孔隙介质弹性波—电磁场耦合效应测井的波场模拟研究
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摘要
本文研究流体饱和孔隙介质中弹性-电磁耦合波场的解析与数值模拟方法。孔隙介质中的弹性波场本身是固相骨架变形与孔隙流体渗流的耦合场,而由于孔隙液体含非均匀分布的带电离子,弹性波场还与电磁波场耦合,这种双重的耦合作用被称为震电效应。探索基于震电效应的勘探和测井新方法,是当前勘探地球物理界,特别是油气勘探开发界关注的研究方向。为了弄清震电波场的激发和传播规律以及各种地层参数对震电波场的影响,需要研究震电波场的模拟方法,特别是可以用于非均匀孔隙地层中震电波场模拟的数值算法。
本文以声电测井为背景,基于井内为流体、井外为流体饱和孔隙地层的介质模型,研究井内点声源激发的井孔声场及其诱导电磁场和井内电流源激发的井孔电诱导声场的解析和数值模拟方法,为开展这声电测井的可行性研究奠定基础。主要内容包括:
在忽略诱导电磁场对声场影响的前提下,先独立计算井中点声源激发的声场,再通过引入赫兹矢量,求解声波诱导电磁场的解析表达式。并将波场推广到非井轴对称情形,结合多极声电测井,采用实轴积分法模拟声波诱导电磁场的全波波形。分析渗透率、孔隙度、弯曲度、流体粘度、矿化度等地层参数对声场及其诱导电磁场波形的影响,并对比声波测井和声-电测井响应对各种地层参数的敏感度。
采用两种不同方法推导井内沿井轴放置的电偶极子激发的电磁场及其诱导声场的解析表达式。一种是直接求解Pride震电耦合方程组;另一种是在忽略诱导声场对电磁场影响的前提下,先独立计算电磁场,再求解其诱导声场。应用实轴积分法模拟电诱导声场的全波波形。通过对两种方法模拟波形的比较,验证忽略诱导声场对电磁场影响的简化计算的可行性。分析井孔电诱导声场波形的特点,以及各种地层参数对井孔电诱导声场波形的影响。
针对井中水平电流环激发的电-声测井,通过直接求解Pride方程组的方式,推导电场及其诱导声场的解析表达式,并采用实轴积分法模拟全波波形。研究这种测井方法在横波速度测井方面的应用价值。
实现了对井内流体、井外为水平分层孔隙地层中弹性波场数值模拟的速度-应力交错网格时域有限差分(FDTD)算法。相比于前人的孔隙弹性波FDTD算法,本文算法考虑惯性力对粘性流体流动的影响,适用于测井频率范围内的模拟。使用目前公认吸收效果最好的完全匹配层(PML)吸收边界条件消除由计算边界的截断带来的人工反射,并采用了不分裂场量的PML处理方式。分别采用两种不同的方式解决不同介质交界面上场量的差分离散化问题。一种是利用界面两侧介质的差分方程和场量在界面上的连续性条件推导出一组显式边界条件差分方程;另一种是在界面上仍采用均匀介质中场量的差分方程形式,但是方程中的参数取界面两侧介质中对应参数的平均值,作者称其为参数平均法。针对不同源频率和地层参数的多个算例,将FDTD算法模拟的波形与实轴积分法计算的结果进行详细地比较,验证FDTD算法的正确性。然后,采用FDTD算法模拟井外水平分层地层中声波测井的全波波形和波场分布情况。
证明了在弹性波的交错网格FDTD算法中应用参数平均法与利用场量的连续性条件进行推导的等价性。另外,在采用统一的方程组表示固体、流体和孔隙介质中弹性波的情况下,将参数平均法扩展到处理任意由上述三种介质组成的交界面问题,实现可以数值模拟含有固体、流体和孔隙介质组成的复杂模型结构中弹性波传播问题的FDTD算法。该算法的实现对孔隙介质中弹性波FDTD算法的规范化和推广使用具有重要意义。
分别提出电-声和声-电测井的有限差分模拟方法,实现对水平分层孔隙地层中井孔电诱导声场和声诱导电场的数值模拟。电-声测井的有限差分模拟,采用先计算电场再计算其诱导声场的简化计算方法。考虑到低频电偶极子激发的电磁场可看成似稳场,将问题转化为求解关于电位的泊松方程。采用有限差分法模拟电位点的空间分布。在电场已知的前提下,对FDTD模拟点声源激发声波测井的算法进行修改,实现这种由井外孔隙地层中电场空间的不均匀分布和时变特性所诱导的声场响应的模拟。声-电测井的有限差分模拟,在不考虑诱导电场对声场影响的前提下,先利用声波测井的FDTD模拟算法获得不同时刻声场的空间分布。与空间位置固定的电流源产生的电磁场不同,孔隙介质中声波诱导的电磁场是由空间波动的电流源产生的。诱导电场可看作是低频似稳场,将问题转化为在不同时刻模拟这种空间波动电流源导致的井内外电位分布,进而采用有限差分法模拟电场。在井外为均匀孔隙地层情况下,分别将有限差分模拟的声诱导电场和电诱导声场波形与实轴积分法计算的结果进行比较,验证有限差分算法的正确性。采用有限差分算法模拟井外孔隙地层中存在水平介质分界面的电-声测井诱导声场和声-电测井诱导电场的全波波形和波场分布,并分析电诱导声场和声诱导电场的波场特点。
本文针对测井问题提出的震电波场模拟算法,对于震电测井仪器设计具有指导意义,对于震电勘探和天然地震诱导电磁场模拟具有参考意义。
Analytical and numerical methods are proposed to model the coupled poroelastic (or acoustic) and electric fields in fluid-saturated porous media. The poroelastic field itself is a coupled field of the solid strain and the pore fluid filtration, and it is coupled with the electromagnetic field due to the net charged ions in the pore fluid. This double coupling is called as seismoelectric effects. Seismoelectric exploration and seismoelectric well logging have caught the attention of geophysics and petroleum research engineers due to their potential usage in petroleum exploration. It is important to have a thorough understanding of the characteristics of the seismoelectric wave-fields and their relations to the formation parameters. This cannot be achieved without modeling the seismoelectric wave-fields correctly and efficiently. Numerical methods that can be applied in heterogeneous porous media are especially important and must be explored. But the simulation of seismoelectric wave-fields is a challenging task due to the difficulty in solving the coupling field.
