Power-law Length Distribution Research Articles

A fractal power-law approach for investigating the interactions between reservoir microstructure and gas migration

As one of the most significant clean energy sources, the migration characteristics and extraction processes of coalbed methane (CBM) have been extensively studied. The structural distribution and evolutionary characteristics of reservoir fractures, as the main conduit for gas migration, significantly affect the permeability and gas production. However, few models have been able to quantitatively and accurately explore reservoir micro–macro interactions under coupled thermal-fluid–solid effects. This work develops a new highly coupled model based on the widely adopted power-law function to quantify reservoir thermal conduction effect, gas pressure evolution, reservoir deformation, in situ stress, the adsorption–desorption effect, and reservoir microstructure evolution. Three parameters are adopted to quantitatively characterize the reservoir structure: (1) fracture power index αf (to characterize the fracture density), (2) fracture length ratio rf (to characterize the fracture size), and (3) the maximum fracture length l. The results demonstrate that the fractal network is a special kind of network in the power-law length distribution. The proposed power-law seepage model is able to accurately characterize the evolution of reservoir microstructure and the impact of microevolution on extraction under multi-field coupling effects, compared to the traditional power-law model. The proposed model can provide a good theoretical and practical support for the study of CBM migration and extraction.

Multiscale study of statistical fractal DFN model and corresponding seismic images

The work addresses the development of a statistical model of a discrete fracture network (DFN) using core, outcrop, and seismic data. We consider the DFN model defined by the fractal distribution of fracture centers and a power-law distribution of fracture lengths. We perform the statistical analysis of model realization on different spatial scales to investigate the possibility to evaluate the corresponding correlation fractal dimension and power exponent. Reproducing the statistical parameters estimated from seismic images is investigated by comparison with the parameters of the original model. We conclude that the analysis of the finite subdomain of the considered model of fractures system makes it possible to estimate the model's statistical characteristics with a decrease in the system's linear size by about one order of magnitude. On the other hand, the system built on the base of seismic images does not reproduce the power-law probability density of fracture length distribution. Also, the distribution of fracture centers in this system does not reflect the fractal nature and is uniform. The results obtained in this work show that data obtained from outcrops and core samples can be used to build a statistical DFN model. On the other hand, an explicit DFN model cannot be recovered based on seismic data.

AN ANALYTICAL MODEL FOR EFFECTIVE THERMAL CONDUCTIVITY OF THE MEDIA EMBEDDED WITH FRACTURE NETWORKS OF POWER LAW LENGTH DISTRIBUTIONS

The effective thermal conductivity of porous media is of steady interest in the design of new materials. In this study, an analytical model for dimensionless effective thermal conductivity of media embedded with fracture networks of the power law length distributions is proposed. It is found that the proposed dimensionless effective thermal conductivity is a function of micro-structural parameters of media, such as the porosity [Formula: see text] and fracture orientation (dip [Formula: see text] and azimuth [Formula: see text]). The present results show that the dimensionless effective thermal conductivity of the media increases with the increase in the ratio [Formula: see text] and the power law exponent [Formula: see text] at [Formula: see text]. Inversely, when [Formula: see text], it decreases with the increase in the power law exponent [Formula: see text]. In addition, the dimensionless effective thermal conductivity is gradually independent of the orientation as the ratio [Formula: see text]. The present model may provide a significant insight into the mechanism of heat transfer in the media embedded with fracture networks of power law length distributions.

