Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
An Improved Reconstruction Method for Porous Media Based on Multiple-point Geostatistics ZHANG Ting1,2, LU DeTang1,2, LI DaoLun1,2 1.Department of Modern Mechanics, University of Science and Technology of China, P.R. China, 230027 2.Research Center of Oil and Natural Gas, University of Science and Technology of China, P.R. China, 230027 [email protected] Abstract The three-dimensional reconstruction of porous media is of great significance to the research of mechanisms of fluid flow in porous media. The real three-dimensional structural data of porous media are helpful to describe the irregular topologic structures quantificationally in porous media, so a method using real volume data and multiple-point geostatistics to reconstruct three-dimensional structures of porous media is proposed. A 3D training image of porous media is generated from volume data obtained by micro-CT scanning with the resolution of micron. According to the probability of each pattern occurring in the three-dimensional training image, states of pixels to be simulated are drawn and the topologic structures of porous media can be predicted by using MPS. This method is tested on the three-dimensional reconstruction of sandstone. Experimental results show that the reconstructed porous structures are similar to those of real volume data. Key words Multiple-point geostatistics, Pore, Training image, Multiple grid, Variogram 1 Introduction The structural characterization and prediction of transport properties in porous materials is of great importance in various fields such as catalysis, oil recovery, aging of building materials, study of hazardous waste repositories, etc. These transport properties critically depend on the geometry and topology of the pore space, the physical relationship between rock grains and the fluids, and the conditions imposed by the flow process. Porous structural information must be available in order to predict the fluid flow properties quantificationally [1]. The evolution of the modeling approaches for the representation of the porous structure is a result of advances in theoretical and experimental techniques as well as in computational resources [2]. The idea of representing the pore space as a two- or three-dimensional network emerged from the pioneering work by Fatt in 1950s [3]. Most networks are based on a regular lattice, for example, typically a cubic lattice with a coordination number of six. It is possible to vary the coordination number by eliminating throats from the network [4]. However, the network is still based on a regular topology, but the real porous media have more irregular structures. To overcome this limitation, Voronoi networks and Delaunay triangulations have been used to allow a variable coordination number [5,6]. Another solution proposed by Bryant [7] is to map the true pore structure of a medium onto a network, so more completely the pore structures can be retained. Because the complexity of the pore space morphology, the pore bodies and throats are usually represented by simplified shapes. Pore bodies have been represented by spheres or cubes, while pore throats have been represented by cylinders or other ducts with non-circular cross-sections [8]. But because the real porous structures are quite complex, the structures based on regular shapes cannot accurately describe the irregular geometry and topology of pore space, which has become an obstacle for the study of transport properties in porous media. With the development of experimental technology and equipments, some methods including serial sectioning, focused ion beam, laser scanning confocal microscopy and x-ray computed tomography [9,10] have been used to create 3D pore space images. Two-dimensional thin cross-sections are, in contrast to 3D images generated by direct imaging, often readily available at high resolution. Geometrical properties can be measured from these cross-sections and used to generate a 3D image with the same statistical properties [1]. 653 There are mainly three kinds of methods to reconstruct 3D pore space images using 2D images. The first way is to reconstruct the porous medium by modeling the geological process by which it was made based on the information extracted from 2D images [11]. Although the process-based method can reproduce the long-range connectivity, there are many systems for which the process-based reconstruction is very difficult to apply. For example, for many carbonates it would be very complex to use a process-based method that mimics the geological history involving the sedimentation of irregular shapes followed by significant compaction, dissolution and reaction [1]. The second method uses statistical techniques to regenerate 3D images from 2D images [12]. But the method fails to reproduce the long-range connectivity of the pore space, especially for low porosity materials and particulate media, such as grain or sphere packs because only the low-order information is used. The third method was proposed by Okabe, using MPS (multiple-point geostatistics) to reconstruct porous media [1,10]. In this method, a pseudo-3D training image is obtained only by rotating a 2D XY plane image. Measured statistics on the XY plane are transformed to the YZ and the XZ planes with an assumption of isotropy in orthogonal directions. At