Rectangular Fracture Research Articles

Fracture energy of birch in tension perpendicular to grain: experimental evaluation and comparative numerical simulations

The present work has experimentally determined the specific fracture energy of the hardwood species silver birch (Betula pendula), which in recent times has caught increased attention for utilization in structural applications. The single-edge-notched beam loaded in three-point-bending was utilized for evaluating the fracture energy with the work-of-fracture method. In addition to birch, Norway spruce (Picea abies) was utilized as a reference material. The effect of two different geometries of the fracture area for each species was evaluated—one triangular and one rectangular fracture area. It should be noted that the geometry of the fracture area did influence the evaluated fracture energy, and this influence was not consistent between species. This was likely in part due to manufacturing difficulties with the triangular fracture area. In addition to the experimental testing, a numerical 2d-model including linear strain-softening behavior was used for comparative simulations. The numerical 2d-models showed reasonable agreement with the experimental results regarding the global load vs. displacement response, despite their relative simple nature. The specific fracture energy for the spruce specimens was evaluated to 221 J/m2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {m}^2$$\\end{document} and for the birch specimens to 656 J/m2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {m}^2$$\\end{document}. Consequently, the present work implies a marked increase in specific fracture energy for birch, compared to spruce. This increase in specific fracture energy could potentially have a large influence on the failure behavior of birch when used in structural applications which is something that needs to be considered in future work.

Numerical investigation of proppant transportation characteristics in hydraulically fractured wedge fractures

Hydraulic fracturing creates multiple induced fractures and micro-fractures, forming a complex fracture network in the reservoir. The study of the transport and distribution of the proppant within the fracture network is critical to the design and evaluation. However, existing simulation studies of proppant transport tend to be overly idealized and neglect the inhomogeneity of fracture widths that occur after fracturing. To address these issues, this study employs computational fluid dynamics (CFD) to study the transportation of fracturing fluid and proppant within a fracture network. The flow dynamics of solid-liquid two-phase flow in fractures are simulated using the Euler-Euler multiphase flow model. Considering the actual variables in field construction and the inherent inhomogeneity in realistic fracture structures, a three-dimensional model was established to capture the gradual variation in fracture width. The accuracy of this model was verified through a comparative analysis with physical experiments. On this basis, an investigation was conducted to explore the impact of particle size, particle density, particle volume concentration, and injection velocity on proppant transportation. The results demonstrate that, in contrast to conventional rectangular fractures, sandbanks formed from wedge fractures exhibit a lower height, which facilitates improved transportation into deeper fractures. Furthermore, particle concentration primarily influences distal fractures, with proppant particle size being second. The injection velocity has a significant impact on the height of the sandbank located in proximity to the fracture inlet. The research findings provide a deeper understanding of the transport and distribution of proppants within wedge fractures, thereby establishing a theoretical basis for the analysis and engineering guidance in on-site hydraulic fracturing construction.

An analytical method to estimate the accuracy representation index of circular and elliptical disc models for rectangular fractures with application to seepage analysis

Constructing a suitable discrete fracture network (DFN) to represent rock fractures proves vital for tackling engineering projects involving rock masses. Among the commonly used models for constructing DFN, the circular disc model (CDM) and elliptical disc model (EDM) stand out. The selection between these models has been facilitated by the introduction of the Accuracy Representation Index (ARI), which quantitatively assesses their suitability. The existing methods for calculating ARI based on Monte Carlo simulations are time-consuming. In order to quickly estimate the ARI for CDM and EDM concerning any length-width ratio (kr) of rectangular fracture, the paper derives analytical formulas based on the geometric relationships between circles and ellipses with rectangles. These formulas show that: (a) for the CDM, ARI is only related to kr, and its value decreases as kr increases; (b) for the EDM, ARI is dependent on the relationship between kr and the long-short axis length ratio (ke) of the ellipse, and it has been theoretically proven that ARI reaches its maximum value of 0.91 when kr is equal to ke. Moreover, we apply ARI to the study of rock mass seepage characteristics, and the results show that ARI could reflect the error rate of permeability of the CDM and EDM for rock masses with rectangular fractures.

