Fast FILTERSIM Simulation with Score-based Distance Function

Transcription

1 Fast FILTERSIM Simulation with Score-based Distance Function Jianbing Wu (1), André G. Journel (1) and Tuanfeng Zhang (2) (1) Department of Energy Resources Engineering, Stanford, CA (2) Schlumberger Doll Research, Boston, MA May, 2007 Abstract By using pattern-to-pattern relations taken from a training image, the FILTERSIM simulation algorithm can reproduce that training image complex patterns and structures. Training patterns are identified through a vector of filter score values. FILTERSIM can handle both continuous and categorical variables. A search for similar patterns using a score distance function is proposed. Both the local data events (dev) even if incomplete and the prototypes (prot) are summarized by their respective filter score vectors, resulting in a significant data dimension reduction and faster search. When using a multiple grid simulation, the coarse grids are only partially informed, hence filters cannot be applied directly. A dual template concept is applied on the penultimate coarse grid to fill in the non-informed data locations, allowing calculation of the filter scores. This score based-distance calculation accelerates significantly the FILTERSIM with a factor up to 10 in 3D applications. 1

2 1 Introduction The multiple-point (mp) simulation technique initially proposed by Journel (1992), Srivastava (1994) combines the strengths of variogram-based algorithms (Journel, 1974; Deutsch and Journel, 1998; Lantuéjoul, 2002) and object-based algorithms (Haldorsen and Chang, 1986; Deutsch and Wang, 1996; Lia et al., 1997; Viseur et al., 1998). The mp concept allows better reproduction of the geological structures by simulating point values or templates of point values conditioned to a pattern-type data event consisting of multiple locations. Instead of using mere 2-point statistics from a variogram model, the mp approach borrows multiplepoint (>> 2) statistics from a training image. This allows to reproduce into the simulated reservoir geological structures present in the training image beyond its variogram. The first practical mp program SNESIM was implemented by Strebelle (2000). It stores all pre-scanned conditional probabilities in a search tree data base. During simulation, these local conditional probabilities can be retrieved from the search tree very fast. The hard data are relocated to the simulation grid to ensure exactitude of data reproduction; and any soft information (such as seismic) is integrated with the probability issued from the search tree through the Tau model proposed by Journel (2002). SNESIM can reproduce simple geological structures depicted by a training image, however, it fails to reconstruct complex geological structures with pronounced curvilinearity, such as thin and highly sinuous channels. Moreover, SNESIM does not work for continuous variables. FILTERSIM (Zhang, 2006) was developed to circumvent the above limitations. The training patterns extracted from a given image are identified by a vector of a few filter score values. Those patterns are then classified and grouped into pattern classes, each class being summarized by an average pattern or prototype in the filter score space. Instead of saving all the probabilities into a search tree, FILTERSIM only saves the training pattern id (the pattern central location) in memory, hence it is RAM efficient. At each node along the simulation path, one pattern is sampled from the class whose prototype filter signature is closest to that of the conditional local data event; this pattern is then pasted onto the simulation grid. The simulation process is thus one of pattern simulations (Wu et al., 2006), simulating all nodal values of a template simultaneously. This pattern simulation approach, adopted from image construction (Bonet, 1997; Paget and Longstaff, 1998; Wei, 2001), allows for a better reproduction of the within-template structures. Because the training patterns are summarized by a few filter score vectors, there is a large dimension reduction from the number of nodes in the template 2

