Investigating Spreading Properties of the 4th Rank Gaussian Dispersivity Tensor Under Anisotropic Geometrical Symmetries

Abstract:

Expansion of the one-dimensional form of the advection-dispersion equation (ADE) to two- and three-dimensions gave rise to the 4^th rank Gaussian dispersivity tensor. The tensorial representation of dispersivity has a total of 81 terms (36 non-zero terms are possible) in 3D that can be used to describe the intrinsic Gaussian spreading characteristics of a dissolved contaminant plume within a porous medium. The most common symmetrical assumption is isotropic conditions which simplifies the 3D dispersivity tensor to a total of 12 non-zero terms that are dependent on values of longitudinal and transverse dispersivity defined from spatial and temporal plume data. However, natural geologic media rarely exhibit isotropic symmetrical conditions due to processes associated with sediment deposition and diagenesis, metamorphism, and tectonic deformation. In this study, we investigate spreading properties of the dispersivity tensor using crystallographic symmetries intended to capture anisotropy inherent in a porous medium. The symmetries investigated, listed in order of low to high symmetric complexity, include hexagonal, tetragonal, orthorhombic, monoclinic, and triclinic conditions which are represented in 3D by 12, 12, 12, 20, and 36 non-zero terms, respectively, that are not clearly defined like in the isotropic case. Symmetric complexity increases with the number of terms that need to be defined: the hexagonal case contains 6 independent terms used to determine the 12 non-zero terms in the tensor, whereas in the orthorhombic case all 12 terms are independent and must be defined. For the hexagonal and orthorhombic cases, the influence of each non-zero term on spreading rates along the three principal Gaussian growth directions is quantitatively defined and illustrated by examples of plume geometry. Representative geological settings are identified for several of the symmetries.