Impact of horizontal spatial clustering in two-dimensional fracture networks on solute transport

This study explores the prevalence of fracture clustering in natural networks and its influence on fluid flow and advective transport. Fracture networks from a compilation of previously published maps that encompass igneous, metamorphic and sedimentary rock types, various geological and tectonic settings, and a broad distribution of length scales are analyzed using a two-point correlation technique to characterize the spatial scaling between fracture barycenters. The correlation dimension (Dc) ranges from 1.47 to 1.89 with a median of 1.8. These results suggest that fracture networks often exhibit spatial clustering and are rarely Poissonian as commonly assumed. Synthetic Discrete Fracture Networks (DFNs) with constant and variable transmissivity applied to all fractures are generated for three clustering (Dc = 1.6, Dc = 1.8, and Poissonian) and power law length exponent (α = 1.4, α = 1.8, and α = 2.2) scenarios. Median particle breakthrough times are shown to increase by 10–70% as spatial clustering transitions from Poissonian to highly clustered (Dc = 1.6). Median particle breakthrough times also increase, by a factor of 1.2–3.4 (20–240%), when the power law length exponent increases from α = 1.4 to α = 2.2. As clustering increases (i.e., Dc reduces), breakthrough curves become highly variable and exhibit features indicative of enhanced solute retention. The increase in median breakthrough times with clustering persists with different network size and transmissivity distribution. These results indicate that spatial clustering and length exponent both significantly influence transport. The use of power-law length distributions in generating fracture networks is relatively common, whereas the spatial organization of fractures is rarely included in transport studies of fractured rock. Structural heterogeneity introduced by spatially clustered organization of fractures leads to substantial complexity in flow paths which is not captured by Poissonian DFNs.

A MULTI-FIELD COUPLED SEEPAGE MODEL FOR COAL SEAM WITH FRACTURES OF POWER LAW LENGTH DISTRIBUTIONS

In the process of gas mining, the fracture distribution with power law length and the pore structure with adsorption effect have an important influence on the coal seam permeability. In recent years, the research on the internal structure of coal seam and the fluid flow mechanism has attracted a large number of researchers. In this paper, by considering the coal matrix deformation caused by adsorption, a pore-fracture model coupled with the multi-field effects and with power law length distribution of fractures in coal seam is established based on the fractal theory for porous media. In this work, we study the influences of the power law exponent [Formula: see text] of fracture length and the ratio [Formula: see text] of the minimum to maximum fracture lengths on the permeability of coal seam and the evolution mechanism of permeability with the structural and mechanical parameters of coal seam. It is found that the permeability of coal seam is inversely proportional to [Formula: see text], directly proportional to [Formula: see text], and to Langmuir volume constant and Langmuir volume strain constant. Compared with other factors, the power law component [Formula: see text] of fractures has the most significant effect on the coal seam permeability.

Equivalent Permeability Distribution for Fractured Porous Rocks: The Influence of Fracture Network Properties

Equivalent fracture models are widely used for simulations of groundwater exploitation, geothermal reservoir production, and solute transport in groundwater systems. Equivalent permeability has a great impact on such processes. In this study, equivalent permeability distributions are investigated based on a state-of-the-art numerical upscaling method (i.e., the multiple boundary method) for fractured porous rocks. An ensemble of discrete fracture models is created based on power law length distributions. The equivalent permeability is upscaled from discrete fracture models based on the multiple boundary method. The results show that the statistical distributions of equivalent permeability tensor components are highly related to fracture geometry and differ from each other. For the histograms of the equivalent permeability, the shapes of k x x and k y y change from a power law-like distribution to a lognormal-like distribution when the fracture length and the number of fractures increase. For the off-diagonal component k x y , it is a normal-like distribution and its range expands when the fracture length and the number of fractures increase. The mean of diagonal equivalent permeability tensor components increases linearly with the fracture density. The analysis helps in generating stochastic equivalent permeability models in fractured porous rocks and reduces uncertainties when applying equivalent fracture models.

SEEPAGE PROPERTIES OF ROCK FRACTURES WITH POWER LAW LENGTH DISTRIBUTIONS

Fractures with power law length distributions abound in nature such as carbonate oil and gas reservoirs, sandstone, hot dry rocks, etc. The fluid transport properties and morphology characterization of fracture networks have fascinated numerous researchers to investigate for several decades. In this work, the analytical models for fracture density and permeability are extended from fractal fracture network to general fracture network with power law length distributions. It is found that the fracture density is related to the power law exponents [Formula: see text] and the area porosity [Formula: see text] of fracture network. Then, a permeability model for the fracture length distribution with general power law exponent [Formula: see text] and the power law exponent [Formula: see text] for fracture length versus aperture is proposed based on the well-known cubic law in individual fracture. The analytical expression for permeability of fractured networks is found to be a function of power law exponents [Formula: see text], area porosity [Formula: see text] of fracture network, and the micro-structural parameters (maximum fracture length [Formula: see text], fracture azimuth [Formula: see text] and fracture dip angle [Formula: see text]). The present model may shed light on the mechanism of seepage in fracture networks with power law length distributions.

