Pore space reconstruction using multiple-point statistics
Hiroshi Okabe a,b ,*,Martin J.Blunt a
a
Department of Earth Science and Engineering,Imperial College London,SW72AZ,UK b
Japan National Oil Corporation,2-2-2Uchisaiwai-cho,Chiyoda-ku 100-8511,Japan
Received 11August 2003;accepted 25August 2004
Abstract
The reconstruction of porous media is of great interest in a wide variety of fields,including earth science and engineering,biology,and medicine.To predict multipha flow through geologically realistic porous media it is necessary to have a three-dimensional (3D)reprentation of the pore space.Multiple-point statistics were ud,bad on two-dimensional (2D)thin-ctions as training images,to generate 3D pore space reprentations.The method was borrowed from geostatistical techniques that u pixel-bad reprentations to reproduce large-scale patterns.Thin-ction images can provide multiple-poin
t statistics,which describe the statistical relation between multiple spatial locations.Assuming that the medium is isotropic,a 3D image can be generated that prerves typical patterns of the void space en in the thin ctions.The method is tested on Fontainebleau and Berea sandstones for which 3D images from micro-CT scanning are available.The u of multiple-point statistics predicts long-range connectivity of the structures (measured by local percolation probability)better than standard two-point statistics methods.The lection of multiple-point statistics is a key issue and is discusd in detail.D 2004Elvier B.V .All rights rerved.
Keywords:Pore space reprentation;Multiple-point statistics;Reconstruction;Long-range connectivity;Percolation probability
1.Introduction
桑葚是热性还是凉性Transport properties such as relative permeability and capillary pressure functions define the flow behavior of porous media.The functions critically depend on the geometry and topology of the pore
乒乓球大满贯得主space,the physical relationship between rock grains and the fluids,and the conditions impod by the flow process.A quantitative prediction of petrophysical properties in disordered media,such as dim
entary rock,frequently employs reprentative microscopic models of the microstructure as input.Pore structural information must be available in order to predict fluid flow properties using network models (Blunt et al.,2002)or other approaches,such as the lattice-Boltzmann method (Chen and Doolen,1998).
The numerical reconstruction of 3D materials,such as porous and composite media,has various potential
0920-4105/$-e front matter D 2004Elvier B.V .All rights rerved.doi:10.1016/j.petrol.2004.08.002
*Corresponding author.Prent address:Oil and Gas Tech-nology Rearch and Development,Japan Oil,Gas and Metals National Corp.,1-2-2Hamada,Mihama-ku,Chiba 261-0025,Japan.Tel.:+81432769263;fax:+81432764063.
E-mail address:jp (H.Okabe).Journal of Petroleum Science and Engineering 46(2005)121–
137
/locate/petrol
applications in earth science and engineering,biology, and medicine.Effective reconstruction methods allow generation of realistic structures and subquent analys can be performed on the images to compute macroscopic parameters,such as transport and mechanical properties.
Several methods have been propod to generate 3D pore space images.A ries of2D ctions can be combined to form a3D image.This is a laborious operation limited by the impossibility of preparing cross ctions with a spacing of less than about10A m (Dullien,1992).However,recent advances,such as the u of a focud ion beam(Tomutsa and Radmilovic,2003)allow higher resolut
ion images (sub-micron size)to be constructed.Another approach is to u non-destructive X-ray computed micro-tomography(Dunsmuir et al.,1991;Spanne et al., 1994;Coles et al.,1998)to image a3D pore space directly at resolutions of around a micron.However, this resolution is not sufficient to image the sub-micron size pores that are abundant in carbonates, which can be imaged by2D techniques such as scanning electron microscopy(SEM).The sub-micron structures of real rocks have been studied using lar scanning confocal microscopy(Fredrich,1999).It has,however,limited ability to penetrate solid materials.In the abnce of higher resolution3D images,reconstructions from readily available2D microscopic images are the only viable alternative.
