Physica A: Statistical Mechanics and its Applications
Pore space morphology analysis using maximal inscribed spheres
Introduction
Fluid transport in porous permeable rock is determined by the void space geometry and connectivity, and the solid surface/fluid chemistry. The ever-changing distribution of fluids in the pores of a gas- and oil-bearing rock must be understood to develop a successful hydrocarbon recovery or environmental remediation process. Although the length-scale of an oil field is measured in kilometers, the ultimate success of an oil and gas recovery scheme is the net result of countless displacement events at a scale measured in microns. The process-dependent redistribution of reservoir fluids during production and injection determines how much of the initial hydrocarbons will be recovered and how much will be left trapped. Recent advances in micro-imaging of natural rocks [1], [2], combined with advances in pore-level flow modeling [3], [4], allow researchers and engineers to gain better insights into pore-level displacement mechanisms. In particular, credible predictions of the impact of the rock wettability and fluid properties on the relative permeabilities and capillary pressures, as well as on the trapped oil and gas saturations, are now possible [5], [6], [7], [8], [9].
A microscopic image of rock is a three-dimensional array of cubic atoms or voxels. Each voxel is assigned a non-zero value if it is attributed to the pore space and zero otherwise. A group of neighboring voxels can make a loosely-defined “pore throat” or a “pore body.” The pore throats control fluid flow, whereas the pore bodies provide fluid storage.
One of the difficulties in studying the microscopic structure of a porous medium is the absence of an elegant mean-field theory, such as the multiple-continua model. As a consequence, the microscopic rock models require storage and processing of huge amounts of data to characterize a tiny piece of rock. For example, a -resolution image of a cubic rock sample 2.5-mm on the side consists of 125 million voxels. In comparison, a reservoir simulation with an equal number of gridblocks would be a challenge even for a high performance parallel computer.
The number and size of the voxels in an image are limited by the resolution and viewing angle of the imaging device. The image itself is an interpretation of the reflection, absorption, attenuation, and diffraction patterns of electromagnetic waves. Each such interpretation is a solution of a series of inverse problems. The inversion errors are then combined with the uncertainty produced by segmentation, that is, assignments of the voxels, which are part solid and part void space. The representativeness of a digital image and the minimal requirements to the resolution are usually determined empirically.
In this context, attempts to develop efficient procedures of computer reconstruction of sedimentary rocks seem to be promising [11], [10]. If such a procedure adequately reproduced a rock, it would be possible to create the corresponding digital image with theoretically arbitrarily high resolution. One has to bear in mind, though, that indefinite refinement of the resolution of the computer-generated rock images may be physically unrealistic.
The literature on processing images of natural rocks can be split into two periods. The earlier period is characterized by the development of basic concepts of mathematical morphology. A systematic presentation of early development and results of the theory of mathematical morphology was given in the monographs by Matheron in 1975 [12] and Serra in 1982 [13]. The concepts of skeleton and medial axis were inspired by early works of Motzkin [14] and Blum [15], and played a pivotal role in all subsequent investigations.
More recently, the revolutionary progress in imaging techniques [2], [16], [17], [18] and computing power induced a new wave of morphological studies. Pore-network modeling made it possible to gain fundamental understanding of multiphase fluid flow in porous media. In conjunction with image analysis, it became possible to build models of porous media capable of predicting the fluid transport properties of rock, so important in oil industry [3], [5], [6], [8], [19], [20], [21], [22]. Surveys of pore-network modeling are presented in Refs. [4], [23], [24], [25].
Even if we concentrate purely on geometric issues and neglect errors introduced by image processing and interpretation, extraction of pore networks from the microscopic 3D images of rocks still poses challenging problems. Many “skeletonization” algorithms are based on thinning methods [17], [18], [26], [27], [28], [29], [30], [31], [32]. Thinning relies on removing the “redundant” elements of an image, while preserving certain topological properties of the entire pore space. However, intuitive extension of topological methods to digital images may be insufficient for rigorous analysis. The difficulties of porting the basic topological concepts [33], such as connectivity, Euler–Poincaré characteristic, etc., to discrete digital images are well known [34]. Tests of thinning algorithms on simple computer-generated images show that a refinement of the resolution can lead to less accurate results, which is an unwanted side effect. Simple illustrative examples demonstrate that the result of thinning may be unstable with respect to the choice of the starting point.
A characterization of the pore space geometry without application of thinning algorithms was proposed in Refs. [35], [36], [37], where some elements of the algorithms proposed here were developed. In particular, the characterization of skeleton as the set of centers of the maximal balls was employed. In Refs. [38], [39], [40], [41], the capillary pressure curves were computed without pore network extraction. We also elaborate on this approach. A skeletonization method based on a complete catalog of shape primitives for 2D and 3D objects and suitable for higher dimensions was developed in Refs. [27], [29], [42], [43].
In this paper, we focus on the analysis of pore space geometry. The image itself may have originated from computer tomography, a computer model of rock deposition, or any other source. We investigate how the results of image analysis are affected by image resolution and other variations of the input data. Our approach is based on the tools and concepts of mathematical morphology, as well as on an efficient implementation of the object-oriented algorithm design. Thus, all computer code used in this research has been written in C++ using standard template library (STL) [44] for data storage. The MIS method has been successfully applied to analysis of 3D geometry of the pore space of chalk [45]. The nanometer resolution images were obtained using focused ion beam (FIB) technology [2]. The capillary pressure curves obtained by MIS-calculations are in a good agreement with laboratory measurements.
This paper has the following structure. In the next section, we briefly overview the salient concepts of mathematical morphology. Then we describe how these concepts are implemented as objects in the algorithm. In the last section, we illustrate our approach with computations performed for digital images of both computer-generated and natural rocks. In Appendix, we briefly describe the cluster search and connectivity algorithms used in this study.