In this thesis, the acoustic field and its converted electromagnetic field of the seismoelectric logs excited by a point acoustic source in the fluid-filled borehole are modeled. In addition, the converted acoustic field of the electroseismic logs excited by an electric source in the borehole is simulated.
The first two chapters are devoted to the semi-analytical methods for simulation of seismoelectric and electroseismic logging. By supposing the acoustic field is not influenced on its induced electromagnetic field, the analytical expressions for the induced electromagnetic field are derived by introducing Hertz vectors in Chapter 2. The full waveforms of the acoustic field and the electromagnetic field in the borehole are calculated when the borehole acoustic sources are respectively a monopole, a dipole and a quadrupole. The influence of formation parameters, such as the permeability, porosity, tortuosity, pore fluid viscosity and salinity, on the waveforms are investigated,
In Chapter 3, two different methods are proposed to derive the analytical expressions for the electromagnetic field and its induced acoustic field of the electroseismic logs excited by a vertical electric dipole in the borehole. In the coupled method, the EM field and the acoustic field are modeled using Pride’s model, which couples Maxwell’s equations and Biot’s equations. In the uncoupled method, the EM field is uninfluenced by the converted acoustic field, resulting in separate acoustic formulation with an electrokinetic source term derived from the primary EM field. The difference of the transient full waveforms between the above two methods is remarkably small for all examples, thus confirming the validity of using the computationally simpler uncoupled method. The waveform characteristics of the electric-induced acoustic field and the influence of formation parameters on the waveforms are investigated.
For the modeling of the electroseismic logs excited by a horizontal current loop, the analytical expressions for the electric field and its induced acoustic field are derived by solving Pride’s equations. In this logging problem, the transverse electric wave is coupled with the induced shear horizontal wave. Numerical simulations of the logs are calculated in different fluid-saturated porous formations, and the shear wave velocities are directly extracted from the simulated full waveforms. This logging method has potential application value in shear velocity logging.
In the two subsequent chapters, I adapt the finite deference (FD) method to cater for acoustic, seismoelectric, and electroseismic wave propagation problems in heterogeneous porous media, with special efforts to treat the inner and outer boundary conditions.
