Dry Sediment Properties

In order to relate the velocities in marine sediments without gas hydrates to porosity, saturation, mineralogy, and effective pressure, I use a modified Hashin-Shtrikman-Hertz-Mindlin theory first introduced by Dvorkin and Nur 1996. This theory first calculates the effective bulk and shear moduli at critical porosity ($\phi_{\rm c} \: \simeq \: 40$%) using the Hertz-Mindlin theory Mindlin (1949). Critical porosity separates the mechanical and acoustic behavior into two distinct regions Nur et al. (1995): for porosities lower than $\phi_{\rm c}$, the mineral grains are load bearing, while for porosities greater than $\phi_{\rm c}$, the sediment becomes a suspension, with the fluid phase load-bearing. The effective moduli at critical porosity are given by:

$$\begin{eqnarray} & \rm K_{\rm HM}& = \: {\left [{{\rm n^2\:(1-\phi_{\rm c})^2 \:... ...2\:\rm G^2}\over{2\:\pi^2\:(1-\nu)^2}}\:\rm P \right ]}^{1\over3},\end{eqnarray}$$ (8)

where $\rm K$ and $\rm G$ are the bulk and shear moduli of the mineral making up the rock. The Poisson's ratio is given by $\nu$, $\rm P$ is the effective pressure and $\rm n$ is the average number of grain contacts, taken to be 8.5 Murphy (1982). The effective pressure $\rm P$ is calculated as follows:

$$\rm P\:=\:(1-\phi)\:(\rho_{\rm s} \:-\:\rho_{\rm f}) \: \rm g \:\rm h$$ (9)

where $\rho_{\rm s}$ and $\rho_{\rm f}$ are the solid and fluid density, respectively; the depth below the seafloor is given by $\rm h$, and $\rm g$ is the gravity acceleration.

If the sediment rock consists of a mixed mineralogy, the bulk and shear moduli $\rm K$ and $\rm G$ of the rock can be determined using a Hill's average formula:

$$\begin{eqnarray} &\rm K&\:=\:{1\over2} \left [ \sum_{\rm i=1}^{\rm m} \:\rm f_{\... ...m} \: {\rm f_{\rm i} \over \rm G_{\rm i}} \right )}^{-1} \right ],\end{eqnarray}$$ (10)

where $\rm m$ is number of different mineral components, $\rm f_{\rm i}$ is the volumetric fraction of the ith component in the rock, and $\rm K_{\rm i}$ and $\rm G_{\rm i}$ are the bulk and shear moduli of the ith component, respectively.

Subsequently, the dry moduli of the solid phase can be calculated for porosities above and below the critical porosity $\phi_{\rm c}$using a modified Hashin-Strikam upper and lower bound Dvorkin and Nur (1996); Ecker et al. (1996b).

Porosity $\phi$ below critical porosity $\phi_{\rm c}$:

For sediment porosities below the critical porosity, the dry moduli are determined by the following equations:

$$\begin{eqnarray} &\rm K_{\rm dry}& = {\left [ {{\phi/\phi_{\rm c}}\over{\rm K_{\... ...{\rm HM}}\over{\rm K_{\rm HM}\: + \: 2 \: \rm G_{\rm HM}}} \right)\end{eqnarray}$$ (11)

Porosity $\phi$ above critical porosity $\phi_{\rm c}$:

If the porosity is above the critical porosity of 40%, the dry moduli can be calculated as follows:

$$\begin{eqnarray} &\rm K_{\rm dry}& =\:{\left [ {{(1-\phi)/(1-\phi_{\rm c})}\over... ...i_{\rm c})/(1-\phi_{\rm c})}\over{\rm Z}} \right ]}^{-1} - \:\rm Z\end{eqnarray}$$ (12)