﻿ 基于三维孔隙网络模型的纵波频散衰减特征分析
 地球物理学报  2021, Vol. 64 Issue (12): 4618-4628 PDF

1. 中国石油勘探开发研究院, 北京 100083;
2. 清华大学周培源应用数学研究中心, 北京 100084

Attenuation and dispersion characteristics of P-waves based on the three-dimensional pore network model
WEI LeLe1, GAN LiDeng1, XIONG FanSheng2, SUN WeiTao2, DING Qian1, YANG Hao1
1. Geophysical Department, RIPED, PetroChina, Beijing 100083, China;
2. Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, China
Abstract: When propagating in a porous medium containing fluid, the wave field will undergo dispersion and attenuation. Such phenomena are related to the petrophysical properties of the porous medium, including porosity, permeability, fluid properties and so forth. In previous work, based on the three-dimensional fracture/soft pore network model, the variation of the aspect ratio of the ellipse section is used to simulate a variety of conditions from flat fractures to soft pores to hard pores, but does not consider the global network space containing both pores and fractures. To better describe the fracture-pore space, we propose three-dimensional fracture-pore network and the method to calculate the permeability. Through volume averaging, we derive the wave equations of these two three-dimensional pore network models. Then, the expressions of dispersion/attenuation curves of P-waves are obtained by plane wave analysis. And by using numerical simulation, many influence factors are analyzed for the attenuation and dispersion of P-waves, such as total porosity, fracture porosity, fracture aspect ratio, fracture density and pore fluid viscosity. Results show that based on the 3D fracture-pore network model, the effects of total porosity and fracture parameters on the dispersion/attenuation of fast P-waves and the dispersion/attenuation trend of slow P-waves are similar to those of the 3D fracture/soft pore network model. Specifically, the attenuation and dispersion of fast P-waves occur in high-frequency bands. The change of porosity mainly affects the peak value of inverse quality factor curve. The fracture density mainly controls the range of significant change in velocity. And the fracture aspect ratio has a significant effect on the P-wave velocity and characteristic frequency.
Keywords: 3D fracture/soft pore network model    3D fracture-pore network model    Wave equation    Dispersion and attenuation
0 引言

Biot(1955)建立了完全饱和孔隙介质中波传播的动力学方程.在Biot理论中，黏性的孔隙流体和固体骨架之间的相互作用导致波的速度频散和衰减.Geertsma和Smit(1961)应用Biot理论推导出纵波的速度频散和衰减的近似解.然而，越来越多的研究发现，Biot理论的这种宏观的流体流动机制很难解释实际生产中地震波的高频散和高衰减现象(Carcione et al., 2010).

Dvorkin和Nur(1993)认为，在含有流体的岩石中，波的速度频散和衰减除了受Biot机制的影响，还受到喷射流动机制的影响.他们将上述两种机制统一起来，提出了BISQ模型，并推导了纵波相速度和品质因子的表达式，研究了渗透率等对纵波速度频散和衰减的影响.

Sun(2021)提出了一种BIPS(Biot-Patchy-Squirt) 模型，用于描述非混相流体饱和裂缝孔隙弹性介质中的波的频散衰减特性.研究结果表明，BIPS模型与实验室数据吻合较好，这一发现将推动上述三种频散衰减机制在波速预测中的潜在应用.

Chapman等(2002)提出了在微观尺度上建立孔隙弹性介质模型的方法，将模型设置成规则的立方体网格，每个网络节点代表孔隙/裂缝空间.但是此模型并未建立孔隙介质渗透率与裂缝参数等之间的几何关系.Tang(2011)Tang等(2012)提出了含裂缝孔隙介质的波动方程模型，在推导过程中直接使用了Johnson等(1987)提出的动态渗透率模型，并没有在渗透率与裂缝参数等影响因素之间建立关系，也没有给出波频散衰减与渗透率之间的关系.Song等(2020)假定裂缝的形状可以是矩形的，建立了一个介质模型来估计饱和多孔岩石中包含一个随机分布的、无限小厚度的矩形裂缝时的纵波衰减和速度频散特征.

1 三维孔隙网络模型 1.1 三维裂缝/软孔隙网络模型

 图 1 三维裂缝/软孔隙网络模型 Fig. 1 3D fracture/soft pore network model

(1) 微裂缝的纵横比：a=R2/R1.