3 to the number of filters. FILTERSIM can handle continuous variables or a large number of categories. 2 FILTERSIM Algorithm Recall In FILTERSIM, a J-node search template T J is used to define the patterns from a given training image. The template T J is spanned by a rectangular box of size n x n y n z, where n x, n y and n z are the template sizes in each X/Y/Z direction, and J = n x n y n z. Each node u j of the template T J is defined as u 0 + h j, where h j = (x, y, z) j is the offset vector from the template center u 0. The offset coordinates x, y, z are integer values. 2.1 Filter and Score A filter is a set of weights associated with the nodes of the search template T J = {u 0 + h j, j = 1,..., J}. The filter value or weight associated with the template location u j is f j. Hence, a J-nodes filter is {f(h j ); j = 1,..., J}. That filter is applied to each local training patten pat centered at location u in the training image. This results in a weighted averaged value S T (u), called score value: S T (u) = n f(h j ) pat(u + h j ), (1) j=1 where pat(u + h j ) is the pattern nodal value. Clearly, one filter is not enough to capture the information carried by any given training pattern. A set of F filters ({f i (h j ); j = 1,, J}, i = 1,, F ) should be designed to capture the diverse characteristics of a training pattern. These F filters create a vector of F scores summarizing each training pattern, Eq. 1 is rewritten as: n ST i (u) = f i (h j ) pat(u + h j ), i = 1,..., F (2) j=1 Note that when representing a training pattern with a score vector, the data dimension is reduced significantly from the template size J (= n x n y n z ) to F. For example a 3D pattern of size is reduced into the 9 default filter scores proposed in FILTERSIM. 3

4 Those filters can be either provided as default by FILTERSIM (Zhang, 2006), or input through a data file as user-defined (Wu et al., 2006). For a continuous training image (TI), the F filters are directly applied to the continuous values constituting each training pattern. For a categorical training image with K categories, the training image is first transformed into K sets of binary indicators I k (u), k = 1,..., K, u TI: { 1 if u belongs to k I k (u) = th category 0 otherwise (3) A K-categories pattern is thus represented by K sets of binary patterns, each indicating the presence/absence of a single category at a certain location. The F filters are applied to each one of the K binary patterns resulting in a total of F K scores S i,k (i = 1,, F, k = 1,, K). A continuous training image can be seen as a special case of a categorical training image with a single category K = 1. Hence, the total number of filter scores can be written generally as F K. 2.2 Pattern Classification After applying the F filters over a K-category training image, each training pattern can be characterized by a unique filter score vector of size F K (K = 1 for continuous variables). Similar training patterns will have similar F K scores. Hence by partitioning the filter score space, similar patterns can be grouped together. The current available partition methods are either cross classification or K-Mean clustering classification (Wu et al., 2006). After partition, each pattern class (whether parent or child) is characterized by an averaged pattern, named prototype (prot), defined as the point-wise average of all training patterns falling into that class. A prototype has the same size (number of nodes) as the filter template, and is used as the pattern class ID (identification number). For a continuous training image, the prototype associated with search template T J is calculated as: prot(h j ) = 1 c c pat(u i + h j ), j = 1,, J (4) i=1 4

5 where h j is the j th offset location in the search template T J, c is the number of training replicates within that prototype class; u i (i = 1,..., c) is the center of a specific training pattern belonging to that class. For a categorical variable, Eq. 4 is applied to each of the K sets of binary indicator values obtained through Eq. 3. Hence a categorical prototype consists of K proportion maps, each map giving the probability of a certain category to prevail at a template location u i + h j : prot(h j ) = { prot k (h j ), k = 1,, K }, (5) where prot k (h j ) = Prob (z(u 0 + h j ) = k), and u 0 is the central location of template T J. 2.3 Sequential Simulation The classic sequential simulation paradigm (Deutsch and Journel, 1998) is extended for FILTERSIM pattern simulation. At each node u along the random path visiting the simulation grid G, a search template T J of same size as the filter template is used to extract the conditioning data event dev(u). The prototype prot closest to that data event dev(u), based on some distance function, is found. Next a pattern pat is randomly drawn from that closest prototype class, and is pasted onto the simulation grid G. The inner part of the pasted pattern is frozen as hard data, and will not be revisited during simulation on the current (multiple) grid. Similar to SNESIM, the multiple-grid simulation concept (Tran, 1994) is applied to capture the large scale structures of the training image. The multiple-grid FIL- TERSIM approach is summarized in Algorithm 1. The distance function used to find the prototype closest to a given data event dev is defined as: d(u 0 ) = J ω j dev(u + h j ) prot(u 0 + h j ) (6) j=1 where u 0 is the center node location of the prototype; J is the total number of nodes in the search template T J ; ω j is the weight associated to each template node; u is the center node of the data event being compared to the prototype; 5