2D high-resolution images provide important geo-metrical properties such as the porosity and typical pore patterns.Bad on the information extracted from 2D images,one promising way is to reconstruct the porous medium by modeling the geological process by which it was made(Bryant and Blunt,1992;Bakke and Øren,1997;Pilotti,2000).Although the process-bad reconstruction is general and it is possible to reproduce the long-range connectivity,there are many systems for which the process-bad reconstruction is very difficult to apply.For example,for many carbonates it would be very complex to u a process-bad method that mimics the geological history involving the dimentation of irregular shapes followed by signifi-cant compaction,dissolution and reaction(Lucia,
1999).In the cas it is necessary to find another approach to generate a pore space reprentation.One method is to u statistical techniques to produce a3D image from2D image analysis(Quiblier,1984;Adler et al.,1990,1992).Traditionally porosity and two-point statistics have been ud to achieve this.The methods have been extended to include other geo-metrical properties to improve the quality of the reconstructed images(Roberts,1997;Roberts and Torquato,1999;Yeong and Torquato,1998a,b;Man-wart et al.,2000;Talukdar et al.,2002).The images, however,often fail to reproduce the long-range connectivity of the pore space,especially for low porosity materials and particulate media,such as grain or sphere packs becau of the low-order information ud(Hazlett,1995,1997;Coles et al.,1998;Levitz, 1998;Biswal et al.,1999;Kainourgiakis et al.,2000;Øren and Bakke,2002,2003).
A multiple-point statistics technique first intro-duced in geostatistics to reprent connected geo-logical bodies,such as sand channels,at the field scale is ud to reconstruct pore space images(Caers, 2001;Strebelle et al.,2003).Conceptually the problem is similar:pore spaces that have a high degree of interconnectivity must be generated.A key aspect is the proper lection of the multiple-point statistics to reproduce satisfactory images.Fontaine-bleau and Berea sandstone are ud as test cas becau they have been characterized thoroughly by other means.Fig.1shows an image of
Fontainebleau Fig.1.A3D pore space reprentation of Fontainebleau sandstone. The pore space is shown in gray.This image was generated using micro-CT scanning with a resolution of7.5A m.
H.Okabe,M.J.Blunt/Journal of Petroleum Science and Engineering46(2005)121–137 122
月之花语
sandstone obtained by micro-CT scanning (Dunsmuir et al.,1991).
2.Methodology
2.1.Multiple-point statistics method
Multiple-point statistics method requires a denly and regularly sampled training image describing the geometries expected to be exist in the real structure
(Caers,2001;Strebelle et al.,2003).For example,photographs of outcrops at the field scale and micro-scope images at the pore scale can be ud as training images.
2.1.1.Borrowing multiple-point statistics from train-ing images
The training image of Fig.2(thin-ction image)is scanned using a template t compod of n t locations u a and a central location u :u a ¼u þh a
欢呼的反义词
a ¼1;N ;n t
ð1Þ夏的组词
where the h a are the vectors describing the template.For example,in Fig.3,h a are the 80vectors of the square 9Â9template.The template is ud to scan the training image and collect the data event at each location u .The data event (or the pattern),for example shown in Fig.4,is defined by dev u ðÞ¼i u ðÞ;i u þh a Þ;
a ¼1;N ;n t ðg
f ð2Þ
where i (u )is the data value at the point within the template.Each point in the template has a number to identify the pattern and to store the pattern in the memory.The t of all data events scanned from the training image results in a training data t
Set ¼dev u j Á;j ¼1;N ;N t ÀÉÈð3Þwhere Set refers to the training data t constructed with template t .N t is the number of different
CPRP
central
Fig.2.Thin-ctions as training images taken from 3D micro-CT images.The pore space is shown white and the grain black.(a)Fontainebleau sandstone with a porosity,/=0.1364(3002pixels).(b)Berea sandstone with /=0.1773(1282pixels).The resolutions of the images are 7.5and 10A m,
respectively.
祖逖
Fig.3.A 9Â9template to capture multiple-point statistics.The training image is scanned and each occurrence of any possible pattern of void space and grain is recorded.
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locations of template t over the training image.The lection of the template geometry is important and various template sizes n t should be tried in order to reproduce better structures.