(2) 裂缝的孔隙度：

 (1)

(3) 单元体内的裂缝数密度：

 (2)

1.2 三维裂缝-孔隙网络模型

 图 2 三维裂缝-孔隙网络模型 Fig. 2 3D fracture-pore network model

 (3)

(1) 微裂缝的纵横比a：裂缝椭圆截面的长短轴半径之比的倒数；

(2) 总孔隙度：ϕ=ϕf+ϕpore

 (4)

 (5)

(3) 单元体内的裂缝数密度：表达式同公式(2).

RiRj表示正圆球形孔隙ij的半径，η为流体黏度，则流体传导系数为，这里r取(Ri+Rj)/2.

 (6)

 (7)

 (8)

 (9)

 (10)

 (11)

2 基于体积平均法的波动方程推导 2.1 体积平均法

 (12)

ψf为孔隙介质中与流体有关的物理量，规定在流体的外部，ψf的值为0.定义在整个区域Ω上对ψf作体积平均的方法为

 (13)

 (14)

 (15)

2.2 波动方程推导

uU分别表示固体、流体质点的位移向量；各物理量上方的一点表示对时间求导；以上下标中的sf分别表示固体骨架、孔隙介质中所含的流体.

 (16)

 (17)

 (18)

 (19)

 (20)

 (21)

 (22)

 (23)

 (24)

 (25)

 (26)

2.3 频散/衰减表征

 (27)

3 频散/衰减特征分析

3.1 基于三维裂缝/软孔隙网络模型的频散衰减特征

3.1.1 裂缝孔隙度的影响

 图 3 快纵波频散曲线随裂缝孔隙度的变化 Fig. 3 Velocity dispersion changing with fracture porosity
 图 4 快纵波衰减曲线随裂缝孔隙度的变化 Fig. 4 Inverse quality factor versus fracture porosity

3.1.2 裂缝纵横比的影响

 图 5 快纵波频散曲线随裂缝纵横比的变化 Fig. 5 Velocity dispersion versus aspect ratio of fracture
 图 6 快纵波衰减曲线随裂缝纵横比的变化 Fig. 6 Attenuation changing with aspect ratio of fracture

3.1.3 裂缝数密度的影响

 图 7 快纵波频散曲线随裂缝数密度的变化 Fig. 7 Velocity dispersion varying with fracture density
 图 8 快纵波衰减曲线随裂缝数密度的变化 Fig. 8 Attenuation varying with fracture density

3.1.4 孔隙流体黏度的影响

 图 9 不同孔隙流体黏度下的快纵波频散曲线 Fig. 9 Velocity dispersion curve as function of fluid viscosity
 图 10 不同孔隙流体黏度下的快纵波衰减曲线 Fig. 10 Attenuation curve under different fluid viscosity
3.1.5 慢纵波频散衰减曲线

 图 11 慢纵波频散曲线 Fig. 11 Velocity dispersion curve of slow P-waves
 图 12 慢纵波衰减曲线 Fig. 12 Attenuation curve of slow P-waves

3.2 基于三维裂缝-孔隙网络模型的频散衰减特征

3.2.1 总孔隙度的影响

 图 13 快纵波频散曲线随总孔隙度的变化 Fig. 13 Velocity dispersion curve as function of total porosity
 图 14 快纵波衰减曲线随总孔隙度的变化 Fig. 14 Attenuation curve as function of total porosity
3.2.2 裂缝参数的影响

3.2.3 慢纵波频散衰减曲线

 图 15 慢纵波频散曲线 Fig. 15 Velocity dispersion curve of slow P-waves
 图 16 慢纵波衰减曲线 Fig. 16 Attenuation curve of slow P-waves
4 结论

(1) 随着裂缝孔隙度的增大，快纵波速度表现出更明显的频散现象.随着裂缝孔隙度的减小，快纵波速度增大，逆品质因子曲线的峰值下降，特征频率(逆品质因子曲线的峰值对应的频率)向低频方向移动.

(2) 随着裂缝纵横比的增大，快纵波速度增大，特征频率向低频方向移动，而频散幅值(速度极大值与极小值的差)和逆品质因子曲线的峰值的大小未表现出规律性的变化.

(3) 裂缝数密度的变化几乎不会影响快纵波速度、频散幅值和逆品质因子曲线的峰值，但对特征频率影响显著.随裂缝数密度的增大，特征频率向低频方向移动.

