Normal Compliances Research Articles

Stress‐induced anisotropy in Gulf of Mexico sandstones and the prediction of in situ stress

AbstractThe strong sensitivity of velocity to stress observed in many sandstones originates from the response of stress‐sensitive discontinuities such as grain contacts and microcracks to a change in effective stress. If the change in stress is anisotropic, then the change in elastic wave velocities will also be anisotropic. Characterization of stress‐induced elastic anisotropy in sandstones may enable estimation of the in situ three dimensional stress tensor with important application in solving problems occurring during drilling, such as borehole instability, and during production, such as sanding and reservoir compaction. Other applications include designing hydraulic fracture stimulations and quantifying production‐induced stresses which may lead to rock failure. Current methods for estimating stress anisotropy from acoustic anisotropy rely on third‐order elasticity, which ignores rock microstructure and gives elastic moduli that vary linearly with strain. Elastic stiffnesses in sandstones vary non‐linearly with stress. Using P‐ and S‐wave velocities measured on Gulf of Mexico sandstones, this non‐linearity is found to be consistent with a micromechanical model in which the discontinuities are represented by stress‐dependent normal and shear compliances. Stress‐induced anisotropy increases with increasing stress anisotropy at small stress but then decreases at larger stresses as the discontinuities close and their compliance decreases. When the ratio of normal‐to‐shear compliance of the discontinuities is unity, the stress‐induced anisotropy is elliptical, but for values different from unity, the stress‐induced anisotropy becomes anelliptic. Although vertical stress can be obtained by integrating the formation's bulk density from the surface to the depth of interest, and minimum horizontal stress can be estimated using leak‐off tests or hydraulic fracture data, maximum horizontal stress is more difficult to estimate. Maximum horizontal stress is overpredicted based on third‐order elasticity using measured shear moduli, with estimates of pore pressure, vertical stress and minimum horizontal stress as input. The non‐linear response of grain contacts and microcracks to stress must be considered to improve such estimates.

Craniectomía descompresiva y compliance intracraneal en el traumatismo craneoencefálico grave pediátrico

ObjectiveTo show the effect of primary decompressive craniectomy on ICP, CPP, and intracranial compliance in a group of pediatric patients who suffered from severe head trauma. DesignDescriptive study. FrameworkPediatric intensive care. PatientsPediatric patients who underwent primary decompressive craniectomy due to severe head trauma, admitted to the pediatric intensive care unit of the “Roberto Rodríguez Fernandez” general teaching hospital in Morón, Ciego de Avila, Cuba between January 2003 and December 2017. ResultsDiffuse brain injury Grade IV, was the most frequent tomographic diagnosis at admission 18 (60%), with a predominance of shifts greater than 6mm in 24 cases (80%) and absent cisterns 15 (50%). Eighteen cases (60%) had normal ICP for their age after craniectomy, 20 (66.7%) cases showed normal cerebral perfusion pressure. There was a correlation between normal ICP and normal intracranial compliances. The pulse wave amplitude was used as a neuromonitoring method and qualitative representation of compliances. Twenty-two cases survived (73.3%), 18 (60%) of them without sequelae. Eight cases died (26.7%). ConclusionsPrimary DC was useful for the control of ICP and maintenance of CPP in the management of pediatric patients with severe TBI in the present study. The favorable effect of the procedure for the increase of intracranial compliance has been verified.

Wave reflection and transmission coefficients for layered transversely isotropic media with vertical symmetry axis under initial stress

SUMMARYThe stress-dependent wave reflection and transmission (R/T) coefficients for the layered transversely isotropic media with a vertical symmetry axis (VTI) are seldom investigated in the published literature. To fill this gap, we propose the exact formulas for the plane wave R/T coefficients on the welded and non-welded boundaries between two distinct VTI half-spaces under the effect of initial stress. The theory of acoustoelasticity is used to describe the influence of initial applied stress on the overall elastic properties of VTI media and to represent two different boundary conditions. The normal and tangential compliances are used to characterize the discontinuity of non-welded boundary based on the linear-slip model, and their stress dependences are ideally considered according to the effect of stress on fracture aperture. Then the plane-wave displacement equations are substituted into the boundary conditions to yield the analytic formulas for frequency-independent R/T coefficients for the welded interface and frequency-dependent R/T coefficients for the non-welded interface. The stress-dependent wave slowness vector and polarization vector embedded in R/T coefficients can be directly computed with the Christoffel equation given by the acoustoelastic equation. Modelling results graphically show the effects of initial stress on the angle-dependent wave velocities, Thomsen elastic anisotropy parameters, fracture compliances, the R/T coefficients and seismic reflection responses for welded and non-welded interfaces in detail. The R/T coefficients are more sensitive to initial stress at relatively large incidence angles for the designed two-layer model with welded or non-welded boundary. The proposed R/T coefficient formulas and modelling results are relevant to in situ stress detection, fracture characterization, and exploration for oil and gas in shale reservoirs in high-stress fields.

