecture. Seismic data affords the opportunity to identify geologic patterns and features at a resolution on the order of 10’s of feet, while well logs and conceptual geologic models provide information at a resolution on the order of one foot. Both the large-scale distribution of geologic features and their internal fine-scale architecture influence reservoir performance. Development and application of modeling techniques that incorporate both large-scale information derived from seismic and fine-scale information derived from well logs, cores, and analog studies represents a significant opportunity to improve reservoir performance predictions. In this paper we present a practical new geostatistical approach for solving this difficult data integration problem and apply it to an actual, prominent reservoir. Traditional geostatistics relies upon a variogram to describe geologic continuity. However, a variogram, which is a two-point measure of spatial variability, cannot describe realistic, curvilinear or geometrically complex patterns. Multiple-point geostatistics uses a training image instead of a variogram to account for geological information. The training image provides a conceptual description of the subsurface geological heterogeneity, containing possibly complex multiple-point patterns of geological heterogeneity. Multiple-point statistics simulation then consists of anchoring these patterns to well data and seismic-derived information. This work introduces a novel alternative approach to traditional Bayesian modeling to incorporate seismic. The focus in this paper lies in demonstrating the practicality, flexibility and CPU-advantage of this new approach by applying it to an actual deep-water turbidite reservoir. Based on well log interpretation and a global geological understanding of the reservoir architecture, a training image depicting sinuous sand bodies is generated using a non-conditional object-based simulation algorithm. Disconnected sand bodies are interpreted from seismic amplitude data using a principal component cluster analysis technique. In addition, a map of local sand probabilities obtained from a principal component proximity transform of the same seismic is generated. Multiple-point geostatistics then simulates multiple realizations of channel bodies constrained to the local sand probabilities, partially interpreted sand bodies and well-log data. The CPU-time is comparable to traditional geostatistical methods.

MS-CCSIM

Pattern-based spatial modeling relies on training images as basic modeling component for generating geostatistical realizations. The methodology recognizes that working with the unit of a pattern aids its reproduction, particularly for large systems. In this paper improvements are made, in terms of both the computation time and conditioning, of a pattern-based simulation method that relies on the cross-correlation-based simulation (CCSIM), introduced by Tahmasebi et al. (2012). The extension lies on the use of a multi-scale (MS) representation of the training image along a pattern projection strategy that is markedly different from the traditional multi-grid methods employed in the current methodologies, and proposes acceleration of the method by carrying out most of the computations in the Fourier space. In the proposed multi-scale representation, we transform the high-resolution training image into a pyramid of consecutively up-gridded views of the same image. The pyramid allows for rapid search of the patterns that can be superimposed over a shared overlap area with previously simulated patterns. A second advantage of the multi-scale view lies in data conditioning by means of a new hard data-relocation algorithm and the use of a co-template for looking for conditioning points ahead of the raster path employed in CCSIM. Using synthetic and real-field multi-million cell examples with sparse, as well as dense datasets, we investigate quantitatively how the improved algorithm performs with respect to CCSIM, as well as the traditional MP simulation algorithms. Computational improvement to pattern simulation using multi-scale search.Improved conditioning in raster-path methods using co-template.Multi-million cell real field application in seconds.

Identifying tails, bounds and end-points of random variables

Abstract The characterization of tails of random variables is of major concern in a safety analysis such as a structural reliability analysis or a quantitative risk analysis of an engineering system. One of the important questions raised is whether the tail is bounded or unbounded. Therefore, in a statistical analysis of a given data set, it makes sense to use only the extreme small or large data in the tail modelling. This raises the important issue of the selection of thresholds above which “tail behaviour” of the data can be justified. In general, thresholds close to the central data will bias the estimation towards the central values which are not informative for the tail. Too extreme thresholds will result in high estimation variances. In this paper we propose to use a finite sample mean square error (MSE) to select such thresholds and to estimate tail characteristics. Estimators for the extreme value index, the end-point and extreme quantiles are based on the so-called generalized quantile plot. This plot is used to discern between bounded and unbounded tail behaviour. A semi-parametric bootstrap technique is used to estimate the MSE at each threshold and to select the optimal threshold at which the MSE is minimized. Confidence limits are obtained using the sampling distribution of estimators at the optimal threshold. In a verification study and an application to wall thickness values of tubes, the MSE-criterion is applied to various extremal properties such as end-points or extreme quantiles and to other parameters that are critically dependent on the tail behaviour of a random variable such as reliability index.