A numerical scheme for heat transfer in fluid flowing in three-dimensional fractures

We present a numerical scheme using a finite volume method (FVM) to simulate the heat transfer in fluid flowing in a three-dimensional fracture in a rock mass. Heat conduction in the rock mass and heat exchange between the fluid and the rock mass are considered, but fracture deformation is not. This scheme could approximate the thermal energy extraction process from a geothermal system when the fractures have been established and the fluid pressure and temperature variations are not high enough to cause a significant change of the fracture aperture. The FVM was employed to solve the pressure and temperature of fluid with the same triangular control cells. The heat conduction in the rock mass was simulated with an indirect boundary element method (IBEM), which uses the triangular control cells of the FVM as the discretization boundary elements. The fluid pressure and temperature are coupled in two systems of equations and a sequential coupling iteration procedure was employed to solve the equations. This paper introduces the FVM for the fluid temperature and the sequential iteration procedure for temperature and pressure which was implemented in a code, and the numerical results agree well with an analytical solution for the temperature of fluid flowing through a rectangular fracture. The numerical scheme was then employed to simulate illustration examples of heat transfer between two well holes.

Low-frequency distributed acoustic sensing shape factors for fracture front detection

Accurate knowledge of fracture extents generated in multistage unconventional completions remains elusive. Crosswell low-frequency distributed acoustic sensing (LF-DAS) measurements can determine the time and location of a frac hit. Knowing where and when a frac hit occurs constrains the fracture extent but does not estimate it quantitatively. A recent study on crosswell LF-DAS demonstrated a simple method to rapidly determine the instantaneous fracture propagation rate when a frac hit occurs. This method, the zero strain-rate location method (ZSRLM), is based on laboratory experiments and numerical modeling assuming a radial fracture geometry. The method estimates a fracture propagation velocity that is used to extrapolate the final fracture extent. The propagation rate is calculated based on dynamic estimates of the nearest distance from the fiber to the front of a propagating fracture. The ZSRLM is adapted to estimate the distance to the fracture front based on rectangular fracture geometries. A 3D displacement discontinuity method program generates crosswell LF-DAS strain-rate waterfall plots considering a single, rectangular fracture of constant height. Over 150 different simulations were conducted varying formation mechanical properties, fracture height, and the vertical and horizontal offset between the treatment and monitor well. For each simulated case, the ZSRLM is used to estimate the distance to the fracture front based on the simulated waterfall plots. The difference between the estimated and actual distance to the front is corrected by a shape factor. The relationship among the shape factor, fracture height ratio, and vertical offset ratio is determined. Using a shape factor improves the performance of the ZSRLM by up to a factor of two for rectangular fractures. The updated ZSRLM is applied to extrapolate final fracture extents in two field cases: a single cluster stage in the Montney Formation and a multicluster stage of an Austin Chalk completion.

Simulation Method and Application of Three-Dimensional DFN for Rock Mass Based on Monte-Carlo Technique

In this study, the authors simulate a polygonal discrete fracture network (DFN) in rock masses. The probability models of the relevant geological parameters, including the orientation, trace length, volume density, and coordinates of the centroid, are firstly developed as fractures are in the shape of rectangles. In the process, the probability distribution of rectangular fractures with side lengths as random variables is introduced and described in terms of mean trace lengths on the basis of the probability model of disk-shaped fracture with the diameter as the random variable. The relationship between the volume density and the linear density of rectangular fractures is given for a negative exponential distribution. Following this, the coordinates of the vertices of fractures are derived based on spatial algebraic geometry, and the data for the three-dimensional DFN model are generated using the Monte-Carlo technique. The resulting three-dimensional DFN is visualized by calling the Open GL graphics database in the environment of Visual C, and the process of implementation of the DFN simulation is given. Finally, the validity of the simulation is verified by applying it to engineering practice.