Novel bi-layer beam elements for elastic fracture analysis of delaminated composite beams

A general finite element method is proposed for linear elastic fracture analysis of delaminated bilayer composite beams. An interface deformable bilayer beam model is adopted in modeling the virgin portions of bilayer beams, and it is capable of guaranteeing the continuity condition of internal forces at the tip of delamination because the bilayers are modeled as two separate shear deformable sub-beams and both the interface normal and shear compliances are introduced to account for interface deformation. The stiffness matrices of bilayer beams using the linear and quadratic interpolation polynomials are both derived using the principle of minimum potential energy. Then, a finite element program is developed by assembling the element stiffness matrices of virgin and delaminated portions of bilayer beams and applying the corresponding equivalent nodal forces. Two typical fracture specimens (i.e., single leg bending and end-notched flexure bilayer beams) are analyzed using the proposed bilayer beam elements. The computed displacement and energy release rate are compared well with those of analytical solutions, and the mode components of energy release rate for single leg bending specimen are also obtained using the virtual crack closure technique. It demonstrates that the proposed novel bilayer beam elements are capable of effectively and efficiently predicting the displacement and energy release rate (including its mode components) of delaminated bilayer beams. The developed bilayer beam element is general in nature and accurate and efficient in particular, because it can achieve the same accuracy as the high order beam theory and be easily implemented in the existing commercial finite element software.

Elastic properties of fractures in transversely isotropic media

A fracture may be characterized by its normal and shear compliances that relate the discontinuity in displacement of the fracture faces to an applied traction. To assess the feasibility of using elastic waves for monitoring natural and hydraulic fractures in anisotropic shale reservoirs, an understanding of how these compliances relate to the properties of the background rock and any fracture infill material is required. Three models of vertical fractures in a VTI (transversely isotropic medium with vertical axis of rotational symmetry) background suitable for feasibility studies are discussed. The first two models are suitable for open vertical fractures and represent the fractures as aligned low aspect ratio elliptic cylindrical inclusions with long axis of the ellipse either horizontal or vertical. In these models the horizontal and vertical shear compliance differ in a way that depends on the anisotropy of the background medium and the orientation of the fractures. The third model is suitable for estimating the properties of hydraulic fractures containing proppant. In this model, proppant is represented as a random packing of identical elastic spheres subject to a uniaxial compression due to the normal stress acting on the fracture. The results are expected to help improve the reliability of fracture models used for assessing the feasibility of elastic waves for characterizing natural and hydraulic fractures.

Assessing Crack‐Induced Compliance in Low Porosity Rocks Damaged by Thermo‐Hydro‐Chemo‐Mechanical Processes

AbstractFracture geometry, density, and distribution control fluid and thermal transport in low porosity rocks, as well as their seismic attributes. The creation of cracks throughout the matrix can result in large and disparate reductions in P and S wave velocities with negligible porosity development, which breaks the classic velocity‐porosity relationship. Properly interpreting these signatures in seismic surveys from stimulated geothermal, oil, and gas reservoirs or seismogenic areas requires a rock physics model that uses realistic elastic properties of fractures. Conventional models for cracked rocks employ specific inclusion shapes such as spheroids with small aspect ratios. However, all these exhibit the same normal‐to‐shear compliance ratio—regardless of aspect ratio—and therefore impact the bulk and shear moduli in the same qualitative way. As such they are inadequate for describing many of the complementary evolutions of P and S wave velocities that transpire with the generation of cracks. Using a differential model of dimensionless compliances, we analyze published velocities for granite, basalt, and limestone to quantify typical compliance ratios and find that they are often different from those assumed in conventional crack models, which then affects crack density estimates. We also investigate how the total normal and shear compliances change due to certain thermo‐hydro‐chemo‐mechanical processes that generate large crack densities, and conclude that the final ratios often still deviate from that of conventional models. Finally, we demonstrate how the model can be used to describe scatter in both P and S wave velocity‐porosity trends. Implications for energy extraction and seismic monitoring are discussed.

