Fracture Length Distribution Research Articles

Statistical Analysis of Fracture-Length Distribution Sampled Under the Truncation and Censoring Effects

The present paper addresses statistical analysis and estimation of fracture- length distributions at scales influenced by the truncation and censoring effects. The computational method employed here uses fracture-length distributions of a given set of measurements and information about observational constraints (i.e., window of observation) to estimate the probability density of the truncated and censored parts of fracture data sets. The results are benchmarked against power-law based maximal likelihood estimations commonly used for the same purpose. The relationship between the accuracy of estimates and size of the window of observation is studied. The utility of employing statistical models with arbitrary probability distributions of fracture lengths in order to provide a valid statistical model approximation is also considered. A verification of the suggested approximation using the Kolmogorov-Smirnov test applied to truncated and censored data is proposed. Numerical computations show that the proposed method can represent an essential improvement compared to other commonly employed techniques.

Fracture network evaluation program (FraNEP): A software for analyzing 2D fracture trace-line maps

Fractures, such as joints, faults and veins, strongly influence the transport of fluids through rocks by either enhancing or inhibiting flow. Techniques used for the automatic detection of lineaments from satellite images and aerial photographs, LIDAR technologies and borehole televiewers significantly enhanced data acquisition. The analysis of such data is often performed manually or with different analysis software. Here we present a novel program for the analysis of 2D fracture networks called FraNEP (Fracture Network Evaluation Program). The program was developed using Visual Basic for Applications in Microsoft Excel™ and combines features from different existing software and characterization techniques. The main novelty of FraNEP is the possibility to analyse trace-line maps of fracture networks applying the (1) scanline sampling, (2) window sampling or (3) circular scanline and window method, without the need of switching programs. Additionally, binning problems are avoided by using cumulative distributions, rather than probability density functions. FraNEP is a time-efficient tool for the characterisation of fracture network parameters, such as density, intensity and mean length. Furthermore, fracture strikes can be visualized using rose diagrams and a fitting routine evaluates the distribution of fracture lengths. As an example of its application, we use FraNEP to analyse a case study of lineament data from a satellite image of the Oman Mountains.

A methodology for pseudo-genetic stochastic modeling of discrete fracture networks

Stochastic simulation of fracture systems is an interesting approach to build a set of dense and complex networks. However, discrete fracture models made of planar fractures generally fail to reproduce the complexity of natural networks, both in terms of geometry and connectivity. In this study a pseudo-genetic method is developed to generate stochastic fracture models that are consistent with patterns observed on outcrops and fracture growth principles. The main idea is to simulate evolving fracture networks through geometric proxies by iteratively growing 3D fractures. The algorithm defines heuristic rules in order to mimic the mechanics of fracture initiation, propagation, interaction and termination. The growth process enhances the production of linking structure and impacts the connectivity of fracture networks. A sensitivity study is performed on synthetic examples. The method produces unbiased fracture dip and strike statistics and qualitatively reproduces the fracture density map. The fracture length distribution law is underestimated because of the early stop in fracture growth after intersection.

Fracture model of the Upper Freeport coal: Marshall County West Virginia pilot ECBMR and CO2 sequestration site

A model discrete fracture or cleat network was developed for the Pennsylvanian Upper Freeport coal seam at a carbon sequestration pilot site in southeastern Marshall Co., West Virginia, U. S. A. The model reservoir cleat network was developed using 3D seismic post-stack processing and interpretation workflows combined with observations of fracture systems in the surrounding region. The Upper Freeport coal is unminable in this area and served as a test for the feasibility of combined enhanced coalbed methane recovery (ECBMR) operations with CO2 storage in unminable coals. Well log-derived subsurface structure maps were used to calibrate 3D seismic time-to-depth conversion. Post-stack processing was used to enhance and identify subtle discontinuities in the seismic data that could be related to field-scale faults, fracture zones or stratigraphic heterogeneity. The cleat model was developed using power law fracture length and aperture distributions. Spatial variations of cleat intensity in the model were controlled using measures of localized structural discontinuity extracted from 3D seismic attribute workflows. The cleat model identifies a permeable fracture facies within the reservoir most likely to be impacted by CO2 injection and migration. Areas of concentrated seismic discontinuity bear some spatial association to areas of positive vertical deformation observed in a tiltmeter survey of the site and attributed to CO2 injection. The model could help focus time-lapse processing and interpretation in regions estimated to have higher permeability and porosity and higher probability for change of acoustic properties in response to CO2 injection and methane production. The approach developed here has general applicability to identification of fracture facies and development of unconventional naturally fractured reservoirs in other lithologies and basins.

