基于Kelvin-Voigt黏弹性骨架的含非饱和流体孔隙介质BISQ模型(英文)

本文综合考虑了在波传播过程中孔隙介质的三种重要力学机制——"Biot流动机制一squirt流动机制-固体骨架黏弹性机制",借鉴等效介质思想,将含水饱和度引入波动力学控制方程,并考虑了不同波频率下孔隙流体分布模式对其等效体积模量的影响,给出了能处理含粘滞性非饱和流体孔隙介质中波传播问题的黏弹性Biot/squirt(BISQ)模型。推导了时间-空间域的波动力学方程组,由一组平面谐波解假设,给出频率-波数域黏弹性BISQ模型的相速度和衰减系数表达式。基于数值算例分析了含水饱和度、渗透率与频率对纵波速度和衰减的影响,并结合致密砂岩和碳酸盐岩的实测数据,对非饱和情况下的储层纵波速度进行了外推,碳酸盐岩储层中纵波速度对含气饱和度的敏感性明显低于砂岩储层。     

双相介质中地震波的频率-空间域数值模拟

本文利用优化的25点频率-空间域有限差分算法对基于BISQ模型双相各向同性介质中的地震波进行了数值模拟.通过与经典的Biot模型理论模拟结果进行对比,分析了Biot流动（宏观流体流动）和Squirt流动（微观流体流动）耦合作用对地震波在孔隙介质中传播特性的影响.数值模拟在地震频段进行,结果显示：在理想相界和黏滞相界情况下,Squirt流动机制都比Biot流动机制产生了更大的速度频散和能量衰减.其中,在Biot流动和Squirt流动耦合作用下的快P波的速度和振幅小于仅考虑Biot流动影响下快P波速度和振幅,而且慢P波的衰减也更加强烈.本文还研究了地震波在双层双相各向同性介质分界面处的反射和透射特征,双相介质中波的反射与透射现象类似于单相介质的情况.模拟结果表明,利用优化25点频率-空间域有限差分法模拟双相孔隙介质中的地震波场是可行的,这为开展双相孔隙介质全波形反演问题的研究提供了可能.     

基于Remez迭代算法求解差分系数的双相介质自适应有限差分正演模拟

利用传统有限差分方法对基于Biot理论的双相介质波动方程进行数值求解时,由于慢纵波的存在,数值频散效应较为明显,影响模拟精度.相对于声学近似方程及普通弹性波方程,Biot双相介质波动方程在同等数值求解算法和精度要求条件下,其地震波场正演模拟需要更多的计算时间.本文针对Biot一阶速度-应力方程组发展了一种变阶数优化有限差分数值模拟方法,旨在同时提高其正演模拟的精度和效率.首先结合交错网格差分格式推导Biot方程的数值频散关系式.然后基于Remez迭代算法求取一阶空间偏导数的优化差分系数,并用于Biot方程的交错网格有限差分数值模拟.在此基础上把三类波的平均频散误差参数限制在给定的频散误差阈值和频率范围内,此时优化有限差分算子的长度就能自适应非均匀双相介质模型中的不同速度区间.数值频散曲线分析表明:基于Remez迭代算法的优化有限差分方法相较传统泰勒级数展开方法在大波数范围对频散误差的压制效果更明显;可变阶数的优化有限差分方法能取得与固定阶数优化有限差分方法相近的模拟精度.在均匀介质和河道模型的数值模拟实验中将本文变阶数优化有限差分算法与传统泰勒展开算法、最小二乘优化算法进行比较,进一步证明其在复杂地下介质中的有效性和适用性.     