6 Algorithm 1 Multiple grid FILTERSIM simulation 1: for Each multiple grid G do 2: Re-scale the geometries of both search template and filter configuration 3: Create score values with the given F filters 4: Group all training patterns into prototype classes in the score space 5: Relocate hard conditioning data on the simulation grid G 6: Define a random path on the simulation grid G 7: for Each node u along the random path do 8: Extract the conditioning data event dev centered at u 9: Find the prototype prot closest to dev(u) 10: Sample a pattern pat from prot with servo-system control 11: Paste pat to the realization being simulated, and freeze the nodes within a central inner patch 12: end for 13: end for h j is the node offset in the search template T J. Note that the data event dev may not be fully informed, thus only those informed nodes are retained for the distance calculation. 3 Motivation In the FILTERSIM sequential simulation procedure presented in Section 2.3, the class prototype closest to the local conditioning data event (dev) is found through a pixel-based distance function (Eq. 6). For a K-category training image (with K = 1 for a continuous variable) and a J-node search template, the data dimension (or total number of pixels) of both data event and prototype is N d = K J. Given N p prototypes, at each node of the simulation grid there will be N p executions of Eq. 6; hence the total number of pixel-wise distance calculations between the data event and all prototypes is N d N p = K J N p. For 3D simulations with a large and diverse training image, N p could be of the order of 10 3 to 10 6, therefore the J K N p distance calculations will be very CPU-demanding. When repeated over the large number of nodes of a 3D simulation grid; the whole FILTERSIM simulation would become extremely slow. 6

7 The idea is thus to speed up FILTERSIM by decreasing the product N d N p = K J N p : Reduce N p : The number N P of prototypes is determined by the partition method. The 3 default filters used for each of the three X/Y/Z directions work well: taken together they are successful in reproducing patterns/structures from most training images. For simple training geometries, the last directional curvature filter might be redundant with the first two default filters. Simple training patterns could be classified with only F < F filters. One can use the user-defined filter option to remove redundant filters. By reducing the number F of filters to F (< F ), the N p value will decrease significantly for 3D multiple categories simulation. However, reducing F might result in poorer simulation quality. Sensitivity analysis on the number and types of filters retained should be done prior to the final simulation. For example, a 2D FILTERSIM run is performed first with all default 6 filters applied to the four facies 2D training image of Figure 1. The simulation grid is of size The target facies proportions are taken from the training image at 0.45, 0.2, 0.2 and 0.15 for mud background, sand channel, levee and crevasse, respectively. The number of multiple grid is 3. Fast cross partition is used with 3 bins for parent pattern classification and 2 bins for further children classification. The search template size is 11 11, and the patch template size is 7 7. Figure 2(a) gives one unconditional simulation. The total FILTERSIM running time is 270 seconds on a IBM notebook with 1.3GHz CPU processor. With the help of the user-defined filter option, one unconditional FIL- TERSIM simulation is performed with only 4 filters: the 2 default directional average filters and the 2 default directional gradient filters, dropping the 2 curvature filters. The other parameter settings are exactly the same as for the run with 6 default filters. Figure 2(b) shows one such FILTER- SIM realization. It is seen that the channel continuity and facies attachment sequence from the training image are reasonably well reproduced in both plots in Figure 2, but there are more noisy spots (levee and crevasse) in the realization simulated with only 4 input filters. Those noisy spots can be removed by post-processing. In terms of CPU time, the new simulation with 4 filters only costs 40 seconds, much faster than the simulation with 6 filters (270 seconds). The acceleration is due to a reduction of the total number of parent prototypes. The speedup when using fewer input filters would be even more significant for 3D simulations. 7

8 Figure 1: Four facies 2D training image (blue: mud background; cyan: sand channel; yellow: levee; brown: crevasse) (a) Simulation with 6 filters (b) Simulation with 4 filters Figure 2: Two unconditional FILTERSIM realizations using different number of filters (left: all 6 default filters; right: 2 average filters and 2 gradient filters only) 8