every voxel (volume pixel) in order to assign pore space or grain state, three principal orthogonal planes XY, XZ, and YZ intersecting this voxel are used to find conditional data on these planes. But the main limitation of this method is that the pseudo-3D training image generated from a 2D horizontal plane doesn’t include real vertical information, failing to reflect the characteristics of anisotropy widely existing in real porous media. To overcome the disadvantages of Okabe’s method, we propose an improved method, based on MPS and real volume data which are obtained by x-ray micro-CT scanning, to reconstruct porous media. The 3D image obtained from real volume data includes both the horizontal and vertical information of porous media and is used as the training image. Meanwhile, the multiple-grid method is used to capture large-scale patterns in the training image and reproduce these patterns in reconstructed images. Experimental results show that our method is applicable in the fields of reproducing the long-range connectivity and describing the complex characteristics of pore space. 2 Basic concepts of MPS 2.1 Data template and data event The training image is scanned using a data template τn that comprises n locations uα and a central location u. The uα is defined as: uα=u+hα(α=1,2,…,n), where the hα are the vectors describing the data template. For example, in figure 1(a), hα are the 80 vectors of the square 9×9 template. In figure 1(b), hα are the 26 vectors of the cube 3×3×3 template with a blue center u. (a) (b) Figure 1 Data template.(a)2D data template;(b)3D data template. Consider an attribute S that has K possible states {sk; k =1,2,…,K}. A data event dn of size n centered at location u constituted by n vectors uα in τn is defined as: dn={S(uα)= skα ; α=1,2,…,n} (1) where S(uα) is the state at the location of uα within the template, and kα is any number from 1 to K. dn actually means that n values S(u1)…S(un) are jointly in the respective states sk1 … skn . Figure 2 illustrates a data event captured by a 5 × 5 data template. Scanning a training image using a data template is to get the probabilities of occurrences of the data events dn, i.e., probabilities of the n vectors u1,…, un within the τn jointly in the respective states 654 sk1 , …, skn : Prob{dn}=Prob{S( uα ) = skα ; α=1,2,…,n} (2) (a) (b) (c) Figure 2 Illustration of a data event captured by a 5×5 template. (a) 5×5 data template;(b) 15×15 training image;(c)a data event. In the process of scanning a training image using a given data template, it is a replicate when a data event in the training image has the same geometric configuration and the same data values as dn associated with τn. Under the hypothesis of stationarity, i.e., the statistics are location-independent, the probability of occurrences of the data events dn is the ratio of replicate number c(dn) found in the training image and the size of effective training image denoted by Nn: Prob{ S(uα) = skα ;α=1,2,…,n} ≈ c(d n ) Nn (3) At any unsampled node u , we need to evaluate the cpdf(conditional probability distribution function) that the unknown attribute value S(u) takes anyone of K possible states sk given n nearest data denoted by S(uα)= skα (α=1,2,…,n). According to the Bayesian relation, the above cpdf is defined as: Prob{S(u)=sk|dn}= Prob{S (u ) = sk and S (uα ) = skα ;α = 1,L, n} (4) Prob{S (uα ) = skα ;α = 1,L , n} where the denominator of relation (4) is the probability of conditional data event and can be inferred by relation (3); the numerator is the probability of occurrences of the conditional data event and u being the state sk at the same time. The numerator can be obtained by the ratio denoted by ck(dn)/ Nn, where ck(dn) is the number of those replicates, among the c(dn) previous ones, associated to a central value S(u) equal to sk. The conditional probability can be defined as: c (d ) (5) Prob{S (u ) = sk | S (uα ) = skα ; α = 1,L , n} ≈ k n c( d n ) Based on the relation (5), the state of u can be drawn using Monte Carlo methodology. Because relation (5) adopts the idea of probability method, the drawn states of u are random, which can reflect prior probability models existing in the training image. 2.2 Accelerating reconstruction by search tree If the training image has to be scanned by data template anew at each unsampled node u to get its cpdf, then it will be extremely CPU-demanding. To avoid the repetitive scanning of the training image and reduce CPU time, the cpdf obtained from training images is stored in a data structure called “search tree” [13]. The search tree stores the numbers of occurrences c(dn) and ck(dn) found in the training image, from which the relation (5) can be calculated. The construction of that search tree is fast since it requires scanning the training image one single time prior to the image simulation. During the simulation, the required local cpdf at any unsampled node u is retrieved directly from the previous search tree, so the whole simulation process is greatly accelerated. 