Download An Improved Reconstruction Method for Porous Media Based on Multiple-point Geostatistics

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].
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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
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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
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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
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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
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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
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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).
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