Delineation of facies in the subsurface and quantification of uncertainty in their boundaries are significant steps in mineral resource evaluation and reservoir modeling, which impact downstream analyses of a mining or petroleum project. This paper investigates the ability of nonparametric geostatistical simulation algorithms (sequential indicator, single normal equation and filter-based simulation) to construct realizations that reproduce some expected statistical and spatial features, namely facies proportions, boundary regularity, contact relationships and spatial correlation structure, as well as the expected fluctuations of these features across the realizations. The investigation is held through a synthetic case study and a real case study, in which a pluri-Gaussian model is considered as the reference for comparing the simulation results. Sequential indicator simulation and single normal equation simulation based on over-restricted neighborhood implementations yield the poorest results, followed by filter-based simulation, whereas single normal equation simulation with a large neighborhood implementation provides results that are closest to the reference pluri-Gaussian model. However, some biases and inaccurate fluctuations in the realization statistics (facies proportions, indicator direct and cross-variograms) still arise, which can be explained by the use of a single finite-size training image to construct the realizations.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Alabert, F. (1987). Stochastic imaging of spatial distributions using hard and soft information. Master’s thesis, Department of Applied Earth Sciences, Stanford University, Stanford, pp. 332.
Al-Mudhafar, W. J. (2018). Multiple-point geostatistical lithofacies simulation of fluvial sand-rich depositional environment: a case study from Zubair formation/South Rumaila oil field. SPE Reservoir Evaluation & Engineering, 21(1), 39–53.
Armstrong, M., Galli, A., Beucher, H., Le Loc’h, G., Renard, D., Renard, B., et al. (2011). Plurigaussian Simulations in Geosciences (p. 187). Berlin: Springer.
Beucher, H., & Renard, D. (2016). Truncated Gaussian and derived methods. Comptes Rendus Géoscience, 348, 510–519.
Boisvert, J. B., Pyrcz, M. J., & Deutsch, C. V. (2007). Multiple-point statistics for training image selection. Natural Resources Research, 16(4), 313–321.
Boisvert, J. B., Pyrcz, M. J., & Deutsch, C. V. (2010). Multiple point metrics to assess categorical variable models. Natural Resources Research, 19, 165–175.
Chautru, J. M., Meunier, R., Binet, H., & Bourges, M. (2015). Geobodies stochastic analysis for geological model parameter inference. Petroleum Geostatistics 2015 (pp. 293–297). Houten: European Association of Geoscientists & Engineers.
Chilès, J. P., & Delfiner, P. (2012). Geostatistics: Modeling spatial uncertainty (p. 699). New York: Wiley.
De Iaco, S. (2013). On the use of different metrics for assessing complex pattern reproduction. Journal of Applied Statistics, 40(4), 808–822.
De Iaco, S., & Maggio, S. (2011). Validation techniques for geological patterns simulations based on variogram and multiple-point statistics. Mathematical Geosciences, 43, 483–500.
Dubrule, O. (2017). Indicator variogram models: do we have much choice? Mathematical Geosciences, 49(4), 441–465.
Emery, X. (2004). Properties and limitations of sequential indicator simulation. Stochastic Environmental Research and Risk Assessment, 18(6), 414–424.
Emery, X. (2007). Simulation of geological domains using the plurigaussian model: New developments and computer programs. Computers & Geosciences, 33(9), 1189–1201.
Emery, X. (2008). Statistical tests for validating geostatistical simulation algorithms. Computers & Geosciences, 34(11), 1610–1620.
Emery, X., Arroyo, D., & Porcu, E. (2016). An improved spectral turning-bands algorithm for simulating stationary vector Gaussian random fields. Stochastic Environmental Research and Risk Assessment, 30, 1863–1873.
Emery, X., & Lantuéjoul, C. (2006). TBSIM: A computer program for conditional simulation of three-dimensional Gaussian random fields via the turning bands method. Computers & Geosciences, 32(10), 1615–1628.
Emery, X., & Lantuéjoul, C. (2011). Geometric covariograms, indicator variograms and boundaries of planar closed sets. Mathematical Geosciences, 43(8), 905–927.
Emery, X., & Lantuéjoul, C. (2014). Can a training image be a substitute for a random field model? Mathematical Geosciences, 46(2), 133–147.
Emery, X., & Ortiz, J. M. (2011). A comparison of random field models beyond bivariate distributions. Mathematical Geosciences, 43(2), 183–202.
Emery, X., & Silva, D. A. (2009). Conditional co-simulation of continuous and categorical variables for geostatistical applications. Computers & Geosciences, 35(6), 1234–1246.
Galli, A., Beucher, H., Le Loc’h, G., & Doligez, B. (1994). The pros and cons of the truncated Gaussian method. In M. Armstrong & P. A. Dowd (Eds.), Geostatistical simulations (pp. 217–233). Dordrecht: Kluwer.
Journel, A. G., & Alabert, F. G. (1990). New method for reservoir mapping. Journal of Petroleum Technology, 42(2), 212–218.
Journel, A. G., & Gómez-Hernández, J. J. (1993). Stochastic imaging of the Wilmington clastic sequence. SPE Formation Evaluation, 8(1), 33–40.
