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The poroelastic response of fluid saturated porous rock due to stress variations is of interest in geophysics and geomechanics as it has practical applications in reservoir depletion, fluid injection, time-lapse monitoring, and carbon dioxide sequestration. The effective stress in a poroelastic medium relates to applied pressure and pore pressure, with the Biot parameter (α) as a scaling factor of the pore pressure. This work offers an independent derivation of the tensor characteristics of α through elastic moduli, a microscopic effective medium derivation, and frequency-dependent behavior of α for an anisotropic medium. We derived simplified equations for isotropic rock subjected to confining pressure and pore pressure, isotropic rock under uniaxial stress considering the nonlinear part of elastic constants, and an equation of α for the frequency-dependent case. In the effective medium derivation, we assumed that the rock contains both isolated pores and connected pores saturated with liquid. We use the GSA method to Barnett shale core samples to link ultrasonic velocities with mineral composition and porosity data. We also use the GSA method in subsequent chapters to estimate the effective properties of a rock. We corroborate our theoretical formulations by applying those equations to experimental data for different scenarios such as changes in confining pressure, pore pressure, and uniaxial stress. We calculated the Biot tensor for sandstone and shale. We found excellent agreement between theoretical prediction and experimental data. It is known that α varies significantly for changes in porosity and rock microstructure in isotropic rock. We also see as much as a 21% difference between horizontal and vertical components of α for transversely isotropic (TI) rock for changes in uniaxial stress. We then estimated the frequency-dependent Biot tensor for TI models using numerical calculations. We noticed significant differences between vertical (α33) and horizontal (α11) components of α, especially at the surface seismic frequency band. However, uniaxial stress and horizontally aligned microstructure influence the elastic moduli and Biot tensor contrarily. In general, anisotropy due to uniaxial stress shows lower α33 and higher α11. The anisotropy due to microstructure shows the opposite.



Biot tensor, stress, anisotropy, rock physics


Portions of this document appear in: Morshed, S. M., Chesnokov, E. M., and Vikhoreva, A. A., 2021, Biot effective stress parameter in poroelastic anisotropic media: Static and dynamic case: Geophysical Prospecting, v. 69, no. 3, p. 530-541; and in: Ghosh, Avradip, and Sharif Morshed. "A Green's function approach to the study of effective anisotropic properties of the Barnett Shale." Geophysical Prospecting 69, no. 5 (2021): 968-983.