Fast stochastic simulation of patterns using a multi-scale search

Transcription

1 1 Fast stochastic simulation of patterns using a multi-scale search Pejman Tahmasebi and Jef Caers Department of Energy Resources Engineering Stanford University Abstract Pattern-based spatial modeling relies on the pattern as the basic modeling component for generating geostatistical realizations. These methodologies recognize that working with the unit of a pattern aids both data conditioning (an advantage shared with pixel-based methods) and better pattern reproduction (an advantage shared with Boolean models). Since the first paper on conditional simulation with patterns by Arpat and Caers (simpat, 2005, 2007), a series of methodologies for simulation have been developed attempting to overcome the main limitation of simpat: CPU time and variability of the generated realizations. In this paper we extend on the ideas of ccsim, a method for pattern simulation relying on the cross-correlation as measure for pattern similarity. The extension lies on the use of a multi-scale representation of the training image along a pattern projection strategy that is markedly different from the traditional multi-grid employed in current methodologies. In this multi-scale representation, we transform the high-resolution training image into a pyramid of consecutively up-gridded views of the same training image. This pyramid allows for rapid search of patterns to interlock with the previously simulated patterns. A second advantage of this multi-scale view lies in data conditioning. Using multi-million cell examples with sparse as well as dense datasets, we illustrate how this method can simulate such large grids in a matter of seconds. Introduction Since more than two decades from its inception, research in multiple-point geostatistics has generated a prolific development in new algorithms whose aim is to capture and reproduce patterns from training images, anchored to local hard and soft data. Regardless of these developments, the wide-spread application in practice is still lacking, even in reservoir modeling, where most of these techniques have received considerable attention. We conjecture that, next to the availability of training images (see discussion in Honarkhah and

2 2 Caers, 2012), two main reasons lie at the origin of this: quality of pattern reproduction and CPU time. In this paper, we present a method that can simulate multi-million cell grids in a matter of second, and, provide a quantum improvement in terms of pattern reproduction over existing algorithms, while maintaining accuracy in hard data conditioning. In his original and seminal algorithm, Strebelle (2002) used a search tree in a single normal equation simulation (snesim) to accelerate the calculation of local conditional probabilities rendering the application of multiple-point geostatistics practically possible. snesim has considerable memory requirements for simulating multiple-facies, although this issue has been addressed in for example Straubhaar et al. (2011). However, these solutions do not address the difficulty in reproducing thin features or very low proportion patterns (curvi-linear shale barriers, fractures etc). Pattern-based methods, proposed originally by Arpat and Caers (2005, 2007), aim to overcome these RAM limitations, as well as improve pattern reproduction quality. These methods basically consider stochastic simulation as a randomized puzzle where patterns are puzzle pieces that need to interlock (and be constrained to data) to generate a realization. In the same context, Zhang et al. (2006, filtersim) provided a filter-based selection of patterns to speed-up the time-consuming search of simpat, however at the cost of lesser pattern reproduction. Honarkhah and Caers (2010, dispat) relied on a distance-based pattern clustering to rapidly search for patterns without the approximations made in filtersim. Tahmasebi et al. (2012) recognized that using a cross-correlation as a distance as well as simulating along a raster path (instead of random path) provided great improvements over filtersim. In this special issue, this ccsim (cross-correlation simulation) is indeed quantitatively proven to perform better than dispat (see Tan et al, this issue). As an extension, Tahmasebi and Sahimi (2012) used a frequency domain to provide more acceleration on large grids. In this paper, we present a new pattern-based method that achieves two goals 1) extremely fast CPU performance and 2) improved pattern reproduction (an accompanying paper by Tan et al, this issue, goes in great detail to provide a quantitative assessment of this statement). Our proposed method is cast on the ccsim framework, an algorithm that will be reviewed first. To accelerate ccsim, we propose a multi-scale method for searching and pasting patterns in the training image. Based again on a multi-scale representation of the training image and simulation grid, a new method for hard data conditioning is established. A comparison (on CPU and pattern reproduction) by means of extensive, categorical, continuous, conditional and unconditional simulation on multi-million cell grid is performed between the existing ccsim and the new ms-ccsim (ms=multi-scale).

