Continuous time random walk model with advection and diffusion as two distinct dynamical origins

Abstract

Modeling the solute transport in geological porous media is of both theoretical interest and practical importance. Of several approaches, the continuous time random walk method is a most successful one that can be used to quantitatively predict the statistical features of the process, which are ubiquitously anomalous in the case of high Péclet numbers and normal in the case of low Péclet numbers. It establishes a quantitative relation between the spatial moment of an ensemble of solute particles and the waiting time distribution in the model. However, despite its success, the classical version of this model is a " static” one in the sense that there is no tuning parameter in the waiting time distribution that can reflect the relative strength of advection and diffusion which are two mechanisms that underlie the transport process, hence it cannot be used to show the transition from anomalous to normal transport as the Péclet numbers decreases. In this work, a new continuous time random walk model is established by taking into account these two different origins of solute particle transport in a geological porous medium. In particular, solute transitions due to advection and diffusion are separately treated by using a mixture probability model for the particle’s waiting time distribution, which contains two terms representing the effects of advection and diffusion, respectively. By varying the weights of these two terms, two limiting cases can be obtained, i.e. the advection-dominated transport and the diffusion-dominated transport. The values of scaling exponent β of the mean square displacement versus time, <inline-formula><tex-math id="M1">\begin{document}${\left( {\Delta {x} } \right)^2} \sim {t^{\rm{\beta }}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20190088_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20190088_M1.png"/></alternatives></inline-formula>, are derived for both cases by using our model, which are consistent with previous results. In the advection dominant case with the Péclet number going to infinity, the scaling exponent β is found to be equal to <inline-formula><tex-math id="M2">\begin{document}$3 - {\rm{\alpha }}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20190088_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20190088_M2.png"/></alternatives></inline-formula> where <inline-formula><tex-math id="M3">\begin{document}${\rm{\alpha }} \in \left( {1,2} \right)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20190088_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20190088_M3.png"/></alternatives></inline-formula> is the anomaly exponent in the advection-originated part of the waiting time distribution that <inline-formula><tex-math id="M4">\begin{document}${{\rm{\omega }}_1}\left( {t} \right) \sim {{t}^{ - 1 - {\rm{\alpha }}}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20190088_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20190088_M4.png"/></alternatives></inline-formula>. As the Péclet number decreases, the diffusion-originated part of the waiting time distribution begins to have a stronger influence on the transport process and in the limit of the Péclet number going to 0 we observe a gradual transition of β from <inline-formula><tex-math id="M5">\begin{document}$3 - {\rm{\alpha }}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20190088_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20190088_M5.png"/></alternatives></inline-formula> to 1, indicating that the underlying transport process changes from anomalous to normal transport. By incorporating advection and diffusion as two mechanisms giving rise to solute transport in the continuous time random walk model, we successfully capture the qualitative transition of the transport process as the Péclet number is varied, which is, however, elusive from the classical continuous time random walk model. Also established are the corresponding macroscopic transport equations for both anomalous and normal transport, which are consistent with previous findings as well. Our model hence fully describes the transition from normal to anomalous transport in a porous medium as the Péclet number increases in a qualitative and semi-quantitative way.