Fracture Compliance Research Articles

Numerical experiments of fracture-induced velocity and attenuation anisotropy

SUMMARY FracturesarecommonintheEarth’scrustduetodifferentfactors,forinstance,tectonicstresses and natural or artificial hydraulic fracturing caused by a pressurized fluid. A dense set of fractures behaves as an effective long-wavelength anisotropic medium, leading to azimuthally varying velocity and attenuation of seismic waves. Effective in this case means that the predominant wavelength is much longer than the fracture spacing. Here, fractures are represented by surface discontinuities in the displacement u and particle velocity v as [κ ·u + η ·v], where the brackets denote the discontinuity across the surface, κ is a fracture stiffness and η is a fracture viscosity. We consider an isotropic background medium, where a set of fractures are embedded. There exists an analytical solution—with five stiffness components—for equispaced plane fractures and an homogeneous background medium. The theory predicts that the equivalent medium is transversely isotropic and viscoelastic. We then perform harmonic numerical experiments to compute the stiffness components as a function of frequency, by using a Galerkin finite-element procedure, and obtain the complex velocities of the medium as a function of frequency and propagation direction, which provide the phase velocities, energy velocities (wavefronts) and quality factors. The algorithm is tested with the analytical solution and then used to obtain the stiffness components for general heterogeneous cases, where fractal variations of the fracture compliances and background stiffnesses are considered.

Pressure-dependent seismic velocities based on effective compliance theory and an asperity deformation model

Seismic velocities of rocks depend strongly on confining pressure, which is often explained by the fracture compliances changes within the rocks. It is important to have an accurate model describing the relations between confining pressure and seismic velocities for applications such as time-lapse reservoir characterization. We propose a new model to address this problem by combining the existing effective compliance theory with new solutions for the pressure dependence of fracture compliances. Specifically, we assume the fracture contact surface can be represented by a set of elastic hemispheres with radii following power-law distribution, and the pressure dependence of seismic velocities can be expressed through pressure-dependent normal and tangential fracture compliances that are derived from Hertzian contact theory. Joint data fittings of P- and S-wave velocity laboratory data show that our model is accurate. We also implement fluid substitution using our model to explain the similar stress-induced velocity variations of fluid-saturated fractured rocks.

Fracture incidence and compliance to treatment in the daily clinical practice: 24-month final results of the observational Italian Study on Severe Osteoporosis (ISSO)

Fracture incidence and compliance to treatment in the daily clinical practice: 24-month final results of the observational Italian Study on Severe Osteoporosis (ISSO)

Impact of Fractures on CO2Storage Monitoring: Keys for an Integrated Approach

The monitoring of CO2 storage in fractured reservoirs (depleted hydrocarbon fields or brine aquifers) requires the study of the impact of fracturation and fluid substitution on seismic data. Seismic data can provide information about the additional compliance due to the fractures and the fluids through the analysis of seismic azimuthal anisotropy with an appropriate rock physics model. We introduce a rock physics model built in collaboration with geologists, providing a realistic description of fractured media. This model concerns fractured geological media in the presence of fluids characterized by some degree of matrix porosity, the presence of pore fluids, connected and/or non-connected fractures, the presence of several fracture sets, and an inherent seismic anisotropy. The direct application of this rock model shows that the P -wave anisotropy value measured through seismic data can be explained by several sets of different parameters such as the fracture density, the pore fluid compliance or the porosity. The presence of inherent layer-induced anisotropy can also modify the P -wave anisotropy and thus the interpretation of this value in terms of fluid substitution in a fractured porous medium. As far as fluid substitution monitoring is concerned, if seismic data are acquired before and after this substitution, a change in the P -wave anisotropy value can be linked to the modification of the compliance of the fluid content in the same medium exhibiting the same fracture network and the same porosity. This relative value can only be correctly interpreted in terms of fluid substitution provided we have some constraints on a few of the parameters involved in the P -wave anisotropy value such as the porosity, and a rough idea of the level of normalized fracture compliance. Then, a multidisciplinary approach is mandatory to constrain these parameters. For instance, borehole and outcrop geological information can give the upper limit of the fracture density expected at depth in the same formation. Furthermore, rock mechanics helps in understanding the fracturation state at depth to identify the predominant fractures in regard to the interpretation of seismic anisotropy in terms of fluid substitution inside the fracture network.

