Flow interpretation implications for poro-elastic modeling

The basis of poro-elastic modeling is a set of assumptions about the nature of the porous media and the fluid that fill its void spaces. The basic assumptions are: (1) the solid matrix is homogenous, elastically deformable, and chemically inert with respect to the fluid, (2) the fluid is single phase and Newtonian, (3) the flow is in the laminar range, (4) there is an explicit one-to-one relationship between porosity and permeability, and (5) the fluid flow is governed by Darcy's law. There are two basic interpretations of Darcy's law. The primary interpretation in poro-elastic acoustics is to assume that Darcy's law is a restatement of Poisseuille's law for porous media applications. This restatement can be extended for bundles of capillary tubes, and non-straight tubes. This gives rise to a linear equation that predicts the acoustic dispersion, and attenuation for acoustic waves in a porous media. Unfortunately, simulations of fluid flow in Hele-Shaw cells suggest that for porous media consisting of face-centered-cubically packed spheres the Poisseuille law treatment is incorrect. Both the distribution of fluid velocity, and viscous drag are incorrect. The second interpretation is to assume that Darcy's law is a macro-scale statistically derived flow relationship derived from underlying micro-scale processes. This methods gives rise to a model that inherently includes flow geometry, and variations in permeability and porosity. The statistical model allows for the definition of permeability such that the tortuosity of the fluid paths are explicitly included in both the permeability and the inertial effects. Additional fluid dynamic phenomena, such as inertial effects, internal friction and local accelerations among others can be included in the calculation. Two quantities of interest in understanding porous media flow characteristics are the dispersivity of the porous media, and the permeability. The dispersivity determines the spreading of a definite fluid portion, and the permeability determines the average flux of a fluid through the porous media. The Poisseuille law treatment leads to models that can explain longitudinal dispersion in porous media flows, but cannot explain transverse dispersion. The macro-scale statistical model can explain both phenomena at a cost of complexity. Thus the Poisseuille law treatment will give rise to less dispersive estimates of the momentum transfer, and thereby greater overall acoustic effects caused by momentum transfer by the fluid. The Poisseuille law treatment or the statistical treatment of Darcy's Law can employed within the consolidation model framework to predict compressional phase speeds, shear phase speeds, and attenuations. The predictions from either method are similar. Both methods predict a non-linear dependence on frequency, frequency dependent phase speeds, and two compressional waves. The overall shape of the sound speed dispersion relationship for each of the fluid flow models is similar, although a significant difference is obvious. The most significant difference in the predictions from the two interpretations is the velocity difference between high-frequency and low-frequency phases speeds is larger by roughly a factor of two for the Poisseuille law interpretation. This discrepancy has implications for the interpretation of acoustical inversions based on poro-elastic models.