流体饱和多孔介质弹性波方程正反演方法研究
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摘要
流体饱和多孔介质(Fluid-Saturated Porous Media)是由包含固相和液相的双相介质组成。Biot流体饱和多孔介质理论认为,地下介质是由弹性多孔固体及充满孔隙空间的具有粘滞性的可压缩流体组成。与单相弹性介质比较,流体饱和多孔介质更接近实际地层介质,流体饱和多孔介质弹性波方程中包含了更多描述地层性质的参数,故在地球物理勘探、地震工程等领域有广泛应用。因此,深入研究流体饱和多孔介质弹性波方程的正反演方法不仅有较高的理论价值也有着广泛的工程意义。
小波分析是近年来国际上公认的前沿研究领域,它既包含有丰富的数学理论,又是工程应用中强有力的方法和工具,给许多相关领域带来了崭新的思想。同伦方法是求解非线性算子方程的一种大范围收敛的方法,在许多领域有着十分重要的应用。水平集方法是一种新颖的求解几何曲线演化的方法,将曲线演化转化成一个纯粹的求偏微分方程数值解问题。本文将小波分析理论结合传统算法进行正演模拟,将小波多尺度方法、同伦方法和水平集方法引入反演过程中,开展了一系列兼有计算量小、抗噪能力强、收敛范围广、程序易于实现的数值反演方法研究。
首先,以Biot流体饱和多孔介质理论为基础,充分利用紧支撑正交小波良好的特性和有限差分法操作简单、程序易于执行的优点,设计了小波有限差分法。针对二维地震勘探模型,模拟了弹性波在均匀介质和双层介质中的传播,给出了相应的地震记录,为随后的介质参数反演研究奠定了坚实的基础。
其次,将可以不断提高分辨率的小波多尺度反演思想引入到参数反演过程中,结合大范围收敛的同伦方法,并对同伦参数的选取进行了改进,设计了小波自适应同伦反演方法。利用该方法求解地震勘探中二维流体饱和多孔介质弹性波方程反问题,数值模拟结果和抗噪性实验均表明了方法的有效性。
随后,针对分布式参数识别问题,引入了水平集方法,并将正则化作用于水平集函数,重新初始化过程保证了水平集函数在演化过程中的稳定性。通过数值实验,我们验证了水平集正则化方法对含有噪声数据的反演问题具有很好的效果。
最后,将小波自适应同伦反演方法引入时间推移地震勘探算法研究中,设计了局域化反演算法,通过使用小波自适应同伦法对基础模型反演成像,可以划定一个包含油藏储层的求解区域,在此相对较小的区域内使用自适应同伦法进行监测模型反演。局域化反演算法不仅提高了计算的精度,还减少了计算量,节省了计算时间。
Fluid-saturated porous media are two-phase media composed of the solid and?uid. Biot theory considered that the subsurface solid is a porous elastic solid con-taining a compressible viscous ?uid. Compared with the single-phase medium theory,?uid-saturated porous media theory can describe the subsurface media more preciselyand the fluid-saturated porous medium elastic wave equation can describe more in-formation than ever. So, fluid-saturated porous media theory can be used widely ingeophysics exploration and earthquake engineering. Therefore, further research of theforward and inverse methods has theory and engineering significance for the elasticwave equation in fluid-saturated porous media.
Wavelet analysis is an international forward research field in recent years. It notonly contains abundant mathematical theories, but also is a powerful method and toolin engineering and it brings new ideas to many fields. The homotopy method is awidely convergent method for solving the nonlinear operator equation and also hasimportant applications in many fields. Level set method has become an novel toolfor the computation of evolving boundaries and interfaces. Instead of tracking thesurfaces, we solve the resulting partial differential equation. Therefore, in this thesiswe compose the wavelet analysis and finite-difference method into the forward simu-lation. Then we draw the wavelet analysis, the adaptive homotopy method and levelset method into the inversion process, and carry out a series of studies of numericalinversion methods, which have the abilities of saving computational costs, noise sup-pression, global convergence and easy realization of the procedure consequently.
Firstly, based on the Biot theory, a wavelet finite-difference method is proposedby combing the good properties of wavelet analysis and traditional finite-differencemethod. For the wavelet multi-resolution method, it owns a quick decomposition andreconstruction algorithm and the finite-difference method is simple and may be read-ily implemented. This algorithm is applied to simulate the propagation of 2-D elasticwave in fluid-saturated porous media for the homogenous and two-layer models. Theresulting synthetic seismograms are shown.
Secondly, the principle of wavelet multi-scale inversion is introduced to the gen- eral estimation problem. Basing on this principle and the homotopy method modifiedthe homotopy parameter adaptively, we design a wavelet adaptive homotopy inversionmethod. Numerical simulations are carried out for the inverse problem of two dimen-sional elastic wave equation in fluid-saturated porous media in seismic prospecting.The numerical results and tests of noise suppression indicate the method’s effective-ness
Thirdly, apply a level set regularization method to the inverse problem for 2-Dwave equation in Fluid-saturated media. we utilized the level set function to presentthe discontinuous parameter, and the regularization functional is applied to the levelset function. Numerical experiments show that the method can recover coefficientswith rather complicated geometry of noise in the observation data.
