A framework for assessing the 4th rank dispersivity tensor under anisotropic axial symmetries

Abstract

Multidimensional expansion of the advection-dispersion equation necessitated the representation of dispersivity as a 4th rank tensor. This tensorial form of dispersivity has 81 terms in three dimensions with a maximum of 36 independent terms that may be used to describe Fickian spreading of a dissolved contaminant plume according to intrinsic properties of a porous medium. The complexity of the 4th rank tensor has led to the common practice of simplifying the tensor to only two or three independent terms by assuming isotropic conditions, although isotropic porous media are uncommon in nature as many natural geologic systems exhibit pronounced anisotropy. A broad set of crystallographic symmetries are investigated for application to the dispersivity tensor. Listed in order of high to low symmetry, these symmetries include isotropic, hexagonal, tetragonal, orthorhombic, monoclinic, and triclinic. A framework is developed for each of these symmetries to identify and quantify connections between individual dispersivity terms with principal values of the 2nd rank dispersion tensor. A numerical method allowing for visualization of resultant multi-Gaussian densities for any of these axial symmetries is also presented. Lastly, conservative particle transport in lattice networks is used to parameterize the full dispersivity tensor for hexagonal, tetragonal, and orthorhombic symmetries.

Résumé

L’expansion multidimensionnelle de l’équation d’advection-dispersion a nécessité la représentation de la dispersivité comme un tenseur de rang 4ème. Cette forme tensorielle de la dispersivité comporte 81 termes en trois dimensions avec un maximum de 36 termes indépendants qui peuvent être utilisés pour décrire la propagation selon la loi de Fick d’un panache de contaminants dissous selon les propriétés intrinsèques d’un milieu poreux. La complexité du tenseur de rang 4ème a conduit à la pratique courante de simplifier le tenseur à seulement deux ou trois termes indépendants en supposant des conditions isotropes, bien que les milieux poreux isotropes soient rares dans la nature car de nombreux systèmes géologiques naturels présentent une anisotropie prononcée. Un large éventail de symétries cristallographiques est. étudié pour l’application du tenseur de dispersivité. Classées par ordre de symétrie élevée à faible, ces symétries comprennent les symétries isotrope, hexagonale, tétragonale, orthorhombique, monoclinique et triclinique. Un cadre est. développé pour chacune de ces symétries afin d’identifier et de quantifier les connexions entre les termes de dispersivité individuels et les valeurs principales du tenseur de dispersion de second rang. Une méthode numérique permettant la visualisation des densités multi-Gaussiennes résultantes pour n’importe laquelle de ces symétries axiales est. également présentée. Enfin, le transport conservatif de particules dans les réseaux maillés est. utilisé pour paramétrer le tenseur de dispersivité complet pour les symétries hexagonale, tétragonale et orthorhombique.

Resumen

La expansión multidimensional de la ecuación advección-dispersión hizo necesaria la representación de la dispersividad como un tensor de 4° orden. Esta forma tensorial de la dispersividad tiene 81 términos en tres dimensiones con un máximo de 36 términos independientes que pueden utilizarse para describir la dispersión Fickiana de una pluma de contaminante disuelto según las propiedades intrínsecas de un medio poroso. La complejidad del tensor de 4° orden ha llevado a la práctica común de simplificar el tensor a sólo dos o tres términos independientes asumiendo condiciones isotrópicas, aunque los medios porosos isotrópicos son poco comunes en la naturaleza ya que muchos sistemas geológicos naturales muestran una pronunciada anisotropía. Se investiga un amplio conjunto de simetrías cristalográficas para su aplicación al tensor de dispersividad. Enumeradas de mayor a menor simetría, estas simetrías incluyen la isotrópica, hexagonal, tetragonal, ortorrómbica, monoclínica y triclínica. Se desarrolla un método para cada una de estas simetrías para identificar y cuantificar las conexiones entre los términos individuales de dispersividad con los valores principales del tensor de dispersión de 2° orden. También se presenta un método numérico que permite visualizar las densidades multigaussianas resultantes para cualquiera de estas simetrías axiales. Por último, se utiliza el transporte conservativo de partículas en redes reticulares para parametrizar el tensor de dispersividad completo para las simetrías hexagonal, tetragonal y ortorrómbica.