Fast-running model for high-volume hydraulic fracturing

We develop a rapid simulator of hydraulic fracture growth in complex geologic layered conditions and extensively drilled multistage fractured wells in a reservoir. The simulator allows running independent simulations for thousands hydraulic fractures within a minute on modern personal computers. The model is meshless and has a simplified rectangular fracture shape; however, coupling all conventional hydraulic fracture mechanics equations makes the model accurate enough. We demonstrate it via comparison with exact analytical solutions in limiting toughness-, viscosity-, and leakoff-dominated regimes, as well as via comparison with other commercial simulators on real field cases. Field validation of the model by temperature log measurements is also provided.To demonstrate the practical implications of the model for reservoir engineering, we show several examples of numerical simulations for a huge number of hydraulic fractures in a reservoir, which interact with each other due to the closeness of stimulated wells and perforation clusters. Rapid simulations allow obtaining the field-scale simulations of fracture growth in the range of a minute and efficiently building distributions of asymmetric fracture growth both in length and in height. We show that the stress shadow effect causes undesired outbreaks to adjacent gas- or water-saturated geological layers if the fracture placement in a reservoir is dense enough. Even small spatial misalignment of stages in the offset wells may result in changes of geometry of transverse fractures growing from neighbored wells. Vertical true vertical depths mismatch in offset wells; their deviations in the dip and azimuth angles are shown to cause the asymmetry of fracture growth as well, which hardly can be predicted in field development workflows without rigorous fracture simulations. Examples of longitudinal fracturing in horizontal wells demonstrate the strongest distortion of conventionally assumed lateral symmetry and height growth similarity of fractures created in one horizontal well. Our simulations also show that these results are affected by the viscosity of pumped fracturing fluids. The presented model of field-scale interactive fracture growth in a reservoir enables finding the best fracture placement characteristics and pumping schedule for the needs of field development with hundreds to thousands of fractures in a short time frame.

A 3-D numerical simulation-based heat production performance research for enhanced geothermal system with two horizontal wells and rectangular multiparallel fractures

Parallel multifracture model with horizontal wells is an important structural form of enhanced geothermal system (EGS) to mine the hot dry rock, for its superiority in connecting injection well and production well. In this work, the numerical simulation study of the EGS heat production performance is carried out. Firstly, a 3-D large-scale model of two horizontal wells and multiparallel rough fractures is established, in which the fracture aperture are millimeter scale. Then, the flow and heat transfer process of water in fractured rock mass is simulated. Finally, sensitivity analysis of the EGS heat production performance to various operational and formation parameters is conducted, in which the well spacing, injection temperature, circulation flow rate, fracture aperture, fracture spacing and thermal conductivity of rock mass are considered. Results indicate that there are flow short circuit and blind angles in the velocity field, and the thermal short circuit exists in the temperature field of rock mass. EGS with four rough fractures can supply heat generation power of 9.11 MW–18.05 MW with an average value of 14 MW within 20 years, and the well spacing has the greatest influence on heat production performance of EGS while fracture aperture has least effect.

Efficient Modeling of Hydraulic Fractures with Diamond Boundary in Shale Gas Reservoirs