Finally, the wavelet adaptive homotopy method is introduced to the time-lapseseismic inversion. we design a local inversion algorithm. According to the inversionresults of base model, we can obtain a small domain containing the oil reservoir, thenin this domain we apply the adaptive homotopy method to inverse the monitor model.The results show that this algorithm can improve the precision and decrease the runtime.
小波分析是近年来国际上公认的前沿研究领域,它既包含有丰富的数学理论,又是工程应用中强有力的方法和工具,给许多相关领域带来了崭新的思想。同伦方法是求解非线性算子方程的一种大范围收敛的方法,在许多领域有着十分重要的应用。水平集方法是一种新颖的求解几何曲线演化的方法,将曲线演化转化成一个纯粹的求偏微分方程数值解问题。本文将小波分析理论结合传统算法进行正演模拟,将小波多尺度方法、同伦方法和水平集方法引入反演过程中,开展了一系列兼有计算量小、抗噪能力强、收敛范围广、程序易于实现的数值反演方法研究。
首先,以Biot流体饱和多孔介质理论为基础,充分利用紧支撑正交小波良好的特性和有限差分法操作简单、程序易于执行的优点,设计了小波有限差分法。针对二维地震勘探模型,模拟了弹性波在均匀介质和双层介质中的传播,给出了相应的地震记录,为随后的介质参数反演研究奠定了坚实的基础。
其次,将可以不断提高分辨率的小波多尺度反演思想引入到参数反演过程中,结合大范围收敛的同伦方法,并对同伦参数的选取进行了改进,设计了小波自适应同伦反演方法。利用该方法求解地震勘探中二维流体饱和多孔介质弹性波方程反问题,数值模拟结果和抗噪性实验均表明了方法的有效性。
随后,针对分布式参数识别问题,引入了水平集方法,并将正则化作用于水平集函数,重新初始化过程保证了水平集函数在演化过程中的稳定性。通过数值实验,我们验证了水平集正则化方法对含有噪声数据的反演问题具有很好的效果。
最后,将小波自适应同伦反演方法引入时间推移地震勘探算法研究中,设计了局域化反演算法,通过使用小波自适应同伦法对基础模型反演成像,可以划定一个包含油藏储层的求解区域,在此相对较小的区域内使用自适应同伦法进行监测模型反演。局域化反演算法不仅提高了计算的精度,还减少了计算量,节省了计算时间。
Fluid-saturated porous media are two-phase media composed of the solid and?uid. Biot theory considered that the subsurface solid is a porous elastic solid con-taining a compressible viscous ?uid. Compared with the single-phase medium theory,?uid-saturated porous media theory can describe the subsurface media more preciselyand the fluid-saturated porous medium elastic wave equation can describe more in-formation than ever. So, fluid-saturated porous media theory can be used widely ingeophysics exploration and earthquake engineering. Therefore, further research of theforward and inverse methods has theory and engineering significance for the elasticwave equation in fluid-saturated porous media.
Wavelet analysis is an international forward research field in recent years. It notonly contains abundant mathematical theories, but also is a powerful method and toolin engineering and it brings new ideas to many fields. The homotopy method is awidely convergent method for solving the nonlinear operator equation and also hasimportant applications in many fields. Level set method has become an novel toolfor the computation of evolving boundaries and interfaces. Instead of tracking thesurfaces, we solve the resulting partial differential equation. Therefore, in this thesiswe compose the wavelet analysis and finite-difference method into the forward simu-lation. Then we draw the wavelet analysis, the adaptive homotopy method and levelset method into the inversion process, and carry out a series of studies of numericalinversion methods, which have the abilities of saving computational costs, noise sup-pression, global convergence and easy realization of the procedure consequently.
Firstly, based on the Biot theory, a wavelet finite-difference method is proposedby combing the good properties of wavelet analysis and traditional finite-differencemethod. For the wavelet multi-resolution method, it owns a quick decomposition andreconstruction algorithm and the finite-difference method is simple and may be read-ily implemented. This algorithm is applied to simulate the propagation of 2-D elasticwave in fluid-saturated porous media for the homogenous and two-layer models. Theresulting synthetic seismograms are shown.
Secondly, the principle of wavelet multi-scale inversion is introduced to the gen- eral estimation problem. Basing on this principle and the homotopy method modifiedthe homotopy parameter adaptively, we design a wavelet adaptive homotopy inversionmethod. Numerical simulations are carried out for the inverse problem of two dimen-sional elastic wave equation in fluid-saturated porous media in seismic prospecting.The numerical results and tests of noise suppression indicate the method’s effective-ness
Thirdly, apply a level set regularization method to the inverse problem for 2-Dwave equation in Fluid-saturated media. we utilized the level set function to presentthe discontinuous parameter, and the regularization functional is applied to the levelset function. Numerical experiments show that the method can recover coefficientswith rather complicated geometry of noise in the observation data.
Finally, the wavelet adaptive homotopy method is introduced to the time-lapseseismic inversion. we design a local inversion algorithm. According to the inversionresults of base model, we can obtain a small domain containing the oil reservoir, thenin this domain we apply the adaptive homotopy method to inverse the monitor model.The results show that this algorithm can improve the precision and decrease the runtime.
引文
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