Effects of petrophysical parameters on attenuation and dispersion of seismic waves in the simplified poroelastic theory: Comparison between the Biot's theory and viscosity‐extended theory

The simplified macro‐equations of porous elastic media are presented based on Hickey's theory upon ignoring effects of thermomechanical coupling and fluctuations of porosity and density induced by passing waves. The macro‐equations with definite physical parameters predict two types of compressional waves (P wave) and two types of shear waves (S wave). The first types of P and S waves, similar to the fast P wave and S wave in Biot's theory, propagate with fast velocity and have relatively weak dispersion and attenuation, while the second types of waves behave as diffusive modes due to their distinct dispersion and strong attenuation. The second S wave resulting from the bulk and shear viscous loss within pore fluid is slower than the second P wave but with strong attenuation at lower frequencies. Based on the simplified porous elastic equations, the effects of petrophysical parameters (permeability, porosity, coupling density and fluid viscosity) on the velocity dispersion and attenuation of P and S waves are studied in brine‐saturated sandstone compared with the results of Biot's theory. The results show that the dispersion and attenuation of P waves in simplified theory are stronger than those of Biot's theory and appear at slightly lower frequencies because of the existence of bulk and shear viscous loss within pore fluid. The properties of the first S wave are almost consistent with the S wave in Biot's theory, while the second S wave not included in Biot's theory even dies off around its source due to its extremely strong attenuation. The permeability and porosity have an obvious impact on the velocity dispersion and attenuation of both P and S waves. Higher permeabilities make the peaks of attenuation shift towards lower frequencies. Higher porosities correspond to higher dispersion and attenuation. Moreover, the inertial coupling between fluid and solid induces weak velocity dispersion and attenuation of both P and S waves at higher frequencies, whereas the fluid viscosity dominates the dispersion and attenuation in a macroscopic porous medium. Besides, the heavy oil sand is used to investigate the influence of high viscous fluid on the dispersion and attenuation of both P and S waves. The dispersion and attenuation in heavy oil sand are stronger than those in brine‐saturated sandstone due to the considerable shear viscosity of heavy oil. Seismic properties are strongly influenced by the fluid viscosity; thus, viscosity should be included in fluid properties to explain solid–fluid combination behaviour properly.     

Biot dispersion for P‐ and S‐wave velocities in partially and fully saturated sandstones

Ultrasonic compressional‐ and shear‐wave velocities have been measured on 34 samples of sandstones from hydrocarbon reservoirs. The sandstones are all of low clay content, high porosity, and cover a wide range of permeabilities. They were measured dry and brine‐saturated under hydrostatic effective stresses of 10, 20 and 40 MPa. For eight of the sandstones, ultrasonic velocity measurements were made at different partial water saturations in the range from dry to fully saturated. The Gassmann–Biot theory is found to account for most of the changes in velocities at high effective stress levels when the dry sandstones are fully saturated with brine, provided the lower velocities resulting when the dry sandstone initially adsorbs small amounts of moisture are used to determine the elastic properties of the ‘dry’ sandstone. At lower effective stress levels, local flow phenomena due to the presence of open microcracks are assumed to be responsible for measured velocities higher than those predicted by the theory. The partial saturation results are modelled fairly closely by the Gassmann–Biot theory, assuming heterogeneous saturation for P‐waves.     

Plane-layered models for the analysis of wave propagation in reservoir environments1

The long-wavelength propagation and attenuation characteristics of three geological structures that frequently occur in reservoir environments are investigated using a theoretical model that consists of a stack of fine and viscoelastic plane layers, with the layers being either solid or fluid. Backus theory properly describes fine layering and a set of fluid-filled microfractures, under the assumption that interfaces between different materials are bonded. The effects of saturation on wave attenuation are modelled by the relative values of the bulk and shear quality factors. The anisotropic quality factor in a fine-layered system shows a variety of behaviours depending on the saturation and velocities of the single constituents. The wave is less attenuated along the layering direction when the quality factors are proportional to velocity, and vice versa when inversely proportional to velocity. Fractured rocks have very anisotropic wavefronts and quality factors, in particular for the shear modes which are strongly dependent on the characteristics of the fluid filling the microfractures. When the size of the boundary layer is much smaller than the thickness of the fluid layer, the stack of solid-fluid layers becomes a layered porous media of the Biot type. This behaviour is caused by the slip-wall condition at the interface between the solid and the fluid. As in Biot theory, there are two compressional waves, but here the medium is anisotropic and the slow wave does not propagate perpendicular to the layers. Moreover, this wave shows pronounced cusps along the layering direction, like shear waves in a very anisotropic single-phase medium.     