Field measurements of normal and shear fracture compliance

ABSTRACTField estimates of the normal and shear compliance of water saturated fractures were obtained from a seismic experiment carried out on a wave‐cut platform of Upper Caithness Flagstone on the North coast of Scotland. The rocks are cut by two orthogonal sets of vertical fractures. Vertical geophones were glued to the rock surface and seismic waves were generated by striking the surface with a sledge hammer. First arrival traveltimes were obtained for source‐receiver distances up to 30 m. Differences of compressional velocity with direction across the platform were interpreted as resulting from the compliance of the fractures intersected by the transmitted waves. The average normal compliance for fractures at this site is 4.0 × 10−12± 1.0 × 10−12 m/Pa. There is much less certainty about the precise value of shear compliance. However, the normal/shear compliance ratio is determined to be ≤0.1.

Seismic modelling of a fractured carbonate reservoir in Abu Dhabi, United Arab Emirates

ABSTRACT The problem of identifying and quantifying the brittle deformation of carbonate reservoirs both in the United Arab Emirates (UAE) and elsewhere is addressed. Naturally occurring fractures may substantially increase or decrease the permeability and porosity of reservoirs, and therefore knowledge of location, orientation, density and connectivity of fractures is required to optimise hydrocarbon production. A rock containing parallel fractures can be seismically anisotropic, provided the vertical and horizontal extent and spacing of the fractures is small compared to the seismic wavelength. Seismic anisotropy may be detectable from attributes of pre-stack 3-D seismic data including reflection amplitude variation with offset and azimuth (AVOA). However, in carbonates seismic-velocity anisotropy can result from many different factors, including present-day horizontal stress anisotropy, sedimentological features such as clinoforms, and geological structure. We present a methodology for determining whether a proposed reservoir-fracture model is consistent with the observed seismic data. The approach includes modelling the seismic anisotropy where an essential input parameter is the compliance of the fractures. Since so little is known about this key parameter, we determine an upper bound to fracture compliance from well data and existing laboratory and field data and consequently obtain an upper bound to the seismic anisotropy that might be detected. We apply our method to data from an onshore carbonate oilfield in Abu Dhabi, United Arab Emirates, where analysis of core, log and 3-D post-stack seismic data indicates that open or partially open fractures may be pervasive and could have a dominant influence on reservoir production. Due to poor AVOA data quality our results are inconclusive. However, this case study is a demonstration of the methodology that could be applied elsewhere.

Fatigue Behavior and Damage Assessment of Stainless Steel/Aluminum Composites

Fatigue crack growth and related damage mechanisms were investigated experimentally in a stainless steel/aluminum laminated composite with middle through thickness crack, and two different fracture mechanics approaches applied to the composite to reveal their differences under fatigue loading. The laminated composite material, which has a unidirectional continuous AISI 304 stainless steel as fibers and Al 1060 as matrix, was produced by using diffusion bonding. Fatigue tests were conducted in accordance with ASTM E 647. The relationships between fatigue crack growth rate (da/dN), stress intensity factor (ΔK), and strain energy release rate (ΔG) were determined; and damage behavior was discussed. Both linear elastic fracture mechanics (LEFM) and compliance method were used, and the results were compared with each other. It is found that as the crack propagates, the LEFM overestimates the ΔG values. Interlaminar and fiber/matrix interface damage were evaluated by fractographic examination.

The relationship between bisphosphonate adherence and fracture: Is it the behavior or the medication? Results from the placebo arm of the fracture intervention trial

Medication compliance may be a surrogate for factors that improve health outcomes such as fractures. Little is known about the size of this potential “healthy adherer” effect. We evaluated the hypothesis that compliance with placebo is associated inversely with bone loss and fractures among women participating in the Fracture Intervention Trial (FIT). Compliance with placebo and alendronate was evaluated using daily medication diaries. Women were defined as having high compliance if they took 80% or more of dispensed study medication. Change in bone mineral density (BMD) was assessed using mixed models comparing women with high versus lower compliance with placebo. Cox proportional-hazards models analyzed the association between placebo compliance and various types of fractures. Among 3169 women randomized to placebo, 82% had high compliance. Compared with women with lower placebo compliance, bone loss at the total hip was lower in compliant placebo-treated women (−0.43%/year versus −0.58%/year, p = .04). Among placebo-treated women, there were 46 hip, 110 wrist, 77 clinical vertebral, and 492 total clinical fractures. Compared with women with lower placebo compliance, women with high placebo compliance had a nonsignificant reduced risk for hip fracture [adjusted hazard ratio (HR) = 0.67, 95% confidence interval (CI) 0.30–1.45]. This trend was not observed for other fractures. Medication compliance may be a proxy for factors that confers benefit on reducing hip fracture (but not other types of fractures) independent of the effect of the medication itself. Nonrandomized studies of interventions designed to maintain or improve bone density and/or hip fracture may need to consider medication compliance as a confounder to better estimate true intervention effects. © 2011 American Society for Bone and Mineral Research.