The influence of 2D periodic surface texture on the partial slip problem for elastic bodies

We consider contact interaction of elastic bodies, one of which has a 2D texture, described by a sinusoidal function of 2n power, in partial slip conditions within two material models. The first approach based on the Cattaneo-Mindlin theory for linear elastic half-plane, the second is an elastic foundation model (EFM) which takes into account normal and tangential compliances. Analytical expressions for determining the size of the contact zone and the stick and slip zones for both models are obtained, as well as the expressions for the normal and tangential tractions. It is shown that for relatively small profile heights, both models give close results for the stick and slip zones sizes when the parameters of the EFM are selected from the developed half-plane solution. We analyze the influence of the geometric parameters (shape, profile height, and distance between the grooves) of the texture profile on the stick zone size while the tangential force increases. The EFM model allows capturing the dependence of the stick zone width on the profile height.

Determining the transverse isotropic rocks’ static elastic moduli with cylindrical plugs: Shortfalls, challenges, and expected outcomes

Cylindrical-shaped plugs can be tested using a Hoek-cell-like apparatus that allows for efficient and inexpensive measurements of a rock’s static elastic properties. However, when it comes to transverse isotropic material, this approach has a natural limitation due to the isotropic radial stresses; particular attention to the boundary conditions and the proper design of pressurization steps is warranted. Typical attempts to constrain the complete set of compliances ([Formula: see text]), using multiple plugs of different orientations, are impeded by the heterogeneity and pressure-dependent elasticities inherent to sedimentary rocks. Through stepwise pressure increases, we can constrain four normal compliances [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], describing two Young’s moduli and three Poison’s ratios, using a single horizontal plug drilled parallel to the rock’s isotropic plane, contrary to the common assumption that both horizontal and vertical plugs are needed. The measurement of the shear modulus [Formula: see text] needs to be obtained using a plug that is drilled oblique to the isotropic plane; replicating the in situ stress environment is not possible using this approach. Finally, the specimen’s anisotropic plane’s geometry is elliptical under isotropic radial stress; this causes a discrepancy between the strain gauge’s contraction and the actual strain. We have developed an iterative inversion approach to account for this issue and calculate the exact strains useful for inferring [Formula: see text] from measurements reported by strain gauges. The example included in this writing indicates that, without correction, inferred values of [Formula: see text] may suffer errors of 20%.

Effect of surface relief on sliding friction of viscoelastic bodies

A model for calculating the sliding friction force between a surface with regular relief and a viscoelastic foundation in dry conditions is suggested. The Winkler type viscoelastic foundation with both normal and tangential compliances is considered. The local Coulomb friction law is assumed between the surfaces. The total friction is calculated including the contribution of both local adhesive friction and hysteretic losses in the viscoelastic material when elements of the surface relief cyclically deform the material during sliding. The contact problem is solved by the strip method in which the area of interaction is divided into strips parallel to the direction of sliding, and a 2D contact problem is solved in each strip in the closed form. Two types of periodic profile of the moving surface are considered—a profile with trapezoidal asperities and a doubly periodic sinusoidal profile. The model is applied to analyze the influence of the parameters of surface relief on the coefficient of sliding friction for various values of external load and sliding velocity.