3 3 A review of cross-correlation-based simulation (ccsim) Unconditional ccsim ccsim extends on the idea of simpat in two ways: 1) the similarity between any pattern and the existing data-event is based on a cross-correlation over a smaller overlap region and 2) a raster path (similar to Daly, 2005; Stien & Kolbjornsen, 2011) is used instead of a random path. Figure 1 and 2 explain the basic concepts of the algorithm. Following a raster path starting at the origin, Figure 1A, a random pattern is selected from the training image, see Figure 1E. Next, the template (which is always square) is shifted to the right, Figure 1B, leaving a small area of overlap (size h, see also Figure 2). It is the overlapping area that, just as in a real puzzle, allows deciding on what the best interlocking pattern is. The criterion for distance used is the crosscorrelation between the first pattern pat 1 and any other pattern pat k in the training image summed over the overlap areas = d( pat, pat ) pat pat (1) 1 k 1, i k, j i A j A A pattern is a vector of grid cell values pat = { pat, pat, K, pat } n T = number of pixels in the template T (2) k k,1 k,2 k, n T A threshold t allows selecting any pattern in the training image that has a distance less than t in Eq. (1) (otherwise one would select the pattern in the training image right next to pat 1 ). Notice that the calculation in Eq (1) is over the overlap area A only, not over the entire pattern (as in dispat), which provides a considerable speed-up. After the first line is completed, most of the pattern cross-correlation calculations involve the geometry shown in Figure 2A, where the shaded area is used to find interlocking patterns (tantamount to finding puzzle pieces with only a bottom and left interlocking component). The above algorithm constitutes the basic idea for unconditional simulation. The randomization is mostly due to the selection of the first pattern and the threshold t. The algorithm requires specification of three parameters: the size of the template (n T ), the size of the overlap area A (h) and the threshold (t). The size of the template is determined in the same way as in dispat, namely using the so-called elbow plot (pattern entropy versus template size). Tahmasebi and Sahimi (2012) found through sensitivity analysis that a reasonable choice is to take h between 1/5th and 1/6th of the template size T. In similar vein, a reasonable choice is for t to be 20% larger than the minimum distance in Eq (1) or

4 4 t = 1.2 min patm, ipatk, j k, m i A j A (3) Figure 1: concept of raster path and overlap region. Figure 2: (A) configuration of template, overlap area A, (B) when hard data are present, (C) splitting of the template.

5 5 Conditional ccsim For conditional simulation, a two-step search similar to simpat and dispat is used. Consider the situation in Figure 2B. First, one collects all the patterns k into a set of patterns: set = { pat pat t k : d( pat previous, pat < k) < t} (4) namely, find the set of patterns that interlocks the previously simulated patterns in Figure 2B. Then, for any given hard data within the template T (see Figure 2B), one selects a pattern from set pat<t those that match the hard data in the data event dev T. It is likely (in particular with dense hard data and inconsistent training image) that no such pattern can be found. In that case, the template T is split into a smaller template, see Figure 2C. The two-step operation is now repeated with this smaller template (similar to dropping far away hard data in snesim). This operation is continued until a matching pattern is found. Note that conditioning is different from Markov mesh models (MMM, see Kjønsberg & Kolbjørnsen, 2008), where an additional kriging is needed to account for any hard data ahead in the raster path. In MMM one simulates pixel values, not patterns. Simulating with patterns allows the algorithm to look for hard data ahead of the raster path. The newly proposed multi-scale approach enhances this looking-ahead for hard data even more, as explained next. Multi-scale modeling Multi-grid vs multi-scale simulation In many MPS algorithms (e.g. snesim, dispat), a multi-grid simulation is used to enhance longrange pattern reproduction. The multi-grid idea starts from a sparse but coarse grid and then successively simulates on progressively finer grids, see Figure 3. It was originally introduced in the context of covariance-based models (Tran, 1994). Later, the same idea was introduced to enhance the reproduction of higher-order statistics borrowed from training images (Caers and Journel, 1998). Simulating on multiple-grids is not without its issues. First, since the first grid to be simulated is coarse and sparse, there is the issue of how to deal with irregularly spaced or clustered data. Some method of data relocation or allocation needs to be introduced and this may result in poor conditioning (Honarkhah, 2011), locally biased pattern reproduction or reduction in uncertainty over multiple realizations (Caers, 2012). The second issue is that of global pattern reproduction. During coarse grid simulation, patterns are constructed from a much sparser template. This sparseness is a source of randomness in the patterns, which may induce