9 (a) Channel values 1 and levee values 2 (b) Channel values 2 and levee values 1 Figure 3: Two unconditional FILTERSIM realizations by treating the categorical training image Figure 1 as a continuous training image using different categorical coding Reduce K: For a K-categories simulation, the K value is a multiplicative factor, hence very significant. In the new FILTERSIM implementation, one can treat a categorical training image (TI) as a continuous training image for simulation (thus reducing K to 1) when conditioning to no soft data. Or one can treat a categorical TI as a continuous TI only during the pattern partition stage (reducing N P value significantly), and perform normal FILTERSIM simulation with categorical variable, allowing for soft data conditioning. These two options will speed up FILTERSIM simulation, however, at the risk of poorer quality. Also notice that the categorical coding is critical to the final simulations. For instance, the FILTERSIM run in Figure 2(a) was repeated by treating the 4 facies training image of Figure 1 as a continuous training image. Figure 3(a) shows one unconditional realization with the original facies coding: 0 for mud background, 1 for sand channel, 2 for levee and 3 for crevasse. Figure 3(b) shows the realization using the same parameters but switching the codes for channel and levee. Both realizations took only 19 second on the IBM notebook, much faster than 270 seconds for the simulation of Figure 2(a). However, these two realizations have poorer 9

10 quality than the realization of Figure 2(a); and these two realizations differ significantly from each other by only switching the facies coding. Reduce N d : Given an input search template size, one can not decrease the J value. However, it is still possible to reduce the data dimension N d of both data event and the prototypes. Recall that with F filters, the data dimension of training patterns is reduced significantly from J to F K. If we could apply the same filters to both the data event and the prototypes, then their data dimension is reduced from N d to F K. Hence the total number of pixel-wise distance calculations would be reduced significantly from N d N p to F K N p, given that (F K) << N d. Reducing the numbers N p and K values speeds up FILTERSIM simulations, but with the risk of poorer simulation quality. If the pattern classification works efficiently, both data events and prototypes are well characterized by their respective score vectors. Thus reducing data dimension N d value should not result in poorer simulation quality. 4 Score-based Distance Function In order to apply filters to both the local data event dev and any prototype prot, both dev and prot must be fully informed. By definition prototypes are always fully informed, hence filters can be applied directly on these prototypes. With the multiple grid simulation approach (Algorithm 1, Section 2.3), the local conditioning dev passed by a coarse grid to the next finer grid does contain non-informed nodes, hence filters cannot be applied directly on the dev on that next finer grid. In order to apply filters on the conditional dev, one needs to first fill in all the uninformed locations in the dev at the grid level being simulated. The challenge here is how to fill in those uninformed locations. 4.1 Fill-in with dual template One may think of some simple and straightforward ways to fill in the uninformed locations from coarse grid to the next finer grid, such as linear interpolation, kriging or sequential Gaussian simulation conditional to the previously simulated data; however those methods will introduce additional information not consistent with the mp statistics and patterns of training image. 10

11 A much better and consistent approach is that of the dual template concept proposed by Arpat (2004). Recall that in multiple grid simulation, the input search template is expanded at each coarse grid by inserting some empty or null nodes. The coarse search template has a large spatial extension but the same number of nodes as the original fine grid search template. A dual template has the same spatial coverage as the coarse search template, but includes all the nodes of the next finer grid. In Figure 4, the plot (a) gives the input fine grid search template T 1 of size 3 3; plot (b) gives the search template T 2 in the 2 nd coarse grid; and plot (c) gives the dual template T 2D of that 2 nd coarse grid. The 2 nd grid search template T 2 has the same number of nodes as the fine grid search template T 1, but with a larger spatial extension; the dual template T 2D has the same spatial coverage as that 2 nd search template but includes all nodes of the finer grid. Figure 4: Search template in the fine grid 1 and coarse grid 2, and the dual search template in grid 2 When simulating on the 2 nd coarse grid, after selection of a training pattern from the closest prototype class, one can retain from the training image all nodal values of that 2 nd grid dual template (Figure 4.c) and pass that information as conditioning data to the next finer grid. When simulating over that finer grid, the fine search template at any node u is full, hence filters can be applied to it. Recall that each pattern pat(u+h j ) in the training image is uniquely identified by its center location u (also known as pattern ID) in that training image and the corresponding search template T, where h j is the offset vector of the j th node from the center of that search template. Because the dual template T 2D and the coarse search template T 2 have same central location, the dual pattern pat D is also uniquely defined by the same pattern ID u and the dual search template T 2D. Therefore, there is almost no additional CPU cost in extracting a dual training pattern. Thus using the dual template T 2D to fill in the uninformed locations in the next finer grid is extremely fast. Because these fill-in nodal values come from the training image, they are fully consistent with the pattern drawn using the 11