Kosterlitz-Thouless scaling at many-body localization phase transitions

We propose a scaling theory for the many-body localization (MBL) phase transition in one dimension, building on the idea that it proceeds via a 'quantum avalanche'. We argue that the critical properties can be captured at a coarse-grained level by a Kosterlitz-Thouless (KT) renormalization group (RG) flow. On phenomenological grounds, we identify the scaling variables as the density of thermal regions and the lengthscale that controls the decay of typical matrix elements. Within this KT picture, the MBL phase is a line of fixed points that terminates at the delocalization transition. We discuss two possible scenarios distinguished by the distribution of rare, fractal thermal inclusions within the MBL phase. In the first scenario, these regions have a stretched exponential distribution in the MBL phase. In the second scenario, the near-critical MBL phase hosts rare thermal regions that are power-law distributed in size. This points to the existence of a second transition within the MBL phase, at which these power-laws change to the stretched exponential form expected at strong disorder. We numerically simulate two different phenomenological RGs previously proposed to describe the MBL transition. Both RGs display a universal power-law length distribution of thermal regions at the transition with a critical exponent $\alpha_c=2$, and continuously varying exponents in the MBL phase consistent with the KT picture.

Seismic Attenuation and Stiffness Modulus Dispersion in Porous Rocks Containing Stochastic Fracture Networks

AbstractUnderstanding seismic attenuation and modulus dispersion mechanisms in fractured rocks can result in significant advances for the indirect characterization of such environments. In this paper, we study attenuation and modulus dispersion of seismic waves caused by fluid pressure diffusion (FPD) in stochastic 2‐D fracture networks, allowing for a state‐of‐the‐art representation of natural fracture networks by a power law length distribution. To this end, we apply numerical upscaling experiments consisting of compression and shear tests to our samples of fractured rocks. The resulting P and S wave attenuation and modulus dispersion behavior is analyzed with respect to the density, the length distribution, and the connectivity of the fractures. We focus our analysis on two manifestations of FPD arising in fractured rocks, namely, fracture‐to‐background FPD at lower frequencies and fracture‐to‐fracture FPD at higher frequencies. Our results indicate that FPD is sensitive not only to the fracture density but also to the geometrical characteristics of the fracture length distributions. In particular, our study suggests that information about the local connectivity of a fracture network could be retrieved from seismic data. Conversely, information about the global connectivity, which is directly linked to the effective hydraulic conductivity of the probed volume, remains rather difficult to infer.

Modeling flow in naturally fractured reservoirs: effect of fracture aperture distribution on dominant sub-network for flow

Fracture network connectivity and aperture (or conductivity) distribution are two crucial features controlling flow behavior of naturally fractured reservoirs. The effect of connectivity on flow properties is well documented. In this paper, however, we focus here on the influence of fracture aperture distribution. We model a two-dimensional fractured reservoir in which the matrix is impermeable and the fractures are well connected. The fractures obey a power-law length distribution, as observed in natural fracture networks. For the aperture distribution, since the information from subsurface fracture networks is limited, we test a number of cases: log-normal distributions (from narrow to broad), power-law distributions (from narrow to broad), and one case where the aperture is proportional to the fracture length. We find that even a well-connected fracture network can behave like a much sparser network when the aperture distribution is broad enough (α ≤ 2 for power-law aperture distributions and σ ≥ 0.4 for log-normal aperture distributions). Specifically, most fractures can be eliminated leaving the remaining dominant sub-network with 90% of the permeability of the original fracture network. We determine how broad the aperture distribution must be to approach this behavior and the dependence of the dominant sub-network on the parameters of the aperture distribution. We also explore whether one can identify the dominant sub-network without doing flow calculations.