(4) 当孔隙充填油、水、气时，油饱和砂岩的快纵波速度明显大于水饱和砂岩，水饱和砂岩明显大于气饱和砂岩.气饱和时的快纵波频散现象最不明显，且逆品质因子曲线的峰值最小.

References
 Ba J, Carcione J M, Cao H, et al. 2012. Velocity dispersion and attenuation of P waves in partially-saturated rocks: wave propagation equations in double-porosity medium. Chinese Journal of Geophysics (in Chinese), 55(1): 219-231. DOI:10.6038/j.issn.0001-5733.2012.01.021 Ba J, Xu W H, Fu L Y, et al. 2017. Rock anelasticity due to patchy saturation and fabric heterogeneity: a double double-porosity model of wave propagation. Journal of Geophysical Research: Solid Earth, 122(3): 1949-1976. Biot M A. 1955. Theory of propagation of elastic waves in a fluid-saturated porous solid. Ⅰ. low-frequency range. The Journal of the Acoustical Society of America, 28(182): 168-191. Carcione J M, Helle H B, Pham N H. 2003. White's model for wave propagation in partially saturated rocks: comparison with poroelastic numerical experiments. Geophysics, 68(4): 551-566. Carcione J M, Morency C, Santos J E. 2010. Computational poroelasticity-a review. Geophysics, 75(5): 75A229-75A243. DOI:10.1190/1.3474602 Chapman M, Zatsepin S V, Crampin S. 2002. Derivation of a microstructural poroelastic model. Geophysical Journal International, 151(2): 427-451. DOI:10.1046/j.1365-246X.2002.01769.x Dutta N C, Odé H. 1979a. Attenuation and dispersion of compressional waves in fluid-filled porous rocks with partial gas saturation (White model)-Part Ⅰ: Biot theory. Geophysics, 44(11): 1777-1788. DOI:10.1190/1.1440938 Dutta N C, Odé H. 1979b. Attenuation and dispersion of compressional waves in fluid-filled porous rocks with partial gas saturation (White model)-Part Ⅱ: Results. Geophysics, 44(11): 1789-1805. DOI:10.1190/1.1440939 Dvorkin J, Nur A. 1993. Dynamic poroelasticity: a unified model with the squirt and the Biot mechanisms. Geophysics, 58(4): 524-533. DOI:10.1190/1.1443435 Geertsma J, Smit D C. 1961. Some aspects of elastic wave propagation in fluid-saturated porous solids. Geophysics, 26(2): 169-181. DOI:10.1190/1.1438855 Guo M Q, Ba J, Ma R P, et al. 2018. P-wave velocity dispersion and attenuation in fluid-saturated tight sandstones: characteristics analysis based on a double double-porosity structure model description. Chinese Journal of Geophysics (in Chinese), 61(3): 1053-1068. DOI:10.6038/cjg2018L0678 He R F, Ba J, Du Q Z, et al. 2020. Seismic wave propagation theory and fluid prediction methods in two-phase media of reservoirs. Progress in Geophysics (in Chinese), 35(4): 1379-1390. DOI:10.6038/pg2020DD0177 Johnson D L, Koplik J, Dashen R. 1987. Theory of dynamic permeability and tortuosity in fluid-saturated porous media. Journal of Fluid Mechanics, 176: 379-402. DOI:10.1017/S0022112087000727 Ling Y. 2021. Well-seismic dispersion correction and its application: a case study of deep-water sandstone in West Africa. Progress in Geophysics (in Chinese), 36(4): 1554-1559. DOI:10.6038/pg2021EE0312 Mavko G M, Nur A. 1978. The effect of nonelliptical cracks on the compressibility of rocks. Journal of Geophysical Research: Solid Earth, 83(B9): 4459-4468. DOI:10.1029/JB083iB09p04459 Mavko G M, Nur A. 1979. Wave attenuation in partially saturated rocks. Geophysics, 44(2): 161-178. DOI:10.1190/1.1440958 Ouyang F, Zhao J G, Li Z, et al. 2021. Modeling velocity dispersion and attenuation using pore structure characteristics of rock. Chinese Journal