AbstractThe effects of hydraulic fractures with complex boundaries on the well performance of a shale gas well, considering a more realistic corner point geological model, have rarely been studied previously. In this study, the nonintrusive embedded discrete fracture model (EDFM) method was employed to investigate the effects of different boundary shapes of hydraulic fracture, coupling a network of sophisticated natural fractures discrete fracture network (DFN) on well performance. First, by implementing the powerful EDFM technology, concepts of two categories (rectangle and diamond) of hydraulic fracture with different boundaries were designed. Next, the geometric equations defining vertices of multiple rectangular- or diamond-shaped hydraulic fractures in arbitrary coordinate systems were derived. Subsequently, the horizontal well with multistaged hydraulic fractures and sophistically oriented 3D natural fractures was inputted into the reservoir model to perform history matching. After history matching, the results were further analyzed to compare the production forecast from the two categories. The results show that 20-year cumulative gas productions for rectangle- and diamond-shaped fractures are approximately 1.237×108 m3 and 1.486×108 m3, respectively. In other words, the diamond category can produce 20.1% more gas than the rectangle category. For cumulative water production, the diamond category produces 3.8×104 m3, as against the 3.0×104 m3 produced by the rectangle category (or 26.7% more). This implies that the diamond-shaped fractures have the potential to reach the far field region of the reservoir away from the wellbore. This means that more intersections with natural fractures DFN can be achieved, and more drainage area is unlocked. The visualization of pressure distributions and drainage volume was easily shown, and these results further confirm that the extent of fluid drainage by the diamond fracture is larger compared to that by the rectangular fracture given the same total surface area.

Experimental investigation on backflow of power-law fluids in planar fractures

In hydrofracturing, we model the backflow of a non-Newtonian fluid in a single flat-walled fracture of planar geometry and support our conceptualization with laboratory experiments. We consider a power-law fluid, a spatially homogeneous fracture aperture, and its variation in time depending on the internal fluid pressure and the elastic relaxation of the walls. The relationship between the latter quantities may be linear, akin to a Winkler soil, or nonlinear, due to the progressive softening or stiffening of the boundary associated with the properties of the surrounding rock. The result is an integrodifferential problem that generally admits a closed-form solution, albeit implicit for some quantities. In particular, a comparison is conducted between the drainage time in the present configuration and point drainage in radial geometry. The approach is generalized by introducing a leak-off, i.e., a loss of fluid at the fracture boundaries that accelerates the fracture closure, when compared to the no leak-off case. To validate the theoretical results, 14 experiments are conducted with an ad hoc replica of a rectangular fracture of aspect ratio 2.5–2.7, with a maximum height of ≈2 mm; the elastic reaction of the walls is due to o-rings, also sealing the fracture without adding friction disturbances. Fluids with different rheology, both Newtonian and shear-thinning, are associated with different boundary conditions of external pressure and overload. The match between theory and experiments is fairly good, with discrepancies of a few percent essentially due to the approximations of the theoretical model, and, for shear-thinning fluids, to the simplified constitutive equation.

A semianalytical model for transient pressure analysis of a horizontal well with non-uniform fracture geometry and shape-dependent conductivity in tight formations

Hydraulic fracture geometry greatly affects production performance of the fractured horizontal wells in a tight formation; however, limited attempts have been made to evaluate the performance of wells with non-uniform fracture geometries which are more close to the reality. In this work, a new slab source function was formulated and employed to accurately and efficiently evaluate the transient flow behaviour of a horizontal well with multiple non-uniform fracture geometries in a tight formation. More specifically, such a slab source function in the Laplace domain can assign a geometrical width for the source, which is usually neglected in the existing studies. Then, a semianalytical method is employed to solve the transient flow problem by dividing the non-uniform fracture into several segments, each of which can be assumed as a uniform-flux slab source. Thus, transient pressure behaviour considering complex fracture geometry can be taken into account with the slab source function using the superposition principle and the Stehfest inverse algorithm. The slab source function was validated with numerical and semianalytical solutions and then extended to field application. Also, effects of both the partially propped fractures and near-wellbore fracture damage are well examined and analyzed. It is found that the width of the fracture only affects the early time transient pressure response, while the pressure response at later time is identical to that of the line source solutions. The fracture geometry imposes a significant impact on the well response at the early and intermediate time period and the fracture with concave geometry results in the lowest pressure response, while the rectangular fracture has the largest pressure response. Both the length of the propped fractures and the damaged fracture zone mainly dominate the intermediate-stage pressure behaviour, while the width of the damaged zone will impose an important impact on the transient pressure response for the early and intermediate time period.

A Method to Represent a Well in a Three-Dimensional Discrete Fracture Network Model.

