A mathematical and computational study of the dispersivity tensor in anisotropic porous media

Highlights

• Dispersion in anisotropic porous media.

• A constructive derivation of the fourth-order dispersivity tensor.

• Effects of fluid velocity and molecular diffusion on dispersivity.

Abstract

Dispersive transport in porous media is usually described through a Fickian model, in which the flux is the product of a dispersion tensor times the concentration gradient. This model is based on certain implicit assumptions, including slowly varying conditions. About fifty years ago, it was first suggested that the parameterization of the second-order dispersion tensor for anisotropic porous media involves a fourth-order dispersivity tensor. However, the properties of the dispersivity tensor have not been adequately studied. This work contributes to achieving a better grasp of dispersion in anisotropic porous media through a number of ways. First, with clearly stated assumptions and from first principles, we use the method of moments to derive a mathematical formula for the fourth-order dispersivity tensor, and show that it is a function of pore geometry, fluid velocity, and pore diffusion. Second, by using pore-scale flow and transport simulations through orderly and randomly packed 2-D and 3-D porous media, we evaluate the effects of the three factors on dispersivity. Different relationships with the Peclét number are observed for the longitudinal and transverse dispersivities and for orderly and randomly packed media. Third, we discuss the limitations of 2-D periodic media with simple structures in computing transverse dispersivity, which is more accurately predicted in the 3-D periodic media and 2-D randomly packed media. Fourth, we exhibit through numerical simulations that the method of moments can, computational limitations notwithstanding, be extended to stationary porous media.

Introduction

Dispersion in porous media is among the most significant topics in the study of solute transport. The classical Advection–Dispersion model assumes that dispersion follows Fick’s Law at the macroscopic level, such that dispersive flux is proportional to the concentration gradient [1]. The properties of dispersion coefficients, especially the longitudinal dispersion coefficient, have been studied extensively and at many scales. Representative literature includes: [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. At the Darcy or laboratory scale, the most widely applied parameterization of dispersion in porous media was proposed by Scheidegger [19]. In its simplest form, Scheidegger’s parameterization [19] states that the dispersion coefficient is the sum of a mechanical-dispersion term and a pore-diffusion term, and that the mechanical dispersion coefficient is a linear function of mean fluid velocity through the definition of dispersivity.

Even though most early work on dispersion focused on longitudinal dispersion, recent studies (e.g., [20], [21]) have revealed that transverse dispersion and the mixing it describes can be more important to chemical reactions and biodegradation of groundwater contaminants than longitudinal dispersion. In practical applications, the transverse dispersion coefficient is also assumed to be a linear function of mean fluid velocity following Scheidegger’s parameterization [1], [22]. A widely used estimation of transverse dispersivity is $3/16\hat{\ell }$ (where $\hat{\ell }$ is the pore length [L]) [23]. However, transverse dispersion may not necessarily follow the same properties as the longitudinal dispersion. For example, according to [7], field-scale transverse dispersivity diminishes in proportion to ${\mathit{Pe}}^{-1}$ instead of solely depending on pore geometry as generally applied.

Previous studies also hypothesized that the dispersivity of an anisotropic porous medium should be a higher-order tensor (see Section 2 for more details). The higher-order tensorial form of dispersivity has been subjected to very few studies. We are not aware of any study that provides a solid proof (starting with well-defined assumptions) to the tensorial form of dispersivity. Although the complexity of a higher-order tensor may impede its application in practical groundwater modeling, a comprehensive study will allow us to improve our understanding of the basic properties of dispersion and to develop better parameterizations, particularly since new upscaling theories and computational tools may allow us to revisit this important topic.

The objective of this paper is to revisit dispersion in anisotropic porous media and to explore the tensorial properties of dispersivity from a scientific standpoint. This study utilizes the method of moments [24], a rigorous approach based on a few well-defined assumptions. In this approach, large-scale heterogeneity is neglected through the discretization of the spectrum of medium properties, which is mathematically equivalent to using a periodic approximation of heterogeneity. We focus on dispersion at large time, when the system is at local equilibrium and changes in the system are gradual. Local equilibrium indicates a status that all the velocity fields, both the high and low velocities, are equally sampled by solute mass in a statistical sense (refer to [9] for more information of local equilibrium). Our mathematical analysis demonstrates that in an anisotropic medium with a mean flow that has an angle with the coordinates, the mechanical dispersion tensor ${\widehat{D}}_{\mathit{mn}}$ can be expressed through a fourth-order dispersivity tensor. A constructive derivation in Section 4 yields a mathematical formula of the dispersivity tensor in periodic porous media starting from first principles under well-defined assumptions. The formula indicates the dependency of the dispersivity on flow conditions as well as pore geometry. Furthermore, we design a numerical experiment to explicitly calculate the fourth-order dispersivity tensor based on pore-scale simulations of the flow and transport, and to evaluate the effects of fluid velocity and molecular diffusion. The results provide a comprehensive understanding of the physical basis and properties of the dispersivity tensor; to the best of our knowledge, this is the first of its kind. Finally, through the comparison of numerical results between 2-D and 3-D models, and between orderly and randomly packed porous media, we demonstrate the effect of pore geometry on dispersivity, providing insights for future pore-scale simulations of dispersion in porous media.