Comment on “Frequency-dependent dispersion in porous media”

In a recent paper, Valdès-Parada and Alvarez-Ramirez [Phys. Rev. E 84, 031201 (2011)] used the technique of volume averaging to derive a “frequency-dependent” dispersion tensor, ${\mathsf{\text{D}}}_{\gamma }^{*}$, the goal of which is to describe solute transport in porous media undergoing periodic processes. We describe two issues related to this dispersion tensor. First, we demonstrate that the definition of ${\mathsf{\text{D}}}_{\gamma }^{*}$ is erroneous and derive a corrected version, ${\mathsf{\text{D}}}_{\gamma }^{*c}$. With this modification, the approach of Valdès-Parada and Alvarez-Ramirez becomes strictly equivalent to the one devised by Moyne [Adv. Water Res. 20, 63 (1997)]. Second, we show that the term “frequency-dependent dispersion” is misleading because ${\mathsf{\text{D}}}_{\gamma }^{*}$ and ${\mathsf{\text{D}}}_{\gamma }^{*c}$ do not depend on the process operating frequency, $\chi$. The study carried out by Valdès-Parada and Alvarez-Ramirez represents a spectral analysis of the relaxation of ${\mathsf{\text{D}}}_{\gamma }^{*}$ towards its steady-state, independent of any periodic operation or excitation.

Figure 1

Logarithmic plots illustrating the temporal behavior of ${\mathsf{\text{D}}}_{\gamma ,xx}^{*c}/{\mathcal{D}}_{\gamma }$ [implicitly weighted by $u\left(\tau \right)$] for the unit cell presented in Ref. [6] and ${\varepsilon }_{\gamma }=0.56$. Results were computed using comsol multiphysics 4.2. In the short-time regime, ${\mathsf{\text{D}}}_{\gamma ,xx}^{*c}/{\mathcal{D}}_{\gamma }$ tends towards unity because we have assumed zero initial conditions for ${\mathbf{B}}_{\gamma }^{\prime }(\mathbf{r},t)$. In the long-time regime, ${\mathsf{\text{D}}}_{\gamma ,xx}^{*c}/{\mathcal{D}}_{\gamma }<1$ when tortuosity effects dominate dispersion effects and ${\mathsf{\text{D}}}_{\gamma ,xx}^{*c}/{\mathcal{D}}_{\gamma }>1$ when dispersion effects dominate tortuosity effects.