Poroelastic backus averaging for anisotropic layered fluid- and gas-saturated sediments

A homogeneous anisotropic effective-medium model for saturated thinly layered sediments is introduced. It is obtained by averaging over many layers with different poroelastic moduli and different saturating fluids. For a medium consisting of a stack of vertically fractured horizontal layers, this effective medium is orthorhombic. We derive the poroelastic constants that define such media in the long-wavelength limit as well as the effective large scale permeability tenser. The permeability shows strong anisotropy for large porosity fluctuations. We observe pronounced effects that do not exist in purely elastic media. At very low frequencies, seismic waves cause interlayer flow of pore fluid across interfaces from more compliant into stiffer layers. For higher frequencies, the layers behave as if they are sealed, and no fluid flow occurs. The effective-medium velocities of the quasi-compressional waves are higher in the no-flow than in the quasi-static limit. Both are lower than the high-frequency, i.e., ray-theory limit. Partial saturation affects the anisotropy of wave propagation. In the no-flow limit, gas that is accumulated primarily in the stiffer layers reduces the seismic anisotropy; gas that is trapped mainly in layers with a more compliant frame tends to increase the anisotropy. In the quasi-static limit, local flow keeps the anisotropy constant independent of partial saturation effects. For dry rock, no-flow and quasi-static velocities are the same, and the anisotropy caused by layering is controlled only by fluctuations of the layer shear moduli. If the shear stiffness of all layers is the same and only the compressive stiffness or saturation varies, only the ray-theory velocity exhibits anisotropy.