Baseline Model for Water Saturated Sediments

The model for water saturated sediments is based on the rock physics model of Dvorkin et al. (1999). This model relates the stiffness of the sediment dry frame to porosity, mineralogy and effective pressure. The effect of water saturation is modeled by Gassmann's (1951) equations. The porosity at which a granular composite ceases to be a suspension and becomes grain supported is called the critical porosity (j)c. For a dense random packing of nearly identical spheres, (j)c is approximately 0.36-0.40 (Nur et al., 1998). Laboratory experiments have shown that the elastic properties of porous materials are best modeled as mixtures with endmembers of critical porosity and solid material or critical porosity and void space instead of simple mixtures of solid material and void space (Nur et al., 1998). The baseline model for water saturated sediments uses the effective moduli of a dense random packing of identical elastic spheres at critical porosity as its starting point.

The effective bulk KHM and shear GHM moduli of the dry rock frame at <J)C are calculated from the Hertz-Mindlin contact theory (Mindlin, 1949):

n2(Wc)2G2 1

where P is the effective pressure; G and V are the shear modulus and Poisson's ratio of the solid phase, respectively; and n is the average number of contacts per grain in the sphere pack (about 8-9: Dvorkin and Nur, 1996; Murphy, 1985). The effective pressure is calculated as the difference between the lithostatic and hydrostatic pressure:

where pb is sediment bulk density; pw is water density; g is the acceleration due to gravity; and D is depth below sea floor.

For porosity <j) less than (j)c, the bulk (KDry) and shear (GDry) moduli of the dry frame are calculated via the modified lower Hashin-Shtrikman bound (Dvorkin and Nur, 1996):

which represents the weakest possible combination of solid and critical porosity material (see Figure 1). K is the bulk modulus of the solid phase. For porosity above critical, KDry and GDry are calculated via the modified upper H-S bound:

which represents the strongest possible combination of critical porosity material and void space (Figure 1).

Figure 1. Schematically going from zero to 100% porosity through critical porosity.

If the sediment is saturated with pore fluid of bulk modulus Kf, the shear modulus GSat and the bulk modulus KSat are calculated from Gassmann's (1951) equations as

<PKDry-{l + (t>)KfKDry/K + Kf (1 -MKf + QK-KfK^/K

Once the elastic moduli are known, the elastic wave speeds are calculated from

where pB is bulk density.

In the common case of mixed mineralogy, the effective elastic constants of the solid phase can be calculated from those of the individual mineral constituents using Hill's (1952) average formula:

2 ¡=i ¿=i where m is the number of mineral constituents; /, is the volumetric fraction of the i-th constituent of the solid phase; and Kt and Gi are the bulk and shear moduli of the ¿-th constituent, respectively. The solid phase density is calculated as m

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