In discrete fracture network (DFN) modeling, fractures are randomly generated and placed in the model domain. The rock matrix is considered impermeable. Small fractures and isolated fractures are often ignored to reduce computational expense. As a result, the rock matrix between fractures could be large and intersections may not be found between a well introduced in the model and the hydraulically connected fracture networks (fracture backbones). To overcome this issue, this study developed a method to conceptualize a well in a three-dimensional (3D) DFN using two orthogonal rectangular fractures oriented along the well's axis. Six parameters were introduced to parameterize the well screen and skin zone, and to control the connectivity between the well and the fracture backbones. The two orthogonal fractures were discretized using a high-resolution mesh to improve the quality of flow and transport simulations around and along the well. The method was successfully implemented within dfnWorks 2.0 (Hyman et al. 2015) to incorporate a well in a 3D DFN and to track particles leaving an injection well and migrating to a pumping well. Verification of the method against MODFLOW/MODPATH found a perfect match in simulated hydraulic head and particle tracking. Using three examples, the study showed that the method ensured the connectivity between wells and fracture backbones, and honored the physical processes of flow and transport along and around wells in DFNs. Recommendations are given for estimating the values of the six introduced well parameters in a real-world case study.

OPTIMASI PANJANG HYDRAULIC FRACTURE PADA RESERVOIR NON-KONVENSIONAL DENGAN METODE UNIFORM CONDUCTIVITY RECTANGULAR FRACTURE

Energy needs in the future will continue to grow along with the growth of the population. Renewable and non-renewable energy sources continue to develop with various innovations. However, energy consumption from non-renewable energy such as coal, oil, and natural gas still dominates. Therefore, one of the potential non-renewable energy sources that can be optimized at present is unconventional oil and gas reserves. Unconventional oil and gas are oil and gas that comes from sourcerock, low permeability reservoirs, such as shale oil, shale gas, tight sand gas, coal bed methane, and methane-hydrate. To produce oil and gas from the tight sand gas reservoir, the hydraulic fracture method is a commonly used method. A hydraulic fracture is a well stimulation technique in which rock is fractured by a pressurized liquid. The process involves the high-pressure injection of fracking fluid into the wellbore to create crack in the deep rock formation through which natural gas, petroleum and brine will flow more freely. When the hydraulic pressure is removed from the well, small grains of hydraulic fracturing proppants hold the fracture open. Well log data such as gamma ray log, SP log, density log, resistivity log and so on will be processed and produce shale volume, porosity, permeability, and water saturation. Procced data from well log will be validated by core data. These data will be input into a reservoir model. A hydraulic fracture design will be made in the reservoir model with a certain length, width, and permeability using the uniform conductivity rectangular fracture method. The simulation will continue by using different length fracture design so that the optimum fracture length value is obtained.
 Keywords: Hydraulic Fracture, Reservoir Modelling, Reservoir Simulation

Well productivity evaluation in deformable single-fracture media

Evaluations and interpretations of reservoir productivity are frequent in geothermal, groundwater and hydrocarbon research and applications. In this study, we consider the closure of fractures around production wells due to compaction that can affect the productivity value, i.e. the ability of subsurface formations for transporting the desired fluid to a borehole. We introduce analytical tools to evaluate and predict changes in the productivity of a deformable fractured porous media.We propose analytical models for three geometries: a rectangular fracture with zero/non-zero orientation and a circular fracture with zero orientation to maximum horizontal stress. An advanced numerical model is utilised to evaluate the impact of spatial variation of fracture aperture induced by the fracture deformation on well productivity. The developed analytical solutions using a uniform fracture aperture always either over- or underestimate the production rate. Hence, an equivalent aperture model is developed for the fracture with aperture distribution under variable contact stresses to circumvent this problem.The proposed equivalent aperture model reduces the average and maximum errors of production-rate prediction from 28% to 0.6% and from 116% to 25%, respectively. We further employ the proposed model for sensitivity analyses to illustrate the impacts of in-situ and human-controlled parameters on productivity reduction. These analyses present that the interactions among initial reservoir pressure, fracture orientation, fracture stiffness, and well pressure control productivity reduction behaviours and the maximum productivity reduction values.