12 coarse template T 2 ; this as opposed to an in-fill done, say, with kriging. The next question is which coarse grid should be filled in using dual template training values if one considers simulation with more than 2 multiple grids. Consider the three following observations: In the original dual template applications by Arpat (2004), the final simulated realizations are much too constrained (looking all similar) if the dual template concept is applied starting from the coarsest grid and if the number of multiple grids is 3. Figure 5 gives a final unconditional categorical FILTERSIM simulation with 3 multiple grids, and its two intermediary results on the two coarse grids. It shows that the final simulation is almost completely controlled by the simulation in the penultimate grid; the simulation on the finest grid appears as just a fine tuning. The same observation can be made from a continuous FILTERSIM simulation. For most 3D simulations, more than 85% of the CPU time is spent on the fine grid simulation that has the largest number of nodes (actually all nodes). (a) 3 rd coarse grid (b) 2 nd coarse grid (c) Finest grid Figure 5: Unconditional categorical FILTERSIM simulation progressing along 3 multiple grids Based on these observations, in our FILTERSIM implementation, the dual template is applied only to the penultimate coarse grid to fill in the data search templates of the final and finest grid. For example, the dual template fill-in approach was applied on the intermediary result of Figure 5(b), resulting in the simulated realizations of Figure 6. 12

13 Comparing Figure 6 to Figure 5(c), there appear only minor differences between the two realizations. Figure 6 has some additional noise. This observation confirms the observation that the fine grid simulation is but a last stage fine tuning. But, in terms of CPU time, the original simulation of Figure 5(c) costs around 90 seconds; while the total CPU time for the result in Figure 6 is only 35 seconds. The potential to speed up the FILTERSIM simulation with the dual template concept is thus very promising. Figure 6: Realization after fill-in on the penultimate coarse grid using the dual template 4.2 Simulation on the finest grid After fill-in with a dual template applied on the penultimate grid, the finest simulation grid is now fully informed with data values. Therefore, during simulation on that finest grid, the filters can be applied directly to any local conditional data event yielding its filter score vector S dev. This process reduces the data dimension from N d to F K with K = 1 for a continuous variable. Similarly, the same filters are applied on all the fine grid prototypes yielding their filter score vectors S prot. During simulation at that finest grid, the prototype prot closest to the local conditional data event dev can now be found based on the difference of their score values. The original pixel wise distance function Eq. 6 is modified as: F K d(u 0 ) = S i dev (u) S i prot(u 0 ), (7) where u 0 is the central location of each class prototype. i=1 13