Lantuéjoul, C. (1994). Non conditional simulation of stationary isotropic multigaussian random functions. In M. Armstrong & P. A. Dowd (Eds.), Geostatistical Simulations (pp. 147–177). Dordrecht: Kluwer Academic.
Lantuéjoul, C. (2002). Geostatistical simulation, models and algorithms (p. 256). Berlin: Springer.
Le Loc’h, G., & Galli, A. (1997). Truncated plurigaussian method: Theoretical and practical points of view. In E. Y. Baafi & N. A. Schofield (Eds.), Geostatistics Wollongong’96 (pp. 211–222). Dordrecht: Kluwer Academic.
Leuangthong, O., McLennan, J. A., & Deutsch, C. V. (2004). Minimum acceptance criteria for geostatistical realizations. Natural Resources Research, 13(3), 131–141.
Lowell, J. D., & Guilbert, J. M. (1970). Lateral and vertical alteration-mineralization zoning in porphyry ore deposits. Economic Geology, 65, 373–408.
Madani, N., & Emery, X. (2015). Simulation of geo-domains accounting for chronology and contact relationships: Application to the Río Blanco copper deposit. Stochastic Environmental Research and Risk Assessment, 29(8), 2173–2191.
Madani, N., & Emery, X. (2017). Plurigaussian modeling of geological domains based on the truncation of non-stationary Gaussian random fields. Stochastic Environmental Research and Risk Assessment, 31(4), 893–913.
Madani, N., Naderi, A., Biranvand, B., & Keshavarz, N. (2018). Lithofacies uncertainty modeling in a siliciclastic reservoir setting by incorporating geological contacts and seismic information. Journal of Petroleum Exploration and Production Technology. https://doi.org/10.1007/s13202-018-0531-7.
Maleki, M., Emery, X., & Mery, N. (2017). Indicator variograms as an aid for geological interpretation and modeling of ore deposits. Minerals, 7(12), 241.
Mariethoz, G., & Caers, J. (2014). Multiple-point geostatistics: Stochastic modeling with training images (p. 376). New York: Wiley.
Matheron, G. (1989). Estimating and choosing: An essay on probability in practice (p. 141). Berlin: Springer.
Matheron, G., Beucher, H., Galli, A., Guérillot, D., & Ravenne, C. (1987). Conditional simulation of the geometry of fluvio-deltaic reservoirs. In: 62nd Annual technical conference and exhibition of the society of petroleum engineers, pp. 591–599. SPE Paper 16753, Dallas.
Modis, K., & Sideri, D. (2013). Geostatistical simulation of hydrofacies heterogeneity of the West Thessaly aquifer systems in Greece. Natural Resources Research, 22(2), 123–138.
Oriani, F., & Renard, P. (2014). Binary upscaling on complex heterogeneities: The role of geometry and connectivity. Advances in Water Resources, 64, 47–61.
Renard, P., & Allard, D. (2013). Connectivity metrics for subsurface flow and transport. Advances in Water Resources, 51, 168–196.
Rongier, G., Collon, P., Renard, P., Straubhaar, J., & Sausse, J. (2016). Comparing connected structures in ensemble of random fields. Advances in Water Resources, 96, 145–169.
Rossi, M. E., & Deutsch, C. V. (2014). Mineral resource estimation (p. 332). New York: Springer.
Strebelle, S. (2002). Conditional simulation of complex geological structures using multiple-point statistics. Mathematical Geology, 34(1), 1–22.
Tan, X., Tahmasebi, P., & Caers, J. (2014). Comparing training-image based algorithm using an analysis of distance. Mathematical Geosciences, 46(2), 149–169.
Yunsel, T., & Ersoy, A. (2011). Geological modeling of gold deposit based on grade domaining using plurigaussian simulation technique. Natural Resources Research, 20(4), 1–19.
Zhang, T., Switzer, P., & Journel, A. (2006). Filter-based classification of training image patterns for spatial simulation. Mathematical Geology, 38(1), 63–80.
The first author acknowledges the Nazarbayev University for financial supporting. The second and third authors acknowledge the support of the Chilean Commission for Scientific and Technological Research (CONICYT), through projects CONICYT/FONDECYT/POSTDOCTORADO/N°3180655 and CONICYT PIA Anillo ACT1407, respectively. The examples presented in “Essential Statistical and Geometrical Characteristics of Facies” section were provided by Dr. Bijan Biranvand from the Department of Petroleum Geology, Research Institute of Petroleum Industry, Tehran, Iran. The data for the case study were provided by CODELCO—Chile. Constructive comments by two anonymous reviewers are gratefully acknowledged. Also, we deeply thank Dr. Carranza for the valuable comments on the final version of manuscript.
About this article
Cite this article
Madani, N., Maleki, M. & Emery, X. Nonparametric Geostatistical Simulation of Subsurface Facies: Tools for Validating the Reproduction of, and Uncertainty in, Facies Geometry. Nat Resour Res 28, 1163–1182 (2019). https://doi.org/10.1007/s11053-018-9444-x
- Geological uncertainty
- Pluri-Gaussian model
- Sequential indicator simulation
- Single normal equation simulation
- Filter-based simulation
- Statistical fluctuations