A procedure to estimate the accuracy of circular and elliptical discs for representing the natural discontinuity facet in the discrete fracture network models

Abstract This paper develops a procedure to estimate the accuracies of circular and elliptical disc models with respect to the natural discontinuity facet that represents various irregular shapes. The core of calculating the Accuracy Representation Index (ARI) was to obtain the area of the overlap between a fracture shape and an ellipse or a circle. The point radial method and Monte Carlo simulation were adopted to calculate the overlap area. Moreover, several rectangular fractures were generated, circular and elliptical disc models were respectively used to represent them and their representation accuracies were calculated. For the elliptical disc model, one important conclusion is that the ARI reaches the maximum when the elliptical disc has the same aspect ratio and rotation angle as the rectangular discontinuity. Furthermore, this method is applied to an actual slope rock mass of an open-pit mine, and the results showed that the simulation effect of the elliptical disc model was superior to that of the circular disc model, and the mean values of ARI for the elliptical and circular disc models were 0.898 and 0.739, respectively. Essentially, the optimum long-short axes ratio and the rotation angle of an elliptical disc model to represent the fracture can be obtained.

Zonal Disintegration Effect of Deep Straight Wall Semicircular Arch with Different Roadway Heights

In order to study the zonal disintegration effect of deep straight wall semicircular arch roadway with different roadway heights, FLAC3D numerical simulation software was used to systematically study the characteristics of plastic zone, surrounding rock deformation and surrounding rock abutment pressure distribution. The results show that: the rupture zone at the floor of straight wall semicircular arch roadway develops layer by layer in the form of “” and the displacement isoline at the roof is “eggshell shape”. No matter what the value of roadway height is, a rectangular fracture zone will be formed; the surrounding rock deformation in Z direction of straight wall semicircular arch roadway exists in the form of “inverted triangle” at the floor, and the height of “inverted triangle” is constant no matter what the height of roadway is, that is, the area is certain; after the excavation of the roadway, the stress concentration phenomenon occurs on the left and right sides of the straight wall semicircular arch roadway; the phenomenon mostly occurs on both sides of the arch.

Study of Oil-Water Two-Phase Stratified Flow in Horizontal Fractures

The relative permeability of oil-water two-phase flow is an important parameter in fractured petroleum reservoirs. It is widely accepted that the sum of relative permeabilities is less than 1. In this study, a series of experiments have been conducted on six rectangular fractures for oil-water two-phase flows. Analytical investigations of the effects of flow rate, aspect ratio, and fracture size on the relative permeability of oil-water two-phase are analysed. Basic fluid flow equations are combined to develop a new analytical model for water-oil two-phase flow in a horizontal fracture. The simulation results predicted by this model are in good agreement with the experimental data. The relative permeability is a function of flow ratio, viscosity ratio, aspect ratio and saturation. It increases as aspect ratio increases if the fracture depths are the same, while it decreases as aspect ratio increases if the fracture widths are identical. Both experiment and model indicate that the sum of relative permeabilities of oil and water is greater than 1 in some cases, different from the accepted view.

2-D Darcian flow in vicinity of permeable fracture perturbing unidirectional flow in homogeneous formation

