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COSMO – S tochastic Mine Planning Laboratory Department of Mining and Materials Engineering

COSMO – S tochastic Mine Planning Laboratory Department of Mining and Materials Engineering. SCRF26 - 2013. High-order stochastic simulations and some effects on flow through heterogeneous media Roussos Dimitrakopoulos. Outline. Introduction Spatial cumulants, examples and interpretations

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COSMO – S tochastic Mine Planning Laboratory Department of Mining and Materials Engineering

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  1. COSMO – Stochastic Mine Planning Laboratory Department of Mining and Materials Engineering SCRF26 - 2013 High-order stochastic simulations and some effects on flow through heterogeneous mediaRoussos Dimitrakopoulos

  2. Outline • Introduction • Spatial cumulants, examples and interpretations • High-order simulation & estimating conditional non-Gaussian distributions • Examples: Data driven vs TI driven, validation, matrix completion and TIs • Application & comparisons • Conclusions

  3. Introduction Going further beyond two-point geostatistics • Second and high(er) order models

  4. 1.2 1.2 0.8 0.8 0.4 0.4 0 0 10 10 20 20 30 30 40 40 Limits of Traditional Geostatistics Very different patterns 2 3 1 may share the Variograms EW Variograms NS 3 3 2 (h) 1 (h) 2 same variogram 1 lags lags Widely different patterns, yet same statistics up to order 2 Source: SCRF

  5. Second and High-order Geostatistics • Multiple-point (MP) geostatistics • Not enough data to accurately infer high-order statistics or patterns? - use training images • SNESIM, FILTERSIM, SIMPAT, … h x h1=h2=…=h3=1 h2 h1 x • Variogram and covariances two-point variances • High-order joint neighbourhoodsof n points

  6. Definitions Spatial moments & cumulants • Concepts • Definitions • Spatial templates

  7. Spatial Cumulants • First-order cumulant (the meanm) of a 3D stationary random function (RF)Z(x) • Second-order cumulant (the covariance)

  8. Spatial Cumulants • Third-order cumulant (zero-mean RF) • Fourth-order cumulant (zero-mean RF)

  9. Example: 2D Binary Image Third-order cumulant Original image h2 h1 Fifth-order cumulant Fourth-order cumulant h2 h2 h4 h1 h1 h3 h4

  10. Example 1: 2D binary image Third-order cumulant Original image h2 h1 Third-order cumulant

  11. Example: 2D Binary Image Fourth-order cumulant map h2 h1 h4

  12. This is Not ... Not the best student Roussos

  13. High-order Simulation Simulation based on high-order spatial cumulants • Estimating conditional distributions • Examples

  14. High-order Simulation Sequential

  15. High-order Simulation • Multivariate Legendre series • The conditional density of Z0 given Z1=a1,…,Zn=an is given by Order of the approximation Legendrecumulants Legendre polynomials = g(ci0i1…in) and ci1i2…in = cum(Xi00,Xi11,…,Xinn)

  16. High-order Simulation and MPS Legendre series Legendre series without using the first cumulants c1, c2 and c3 of the true distribution (orders 1, 2 & 3).

  17. Calculating Cumulants when Simulating u0+h2 u0+h2 u0+h3 h2 h2 h3 u0+h1 u0+h1 h1 h1 u0 h4 u0+h4 u0 h3 h5 u0+h3 u0+h5 Node to Simulate 2, order = 6, calculate up to order 4 from data, and the rest from a Training Image Node to Simulate 1, order = 6, calculate cumulants from data

  18. Examples • High-order simulations (HOSIM) • Simulations are data driven • Simulation and validation of a fully known “fluvial reservoir” • Data driven training images

  19. High-order Simulations are Data Driven Exhaustive 125 Samples Training Image (TI) 3rd order cumulant Histograms Data Realization Variograms Realizations

  20. Simulating a 3D `Fluvial Reservoir` Exhaustive image and 500 sample data

  21. Simulating a 3D `Fluvial Reservoir` Realizations using different terms

  22. Simulating a 3D `Fluvial Reservoir` Histogram and variograms of two realizations

  23. Simulating a 3D `Fluvial Reservoir` Third-order cumulant maps Data Realization 1 Realization 2

  24. Simulation of a 3D “fluvial reservoir” Fourth-order cumulant maps Data Realization 1 Realization 2

  25. Matrix Completion: Data-based TIs Exhaustive 100 Samples Conventional Training Image (TI) MSE plot of HOSIM+MC & HOSIM simulations HOSIM + MC realization HOSIM + conventional TI realization Histogram and covariance of HOSIM+MC realization, Exhaustive Image

  26. Application Some implications for reservoir forecasting • Incompressible Two-Phase Flow

  27. Application Geological heterogeneity representation: Permeability simulation Exhaustive image 32 samples

  28. Application • Phase saturation equation • Phase velocity equation: Darcy’s law • Closure relations

  29. Application Realization 3 Realization 2 Realization 1 HOSIM realizations SGS realizations Connectivity

  30. Application HOSIM realizations SGS realizations Oil recovery Water recovery up to 20% <1% Error

  31. Application Water saturation profiles (0.75 PVI) Exhaustive image HOSIM realizations SGS realizations

  32. Conclusions • High-order simulation: • Uses no- preprocessing • Generates complex spatial patterns • Reproduces bimodal data distributions, high-order spatial cumulants of data • Data driven (not training image driven) • Reconstructs the lower-order spatial complexity in data

  33. Examples Mixture of Gaussians Bivariate lognormal L1,1 . . L1,12 . . . . L12,1 . . L12,12 L1,1 . . L12,1 . . L12,12

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