Shape factor for dual-permeability fractured reservoir simulation: Effect of non-uniform flow in 2D fracture network

The flow properties of naturally fractured reservoirs are dominated by flow through the fractures. In a previous study we showed that even a well-connected fracture network behaves like a much sparser network when the aperture distribution is broad enough: i.e., most fractures can be eliminated while leaving a sub-network with virtually the same permeability as the original fracture network. In this study, we focus on the influence of eliminating unimportant fractures which carry little flow on the inferred characteristic matrix-block size. We model a two-dimensional fractured reservoir in which the fractures are well-connected. The fractures obey a power-law length distribution, as observed in natural fracture networks. For the aperture distribution, because information from the subsurface is limited, we test a number of cases: log-normal distributions (from narrow to broad), and power-law distributions (from narrow to broad). The matrix blocks in fractured reservoirs are of varying sizes and shapes; we adopt the characteristic radius and the characteristic length to represent the characteristic matrix-block size. We show how the characteristic matrix-block sizes increase from the original fracture network to the dominant sub-network. This suggests that the matrix-block size, or shape factor, used in dual-porosity/dual-permeability waterflood or enhanced oil recovery (EOR) simulations or in homogenization should be based not on the entire fracture population but on the sub-network that carries almost all of the injected fluid (water or EOR agent).

Exact Length Distribution of Filamentous Structures Assembled from a Finite Pool of Subunits.

Self-assembling filamentous structures made of protein subunits are ubiquitous in cell biology. These structures are often highly dynamic, with subunits in a continuous state of flux, binding to and falling off of filaments. In spite of this constant turnover of their molecular parts, many cellular structures seem to maintain a well-defined size over time, which is often required for their proper functioning. One widely discussed mechanism of size regulation involves the cell maintaining a finite pool of protein subunits available for assembly. This finite pool mechanism can control the length of a single filament by having assembly proceed until the pool of free subunits is depleted to the point when assembly and disassembly are balanced. Still, this leaves open the question of whether the same mechanism can provide size control for multiple filamentous structures that are assembled from a common pool of protein subunits, as is often the case in cells. We address this question by solving the steady-state master equation governing the stochastic assembly and disassembly of multifilament structures made from a shared finite pool of subunits. We find that, while the total number of subunits within a multifilament structure is well-defined, individual filaments within the structure have a wide, power-law distribution of lengths. We also compute the phase diagram for two multifilament structures competing for the same pool of subunits and identify conditions for coexistence when both have a well-defined size. These predictions can be tested in cell experiments in which the size of the subunit pool or the number of filament nucleators is tuned.

Thin-finger growth and droplet pinch-off in miscible and immiscible displacements in a periodic network of microfluidic channels

We report the results of experimental and numerical studies of two-phase flow in a periodic, rectangular network of microfluidic channels. This geometry promotes the formation of anisotropic, dendrite-like structures during viscous fingering experiments. The dendrites then compete with each other for the available flow, which leads to the appearance of hierarchical growth pattern. Combining experiments and numerical simulations, we analyze different growth regimes in such a system, depending on the network geometry and fluid properties. For immiscible fluids, a high degree of screening is present which results in a power-law distribution of finger lengths. Contrastingly, for miscible fluids, strong lateral currents of displaced fluid lead to the detachment of the heads of the longest fingers from their roots, thus preventing their further growth.

