Alain Galli

A comparison of seven geostatistically based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow

This paper describes the first major attempt to compare seven different inverse approaches for identifying aquifer transmissivity. The ultimate objective was to determine which of several geostatistical inverse techniques is better suited for making probabilistic forecasts of the potential transport of solutes in an aquifer where spatial variability and uncertainty in hydrogeologic properties are significant. Seven geostatistical methods (fast Fourier transform (FF), fractal simulation (FS), linearized cokriging (LC), linearized semianalytical )LS), maximum likelihood (ML), pilot point (PP), and sequential self-calibration (SS)) were compared on four synthetic data sets. Each data set had specific features meeting (or not) classical assumptions about stationarity, amenability to a geostatistical description, etc. The comparison of the outcome of the methods is based on the prediction of travel times and travel paths taken by conservative solutes migrating in the aquifer for a distance of 5 km. Four of the methods, LS, ML, PP, and SS, were identified as being approximately equivalent for the specific problems considered. The magnitude of the variance of the transmissivity fields, which went as high as 10 times the generally accepted range for linearized approaches, was not a problem for the linearized methods when applied to stationary fields; that is, their inverse solutions and travel time predictions were as accurate as those of the nonlinear methods. Nonstationarity of the “true” transmissivity field, or the presence of “anomalies” such as high-permeability fracture zones was, however, more of a problem for the linearized methods. The importance of the proper selection of the semivariogram of the log10 (T) field (or the ability of the method to optimize this variogram iteratively) was found to have a significant impact on the accuracy and precision of the travel time predictions. Use of additional transient information from pumping tests did not result in major changes in the outcome. While the methods differ in their underlying theory, and the codes developed to implement the theories were limited to varying degrees, the most important factor for achieving a successful solution was the time and experience devoted by the user of the method.

Plurigaussian simulations in geosciences

Simulations are the fastest developing branch in geostatistics, and simulating the acies inside reservoirs and ore bodies is the most exciting part of this. Several methods have been developed to do this (sequential indicator simulations, Boolean methods, Markov chains and plurigaussian simulations). This book focusses on the last type of simulation. It develops the theory required to understand the method together and presents practical examples of applications in mining and petroleum, plus tutorial examples. An accompanying CD-ROM featuring demonstration software and color images complements the printed book.

The Pros and Cons of the Truncated Gaussian Method

The truncated gaussian method [6] [8] has been widely used in the past in order to simulate indicators. In this paper we investigate the potential and limitations of this method, from a geostatistical point of view and also from a practical one. First we present the method in detail including the hypotheses behind it. In particular we show that this method ensures consistency for the model in terms of the indicators variograms and cross variograms. Examples of the types of variograms that can and cannot be fitted using this method are presented. Then we discuss how to take into account vertical non stationarity for the indicators, as this seems to be the usual case (due to sequential evolutions in a sedimentary unit). This point will be illustrated via several examples. We then explain how the method can deal with a complex 3D non stationarity, and discuss the practical problems of using it in this case.

Factorial Kriging Analysis: A Substitute to Spectral Analysis of Magnetic Data

Factorial kriging analysis is a new method which combines kriging analysis and principal component analysis into the framework of geostatistics. The variables are split into principal components corresponding to different frequency ranges.

Rate of Convergence of the Gibbs Sampler in the Gaussian Case

We show that the Gibbs Sampler in the Gaussian case is closely linked to linear fixed point iterations. In fact stochastic linear iterations converge toward a stationary distribution under the same conditions as the classical linear fixed point one. Furthermore the covariance matrices are shown to satisify a related fixed point iteration, and consequently the Gibbs Sampler in the gaussian case corresponds to the classical Gauss-Seidel iterations on the inverse of the covariance matrix, and the stochastic over-relaxed Gauss-Seidel has the same limiting distribution as the Gibbs Sampler. Then an efficient method to simulate a gaussian vector is proposed. Finally numerical investigations are performed to understand the effect of the different strategies such as the initial ordering, the blocking and the updating order for iterations. The results show that in a geostatistical context the rate of convergence can be improved significantly compared to the standard case.

Quantification of Facies Relationships Via Proportion Curves

Proportions curves which were introduced at the end of the 80’s, have proved to be a powerful tool for sequence stratigraphy analysis and for quantifying lithofacies. In this paper we discuss both the qualitative and quantitative aspects via several examples coming either from analogue studies or from real oil fields.

Including a Regional Trend in Reservoir Modelling Using the Truncated Gaussian Method

When studying the geometry of petroleum reservoirs, we have to integrate different kinds of data (wells and seismic) as well as geological knowledge. Usually, wells show the fine vertical variation of lithofacies, but are less precise about horizontal variations. On the another hand, seismic data give a good idea of the horizontal evolution but we lose information on the vertical variability. A realistic reservoir model has to take this into account.

