Reservoir Spatial Heterogeneity Modeling Through Pattern-Based Stochastic Simulation

ECMOR XIV – 14th European Conference on the Mathematics of Oil Recovery Catania, Sicily, Italy, 8-11 September 2014

Mo P33Reservoir Modeling through Fast Wavelet-basedStochastic SimulationH.M. Mustapha* (Schlumberger), S. Chatterjee (National Institute ofTechnology Rourkela) & R. Dimitrakopoulos (Department of Mining andMaterials Engineering, Mc)

SUMMARYStochastic simulation of complex geology is addressed through discrete wavelet transformation (DWT)that handles multiscale spatial characteristics in datasets and training images. The simulation of theproposed approach is performed on the frequency (wavelet) domain. A multiscale, multipoint simulationalgorithm is proposed in this paper in which the scaling image and wavelet images across the scale aresimulated jointly. The inverse DWT reconstructs simulated realizations of an attribute of interest in thespace domain. The proposed algorithm reduces the computational time required for simulating largedomain as compared to spatial domain multipoint simulation algorithm since the simulation is performedin the wavelet domain in which numbers of nodes to be simulated are significantly less as compared tospatial domain nodes. The algorithm is tested with an exhaustive dataset using unconditional simulation intwo-dimensional data set. The realisations generated perform well and reproduce the statistics of thetraining image. The study conducted comparing the spatial domain filtersim multiplepoint simulationalgorithm suggests that the proposed multiscale, multipoint algorithm generates equally good realisationsat much lower computational cost.


Introduction

Stochastic simulation of complex geologic environments involves reconstruction of a geologic environment from a finite number of observations in space and constraining data, such as geophysical and geologic interpretations. This input information may be expressed at different scales. Mineral deposits and petroleum reservoirs are examples of environments where natural complexity controls the spatial attributes of features of interest. Conventional spatial random field models with their first- and second-order spatial statistics are limited in their ability to adequately model spatial complexity (Journel 2007). To address the limits of conventional approaches, higher-order spatial statistics - namely multiple-point (MP) models - were introduced in which templates and training images replace conventional two-point variograms (Strebelle 2002; Zhang el al. 2006). MP methods include a sequential simulation framework and the use of single normal equations (Strebelle 2002), filters and resulting filter scores (Zhang el al. 2006), patterns (Apart and Caers 2007), distance based (Honarkhah and Caers 2010), or wavelet based (Chaterjee et al. 2012). These methods are based on finding repeatable patterns and their probability of occurrence in a training image representing an analog of the studied geologic environment. The use of discrete wavelet transformation (DWT) in stochastic simulation has advantages relative to other stochastic simulation methods. DWT identifies dominant scales in statistically heterogeneous data. The DWT convolves scaled and translated functions generated from a basis function, with the input data to separate local and global information at decreasing scales (resolutions). Joint modeling of local and global information assists in dealing with non-stationary data. Related to the use of DWT in geostatistical modeling is the contribution of Tran et al. (2002), who unconditionally simulated a two-dimensional random process using the Haar wavelet (Mallat 1998). This study has two weaknesses: hard data that are external to the DWT are conditioned, and simulation does not consider dependency of the wavelet coefficients across scales. It has been shown in the technical literature that wavelet coefficients are not independent (Vannucci and Corradi 1999). Gloaguen and Dimitrakopoulos (2009) present a direct conditional cosimulation method that incorporates inter-scale dependency of wavelet coefficients. There are two major limitations in the work of Gloaguen and Dimitrakopoulos (2009): (i) cosimulation of multiple scales for three-dimensional data over a large domain is computationally intensive, and (ii) the wavelet images obtained using conditional cosimulation with multi-scale statistics does not guarantee the coefficients of the same wavelet image at different scales and at the same spatial location will have corresponding high and low values. Wavelet images have mostly high-frequency signals and cosimulating with variogram models that have high nugget values means the frequency hierarchy at different data scales cannot be maintained. This leads to the blurring of generated simulations after the inverse discrete wavelet transformation (IDWT). In this article, a new approach to handle the two limitations of Gloaguen and Dimitrakopoulos (2009) work is proposed. In the proposed algorithm, joint simulations of scaling image and wavelet images across the scales are performed using a multi-point simulation algorithm. The major advantage of this approach is that only a small number of nodes in the wavelet domain need to be simulated, which requires less computational time than any multi-point simulation algorithm in spatial domain. Since, the wavelet coefficients are simulated by pattern matching like other pattern-based multi-point algorithms, wavelet images at different scales will contain the same local highs and lows, which will ultimately reduce the blurring effect on the final image.