Fracture analysis in Tabnak hydrocarbon field of Iran by using fractal geometry and multi-fractal analysis

High gas production from the Dashtak formation of Tabnak hydrocarbon field in Fars province, Iran, indicates the presence of natural fractured reservoir whose production potential is dominated by the structural fracture. The connectivity of fractured media depends upon the power–law exponent and the fracture density. Fracture pattern traces obtained from the outcrops of producing formations of six different stations in Tabnak hydrocarbon field. 2D fracture network maps of Tabnak hydrocarbon field have been analyzed from their scaling properties. The fractal analysis of fracture intensity showed heterogeneous multi-fractal structure characterized by generalized dimensions. Distribution of fracture lengths exhibits power–law behavior with specific exponent. Scaling laws serve to make extrapolations and to study the fracture connectivity related to scale. Fracture distribution model and reservoir productivity can be estimated, which are of great interest in decision making to optimize gas production.

Determination of geometric parameters of fracture networks using 1D data

To simulate a suitable fracture network for hydrogeological modelling, input statistical data of the individual faults, as well as fracture sets, should be determined first using either 2D sections or 1D scanlines. Although the accuracy of this measurement is fundamental, exact determination is rather problematic and is usually possible only at a particular scale. This paper introduces a coupled method for computing length exponent ( E) and spatial density ( D c ), the two most essential parameters for modelling fracture networks. To calculate the length exponent, data sets of at least two independent imaging processes are needed. Utilizing different sensitivity thresholds of the two methods and the well-known analytical form of a fracture length distribution function, its parameters can be calculated. To estimate the spatial density of fracture centres in 3D, the series of intersections should be analysed as a fractional Brownian motion and then calibrated with virtual wells simulated with optional modelling software. The method makes fracture intensity logging possible along scanlines. Based on these approaches, there is no need to import fracture parameters from the outcrop survey or from other parts of the reservoir, because all geometric information of the fracture system refers to the rock body under examination. Using site-specific parameters makes fracture network modelling more reliable.

Flow Dimension and Anomalous Diffusion of Aquifer Tests in Fracture Networks

Characterization and modeling of aquifers is particularly challenging in fractured media, where flow is concentrated into channels that are poorly suited to traditional approaches to analysis. The generalized radial flow model is an alternative method for hydraulic test interpretation that infers an additional parameter, the flow dimension n, to describe the flow geometry. This study was a Monte Carlo analysis of numerical models of aquifer tests in two‐dimensional fractured media, with the objective of understanding the relationships between the parameters of a discrete fracture network (DFN) with lengths distributed as a power law, the flow dimension, and the regime of hydraulic diffusion (e.g., Fickian or non‐Fickian). The diffusion regime of each realization was evaluated using 〈R2〉 ∼ tk, where t is time, 〈R2〉 is the mean squared radius of hydraulic diffusion, and an apparent value of k = 1 indicates normal (Fickian) diffusion and k < 1 indicates anomalous (non‐Fickian) diffusion. For the DFN model, the apparent flow dimension and exponent k depend on both the density and the power of the fracture length distribution, and thus also on the connectivity regime of the fracture network system. Depending on the connectivity regime, the apparent flow dimension stabilizes to less than the Euclidean dimension and the apparent value of k < 1 indicates that hydraulic diffusion is non‐Fickian. These results suggest that the flow dimension and the exponent k may be useful for characterizing flow and transport in fractured media.

