PNG 550
Reactive Transport in the Subsurface

6.4 Dispersion

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Dispersion describes the mixing of a solute due to fluctuations around the average velocity. This is caused by three factors: 1) microscopic heterogeneity, which make the fluid moves faster at the center of the pore and slower at the water grain boundary; 2) variations in pore sizes, in which cases fluid particles will move through larger pores faster; 3) variations in path length, causing some fluid particles going longer paths than others.

Mechanical Dispersion

Mechanical dispersion is a result of variations in flow velocities. Dispersion coefficients in porous media is typically defined as the product of the average fluid velocity and dispersivity $\alpha$:

\begin{equation}\mathbf{D}_{m}=\boldsymbol{\alpha} \cdot \mathbf{u}\end{equation}

where u is the average flow velocity (m/s) and $\alpha$ refers to the dispersivity (m). In systems with more than one direction, the longitudinal dispersivity DL in the principle flow direction is typically higher than DT in the direction transverse to the main flow.

Dispersion is a scale-dependent process with larger dispersivity values observed at larger spatial scales. At the column scale, a typical dispersivity may be on the order of centimeters. At the field scale, apparent dispersivities can vary from 10 to 100 m, as shown in Figure 2.

Longitudinal dispersivity versus scale of observation
Figure 2. Longitudinal dispersivity versus scale of observation 
Gelhar, 1986

Hydrodynamic dispersion

The spreading of the solute mass as a result of diffusion and dispersion is similar to diffusion. This has led to the use of Fick’s First Law to describe the dispersion process as follows:

\begin{equation}\mathbf{J}_{d i s p}=-\mathbf{D}_{h} \frac{\partial C}{\partial x}\end{equation}

where Dh is the hydrodynamic dispersion coefficient defined as the sum of effective diffusion coefficient De and mechanical dispersion coefficient Dm:

\begin{equation}D_{h}=D_{e}+D_{m}=\phi^{m} D_{0}+D_{m}\end{equation}

As such, the hydrodynamic dispersion includes both diffusion and mechanical dispersion processes.