A velocity-stress finite-difference time-domain (FDTD) algorithm is proposed in Chapter 4 to simulate the acoustic fields in a fluid filled borehole embedded in fluid-saturated porous media. This algorithm considers both the low-frequency viscous force and the high-frequency inertial force in poroelastic media, extending its application to a wider frequency range compared to existing algorithms which are only valid in the low-frequency limit. The perfectly matched layer (PML) is applied as an absorbing boundary condition to truncate the computational region. A PML technique without splitting the fields is extended to the poroelastic wave problem. In the FDTD algorithm, two different method are respectively used to discrete the field quantities on the borehole wall between the fluid and porous media. One is to derive the discrete equations by using the boundary conditions. The other is to formulate in the same forms as those in the homogeneous media, but the average of the medium parameters on both sides of the interface are used instead, this method is called as parameter averaging technique. The FDTD algorithm is validated by comparisons against the analytical method in a variety of formations with different velocities and permeabilities. The acoustic logs in a horizontally stratified porous formation are simulated with the proposed FDTD algorithm.
The validity of the parameter averaging technique used for the interface between two different porous media is proved by deriving the discrete equations with average parameters from the continuity conditions on the interface. By using a set of generic equations to express elastic waves in solid, fluid and porous media, this parameter averaging technique is applied to deal with not only the interfaces between two media of the same type but also those between solid, fluid, and porous media. And the 2-D FDTD algorithm in axisymmetric cylindrical coordinates is presented to simulate elastic wave propagation in combined structures with solid, fluid and porous subregions. This algorithm has important significance to normalize and popularize the FDTD method for the modeling of elastic waves in porous media.
In the last chapter, FD algorithms are proposed to simulate the induced acoustic field of the electroseismic logging and the induced electric field of the seismoelectric logging in horizontally stratified porous formations. For the modeling of electroseismic logging, the EM field excited by the electric dipole is calculated separately by FD first, and is considered as a distributed exciting source term in a set of extended Biot’s equations for the induced acoustic wavefield in the formation. This set of equations is solved by a modified FDTD algorithm. For the modeling of seismoelectric logging, the acoustic field excited by the acoustic source is calculated by the FDTD algorithm first, and is considered as a wave-like exciting source term in the Poisson equation for the induced electric potential. This Poisson equation is solved by FD. The FD algorithms are validated by comparisons against the analytical method in homogeneous porous formation. The electroseismic and seismoelectric logs in heterogeneous porous formations with a horizontal interface are simulated with the FD algorithms.
