Critique of Biot-related theories of acoustic waves in porous media

Biot’s theory of porous media is discussed in detail and critically. It is pointed out that the use of two and only two displacement fields has a certain arbitrariness, and that models with additional displacement fields are possible. Biot’s expression for the strain-energy per unit volume is justified in part, but it is pointed out that additional terms might be included. The theory in the low-frequency limit is discussed in detail, and the partitioning of the disturbance into three distinct types of fields is discussed. It is shown that there is sufficient latitude in the choice of coefficients in the Biot low-frequency model that the coefficients can be adjusted to fit all the major parameters associated with the three types of disturbances, but it is conjectured that the model will lead to inconsistencies for prediction of minor parameters. Unless measurements of such minor parameters are known from independent experiments, the model cannot be tested quantitatively. The use of the Biot model at higher frequencies is discussed, and it is shown that in the high-frequency limit there are always two propagating modes where the displacement fields have zero curl. It is also shown that the model predicts the attenuation at high frequencies to be independent of frequency. It is pointed out that the reported observations of the second wave were for situations where the artificial porous medium was perfectly periodic. If the medium is not periodic, it is doubtful that a second propagating wave exits. INTRODUCTION The papers on propagation in a fluid-saturated porous solid which Biot published in the Journal of the Acoustical Society in 1956 ([1], [2]) rank among the most highly cited papers in the history of acoustics. The Google Scholar web site (June 2010) shows 3089 citations for the second paper. [The manner by which Google Scholar counts citations is somewhat superficial; most authors tend to cite both papers as a unit, so the first paper is presumably cited as much or more than the first.] These papers, along with a third [3] in 1962, can be taken as what is called the Biot theory. For marine sediments, this, plus modifications due to Stoll and his colleagues, have come to be be known as the Biot-Stoll theory. [The principal account can be found in a 1970 paper by Stoll and Bryan [4]. An extensive discussion can be found in a 1989 book [5] by Stoll.] The present paper argues that the apparent current whole-hearted acceptance of the Biot theory, and especially of the later modifications associated with Stoll, has been made with insufficient critical thought. The derivations, while appealing, are heuristic, and there is little reason to expect broad applicability. The piecing together of Biot’s assumptions in the present paper is retrospective. Biot wrote a large number of papers on the subject of porous media (many of which were reprinted in a book [6] published by the Acoustical Society of America). The references in the later Biot papers were almost exclusively to earlier papers by Biot, and Biot restated his assumptions differently in subsequent papers and he also revised his notation. The present document is a work in progress and is incomplete. It is being submitted at this time (June 10, 2010) to meet the deadline for inclusion in the ICA conference proceedings. It is intended that a more fuller account will be available by the time of the conference and that it will be submitted for journal publication. SEPARATE FIELDS FOR SOLID AND FLUID What appears to be the principal assumption underlying Biot’s models is that the description of all phenomena of interest can be cast in terms of two (and only two) displacement fields. These were denoted by u and U . In retrospect, these can be defined as the local averages of the particle displacements of the solid and fluid matter, so that u(x) = 1 Vs � � umic(x+ξ )dVs, (1) U (x) = 1 Vf � �� U mic(x+ξ )dVf . (2) Here the integrations are taken over small volumes centered as the observation point x. The subscript “mic” is an abbreviation for microscopic. In the integration for the determination of the locally-averaged solid displacement, the integration only includes the volume Vs occupied by solid material (this restriction being represented by a prime on the integral sign). In the integration for the determination of the locally-averaged fluid displacement, the integration only includes the volume Vf occupied by fluid material (this restriction being represented by a double-prime on the integral sign). The total volume being taken into consideration for this averaging process is