A geostatistical implicit modeling framework for uncertainty quantification of 3D geo-domain boundaries: Application to lithological domains from a porphyry copper deposit

Abstract The spatial modeling of geo-domains has become ubiquitous in many geoscientific fields. However, geo-domains’ spatial modeling poses real challenges, including the uncertainty assessment of geo-domain boundaries. Geo-domain models are subject to uncertainties due mainly to the inherent lack of knowledge in areas with little or no data. Because they are often used for impactful decision-making, they must accurately estimate the geo-domain boundaries’ uncertainty. This paper presents a geostatistical implicit modeling method to assess the uncertainty of 3D geo-domain boundaries. The basic concept of the method is to represent the underlying implicit function associated with each geo-domain as a sum of a random implicit trend function and a residual random function. The conditional simulation of geo-domains is performed through a step-by-step approach. First, implicit trend function realizations and optimal covariance parameters associated with the residual random function are generated through the probability perturbation method. Then, residual function realizations are generated through classical geostatistical unconditional simulation methods and added to implicit trend function realizations to obtain unconditional implicit function realizations. Next, the conditioning of unconditional implicit function realizations to hard data is performed via principal component analysis and randomized quadratic programming. Finally, conditional implicit function simulations are transformed to conditional geo-domain simulations by applying a truncation rule. The proposed method is constructed to honor hard data and stated rules of how geo-domains interact spatially. It is applied to a lithological dataset from a porphyry copper deposit. A comparison with the classical sequential indicator simulation (SIS) method is carried out. The results indicate that the proposed approach can provide a more reliable and realistic uncertainty assessment of 3D geo-domain boundaries than the traditional sequential indicator simulation (SIS) approach.

[1]  Andre G. Journel,et al.  New method for reservoir mapping , 1990 .

[2]  Weihong Zhang,et al.  Feature-driven topology optimization method with signed distance function , 2016 .

[3]  Hong Yi,et al.  A survey of the marching cubes algorithm , 2006, Comput. Graph..

[4]  Xavier Emery,et al.  Simulation of geological domains using the plurigaussian model: New developments and computer programs , 2007, Comput. Geosci..

[5]  M. Powell A View of Algorithms for Optimization without Derivatives 1 , 2007 .

[6]  C. Lajaunie,et al.  Foliation fields and 3D cartography in geology: Principles of a method based on potential interpolation , 1997 .

[7]  G. Marsily Quantitative Hydrogeology: Groundwater Hydrology for Engineers , 1986 .

[8]  Chakib Bennis,et al.  Construction of coherent 3D geological blocks , 2003 .

[9]  L. Hu,et al.  Extended Probability Perturbation Method for Calibrating Stochastic Reservoir Models , 2008 .

[10]  Benoit Noetinger,et al.  The FFT Moving Average (FFT-MA) Generator: An Efficient Numerical Method for Generating and Conditioning Gaussian Simulations , 2000 .

[11]  Guillaume Caumon,et al.  3-D Structural geological models: Concepts, methods, and uncertainties , 2018 .

[12]  Clayton V. Deutsch,et al.  Boundary modeling with moving least squares , 2019, Comput. Geosci..

[13]  Donald Goldfarb,et al.  A numerically stable dual method for solving strictly convex quadratic programs , 1983, Math. Program..

[14]  Francky Fouedjio,et al.  Geostatistical clustering as an aid for ore body domaining: Case study at the Rocklea Dome channel iron ore deposit, Western Australia , 2018 .

[15]  Jeff B. Boisvert,et al.  Stochastic Distance Based Geological Boundary Modeling with Curvilinear Features , 2013, Mathematical Geosciences.

[16]  Saeed Soltani-Mohammadi,et al.  Distance function modeling in optimally locating additional boreholes , 2018 .

[17]  M. Jessell,et al.  Assessing and Mitigating Uncertainty in Three-Dimensional Geologic Models in Contrasting Geologic Scenarios , 2018 .

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

[19]  J. Wellmann,et al.  Structural geologic modeling as an inference problem: A Bayesian perspective , 2016 .

[20]  Clayton V. Deutsch,et al.  Geostatistical Reservoir Modeling , 2002 .

[21]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[22]  J. Chilès,et al.  3D Geological Modelling and Uncertainty: The Potential-field Method , 2005 .

[23]  P. Collon-Drouaillet,et al.  Simulation of 3D karst conduits with an object-distance based method integrating geological knowledge , 2014 .

