Morphological decomposition of sandstone pore–space: fractal power-laws (original) (raw)
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Modeling, description, and characterization of fractal pore via mathematical morphology
Discrete Dynamics in Nature and Society, 2006
The aim of this paper is to provide description of fast, simple computational algorithms based upon mathematical morphology techniques to extract descriptions of pore channels—throats—and bodies and to represent them in 3D space, and to produce statistical characterization of their descriptions. Towards this goal, a model fractal binary pore is considered and is eroded recursively to generate different slices possessing decreasing degrees of porosity. By employing simple morphology-based approach, each slice of this pore space is decomposed into pore-channel, pore-throat, and pore-body, which are abstract structures that summarize the overall connectivity, orientation, and shape of the pore space. We consider the pore slices and their corresponding morphological quantities to stack them to further represent them in 3D space. We further provide a formulation essentially based on set theory to represent these three morphologic quantities to connect them appropriately across slices. Th...
2017
A fractal is an object or a structure that is self-similar in all length scales. Fractal geometry is an excellent mathematical tool used in the study of irregular geometric objects. The concept of the fractal dimension, D, as a measure of complexity is defined. The concept of fractal geometry is closely linked to scale invariance, and it provides a framework for the analysis of natural phenomena in various scientific and engineering domains. The relevance of the power law scaling relationships is discussed. Fractal characteristics of porous media and the characteristic method of the porous media are also discussed. Different methods of analysis on the permeability of porous media are discussed in this chapter.
Journal of Petroleum Exploration and Production Technology, 2018
Wallace sandstone has been extensively used by the construction industry for a long time in Nova Scotia. Apart from oxide analysis and a few strength-related parameter data found on some websites, petrophysical data regarding pore-sized distribution and fractal dimension are lacking. In the petroleum engineering literature, the spontaneous imbibition dynamics mechanism has been modeled where imbibition time has been linked to imbibition rise. One of the models links imbibition time to imbibition rise through a group of parameters that integrate the fractal dimension and sediment tortuosity. Based on the assumption of a bundle of parallel capillary tubes model found in the petrophysical literature, we have used the spontaneous imbibition model to derive an equation that links fractal dimension to porosity and permeability. Using literature source data on Wallace sandstone core samples, we have calculated the fractal dimension and pore size distribution using our equation. Results show that this sandstone has a significant level of heterogeneity. Calculation using another literature source data shows that our equation calculates fractal dimensions that are closer to those reported for the capillary pressure method. Although the assumption of bundle of parallel capillary tubes leads to deviations in calculated fractal dimensions using literature source data with experimentally determined porosity, permeability and fractal dimensions, our equation calculates meaningful values of fractal dimensions.
Dependence of the surface fractal dimension of soil pores on image resolution and magnification
European Journal of Soil Science, 2003
Two recent investigations have reported contradictory trends concerning the effect of image resolution on the surface fractal dimension of soil pores, evaluated via image analysis. In one case, dealing with a preferential flow pathway and an ideal fractal, image resolution had no influence on the estimated fractal dimension, whereas in the other case, involving images of soil thin sections, the surface fractal dimension decreased significantly with image resolution. In the present paper, we try to determine the extent to which these conflicting observations may have been due to the different ways in which image resolution was varied. By narrowing down (up to 400 times) the field of view on progressively smaller portions of a textbook surface fractal, the von Koch island, one causes its apparent surface fractal dimension to decrease significantly. On the other hand, changing the resolution of images of soil thin sections (up to 6 times), while keeping the magnification constant, does not lead to appreciable changes in the surface fractal dimension. These results demonstrate that there is no real conflict in earlier reports, as long as both the resolution and the magnification of images are taken into account in image-based evaluations of surface fractal dimensions of soil pores.
Journal of Physics: Conference Series, 2019
3D Fractal dimension as a complexity parameter of interconnected pore in 3D porous media is investigated. Four sets of 3D digital porous media with the size of 150×150×150 pixels are constructed by randomly depositing spherical grains with radius in the range of 10-20 pixels in an empty cube. Fractal dimension is obtained by means of the box counting method. Interconnected pore in 3D porous media are identified by applying cubical random walk in three different directions. The complexity of the interconnected pore is then described means of tortuosity, which is defined as the ratio between geodesic lengths of the interconnected pore to the sidelength of the medium. To observe the consistency of 3D fractal dimension as complexity parameter of 3D porous media structure, the four sets of samples of 3D porous media were generated with four different porosity. The correlation coefficient obtained from the samples are 0.1999, 0.3962, 0.9012 and 0.5682. Based on the results 3D fractal dimension shows a positive correlation with average tortuosity for each different porosity. It indicates that the higher 3D fractal dimension for 3D porous media, the interconnected pore of the porous media is more complex.
Influence of multifractal scaling of pore geometry on permeabilities of sedimentary rocks
Chaos, Solitons & Fractals, 1995
Scanning electron microscope pictures of thin sections of sedimentary rocks have been digitized, and their pore space geometry analysed using multifractal statistics. It is observed that such analysis can lead to characterization of different sedimentological environments. In particular it is shown that the information dimension D(1) of the multifractal spectrum of the pore space is correlated to air permeability values measured from thb corresponding core samples.