Rock fracture compliance derived from time delays of elastic waves

ABSTRACTThe purpose of this study is to compare the reliability of various methods of estimating normal rock fracture compliance from elastic wave measurements. We compare ultrasonic through‐transmission laboratory measurements for a smooth fracture in a Westerly granite specimen with numerical simulations and analytical solutions. The focus is on deriving compliance from time delays. The influence of specimen and source transducer width was constrained using numerical wave simulations. We find that measured ultrasonic phase delays are better suited to estimate the fracture compliance than group delays. Using the frequency domain instead of the time domain increases the accuracy of the fracture compliance estimates. We further show that for cases where precise phase delay measurements are unavailable, employing first break times in conjunction with numerical simulations can be considered as an alternative.

Fracture characterization at multiple scales using borehole images, sonic logs, and walkaround vertical seismic profile

We present a quantitative forward-modeling methodology to link and interpret several measurements relevant to mechanical properties of fractures such as borehole images, sonic anisotropy logs, and borehole seismic anisotropy. The analysis is applied to a case study from a north African tight gas field using data from a vertical well. Two studies are conducted independently using the same geological fracture data to model fracture-induced anisotropy. In the first study, we use the orientation of the natural and drilling-induced fractures interpreted on the image log to model the azimuthal fracture-induced anisotropy at the sonic scale. The mechanical effects of natural and drilling-induced fractures are treated using different compliance parameters for each fracture type. We show that modeled sonic fast shear azimuths could be biased by the presence of noncompliant fractures in each fracture type, and we propose an empirical selection criterion to reject noncompliant fractures prior to compliance estimation. Then, we estimate the fracture compliances and confirm that natural open fractures have larger compliances than drilling-induced fractures. In the second study, we apply interpreted borehole images toward modeling of the azimuthal vertical seismic profile (VSP) attributes as a function of source azimuthal position. Natural fractures inside a window of height, h, and located at depth, d, are included, and several volume sizes and positions (i.e., h and d) are considered. We find a good agreement between modeled and observed transverse-over-radial displacement trends using natural fractures within windows located at the depth of the VSP receiver, and having window heights on the order of one to two VSP shear wavelengths.

Validation of elastic wave measurements of rock fracture compliance using numerical discrete particle simulations

ABSTRACTWe test various methods of quantifying the compliance of single and multiple rock fractures from synthetic ultrasonic data. The data are generated with a 2D discrete particle scheme which has previously been shown to treat fractures in agreement with linear‐slip theory. Studying single fractures, we find that delays derived from peak amplitudes do not correspond to group delays, as might be expected. This is due to waveform distortion caused by the frequency‐dependent transmission across the fracture. Instead the delays correspond to an expression for phase delays, which we derive from linear‐slip theory. Phase delays are a unique function of compliance, whereas group delays are non‐uniquely related to compliance. We believe that this property of group delays has hindered the wider application of deriving fracture compliances from traveltimes. We further show that transmission coefficients derived from waveform spectra yield more accurate fracture compliances than those obtained from ratios of signal peak amplitudes. We also investigate the compliance of a set of parallel fractures. Fracture compliance can only be determined from transmission coefficients if the fracture spacing is so large that the first arriving pulse is not contaminated by reverberations. In the case of contamination the direct measurement of group or phase delays is not practical. However, we demonstrate that in such cases of strong waveform distortion the coda wave interferometry method is very effective for determining relative fracture compliance. First break delays in the fracture set data are related to those observed in single fracture simulations. This means that fracture set compliance can be estimated from first break data if used together with numerical simulations.

Effect of Inclined Cracks on the Fatigue and Fracture Behavior of Woven Steel/Reinforced Polyethylene Composite

In this study, the fatigue and fracture behavior of woven steel/reinforced high-density polyethylene thermoplastic composite, having a middle tension specimen with inclined through-thickness cracks, under fatigue loading were examined. The composite material was manufactured from bidirectional woven Cr—Ni steel fibers and high-density polyethylene with a fiber volume fraction (Vf) of 0.04. The separate cracks on the sample are oriented at different angles such as 15°, 30°, 45°, 60°, and 75°. According to each crack geometry, fatigue crack growth was investigated. During the fatigue test, crack initiation, fatigue life, fatigue crack growth rate (da/dN), and stress intensity factor ranges (ΔKI, ΔK II, ΔKIeq, ΔKIIeq, and ΔKeq) were examined. In addition, strain energy release rate (ΔG) was checked out, since it was also an effective parameter in crack propagation. Crack growth rate was used as a damage parameter related to the crack geometry. In the results, crack growth rate was controlled through stress intensity factor ranges (ΔK ) and strain energy release rates (ΔG). Graphs of Δa/N, da/dN - ΔK and da/dN - ΔG such as material properties under consideration were plotted on log scale. Material constants were determined using Paris—Erdogan equation. Also, ΔG values were determined using linear elastic fracture method and compliance method, and the results were compared.