Method

The algorithm proposed herein for simulating spatial data in the wavelet domain by jointly simulating the scaling image and wavelets images at different scales. The algorithm has six main steps: (a) wavelet decomposition of training image; (b) generating wavelet domain pattern database; (c) clustering the pattern database and generate class prototype; (d) similarity measure measures between


conditioning data and class prototype to simulate multi-scale data jointly; and (e) up-scaling using inverse wavelet transformation to obtain a spatial domain simulated map of the primary variable. In first step, the training image ti , where ( )ti u as a value of the training image at location u, tiu G

and tiG is the regular Cartesian grid discretizing the training image, is decomposed using selected

wavelet basis function. Throughout this paper, Haar wavelet basis functions are used. The scaling and

wavelet images 1ja and image d-3for 7,...,2,1 image, d-2for 3,2,1,1 iiijw , at scale j-1 can

be experimentally calculated by taking inner products.

The wavelet domain pattern database is generated by scanning the scaling image Ja using a given a

template t. If )(vJa is a value of the scaling image Ja , where J

JGav and J

JGa is the regular

Cartesian grid discretizing the scaling image, then )(vta indicates a specific multiple-point vector of

)(vJa within a template t centered at node v , that is

)}(),....,(),...,(),({)(tnα21t hvhvhvhvv JJJJ aaaaa

(1) where, the αh vectors are the vectors defining the geometry of the tn nodes of template t and

},...,2,1{ tn . The vector 01h represents the central location v of template t. The pattern

database is then obtained by scanning the whole scaling image Ja using template t and stored the

multi-point )(vta vectors in the database.

The aim of this paper is to jointly simulate the scaling image and wavelet images at different scale.

Since, wavelet coefficients are nested in the structure and the value of scaling image )(vJa at

location v is directly related to the values of wavelet images )(vijw at different

scales 1,..,1, JJj ; therefore it is easy to keep track of wavelet coefficients at different scale

corresponding to scaling image coefficient )(vJa at location v . The template is placed at location v

of the scaling image and the multiple-point vector of )(vJa is extracted from scaling image Ja

along with the corresponding wavelet coefficients 1,..,1,),(1 JJjij vw . The updated pattern

vector, which termed now onwards as multi-scale pattern vector, for location v at scaling image can be represented as:

1,..,1,};),(),....,(),...,(),({)( JJjwaaaaa ijJJJJ

MS

tnα21t hvhvhvhvv (2)

where, image d-3for 7,...,2,1 image, d-2for 3,2,1 ii . The multi-scale pattern database,

MSpatdbt is then obtained by scanning the Ja using template t and stored the multi-scale multi-

point vectors )(vtMSa in the database.

During the simulation, the best matched multi-scale pattern corresponding to conditional coarse-scale

multi-point data (template) will be searched from the multi-scale pattern databaseMSpatdbt . Since,

the number of patterns (tPatn ), and the dimension of the patterns


( scale ofnumber is dimension; image trainingis ;11 Jddn J t ) in

MSpatdbt are relatively

large, to search the best matched multi-scale pattern from MSpatdbt is computationally time

consuming. To reduce the computational time of multi-scale joint simulation MSpatdbt is classified

into number of classes. During simulation, the distance from the conditioning data to the class is calculated. Therefore, representatives (prototypes) of each class are calculated from each class.

Instead using whole 11 Jdnt dimension ofMSpatdbt for classification, in this paper only first

tn dimension, which is multi-point information of scaling image, are used. This way computational

time of MSpatdbt classification can be reduced. The multi-scale, multi-point patterns MSat in pattern

database MSpatdbt are classified into different classes using first tn dimension for each pattern.