2.1.2.Pattern reproduction
To reconstruct pore scale structures,binarized thin ction images were ud as training images that have only void or solid.A detailed discussion of the training image is prented later under Rock Sample.The probability of occurrence of any data event dev n associated with the data template t can be inferred from the training image by counting the number c (dev n )of replicates of dev n found in the training image.A replicate should have the same geometry and the same data values as dev n .The multiple-point statistics can be identified to the proportion:Pr(dev n )c c (dev n )/N n where N n is the size of the training image.The key to this algorithm is the determination of local conditional probability distribution functions (cpdf).The probability that the unknown attribute value i (u )takes any of two possible states (void or solid)given n nearest data during the reproduction at any unsampled location u must be evaluated.If multiple-point statistics are available,then the conditioning of i (u )to the single global data event dev n can be considered,and the conditional probability can be identified to the train-ing proportion.The cpdf is inferred directly and
consistently from the training image.The multiple-point statistics,the geometrical structures in other words,are borrowed directly from the training image.This approach can theoretically apply to a 3D field when 3D structural information is available.Since it is difficult or impossible to measure 3D sub-
micron scale data,our only alternative is to u 2D images to measure multiple-point statistics.In order to generate 3D structure from 2D information,measured multiple-point statistics on one plane are rotated 908around each principal axis.In other words,measured statistics on the XY plane are transformed to the XZ and the YZ planes with an assumption of isotropy in orthogonal directions.At every voxel in order to assign void or solid pha,three principal orthogonal planes,XY ,XZ and YZ intercting this voxel are ud to find conditioning data on the planes one by one.Each probability of the pha at this voxel on the different planes are estimated by this process,and then the three measured probabilities are weighted by the number of conditioning data on each plane to obtain a single probability on this voxel.Finally,the pha at the voxel is assigned bad on this weighted probability to generate a 3D image.There is less conditioning data during the initial stage of the reproduction.In this ca,the porosity value can be ud as the probability.
To borrow directly all required cpdf from a training image possibly leads to having excessive amounts of information.In a designated template t of n data variables,inference of a probability distribution function conditional to a data event dev n requires that enough occurrences,which are dependent on the size of the training image,should be found.Each of the 80nodes in the data template (9Â9square shape),except a center point,can take two states (void or solid).This leads to 2
80possible data events which means that a relatively small number of cpdf can be actually inferred from the training image.To alleviate this problem,at each unsampled node,infer the local cpdf by scanning the training image with progressive reduction of the size n of the conditioning data event dev n until a designated minimum number of replicates of dev n are found.To minimize the repetitive process and the calculation time,each unsampled grid node is visited only once using a random path and each simulated value becomes a conditioning datum value.Needless to say,conditioning data are frozen at
their
垃圾分类宣传活动Fig.4.A data event (or a pattern)measured by 9Â9template.The frequency of every possible data event is found by scanning the template (Fig.3)over the training image.
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data location and are ud for the successive simulation.The cycles are iterated until all the grids are simulated.In order to avoid the repetitive scanning of the training image and to store multiple-point statistics effectively,a dynamic data structure called a arch tree is ud to store all training cpdf in advance.The algorithm is explained by a flow chart in Fig.5.
In the prence of large-scale structures,the u of a single limited size template would not suffice to model the large-scale characteristics obrved in the training image of Fig.6.In this figure,both a small template and a template large enough to capture the large-scale structures are applied to the thin ction image.Thus,a large template is necessary to reproduce large-scale structures obrved in the
train-Fig.5.Flow diagram of multiple-point statistics reconstruction of pore space images.
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ing image.The template size can be theoretically expanded to match the largest structure in the training image;however,the template size is limited by memory limitations in the numerical simulation. Although the numerical modeling is always limited at certain points,an alternative approach can be introduced by a sort of multigrid simulation(Caers, 2001;Strebelle et al.,2003).Four different sized templates(Fig.7)are ud to scan the training image, resulting in four different data ts Set t1,Set t2,Set t3 and Set t4.Larger scale templates can simply be expanded from the small-scale template.In a multi-grid system,a simulation is first performed on the coarst grid.Once the coar simulation is finished, the simulated values are assigned to the correct grid locations on the finer grid,and are ud as condition-ing data on the finer grid.When large-scale structures exist in the training image,this multigrid approach is effective to measure the large-scale multiple-point statistics while requiring relatively little memory (Caers,2001;Strebelle et al.,2003).
2.2.Image processing-noi reduction and smoothing
There is noi in3D images generated from2D data,due to insufficient statistics,which is inevitable w
hen stochastic methods are ud.This unrealistic noi in the image can be reduced by image processing.Some rearchers(Ioannidis et al., 1997;Liang et al.,1998)who u
stochastic Fig.6.A small template and a template large enough to capture pore-scale structures.
Fig.7.Expanded templates to capture large-scale structure(from left to right).A form of multigrid simulation is performed with an image first generated using the largest template and then successively smaller templates are ud where the large-scale information acts as conditioning data.
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