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.

Vertically fractured transversely isotropic media: dimensionality and deconstruction

ABSTRACTA vertically fractured transversely isotropic (VFTI) elastic medium is one in which any number of sets of vertical aligned fractures (each set has its normal lying in the horizontal x1, x2‐plane) pervade the medium and the sets of aligned fractures are the only features of the medium disturbing the axi‐symmetry about the x3‐axis implying that in the absence of fractures, the background medium is transversely isotropic (TI). Under the assumptions of long wavelength equivalent medium theory, the compliance matrix of a fractured medium is the sum of the background medium's compliance matrix and a fracture compliance matrix. For sets of parallel rotationally symmetric fractures (on average), the fracture compliance matrix is dependent on 3 parameters − its normal and tangential compliance and its strike direction. When one fracture set is present, the medium is orthorhombic and the analysis is straightforward. When two (non‐orthogonal) or more sets are present, the overall medium is in general elastically monoclinic; its compliance tensor components are subject to two equalities yielding an 11 parameter monoclinic medium. Constructing a monoclinic VFTI medium with n embedded vertical fracture sets, requires 5 TI parameters plus 3×n fracture set parameters. A deconstruction of such an 11 parameter monoclinic medium involves using its compliance tensor to find a background transversely isotropic medium and several sets of vertical fractures which, in the long wavelength limit, will behave exactly as the original 11 parameter monoclinic medium. A minimal deconstruction, would be to determine, from the 11 independent components, the transversely isotropic background (5 parameters) and two fracture sets (specified by 2 × 3 = 6 parameters). Two of the background TI medium's compliance matrix components are known immediately by inspection, leaving nine monoclinic components to be used in the minimal deconstruction of the VFTI medium. The use of the properties of a TI medium, which are linear relations on its compliance components, allows the deconstruction to be reduced to solving a pair of non‐linear equations on the orientations of two fracture sets. A single root yielding a physically meaningful minimum deconstruction yields a unique minimal representation of the monoclinic medium as a VFTI medium. When no such root exists, deconstruction requires an additional fracture set and uniqueness is lost. The boundary between those monoclinic media that have a unique minimal representation and those that do not is yet to be determined.

Seismic characterization of reservoirs containing multiple fracture sets

ABSTRACTNatural fractures in reservoirs play an important role in determining fluid flow during production and knowledge of the orientation and density of fractures is required to optimize production. Variations in reflection amplitude versus offset (AVO) are sensitive to the presence of fractures but current models used to invert the seismic response often make simplified assumptions that prevent fractured reservoirs from being characterized correctly. For example, many models assume a single set of perfectly aligned fractures, whereas most reservoirs contain several fracture sets with variable orientation within a given fracture set. In addition, many authors only consider the azimuthal variation in the small offset amplitude versus offset and azimuth response (the variation in AVO gradient with azimuth), while the effect of fractures on amplitude versus offset and azimuth increases with increasing offset. In this paper, the variation in the reflection coefficient of seismic P‐waves as a function of azimuth and offset due to the presence of multiple sets of fractures with variable orientation within any fracture set is used to determine the components of a second‐rank fracture compliance tensor αij. The variation in the trace of this tensor as a function of position in the reservoir can be used to estimate the variation in fracture density with position in the reservoir and may be used to choose the location of infill wells in the field. The principal axes of αij reveal the most compliant direction within the reservoir and may be used to optimize the trajectory of deviated wells. The determination of the principal axes of αij requires wide azimuth acquisition and the use of the small‐offset amplitude versus offset and azimuth (the azimuthal variation of the AVO gradient) may give misleading results.

Interpreting seismic anisotropy in fractured reservoirs

An upper limit to the compliance of individual fractures within a reservoir can be determined, given estimates of the effective confining pressure at reservoir depths and the mean aperture of open fractures which is usually obtained from core and well log analysis. The value can then be used with elastic wave numerical modelling to determine an upper limit to the seismic velocity anisotropy that would result from any specified reservoir discrete fracture model. This provides a level of confidence when using seismic anisotropy as an indicator of the presence of fractures that could contribute significantly to reservoir permeability. The least certain aspect of the proposed methodology is the estimate of the ratio of normal to shear fracture compliance.