6 6 randomness in the generated realizations at the coarsest grid level that is maintained (by construction) at the finer scale grid simulation, leading to poor pattern reproduction quality. In this paper we propose using an entirely different approach based on resolution theory, common to many methods of mimicking the human visual system (a similar approach was presented in Honarkhah, 2011). Figure 3: simulating on a multi-grid. Starting from a coarse grid and sparse template, the simulation grid is consecutively refined until the final resolution. Multi-Scale Theory Our visual system makes measurements by integrating information around us, either small objects or large scenes. The eye does not limit itself to small or large objects. It has the flexibility to gauge its surroundings without any prior information on what to expect, whether the object is big or small, whether it is square or a line. The visual front-end of the eye has been regarded as a multi-scale geometry engine" (Koenderink, 1984). When an observation is made in space, all information is transmitted to the brain, as not simple one single image, but as a stack at different scales. This is the basic principle in scale-space theory that aims at explaining how the visual front-end performs (Lindeberg, 1994). We propose a multi-scale approach based

7 7 on the psycho-visual model of human optical system. However, pattern-based simulation algorithms are not able to handle all resolutions simultaneously due to the presence of local hard data. Instead, we propose to maintain separate pattern resolutions in the training image. In the same way as humans analyze a structure by first blurring the features to get a big picture representation of the model, and then focusing on details and fine-scale patterns, we search the training image starting from the coarsest resolution and follow a top-down approach. In the next section these principles are implements in the ccsim algorithm. Unconditional multi-scale cross-correlation-based simulation (ms-ccsim) In a multi-scale approach, we first up-grid the training image into a pyramid of multi-resolution views of the same training, see Figure 4. The finest resolution training image is the original training image and termed ti 0. We use bi-cubic interpolation (Lekien & Marsden, 2005) to obtain values for the lower-resolution training images ti 1,..ti G-1, where G is the total number of resolutions considered. The template size for up-gridding remains the same for all resolutions (in 2D 4x4, in 3D 4x4x1 to maintain vertical variability). In case of binary variables, Otsu s thresholding method (Otsu, 1979) is used to turn the continuous valued bi-cubic interpolations back to binary variables (categorical values are basically a vector of binary variables), see Figure 5. Figure 4: procedure for up-gridding by tri-cubic interpolation.

8 8 Figure 5: example application of up-gridding a binary training image using tri-cubic interpolation and Otsu s thresholding. The unconditional ms-ccsim proceeds similarly to the unconditional ccsim, except that the training image as well as overlap area A is now upgridded. Based on the multi-scale representation of the training image, we propose a multi-scale patterns search as depicted in Figure 6. One starts from the lowest resolution training image tig-1 and finds a pattern in tig-1 that interlocks the previous simulated patterns at that resolution. Next, the same location in the next scale g-2 is determined and a search window (red box in Figure 6) is used having the following size Search box size S = 2 ( template size T for grid g-2 Overlap h for grid g-2 ) Note that the template size T as well as overlap size h doubles when going from a resolution g-i to a resolution g-i-1. The idea of using this search box is that there is no longer a need to search for a matching pattern over the entire training image ti g-2, the search is limited to this smaller neighborhood. This is continued until the final resolution ti 0. The pattern found at this resolution is then pasted on the simulation grid.

9 9 Figure 6: principle of the multi-scale search: starting from a coarse resolution version of the training image, a matching pattern is search top-down in the pyramid. Conditional ms-ccsim - Categorical variables In the traditional multi-grid approach, Strebelle (2000) uses a data relocation approach, where the hard data are relocated to their closest grid node in the multi-grid realizations. This ensures that hard data are taken into account during coarse grid simulations, and as a result, the finer grids are informed about the larger scale statistical dependencies of hard data. This works for simple cases, but with locally dense data, conflicts arise, and as a result, hard data are dropped at coarse grid simulations leading to poor conditioning. In this paper, we present a method for hard data conditioning that is adapted to the multi-scale representation of the training image presented above. First, we would like to stipulate very clearly the nature of a multi-resolution grid g. As in any geostatistical approach we basically work with a corner-point representation for each multi-resolution grid (see Deutsch and Journel, 1992), meaning that at every corner of a grid, a cell of certain volume (often the same as the volume of any hard data, which can be tiny) is located, regardless of the resolution g, see Figure 7. The volume v of this cell is the same for all resolutions g.