14 If both the local data event dev and the prototype are well characterized by their filter scores, then Eq. 7 should be able to find the prototype closest to the local dev. Figure 7(a) shows a particular data event (upper left), the closest prototype (upper right) found by the direct pixel-based distance function (Eq. 6), the corresponding data event (low left) after fill-in with dual template, and the closest prototype (lower right) using the score-based distance (Eq. 7). The two prototypes are identical which proves the worth of using the faster score-based distance approach. The search template is of size 9 9, containing J = 81 nodes. However in general, the two closest prototypes found using two different distance functions are only expected to be similar hence providing similar training patterns. For instance, in Figure 7(b), the two closest prototypes found through Eq. 6 and Eq. 7 are similar but not identical. The modified FILTERSIM simulation on the last (finest) grid is described in Algorithm 2. Algorithm 2 Fine grid FILTERSIM simulation with score-based distance 1: During simulation on the penultimate coarse grid G 2, record all sampled pattern IDs along the random path 2: After completing the simulation on the grid G 2, use the dual template to fill in the uninformed locations on the finest grid using the same random path 3: Create score values of the training image using filters 4: Partition the training patterns into classes and retrieve their prototypes 5: Apply filters to all pattern prototypes 6: Define a new random path on the finest simulation grid G 1 7: Relocate hard conditioning data onto the finest grid G 1 8: for Each node u along the random path do 9: Extract the conditioning data event dev centered at u 10: Calculate dev score vector S dev using the same filters 11: Find the prototype prot closest to dev using Eq. 7 12: Randomly draw a pattern pat from prot class 13: Paste pat onto the simulation grid G 1, and freeze the nodes of its inner patch 14: end for 14

15 (a) Identical prototypes (b) similar prototypes Figure 7: Prototypes found by pixel-based distance function and by score-based distance (left: identical prototypes; right: similar prototypes). The gray area indicates the uninformed nodes 15

16 4.3 Boundary processing in finest grid simulation The above proposed approach (Algorithm 2) works well when simulating nodes well within the simulation grid. During the finest grid simulation, when the simulation node is close to the field boundaries, the conditional data event centered at that location might be not fully informed, see Figure 8. Hence one cannot calculate the dev score anymore. Figure 8: Data event centered at boundary location The proposed solution is described as follows: When the data event dev(u) centered at location u over the finest grid has non-data value, find the nearest neighboring node u 2 in the penultimate coarse grid, whose data event dev(u 2 ) obtained by coarse template T 2 is fully informed. Retrieve the training pattern pat(p 2 ) pasted at location u 2, whose pattern ID is p 2. Shift p 2 by a vector (u 2 u) to p. If p is a valid pattern ID, then paste the training pattern defined by template T 1 and ID p onto the simulation grid at location u; otherwise, paste the fine grid pattern pat(p 2 ). Notice that there is no additional cost for finding fine grid patterns pat(p) and pat(p 2 ), because their pattern IDs are already saved in memory (see step 1 in Algorithm 2) 16

17 5 Examples In this section, three examples were run using the score-based (dual template concept) distance function to check its performance in terms of simulation speed and quality. In the first example, the score-based simulation is applied on a 2D continuous grid. The two other examples test 3D FILTERSIM simulation for both continuous and categorical variables. 5.1 Example 1: 2D continuous simulation In this example, FILTERSIM simulation was performed unconditionally on a 2D grid of size , using the 2D continuous training image given in Figure 9(a). The search template size is 9 9 with an inner patch of size 3 3. The partition method is cross partition with 4 bins for parent classification. The number of multiple grids is 3. Figure 9(b) gives the final realization using the pixel-based distance; it took 34 seconds on the IBM notebook. Figure 9(c) gives the result after fill-in with a dual template on the penultimate grid; it took 16 seconds. Figure 9(d) shows one realization using the score-based simulations; it took only 18 seconds. The two realizations with two different distance function are very similar. In terms of speed, FILTERSIM simulation using the score-based distance cuts the CPU cost by half, compared to the pixel-based distance. 5.2 Example 2: 3D continuous simulation For this example, FILTERSIM was run unconditionally with the 3D continuous porosity training image of size shown in Figure 10, which was generated as an unconditional SGSIM realization. The simulation grid size is , the size of search template is and the size of the inner patch is The number of multiple grids is 3, and the pattern classification method is cross partition with 4 bins for parent partition. Figure 11(a) shows a realization using the pixel-based distance, with a total CPU time of 1,255 seconds on the IBM notebook. The number of parent prototypes are 707, 3,688 and 7,389 for the 3 rd, 2 nd and 1 st multiple grids, respectively. The simulation is very slow over the finest (1 st ) grid which involves data event distance checking with more than 7,000 class prototypes. Figure 11(b) gives the result after filling-in with the dual template concept, the total CPU time is 83 seconds. The final unconditional realization using score- 17

