Fracture Orientation Distribution Research Articles

Sampling and Distribution Parameter Analysis for Estimating Three-dimensional Fracture Orientation Distributions from a Circular Sampling Window

Precise evaluation of three-dimensional (3D) fracture orientation distributions is vital to a reliable discrete fracture network (DFN) model for rock mass. This study analyzed the accuracy and dispersion of the evaluated orientation distributions of fracture sets with different fracture geometric parameters. The geometric probability method by two-dimensional (2D) trace data is introduced and compared with the three methods of the Terzaghi family by one-dimensional (1D) orientation data. The Monte Carlo method generated many groups of 3D fracture networks to analyze the influence of uniform orientation distribution, average fracture diameter, and volume density on the evaluated orientation distributions. The results indicated that the piecewise distribution increases the computational complexity of the geometric probability method. This method can be used to effectively evaluate the orientation distributions of fracture sets with an average diameter greater than 0.25 m and volume density greater than 0.2 m−3.

Estimation of Fracture Orientation Distributions from a Sampling Window Based on Geometric Probabilistic Method

Accurate orientation distributions are crucial to generating a reliable discrete fracture network (DFN) model for rock mass, while conventional one-dimensional (1D) and two-dimensional (2D) observation data have significant sampling bias. The study proposes a complete analytical method for estimating the orientation distributions of three-dimensional (3D) fractures in rock mass in conjunction with trace statistics in a sampling window, which is suitable for most continuous distributions by reducing the sampling bias. Traces are divided into three categories to derive the geometric probabilistic relationships between 2D trace statistics and 3D fracture orientation distributions. The moment estimation, number estimation, and normalization error functions are derived, and the distribution parameters are determined by minimizing the total error function. The proposed method is compared with the Terzaghi family methods and validated by multiple sets of stochastic fracture networks with different orientation distributions and sampling windows generated by the Monte Carlo method. The results indicate that the estimated continuous orientation distributions subjected to the error functions from a large single sampling window are well matched with the true distributions after removing the number estimation error functions of trace samples fewer than 20. Daxiagu tunnel is selected as a case study and the distributions estimated by the proposed method are more coincident with the field observation than those fitted by the orientations of the rock outcrops on the excavation face.

Numerical Modeling of Fracture Network Evolution in Organic‐Rich Shale With Rapid Internal Fluid Generation

AbstractWhen low‐permeability and organic‐rich rocks such as shale experience sufficient heating, chemical reactions including shale dehydration and maturation of organic matter lead to internal fluid generation. This may cause substantial pore fluid overpressure and fracturing. In the vicinity of igneous intrusions emplaced in organic‐rich shales, temperatures of several hundred degrees accelerate these processes and lead to intense fracturing. The resulting fracture network provides hydraulic pathways, which allow fluid expulsion and affect hydrothermal fluid flow patterns. However, the evolution of these complex fracture networks and controls on geometry and connectivity are poorly understood. Here, we perform a numerical modeling study based on the extended finite element method to investigate coupled hydromechanical fracture network evolution due to fast internal fluid generation. We quantify the evolution of different initial fracture networks under varying external stresses by analyzing parameters including fracture length, opening, connectivity, and propagation angles. The results indicate a three‐phase process including (1) individual growth, (2) interaction, and (3) expulsion phase. Magnitude of external stress anisotropy and degree of fracture alignment with the largest principal stress correlate with increased fracture opening. We additionally find that although the external stress field controls the overall fracture orientation distribution, local stress interactions may cause significant deviations of fracture paths and control the coalescence characteristics of fractures. Establishing high connectivity in cases with horizontally aligned initial fractures requires stress anisotropy with σV > σH, while the initial orientation distribution is critical for connectivity if stresses are nearly isotropic.

Optimal selection of estimator for obtaining an accurate three-dimensional rock fracture orientation distribution

Advances in estimating the three-dimensional (3D) orientation distribution include i) the stereology-based Fouché estimator (FE); and ii) its recent modification (MFE), which has one less meshing step and has three modified components compared with the FE. One way to achieve an accurate estimate is by selecting the optimum estimator. Here, we developed a recognizer for the optimum estimator to help automatically select the most likely accurate estimator for use in the FE and MFE. Before this, we conducted a systematic comparison of the likelihood of obtaining an accurate estimation to clarify for which patterns FE and MFE are likely to perform accurately. The statistical comparison results showed that the MFE was more likely to perform accurately than the FE for patterns of moderately to strongly aligned fracture sets and for patterns of poorly aligned fracture sets with medium to wide intersection angles to the 1D probe, and thus, the MFE was the optimum estimator. For patterns of poorly aligned fracture sets with narrow intersection angles to the 1D probe, however, the FE was more likely to perform accurately than the MFE, and thus, the FE was the optimum estimator. On the basis of these comparison results, we developed a recognizer for the optimum estimator. This developed recognizer provided a simple, reliable tool for selecting the optimum estimator to accurately produce a 3D orientation distribution estimation, which is beneficial to realistic fracture network modelling. Weakness of the developed recognizer was also disclosed.

