Quantitative geophysical pore‐type characterization and its geological implication in carbonate reservoirs

This paper discusses and addresses two questions in carbonate reservoir characterization: how to characterize pore‐type distribution quantitatively from well observations and seismic data based on geologic understanding of the reservoir and what geological implications stand behind the pore‐type distribution in carbonate reservoirs. To answer these questions, three geophysical pore types (reference pores, stiff pores and cracks) are defined to represent the average elastic effective properties of complex pore structures. The variability of elastic properties in carbonates can be quantified using a rock physics scheme associated with different volume fractions of geophysical pore types. We also explore the likely geological processes in carbonates based on the proposed rock physics template. The pore‐type inversion result from well log data fits well with the pore geometry revealed by a FMI log and core information. Furthermore, the S‐wave prediction based on the pore‐type inversion result also shows better agreement than the Greensberg‐Castagna relationship, suggesting the potential of this rock physics scheme to characterize the porosity heterogeneity in carbonate reservoirs. We also apply an inversion technique to quantitatively map the geophysical pore‐type distribution from a 2D seismic data set in a carbonate reservoir offshore Brazil. The spatial distributions of the geophysical pore type contain clues about the geological history that overprinted these rocks. Therefore, we analyse how the likely geological processes redistribute pore space of the reservoir rock from the initial depositional porosity and in turn how they impact the reservoir quality.     

Effect of stress interactions on anisotropic P-SV-wave dispersion and attenuation for closely spaced cracks in saturated porous media

Generally, local stress induced by individual crack hardly disturbs their neighbours for small crack densities, which, however, could not be neglected as the crack density increases. The disturbance becomes rather complex in saturated porous rocks due to the wave-induced diffusion of fluid pressures. The problem is addressed in this study by the comparison of two solutions: the analytical solution without stress interactions and the numerical method with stress interactions. The resultant difference of effective properties can be used to estimate the effect of stress interactions quantitatively. Numerical experiments demonstrate that the spatial distribution pattern of cracks strongly affects stress interactions. For regularly distributed cracks, the resulting stress interaction (shielding or amplification) shows strong anisotropy, depending on the arrangement and density of cracks. It has an important role in the estimation of effective anisotropic parameters as well as the incident-angle-dependency of P- and SV-wave velocities. Contrarily, randomly distributed cracks with a relative small crack density generally lead to a strong cancellation of stress interactions across cracks, where both the numerical and analytical solutions show a good agreement for the estimation of effective parameters. However, for a higher crack density, the incomplete cancellation of stress interactions is expected, exhibiting an incidence-angle dependency, slightly affecting effective parameters, and differentiating the numerical and analytical solutions.     

龙马溪页岩弹性各向异性与矿物分布之间的关系探讨

岩石弹性各向异性特征是普遍存在的,但导致岩石弹性各向异性的原因复杂且仍然存在一定争议.本研究以龙马溪页岩为例,试图建立页岩弹性各向异性和矿物分布之间关系.研究使用无损超声波探测获取岩石弹性各向异性参数,并使用背散射技术获取岩石矿物分布特征.研究通过引进变异系数来描述矿物或孔裂隙在不同方向的差异性,并通过2个正交方向的变异系数建立微观异质性指标,用于表征微观尺度上矿物或孔裂隙在不同方向的差异程度.微观背散射图像显示,龙马溪组页岩主要由石英和伊利石构成,且这两种矿物和孔裂隙在X和Y面上都有定向排列特征.相应地,它们的变异系数在X面和Y面上均表现出随角度增大而降低的特征;而在Z面,石英、伊利石和孔裂隙变异系数变化不明显,这与Z面上矿物和孔裂隙无明显方向性的特点一致.无损超声波探测结果显示,波速在X和Y面上随角度增加而减小,这与主要矿物和孔裂隙变异系数变化趋势相同;而在Z面,波速变化不大,与主要矿物和孔裂隙变异系数变化不明显的特征一致.以上观测结果说明,宏观波速与矿物的微观变异系数明显相关,暗示岩石弹性各向异性与矿物分布直接相关.     