18 (a) Training image (b) Pixel-based realization (34s) (c) Fill-in with dual template (16s) (d) Score-based realization (18s) Figure 9: A 2D continuous porosity training image (upper left), unconditional continuous FILTERSIM simulation with pixel-based distance (upper right), fillin with dual template (lower left), and realization with score-based distance (low right). The numbers inside the parentheses are the total CPU time. 18

19 Figure 10: 3D continuous training image (size ) based distance calculation is given in Figure 11(c) at a total CPU expense of only 120 seconds. Compared to the pixel-based distance simulation, the new simulation procedure with score-based distance is extremely fast, with an overall speedup factor above 10. In terms of CPU expense cost for simulating the fine grid only, the speedup factor is greater than 31. This clearly shows that the score-based distance option is very promising for 3D simulations. In such case, more than 90% of the CPU time is used for the fine grid simulation. The ratio of number of nodes in the inner patch over number nodes in the search template is = 0.067, which means that 93.3% of the nodes are re-simulated in the finest grid. This is the main reason for the large speedup factor. Compared to the pixel-based simulation in Figure 11(a), Figure 11(c) reproduces better the high value patterns present in the fill-in result (that is the result of the penultimate coarse grid). The score-based distance results in better pattern reproduction than the pixel-based distance. This is because with the pixel-based distance, there are fewer conditioning data (due to non-data nodes) in the data template, which leads to poorer data conditioning. While in the score-based simulation, the fine grid is fully informed after fill-in thus the data event is always full, providing more conditioning data for prototype searching. 19

20 (a) Pixel-based realization (1,225s) (b) Fill-in with dual template (83s) (c) Score-based realization (120s) (d) Score-based realization (slice view) Figure 11: 3D unconditional continuous FILTERSIM realization using pixelbased and score-based distance functions 20

21 Figure 12 shows the variograms of the pixel-based realization of Figure 11(a) and of the score-based realization of Figure 11(c) along both X and Y directions. It is seen that these two simulations have almost the same X and Y variograms, which means that in terms of 2-point statistics simulation with score-based distance is as good as the simulation with pixel-based distance, but at a much smaller CPU cost. (a) Variogram in X direction (b) Variogram in Y direction Figure 12: Variograms of pixel-based realization (Figure 11(a)) and score-based realization (Figure 11(c)) in both X and Y direction 5.3 Example 3: 3D categorical simulation In this example, FILTERSIM is run with the 3D four-facies categorical training image shown in Figure 13(a), which was created by stacking vertically the same 2D four facies training image of Figure 1. Both the training image and the simulation grid are of size The search template size is with an inner patch of size The pattern classification method is cross partition with 4 bins for parent partition. The realizations with pixel-based distance and with score-based distance are given in Figure 13(b) and Figure 13(d), respectively. The intermediary result filled-in with dual template values is given in Figure 13(c). It is seen that the two realizations are reasonably similar. The two disconnected channels at the top of Figure 13(b) (inside the white ellipse) are now connected in Figure 13(d). Both realizations are consistent with the fill-in result shown in Figure 13(c). 21

22 (a) Training image (b) Pixel-based simulation (10,673s) (c) Fill-in with dual template (d) Score-based simulation (2,880s) Figure 13: 3D unconditional categorical FILTERSIM simulations with pixelbased and score-based distances 22