Here, the classification of the database coefficients is performed utilizing the k-means clustering technique. A class representative (prototype) is calculated by averaging the class coefficients. After

classifying the MSpatdbt and prototype calculation, simulation of multi-scale patterns was carried

out. During simulation, the similarity between the conditioning data event and the prototypes of the classes are carried out. In this paper, a sequential simulation algorithm (Goovearts 1998) is used for multi-scale simulation. At each visited node in coarsest scale, a conditioning data event is obtained by placing the same template t used in the multi-scale pattern generation, centering at the node to be simulated. The similarity between the conditioning data and prototypes of classes are calculated by a distance function. It is noted that, since jointly the multi-scale simulation were used, the all nodes in the wavelet images at different scale will be simulated when the coarsest-scale scaling image is simulated. Once the scaling image and all wavelet images have been simulated, the inverse DWT is applied to the simulated coefficients and the space-domain simulated map obtained.

Figure 1 Schematic diagram of multi-scale simulation method proposed in this paper.


Fig. 1 presents the schematic diagram of the sequential multi-scale simulation method proposed in this paper with 3 x 3 template for the first random visiting node. This process is repetitive until all the nodes within the coarsest-scale scaling image are visited.

Example of 2-D unconditional simulation

The proposed algorithm is validated by simulating known continuous two-dimensional. The exhaustive data sets are obtained from different sources. For wavelet decomposition, the Haar basis functions are applied for all cases unless otherwise specified. The results of the proposed method are compared with the corresponding filtersim results. The training image is a blasted rock image from a mining face (Fig. 2(a)). There is some inter-rock void space presented by low pixel values. The size of the training image is 624 x 460. The scale size, template size at coarsest scale, and cluster numbers are selected as 3, 9 x 9, and 35, respectively. The template size for the filtersim algorithm is 17x17 and the cluster number is selected as 200. The unconditionally simulated realisation of both the method and filtersim are presented in Fig 2 (b), and 2 (c), respectively. It is observed from the figure that vertical continuity of low pixel valued void space at right and left extreme of the simulated realisations are well reproduced in the proposed method; however, the continuity of the porous can’t be reproduced by the filtersim algorithm. The histograms (Fig 3 (a)) and variograms (Fig 3(b)) of the training image, 3 unconditionally simulated realisations of the proposed method and filtersim demonstrated that the one- and two-points statistics of the proposed method and filtersim have a good agreement with the training image statistics. The computational time of the proposed method and the filtersim algorithm are 461 and 3852 seconds, respectively. It is observed that for this example the algorithm can generate a single realisation approximately 8 times faster than filtersim.

(a) Training image (b) Realisations of the proposed method

(c) filtersim realisation

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Figure 2 Training imag (a), unconditionally simulated realisations of (a) the proposed method; and (b) the filtersim algorithm of blasted rocks.

(a) Histograms (b) Variograms

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Figure 3 (a) Histograms and (b) variograms of training image, unconditionally simulated realisations of proposed method, and the filtersim algorithm of blasted rocks image.


Example of 2-D conditional simulation

The training image utilized in this example is a slice from the Stanford 3D fluvial data set (Mao and Journel 1999); the image is presented in Fig. 4 (a). The exhaustive image or reference image is another slice from the same 3D fluvial data set. The hard data are randomly sampled from the exhaustive image. The exhaustive image and hard data locations are presented in Fig. 4 (b) and Fig 4 (c), respectively. There are 208 hard data used in this example. The wavelet scale of decomposition is selected as 2. After performing wavelet decomposition of the training image, the coarsest scale training image (which will be used for simulating the coarsest scale image) has a different support size than the finest scale hard data. To perform a simulation using different support sized hard data and training image, the hard data has to fit into the coarsest scale. The same method as proposed by Chatterjee and Dimitrakopoulos (2012) is used here to fit the hard conditioning data at coarsest scale. The fitted data is then used as the sample from the coarsest scale image. The multi-scale database was generated from the training image, using template size of 9 x 9 and Haar wavelet basis functions and the number of cluster for k-means clustering is selected as 25.