Effective elastic properties of randomly fractured soils: 3D numerical experiments

This paper is concerned with numerical tests of several rock physical relationships. The focus is on effective velocities and scattering attenuation in 3D fractured media. We apply the so‐called rotated staggered finite‐difference grid (RSG) technique for numerical experiments. Using this modified grid, it is possible to simulate the propagation of elastic waves in a 3D medium containing cracks, pores or free surfaces without applying explicit boundary conditions and without averaging the elastic moduli. We simulate the propagation of plane waves through a set of randomly cracked 3D media. In these numerical experiments we vary the number and the distribution of cracks. The synthetic results are compared with several (most popular) theories predicting the effective elastic properties of fractured materials. We find that, for randomly distributed and randomly orientated non‐intersecting thin penny‐shaped dry cracks, the numerical simulations of P‐ and S‐wave velocities are in good agreement with the predictions of the self‐consistent approximation. We observe similar results for fluid‐filled cracks. The standard Gassmann equation cannot be applied to our 3D fractured media, although we have very low porosity in our models. This is explained by the absence of a connected porosity. There is only a slight difference in effective velocities between the cases of intersecting and non‐intersecting cracks. This can be clearly demonstrated up to a crack density that is close to the connectivity percolation threshold. For crack densities beyond this threshold, we observe that the differential effective‐medium (DEM) theory gives the best fit with numerical results for intersecting cracks. Additionally, it is shown that the scattering attenuation coefficient (of the mean field) predicted by the classical Hudson approach is in excellent agreement with our numerical results.     

Elastic properties of double-porosity rocks using the differential effective medium model

An approach to determining the effective elastic moduli of rocks with double porosity is presented. The double‐porosity medium is considered to be a heterogeneous material composed of a homogeneous matrix with primary pores and inclusions that represent secondary pores. Fluid flows in the primary‐pore system and between primary and secondary pores are neglected because of the low permeability of the primary porosity. The prediction of the effective elastic moduli consists of two steps. Firstly, we calculate the effective elastic properties of the matrix with the primary small‐scale pores (matrix homogenization). The porous matrix is then treated as a homogeneous isotropic host in which the large‐scale secondary pores are embedded. To calculate the effective elastic moduli at each step, we use the differential effective medium (DEM) approach. The constituents of this composite medium – primary pores and secondary pores – are approximated by ellipsoidal or spheroidal inclusions with corresponding aspect ratios. We have applied this technique in order to compute the effective elastic properties for a model with randomly orientated inclusions (an isotropic medium) and aligned inclusions (a transversely isotropic medium). Using the special tensor basis, the solution of the one‐particle problem with transversely isotropic host was obtained in explicit form. The direct application of the DEM method for fluid‐saturated pores does not account for fluid displacement in pore systems, and corresponds to a model with isolated pores or the high‐frequency range of acoustic waves. For the interconnected secondary pores, we have calculated the elastic moduli for the dry inclusions and then applied Gassmann's tensor relationships. The simulation of the effective elastic characteristic demonstrated that the fluid flow between the connected secondary pores has a significant influence only in porous rocks containing cracks (flattened ellipsoids). For pore shapes that are close to spherical, the relative difference between the elastic velocities determined by the DEM method and by the DEM method with Gassmann's corrections does not exceed 2%. Examples of the calculation of elastic moduli for water‐saturated dolomite with both isolated and interconnected secondary pores are presented. The simulations were verified by comparison with published experimental data.     

Analytical approximations of bulk and shear moduli for dry rock based on the differential effective medium theory

Differential effective medium theory has been applied to determine the elastic properties of porous media. The ordinary differential equations for bulk and shear moduli are coupled and it is more difficult to obtain accurate analytical formulae about the moduli of dry porous rock. In this paper, in order to decouple these equations we first substitute an analytical approximation for the dry‐rock modulus ratio into the differential equation and derive analytical solutions of the bulk and shear moduli for dry rock with three specific pore shapes: spherical pores, needle‐shaped pores and penny‐shaped cracks. Then, the validity of the analytical approximations is tested by integrating the full differential effective medium equation numerically. The analytical formulae give good estimates of the numerical results over the whole porosity range for the cases of the three given pore shapes. These analytical formulae can be further simplified under the assumption of small porosity. The simplified formulae for spherical pores are the same as Mackenzie's equations. The analytical formulae are relatively easy to analyse the relationship between the elastic moduli and porosity or pore shapes and can be used to invert some rock parameters such as porosity or pore aspect ratio. The predictions of the analytical formulae for experimental data show that the formulae for penny‐shaped cracks are suitable to estimate the elastic properties of micro‐crack rock such as granite, they can be used to estimate the crack aspect ratio while the crack porosity is known and also to estimate the crack porosity evolution with pressure if the crack aspect ratio is given.     

