Validation Techniques for Geological Patterns Simulations Based on Variogram and Multiple-Point Statistics

Traditional simulation methods that are based on some form of kriging are not sensitive to the presence of strings of connectivity of low or high values. They are particularly inappropriate in many earth sciences applications, where the geological structures to be simulated are curvilinear. In such cases, techniques allowing the reproduction of multiple-point statistics are required. The aim of this paper is to point out the advantages of integrating such multiple-statistics in a model in order to allow shape reproduction, as well as heterogeneity structures, of complex geological patterns to emerge. A comparison between a traditional variogram-based simulation algorithm, such as the sequential indicator simulation, and a multiple-point statistics algorithm (e.g., the single normal equation simulation) is presented. In particular, it is shown that the spatial distribution of limestone with meandering channels in Lecce, Italy is better reproduced by using the latter algorithm. The strengths of this study are, first, the use of a training image that is not a fluvial system and, more importantly, the quantitative comparison between the two algorithms. The paper focuses on different metrics that facilitate the comparison of the methods used for limestone spatial distribution simulation: both objective measures of similarity of facies realizations and high-order spatial cumulants based on different third- and fourth-order spatial templates are considered.

[1]  Roussos G. Dimitrakopoulos,et al.  A new approach for geological pattern recognition using high-order spatial cumulants , 2010, Comput. Geosci..

[2]  Roussos Dimitrakopoulos,et al.  High-order Statistics of Spatial Random Fields: Exploring Spatial Cumulants for Modeling Complex Non-Gaussian and Non-linear Phenomena , 2009 .

[3]  Alexandre Boucher,et al.  A SGeMS code for pattern simulation of continuous and categorical variables: FILTERSIM , 2008, Comput. Geosci..

[4]  Sebastien Strebelle,et al.  Conditional Simulation of Complex Geological Structures Using Multiple-Point Statistics , 2002 .

[5]  Yuhong Liu,et al.  Using the Snesim program for multiple-point statistical simulation , 2006, Comput. Geosci..

[6]  A. Soares,et al.  Geostatistics Tróia '92 , 1993 .

[7]  R. Dimitrakopoulos,et al.  Two-dimensional Conditional Simulations Based on the Wavelet Decomposition of Training Images , 2009 .

[8]  BoucherAlexandre,et al.  A SGeMS code for pattern simulation of continuous and categorical variables , 2008 .

[9]  Håkon Tjelmeland,et al.  Markov Random Fields with Higher‐order Interactions , 1998 .

[10]  Paul Switzer,et al.  Filter-Based Classification of Training Image Patterns for Spatial Simulation , 2006 .

[11]  Andre G. Journel,et al.  Spatial Connectivity: From Variograms to Multiple-Point Measures , 2003 .

[12]  Clayton V. Deutsch,et al.  GSLIB: Geostatistical Software Library and User's Guide , 1993 .

[13]  J. Caers,et al.  Conditional Simulation with Patterns , 2007 .

[14]  C. Deutsch,et al.  Geostatistics Banff 2004 , 2005 .

[15]  Haakon Tjelmeland,et al.  Directional Metropolis : Hastings Updates for Posteriors with Nonlinear Likelihoods , 2005 .

[16]  L. Hu,et al.  Multiple-Point Simulations Constrained by Continuous Auxiliary Data , 2008 .

[17]  Jef Caers,et al.  Representing Spatial Uncertainty Using Distances and Kernels , 2009 .

[18]  Jef Caers,et al.  Multiple-point Geostatistics: A Quantitative Vehicle for Integrating Geologic Analogs into Multiple Reservoir Models , 2004 .

[19]  R. M. Srivastava,et al.  Multivariate Geostatistics: Beyond Bivariate Moments , 1993 .

[20]  F. Alabert,et al.  Non-Gaussian data expansion in the Earth Sciences , 1989 .

[21]  Daniel M. Tetzlaff,et al.  Stationarity Scores on Training Images for Multipoint Geostatistics , 2009 .

[22]  J. Caers,et al.  Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling , 2010 .

[23]  Colin Daly,et al.  Higher Order Models using Entropy, Markov Random Fields and Sequential Simulation , 2005 .

[24]  Alexandre Boucher,et al.  Considering complex training images with search tree partitioning , 2009, Comput. Geosci..

[25]  Conditional cumulants in a weakly non-linear regime , 2004, astro-ph/0405590.

[26]  Jef Caers,et al.  G s TL: the geostatistical template library in C++ , 2002 .

[27]  R. Dimitrakopoulos,et al.  High-order Stochastic Simulation of Complex Spatially Distributed Natural Phenomena , 2010 .

[28]  Alexandre Boucher,et al.  Applied Geostatistics with SGeMS: A User's Guide , 2009 .