3 Multiple grid The data search neighborhood should not be taken too small, otherwise large-scale structures of the 655 training image cannot be reproduced. On the other hand, if the search neighborhood is too large, the associated data template τn will include a large number of grid nodes, which will increase the CPU-time and the memory taken by the search tree. One solution to capture large-scale structures while considering a data template with a reasonably small number of grid nodes is provided by the multiple-grid method [14]. The multiple-grid method consists in simulating a number of increasingly finer grids. The data template τn associated to the search neighborhood of the fine grid is scaled proportionally to the spacing of the nodes within the grid to be simulated. One search tree needs to be constructed for each simulation grid. When each grid simulation is completed, its simulated values are frozen as conditional data to be used for conditioning on the next finer simulation grid. For example, figure 3 shows a 2D multiple-grid simulation sequence for a simulation grid of size 13×13 nodes using 3 nested increasingly finer grids. Previously simulated nodes are in black. Nodes to be simulated within the current grid are in gray and nodes to be ignored are in white. Figure 4 shows the corresponding 2D data templates of figure 3. In figure 4, the locations of nodes to be simulated are in gray and those to be ignored are in white. Note that the original data template is scaled for each of the multiple grids, with the geometry unchanged. When scanning figure 3(a) using figure 4(a), the horizontal and vertical moving distances are both 4 nodes. The moving distances are respectively 2 nodes and 1 node when scanning figure 3(b) using figure 4(b) and scanning figure 3(c) using figure 4(c). The simulated results generated by the first data template are used as conditional data for the second grid, which are the black nodes shown in figure 3(b). Then scan figure 3(b) using the data template shown in figure 4(b). After the second scanning, the results simulated by the first and second data templates are both used as conditional data for the third grid, which are the black nodes shown in figure 3(c). In the end, the final simulated image can be obtained by scanning the finest grid shown in figure 3(c) using the third data template shown in figure 4(c). (a) (b) (c) Figure 3 2D three grids.(a)first grid;(b)second grid;(c)third grid. (a) (b) (c) Figure 4 2D three-grid data templates.(a)first template;(b)second template;(c)third template. 4 The reconstruction method of porous media based on MPS and Real Volume Data step 1. Using 3D multiple-grid data templates, scan the 3D training image generated from real volume data to build search trees. step 2. Assign original conditional data to the closest grid nodes if original conditional data are available; otherwise, go to the step 3. step 3. Define a random path visiting only once each unsampled node. At each unsampled location u, retain the conditional data actually present within the template τn centered on u. Let n’ be the number of those conditional data, and dn’ the corresponding data event. If no replicate of dn’ can be found in the training image, the conditional probability of u is replaced by the marginal probability; otherwise, retrieve from the search tree the cpdf of occurrence of the data event dn’. Then draw a simulated value for node u from the previous conditional probability using Monte Carlo methodology. The simulated value is then added to the conditional data for the simulation at all subsequent nodes. step 4. Loop the step 3 until all grid nodes are simulated. Then one stochastic image has been 656 generated. 5 Experimental results 5.1 Reconstruct porous media using the proposed method Our proposed method in part 4 was tested on sandstone. The real volume data of sandstone was obtained by x-ray micro-CT scanning with a resolution of 10µm. A 3D sandstone image generated by real volume data was used as a training image (80×80×80 voxels, the porosity=0.1763). The training image can provide reference data to evaluate the simulated results. There are only two states, pore space and grain, existing in sandstone. The exterior image, cross-section image(X=40,Y=40,Z=40) and pore space of the training image are shown respectively in figure 5. The pore space is blue and the grain is gray. (a) (b) (c) Figure 5 Training image.(a)exterior image;(b)cross-section image;(c)pore space. There were no original conditional data used in our experiments. The marginal probability of pore space was assumed to equal the porosity of the training image. Using 3D three-grid data templates to scan the training image, our reconstruction method was used to reconstruct the sandstone images. The exterior image, cross-section image(X=40,Y=40,Z=40) and pore space of the reconstructed image are shown respectively in figure 6. To compare with the reconstruction using three-grid data templates, 3D two-grid data templates and one-grid data template were respectively used to reconstruct sandstone images. The reconstructed images are shown in figure 7 and 8. We can observe that the reconstructed images using three-grid data templates have similar pore space and grain structures with those of the training image and the long-range connectivity of pore space is well reproduced (see figure 6). Compared with the reconstructed pore space using three-grid data templates, the reconstructed pore space using one-grid data template comprises more isolated small porous regions and the long-range connectivity of pore space is not well reproduced (see figure 8). The reproduced long-range connectivity using two-grid