Calculation of elastic properties in lower part of the Kola borehole from bulk chemical compositions of core samples

In-situ elastic properties in deep boreholes are controlled by several factors, mainly by lithology, petrofabric, fluid-filled cracks and pores. In order to separate the effects of different factors it is useful to extract lithology-controlled part from observedin-situ velocities. For that purpose we calculated mineralogical composition and isotropic crack-free elastic properties in the lower part of the Kola borehole from bulk chemical compositions of core samples. We use a new technique of petrophysical modeling based on thermodynamic approach. The reasonable accuracy of the modeling is confirmed by comparison with the observations of mineralogical composition and laboratory measurements of density and elastic wave velocities in upper crustal crystalline rocks at high confining pressure. Calculations were carried out for 896 core samples from the depth segment of 6840–10535m. Using these results we estimate density and crack-free isotropic elastic properties of 554 lithology-defined layers composing this depth segment. Average synthetic P- wave velocity appears to be 2.7% higher than the velocity from Vertical Seismic Profiling (VSP), and 5% higher than sonic log velocity. Average synthetic S-wave velocity is 1.4 % higher than that from VSP. These differences can be explained by superposition of effects of fabric-related anisotropy, cracks aligned parallel to the foliation plain, and randomly oriented cracks, with the effect of cracks being the predominant control. Low sonic log velocities are likely caused by drilling-induced cracking (hydrofractures) in the borehole walls. The calculated synthetic density and velocity cross-sections can be used for much more detailed interpretations, for which, however, new, more detailed and reliable seismic data are required.     

Changes in geophysical properties caused by fluid injection into porous rocks: analytical models

Analytical models are provided that describe how the elastic compliance, electrical conductivity, and fluid‐flow permeability of rocks depend on stress and fluid pressure. In order to explain published laboratory data on how seismic velocities and electrical conductivity vary in sandstones and granites, the models require a population of cracks to be present in a possibly porous host phase. The central objective is to obtain a consistent mean‐field analytical model that shows how each modeled rock property depends on the nature of the crack population. The crack populations are described by a crack density, a probability distribution for the crack apertures and radii, and the averaged orientation of the cracks. The possibly anisotropic nature of the elasticity, conductivity, and permeability tensors is allowed for; however, only the isotropic limit is used when comparing to laboratory data. For the transport properties of conductivity and permeability, the percolation effect of the crack population linking up to form a connected path across a sample is modeled. However, this effect is important only in crystalline rock where the host phase has very small conductivity and permeability. In general, the importance of the crack population to the transport properties increases as the host phase becomes less conductive and less permeable.     

Effective velocities in fractured media: a numerical study using the rotated staggered finite‐difference grid

The modelling of elastic waves in fractured media with an explicit finite‐difference scheme causes instability problems on a staggered grid when the medium possesses high‐contrast discontinuities (strong heterogeneities). For the present study we apply the rotated staggered grid. Using this modified grid it is possible to simulate the propagation of elastic waves in a 2D or 3D medium containing cracks, pores or free surfaces without hard‐coded boundary conditions. Therefore it allows an efficient and precise numerical study of effective velocities in fractured structures. We model the propagation of plane waves through a set of different, randomly cracked media. In these numerical experiments we vary the wavelength of the plane waves, the crack porosity and the crack density. The synthetic results are compared with several static theories that predict the effective P‐ and S‐wave velocities in fractured materials in the long wavelength limit. For randomly distributed and randomly orientated, rectilinear, non‐intersecting, thin, dry cracks, the numerical simulations of velocities of P‐, SV‐ and SH‐waves are in excellent agreement with the results of the modified (or differential) self‐consistent theory. On the other hand for intersecting cracks, the critical crack‐density (porosity) concept must be taken into account. To describe the wave velocities in media with intersecting cracks, we propose introducing the critical crack‐density concept into the modified self‐consistent theory. Numerical simulations show that this new formulation predicts effective elastic properties accurately for such a case.     

