1. SCRF
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

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

4. Limits of Traditional Geostatistics
Very different patterns 2 3 1
may share the
Variograms EW
Variograms NS
0.4
0.8
1.2 10 20 30 40 3 1 2
lags
(h)
same variogram
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, …
h1=h2=…=h3=1 x h x h1 h2
Variogram and covariances two-point variances
High-order joint neighbourhoods of n points

6. Spatial moments & cumulants
Definitions
Spatial moments & cumulants
Concepts
Definitions
Spatial templates
Now, let’s give a closer look to spatial high-order statistics

7. Spatial Cumulants First-order cumulant (the mean m)
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
Original image
Third-order cumulant h2 h1
Fourth-order cumulant
Fifth-order cumulant h2 h2 h4 h1 h1 h3 h4

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

11. Example: 2D Binary Image
Fourth-order cumulant map h1 h2 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
Now, let’s see how spatial high-order statistics can be used in stochastic simulation

14. High-order Simulation
Sequential
Geological models are represented as grids
Samples are used to condition local distributions of possible values at the grid nodes.
The values at the nodes of the grid are simulated from the local conditional distributions
The simulated values are used to conditioning the local distributions of other nodes
Until all nodes are simulated.

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

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

17. Calculating Cumulants when Simulating
u0+h4
u0+h5 h2
u0+h2 h3
u0+h3
u0+h1 h1 u0 h3
u0+h3 h2
u0+h2
u0+h1 h1 u0
The cumulants required for fitting the joint distribution are obtained using a template defined by the locations of the conditioning samples.
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. High-order simulations (HOSIM)
Examples
High-order simulations (HOSIM)
Simulations are data driven
Simulation and validation of a fully known “fluvial reservoir”
Data driven training images
Now, let’s see how spatial high-order statistics can be used in stochastic simulation

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 Realization 2

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

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

26. Some implications for reservoir forecasting
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 1 Realization 2 Realization 3 HOSIM
realizations
SGS
realizations
Connectivity

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

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, L12,12
L1,1 .
L12, L12,12