Explicit use of the Biot coefficient in predicting shear-wave velocity of water-saturated sediments

Predicting the shear‐wave (S‐wave) velocity is important in seismic modelling, amplitude analysis with offset, and other exploration and engineering applications. Under the low‐frequency approximation, the classical Biot–Gassmann theory relates the Biot coefficient to the bulk modulus of water‐saturated sediments. If the Biot coefficient under in situ conditions can be estimated, the shear modulus or the S‐wave velocity can be calculated. The Biot coefficient derived from the compressional‐wave (P‐wave) velocity of water‐saturated sediments often differs from and is less than that estimated from the S‐wave velocity, owing to the interactions between the pore fluid and the grain contacts. By correcting the Biot coefficients derived from P‐wave velocities of water‐saturated sediments measured at various differential pressures, an accurate method of predicting S‐wave velocities is proposed. Numerical results indicate that the predicted S‐wave velocities for consolidated and unconsolidated sediments agree well with measured velocities.     

地震波在孔隙介质中低频衰减现象的粘弹性特征分析及近似（英文）

由于介观尺度的孔隙流体流动,弹性波传播过孔隙岩层时在地震频段表现出较强的频散和衰减。Johnson理论给出了在任意孔隙形状的条件下,部分气水饱和孔隙介质的理论相速度和品质因子的解析解。本文在Johnson模型的基础上,通过对Q值曲线的低频和高频近似,推导了Q值曲线的近似公式,以及基于孔隙介质基本地球物理参数和孔隙斑块几何形态参数T和比表面积S/V的最大衰减Qmin近似公式。通过与理论值的对比,对Qmin近似公式存在的线性误差进行改正,进一步提高了精度。复杂的斑块形态对最大衰减Qmin和过渡频率ftr的都产生一定影响,且对ftr影响更大。因为数值模拟直接求解介观尺度的Biot孔隙介质方程需要极大的计算量,我们使用Zener模型建立了等效粘弹模型,有效地模拟了地震频带内的衰减和频散现象。     

含孔隙、裂隙致密介质中多极子声波的传播特征

非常规油气藏（如致密性地层及蕴藏油气的页岩地层）的重要特征是低孔、低渗,但裂隙或裂缝比较发育.为满足非常规勘探的需求,本文将孔、裂隙介质弹性波传播理论应用于多极子声波测井的井孔声场模拟,重点研究了致密介质中裂隙发育时多极子声波的传播机理以及衰减特征.井孔声场的数值计算结果表明裂隙的存在明显改变了弹性波和井孔模式波的频散、衰减和激发强度,尤其是井壁临界折射纵波的激发谱的峰值随着频率的增加逐渐降低,这与应用经典的Biot理论下的计算结果相反,且裂隙的存在也使得饱含水和饱含气时临界折射纵波激发强度的差异变大.井孔模式波的衰减与地层横波衰减和井壁流体交换有关,井壁开孔边界下致密地层裂隙发育还使得井孔斯通利波和艾里相附近的弯曲波对孔隙流体的敏感性增强,在井壁闭孔边界条件下引起井孔模式波衰减的主要因素是裂隙引起的地层横波衰减造成的,且在截止频率附近弯曲波的衰减与地层的横波衰减一致.数值计算结果为解释非常规油气地层的声学响应特征提供了参考.     