The fractal dimension of pore distribution patterns in variously-compacted soil
Soil and Tillage Research, 1998
Pore-size distribution pattern signi®cantly alters many soil properties affecting water movement and root growth. The distribution is largely in¯uenced by soil compaction but information on how to describe this effect is very limited. In this study we used the fractal dimension to characterize pore distribution patterns in variously-compacted soil. The soil used was an Orthic Luvisol (Lublin Region, Poland). The various soil compaction was obtained by wheel traf®c treatments: unwheeled (L); moderately compacted, 3 tractor passes (MC); strongly compacted, 8 tractor passes (SC). Pore distribution patterns of all pores (>0.3 mm) and water-conducting pores were analyzed with an image analyzer and the two-dimensional fractal dimension was estimated. All pores were analyzed on the drawings obtained from the polished surfaces of soil blocks 8Â9Â2 cm. To analyze the water-conducting pores, soil cores were taken in cylinders of length 20 cm and diameter 21.5 cm from the plots on which methylene blue solution was applied. The pores were analyzed on horizontal cuts at 2 cm depth intervals. Mean values of fractal dimensions for all pores (D p 2) in the horizontal plane of surface soil in L, MC and SC were 1.69, 1.42 and 1.35, respectively. In the vertical plane, the corresponding values were 1.48, 1.35 and 1.29. In L the fractal dimension re¯ected pores of different size ranging from a few tenths of a millimetre to a dozen or so millimetres with rather smooth walls. In MC the contribution of large pores decreased whereas that of medium-sized pores considerably increased forming net-like patterns. However in SC, the largest area was of massive structure with longitudinal cracks and scarce and unevenly-distributed larger pores. The D p 2 was linearly correlated with areal porosity (R0.965) and the arithmetic mean of the areas of pores (R0.914). Mean values of fractal dimensions for the blue staining patterns (D s 2) in the plough layer ranged from 1.06 to 1.12 in L and MC whereas it decreased to 0.94 in SC. The wide range of D s 2 values for 2 cm layers of upper soil in L re¯ected the high variability of pore structures in this treatment. In the subsoil, the D s 2 varied from 1.03 to 1.09 and re¯ected mostly the distribution pattern of earthworm channels. The values of fractal dimensions for roots (D r 2) re¯ected different branching and root growth in variously-compacted soil. This study showed that fractal analysis provides a relevant quanti®cation of the changes of pore and root structure in relation to soil compaction. #
Geophysical Journal International, 2000
The box-counting algorithm is the most commonly used method for evaluating the fractal dimension D of natural images. However, its application may easily lead to erroneous results. In a previous paper (Gonzato et al. 1998) we demonstrated that a crucial bias is introduced by insu¤cient sampling and/or by uncritical application of the regression technique. This bias turns out to be common in many practical applications. Here it is shown that an equally important additional bias is introduced by the orientation, placement and length of the digitized object relative to that of the initial box. Some additional problems are introduced by objects containing unconnected parts, since the discontinuities may or may not be indicative of a fractal pattern. Last, but certainly not least in magnitude, the thickness of the digitized pro¢le, which is implicitly controlled by the scanner resolution versus the image line thickness, plays a fundamental role. All of these factors combined introduce systematic errors in determining D, the magnitudes of which are found to exceed 50 per cent in some cases, crucially a¡ecting classi¢cation. To study these errors and minimize them, a program that accounts for image digitization, zooming and automatic box counting has been developed and tested on images of known dimension. The code automatically extracts the unconnected parts from a digitized shape given as input, zooms each part as optimally as possible, and performs the box-counting algorithm on a virtual screen. The size of the screen can be set to meet the sampling requirement needed to produce stable and reliable results. However, this code does not provide image vectorization, which must be performed prior to running this program. A number of image vectorizing codes are available that successfully reduce the thickness of the image parts to one pixel. Image vectorization applied prior to the application of our code reduces the sampling bias for objects with known fractal dimension to around 10^20 per cent. Since this bias is always positive, this e¡ect can be readily compensated by a multiplying factor, and estimates of the fractal dimension accurate to about 10 per cent are e¡ectively possible.
Estimation of fractal dimension through morphological decomposition
Chaos, Solitons & Fractals, 2004
Set theory based morphological transformations have been employed to decompose a binary fractal by means of discrete structuring elements such as square, rhombus and octagon. This decomposition provides an alternative approach to estimate fractal dimensions. The fractal dimensions estimated through this morphological decomposition procedure by employing different structuring elements are considerably similar. A color-coding scheme is adapted to identify the several sizes of decomposed non-overlapping disks (DNDs) that could be fit into a fractal. This exercise facilitates to test the number-radius relationship from which the fractal dimension has been estimated for a Koch Quadric, which yield the significantly similar values of 1.67 ± 0.05 by three structuring elements. In addition to this dimension, by considering the number of DNDs of various orders (radii) and the mean diameter of disks (MDDs) of corresponding order, two topological quantities namely number ratio (R B) and mean diameter ratio (R L) are computed, employing which another type of fractal dimension is estimated as log R B log R L. These results are in accord with the fractal dimensions computed through number-radius relationship, and connectivity network of the Koch Quadric that is reported elsewhere.