[24]  J. Caers,et al.  Conditional simulation of categorical spatial variables using Gibbs sampling of a truncated multivariate normal distribution subject to linear inequality constraints , 2020, Stochastic Environmental Research and Risk Assessment.

[25]  Margaret Armstrong,et al.  Plurigaussian Simulations in Geosciences , 2014 .

[26]  Guillaume Caumon,et al.  Towards Stochastic Time-Varying Geological Modeling , 2010 .

[27]  Liang Yang,et al.  Assessing and visualizing uncertainty of 3D geological surfaces using level sets with stochastic motion , 2019, Comput. Geosci..

[28]  Florian Wellmann,et al.  GemPy 1.0: open-source stochastic geological modeling and inversion , 2019, Geoscientific Model Development.

[29]  Tobias Frank,et al.  3D-reconstruction of complex geological interfaces from irregularly distributed and noisy point data , 2007, Comput. Geosci..

[30]  Silva Maureira,et al.  Enhanced Geologic Modeling with Data-Driven Training Images for Improved Resources and Recoverable Reserves , 2015 .

[31]  Roland Martin,et al.  Next Generation Three-Dimensional Geologic Modeling and Inversion , 2014 .

[32]  STOCHASTIC GEOLOGICAL MODELLING USING IMPLICIT BOUNDARY SIMULATION , 2014 .

[33]  X. Emery,et al.  Simulation of geo-domains accounting for chronology and contact relationships: application to the Río Blanco copper deposit , 2015, Stochastic Environmental Research and Risk Assessment.

[34]  B. Lévy,et al.  Stochastic simulations of fault networks in 3D structural modeling. , 2010 .

[35]  Jeff B. Boisvert,et al.  Iterative refinement of implicit boundary models for improved geological feature reproduction , 2017, Comput. Geosci..

[36]  G. Caumon,et al.  Stochastic structural modelling in sparse data situations , 2015 .

[37]  J. Chilès,et al.  Geostatistics: Modeling Spatial Uncertainty , 1999 .

[38]  Dean S. Oliver,et al.  Moving averages for Gaussian simulation in two and three dimensions , 1995 .

[39]  H. J. Ross,et al.  Practical Implicit Geological Modelling , 2003 .

[40]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[41]  M. Jessell,et al.  Geodiversity: Exploration of 3D geological model space , 2013 .

[42]  M. Jessell,et al.  Locating and quantifying geological uncertainty in three-dimensional models: Analysis of the Gippsland Basin, southeastern Australia , 2012 .

[43]  F. Horowitz,et al.  Towards incorporating uncertainty of structural data in 3D geological inversion , 2010 .

[44]  Lachlan Grose,et al.  Inversion of Structural Geology Data for Fold Geometry , 2018, Journal of Geophysical Research: Solid Earth.

[45]  Eulogio Pardo-Igúzquiza,et al.  Plurigau: a computer program for simulating spatial facies using the truncated Plurigaussian method , 2003 .

[46]  Clayton V. Deutsch,et al.  A multidimensional scaling approach to enforce reproduction of transition probabilities in truncated plurigaussian simulation , 2013, Stochastic Environmental Research and Risk Assessment.

[47]  Alain Galli,et al.  The Pros and Cons of the Truncated Gaussian Method , 1994 .

[48]  G. Caumon,et al.  ODSIM: An Object-Distance Simulation Method for Conditioning Complex Natural Structures , 2010 .

[49]  G. Mariéthoz,et al.  Multiple-point Geostatistics: Stochastic Modeling with Training Images , 2014 .

[50]  Clayton V. Deutsch,et al.  Cleaning categorical variable (lithofacies) realizations with maximum a-posteriori selection , 1998 .

[51]  J. F. Coimbra Leite Costa,et al.  Boundary simulation – a hierarchical approach for multiple categories , 2021, Applied Earth Science.

[52]  G. Caumon,et al.  Generating variable shapes of salt geobodies from seismic images and prior geological knowledge , 2019, Interpretation.

[53]  M. Jessell,et al.  Towards an integrated inversion of geoscientific data: What price of geology? , 2010 .

[54]  Clayton V. Deutsch,et al.  Modeling multiple geologic domains in mineral deposits with an implicit signed distance function approach , 2016 .

[55]  Christian Lantuéjoul,et al.  Geostatistical Simulation: Models and Algorithms , 2001 .

[56]  Pejman Tahmasebi,et al.  Multiple-point geostatistical modeling based on the cross-correlation functions , 2012, Computational Geosciences.