Biot-Stoll Model

To deseribe wave propagation in marine sediments mathematieally, various simple to eomplex models have been developed which approximate the sediment by a dilute suspension (Wood 1946) or an elastic, water-saturated frame (Gassmann 1951 Biot 1956a, b). The most common model whieh considers the microstructure of the sediment and simulates frequency-dependent wave propagation is based on Biot s theory (Biot 1956a, b). It includes Wood s suspension and Gassmann s elastic frame model as low-frequency approximations and combines acoustic and elastic parameters - P- and 

S-wave velocity and attenuation and elastic moduli - with physical and sedimentological parameters like mean grain size, porosity, density and permeability. 

Based on Biot s fundamental work Stoll (e.g. 1974, 1977, 1989) reformulated the mathematical background of this theory with a simplified uniform nomenclature. Here, only the main physical principles and equations are summarized. For a detailed description please refer to one of Stoll s publications or Biot s original papers. 

The theory starts with a description of the micro structure by 11 parameters. The sediment grains are characterized by their grain density (p ) and bulk modulus (Kp, the pore fluid by its density (p, ), bulk modulus (K, ) and viscosity (r ). The porosity (([)) quantifies the amount of pore space. Its shape and distribution are specified by the permeability (k), a pore size parameter a=dy3 (j)/(l-(j)), d = mean grain size (Hovem and Ingram 1979 Courtney and Mayer 1993), and stracture factor a (0 r 1) indicating 

An elastie wave propagating in water-saturated sediments eauses different displacements of the pore fluid and sediment frame due to their different elastie properties. As a result (global) fluid motion relative to the frame occurs and can approximately be deseribed as Poiseuille s flow. The flow rate follows Darey s law and depends on the permeability and viseosity of the pore fluid. Viscous losses due to an interstitial pore water flow are the dominant damping mechanism. Intergranular frietion or loeal fluid flow can additionally be ineluded but are of minor importanee in the frequeney range eonsidered here. 

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Courtney R.C., Mayer L.A., 1993. Acoustic properties of fine-grained sediments from Emerald Basin toward an inversion for physical properties using the Biot-Stoll model. Journal of the Acoustical Society of America 93 3193-3200... 

In what follows the theoretical background of the most common physical properties and their measuring tools are described. Examples for the wet bulk density and porosity can be found in Section 2.2. For the acoustic and elastic parameters first the main aspects of Biot-Stoll s viscoelastic model which computes P- and S-wave velocities and attenuations for given sediment parameters (Biot 1956a, b, Stoll 1974, 1977, 1989) are summarized. Subsequently, analysis methods are described to derive these parameters from transmission seismograms recorded on sediment cores, to compute additional properties like elastic moduli and to derive the permeability as a related parameter by an inversion scheme (Sect. 2.4). 

Table 2.2 Physical properties of sediment grains, pore fluid and sediment frame used for the computation of attenuation and phase velocity curves according to Biot-Stoll s sediment model (Fig. 2.12).

Fig. 2.16 Comparison of P-wave attenuation and velocity dispersion data derived from ultrasonic transmission seismograms with theoretical curves based on Biot-Stoll s model for six traces of the turbidite layer of gravity core GeoB1510-2. Permeabilities vary in the model curves according to constant ratios K/a = 0.030, 0.010, 0.003 (K = permeability, a = pore size parameter). The resulting permeabilities are given in each diagram. Modified after Breitzke et al. (1996).

Holland, C.W., and Brunson, B.A. 1988. The Biot-Stoll sediment model An environmental assessment. Journal of the Acoustical Society of America, 84 1437-1443. 

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