(a) Training image (b) Reference image (c) Hard conditioning data

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Figure 4 Training imag (a), unconditionally simulated realisations of (a) the proposed method; and (b) the filtersim algorithm of blasted rocks. Fig. 5 presents generated conditionally simulated realisations of the proposed method and the filtersim algorithm. It is observed from the figures that the channels are well reproduced in both algorithms. The performance of the algorithm presented here is verified by comparing the histograms and variograms. Fig. 6 shows the conditionally simulated realisations and exhaustive image histograms and variograms. The histograms are well reproduced by the generated realisations using both algorithms; however, the proportions of low valued data are overestimated by the filtersim algorithm. In addition, the realisations also reproduced the variograms, as shown in Fig. 6 (b).

(b) Realisations of the proposed method

(c) filtersim realisation

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Figure 5 Conditionally simulated realisations of (a) the proposed method; and (b) the filtersim algorithm of Stanford 3D fluvial data set. The computational time of the proposed method (305 sec) is substantially less in comparison with the filtersim algorithm (2019 sec). It is noted that the computational time of the proposed algorithm not only depends on the number of gird points to be simulated but also on the scale of wavelet decomposition. If the scale of decomposition is increased, the computational time will be less. The main reason for reduction of the computational time in the proposed method is that the simulation is


performed in the wavelet domain where the numbers of nodes to be simulated are substantially fewer than the spatial domain nodes. Also, since, pattern database generation and pattern classification are performed in wavelet domain, the size of templates, dimensions of patterns, and numbers of clusters are always less as compared to spatial domain simulation.


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Figure 6 (a) Histograms and (b) variograms of exhaustive image, conditionally simulated realisations of proposed method, and the filtersim algorithm of Stanford 3D fluvial data set.

Example of 3-D conditional simulation

In this example, the proposed technique is applied to a three-dimensional dataset, the same Stanford 3D fluvial data set used in testing the proposed technique in the two-dimensional simulation. In this case, the training image and exhaustive data are the same, with the hard data sampled from the exhaustive image. The training image and hard data used in this example are presented in Fig. 7.

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Figure 7 (a) Training image and (b) 3-D hard conditioning data.

The size of the three-dimensional training image in this example is 100×128×28 pixels. The number of samples in this example is 782. The training image is decomposed to two scales. The wavelet coefficients database is constructed by scanning the training image using selected three-dimensional template. To reduce computational time, clustering of the data is performed by selecting cluster number of 94. The aim in this example is to reconstruct the three-dimensional image by simulating jointly the scaling image and multi-scale wavelet images using the hard data and the training image database. The hard data are fit to the coarsest scale using the same approach as discussed in Chatterjee and Dimitrakopoulos (2012). As the support size of the coarsest scale training image is 4 times that of the

hard data and the dimension of the data is 3, the hard data was multiplied by 8 ( 3*2( 2) ) to fit them into the coarsest scale. The simulated realisations of the proposed method and filtersim algorithm are shown in Fig. 8. The channels in the exhaustive dataset are well reproduced in the realisations generated by the proposed technique as well as filtersim.


(a) the proposed method (b) filtersim

Figure 8 Conditionally simulated realisations of (a) the proposed method; and (b) filtersim algorithm of three dimensional conditional simulation. The first- and second-order statistics of the simulated realisations and the exhaustive data are also calculated and presented in Fig. 9. The statistics of the exhaustive dataset are well reproduced in the simulated realisations by both the algorithms.


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Figure 9 (a) Histograms and (b) variograms of the training image ( ), 5 conditionally simulated realisations of the proposed method ( ) and the filtersim algorithm ( ) of 3-D conditional simulation. The computational times of conditional simulation for a single realisation of a 3-dimensional image using the filtersim algorithm and the proposed method are presented are 952 and 38, respectively. The computational time results reveal that hte proposed algorithm is 25 times faster than the filtersim algorithm. It is revealed from this example that the computational time saved in the 3-dimensional simulation is much better than the 2-dimensional simulation when compared with filtersim. The reason for this being that the computational time in filtersim is directly proportional to the number of nodes to be simulated, i.e. time m , where m is the number of simulated nodes. However, the

computational time of the proposed algorithm is *

(2) j n

mtime , where j is scale of decomposition,

and n is number of dimension. It is clear from the expression that the saving of computational time will be more when the dimension and scale of decomposition are higher.