Параметры акустической неоднородности для неразрушающей оценки влияния технологии изготовления и эксплуатационной поврежденности на структуру металла

In this paper, we propose a non-destructive acoustic method for assessing microstructural alterations produced in metals as a result of processing and exploitation damage. The method is based on measuring ratios of longitudinal and shear ultrasound waves velocities with subsequent normalization of these ratios to their counterparts corresponding to a “baseline state” of the metal. The choice of the baseline state depends of the particular problem and corresponds to either the pre-exploitation metal state for the object to be monitored, or the state of reference samples, for which the strength characteristics fall into acceptable range according to the results of conventional strength-evaluation methods. For the introduced Microstructure-Sensitive Acoustic parameters (MSA-parameters) characterizing alterations in the metal microstructure, we discuss the reasons of the difference between the baseline and current states of the metal in terms of the total values of the normal and tangential compliances imparted to the metal due to the appearance of microstructural heterogeneities. The efficiency of utilization of the proposed acoustic diagnostic parameters is demonstrated using experimental data for two types of metal samples with essentially different microstructure (isotropic and anisotropic heterogeneity). Measurements of the proposed microstructure-sensitive acoustic parameters can be used in a variety of applications: (i) non-destructive quality control of manufacturing technologies in metallurgy and mechanical engineering (ii) evaluation of damage accumulation during exploitation manufactured constructions/samples; (iii) evaluation of mechanical properties, fracture resistance and other physical parameters of metals.

The stress dependence anisotropic elastic properties of shale

The stress dependence anisotropic elastic properties of shale is experimentally evident. It is well explained by the homogenization theory for a medium containing imperfect contact areas (ICA) of which the elastic properties are functions of the effective confining stress. For practical purpose, a minimum number of parameters related to the ICA is considered and calibrated on experimental data. It is observed that the trends of the five transversely isotropic elastic stiffnesses versus the effective confining stress can be well captured by four microscopic parameters: (1) a parameter that characterizes the orientation distribution of the contacts; (2) a ratio between the normal and in-plane compliances of the contacts; (3) the magnitude of the contact compliance at the stress-free condition; and (4) a curvature parameter that defines the dimensionless evolution of the contact compliance versus the effective confining stress. More importantly, these parameters can be fixed for different groups of materials: e.g. wet shale, dry shale, kerogen-rich shale, etc. therefore, a strong predictive capacity of the proposed method is approved. It is then used to predict the effective stiffnesses of shale at a high stress range using the measured data at a low stress range. It can also be used to explain the difference between the elastic properties of in-situ shale and shale samples at a stress-free or a modified stress condition in laboratory. Taking into account the stress dependence contact effect can also offers a capacity to analyze the relationship between its dynamic and static elastic properties.

Seismic AVOA Inversion for Weak Anisotropy Parameters and Fracture Density in a Monoclinic Medium

In shale gas development, fracture density is an important lithologic parameter to properly characterize reservoir reconstruction, establish a fracturing scheme, and calculate porosity and permeability. The traditional methods usually assume that the fracture reservoir is one set of aligned vertical fractures, embedded in an isotropic background, and estimate some alternative parameters associated with fracture density. Thus, the low accuracy caused by this simplified model, and the intrinsic errors caused by the indirect substitution, affect the estimation of fracture density. In this paper, the fractured rock of monoclinic symmetry assumes two non-orthogonal vertical fracture sets, embedded in a transversely isotropic background. Firstly, assuming that the fracture radius, width, and orientation are known, a new form of P-wave reflection coefficient, in terms of weak anisotropy (WA) parameters and fracture density, was obtained by substituting the stiffness coefficients of vertical transverse isotropic (VTI) background, normal, and tangential fracture compliances. Then, a linear amplitude versus offset and azimuth (AVOA) inversion method, of WA parameters and fracture density, was constructed by using Bayesian theory. Tests on synthetic data showed that WA parameters, and fracture density, are stably estimated in the case of seismic data containing a moderate noise, which can provide a reliable tool in fracture prediction.

The velocity-stress finite-difference method with a rotated staggered grid applied to seismic wave propagation in a fractured medium

To enable a mathematical description, geologic fractures are considered as infinitely thin planes embedded in a homogeneous medium. These fracture structures satisfy linear slip boundary conditions, namely, a discontinuous displacement and continuous stress. The general finite-difference (FD) method described by the elastic wave equations has challenges when attempting to simulate the propagation of waves at the fracture interface. The FD method expressed by velocity-stress variables with the explicit application of boundary conditions at the fracture interface facilitates the simulation of wave propagation in fractured discontinuous media that are described by elastic wave equations and linear slip interface conditions. We have developed a new FD scheme for horizontal and vertical fracture media. In this scheme, a fictitious grid is introduced to describe the discontinuous velocity at the fracture interface and a rotated staggered grid is used to accurately indicate the location of the fracture. The new FD scheme satisfies nonwelded contact boundary conditions, unlike traditional approaches. Numerical simulations in different fracture media indicate that our scheme is accurate. The results demonstrate that the reflection coefficient of the fractured interface varies with the incident angle, wavelet frequency, and normal and tangential fracture compliances. Our scheme and conclusions from this study will be useful in assessing the properties of fractures, enabling the proper delineation of fractured reservoirs.

