Biot介质的交错网格差分法波场模拟研究
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摘要
交错网格有限差分技术已经被广泛应用于地震波波场模拟中,本文使用该技术模拟了Biot饱和流体孔隙介质的波场特征。本文将常见的二阶微分Biot波动方程用等效的一阶速度-应力双曲方程表示,其中未知的波场向量包括固相和流体的速度分量和应力分量。由此才能对其使用交错网格差分方法。模拟中震源项采取双相介质中纵波线震源激发的形式。通过该技术本文对各向同性双相介质和各向异性双相介质在单层介质模型和双层介质模型中的波场特征进行了研究。因为慢纵波在饱和黏滞流体的孔隙介质中具有强衰减性,本文也考虑了介质的衰减机制。本文研究的结果显示在两层介质分界面上,当地震波产生反射和投射时能观测到两类纵波和横波,并且在衰减系数大的介质里慢纵波很难见到。本文还对模拟了含天然气水合物的沉积地层的地震波场并研究了其合成记录。
用有限差分技术求解波动方程时会带来不必要的震动,即数值频散。数值频散现象严重影响了模拟效果,降低了模拟精度。本文探讨了引起数值频散的原因,并给出高阶差分方程和通量校正传输(FCT)的办法以消除频散。
为了避免在人工边界上引起的反射。本文将完全匹配层(PML)引入到孔隙介质的地震波传播模拟中来。完全匹配层是一种非物理性的介质,它作为一种吸收边界条件应用在有限差分计算区域的边界上。Biot波动方程的完全匹配层的表达公式与其他的完全匹配层方程有所不同,而且需要特定的衰减因子。数值模拟的结果显示完全匹配层能够非常有效地吸收掉传播进去的地震波。
Staggered-grid difference method is widely used in seismic wave-field simulation. In this article we apply this method on the simulation of Biot's fluid-saturated porous media. Instead of the prevailing second-order differential equations, we consider a first-order velocity-stress hyperbolic system that is equivalent to Biot’s equations. The vector of unknowns in this system consists of the solid and fluid particle velocity components and stress components. Then the staggered-grid difference scheme is available. A P-wave line source in a uniform poroelastic medium is derived in the simulation. With this method the wave-fields of homogeneous and heterogeneous poroelastic media are studied with single-layer and two-layer models. For“slow”compress ional wave is highly attenuated in porous media saturated by a viscous fluid, the attenuation mechanism is also considered. The results of this study suggest that on the interface of two-layer media, when seismic wave are reflected or transmitted two kinds of P-wave and shear wave can be observed, and“slow”compress ional wave is hardly seen in the media that has large damping coefficient. We also try to study the wave-field characteristics and the synthetic record when seismic propagating in gas hydrate sediment.
Finite-difference schemes for numerically solving the wave equation suffer from undesirable ripples, that is numerical dispersion. The numerical dispersion interferes with the seismic modeling seriously and decrease the precision of simulation. Here, the reasons of this phenomenon is discussed. To eliminate the numerical dispersion higher-order finite-difference equation and the flux-corrected transport (FCT) method are introduced.
In this paper, a method is developed to extend the perfectly matched layer (PML) to simulating seismic wave propagation in poroelastic media to avoid the reflection on the artificial boundaries. This nonphysical material is used at the computational edge of a finite-difference algorithm as an absorbing boundary condition to truncate unbounded media. The incorporation of PML in Biot’s equations is different from other PML applications and a loss coefficient in the PML region is required. Numerical results show that the PML attenuates the outgoing waves effectively.
用有限差分技术求解波动方程时会带来不必要的震动,即数值频散。数值频散现象严重影响了模拟效果,降低了模拟精度。本文探讨了引起数值频散的原因,并给出高阶差分方程和通量校正传输(FCT)的办法以消除频散。
为了避免在人工边界上引起的反射。本文将完全匹配层(PML)引入到孔隙介质的地震波传播模拟中来。完全匹配层是一种非物理性的介质,它作为一种吸收边界条件应用在有限差分计算区域的边界上。Biot波动方程的完全匹配层的表达公式与其他的完全匹配层方程有所不同,而且需要特定的衰减因子。数值模拟的结果显示完全匹配层能够非常有效地吸收掉传播进去的地震波。
Staggered-grid difference method is widely used in seismic wave-field simulation. In this article we apply this method on the simulation of Biot's fluid-saturated porous media. Instead of the prevailing second-order differential equations, we consider a first-order velocity-stress hyperbolic system that is equivalent to Biot’s equations. The vector of unknowns in this system consists of the solid and fluid particle velocity components and stress components. Then the staggered-grid difference scheme is available. A P-wave line source in a uniform poroelastic medium is derived in the simulation. With this method the wave-fields of homogeneous and heterogeneous poroelastic media are studied with single-layer and two-layer models. For“slow”compress ional wave is highly attenuated in porous media saturated by a viscous fluid, the attenuation mechanism is also considered. The results of this study suggest that on the interface of two-layer media, when seismic wave are reflected or transmitted two kinds of P-wave and shear wave can be observed, and“slow”compress ional wave is hardly seen in the media that has large damping coefficient. We also try to study the wave-field characteristics and the synthetic record when seismic propagating in gas hydrate sediment.