Conclusions

In this paper, a stochastic multi-scale, multi-point simulation algorithm based on discrete wavelet decomposition was presented. The scaling image and multi-scale wavelet images are simulated jointly by borrowing information from the training image. The main advantage of the proposed technique is that the simulations are performed jointly across all scales at wavelet domain. Therefore, computational time can be easily reduced for large domain. The practical advantages of the proposed method are demonstrated through an unconditional simulation. Although the simulation was performed in the wavelet domain, the statistics of the space-domain data were well reproduced.


The comparative study reveals that the proposed algorithm generates equally good realisations while reducing the computational time of the filtersim algorithm. This helps the proposed algorithm be applicable for spatial modeling of large mining deposits and oil reservoirs. The main advantage of the proposed algorithm is to incorporate the training image and data with different supports. The conditional simulation example presented in this paper demonstrated that with little effort the proposed algorithm can able to handle with multiple support hard data and training image. The proposed algorithm presented in this paper valid for continues variables. Since, inverse wavelet transformation doesn’t guaranteed the generation of integer values in the spatial domain, therefore the proposed algorithm is not suitable for modeling the categorical variables. However, that limitation can be overcome by using the special type of wavelet transformation.

References

1. Arpat, G. and Caers, J. [2007] Conditional simulation with patterns. Mathematical Geology, 39(2), 177-203.

2. Chatterjee, S., Dimitrakopoulos, R. and Mustafa, H. [2012] Dimensional reduction of pattern-based simulation using wavelet analysis. Mathematical Geosciences, 44, 343-374.

3. Chatterjee, S. and Dimitrakopoulos, R. [2012] Multi-scale stochastic simulation with wavelet-based approach, Computers and Geosciences, 45, 177-189.

4. Gloaguen, E. and Dimitrakopoulos, R. [2009] Two dimensional conditional simulation based on the wavelet decomposition of training images. Mathematical Geosciences, 41(7), 679-701.

5. Goovaerts, P. [1998] Geostatistics for Natural Resources Evaluation (Applied Geostatistics Series). Oxford University Press, New York.

6. Honarkhah, M. and Caers, J. [2010] Stochastic simulation of patterns using distance-based pattern modelling. Mathematical Geosciences, 42, 487-517.

7. Journel, A. [2007] Roadblocks to the evaluation of ore reserved–The simulation overpass and putting more geology into numerical models of deposit. In: R. Dimitrakopoulos (Ed.) Orebody modeling and strategic mine planning, AusIIMM, Melbourn, 2nd Edition, Spectrum Series, 14, 29-32.

8. Mallat, S. [1998] A wavelet tour of signal processing. Academic Press, San Diego, CA. 9. Mao, S. and Journel, A.G. [2009] Generation of a reference petrophysical and seismic 3D data

set: The Stanford V reservoir. Stanford Center for Reservoir Forecasting Annual Meeting. http://ekofisk.stanford.edu/SCRF.html (1999), Accessed 23 February 2014.

10. Strebelle, S. [2002] Conditional simulation of complex geological structures using multiple-point statistics, Mathematical Geology, 34(1), 1-21.

11. Tran, T., Mueller, U.A. and Bloom, L.M. [2002] Multi-scale conditional simulation of two-dimensional random processes using Haar wavelets. Proceedings of GAA Symposium, Perth, 56-78.

12. Vannucci, M. and Corradi, F. [1999] Covariance structure of wavelet coefficients: theory and models in a Bayesian perspective. J.R. Statis. Soc., 61, 971-986.

13. Zhang, T., Switzer, P. and Journel, A. [2006] Filter-based classification of training image patterns for spatial simulation. Mathematical Geology, 38(1), 63-80.

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Reservoir Spatial Heterogeneity Modeling Through Pattern-Based Stochastic Simulation