*Oceanography 540--Marine Geological Processes--Winter Quarter 2000*
### Transport Equations for Solutes in Pore Fluids

The diffusive flux of a solute is defined by Fick's Law, a empirical expression
relating the flux of material across a plane to the gradient of concentration
perpendicular to that plane:
(1)

where

*F* is the diffusive flux of component i (M L T)
*c* is the concentration of component i (M L)
*D* is the diffusion coefficient of component i (L T)
- x is position (L)

The sediment-water interface is usually adopted as the datum for the
coordinate system, and so position x=0 moves with time. This imparts a
pseudo-advection of fluid at the sedimentation rate, s. In addition,
there will be upward movement of fluid relative to the solid matrix due
to compaction. However in the deep sea, these terms are much smaller
than the diffusive flux and can be neglected.
The diffusion coefficient of a solute depends on the solute, the temperature
and the ionic environment. Some typical values of the tracer (i.e., no
ion-ion interactions) at 5°C:

The diffusion coefficient is sensitive to temperature, approximately doubling
from 0 to 25°C. Within a porous medium, the additional path length to travel
around particles causes the diffusion coefficient to decrease. This excess
path length is expressed as a tortuosity:

(2)

where dl is the mean length of the path followed to travel a linear distance
dx. The sediment diffusion coefficient, Ds, is then:

(3)

There are a number of empirical laws relating tortuosity to porosity. A
reasonable approximation for deep-sea sediments is to take

(4)

The mass balance within a small volume reflects the fluxes in and out of that
volume and the internal change of the pore water concentration due to chemical
reaction, J:

(5)

(6)

For more information consult reference **(21)**.

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