Dispersion and attenuation of elastic waves due to multiple scattering from inclusions

The propagation of elastic waves in a medium containing many inclusions is considered. Under the assumption that the spatial distribution of inclusions is uniform, a general equation is derived for the determination of the velocity dispersion and attenuation coefficient of the effective waves. A simple example is presented where scatterers are infinitesimally thin cracks. The calculated results show that the attenuation coefficient Q^?1 takes a peak value for the wavelength nearly equal to twice the crack length.     

Burial stress and elastic strain of carbonate rocks

Burial stress on a sediment or sedimentary rock is relevant for predicting compaction or failure caused by changes in, e.g., pore pressure in the subsurface. For this purpose, the stress is conventionally expressed in terms of its effect: “the effective stress” defined as the consequent elastic strain multiplied by the rock frame modulus. We cannot measure the strain directly in the subsurface, but from the data on bulk density and P‐wave velocity, we can estimate the rock frame modulus and Biot's coefficient and then calculate the “effective vertical stress” as the total vertical stress minus the product of pore pressure and Biot's coefficient. We can now calculate the elastic strain by dividing “effective stress” with the rock frame modulus. By this procedure, the degree of elastic deformation at a given time and depth can be directly expressed. This facilitates the discussion of the deformation mechanisms. The principle is illustrated by comparing carbonate sediments and sedimentary rocks from the North Sea Basin and three oceanic settings: a relatively shallow water setting dominated by coarse carbonate packstones and grainstones and two deep water settings dominated by fine‐grained carbonate mudstones and wackestones.     

Dynamic interaction of twin vertically overlapping lined tunnels in an elastic half space subjected to incident plane waves

The scattering of plane harmonic P and SV waves by a pair of vertically overlapping lined tunnels buried in an elastic half space is solved using a semi-analytic indirect boundary integration equation method. Then the effect of the distance between the two tunnels, the stiffness and density of the lining material, and the incident frequency on the seismic response of the tunnels is investigated. Numerical results demonstrate that the dynamic interaction between the twin tunnels cannot be ignored and the lower tunnel has a significant shielding effect on the upper tunnel for high-frequency incident waves, resulting in great decrease of the dynamic hoop stress in the upper tunnel; for the low-frequency incident waves, in contrast, the lower tunnel can lead to amplification effect on the upper tunnel. It also reveals that the frequency-spectrum characteristics of dynamic stress of the lower tunnel are significantly different from those of the upper tunnel. In addition, for incident P waves in low-frequency region, the soft lining tunnels have significant amplification effect on the surface displacement amplitude, which is slightly larger than that of the corresponding single tunnel.     

General energetic analysis of fracture instabilities in elastic media and the elastic field rupture

The singular integral equations method makes it possible to determine a general analytical solution to the problem of a crack subjected to any stresses, including singular ones. The singularity of stresses means that they tend to infinity in the concentration point. In exponential functions describing this relationship, the exponent characterizes the stress curvature growth. Also the energy released by crack opening can be described by a simple analytical formula. The problem is solvable for an exponent greater than −1. The class of all the cracks subject to stresses that exponentially grow to one of the crack ends is divided into three sub-classes. One of these embraces most of crack types, also Griffith’s. The remaining two are a source of microcracks in an elastic medium. The onset of such a stress concentration gives rise to a microcrack which cancels the stress singularity up to that with the exponent of −1/2, ensuring a strong stability of the medium. An analysis of the nucleation of such cracks brought about a concept of elastic field rupture without destruction of interatomic bonds, which has implications relating to the conductivity of metals. A general formula for the crack energy singles out a special crack of unit length, whose energy is constant and independent of stress concentration.     

有效应力变化对致密砂岩孔隙结构及弹性波响应的影响规律

致密砂岩气藏具有裂缝发育和有效应力高的特征,研究不同有效压力下孔、裂隙介质地震波传播特征,有利于地震解释与地下储层的识别.但是前人的研究较少考虑岩石内部微观孔隙结构特征与孔隙、裂隙间流体流动的关系.本文首先通过选取四川盆地典型致密砂岩岩样,在不同有效压力下对岩石样本进行超声波实验测量.然后基于实验测得的纵、横波速度进行裂隙参数反演,得到不同有效压力下致密砂岩样本的裂隙孔隙度.再将裂隙孔隙度和样本岩石物理参数代入双重孔隙介质模型,模拟得到不同有效压力下饱水致密砂岩样本纵横波速度频散和衰减的变化规律.结果表明模型预测的速度频散曲线与纵波速度实验测量结果能够较好的吻合.最后统计分析了致密砂岩裂隙参数,得到了致密砂岩储层裂隙参数随有效压力及孔隙度变化特征.依据实际岩石物理参数建立模型,其裂隙参数三维拟合结果能够较好描述致密砂岩裂隙结构与孔隙度、应力的关联,可为实际地震勘探中预测储层裂缝性质提供基础依据.     

