SCRF 26 2013 Highorder stochastic simulations and some
- Slides: 33
SCRF 26 - 2013 High-order stochastic simulations and some effects on flow through heterogeneous media Roussos Dimitrakopoulos COSMO – Stochastic Mine Planning Laboratory Department of Mining and Materials Engineering
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
Introduction Going further beyond two-point geostatistics • Second and high(er) order models
Limits of Traditional Geostatistics Very different patterns 2 1 may share the Variograms EW 3 2 1 1. 2 0. 8 0. 4 3 1. 2 1 2 0. 8 0. 4 0 Source: SCRF Variograms NS (h) same variogram 3 10 20 lags 30 40 0 10 20 lags 30 40 Widely different patterns, yet same statistics up to order 2
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 1=h 2=…=h 3=1 h 2 x h x • Variogram and covariances two-point variances h 1 • High-order joint neighbourhoods of n points
Definitions Spatial moments & cumulants • Concepts • Definitions • Spatial templates
Spatial Cumulants • First-order cumulant (the mean m) of a 3 D stationary random function (RF) Z(x) • Second-order cumulant (the covariance)
Spatial Cumulants • Third-order cumulant (zero-mean RF) • Fourth-order cumulant (zero-mean RF)
Example: 2 D Binary Image Original image Third-order cumulant h 2 h 1 Fourth-order cumulant h 2 h 1 h 4 Fifth-order cumulant h 4 h 2 h 1 h 3
Example 1: 2 D binary image Original image Third-order cumulant h 2 h 1 Third-order cumulant
Example: 2 D Binary Image Fourth-order cumulant map h 2 h 1 h 4
This is Not. . . Not the best student Roussos
High-order Simulation based on high-order spatial cumulants • Estimating conditional distributions • Examples
High-order Simulation Sequential
High-order Simulation • Multivariate Legendre series • The conditional density of Z 0 given Z 1=a 1, …, Zn=an is given by Order of the approximation Legendre cumulants Legendre polynomials = g(ci 0 i 1…in) and ci 1 i 2…in = cum(Xi 00, Xi 11, …, Xinn)
High-order Simulation and MPS Legendre series without using the first cumulants c 1, c 2 and c 3 of the true distribution (orders 1, 2 & 3).
Calculating Cumulants when Simulating u 0+h 2 u 0+h 3 h 2 h 3 u 0+h 4 h 1 h 4 u 0+h 1 u 0 h 5 u 0+h 5 Node to Simulate 1, order = 6, calculate cumulants from data u 0+h 2 u 0 h 1 u 0+h 1 h 3 u 0+h 3 Node to Simulate 2, order = 6, calculate up to order 4 from data, and the rest from a Training Image
Examples High-order simulations (HOSIM) • Simulations are data driven • Simulation and validation of a fully known “fluvial reservoir” • Data driven training images
High-order Simulations are Data Driven Exhaustive 125 Samples Training Image (TI) Histograms Variograms Realizations 3 rd order cumulant Data Realization
Simulating a 3 D `Fluvial Reservoir` Exhaustive image and 500 sample data
Simulating a 3 D `Fluvial Reservoir` Realizations using different terms
Simulating a 3 D `Fluvial Reservoir` Histogram and variograms of two realizations
Simulating a 3 D `Fluvial Reservoir` Third-order cumulant maps Data Realization 1 Realization 2
Simulation of a 3 D “fluvial reservoir” Fourth-order cumulant maps Data Realization 1 Realization 2
Matrix Completion: Data-based TIs Exhaustive HOSIM + MC realization 100 Samples HOSIM + conventional TI realization Conventional Training Image (TI) MSE plot of HOSIM+MC & HOSIM simulations Histogram and covariance of HOSIM+MC realization, Exhaustive Image
Application Some implications for reservoir forecasting • Incompressible Two-Phase Flow
Application Geological heterogeneity representation: Permeability simulation Exhaustive image 32 samples
Application • Phase saturation equation • Phase velocity equation: Darcy’s law • Closure relations
Application Realization 1 Realization 2 HOSIM realizations SGS realizations Connectivity Realization 3
Application HOSIM realizations SGS realizations Water recovery Oil recovery Error <1% up to 20%
Application Water saturation profiles (0. 75 PVI) Exhaustive image HOSIM realizations SGS realizations
Conclusions • High-order simulation: • Uses no- preprocessing • Generates complex spatial patterns • Reproduces bimodal data distributions, highorder spatial cumulants of data • Data driven (not training image driven) • Reconstructs the lower-order spatial complexity in data
Examples Mixture of Gaussians Bivariate lognormal L 1, 1 . . L 1, 12 . . L 12, 12 L 1, 1. . L 12, 12
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