本文以声电测井为背景,基于井内为流体、井外为流体饱和孔隙地层的介质模型,研究井内点声源激发的井孔声场及其诱导电磁场和井内电流源激发的井孔电诱导声场的解析和数值模拟方法,为开展这声电测井的可行性研究奠定基础。主要内容包括:
在忽略诱导电磁场对声场影响的前提下,先独立计算井中点声源激发的声场,再通过引入赫兹矢量,求解声波诱导电磁场的解析表达式。并将波场推广到非井轴对称情形,结合多极声电测井,采用实轴积分法模拟声波诱导电磁场的全波波形。分析渗透率、孔隙度、弯曲度、流体粘度、矿化度等地层参数对声场及其诱导电磁场波形的影响,并对比声波测井和声-电测井响应对各种地层参数的敏感度。
采用两种不同方法推导井内沿井轴放置的电偶极子激发的电磁场及其诱导声场的解析表达式。一种是直接求解Pride震电耦合方程组;另一种是在忽略诱导声场对电磁场影响的前提下,先独立计算电磁场,再求解其诱导声场。应用实轴积分法模拟电诱导声场的全波波形。通过对两种方法模拟波形的比较,验证忽略诱导声场对电磁场影响的简化计算的可行性。分析井孔电诱导声场波形的特点,以及各种地层参数对井孔电诱导声场波形的影响。
针对井中水平电流环激发的电-声测井,通过直接求解Pride方程组的方式,推导电场及其诱导声场的解析表达式,并采用实轴积分法模拟全波波形。研究这种测井方法在横波速度测井方面的应用价值。
实现了对井内流体、井外为水平分层孔隙地层中弹性波场数值模拟的速度-应力交错网格时域有限差分(FDTD)算法。相比于前人的孔隙弹性波FDTD算法,本文算法考虑惯性力对粘性流体流动的影响,适用于测井频率范围内的模拟。使用目前公认吸收效果最好的完全匹配层(PML)吸收边界条件消除由计算边界的截断带来的人工反射,并采用了不分裂场量的PML处理方式。分别采用两种不同的方式解决不同介质交界面上场量的差分离散化问题。一种是利用界面两侧介质的差分方程和场量在界面上的连续性条件推导出一组显式边界条件差分方程;另一种是在界面上仍采用均匀介质中场量的差分方程形式,但是方程中的参数取界面两侧介质中对应参数的平均值,作者称其为参数平均法。针对不同源频率和地层参数的多个算例,将FDTD算法模拟的波形与实轴积分法计算的结果进行详细地比较,验证FDTD算法的正确性。然后,采用FDTD算法模拟井外水平分层地层中声波测井的全波波形和波场分布情况。
证明了在弹性波的交错网格FDTD算法中应用参数平均法与利用场量的连续性条件进行推导的等价性。另外,在采用统一的方程组表示固体、流体和孔隙介质中弹性波的情况下,将参数平均法扩展到处理任意由上述三种介质组成的交界面问题,实现可以数值模拟含有固体、流体和孔隙介质组成的复杂模型结构中弹性波传播问题的FDTD算法。该算法的实现对孔隙介质中弹性波FDTD算法的规范化和推广使用具有重要意义。
分别提出电-声和声-电测井的有限差分模拟方法,实现对水平分层孔隙地层中井孔电诱导声场和声诱导电场的数值模拟。电-声测井的有限差分模拟,采用先计算电场再计算其诱导声场的简化计算方法。考虑到低频电偶极子激发的电磁场可看成似稳场,将问题转化为求解关于电位的泊松方程。采用有限差分法模拟电位点的空间分布。在电场已知的前提下,对FDTD模拟点声源激发声波测井的算法进行修改,实现这种由井外孔隙地层中电场空间的不均匀分布和时变特性所诱导的声场响应的模拟。声-电测井的有限差分模拟,在不考虑诱导电场对声场影响的前提下,先利用声波测井的FDTD模拟算法获得不同时刻声场的空间分布。与空间位置固定的电流源产生的电磁场不同,孔隙介质中声波诱导的电磁场是由空间波动的电流源产生的。诱导电场可看作是低频似稳场,将问题转化为在不同时刻模拟这种空间波动电流源导致的井内外电位分布,进而采用有限差分法模拟电场。在井外为均匀孔隙地层情况下,分别将有限差分模拟的声诱导电场和电诱导声场波形与实轴积分法计算的结果进行比较,验证有限差分算法的正确性。采用有限差分算法模拟井外孔隙地层中存在水平介质分界面的电-声测井诱导声场和声-电测井诱导电场的全波波形和波场分布,并分析电诱导声场和声诱导电场的波场特点。
本文针对测井问题提出的震电波场模拟算法,对于震电测井仪器设计具有指导意义,对于震电勘探和天然地震诱导电磁场模拟具有参考意义。
Analytical and numerical methods are proposed to model the coupled poroelastic (or acoustic) and electric fields in fluid-saturated porous media. The poroelastic field itself is a coupled field of the solid strain and the pore fluid filtration, and it is coupled with the electromagnetic field due to the net charged ions in the pore fluid. This double coupling is called as seismoelectric effects. Seismoelectric exploration and seismoelectric well logging have caught the attention of geophysics and petroleum research engineers due to their potential usage in petroleum exploration. It is important to have a thorough understanding of the characteristics of the seismoelectric wave-fields and their relations to the formation parameters. This cannot be achieved without modeling the seismoelectric wave-fields correctly and efficiently. Numerical methods that can be applied in heterogeneous porous media are especially important and must be explored. But the simulation of seismoelectric wave-fields is a challenging task due to the difficulty in solving the coupling field.
In this thesis, the acoustic field and its converted electromagnetic field of the seismoelectric logs excited by a point acoustic source in the fluid-filled borehole are modeled. In addition, the converted acoustic field of the electroseismic logs excited by an electric source in the borehole is simulated.