The effect of clay distribution on the elastic properties of sandstones

The shape and location of clay within sandstones have a large impact on the P‐wave and S‐wave velocities of the rock. They also have a large effect on reservoir properties and the interpretation of those properties from seismic data and well logs. Numerical models of different distributions of clay – structural, laminar and dispersed clay – can lead to an understanding of these effects. Clay which is located between quartz grains, structural clay, will reduce the P‐wave and S‐wave velocities of the rock. If the clay particles become aligned or form layers, the velocities perpendicular to the alignment will be reduced further. S‐wave velocities decrease more rapidly than P‐wave velocities with increasing clay content, and therefore Poisson's ratios will increase as the velocities decrease. These effects are more pronounced for compacted sandstones. Small amounts of clay that are located in the pore space will have little effect on the P‐wave velocity due to the competing influence of the density effect and pore‐fluid stiffening. The S‐wave velocity will decrease due to the density effect and thus the Poisson's ratio will increase. When there is sufficient clay to bridge the gaps between the quartz grains, P‐wave and S‐wave velocities rise rapidly and the Poisson's ratios decrease. These effects are more pronounced for under‐compacted sandstones. These general results are only slightly modified when the intrinsic anisotropy of the clay material is taken into account. Numerical models indicate that there is a strong, nearly linear relationship between P‐wave and S‐wave velocity which is almost independent of clay distribution. S‐wave velocities can be predicted reasonably accurately from P‐wave velocities based on empirical relationships. However, this does not provide any connection between the elastic and petrophysical properties of the rocks. Numerical modelling offers this connection but requires the inclusion of clay distribution and anisotropy to provide a model that is consistent with both the elastic and petrophysical properties. If clay distribution is ignored, predicting porosities from P‐wave or S‐wave data, for example, can result in large errors. Estimation of the clay distribution from P‐wave and S‐wave velocities requires good estimates of the porosity and clay volume and verification from petrographic analyses of core or cuttings. For a real data example, numerical models of the elastic properties suggest the predominance of dispersed clay in a fluvial sand from matching P‐wave and S‐wave velocity well log data using log‐based estimates of the clay volume and porosity. This is consistent with an interpretation of other log data.     

孔、裂隙并存地层中的声波测井理论及多极子声场特征

在油、气储层的勘探和开发中观察到的一个现象是储层岩石中普遍存在孔隙和裂隙.随着近年来孔、裂隙介质弹性波动理论的进展,我们可以将此理论应用于测井技术,以此来指导从声波测井中测量孔、裂隙地层的声学参数.本文计算了孔、裂隙地层里充流体井眼中的多极子声场,分析了声场随裂隙介质的两个主要参数(即裂隙密度和裂隙纵横比)的变化特征.井孔声场的数值计算表明裂隙密度可以大幅度地降低井中声波纵、横波的波速和振幅.随着裂隙密度的增加,在测井频段内也可以看到纵、横波速的频散现象(这种频散在孔隙地层中一般是观察不到的).本文还研究了多极子模式波 (即单极的Stoneley波、伪瑞利波以及偶极的弯曲波)随裂隙参数的变化特征.结果表明,这些模式波的振幅激发和速度频散都受裂隙密度的影响.裂隙密度越高影响越大.此外,裂隙还对模式波的传播造成较大的衰减.相对裂隙密度而言,裂隙纵横比是一个频率控制参数,它控制裂隙对声场影响的频率区间.本文的分析结果对裂缝、孔隙型地层的声波测井具有指导意义.     

