A comparative study of Biot's theory and the linear Theory of Porous Media for wave propagation problems


 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.

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Received July 15, 2002 Published online: April 17, 2003

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Schanz, M., Diebels, S. A comparative study of Biot's theory and the linear Theory of Porous Media for wave propagation problems. Acta Mechanica 161, 213–235 (2003). https://doi.org/10.1007/s00707-002-0999-5

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  • Porous Medium
  • Compressional Wave
  • Pore Pressure
  • Geomechanics
  • Pressure Solution