Minimum Scanline-to-Fracture Angle and Sample Size Required to Produce a Highly Accurate Estimate of the 3-D Fracture Orientation Distribution

Accurate estimates of the three-dimensional (3-D) rock fracture orientation distribution are crucial for generating a reliable fracture system model. Although an earlier method by Fouche and Diebolt allows one to estimate such distributions from one-dimensional (1-D) samples, this method does not typically produce highly accurate estimates of the 3-D orientation distribution (HAE3DOD). In this study, the minimum scanline-to-fracture angle (minimum θ) and the minimum orientation sample size (minimum n) required to produce HAE3DOD are investigated. Firstly, the factors significantly influencing minimum θ and minimum n are identified, and the influence clarified. For minimum θ, the possible influencing factors include the orientation concentration parameter (к) and n, while for minimum n, the possible influencing factors include к and θ. Fractures from three selected sites in China provide sufficient data for this investigation. The investigation results reveal that minimum θ varies almost linearly with к and n. Moreover, minimum n varies linearly with к and θ. Variations in the minimum θ and minimum n values are strongly associated with discrepancies in sample density resulting from different values of these factors. To ease the estimation of minimum θ and minimum n, empirical relations that take into account the separate factors are proposed for these two variables. A practical example demonstrates that the proposed relations accurately and efficiently estimate minimum θ and minimum n and can provide sampling guidelines (i.e., the intervals of scanline direction and sample size) for producing HAE3DOD. The potential limitations of the proposed relations are also discussed.

Some Results of Quantitative Analysis of Fracture Orientation Distribution along the Segment Tien Yen-Mui Chua of Cao Bang-Tien Yen Fault Zone, Quang Ninh Province, Viet Nam

The results of quantitative analysis of fracture orientation distribution according to the pair of stereonet windows among 05 different survey sites along the segment Tien Yen-Mui Chua, belong to Cao Bang-Tien Yen fault zone, Quang Ninh province, Viet Nam showed that, the correlation values are over 0.80 corresponding to 50%, over 0.75 corresponding to 30% and over 0.65 corresponding to 20%. These values are quite consistent with over 60% of the fracture number in the direction of NW-SE. Especially, the compatibility is clearly reflected in the determination of the frequency of fracture measurements within the division intervals of 20 degrees for dip direction and 10 degrees for dip angle at 05 different survey sites.

Influence of fracture orientation distribution function on the estimates of gas-hydrate saturation of site 10 at Krishna-Godavari Basin, India

<p>We estimate the gas-hydrate saturation in fractured reservoirs of gas hydrate-rich marine sediments from Krishna-Godavari basin along the eastern margin of India. Effective medium modeling (EMM) is a combination of self-consistent approximation (SCA), differential effective medium (DEM) modeling and smoothing and it deals with the anisotropy associated with clay platelets’ orientation. It proves to be a good choice attributable to its dependency on physical parameters alone. Previous approaches appertain to the implementation of orientation distribution function (ODF) on clay platelets alone assuming a vertical ellipsoidal shape of gas hydrates whilst in the present study, we have implemented ODF on hydrate bearing fractures assigning an ellipsoidal shape to them as well. A bias of ~2-4% in the saturation estimates apparently streamlines the betterness of our method.</p>

Fracture patterns in nonplane strain boudinage—insights from 3‐D discrete element models

We use 3‐D Discrete Element Method simulations to model the evolution of boudin structures in a layered material under nonplane strain conditions. As the models are shortened perpendicular to the layer orientation, they are extended at different rates in the two layer‐parallel directions. The particular emphasis of the study is on the orientation of fractures between the boudin blocks. The results show that the fracture orientation distribution is closely connected to the ratio of the two layer‐parallel extension rates. The anisotropy of the fracture orientation distribution increases systematically from no anisotropy at isotropic layer‐parallel extension to a highly anisotropic distribution in case of uniaxial extension. We also observe an evolution of the anisotropy of fracture orientation distribution with increasing deformation in each individual model from a high‐initial anisotropy towards a value characteristic for the ratio of the layer‐parallel extension rates. The observations about the relation between the strain ratios and the fracture patterns do have the potential to serve as the basis for a new method to analyze strains in naturally boudinaged rocks.

Parametric and non-parametric statistical approaches to the determination of paleostress from dilatant fractures: Application to an Early Miocene dike swarm in central Japan

Several methods have been proposed for determining paleostress states from orientations of dilatant fractures such as dikes and veins. Recently a stochastic inversion method was invented to objectively estimate the principal stress axes and the stress ratio. Whether a fracture is dilated or not is controlled by the balance of the fluid pressure and the normal stress acting on it. The magnitude of normal stress depends on the fracture orientation, which causes anisotropic orientation distribution of dilatant fractures. The inversion method assumes that the orientation distribution of fractures can be approximated by a Bingham distribution, an exponential probability distribution on the unit sphere, of which symmetric axes are interpreted as the principal stress axes. However, it is unknown if the exponential type of distribution function is suitable or not. Here, we examine the distribution functions and propose two improved methods. One method uses the shifted power-law function as the shape of probability distribution, which is more flexible than the Bingham distribution and is applicable to various shapes of orientation distributions. Furthermore, an index of the driving fluid pressure can be estimated with a confidence interval. The other is a non-parametric (distribution-free) method, which can avoid the a priori assumption on the shape of distribution function without significantly losing accuracy or precision. The new methods were applied to an Early Miocene dike swarm formed during the back-arc opening of the Japan Sea. A normal-faulting stress regime with the minimum principal stress axis trending roughly perpendicular to the arc was obtained from the dikes. A moderately high stress ratio and a high fluid pressure were also estimated.