10 10 Figure 7: Definition of the geostatistical grid for simulating a property with volume v on the primal mesh. Figure 8: various possible configurations of hard data within the dual mesh. (A) simplest case of not more than one hard data per dual mesh grid cell, (B) case requiring re-assigning hard data based on distance to the nearest location. (C) case where multiple hard data cannot be assigned to the nearest location: in that case we assign that category with the highest frequency. Consider now a multi-resolution grid g with a number of possible configurations of hard data. The simplest situation arises in Figure 8A, where each cell in a dual multi-resolution mesh g contains no more than one hard datum. In that case any hard data (if present) is assigned to the

11 11 geostatistical cell in the center. If more than one hard datum is present in a single cell of the dual mesh, then hard data are assigned based on the distance between the geostatistical cell location and that hard datum. If however, too many hard data are present and not all hard data can be assigned, then a frequency distribution of the remaining hard data is calculated and the geostatistical cell is assigned the category with highest frequency. In fact the latter operation is tantamount to up-gridding the hard data to the multi-resolution grid. - Continuous variables The pattern conditioning in the case of continuous-valued training image is considerably more challenging than in the binary case, simply because one rarely finds an exact match between a training image pattern and a hard data event. In addition, when applying the multi-scale approach, the original training image is averaged into progressively smoother version of that training image. Hence, it becomes even less likely, at the coarse resolutions, to find a pattern that matches the hard data-event. To alleviate this problem, we apply a histogram transform, on each multi-resolution grid of the original training image, the target histogram being the original training image histogram. The continuous data are then categorized into bins and the data conditioning follows the same methodology as for the categorical case. Examples Unconditional simulation We present various examples of training images with each three unconditional realizations generated from them. Some of the parameters used are summarized in Table 1. Table 1: Summary of template size, overlap size, number of multi-scale levels and CPU time for different examples in this paper. Example Template Number of CPU time (s) h for g=2 size for g=2 resolutions g (Matlab) 2D binary channel D cobbles D channel with crevasse D continuous D binary fracture network D tank data

12 12 Figure 9 shows a 1000x1000 binary channel case: the difficulty lies in the reproduction of long connected sinuous channels. Figure 10 shows a gray-level training image of cobbles: the difficulty lies in reproducing closed curved features. Figure 11 shows a 3D channel training image with crevasse splays: the difficulty lies in creating connected channels in 3D. Figure 12 shows a 3D training image generated with sequential Gaussian simulation (using a Gaussian variogram): the difficulty lies in the smooth spatial continuity that is difficult to reproduce with pattern-based method (filtersim and dispat realization often display an artificial patchiness related to the template size). Figure 13 shows a training image extracted from a tank experiments (Paola et al. 2001; Connell et al., 2012). Tank experiments are laboratory scale experiment simulating sediment deposition in realistic circumstances. After the experiment is completed, the box of sediments is cut into consecutive slices of which photographs are taken. One such photograph is show in Figure 13 (top). From the series of consecutive slices we then constitute a single 3D training image for testing geostatistical algorithms. The patterns seen in these experiments are analogs of real depositional features (notice the scour fill in Figure 13), hence can provide excellent test cases for geostatistical algorithms: the difficulty here lies in reproducing complex sedimentary features represented by a continuous variable. In this particular case we see an excellent reproduction of scour features in Figure 14. Figure 9: 2D binary channel training image on a grid with three unconditional realizations.

13 13 Figure 10: cobble stone training image on a grid with three unconditional realizations. Figure 11: 3D unconditional simulation on a complex 3-facies system.

14 14 Figure 12: Three unconditional realizations for the shown continuous 3D training image. Figure 13: Extracted training image from tank data.

15 15 Figure 14: Three realization by using tank experimental data to construct a training image on a grid. Conditional simulation Figure 15 describes a case for testing the conditioning accuracy of the ms-ccsim. A simple case with two hard data (one channel datum and one background datum) is shown. In this case we can perform rejection sampling, meaning: we perform an unconditional ms-ccsim realization and accept it as it matches the two data. We generate 100 realizations using this rejection sampler. We also generate 100 realizations using the conditional ms-ccsim, of which three are shown in Figure 15. We calculate the ensemble average of both sets of realizations. The ensemble average of the rejection sampler compares well with the conditional method.

16 16 Figure 15: Conditional simulation with two closes hard data. The result of rejection sampling and ensemble average of 100 realizations in a zoom-in view are shown. Using the same training image we set up a case with dense conditioning data: first we generate an unconditional realization and extract a dense data set. Then we generate 100 conditional realizations as shown in Figure 16. The ensemble average shows that conditioning works without creating discontinuities near the data. Figure 17 shows a case with dense continuous data (similar to the cases used in Dimitrakopoulos et al, 2010). The ensemble average shows reasonable conditioning but with some discontinuity in the channel geometry (see enclosed circle). Conditional continuous simulation remains a problem for pattern-based methods as it is difficult to interlock continuous puzzle pieces while at the same time constraining to data.

17 17 Figure 16: Conditional simulation with dense hard data. Reference image which is used for hard data picking, three different realizations, and ensemble average of 100 realizations are shown. Figure 17: Conditional simulation with continuous training image. The hard data is selected in a regular grid pattern from a reference image. One realization and ensemble average are shown.

