Abstract
Wave propagation in porous media is an important topic for example in geomechanics or oil-industry. Especially due to the interplay of the solid skeleton with the fluid the so-called second compressional wave appears. The existence of this wave is reported in the literature not only for Biot's theory (BT) but also for theoretical approaches based on the Theory of Porous Media (TPM – mixture theory extended by the concept of volume fractions). Assuming a geometrically linear description (small displacements and small deformation gradients) and linear constitutive equations (Hooke's law) the governing equations are derived for both theories, BT and the TPM, respectively. In both cases, the solid displacements and the pore pressure are the primary unknowns. Note that this is only possible in the Laplace domain leading to the same structure of the coupled differential equations for both approaches. But the differential equations arising in BT and TPM possess different coefficients with different physical interpretations. Correlating these coefficients to each other leads to the well-known problem of Biot's “apparent mass density”. Furthermore, some inconsistencies are observed if Biot's stress coefficient is correlated to the structure arising in TPM. In addition to the comparison of the governing equations and the identification of the model parameters, the displacement and pressure solutions of both theories are presented for a one-dimensional column. The results show good agreement between both approaches in case of incompressible constituents whereas in case of compressible constituents large differences appear.
Highlights
For a wide range of fluid infiltrated materials, such as water saturated soils, oil impregnated rocks, or air filled foams, the elastic as well as a viscoelastic description of the material behavior is a crude approximation for the investigation of wave propagation in such media
Assuming a geometrically linear description and linear constitutive equations (Hooke’s law) the governing equations are derived for both theories, Biot’s theory (BT) and the Theory of Porous Media (TPM), respectively
The differential equations arising in BT and TPM possess different coefficients with different physical interpretations
Summary
For a wide range of fluid infiltrated materials, such as water saturated soils, oil impregnated rocks, or air filled foams, the elastic as well as a viscoelastic description of the material behavior is a crude approximation for the investigation of wave propagation in such media. Based on the work of von Terzaghi, a theoretical description of porous materials saturated by a viscous fluid was presented by Biot [6] This was the starting point of the theory of poroelasticity or the BT. As a consequence of the work by Bowen [16] and by Ehlers and Kubik [22], it may be stated that even if both approaches are similar the theories are mainly different in the way how the solid-fluid interaction is modelled In both papers, the authors used solid displacements, seepage velocity, and pore pressure as unknowns. The assumption of full saturation is made, e.g., the whole pore space is filled with fluid For such materials the governing equations are given based on the TPM [3], [18] in Sect. Commas (),i denote spatial derivatives and primes ðÞ0 denote the material time derivative with respect to the moving skeleton
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