The first two chapters are devoted to the semi-analytical methods for simulation of seismoelectric and electroseismic logging. By supposing the acoustic field is not influenced on its induced electromagnetic field, the analytical expressions for the induced electromagnetic field are derived by introducing Hertz vectors in Chapter 2. The full waveforms of the acoustic field and the electromagnetic field in the borehole are calculated when the borehole acoustic sources are respectively a monopole, a dipole and a quadrupole. The influence of formation parameters, such as the permeability, porosity, tortuosity, pore fluid viscosity and salinity, on the waveforms are investigated,
In Chapter 3, two different methods are proposed to derive the analytical expressions for the electromagnetic field and its induced acoustic field of the electroseismic logs excited by a vertical electric dipole in the borehole. In the coupled method, the EM field and the acoustic field are modeled using Pride’s model, which couples Maxwell’s equations and Biot’s equations. In the uncoupled method, the EM field is uninfluenced by the converted acoustic field, resulting in separate acoustic formulation with an electrokinetic source term derived from the primary EM field. The difference of the transient full waveforms between the above two methods is remarkably small for all examples, thus confirming the validity of using the computationally simpler uncoupled method. The waveform characteristics of the electric-induced acoustic field and the influence of formation parameters on the waveforms are investigated.
For the modeling of the electroseismic logs excited by a horizontal current loop, the analytical expressions for the electric field and its induced acoustic field are derived by solving Pride’s equations. In this logging problem, the transverse electric wave is coupled with the induced shear horizontal wave. Numerical simulations of the logs are calculated in different fluid-saturated porous formations, and the shear wave velocities are directly extracted from the simulated full waveforms. This logging method has potential application value in shear velocity logging.
In the two subsequent chapters, I adapt the finite deference (FD) method to cater for acoustic, seismoelectric, and electroseismic wave propagation problems in heterogeneous porous media, with special efforts to treat the inner and outer boundary conditions.
A velocity-stress finite-difference time-domain (FDTD) algorithm is proposed in Chapter 4 to simulate the acoustic fields in a fluid filled borehole embedded in fluid-saturated porous media. This algorithm considers both the low-frequency viscous force and the high-frequency inertial force in poroelastic media, extending its application to a wider frequency range compared to existing algorithms which are only valid in the low-frequency limit. The perfectly matched layer (PML) is applied as an absorbing boundary condition to truncate the computational region. A PML technique without splitting the fields is extended to the poroelastic wave problem. In the FDTD algorithm, two different method are respectively used to discrete the field quantities on the borehole wall between the fluid and porous media. One is to derive the discrete equations by using the boundary conditions. The other is to formulate in the same forms as those in the homogeneous media, but the average of the medium parameters on both sides of the interface are used instead, this method is called as parameter averaging technique. The FDTD algorithm is validated by comparisons against the analytical method in a variety of formations with different velocities and permeabilities. The acoustic logs in a horizontally stratified porous formation are simulated with the proposed FDTD algorithm.
The validity of the parameter averaging technique used for the interface between two different porous media is proved by deriving the discrete equations with average parameters from the continuity conditions on the interface. By using a set of generic equations to express elastic waves in solid, fluid and porous media, this parameter averaging technique is applied to deal with not only the interfaces between two media of the same type but also those between solid, fluid, and porous media. And the 2-D FDTD algorithm in axisymmetric cylindrical coordinates is presented to simulate elastic wave propagation in combined structures with solid, fluid and porous subregions. This algorithm has important significance to normalize and popularize the FDTD method for the modeling of elastic waves in porous media.
In the last chapter, FD algorithms are proposed to simulate the induced acoustic field of the electroseismic logging and the induced electric field of the seismoelectric logging in horizontally stratified porous formations. For the modeling of electroseismic logging, the EM field excited by the electric dipole is calculated separately by FD first, and is considered as a distributed exciting source term in a set of extended Biot’s equations for the induced acoustic wavefield in the formation. This set of equations is solved by a modified FDTD algorithm. For the modeling of seismoelectric logging, the acoustic field excited by the acoustic source is calculated by the FDTD algorithm first, and is considered as a wave-like exciting source term in the Poisson equation for the induced electric potential. This Poisson equation is solved by FD. The FD algorithms are validated by comparisons against the analytical method in homogeneous porous formation. The electroseismic and seismoelectric logs in heterogeneous porous formations with a horizontal interface are simulated with the FD algorithms.
引文
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