of Geophysics (in Chinese), 64(3): 1034-1047. DOI:10.6038/cjg2021O0355 Song Y J, Hu H S, Han B. 2020. P-wave attenuation and dispersion in a fluid-saturated rock with aligned rectangular cracks. Mechanics of Materials, 147: 103409. DOI:10.1016/j.mechmat.2020.103409 Sun W T. 2021. On the theory of Biot-patchy-squirt mechanism for wave propagation in partially saturated double-porosity medium. Physics of Fluids, 33(7): 076603. DOI:10.1063/5.0057354 Tang X M. 2011. A unified theory for elastic wave propagation through porous media containing cracks-an extension of Biot's poroelastic wave theory. Science China Earth Sciences, 54(9): 1441-1452. DOI:10.1007/s11430-011-4245-7 Tang X M, Chen X L, Xu X K. 2012. A cracked porous medium elastic wave theory and its application to interpreting acoustic data from tight formations. Geophysics, 77(6): D245-D252. DOI:10.1190/geo2012-0091.1 Wang Y J, Chen S Q, Wang L, et al. 2014. Gas saturation analysis with seismic dispersion attribute based on patchy-saturation model. Oil Geophysical Prospecting (in Chinese), 49(4): 715-722. Whitaker S. 1999. Dispersion in porous media. //The Method of Volume Averaging. Dordrecht: Springer: 124-160. White J E. 1975a. Computed seismic speeds and attenuation in rocks with partial gas saturation. Geophysics, 40(2): 224-232. DOI:10.1190/1.1440520 White J E. 1975b. Low-frequency seismic waves in fluid-saturated layered rocks. The Journal of the Acoustical Society of America, 57(S1): S30. DOI:10.1121/1.1995164 Xiong F S, Sun W T, Ba J, et al. 2020. Effects of fluid rheology and pore connectivity on rock permeability based on a network model. Journal of Geophysical Research: Solid Earth, 125(3): e2019JB018857. DOI:10.1029/2019JB018857 Xiong F S, Gan L D, Sun W T, et al. 2021. Characterization of reservoir permeability and analysis of influencing factors in fracture-pore media. Chinese Journal of Geophysics (in Chinese), 64(1): 279-288. DOI:10.6038/cjg2021N0175 Zhang L, Ba J, Carcione J M, et al. 2020. Differential poroelasticity model for wave dissipation in self-similar rocks. International Journal of Rock Mechanics and Mining Sciences, 128: 104281. DOI:10.1016/J.IJRMMS.2020.104281 Zhang L, Ba J, Carcione J M. 2021. Wave propagation in infinituple-porosity media. Journal of Geophysical Research: Solid Earth, 126(4): e2020JB021266. DOI:10.1029/2020JB021266 Zhao P Q, Ni T L, Zhang J L, et al. 2021. Review of the correlation between seepage field parameters and seismic wave field parameters. Progress in Geophysics (in Chinese), 36(4): 1661-1668. DOI:10.6038/pg2021EE0478 巴晶, Carcione J M, 曹宏, 等. 2012. 非饱和岩石中的纵波频散与衰减: 双重孔隙介质波传播方程. 地球物理学报, 55(1): 219-231. DOI:10.6038/j.issn.0001-5733.2012.01.021 郭梦秋, 巴晶, 马汝鹏, 等. 2018. 含流体致密砂岩的纵波频散及衰减: 基于双重双重孔隙结构模型描述的特征分析. 地球物理学报, 61(3): 1053-1068. DOI:10.6038/cjg2018L0678 何润发, 巴晶, 杜启振, 等. 2020. 储层双相介质地震波传播理论及流体预测方法. 地球物理学进展, 35(4): 1379-1390. DOI:10.6038/pg2020DD0177 凌云. 2021. 井震频散校正及其应用: 以西非深水砂岩为例. 地球物理学进展, 36(4): 1554-1559. DOI:10.6038/pg2021EE0312 欧阳芳, 赵建国, 李智, 等. 2021. 基于微观孔隙结构特征的速度频散和衰减模拟. 地球物理学报, 64(3): 1034-1047. DOI:10.6038/cjg2021O0355 王峣钧, 陈双全, 王磊, 等. 2014. 基于斑块饱和模型利用地震波频散特征分析含气饱和度. 石油地球物理勘探, 49(4): 715-722. 熊繁升, 甘利灯, 孙卫涛, 等. 2021. 裂缝-孔隙介质储层渗透率表征及其影响因素分析. 地球物理学报, 64(1): 279-288. DOI:10.6038/cjg2021N0175 赵平起, 倪天禄, 张家良, 等. 2021. 渗流场参数与地震波场参数的关联关系研究综述. 地球物理学进展, 36(4): 1661-1668. DOI:10.6038/pg2021EE0478