Numerical solution of anisotropic Biot equations of poroelastic fluid-saturated media in quasi-static state for numerical upscaling

In this paper we present the numerical algorithm for quasi-static loading of porous fluidsaturated sample used to solve the numerical upscaling problem for fractured porous fluid-saturated media. Numerical upscaling is aimed to recover homogeneous anisotropic viscoelastic media, which is equivalent to the initial poroelastic media and defined by complex-valued frequency-dependent stiffness tensor. We apply recovered stiffness tensor components to estimate both frequencydependent attenuation and phase velocity of seismic wave. Numerical upscaling procedure includes numerical solution of boundary-value problem for Biot poroelasticity equations for anisotropic fluid-saturated media in the frequency domain for a set of frequencies and different boundary conditions. Numerical solution of Biot system of equations is based on finite-difference approximation of equations in quasi-static form, and for resulting SLAE we apply direct solver. Applied direct solver support effective solution of SLAE for several right-hand vectors essential for numerical upscaling. Presented algorithm realization allows us solve 2D problem on computational grid of 2000 $\times$ 2000 nodes using a single machine, what makes it capable to perform the upscaling for detailed representative fractured porous samples. To demonstrate the applicability of the algorithm we perform several sets of numerical experiments aimed at the investigation of fracture connectivity and microscale anisotropy effects on wave-induced fluid flow attenuation and phase velocity of seismic wave propagating in fractured porous fluid-saturated media.