data templates is better than that using one-grid data template, but worse than that using three-grid data templates (see figure 7). Further experiments prove that the quality of reconstructed images will not be improved obviously when the number of multiple-grid is larger than three. The reason is that on the very coarse grids, a very large scaled template is used to scan the training image. Due to the limited size of the training image, few replicates for such large data events can be found. Hence, the few replicates cannot present enough statistical information to improve the quality of reconstructed images. Table 1 shows the porosity of reconstructed images respectively using one-grid, two-grid and three-grid data templates. The porosity of reconstructed images using three-grid data templates is closest to that of the training image. 5.2 Comparison of variogram Variogram can reflect the relativity and variability of a spatial variable in certain direction. If variogram curves of different images in a direction are similar, then the structures of these images in this direction are similar. Figure 9 shows the variogram of the training image and images respectively using one-grid, two-grid and three-grid data templates in the directions of X, Y and Z. In figure 9, the variogram of the training image is similar to that using three-grid data templates, but differs from the variogram under two other conditions, demonstrating that the reconstructed structures 657 using three-grid data templates are closest to those of the training image. Because the training image is obtained by scanning the real sandstone, the reconstructed images using three-grid data templates have the similar structures with the real sandstone. (a) (b) (c) Figure 6 Reconstructed image using three-grid data templates.(a)exterior image;(b)cross-section image;(c)pore space. (a) (b) (c) Figure 7 Reconstructed image using two-grid data templates.(a)exterior image;(b)cross-section image;(c)pore space. (a) (b) (c) Figure 8 Reconstructed image using one-grid data template.(a)exterior image;(b)cross-section image;(c)pore space. Table 1 the porosity of reconstructed images using one-grid, two-grid and three-grid data templates one-grid data template two-grid data templates three-grid data templates porosity 0.1103 0.1354 0.1743 (a)X-direction (b) Y-direction (c) Z-direction Figure 9 Variogram of the training image, images using three-grid, two-grid and one-grid data templates in the directions of X, Y and Z. 6 Conclusion 658 We have proposed an improved method to reconstruct the porous media using MPS and real volume data. A 3D training image is obtained from the real volume data, so it includes the horizontal and vertical information of real porous media. MPS can capture porous structures from the 3D training image, and then reproduce these structures in reconstructed images. Experimental results show that the porous structures of reconstructed images using our method are similar to those of real volume data. Acknowledgement It is a project supported by National Natural Science Foundation of China (10672159, 10702069) and China 973 Projects (2006CB705805). References [1] Okabe H., Blunt M. J. Pore space reconstruction using multiple-point statistics. Journal of Petroleum Science and Engineering,2005,46:121-137 [2] Kainourgiakis M. E., Kikkinides E. S., Galani A. et al. Digitally Reconstructed Porous Media: Transport and Sorption Properties. Transp Porous Med, 2005, 58:43–62 [3] Fatt I. The network model of porous media I. capillary pressure characteristics. Transactions of AIME, 1956,207(1):144-159 [4] Dixit A. B., Buckley J. S. McDougall S. R., et al. Empirical measures of wettability in porous media and the relationship between them derived from pore-scale modeling. Transport Porous Med, 2000,40(1):27-54 [5] Blunt M., King P. Macroscopic parameters from simulations of pore scale flow. Physical Review A, 1990,42(8):4780-4787 [6] Blunt M, King P. Relative permeability from two- and three-dimensional pore-scale modeling. Transport in Porous Media, 1991,6(8):407-433 [7] Bryant S. L., King P. R., Mellor D. W. Network model evaluation of permeability and spatial correlation in a real random sphere packing. Transport in Porous Media, 1993,11(1):53-70 [8] Riyadh Al-Raoush, Karsten Thompson, Clinton S. Willson. Comparison of Network Generation Techniques for Unconsolidated Porous Media. Soil Sci. Soc. Am. J., 2003,67:1687–1700 [9] Fredrich J. T. 3D imaging of porous media using laser scanning confocal microscopy with application to microscale transport processes. Physics and Chemistry of the Earth. Part A: Solid Earth and Geodesy,1999,24(7): 551-561 [10]Okabe H., Blunt M. J. Prediction of permeability for porous media reconstructed using multiple-point statistics. Physical Review E, 2004,70, 066135:1-9 [11] Bryant S., Blunt M. Prediction of relative permeability in simple porous-media. Physical Review A,1992,46 (4): 2004-2011 [12] Quiblier J. A. A new three-dimensional modeling technique for studying porous-media. Journal of Colloid and Interface Science, 1984,98 (1): 84-102 [13] Strebelle S. Conditional simulation of complex geological structures using multiple-point statistics. Mathematical Geology, 2002,34(1): 1-21 [14] Tran T. Improving variogram reproduction on dense simulation grids. Computers&Geosciences, 1994,20(7): 1161–1168 659