Вычисление нормальной и сдвиговой податливостей трехмерной трещины с учетом контакта между берегами

Вычисление нормальной и сдвиговой податливостей трехмерной трещины с учетом контакта между берегами

Increasing contribution of grain boundary compliance to polycrystalline ice elasticity as temperature increases

ABSTRACTMeasured elastic stiffnesses of ice polycrystals decrease with increasing temperature due to a decrease in grain boundary stiffness with increasing temperature. In this paper, we represent grain boundaries as imperfectly bonded interfaces, across which traction is continuous, but displacement may be discontinuous. We express the additional compliance due to grain boundaries in terms of a second-rank and a fourth-rank tensor, which quantify the effect on elastic wave velocities of the orientation distribution as well as the normal and shear compliances of the grain boundaries. Measurement of the elastic stiffnesses allows determination of the components of these tensors. Application of the method to resonant ultrasound spectroscopy measurements made on ice polycrystals enables determination of the ratio BN/BS of the normal to shear compliance of the grain boundaries, which are found to be more compliant in shear than in compression. The ratio BN/BS is small at low temperatures, but increases as temperature increases, implying that the normal compliance increases relative to the shear compliance as temperature increases.

Review Paper: A semi‐empirical model of strain sensitivity for 4D seismic interpretation

ABSTRACTWe reformulate the original model of Hatchell and Bourne and Røste, Stovas and Landrø that couples fractional velocity change to subsurface strain via a fundamental constant R. The new model combines elastic compressibility of a dual‐porosity system for a sand–shale mixture with horizontal planes of inter‐granular weakness. The majority of observed R‐factor magnitudes from post‐stack 4D seismic data in both the reservoir and overburden can thus be explained. R is predicted to depend strongly on lithology and also initial strain state. The model is also extended to predict the observed angle‐dependence of time‐lapse time‐shifts from pre‐stack data. An expression for the gradient of time‐shift with incidence angle is obtained in terms of the background VP/VS, and also the ratio of tangential to normal compliances BT/BN representing loss or creation of inter‐granular coupling. If accurately estimated from data, this compliance ratio can be used as an additional parameter to assess the post‐production state of the overburden. It is concluded that whilst R remains the over‐arching parameter controlling the magnitude of time‐shifts measured from 4D seismic data, BT/BN is a subtler parameter that may also prove of future value.

Extracting real-crack properties from non-linear elastic behaviour of rocks: abundance of cracks with dominating normal compliance and rocks with negative Poisson ratios

Abstract. Results of examination of experimental data on non-linear elasticity of rocks using experimentally determined pressure dependences of P- and S-wave velocities from various literature sources are presented. Overall, over 90 rock samples are considered. Interpretation of the data is performed using an effective-medium description in which cracks are considered as compliant defects with explicitly introduced shear and normal compliances without specifying a particular crack model with an a priori given ratio of the compliances. Comparison with the experimental data indicated abundance (∼ 80 %) of cracks with the normal-to-shear compliance ratios that significantly exceed the values typical of conventionally used crack models (such as penny-shaped cuts or thin ellipsoidal cracks). Correspondingly, rocks with such cracks demonstrate a strongly decreased Poisson ratio including a significant (∼ 45 %) portion of rocks exhibiting negative Poisson ratios at lower pressures, for which the concentration of not yet closed cracks is maximal. The obtained results indicate the necessity for further development of crack models to account for the revealed numerous examples of cracks with strong domination of normal compliance. Discovering such a significant number of naturally auxetic rocks is in contrast to the conventional viewpoint that occurrence of a negative Poisson ratio is an exotic fact that is mostly discussed for artificial structures.