FROM LÉVY WALKS TO SUPERDIFFUSIVE SHOCK ACCELERATION

In this paper, we present a general scenario for nondiffusive transport and we investigate the influence of anomalous, superdiffusive transport on Fermi acceleration processes at shocks. We explain why energetic particle superdiffusion can be described within the Levy walk framework, which is based on a power-law distribution of free path lengths and on a coupling between free path length and free path duration. A self-contained derivation of the particle mean square displacement, which grows as (Δx {sup 2}) = 2D {sub α} t {sup α} with α > 1, and the particle propagator, is presented for Levy walks, making use of a generalized version of the Montroll-Weiss equation. We also derive for the first time an explicit expression for the anomalous diffusion coefficient D {sub α} and we discuss how to obtain these quantities from energetic particle observations in space. The results are applied to the case of particle acceleration at an infinite planar shock front. Using the scaling properties of the Levy walk propagator, the energy spectral indices are found to have values smaller than the ones predicted by the diffusive shock acceleration theory. Furthermore, when applying the results to ions with energies of a few MeV accelerated atmore » the solar wind termination shock, the estimation of the anomalous diffusion coefficient associated with the superdiffusive motion gives acceleration times much smaller than the ones related to normal diffusion.« less

Evaluation of sampling methods for fracture network characterization using outcrops

Outcrops provide valuable information for the characterization of fracture networks. Sampling methods such as scanline sampling, window sampling, and circular scanline and window methods are available to measure fracture network characteristics in outcrops and from well cores. These methods vary in their application, the parameters they provide and, therefore, have advantages and limitations. We provide a critical review on the application of these sampling methods and apply them to evaluate two typical natural examples: (1) a large-scale satellite image from the Oman Mountains, Oman (120,000 m2 [1,291,669 ft2]), and (2) a small-scale outcrop at Craghouse Park, United Kingdom (19 m2 [205 ft2]). The differences in the results emphasize the importance to (1) systematically investigate the required minimum number of measurements for each sampling method and (2) quantify the influence of censored fractures on the estimation of fracture network parameters. Hence, a program was developed to analyze 1300 sampling areas from 9 artificial fracture networks with power-law length distributions. For the given settings, the lowest minimum number of measurements to adequately capture the statistical properties of fracture networks was found to be approximately 110 for the window sampling method, followed by the scanline sampling method with approximately 225. These numbers may serve as a guideline for the analyses of fracture populations with similar distributions. Furthermore, the window sampling method proved to be the method that is least sensitive to censoring bias. Reevaluating our natural examples with the window sampling method showed that the existing percentage of censored fractures significantly influences the accuracy of inferred fracture network parameters.

ENTROPY-GROWTH-BASED MODEL OF EMOTIONALLY CHARGED ONLINE DIALOGUES

We analyze emotionally annotated massive data from Internet relay chat (IRC) as well as from BBC forum website and model the dialogues between chat participants by assuming that the driving force for the discussion is the entropy growth of emotional probability distribution. This process is claimed to be responsible for a power-law distribution of the discussion lengths observed in the dialogues. We perform numerical simulations based on the noticed phenomenon obtaining a good agreement with the real data. Finally, we propose a method to artificially prolong the duration of the discussion that relies on the entropy of emotional probability distribution.

Relations between the scaling exponents, entropies, and energies of fracture networks

Abstract Fracture networks commonly show power-law length distributions. Thermodynamic principles form the basis for understanding fracture initiation and growth, but have not been easily related to the power-law size distributions. Here we present the power-law scaling exponents and the calculated entropies of fracture networks from the Holocene part of the plate boundary in Iceland. The total number of tension fractures and normal faults used in these calculations is 565 and they range in length by five orders of magnitude. Each network can be divided into populations based on ‘breaks’ (abrupt changes) in the scaling exponents. The breaks, we suggest, are related to the comparatively long and deep fractures changing from tension fractures into normal faults and penetrating the contacts between the Holocene lava flows and the underlying and mechanically different Quaternary rocks. The results show a strong linear correlation (r = 0.84) between the population scaling exponents and entropies. The correlation is partly explained by the entropy (and the scaling exponent) varying positively with the arithmetic average and the length range (the difference between the longest and the shortest fracture) of the populations in each network. We show that similar scaling laws apply to other lineaments, such as streets. We propose that the power-law size distributions of fractures are a consequence of energy requirements for fracture growth.