Transport of conservative solutes in simulated fracture networks: 2. Ensemble solute transport and the correspondence to operator‐stable limit distributions

In networks where individual fracture lengths follow a fractal distribution, ensemble transport of conservative solute particles at the leading plume edge exhibit characteristics of operator‐stable densities. These densities have, as their governing equations of transport, either fractional‐order or integer‐order advection‐dispersion equations. Model selection depends on the identification of either multi‐Gaussian or operator‐stable transport regimes, which in turn depends on the power law exponent of the fracture length distribution. Low to moderately fractured networks with power law fracture length exponents less than or equal to 1.9 produce solute plumes that exhibit power law leading‐edge concentration profiles and super‐Fickian plume growth rates. For these network types, a multiscaling fractional advection‐dispersion equation (MFADE) provides a model of multidimensional solute transport where different rates of power law particle motion are defined along multiple directions. The MFADE model is parameterized by a scaling matrix to describe the super‐Fickian growth process, in which the eigenvectors correspond to primary fracture group orientations and the eigenvalues code fracture length and transmissivity. The approximation of particle clouds by a multi‐Gaussian (a subset of the operator stable) for densely fractured networks with finite variance fracture length distributions can be ascribed to the classical ADE where Fickian scaling rates pertain along orthogonal plume growth directions. Fracture networks show long‐term particle retention in low‐velocity fractures so that coupling of the equations of motion with retention models such as continuous time random walk or multirate mobile/immobile will increase accuracy near the source. Particle arrival times at exit boundaries for multi‐Gaussian plumes vary with spatial density. Generally, arrival times are faster in sparsely fractured domains where transport is governed by a few very long fractures.

Stress effects on permeability in a fractured rock mass with correlated fracture length and aperture

The effect of stress on permeability and fluid flow patterns in fractured rock masses is studied when distributed fracture aperture is correlated with fracture trace length, using a discrete element method (DEM). The basic assumptions are that the rock matrix is impermeable and linearly elastic, and that the fluid flows only in fractures. A new nonlinear algorithm is developed for prediction of normal stress–normal displacement behavior of fractures based on the Bandis model and the correlation between aperture and length. The results show that when small stress ratios ( K=horizontal/vertical stress) are applied at the model boundaries, the overall permeability of the fracture network is generally decreased. However, contribution from a few large fractures of higher hydraulic conductivity prevents drastic reduction of the overall permeability, compared with models that assume uniform fracture apertures. With large values of the stress ratio, both the overall permeability and flow patterns are controlled by a combination of highly conductive larger fractures and fractures with shear slipping and dilation, with much increased overall permeability and shear-induced flow channeling. With increasing stress ratios, it becomes more and more difficult to establish an equivalent permeability tensor and representative elementary volume (REV) of a fractured rock, compared with the unstressed model. These results show significant difference between correlated and non-correlated aperture and fracture length distributions, and highlight more significant scale and stress dependence of hydro-mechanical behavior of fractures rocks when geometric parameters of rock fractures are correlated.

Percolation conditions in fractured hard rocks: A numerical approach using the three‐dimensional binary fractal fracture network (3D‐BFFN) model

We numerically investigate fracture connectivity and percolation conditions in fractured hard rocks using a three‐dimensional binary fractal fracture network (3D‐BFFN) model based on three fractal geometric parameters: the fractal dimensions (D2) of the spatial distribution of fractures, the exponent of the power‐law cumulative fracture length distribution (a), and the maximum fracture length (lmax) normalized by the domain length (L), lmax/L. Numerical results clarify that the percolation threshold in 3D‐BFFN models is strongly controlled by fractal geometric parameters and is independent of any anisotropy in the orientations Θ. In addition, when a < 1.8 and lmax/L < 1.0, percolation seldom occurs independently of D2 and Θ. In the current study the analytical solution of percolation probability (P) is presented as a function of the three fractal parameters within the 3D‐BFFN model. Application of the 3D‐BFFN model to seismogenic fractures determined from the earthquake catalogue in an offshore volcanic region between Miyake‐jima Island (MI) and Kozu‐shima Island (KI) off the Izu Peninsula, Japan, suggests that P is mainly affected by the error involved in determining a during actual surveys. Otherwise, P provides a useful index for determining whether a three‐dimensional domain percolates in fracture networks in fractured hard rocks. The basis of this approach is the observation from fracture network connections that domains with P > 0.55 are percolated domains. The zone of percolation within seismogenic fracture networks between MI and KI reveals that the networks formed from seismic‐swarm‐related seismogenic fractures over a 7‐week period related to the intrusion of a dyke, inferred previously from seismicity and deformation data.