Extracting shear and normal compliances of crack-like defects from pressure dependences of elastic-wave velocities

We present a differential formulation of effective-medium model in which the normal and shear compliances of the high-compliance porosity are explicitly decoupled. This feature of the decoupled-compliance model (DC model) is in contrast to conventional models in which such defect's properties are implicitly assumed and are subject to strong limitations defined by the used particular crack model. Comparison with the DC model makes it possible to reveal such implicit assumptions in the conventional models. Furthermore, for the conventional cracks, our approach gives the same results as the conventional models. The ability of the DC model to incorporate arbitrary defect properties in terms of their normal-to-shear compliance ratio (q-ratio) is used to formulate an analogue of Hashin-Shtrikman constraints on the range of feasible crack-induced variations in the moduli. Comparison of the DC model with experimental pressure dependences of elastic-wave velocities in rocks makes it possible to extract the q-ratio for real crack-like defects. These results demonstrate that properties of real cracks usually noticeably differ from those of popular crack models such as cracks with free faces (e.g., penny-shape) or pure shear cracks. We discuss an example of sandstone with pronouncedly negative Poisson's ratio that is due to the fact that the ratio of normal-to-shear compliances of voids in this rock (q~7–8) is significantly higher than for the conventional cracks (q~2). Ability of the DC model to accurately extrapolate pressure dependences of the moduli from relatively low pressures to several times greater is demonstrated, including the cases, for which the conventional models give huge errors. The introduced parameter q – the ratio of normal-to-shear compliances of voids – provides additional insight into properties of real crack-like defect in rocks.

Extension of the classical linear slip model for fluid‐saturated fractures: Accounting for fluid pressure diffusion effects

AbstractWe develop extended boundary conditions based on the linear slip model that account for the impact of wave‐induced fluid pressure diffusion between a fracture and its embedding background on the stiffening effect of the fluid saturating the fracture. We include these poroelastic effects into the linear slip model through complex‐valued and frequency‐dependent parameters characterizing the mechanical and hydraulic coupling between the two regions. This new set of effective fracture parameters contains generalized normal and tangential compliances, analogous to those defined in the classical formulation of the linear slip model, and an additional parameter related to the coupling between horizontal and vertical deformation of the fracture. Comparisons of the extended and classical linear slip models with a poroelastic thin layer model show that the extended formulation always performs better when modeling the displacement fields induced by an incident P wave as well as the scattering coefficients. We find that the contribution of the additional effective parameter involved in the proposed boundary conditions is significant at low frequencies with respect to the undrained frequency regime of the fracture and large angles of incidence. These extended boundary conditions can be readily incorporated into viscoelastic modeling algorithms simulating the response of a large‐scale fluid‐saturated fracture or multiple noninteracting fractures of this kind. The proposed model is expected not only to improve the estimation of mechanical characteristics of fractures in corresponding inversion schemes but can also be used for extracting information with regard to other practically important parameters, such as the background permeability.

The contact properties of naturally occurring geologic materials: experimental observations

The discrete element method (DEM) is seeing widespread use in geotechnical applications and it is generally acknowledged that the validity of DEM simulations involving this class of materials depends critically on employing realistic contact laws. To support the development of such contact laws for common naturally occurring materials, we have conducted grain-to-grain contact experiments on several quartz sands, magnesite (limestone), crushed and ball-milled gneiss and ooids (precipitated calcium carbonate spheroids) and two reference materials (glass beads and the synthetic Delrin). Here we present the results of normal and shear contact experiments on these materials, with emphasis on the range of behavior encountered and aspects of behavior that deviate from current thinking on the topic. The development of contact laws based on the results presented herein, macroscopic frictional sliding and the implementation of the contact laws in DEM simulations are addressed in separate publications. The present experiments primarily examined the cyclic loading response, which allowed for a precise determination of normal and shear contact stiffness and loss compliances. Deformation generally consisted of elastic, anelastic and permanent components and the latter could be observed in shear prior to macroscopic sliding. Normal and shear stiffness ranged from 0.2–2.0 to 0.1– $$1.0\,\hbox {MN\,m}^{-1}$$ , respectively although the ratio of shear to normal stiffness for specific materials varied from 0.3 to $${\approx }1$$ . Frictional loss was observed to varying degrees in all of the contacts examined and internal friction in shear was found to be constant prior to the onset of permanent deformation. Internal friction ranged from barely perceptible to 0.1 for normal contacts and from 0.02 to 0.3 in shear. The modeling implications of the findings are examined as well.