Dislocation arrangement in small crystal volumes determines power-law size dependence of yield strength

It is by now well-known that micron-sized metallic crystals exhibit a smaller-being-stronger size effect: the yield strength σ varies with specimen size D approximately as a power-law σ∼D−m, and the exponent m has been found to vary within a range of ∼0.3–1.0 for different metals. However, little is known about why such a power-law comes into play, and what determines the actual value of the exponent m involved. This work shows that if the yield strength is determined by the Taylor interaction mechanism within the initial dislocation network, then for the size dependence of strength to be of the power-law relation observed, it is necessary for the mesh lengths L of the dislocation network to be power-law distributed, i.e. p(L)∼L−q. In such a case, the exponent m of the size effect is predicted to be inversely proportional to the sum of q the exponent of the mesh-length distribution and n the exponent of the dislocation velocity vs. stress law. To verify these predictions, compression experiments on aluminum micro-pillars with different pre-strains from 0% to 15% were carried out. The different pre-strains led to different initial dislocation networks, as well as different exponent m in the size dependence of strength. Box-counting analyses of transmission electron micrographs of the initial dislocation networks showed that the 2-D projected dislocation patterns were approximate fractals. On increasing pre-strain, the exponent m for the size dependence of strength was found to decrease while the fractal dimension of the initial dislocation patterns increased, thus verifying the inverse relationship between the two quantities. These findings show that the commonly observed power-law scaling of strength with size is due to an approximate power-law distribution of the initial dislocation mesh lengths, which also appears to be a robust feature in deformed metals. Furthermore, for a given metal, it is the exponent q of the initial mesh-length distribution which determines the value of the exponent m in the size dependence of strength.

A Simple Formula To Estimate 2D Fracture Connectivity

Summary A simple expression is derived for average fracture intersections per fracture, λ, as the product of fracture density, average length, and angular standard deviation in 2D fracture models. The calculated λ value quite accurately estimates the observed λ values for a variety of cases including one- or two-fracture sets, truncated or not-truncated fractures, and log-normal or power law length distribution. The formula fails to estimate the number of fracture intersections accurately when fractures are clustered. Fracture clusters could either be fault-related fracture corridors or highly fractured layers. In either case, it is necessary to calculate λ for each fracture corridor or each fractured layer with a different fracture density. The relationship between λ and fracture connectivity was investigated using several stochastic fracture models. λ seems to be a reasonably good estimator of fracture connectivity, which we define as the percentage of fractures within the largest interconnected fracture aggregate. The relationship between the number of fracture intersections and percolation threshold is, however, weak because percolation threshold is dependent on the relative length of fractures with respect to the distance between two opposite ends of a fractured medium.

