1 / 33

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

cberger
Télécharger la présentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related