The effect of mineralization on the ratio of normal to tangential compliance of fractures

ABSTRACTThe effect of a fracture on the propagation of seismic waves can be represented in terms of the normal compliance BN and tangential compliance BT of the fracture. If BN/BT= 1 for all fractures, the effective elastic stiffness tensor of an isotropic background containing an arbitrary orientation distribution of fractures is orthotropic (i.e., has three orthogonal planes of mirror symmetry) in the long‐wave limit. However, deviations from orthotropy may occur if BN/BT differs significantly from unity and this can cause the principal axes of the P‐wave NMO ellipse and of the variation in the PP‐reflection amplitude as a function of azimuth, to deviate from the fast and slow polarization direction of a vertically propagating S‐wave. Simple models of a fracture in terms of a planar distribution of cracks suggest that BN/BT≈ 1 for dry fractures. However, naturally occurring fractures often exhibit mineralization in the form of bridges between opposing faces of the fracture. The presence of such bridges leads to significant departures of BN/BT from unity.

Seismic anisotropy of fractured rock

A simple method for including the effects of geologically realistic fractures on the seismic propagation through fractured rocks can be obtained by writing the effective compliance tensor of the fractured rock as the sum of the compliance tensor of the unfractured background rock and the compliance tensors for each set of parallel fractures or aligned fractures. The compliance tensor of each fracture set is derivable from a second rank fracture compliance tensor. For a rotationally symmetric set of fractures, the fracture compliance tensor depends on only two fracture compliances, one controlling fracture compliance normal, the other, tangential, to the plane of the fractures. The stiffness tensor, which is more useful in the consideration of elastic wave propagation through rocks, can then be obtained by inversion. The components of the excess fracture compliance tensor represent the maximum amount of information that can be obtained from seismic data. If the background rock is isotropic and the normal and shear compliance of each fracture are equal, although different from those of other fractures, the effective elastic behavior of the fractured rock is orthorhombic for any orientation distribution of fractures. A comparison of the theory with recent ultrasonic experiments on a simulated fractured medium shows near equality of the normal and shear compliance for the case of air‐filled fractures.

Radionuclide transport in fractured rock a new model: Application and discussion

A model for predicting the concerntration of radionuclides transported through fractured rock was developed in several earlier papers as part of the CEC programme on management and storage of radioactive waste. In this report, we examine that theory in more depth and develop expressions for the system parameters in terms of the rock statistics. A review of related work on fracture statistics is given and its relevance to the present work highlighted. In particular, we note that the connectivity, i.e. the average number of intersections per fracture, plays a vital part in determining the ‘cross section’ and redistribution function at intersections. Expressions are developed for these cross sections in terms of fracture orientation distributions. For a single set of fractures, i.e. fractures of one particular type, we have obtained numerical solutions and have demonstrated how the concentration profile becomes more dispersive as the orientation distribution becomes more isotropic. The method shows considerable promise and has many advantages when compared with the classical advective-dispersive model. In particular, it enables microscopic rock parameters, e.g. fracture length and orientation, to be employed directly and avoids the effective continuum approximation common to many approximations. It also explains the scaledependent effect and can account in a consistent manner for dead-end pores and channelling.

Permeability estimation from velocity anisotropy in fractured rock

Cracks in a rock mass subjected to a uniaxial stress will be preferentially closed depending on the angle between the fracture normal vectors and the direction of the applied stress. If the prestress fracture orientation distribution is isotropic, the effective elastic properties of such a material after application of the stress are then transversely isotropic due to the overall alignment of the cracks still open. Velocity measurements in multiple directions are used to invert for the probability density function describing orientations of crack normals in such a rock. This is accomplished by expanding the crack orientation distribution function into generalized spherical harmonics. The coefficients in this expansion are functions of the crack density and the crack aspect ratio distribution. The information on fracture distribution obtained from the velocity inversion allows an estimation of the anisotropic permeability of the fractured rock system. Permeability estimates are based on the number of cracks open of each aspect ratio, and the contribution of a given crack is weighted by the cosine of the angle between the crack and the direction of the applied pressure gradient. This approach yields a prediction of permeability as a function of the angle from the uniaxial stress axis. The inversion for crack orientation is applied to ultrasonic velocity measurements on Barre granite, and permeability predictions for this sample are presented. The inversion results are good and reproduce velocity measurements well, and the permeability predictions show some of the expected trends. Initial comparisons of the predictions with available permeability data, however, show deviations, suggesting that further information on partial crack closure and connectivity of cracks should be included into the permeability model.