The diffusion and telegraph equations in diagenetic modelling

Abstract

This paper considers the practical aspects of differentiating between the solutions of the Diffusion and Telegraph Equations when they are used to model molecular diffusion in sediment porewaters and the diffusive bioturbation of solids. If molecular diffusion is the only transport mechanism in porewater, then the results from a simple random-walk model coupled to the hydrodynamic or kinetic theories of diffusion indicate that the solute profiles predicted by these two equations differ measurably only for periods up to 10 −10 s after the introduction of a transient and for spatial scales less than 10 −6 cm. In addition, the distinct propagating front predicted by the Telegraph Equation moves so fast and is so attenuated as to be unmeasurable. In this situation, the Telegraph Equation offers no practical advantage over the Diffusion Equation for the description of diagenetic porewater profiles. These findings also hold when advection due to burial and chemical reaction are included in the model. The time scales associated with bioturbation of solids are sufficiently long compared to normal sampling times that the profiles of some transients, both in deep-sea and near-shore sediments, should exhibit behavior characteristic of the Telegraph Equation if mixing is diffusive (local).