Multiscale rock-physics templates for gas detection in carbonate reservoirs

The heterogeneous distribution of fluids in patchy-saturated rocks generates significant velocity dispersion and attenuation of seismic waves. The mesoscopic Biot–Rayleigh theory is used to investigate the relations between wave responses and reservoir fluids. Multiscale theoretical modeling of rock physics is performed for gas/water saturated carbonate reservoirs. Comparisons with laboratory measurements, log and seismic data validate the rock physics template. Using post-stack and pre-stack seismic inversion, direct estimates of rock porosity and gas saturation of reservoirs are obtained, which are in good agreement with oil production tests of the wells.     

介观尺度孔隙流体流动作用对纵波传播特征的影响研究——以周期性层状孔隙介质为例

介观尺度孔隙流体流动是地震频段岩石表现出较强速度频散与衰减的主要作用.利用周期性层状孔隙介质模型,基于准静态孔弹性理论给出了模型中孔隙压力、孔隙流体相对运动速度以及固体骨架位移等物理量的数学解析表达式,同时利用Biot理论将其扩展至全频段条件下,克服了传统White模型中介质分界面处流体压力不连续的假设. 在此基础上对准静态与全频段下模型介质中孔隙压力、孔隙流体相对运动速度变化形式及其对弹性波传播特征的影响进行了讨论,为更有效理解介观尺度下流体流动耗散和频散机制提供物理依据.研究结果表明,低频条件下快纵波孔压在介质层内近于定值,慢纵波通过流体扩散改变总孔隙压力, 随频率的增加慢波所形成的流体扩散作用逐渐减弱致使介质中总孔压逐渐接近于快纵波孔压,在较高频率下孔压与应力的二次耦合作用使总孔压超过快纵波孔压.介质中孔隙流体相对运动速度与慢纵波形成的流体相对运动速度变化形式一致;随频率的增加孔隙流体逐渐从排水的弛豫状态过渡到非弛豫状态,其纵波速度-含水饱和度变化形式也从符合孔隙流体均匀分布模式过渡到斑块分布模式,同时介质在不同含水饱和度下的衰减峰值与慢纵波所形成的孔隙流体相对流动速度具有明显的相关性.     

Viscoelastic approximation of poroelastic media for wave scattering problems

Herein, we propose a simplified and approximate method for solving dynamic problems in poroelastic media. The method is based on the definition of equivalent-viscoelastic materials that have the same wave numbers as poroelastic media. The viscoelastic approximation was applied to study the scatter of plane compressional waves by a spherical poroelastic inhomogeneity. In this wave scattering problem, the poroelastic and equivalent-viscoelastic solutions gave almost identical results far away from the inhomogeneity. The proposed method extends our existing numerical and analytical methods for poroelastic media. It is useful to derive approximate analytical solutions which can be applied for assessing the damping characteristics of small amplitude waves in saturated soils.     

On the applicability of the Biot solution to the problems of elastic propagation in saturated porous media

In this paper, the solution of the system of homogeneous Biot equations, which was derived by Biot for the displacement vectors of plane monochrome elastic waves propagating in a homogeneous infinite two-phase medium, is expanded to the case where the propagation area of the elastic waves is limited and the wavefront is a piecewise smooth curved surface. It is shown that the arbitrary system of homogeneous Biot equations for the displacement vectors of the solid and liquid phases can be reduced to three different equations pertaining to the class of Helmholtz equations. From this, irrespective of the geometry of the seismic wavefront and the boundaries of the studied two-phase medium, there is the following. (1) Each displacement vector (of the solid and liquid phase) splits into three independent vectors satisfying three different Helmholtz equations. Two of these vectors correspond to the two types of compressional waves, namely, fast waves (waves of the first kind) and slow waves (waves of the second kind). The third vector describes shear waves. (2) The similar (related to the same wave type) components of the displacement vector in the solid and liquid phases satisfy the same Helmholtz equation and are linked with each other through a corresponding scalar factor that is expressed in terms of the coefficients of the Biot equations. Taking into account the established properties of the displacement vectors in the solid and liquid phases seems to be helpful in the problems dealing with calculation of elastic fields of arbitrary sources in piecewise-homogeneous two-phase media.     