23 In terms of CPU expense, the pixel-based simulation took 10,673 seconds, while the simulation with score-based distance function took only 2,880 seconds. The overall speedup factor is about 3.5, and the simulation acceleration for the finest grid simulation is more than 50. The total CPU times for score calculation is about 290 seconds, with 53, 93 and 146 seconds for each multiple grid. The total CPU time for pattern classification is about 820 seconds, and 29, 213 and 517 seconds for each coarse grid. Notice here that the ratio of number of nodes in the patch template over that in the search template is = 0.243, which means that only 73.7% of grid nodes are re-simulated in the finest grid. 6 FILTERSIM Applications with Region and Scorebased Distance The region concept (Wu, 2007, in the same report) can be used together with the score-based distance developed in Section 4. In this example, FILTERSIM is run with the region concept and a score-based distance function. The simulation grid is of size , it is divided into 7 pseudo regions (Figure 14) with each region associated with different rotation and scaling constraints. The region settings are tabulated in Table 1. Figure 14: Seven pseudo regions for local rotation and affinity constraints ( N means scaling the original geo-bodies by a factor N; θ 0 means rotating the geobodies by θ degree) 23

24 region index affinity (f x, f y ) (2, 2) (1, 1) (0.5, 0.5) (1, 1) (0.5, 0.5) (1, 1) (0.5, 0.5) rotation angle Table 1: Affinity factors and rotation angles for seven pseudo regions FILTERSIM is run unconditionally using the 2D continuous training image given in Figure 9(a). The search template size is 11 11, the inner patch size is 3 3, and the number of multiple grids is 3. Cross partition is used for pattern classification, using 4 bins for the parent partition. In Figure 15, plot (a) gives the simulation on the penultimate coarse grid, with the uninformed nodes in gray; plot (b) is the intermediary result after fill-in with dual template data: note the artifact of some small patches; plot (c) is the final simulation using the score-based distance function; and plot (d) gives the final simulation using the pixel-based distance function. Continuities of high value zones across the region boundaries is reasonably well preserved in both final realizations. The final score-based simulation of Figure 15(c) is more consistent with the simulations on the penultimate coarse grid or after fill-in. The high value areas in region R2 of Figure 15(a) are not preserved in the realization with pixel-based distance of Figure 15(d): notice the low value area inside the black ellipse. This is because the previously simulated values in the coarser grid are re-simulated in the next finer grid, and those values could be replaced by a totally different pattern if using pixel-based distance on local data events not fully informed. This rarely happens for simulation using score-based distance because the local data event is always fully informed. As for CPU expense, it took 46 seconds to complete the simulation over the penultimate coarse grid in the IBM notebook, and only 1 additional second to fill in the uninformed locations. The total CPU time for simulation on the final fine grid using score-based simulation is 57 seconds, only half of the CPU time cost (109 seconds) it took for the final simulation using the pixel-based distance. 24

25 (a) Simulation on penultimate grid (46s) (b) Fill-in with dual template (47s) (c) Score-based simulation (57s) (d) Pixel-based simulation (109s) Figure 15: 3D unconditional FILTERSIM simulations with pixel-based and with score-based distances (the white lines are the region boundaries, and the numbers inside the parentheses are the total CPU time) 25

26 7 Conclusions From the previous example runs and discussions, the following conclusions can be drawn: FILTERSIM works with both continuous and categorical variables, and is able to capture complex patterns and structures from a training image. The dual template concept can be used to fill in uninformed locations of the penultimate coarse grid. The filters can then be applied to both the local conditioning data event (after fill-in) and the prototypes. The prototype closest to the local data event can be found based on a score distance. The fine grid simulation with the score-based distance is extremely fast. In 3D simulation, the speedup factor can be better than 5. Simulation with score-based distance is equally good as the simulation using the pixel-based distance function, and provides final realizations more consistent with the simulation obtained on the penultimate coarse grid. The new score-based distance function can be combined with user-defined filters, K-Mean partition method and region simulation concept. Acknowledgments This work was supported by Stanford Center for Reservoir Forecasting (SCRF) and Schlumberger Doll Research (SDR) through the summer internship program. Special thanks go to Dr. Claude Signer and Dr. David McCormick from SDR, for their supports and suggestions. References Arpat, G. B.: 2004, Sequential simulation with patterns, Phd thesis, Stanford University, Stanford, CA. Bonet, J. S. D.: 1997, Multiresolution sampling procedure for analysis and synthesis of texture images, Computer Graphics 31(Annual Conference Series),

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