Fast Estimation of Connectivity in Fractured Reservoirs Using Percolation Theory

Summary Investigating the impact of geological uncertainty (i.e., spatial distribution of fractures) on reservoir performance may aid management decisions. The conventional approach to address this is to build a number of possible reservoir models, upscale them, and then run flow simulations. The problem with this approach is that it is computationally very expensive. In this study, we use another approach based on the permeability contrasts that control the flow, called percolation approach. This assumes that the permeability disorder of a rock can be simplified to either permeable or impermeable. The advantage is that by using some universal laws from percolation theory, the effect of the complex geometry which influences the global properties (e.g., connectivity or conductivity) can be easily estimated in a fraction of a second on a spreadsheet. The aim of this contribution is to establish the percolation framework to examine the connectivity of fracture systems at a given finite observation scale in 2D and 3D. In particular, we use numerical simulation to show how the scaling laws of the connectivity derived originally for constant-length isotropic systems can be expanded to cover more realistic cases including fracture systems with anisotropy and fracture-length distribution. Finally, the outcrop data of mineralized fractures exposed on the southern margin of the Bristol Channel Basin was used to show that the predictions from the percolation approach are in agreement with the results calculated from field data but can be obtained very quickly. As a result, this may be used for practical engineering purposes for decision making.

Fracture aperture, length and pattern geometry development under biaxial loading: a numerical study with applications to natural, cross-jointed systems

Abstract Fracture mechanics modelling of fracture pattern development was used to analyse pattern geometry and population statistics for natural opening-mode fractures. Orthogonal fracture network geometries were generated under biaxial extension loading conditions from a slightly anisotropic initial strain state. Fracture statistics were analysed by grouping all fracture orientations into one population for these unique orthogonal pattern geometries. Fracture aperture distributions resembled negative exponential curve shapes, consistent with published observations for stratabound fractures in sedimentary rock. Fracture length distributions had a strongly power-law shape, and showed that longer fractures grew first and reached their fullest extent before shorter fractures began propagating. The power-law shape of the length distribution was first established by the growth of the longest fractures in the population, followed by the later growth of shorter fractures that extended the power-law shape to smaller sizes. The shortest fracture length at which the power-law distribution was truncated varied with the magnitude of the applied strain. Other variations in fracture pattern results were tied to mechanical layer thickness and subcritical crack growth propagation properties of the fractured media.

Flow in multiscale fractal fracture networks

Abstract The paper aims at defining the flow models, including equivalent permeability, that are appropriate for multiscale fracture networks. As a prerequisite of the flow analysis, we define the scaling nature of fracture networks that is likely quantified by power-law length distributions whose exponent fixes the contribution of large fractures versus small ones. Despite the absence of any characteristic length scale of the power-law model, the flow structure appears to contain three length scales at the very maximum: the connecting scale, the channelling scale, and the homogenization scale, above which the equivalent permeability tends to a constant value. These scales, including their existence, depend on the fracture length distribution and on the transmissivity distribution per fracture. They are basic in defining the flow properties of fracture networks.

Tracer-Test Simulation on Discrete-Fracture- Network Models for Characterizing Fractured Reservoirs

