Accelerating simulation for the multiple-point statistics algorithm using vector quantization

Multiple-point statistics (MPS) is a prominent algorithm to simulate categorical variables based on a sequential simulation procedure. Assuming training images (TIs) as prior conceptual models, MPS extracts patterns from TIs using a template and records their occurrences in a database. However, complex patterns increase the size of the database and require considerable time to retrieve the desired elements. In order to speed up simulation and improve simulation quality over state-of-the-art MPS methods, we propose an accelerating simulation for MPS using vector quantization (VQ), called VQ-MPS. First, a variable representation is presented to make categorical variables applicable for vector quantization. Second, we adopt a tree-structured VQ to compress the database so that stationary simulations are realized. Finally, a transformed template and classified VQ are used to address nonstationarity. A two-dimensional (2D) stationary channelized reservoir image is used to validate the proposed VQ-MPS. In comparison with several existing MPS programs, our method exhibits significantly better performance in terms of computational time, pattern reproductions, and spatial uncertainty. Further demonstrations consist of a 2D four facies simulation, two 2D nonstationary channel simulations, and a three-dimensional (3D) rock simulation. The results reveal that our proposed method is also capable of solving multifacies, nonstationarity, and 3D simulations based on 2D TIs.

Figure 1

A TI of size $6\times 6$ and a template of size 4.

Figure 2

A search tree-structured database.

Figure 3

A list-structured database.

Figure 4

A simulation domain of size $6\times 6$.

Figure 5

Multigrid strategy for a simulation domain of size $8\times 8$. (a) The coarsest grid. (b) The moderate grid. (c) The finest grid.

Figure 6

A 2D plane to represent three states.

Figure 7

A list represented by the proposed representation method.

Figure 8

The codebook $Boo{k}_0$.

Figure 9

The codebook $Boo{k}_{10}$.

Figure 10

The final codebook Book.

Figure 11

The final tree-structured codebook $Boo{k}_{\mathrm{tree}}$.

Figure 12

CVQ in the proposed method. (a) The training procedure. (b) The simulation procedure.

Figure 13

A 2D stationary channelized TI.

Figure 14

The first two simulated realizations generated by full search VQ with codebooks of size 128.

Figure 15

The first two simulated realizations generated by TSVQ with codebooks of size 128.

Figure 16

The first two simulated realizations generated by TSVQ with codebooks of size 256.

Figure 17

The first two simulated realizations generated by S-IMPALA.

Figure 18

The first two simulated realizations generated by SNESIM.

Figure 19

The first two simulated realizations generated by FDS.

Figure 20

The first two simulated realizations generated by CCSIM.

Figure 21

The first two simulated realizations generated by MS-CCSIM.

Figure 22

A 2D application with four facies simulation. (a) The TI. (b), (c) Two realizations.

Figure 23

A 2D application with spatial transformation of patterns. (a) Rotation. (b) Affinity. (c) The TI. (d) A simulated realization.

Figure 24

A 2D application with one secondary variable. (a) The first variable for a nonstationary TI. (b) Secondary variable describing the nonstationarity of the first variable in the TI. (c) A simulated realization for the first variable. (d) Secondary variable describing the nonstationarity of the first variable in the simulation domain.

Figure 25

A rock slice.

Figure 26

A simulated realization and a CT sample. (a) 3D exterior view of the simulated realization. (b) A cross section of the simulated realization. (c) The perspective image of the simulated realization. (d) 3D exterior view of the CT sample. (e) A cross section of the CT sample. (f) The perspective image of the CT sample.

Figure 27

Comparison of ACFs for TI, CT sample, and simulated realization.

Figure 28

Comparison of LPFs for TI, CT sample, and simulated realization.