Pejman Tahmasebi and Jef Caers Stanford Center for Reservoir Forecasting MS‐CCSIM: accelerating pattern‐based geostatistical simulation categorical variables using a multi‐scale search in Fourier space Pejman Tahmasebi and Jef Caers

Pattern based multiple point geostatistics Advantages Disadvantages Fast Better Pattern reproduction Conditioning Less variability Pattern patchiness CPU time

Using adaptive recursive template splitting Conditional CCSIM Multiple-point geostatistical modeling based on the cross-correlation functions, Computational Geosciences, 2012 MS‐CCSIM: accelerating pattern‐based geostatistical simulation categorical variables using a multi‐scale search in Fourier space, Computers & Geosciences, 2014 TI Simulation Grid with Hard Data Using adaptive recursive template splitting

TI TI Realization HD Realization 1 Realization 2 (WCA dataset from Chevron) TI Realization

Limitations of CCSIM CPU Make it does not work fast for multi-million cell 3D grids and TIs Conditioning Does not take into account of data ahead of the path point data

i) Fourier Acceleration CPU time CCSIM uses a cross-correlation function as a measure for doing the pattern matching Cross-correlation acts as a convolution Computation time can be reduced by transferring the data to the Fourier space by using the convolution theorem Transferring the TI and the Overlap areas to the Fourier space Calculating the cross-correlation Inverse Fourier transforming the result to the Spatial domain

ii) Constructing of training image at different scales CPU time ii) Constructing of training image at different scales Scale x y x Original TI The rescaled TI can be obtained by using an interpolation method

Start on a grid (G) with the size of MS0 CPU time MS-CCSIM Algorithm MS 2 MS 1 MS 0 Start on a grid (G) with the size of MS0 search boundary final search boundary Find the pattern location Find the pattern location Project the location Move to finest scale Select the similar pattern and insert in G END

Unconditional Simulation Realization 1 Realization 2 Realization 3 4.8(s) 1000x1000 Unconditional Simulation 4.8(s) 1000x1000 50(s) 180x140x30 450 (s) 200x200x50

CPU time comparison 3D Simulation

MS-CCSIM Method for Conditional Simulation Conditioning A B C Larger simulation grid cell cell value is assigned the category with highest frequency

Accounting for the data ahead of the raster path Conditioning primary pattern co-pattern point data co-template primary template Simulation Grid TI another solution: Using Kriging as an auxiliary variable

Using multiple raster path Hard Data Conditioning Using multiple raster path Reproducing better long-range channels using co-template and multiple-path using single path using multiple-path

Conditional Simulation examples Local ANODI Global ANODI ccsim ms-ccsim Rejection sampling Two Neighborhood Shale-Sand HD TI Rejection Sampling ensemble average ms-ccsim ensemble average Conditional Realization

Sensitivity Analysis (Tank Dataset) Number of Candidates Candidate: Categorical: 3-10 Continuous: 3-30 CPU time Quality Variability TI (200x500) Candidate = 4 Candidate = 40 Candidate = 400

Sensitivity Analysis Template Size CPU time Quality Variability Realizations T = 10 T = 20 T = 50 Ensemble Average

Sensitivity Analysis CPU time Quality Overlap Size OL: (1/4 – 1/5)*Template size Variability OL = 3 OL = 8 OL = 15

Conclusion MS-CCSIM can give a tremendous acceleration to geostatistical simulation Proposed data relocation and Co-template can be used for any hard data distribution Hereafter, using very rich, large and complex process based training image will be possible!

The code are available online ! https://github.com/SCRFpublic/MS_CCSIM