This article, written by Technology Editor Dennis Denney, contains highlights of paper SPE 94344, “Tracer-Test Simulation on Discrete-Fracture-Network Models for the Characterization of Fractured Reservoirs,” by A. Lange, J. Bouzian, and B. Bourbiaux, Inst. Français du Pétrole, prepared for the 2005 SPE Europec/EAGE Annual Conference, Madrid, Spain, 13–16 June. Fractures affect flow dynamics within a reservoir, causing uncertainty in field-productivity and -reserves assessment. Geologically realistic models of the fault/fracture network can be constructed, and these simplified models can be used as field-scale simulations of multiphase production. A 3D discrete-fracture-network (DFN) simulator enables the critical intermediate step of validating the geometry of the geological fault/fracture network and characterizing it in terms of flow properties. An extension of this simulator to compute tracer tests is detailed in the full-length paper. Compared with pressure tests, tracer tests may provide additional information about the reservoir heterogeneity because of the convective nature of these tests. Introduction The increasing number of mature fields in which fractures caused unexpected production features gave rise to extensive efforts in better characterizing and integrating fracture properties into field-scale simulation models for production assessment and optimization. A workflow for integrating the multiscale fracture-flow properties in reservoir simulators was proposed earlier. This workflow consists of the following. - Geologically realistic models of the fault and fracture network are constructed from seismic, well, and outcrop data. - Models are validated from dynamic field information such as well tests, interference tests, and production logs. - An equivalent simulation model applicable at reservoir scale is constructed with innovative flow scale-up procedures. Multiphase field production is simulated at reservoir scale with this equivalent-model. Using this workflow, field-simulation models remain interpretable in geological terms and enable explanation of unexpected production features, such as early water breakthroughs. However, validation of the geological fault/fracture-network model in terms of geometry and flow properties remains a critical step. Fracture conductivities must be characterized, and uncertainties on some geometrical parameters may have a large effect on the flow properties of the fractured medium. For instance, small variations in the fracture-length distribution may lead to large differences in the fracture-network connectivity, thus affecting the flow properties of the medium. A 3D DFN flow simulator was developed to fulfill this validation and characterization step. This tool enables simulating fluid flow on DFN realizations obtained from the geological model. Well tests, interference tests, and production logs then can be simulated on geologically realistic fracture networks and calibrated from available field data for validating the geometry of the geological model and characterizing the fracture conductivities. The DFN flow simulator with a dual-permeability option can be used with any type of fractured reservoir, whatever the scale, density, and connectivity of fractures. Model This simulator uses the dual-medium concept applied to an explicit representation of the geological model of the fault/fracture network and matrix medium. The transport and dispersion of nonreactive tracers are modeled by introducing a convection/dispersion equation into the existing dual-porosity flow model. A hybrid scheme discretizes the convective term implicitly within densely fractured regions for stability purposes and explicitly by use of a double-upstream scheme everywhere else for reducing spurious numerical dispersion.

Laslett's line segment problem

Our problem is to estimate the length distribution of fractures in a rock surface from a geological map. We do not fully observe the fractures because part of the rock surface is covered by vegetation, soil and water. The uncovered region is very irregular and, as a result, we tend to observe several pieces of a single fracture. It is quite impossible to decide from the map if two pieces belong to the same underlying fracture. Under the assumption that the observed pieces are independent, we derive the nonparametric maximum likelihood estimator of the length distribution of the underlying fractures. The assumption is clearly false, but our approach is justified by proving consistency of the estimator without appealing to the independence. We apply our estimator to the geological data.

Influence of spatial correlation of fracture centers on the permeability of two‐dimensional fracture networks following a power law length distribution

Observations of outcrops of fractured media show that fractures are correlated at all scales of the medium and that there are also fractures of sizes ranging from the microscale to the macroscale. Recent studies have shown that the correlation pattern is often a fractal characterized by its dimension Dc and that the fracture length distribution is a power law of characteristic exponent −a. We study the influence of these long‐range heterogeneities on the equivalent medium permeability. Fracture correlation and fracture length are present at all scales of the medium and may thus influence the hydraulic properties and their scaling at all scales. We have numerically investigated the effect of the fractal correlation pattern and of the power law length distribution on the equivalent permeability of fracture networks. The correlation and the length distribution have opposite effects on connectivity and permeability. Increasing correlation lets connectivity and permeability decrease and conversely for the length distribution. In most cases, one of the characteristics (i.e., length or correlation) supersedes the other. There is only a small area defined by a > 2 and a ≤ Dc + 1 where both characteristics have a simultaneous influence on permeability. In this area the influence of the correlation remains low, more precisely a scale increase of at least 5 orders of magnitude is required to produce a permeability increase of 1 order of magnitude. Although the correlation pattern does not have much influence on the equivalent permeability, its influence on the flow pattern increases from negligible for networks at threshold to important for dense networks.

Fissure eruption of flood basalts from statistical analysis of dyke fracture length