Section snippets
Basic concepts of mathematical morphology
The fundamental concepts of mathematical morphology, such as skeleton, medial axis, thinning, etc., are widely used in the literature on image processing. These concepts are usually illustrated in two dimensions where, for example, the medial axis of a channel-like structure is a curve passing through the points “in the middle” of that channel. Although even in 2D an intuitively clear definition can produce unexpected geometric features, in 3D the picture is much more complicated. It can be
Voxel objects: Computing the skeleton
A discrete image is a set of voxels in space. Image processing algorithms operate on individual voxels to extract information about the entire image. Thus, a voxel object is the basic element of the pore space analysis. In addition to the three coordinates, its properties also include the radius of the maximal inscribed ball centered at the voxel. Depending on the task, this radius can later be complemented or replaced with the maximal radius of an inscribed ball covering this voxel,
Stick and ball diagram
Evaluation of the average coordination number is an important part of pore space analysis. Clearly, the result depends on how the pore bodies and pore throats are defined. As examples in Figs. 1 and 2 show, just the skeleton itself may carry very little information about the pore space structure. In this section, we introduce definitions of pore bodies and pore throats through the maximal inscribed balls and demonstrate on examples that they lead to a robust description of the pore space. In
MIS-calculation of the dimensionless capillary pressure
When the pore space is shared between two immiscible fluids in equilibrium, the wetting fluid occupies the corners of large pores and small pores, while the non-wetting fluid occupies the central parts of the pores it invaded. The interface between these two fluids is a surface whose curvature is determined by the capillary pressure. Although in reality the fluid interfaces are not spherical, they can be approximated by spheres. The strongest deviation from the spherical shape of the interface
Conclusions
We have presented a new robust approach to study the pore space morphology. Our approach is based on the fundamental concepts of mathematical morphology, summarized, e.g. in Ref. [13]. Based on the maximal radii analysis, we have introduced the concepts of pore body and pore throat. An efficient and stable algorithm distinguishing between the introduced pore bodies and pore throats has been developed and verified. This algorithm also leads to establishing the respective volumes and connectivity
Acknowledgments
This research was partially supported by the Assistant Secretary for Fossil Energy, Office of Natural Gas and Petroleum Technology, through the National Petroleum Technology office, Natural Gas and Oil Technology Partnership under US Department of Energy contract no. DE-AC03-76SF00098 to Lawrence Berkeley National Laboratory. Partial support was also provided by gifts from Chevron and ConocoPhillips to UC Oil, Berkeley. We are thankful to Schlumberger for providing the synchrotron images of
References (64)
Flow in porous media—pore-network models and multiphase flow
Current Opin. Colloid Interface Sci.
(2001)- et al.
Detailed physics, predictive capabilities and upscaling for pore-scale models of multiphase flow
Adv. Water Resour.
(2002) - et al.
Investigating 3D geometry of porous media from high resolution images
Phys. Chem. Earth (A)
(1999) - et al.
Building skeleton models via 3-d medial surface/axis thinning algorithm
Graphical Models and Image Process.
(1994) - et al.
A 3D 6-subiteration thinning algorithm for extracting medial lines
Pattern Recog. Lett.
(1998) - et al.
New algorithms in 3D image analysis and their application to the measurement of a spatialized pore size distribution in soils
Phys. Chem. Earth, Part A: Solid Earth and Geodesy
(1999) - et al.
DXSoil, a library for 3D image analysis in soil science
Comput. Geosci.
(2002) - et al.
A reconstruction technique for three-dimensional porous media using image analysis and Fourier transform
J. Petroleum Sci. Eng.
(1998) - et al.
Modelling two-phase equilibrium in three-dimensional porous microstructures
Int. J. Multiphase Flow
(2000) Skeletons in n dimensions using shape primitives
Pattern Recog. Lett.
(2002)
On the geometry and topology of 3D stochastic porous media
J. Colloid Interface Sci.
Microscopic imaging of porous-media with X-ray computer-tomography
SPE Form. Eval.
Verification of a complete pore network simulator of drainage and imbibition
SPE J.
Impact of wettability on two-phase flow characteristics of sedimentary rock: quasi-static model
Water Resour. Res.
Process based reconstruction of sandstones and prediction of transport properties
Transp. Porous Media
Random Sets and Integral Geometry
Image Analysis and Mathematical Morphology
Sur quelque proprietés caractéristiques des ensemples bornés non convexes
Atti Acad. Naz. Lincei
An associative machine for dealing with the visual field and some of its biological implications
Prediction of petrophysical parameters based on digital core images
SPE Reservoir Eval. Eng.
Pore and throat size distributions measured from synchrotron x-ray tomographic images of Fontainebleau sandstones
J. Geophys. Res.—Solid Earth
Relative permeabilities from two- and three-dimensional pore-scale metwork modeling
Transp. Porous Media
Pore network modeling of wetting
Phys. Rev. E
NetSimCPP: object-oriented pore network simulator
Percolation theory and its application to groundwater hydrology
Water Resour. Res.
Cited by (392)
Study on the characterization method and evolution law of the three-dimensional pore structure in frozen loess under loading
2024, Cold Regions Science and TechnologyStudy of air flow and heat transfer in soybean piles based on CT
2024, Journal of Food EngineeringA new dual-scale pore network model with triple-pores for shale gas simulation
2024, Geoenergy Science and EngineeringUltrasonic intensification of flocculation filtration for kaolinite: Unraveling mechanisms and performance gains
2024, Separation and Purification TechnologyControl mechanism of trichloroethylene back diffusion by microstructure in a low permeability zone
2024, Journal of Hazardous MaterialsA continuum theory of diffusive bubble depletion in porous media
2024, Advances in Water Resources