Laboratory estimates of normal and shear fracture compliance

ABSTRACTLaboratory estimates of the normal (Bn) and shear (Bt) compliance of artificial fractures in samples of Jurassic and Carboniferous limestone under wet and dry conditions are presented. The experiments were performed over a range of confining pressures (from 5 MPa up to 60 MPa), at ultrasonic frequencies in a Triaxial Hoek cell, using the pulse‐echo reflection technique. The results of this study confirm that the Bn/Bt ratio of a fracture is dependent on the fluid fill. A value of Bn/ Bt of less than 0.05 was obtained for our wet (honey saturated) sample which is consistent with the prediction that this ratio should be close to zero for fluid saturated fractures. Values of Bn/Bt for the dry sample are significantly higher and increase with confining pressure from 0.2 to 0.5. It is suggested that a Bn/Bt ratio of 0.5 is probably a more representative value to use in modelling studies of gas filled fractures than the common assumption that Bn ≈ Bt.

Multiple cracks in VTI rocks: Effective properties and fracture characterization

The existing fracture-characterization techniques are based on assuming the unfractured host rock either to be isotropic or the magnitudes of both the background and crack-induced anisotropies to be small. I relax both assumptions and examine the effective media caused by fractures with realistic (not small) crack densities in a strongly anisotropic, primarily transversely isotropic (TI) host rock. The analysis of penny-shaped cracks in the noninteraction approximation (NIA) reveals the dependence of their excess fracture compliance tensors on the orientation of the background symmetry axis. As a result of this dependence, the excess fracture compliance tensor generally becomes rotationally noninvariant even when the cracks are circular. One of the consequences of this complication (compared to the background isotropy) is a reduction of symmetry from TI to monoclinic resulting from the presence of a single oriented fracture set. Verti–cal dry cracks in a vertically transversely isotropic (VTI) host constitute an important exception to this general rule.The effective symmetry for this arrangement is approximately orthorhombic (or orthotropic) even in the presence of multiple fracture sets that have arbitrary azimuths. I perform finite-element simulations on the so-called digital rocks to verify both the proximity of effective symmetry to orthotropy and the accuracy of the NIA up to the crack density of 0.15. Multiple sets of dry vertical cracks in a VTI host not only result in nearly orthorhombic effective symmetry but also their cumulative influence is equivalent to that of just two orthogonal (or principal) fracture sets. The possibility of replacing multiple fracture sets with two orthogonal ones paves the way for their characterization. The inverse problem of estimating the parameters of two orthogonal crack systems in a VTI background from the effective elasticity, however, is known to be nonunique. I suggest overcoming its ambiguity by combining 3D, wide-azimuth, multicomponent seismic data with sonic logs.

The scaling of fracture compliance

Abstract The seismic visibility of fractures depends on the magnitude of their normal and shear compliance and how these quantities vary with the scale of the fracture and depth of burial. Reliable estimates of fracture compliance as a function of confining pressure, in the range 10 -13 –10 -14 m Pa -1 , have been obtained from laboratory measurements on core samples. The possibility of fractal scaling of fracture parameters has been proposed, in which case fracture compliance might be expected to increase with the scale of the fracture. Laboratory and field estimates of fracture compliance are presented covering a range of fracture sizes. Compliance is shown to increase with the scale of the fractures. Results obtained are broadly consistent with the magnitudes predicted from linear slip theory, in which the displacement discontinuity across a partially sealed interface is linearly related to the traction on the interface.

A field investigation of fracture compliance

ABSTRACTA field measurement of fracture compliance is described. The aim was to determine how compliance scales with fracture size and, specifically, how laboratory measurements of fracture compliance compared with field estimates from sonic and seismic data. A test site was constructed, consisting of three 40 m vertical boreholes drilled in the floor of a Carboniferous Limestone quarry. Detailed knowledge of the rocks in the test area was obtained from core analysis, wireline logging and local area fracture mapping. Seismic cross‐hole surveys were performed using a sparker source with a dominant frequency of 2000 Hz and hydrophone receivers. The rocks had a compressional‐wave velocity anisotropy of 10%, which was attributed to the presence of predominantly horizontal, partially open fractures. Estimates of normal fracture compliance within a range from 2.5 × 10−13 m/Pa to 3.5 × 10−12 m/Pa were obtained from both the cross‐hole data and the sonic‐log data. This is an order of magnitude greater than values obtained from laboratory experiments which are reported elsewhere.