Wave propagation analysis of porous rocks with the thermal activated relaxation mechanism

Wave attenuation and phase velocity dispersion in the temperature domain are more complicated than those in the frequency domain. To describe wave propagation properties in the temperature domain, a so-called thermal activation mechanism model is built on the experimental result that increasing the temperature or decreasing frequency could obtain similar results on the attenuation. A rheological model (the Zener model) is employed to describe viscoelastic attenuation in saturated porous rocks. The Arrhenius relation is introduced to describe the thermal activation mechanism. The wave propagation model with thermal effects in porous media is then obtained, and 1-D P-wave and S-wave propagation characteristics are analyzed in numeric process, respectively.Two attenuation mechanisms are found in this model, the Biot loss and the thermal activation relaxation. The thermal relaxation attenuation peak and the Biot attenuation peak are observed in both frequency and temperature spectra. These two peaks move towards each other when the temperature increases on frequency spectra. The thermal relaxation peak shifts towards higher frequencies while the Biot peak shifts towards lower frequencies. At some temperature, these two peaks will superpose. The combination of the thermal relaxation and the Biot loss leads to the complexity of wave velocity curves. Similar phenomena could be observed on temperature spectra. The thermal relaxation features may relate to a so-called “local heat transfer” mechanism. These two peaks in the temperature domain have been observed in the experiments by other investigators. The characteristics of velocity and attenuation are more remarkable for high porosity rock samples. The model is helpful for the understanding of wave propagation in the temperature domain.     

Slowness domain offset and traveltime approximations in layered vertical transversely isotropic media

Considering horizontally layered transversely isotropic media with vertical symmetry axis and all types of pure‐mode and converted waves we present a new wide‐angle series approximation for the kinematical characteristics of reflected waves: horizontal offset, intercept time, and total reflection traveltime as functions of horizontal slowness. The method is based on combining (gluing) both zero‐offset and (large) finite‐offset series coefficients. The horizontal slowness is bounded by the critical value, characterised by nearly horizontal propagation within the layer with the highest horizontal velocity. The suggested approximation uses five parameters to approximate the offset, six parameters to approximate the intercept time or the traveltime, and seven parameters to approximate any two or all three kinematical characteristics. Overall, the method is very accurate for pure‐mode compressional waves and shear waves polarised in the horizontal plane and for converted waves. The application of the method to pure‐mode shear waves polarised in the vertical plane is limited due to cusps and triplications. To demonstrate the high accuracy of the method, we consider a synthetic, multi‐layer model, and we plot the normalised errors with respect to numerical ray tracing.     

P-wave dispersion and attenuation in fractured and porous reservoirs – poroelasticity approach

Natural fractures in hydrocarbon reservoirs can cause significant seismic attenuation and dispersion due to wave induced fluid flow between pores and fractures. We present two theoretical models explicitly based on the solution of Biot's equations of poroelasticity. The first model considers fractures as planes of weakness (or highly compliant and very thin layers) of infinite extent. In the second model fractures are modelled as thin penny-shaped voids of finite radius. In both models attenuation is a result of conversion of the incident compressional wave energy into the diffusive Biot slow wave at the fracture surface and exhibits a typical relaxation peak around a normalized frequency of about 1. This corresponds to a frequency where the fluid diffusion length is of the order of crack spacing for the first model and the crack diameter for the second. This is consistent with an intuitive understanding of the nature of attenuation: when fractures are closely and regularly spaced, the Biot's slow waves produced by cracks interfere with each other, with the interference pattern controlled by the fracture spacing. Conversely, if fractures are of finite length, which is smaller than spacing, then fractures act as independent scatterers and the attenuation resembles the pattern of scattering by isolated cracks. An approximate mathematical approach based on the use of a branching function gives a unified analytical framework for both models.     