Geostatistical Models of Secondary Oil Migration Within Heterogeneous Carrier Beds: A Theoretical Example

This paper critically evaluates the utility of two different geostatistical methods in tracing long-distance oil migration through sedimentary basins. Geostatistical models of petroleum migration based on kriging and the conditional simulation method are assessed by comparing them to known oil migration rates and directions through a numerical carrier bed. In this example, the numerical carrier bed, which serves as ground truth in the study, incorporates a synthetic permeability field generated using the method of turning bands. Different representations of lateral permeability heterogeneity of the carrier bed are incorporated into a quasi-three-dimensional model of secondary oil migration. The geometric configuration of the carrier bed is intended to represent migrat on conditions within the center of a saucer-shaped intracratonic sag basin. In all of the numerical experiments, oil is sourced in the lowest 10% of a saucer-shaped carrier bed and migrates 10-14 km outward in a radial fashion by buoyancy. The effects of vertical permeability variations on secondary oil migration were not considered in the study. Two sets of numerical experiments are presented in this study. In the first set, computed oil migration patterns and rates through the numerical carrier bed are compared to kriged and optimal estimations based on a suite of 50 conditional simulations to determine the utility of each geostatistical approach. The geostatistical models also are compared to a much simpler oil migration simulation using a homogeneous carrier bed, an assumption typically made in cross sectional models of oil migration. The permeability realizations generated by kriging and the conditional simulation method were constructed using 50 randomly placed sampling points. A second set of numerical experiments was made to determine how data sample size (20, 35, 50, 100, and 250 randomly selected points) influences p edicted oil migration patterns using kriged permeability fields. Oil migration patterns through the synthetic carrier bed were quantitatively compared to geostatistical representations of oil transport by computing the degree of correlation (r2) of oil lens thickness at 5, 10, and 15 m.y. after the onset of migration. Quantitative results indicate that both kriging and the conditional simulation method produced realistic representations of spatial oil lens patterns and first arrival times. Both kriging and the conditional simulation method produced results far more realistic than the simulation with a homogeneous carrier bed. After 15 m.y., the spatial correlations (r2) between the known oil lens patterns and that of the kriged and ensemble average of the 50 conditional simulations were 0.67 and 0.74, respectively. Although the conditional simulation method produced the best results, it did so at a significant computational expense. Computed oil migration patterns based on different kriged permeability maps that incorporated varying amounts of control data points did not improve signif cantly when more than 100 randomly placed wells were used. The study provides compelling evidence that secondary oil migration is an inherently three-dimensional process. For well-studied basins, geostatistical representations of long-range oil migration could be of value in ongoing exploration strategies. End_Page 1679------------------------------

Scenario Reduction Applied to Geostatistical Simulations

Having a large number of geostatistical simulations of a mineral or petroleum deposit provides a better idea of its upside potential and downside risk; however, large numbers of simulated realizations of a deposit may pose computational difficulties in subsequent decision-making phases. Hence, depending on the specific case, there can be a need to select a representative subset of conditionally simulated deposit realizations. This paper examines and extends an approach developed by the stochastic optimization community based on stochastic mathematical programming with recourse and is discussed here in the context of mineral deposits while it is possibly suitable for other earth science applications. The approach is based on measuring the “distance” between simulations and the introduced distance measure between simulated realizations of a mineral deposit is based on the metal above a given set of cutoff grades while a pre-existing mine design is available. The approach is tested on 100 simulations of the Walker Lake data with promising results.

Geostatistics Rio 2000

Preface R. Olea. Introduction M. Armstrong, et al. Petroleum Papers. Using Quantitative Outcrop Databases As a Guide for Geological Reservoir Modelling R. Eschard, et al. Quantification of facies relationships via proportion curves C. Ravenne, et al. Geologic Modelling of External and Internal Reservoir Architecture of Fluvial Depositional Systems P.E. Patterson, et al. Impact of Seismic Constraints on a Stochastic Reservoir Model and Fluid Flow Simulation I. Azpiritxaga, et al. Practical Workflows for Reservoir Modelling J.M. Yarus, et al. Mining Papers. Simulating the Geometry of a Granite-hosted Uranium Orebody T. Skvortsova, et al. Simulation of Weathered Iron Ore Facies: Integrating Leaching Concepts and Geostatistical Models D.T. Ribeiro, R.M. Carvalho. Geostatistical Simulation of Structurally Controlled Low Grade High Tonnage Gold Ores: A Strategy for Targeting Genuine Enriched Zones A.H.M. Silva, et al. Geostatistical Framework for Modelling Clay Deposits: Nova Veneza Case Study in Southern Brazil R.L. Stangler, et al. Risk Assessment of Reserve Calculation during Milestones of a Mine Life F. Silva, et al. Successful Incorporation of Geological Controls into Reserve Evaluation: Recent Examples from Giant Copper Mines in Chile N. Champigny, et al. Geological Conditions for a Correct Geostatistical Evaluation: Example 177 from the Elatsite Copper Deposit in Bulgaria J. Todorov, et al. Evaluation of Kriging and Cokriging for Asbestos Ore Reserve Estimation at the Cana Brava Mine, Goias, Brazil R.P. Conde, J.K. Yamamoto. Geostatistical Evaluation of the Mineral Resources of the Cajati Mine, State of Sao Paulo, Brazil G. Barros, J.K. Yamamoto. Variographic Trends of Gold in the Alluvial Sediments Associated with the QuartzLodes in the Princesa Isabel Region, Paraiba, Brazil S.R. Lanusse, T.R. Gopinath.