Surface waves in a micropolar elastic solid containing voids

The present paper investigates the effect of voids on the propagation of surface waves in a homogeneous micropolar elastic solid medium which contains a distribution of vacuous pores (voids). The general theory for surface wave propagation in micropolar elastic media containing voids has been presented. Particular cases of surface waves (Rayleigh’s, Love’s and Stoneley’s) in micropolar media which contain vacuous pores have been deduced from the above general theory. Discussions have been made in each case to highlight the effect of voids and micropolar character of the material medium separately. Their joint effect has also been studied in details. Modulation of Rayleigh wave velocity has been studied numerically. It is observed that Love waves are not affected by the presence of voids.     

Nonlinear elastic behavior of fiber-reinforced soil under cyclic loading

Experimental investigations and modeling of nonlinear elasticity of fiber-reinforced soil under cyclic loading at small strain are conducted in this paper. The investigations include three aspects. First, cyclic shear tests are conducted using conventional triaxial apparatus. Twenty-seven specimens with three different fiber contents are employed to conduct triaxial cyclic shear tests under different confining pressure and loading repetition. Effects of geofiber, confining pressure and loading repetition on elastic shear modulus of reinforced soil are studied and analyzed. Second, a hyperbolic function is introduced to describe the nonlinear stress–strain skeletal curve under cyclic loading. Nonlinear elastic modulus is expressed as a function of shear strain and two variables A and B that are related to the initial tangential modulus and ultimate cyclic loading stress, respectively. In the present paper, variables A and B both are further assumed to be functions of geofiber content, confining pressure and loading repetition. Finally, eight constitutive coefficients of the nonlinear elastic model are calibrated using stress–strain curves from cyclic triaxial shear tests. The calibration of parameters is conducted using the technique of the linear regression for multiple variables. Impacts and effects of geofiber, confining pressure and loading repetitions on soil nonlinear elastic behavior are discussed.     

多重孔岩石微分等效介质模型及其干燥情形下的解析近似式

经典的微分等效介质(DEM)理论可用于确定多孔介质的弹性性质,但由于缺乏多重孔DEM方程,其估计的多重孔岩石的等效弹性模量依赖于包裹体(即不同孔隙纵横比的孔或缝)的添加顺序.本文首先从Kuster-Toksöz理论出发建立了Zimmermann和Norris两种形式的多重孔DEM方程.Norris形式的多重孔DEM方程预测的等效弹性模量总是位于Hashin-Shtrikman上下限内,而Zimmermann形式的多重孔DEM方程有时会越界.然后,通过使用干燥岩石模量比的解析近似式,对两个相互耦合的Norris形式DEM方程进行解耦得到干燥多重孔岩石的体积和剪切模量解析式.用全DEM方程的数值解对解析近似式的有效性进行了测试,解析公式的计算结果在整个孔隙度分布区间与数值解吻合良好.对实验室测量数据在假设岩石含有双重孔隙的情形下用双重孔DEM解析公式对岩石的弹性模量进行了预测,结果表明,解析式准确地预测了弹性模量随孔隙度的变化.双重孔(即软、硬孔)DEM解析模型可用来反演各孔隙类型的孔隙体积比,它可以通过实验室测量与理论预测之间的平方误差最小反演得到.砂岩样品的反演结果揭示,软孔的孔隙体积百分比与粘土含量没有明显的相关性.     

Limitations on linear elastic procedures in performance assessment of regular frames

Linear elastic analysis procedures are employed exclusively in the traditional seismic design of new structures and widely employed in the seismic assessment of existing structures. It is also a convenient tool for the initial checking of deformations in displacement‐based design. The limitations that should be imposed on linear elastic procedures have been evaluated in this study by comparing the deformation‐based response quantities obtained from response spectrum analysis with those from the nonlinear time history analysis. Both procedures were applied to different design variants of 5, 8, and 12 story moment frames, subjected to 20 strong motion components exhibiting a variety of intensities. Member plastic rotations and interstory drift ratios were employed as the basic response parameter in performance assessment. It has been found that average column demand to capacity ratio (DCR) (the ratio of flexural demand from linear elastic analysis to flexural capacity) and average beam DCR at the critical story are the most effective parameters in determining the validity range of linear elastic procedures in regular moment frames. Limiting values for these response parameters are proposed. Furthermore, amplification factors for member rotation demands predicted by the linear procedures are suggested for moment frames when these limiting values are exceeded. These factors ensure that the amplified linear elastic rotations are not smaller than 84 percentile (mean – 1sigma) of the rotations obtained from nonlinear time history analysis. Copyright © 2015 John Wiley & Sons, Ltd.