Finite-difference schemes for numerically solving the wave equation suffer from undesirable ripples, that is numerical dispersion. The numerical dispersion interferes with the seismic modeling seriously and decrease the precision of simulation. Here, the reasons of this phenomenon is discussed. To eliminate the numerical dispersion higher-order finite-difference equation and the flux-corrected transport (FCT) method are introduced.
In this paper, a method is developed to extend the perfectly matched layer (PML) to simulating seismic wave propagation in poroelastic media to avoid the reflection on the artificial boundaries. This nonphysical material is used at the computational edge of a finite-difference algorithm as an absorbing boundary condition to truncate unbounded media. The incorporation of PML in Biot’s equations is different from other PML applications and a loss coefficient in the PML region is required. Numerical results show that the PML attenuates the outgoing waves effectively.
引文
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[3]. Wyllie MR J, Gregory A R, GardnerL W. Elastic wave velocities in Heterogeneous and porous media. Geophysics, 1956, 21: 41~70
[4]. Gassmann F. Elastic waves through a packing of spheres. Geophysics, 1951, 16: 673~685
[5]. Biot M. A. Theory of propagation of elastic waves in a fluid-saturated porous solid. 1. Low -frequency range: J. Acoust. Soc. Am., 1956, 28, 168~178
[6]. Biot M. A. Theory of propagation of elastic waves in a fluid-saturated porous solid. 1. Higher-frequency range: J. Acoust. Soc. Am., 1956, 28, 179~191
[7]. Biot M. A. Mechanics deformation and acoustic propagation in porous media: J. Appl. Phys., 1962, 33, 1482~1498
[8]. Biot M. A. Generalized theory of acoustic propagation in porous dissipative media: J. Acoust. Soc. Am., 1962, 34, 1254~1264
[9]. X. Zhu, G. A. McMechan. Numerical simulation of seismic responses of poroelastic reservoirs using Biot theory. Geophysics, 1991, 56(3): 328~339
[10]. Siamak Hassanzadeh. Acoustic modeling in fluid-saturated porous media. GEOPHYSICS, 1991, 56(4): 424~435
[11]. 刘仲一,韩其玉. 多孔介质声波传播. 石油大学学报(自然科学版),1994,18(6):30~35
[12]. 牛滨华,吴有校,孙春岩. 裂隙含流体、气体各向异性介质波场数值模拟①. 长春地质学院学报,1994,24(4):454~460
[13]. 徐文骏,郭建. 流体饱和多孔介质中弹性波的数值模拟. 地球物理学报,1994,37 (Supp.1):424~433
[14]. 郭宝琦,董晓丽,包洪洋. 流体饱和多孔隙固体中波动方程的数值模拟. 哈尔滨工业大学学报,1994,26(5):5~7
[15]. N. Dai, A. Vafidis, E. R. Kanasewich. Wave propagation in heterogeneous, porous media: a velocity-stress, finite-difference method. Geophysics, 1995, 60(2): 327~340
[16]. 石玉梅. 流体饱和多孔隙介质中弹性波运动方程的解—伪谱法. 西南石油学院学报,1995,17(2)34~37
[17]. 赵成刚, 杜修力 ,崔 杰. 固体、流体多相孔隙介质中的波动理论及其数值模拟的进展. 力学进展,1998,28(1):83~92
[18]. 邵秀民,蓝志凌. 流体饱和多孔介质波动方程的有限元解法. 地球物理学报,2000,43(2):264~278
[19]. Borge Arntsen, Jose M. Carcione. Numerical simulation of the Biot slow wave in water-saturated Nivelsteiner Sandstone. GEOPHYSICS, 2001, 66(3): 890~896
[20]. Stephane Garambois, Michel Dietrichz. Seismoelectric wave conversions in porous media: Field measurements and transfer function analysis. GEOPHYSICS, 2001, 66(5): 1417~1430
[21]. 杨宽德,杨顶辉,王书强. 基于 Biot-Squirt 方程的波场模拟. 地球物理学报,2002,45(6):853~861
[22]. 杨宽德,杨顶辉,王书强. 基于横向各向同性 BISQ 方程的弹性波传播数值模拟. 地震学报,2002,24(6):599~606
[23]. 孟庆生,何樵登,朱建伟,王德利. 基于 BISQ 模型双相各向同性介质中. 吉林大学学报(地球科学版),2003,33(2):217~221
[24]. 刘洋,魏修成. 双相各向异性介质中弹性波传播有限元方程及数值模拟. 地震学报,2003,25(2):154~162
[25]. 刘洋,李承楚. 双相各向异性介质中弹性波传播伪谱法数值模拟研究. 地震学报,2000,22(2):132~138
[26]. 孙卫涛,杨慧珠. 双相各向异性介质弹性波场有限差分正演模拟. 固体力学学报,2004,25(1):21~28
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