제3기 결정질 응회암에서 발달하는 미세균열의 길이 분포에 대한 통계적 분석

제3기 결정질 응회암에서 발달하는 미세균열 모집단의 길이분포에 대한 스케일링 성질을 조사하였다. 15개 방향각 및 5개 그룹 (I~V)에 대한 길이범위의 분포도에서 미세균열의 방향성에 따른 평균길이의 체계적인 변화가 나타난다. 분포도는 거의 남-북방향을 경계로 하여 좌우 대칭형태를 취하는 것이 특징이다. 미세균열의 모집단에 대한 길이-누적빈도 도표의 전 영역은 상관곡선의 분포양상에 의하여 3개 구간으로 구분할 수 있다. 특히, 5개 그룹에 대한 각 도표의 선형의 중앙구간은 멱함수 분포를 지시한다. 5개 그룹에 대한 중앙의 선형구간의 빈도비는 46.6%~67.8T의 범위이다. 한편 각 그룹에 대한 선형의 중앙구간의 기울기는 그룹 V(<TEX>$N60{\sim}90^{\circ}E$</TEX>, -2.02) > 그룹 IV(<TEX>$N20{\sim}60^{\circ}E$</TEX>, -1.55) > 그룹 I(<TEX>$N60{\sim}90^{\circ}W$</TEX>, -1.48), 그룹 II(<TEX>$N10{\sim}60^{\circ}W$</TEX>, -1.48) > 그룹 III(<TEX>$N10^{\circ}W{\sim}N20^{\circ}E$</TEX>, -1.06)의 순으로 나타난다. 거의 멱함수의 길이분포를 따르는 부집단(5개 그룹)에서는 지수(-1.06~-2.02)의 범위가 넓다. 5개 그룹간의 이러한 지수의 상대적인 차이는 방향성 효과의 중요성을 강조한다. 또한, 곡선의 하부에서의 기울기의 분리는 보다 긴 미세균열의 급격한 발달을 대변하며, 멱함수 지수의 감소로 반영된다. 특히, 이러한 분포양식은 <TEX>$N10{\sim}20^{\circ}E,\;N10{\sim}20^{\circ}W$</TEX> 및 <TEX>$N60{\sim}70^{\circ}W$</TEX>의 방향각에 대한 도표에서 볼 수 있다. 이들 3개 방향각은 연구지역 일대에서 발달한 단층의 주방향과 부합한다. 15개 방향각에 대한 길이-누적빈도 도표의 개개 특성을 보여주는 분포도를 작성하였다 상기한 도표들을 3개 그룹(A, B and C)의 범주에 따라 배열함으로서 이들 그룹간 길이-빈도 분포의 차이를 용이하게 도출할 수 있다. 분포도는 미세균열 조들에 대한 개별적인 분리의 중요성을 보여준다. 관계도에서, 보다 짧은 미세균열의 출현빈도는 그룹A > 그룹 B > 그룹 C의 순서를 보인다. 이들 3가지 유형의 분포양상은 미세균열이 성장하는 동안 발생한 과정들에 대한 중요한 정보를 드러낼 수 있다. The scaling properties on the length distribution of microcrack populations from Tertiary crystalline tuff are investigated. From the distribution charts showing length range with 15 directional angles and five groups(I~V), a systematic variation appears in the mean length with microcrack orientation. The distribution charts are distinguished by the bilaterally symmetrical pattern to nearly N-S direction. The whole domain of the length-cumulative frequency diagram for microcrack populations can be divided into three sections in terms of phases of the distribution of related curves. Especially, the linear middle section of each diagram of five groups represents a power-law distribution. The frequency ratio of linear middle sections of five groups ranges from 46.6% to 67.8%. Meanwhile, the slope of linear middle section of each group shows the order: group V(<TEX>$N60{\sim}90^{\circ}E$</TEX>, -2.02) > group IV(<TEX>$N20{\sim}60^{\circ}E$</TEX>, -1.55) > group I(<TEX>$N60{\sim}90^{\circ}W$</TEX>, -1.48), group II(<TEX>$N10{\sim}60^{\circ}W$</TEX>, -1.48) > group III(<TEX>$N10^{\circ}W{\sim}N20^{\circ}E$</TEX>, -1.06). Five sub-populations(five groups) that closely follow the power-law length distribution show a wide range in exponents( -1.06 - -2.02). These differences in exponent among live groups emphasizes the importance of orientation effect. In addition, breaks in slope in the lower parts of the related curves represent the abrupt development of longer lengths, which is reflected in the decrease in the power-law exponent. Especially, such a distribution pattern can be seen from the diagram with <TEX>$N10{\sim}20^{\circ}E,\;N10{\sim}20^{\circ}W$</TEX> and <TEX>$N60{\sim}70^{\circ}W$</TEX> directional angles. These three directional angles correspond with main directions of faults developed around the study area. The distribution chart showing the individual characteristics of the length-cumulative frequency diagrams for 15 directional angles were made. By arraying above diagrams according to the categories of three groups(A, B and C), the differences in length-frequency distributions among these groups can be easily derived. The distribution chart illustrates the importance of analysing microcrack sets separately. From the related chart, the occurrence frequency of shorter microcracks shows the order: group A > group B > group C. These three types of distribution patterns could reveal important information on the processes occurred during microcrack growth.