Geostatistical History Matching Under Training-Image Based Geological Model Constraints

This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the authors(s).

Geological realism in hydrogeological and geophysical inverse modeling: A review

Abstract Scientific curiosity, exploration of georesources and environmental concerns are pushing the geoscientific research community toward subsurface investigations of ever-increasing complexity. This review explores various approaches to formulate and solve inverse problems in ways that effectively integrate geological concepts with geophysical and hydrogeological data. Modern geostatistical simulation algorithms can produce multiple subsurface realizations that are in agreement with conceptual geological models and statistical rock physics can be used to map these realizations into physical properties that are sensed by the geophysical or hydrogeological data. The inverse problem consists of finding one or an ensemble of such subsurface realizations that are in agreement with the data. The most general inversion frameworks are presently often computationally intractable when applied to large-scale problems and it is necessary to better understand the implications of simplifying (1) the conceptual geological model (e.g., using model compression); (2) the physical forward problem (e.g., using proxy models); and (3) the algorithm used to solve the inverse problem (e.g., Markov chain Monte Carlo or local optimization methods) to reach practical and robust solutions given todays computer resources and knowledge. We also highlight the need to not only use geophysical and hydrogeological data for parameter estimation purposes, but also to use them to falsify or corroborate alternative geological scenarios.

History matching and uncertainty quantification of facies models with multiple geological interpretations

Uncertainty quantification is currently one of the leading challenges in the geosciences, in particular in reservoir modeling. A wealth of subsurface data as well as expert knowledge are available to quantify uncertainty and state predictions on reservoir performance or reserves. The geosciences component within this larger modeling framework is partially an interpretive science. Geologists and geophysicists interpret data to postulate on the nature of the depositional environment, for example on the type of fracture system, the nature of faulting, and the type of rock physics model. Often, several alternative scenarios or interpretations are offered, including some associated belief quantified with probabilities. In the context of facies modeling, this could result in various interpretations of facies architecture, associations, geometries, and the way they are distributed in space. A quantitative approach to specify this uncertainty is to provide a set of alternative 3D training images from which several geostatistical models can be generated. In this paper, we consider quantifying uncertainty on facies models in the early development stage of a reservoir when there is still considerable uncertainty on the nature of the spatial distribution of the facies. At this stage, production data are available to further constrain uncertainty. We develop a workflow that consists of two steps: (1) determining which training images are no longer consistent with production data and should be rejected and (2) to history match with a given fixed training image. We illustrate our ideas and methodology on a test case derived from a real field case of predicting flow in a newly planned well in a turbidite reservoir off the African West coast.

Efficient gradual deformation using a streamline-based proxy method

Abstract The application of state-of-the-art history matching methods to large heterogeneous reservoirs is hampered by two main problems: (1) reservoir models should be constrained jointly to dynamic data and a large variety of geological continuity information, and (2) CPU demand should not be prohibitive for large models. This paper proposes a method contributing to alleviating these two main concerns. By combining three existing ideas, namely gradual deformation, multiple-point geostatistics and a fast streamline-based history matching method, an algorithm is proposed that could honor a large variety of geological scenarios and is obtained with a limited amount of flow simulations. The method expands on the traditional gradual deformation methodology in three ways: (1) multiple-point geostatistics is used to generate models that are geologically more realistic than traditional variogram-based models, (2) a streamline simulator is used to define “zones-of-influence” of producers in order to locally deform an initial model toward jointly matching a large amount of wells and most importantly (3) the number of flow calculations is limited by defining a proxy to the streamline simulator in terms of streamline-based harmonic averages. Using synthetic examples of increasing complexity, the methods efficiency and generality is assessed.

Adding Local Accuracy to Direct Sequential Simulation

Geostatistical simulations are globally accurate in the sense that they reproduce global statistics such as variograms and histograms. Kriging is locally accurate in the minimum local error variance sense. Building on the concept of direct sequential simulation, we propose a fast simulation method that can share these opposing objectives. It is shown that the multiple-point entropy of the resulting simulation is related to the univariate entropy of the local conditional distributions used to draw simulated values. Adding local accuracy to conditional simulations does not detract much from variogram reproduction and can be used to increase multiple-point entropy. The methods developed are illustrated using a case study.