Flood basalt emplacement results from eruption of dyke swarms. Several lines of evidence suggest, however, that a large number of dykes in large igneous provinces have never erupted, or have done so over too short a length for any evidence of eruption to be found in the field. We have mapped 1025 dykes and dyke segments from two swarms in western Ethiopia, most of which are of basaltic composition, the others of silicic composition. The dykes cut across the basal breccia of the trap series, and are observed at a structural level c. 1000 m below the current top of the lava pile on the nearby volcanic plateau. We test the hypothesis that at such a depth, some of the basaltic dykes may have erupted whereas some others may not. Due to higher buoyancy within the trap series, silicic dykes are likely to have been systematically eruptive. The area is flat, which has allowed analysis of dyke length distribution at constant structural level, a useful approach for the characterisation of other fracture types such as tension fractures and faults. We found that dyke length follows a power-law distribution. When en échelon dyke segments are counted as single dykes (having as their length the accumulated length of the individual segments), the population exponent, 1.8, is in the mean of the exponents usually obtained for fracture data sets. This exponent correctly fits the shortest dykes considered (1–9 km). A much steeper slope (3.3) is obtained for the distribution of intermediate lengths (9–10 to 10–20 km), and is interpreted as a possible consequence of the eruption of those dominantly basaltic dykes that are more than 9–10 km long at the observed structural depth. The longest, silicic dykes (10–20 to 50 km) have the same population exponent as the shortest, basaltic dykes. This is explained by the homogeneous propagation mode of both populations, either dominantly horizontal or dominantly vertical. However, the silicic dyke population exhibits a slightly larger population coefficient, for which we discuss the possible influence of the Precambrian fabric. Analysis of fracture length distribution appears to be a promising tool for estimating how much of a dyke swarm is eruptive and better understanding the mechanisms of fissure eruption in large igneous provinces.

Estimation of 2‐D and 3‐D Fracture Densities from 1‐D Data — Experimental and Field Results

Abstract 2‐D and 3‐D densities of fractures are commonly used in mining safety design, natural gas and oil production in fractured reservoirs, and the characterization of subsurface flow and transportation systems in fractured rocks. However, many field data sets are collected in 1‐D frequency (f) (e.g., scanlines and borehole data). We have developed an ARC/INFO‐based technology to calculate fracture frequency and densities for a given fracture network. A series of numerical simulations are performed in order to determine the optimal orientation of a scanline, along which the maximum fracture frequency of a fracture network can be obtained. We calculated the frequency (f) and densities (both D1 and D2) of 36 natural fracture trace maps, and investigated the statistical relationship between fracture frequency and fracture density D1, i.e. D1=1.340f+ 0.034. We derived analytical solutions for converting dimensional density (D1) to non‐dimensional densities (D2 and D3) assuming that fracture length distribution follows an exponential or power law. A comparison between observed and calculated results based on the equations we developed shows that (1) there exists a linear relationship between fracture frequency and fracture density (D1), and this relationship can be used to estimate fracture density (D1) if the fracture frequency is determined from a scanline survey or from borehole data; (2) the analytical solutions we derived can accurately determine the non‐dimensional 2‐D fracture density (D2) in practice and 3‐D fracture density (D3) in theory if the fracture length distribution function is assumed.

Use of multifractal seismic waveform parameters to characterize the hydraulic properties of fractured media: numerical experiments

SUMMARY The multifractal characteristics of seismic waveforms that have propagated through fractured media were studied using numerical experiments. The study suggests that multifractal waveform parameters (i.e. information dimension, correlation dimension, curvature and range indices) bear correlations with the hydraulic properties of the fractured media. The characteristic waveform parameters were estimated from a multifractal analysis of 1-D synthetic seismograms generated using a variation of the reflectivity method. Waveforms from fractured zones were obtained by a superposition of reflections from horizontal fractures with a varying distribution of fracture length, aperture and spacing. The constructive and destructive interferences of the composite reflections from the fractures in the fractured layer result in waveforms that possess multifractal attributes. The characteristic parameters extracted from the waveforms are shown here to correlate with the hydraulic properties of the fractured rockmass, which include the fracture permeability, the discontinuity index and the fracture density parameter.

Percolation conditions in binary fractal fracture networks: Applications to rock fractures and active and seismogenic faults

The fracture network patterns in crystalline rock are treated using a binary fractal model described by three fractal geometric parameters: the fractal dimension D of the spatial distribution of fractures, the fractal dimension a of the fracture length distribution, and the maximum fracture length (lmax) normalized by the domain length (L), lmax/L. We present a percolation threshold relationship describing the boundary between percolating and nonpercolating conditions in random binary fractal fracture networks (RBFFNs) and apply this to observations of natural fractures. Our results indicate that the three fractal parameters identified indeed control the connectivity of fracture networks, and hence the migration of fluids through rock fractures, and active and seismogenic faults. Moreover, our numerical approach may be applied in the future to investigating fluid migration associated with seismicity based on fractal parameters determined from earthquake catalog data.