Full frequency-range transient solution for compressional waves in a fluid-saturated viscoacoustic porous medium1

An analytical transient solution is obtained for propagation of compressional waves in a homogeneous porous dissipative medium. The solution, based on a generalization of Biot's poroelastic equations, holds for the low- and high-frequency ranges, and includes viscoelastic phenomena of a very general nature, besides the Biot relaxation mechanism. The viscodynamic operator is used to model the dynamic behaviour associated with the relative motion of the fluid in the pores at all frequency ranges. Viscoelasticity is introduced through the standard linear solid which allows the modelling of a general relaxation spectrum. The solution is used to study the influence of the material properties, such as bulk moduli, porosity, viscosity, permeability and intrinsic attenuation, on the kinematic and dynamic characteristics of the two compressional waves supported by the medium. We also obtain snapshots of the static mode arising from the diffusive behaviour of the slow wave at low frequencies.     

含混合裂隙、孔隙介质的纵波衰减规律研究

地下多孔介质中的孔隙类型复杂多样,既有硬孔又有扁平的软孔.针对复杂孔隙介质,假设多孔介质中同时含有球型硬孔和两种不同产状的裂隙(硬币型、尖灭型裂隙),当孔隙介质承载载荷时,考虑两种不同类型的裂隙对于孔隙流体压力的影响,建立起Biot理论框架下饱和流体情况含混合裂隙、孔隙介质的弹性波动方程,并进一步求取了饱和流体情况下仅由裂隙引起流体流动时的含混合裂隙、孔隙介质的体积模量和剪切模量,随后,在此基础上讨论了含混合裂隙、孔隙介质在封闭条件下地震波衰减和频散的高低频极限表达式.最后计算了给定模型的地震波衰减和频散,发现地震波衰减曲线呈现"多峰"现象,速度曲线为"多频段"频散.针对该模型分析讨论了渗透率参数、裂隙纵横比参数以及流体黏滞性参数对于地震波衰减和频散的影响,表明三个参数均为频率控制参数.     

A wave propagation model with the Biot and the fractional viscoelastic mechanisms

Energy loss in porous media containing fluids is typically caused by a variety of dynamic mechanisms.In the Biot theory,energy loss only includes the frictional dissipation between the solid phase and the fluid phase,resulting in underestimation of the dispersion and attenuation of the waves in the low frequency range.To develop a dynamic model that can predict the high dispersion and strong attenuation of waves at the seismic band,we introduce viscoelasticity into the Biot model and use fractional derivatives to describe the viscoelastic mechanism,and finally propose a new wave propagation model.Unlike the Biot model,the proposed model includes the intrinsic dissipation of the solid frame.We investigate the effects of the fractional order parameters on the dispersion and attenuation of the P-and S-waves using several numerical experiments.Furthermore,we use several groups of experimental data from different fluid-saturated rocks to testify the validity of the new model.The results demonstrate that the new model provides more accurate predictions of high dispersion and strong attenuation of different waves in the low frequency range.     

三相周期成层Patchy模型中地震波频散和衰减

本文基于White周期层状模型,将传统两相流体（气,水）模型扩展至三相流体（油、气、水）周期层状模型,在Biot理论框架下将其延伸至高频,研究介观尺度地震频带下纵波在三相Patchy模型中的传播规律.应用孔隙介质弹性波波动方程解耦方法,结合等效边界条件,得到Biot1956年理论框架下的纵波速度频散与衰减的解析解.利用频率域有限差分方法,将Biot1941年、1956年理论方程中的微分算子离散化,正演得到纵波速度频散与衰减的数值解,结合由Biot-Gassmann-Hill公式和Biot-Gassmann-Wood公式计算得到的纵波速度上下限,证明了三相流体模型解析解的正确性.随后,研究了三种流体排列顺序对于纵波速度频散和衰减的影响.从受力分析角度来看,油层抵消部分水流对气层的力,使其受力减小,反之水流促进油流入气层,气层受到叠加力,因此改变流体顺序后气层受力情况不同,造成纵波频散和衰减结果有所差异.最后研究了含气饱和度变化对于纵波速度频散和衰减影响.