18 18 Figure 18 and 19 show two challenging cases of conditioning to thin features. Given the complexity of the patterns, the conditioning data as well as the size of the models, standard MPS algorithms fail to produce meaningful results. In Figure 18, we have a training image generated using an ant tracking algorithm on seismic data (Pederson et al. 2002). The result is a delineation of lineaments (faults/fractured zones) that are deemed relevant for an area where no such seismic data is available. In this area however, wells have been drilled and fractured zones interpreted from well-bore an imaging data, as shown in Figure 18 (red color indicated presence of fractures, blue indicated absence). We generate 50 conditional realization of which one is shown in Figure 18. Figure 18: Fracture training image generated using an ant tracking algorithm on seismic data (upper left), hard data (upper right) and one realization (down). To assess the conditioning accuracy, we calculate the ensemble average, shown in Figure 19. A high probability indicated presence of fractures. The zoom shows clearly that the conditional simulation extrapolate from the interpreted fracture zones using the pattern of the training image in Figure 18. However, we can still observe some patchiness in the ensemble average related to the template size; hence some room for improvement is still available. Figure 20

19 19 shows a process-based model of a sand/shale sequence consisting of shale drapes. In reservoirs such shale drapes may compartmentalize the system and affect performance considerably. Figure 20 shows that these shale drapes have a complex 3D geometry that is impossible to reproduce with existing MPS techniques. A single conditional model, conditioned to the 8 wells is shown in Figure 20. The shale drapes intersect at the location where shale is presence in the well, with reasonable, but not perfect, reproduction of the shale drape continuity. Figure 19: Ensemble average of 50 realizations on a grid. Algorithm performance In this section the overall CPU performance of ms-ccsim and ccsim is compared. First, Table 1 shows absolute CPU times for the ms-ccsim algorithm for the cases shown above. All models are generated with less than 1 minute CPU, some only a few seconds, even for multi-million cell models. This performance is several orders of magnitude faster than any existing MPS code. Next, the impact of the multi-scale approach is illustrated in Figure 21. According to this analysis, it is clear that by increasing the size of training image and simulation grid, the superior performance of ms-ccsim through multi-scale search becomes evident.

20 20 Figure 20: A process-based model of a sand/shale sequence consisting of shale drapes (top), hard data (left down) and one conditional realization (right down). Conclusions There is natural logic to use pattern-based approaches for simulating large grids. In multimillion cell models, it is simply inefficient to simulate every single grid cell one at a time whether using a random or raster path. The result is that pixel-based models are too CPU demanding for such cases and secondly that the pattern-reproduction is unacceptably poor. In this paper we build further on the idea of simulating patterns, more specifically on the ccsim algorithm. The main contribution lies in using a multi-scale search for finding a pattern interlocking optimally with previously simulating patterns along a raster path. The multi-scale search provides two main advantages: it speeds up the algorithm such that multi-million models can be simulated in seconds and it improves conditioning to hard data. A large set of complex example of diverse nature testifies of the ability of the new ms-ccsim algorithm. Our further work will focus on improving the pattern reproduction for continuous cases as well as investigate ways of conditioning to secondary data.

21 21 Figure 21: CPU-time comparison for (a) 2D examples and (b) 3D examples. Acknowledgement We would like to thank Dr. Tomomi Yamada of JAPEX for donating the datasets of Figure 18. We would like to thank Dr. Hongmei Li of ExxonMobil for donating the shale drape dataset of Figure 20. We would also like to thank Dr. Chris Paola of the University of Minnesota for the tank experiment data and the many interesting discussions about the use of geostatistics in modeling sedimentary systems. References Arpat, G.B., and Caers, J., 2005, A multiple-scale, pattern-based approach to sequential simulation, Geostatistics Banff 2004, Quantitative Geology and Geostatistics, 14(1): Arpat, G.B., and Caers, J., 2007, Stochastic simulation with patterns, Mathematical Geology, 39(2): Bar-Joseph, Z., El-Yaniv, R., Lischinski, D., and Werman, M., (2001, Texture mixing and texture movie synthesis using statistical learning, IEEE Transaction on Visual and Computer Graphic, 7(2): Caers, J., and Journel, A.G., 1998, Stochastic reservoir simulation using neural networks trained on outcrop data, In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, New Orleans, LA, pp Connel, S.D., Wonsuck, K., Smith, G. and Paola, C., 2012, Stratigraphic Architecture of An Experimental Basin With

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