A thin planar fracture with a highly permeable porous filling disturbs an incident unidirectional one-phase flow in a formation/aquifer and makes it essentially 2-D. The main objective of this paper is to get an analytical evaluation of the quantity of fluid intercepted, conveyed, and released back by the fracture and assessment of the near-fracture zone where 2-D effects are essential. The impact of the fracture is modelled by the Robin (third-type) boundary condition imposed along a mathematical cut. Flow in the fracture is 1-D and along the fracture axis. For a fracture of a rectangular shape, we use a conformal mapping of a physical plane with a cut onto the exterior of a unit disc in a reference plane and transform the Robin boundary condition into an infinite system of linear equations. Using the Kantorovich–Krylov theory, we rigorously prove that the system is entirely regular, i.e. it can be solved by the method of reduction (truncation to a finite system). A method of iterations is used to solve a reduced system. Convergence with respect to the truncation order is investigated. For non-rectangular (curvilinear) planar fractures with a straight axis, we use an inverse approach of Pilatovskii, viz., specifying the flow rate function along the fracture axis, i.e. the quantity of fluid which is intercepted by the fracture from the ambient rock. The fracture geometry is obtained as a part of solution. Pilatovskii’s function is involved as a weight of a singular integral with the Cauchy kernel. The corresponding integro-differential equation is a special case of the Prandtl equation for a thin airfoil in an ideal fluid flow with the circulation along the airfoil cord as a weight. The derivative of the Pilatovskii function is expanded into a series of Chebyshev’s orthonormal functions. This series is truncated, giving, in particular, the Pilatovskii fracture as a limit. Flow nets for both rectangular and curvilinear fractures are plotted. The dependences of the capture zone (quantity of intercepted–released fluid) on the ratios of permeabilities of the bulk rock-soil/fracture’s filling and fracture’s length/thickness, as well as the flow nets, are of interest to reservoir engineers and groundwater hydrologists working with Darcian flows in heterogeneous porous media.

Methods for Estimating Fracture Abundance and Size From Borehole Observations

SummaryThis study develops deterministic, exact equations relating fracture frequency (P10), density (P32), and length and width dimensions of rectangular and elliptical fractures from borehole observations. These general equations are applicable to both image logs and cores. A Monte Carlo type of simulation model generated the stochastic data used to derive the equations. The equations use five basic parameters relating frequency to density: borehole diameter, fracture length, fracture width, fracture angle with borehole axis (β), and the rotation angle of the long fracture axis within the fracture plane (γ). For both general equations, density corrections can be applied to individual fractures to find density. Fracture porosity is calculated on a per–fracture basis by applying aperture to density corrections. Both equations match the model to within the small standard deviation between simulations. In addition, the elliptical equation generally agrees with the existing exact theory.The study also develops methods for calculating fracture height (width) and length for rectangular fractures. Fracture height and length are calculated by observing the borehole–enclosed height and length and comparing them with the borehole–enclosed area. The calculation of fracture size is extended to estimate block height and length (block–face size.)These relationships cover a wide range of fracture size and orientation vs. borehole diameter. The theory is valid from small boreholes to tunnels. The general equations consider fractures as planar objects with length and width dimensions and negligible aperture compared with the other dimensions. Fracture aperture is applied, on a per–fracture basis, after the density correction to calculate fracture porosity. In addition to density and porosity calculations, an existing method for fracture–frequency prediction is improved by applying the general relationships.The methods described here are demonstrated using an image log from a vertical well from British Columbia, Canada. In this well, an image log was run over the Triassic section, including the zone of interest, the Montney Formation. Although the average fracture size in the Montney was very large, possibly on the order of tens of meters, they had a much smaller block–face size, on the order of a few meters, which could explain some of the production aspects from this formation.

Effective Permeability of Heterogeneous Fractured Porous Media

We compute the effective permeability of a model fractured porous medium in which rectangular fractures with finite thickness are distributed in the matrix with random orientations. The local permeability of the porous matrix is spatially distributed according to a fractional Brownian motion, which induces long-range correlations in the permeabilities. Using numerical solution of the Stokes’ equation, the effective macroscopic permeability is calculated as a function of a variety of factors, including the Hurst exponent that characterizes the type of the correlations, and the number density and width of the fractures. The results indicate that the correlations give rise to non-trivial effects on the effective permeability and that the assumption of a uniform matrix in a fractured porous medium leads to overall permeabilities that differ significantly from those in which the matrix is heterogeneous.