摘要

对流弥散方程的多维扩展需要将弥散度表示为四阶张量。根据多孔介质的内在特性，这种弥散度的张力在三个维度上具有81个项，最多36个独立项，可用于描述溶解的污染物羽流的Fickian扩散。四阶张量的复杂性导致了通过假设各向同性条件将张量简化为两个或三个独立术语的大量实践，尽管各向同性多孔介质在自然界并不常见，因为许多天然地质系统表现出明显的各向异性。 研究了广泛的晶体学对称性，以应用于弥散张量。这些对称性按高到低对称性的顺序列出，包括各向同性，六角形，四方形，正交形，单斜晶系和三斜晶系。 为每个对称性开发了一个框架，以识别和量化单个弥散量之间的连接，并使用二阶弥散张量的主值。还提出了考虑轴向对称性中的任何可视化产生的多高斯密度的数值方法。最后，晶格网络中的保守粒子传输用于参数化六角形，四方形和正交对称性的全部弥散张量。

Resumo

A expansão multidimensional da equação advecção-dispersão exigiu a representação da dispersividade como um tensor de 4ª ordem. Esta forma tensorial de dispersividade tem 81 termos em três dimensões com um máximo de 36 termos independentes que podem ser usados para descrever o espalhamento Fickiano de uma pluma contaminante dissolvida de acordo com as propriedades intrínsecas de um meio poroso. A complexidade do tensor de 4ª ordem levou à prática comum de simplificar o tensor para apenas dois ou três termos independentes, assumindo condições isotrópicas, embora meios porosos isotrópicos sejam incomuns na natureza, pois muitos sistemas geológicos naturais exibem anisotropia pronunciada. Um amplo conjunto de simetrias cristalográficas é investigado para aplicação ao tensor de dispersividade. Listadas em ordem de simetria alta a baixa, essas simetrias incluem isotrópica, hexagonal, tetragonal, ortorrômbica, monoclínica e triclínica. Uma estrutura é desenvolvida para cada uma dessas simetrias para identificar e quantificar conexões entre os termos de dispersividade individuais com valores principais do tensor de dispersão de 2ª ordem. Um método numérico que permite a visualização das densidades multigaussianas resultantes para qualquer uma dessas simetrias axiais também é apresentado. Por fim, o transporte conservativo de partículas em redes reticulares é usado para parametrizar o tensor de dispersividade total para simetrias hexagonais, tetragonais e ortorrômbicas.

Appendices

Appendix 1: Dispersion Values

For the 2D triclinic case,

$$\begin{array}{l}{\lambda }_{1}=\frac{1}{2}\left({a}_{11}{V}_{1}+{a}_{12}{V}_{1}+{a}_{13}{V}_{2}+{a}_{14}{V}_{2}+{a}_{21}{V}_{1}+{a}_{22}{V}_{1}+{a}_{23}{V}_{2}+{a}_{24}{V}_{2}\right)+\frac{1}{2}{(a}_{11}^{2}{V}_{1}^{2}+2{a}_{11}{a}_{12}{V}_{1}^{2}+\\ 2{a}_{11}{a}_{13}{V}_{1}{V}_{2}+2{a}_{11}{a}_{14}{V}_{1}{V}_{2}-2{a}_{11}{a}_{21}{V}_{1}^{2}-2{a}_{11}{a}_{22}{V}_{1}^{2}-2{a}_{11}{a}_{23}{V}_{1}{V}_{2}-2{a}_{11}{a}_{24}{V}_{1}{V}_{2}+2{a}_{12}^{2}{V}_{1}^{2}+\\ \begin{array}{l}2{a}_{12}{a}_{13}{V}_{1}{V}_{2}+2{a}_{12}{a}_{14}{V}_{1}{V}_{2}-2{a}_{12}{a}_{21}{V}_{1}^{2}-2{a}_{12}{a}_{22}{V}_{1}^{2}-2{a}_{12}{a}_{23}{V}_{1}{V}_{2}-2{a}_{12}{a}_{24}{V}_{1}{V}_{2}+{a}_{13}^{2}{V}_{2}^{2}+\\ 2{a}_{13}{a}_{14}{V}_{2}^{2}-2{a}_{13}{a}_{21}{V}_{1}{V}_{2}-2{a}_{13}{a}_{22}{V}_{1}{V}_{2}-2{a}_{13}{a}_{23}{V}_{1}^{2}-2{a}_{13}{a}_{24}{V}_{2}^{2}+{a}_{14}^{2}{V}_{2}^{2}-2{a}_{14}{a}_{21}{V}_{1}{V}_{2}-\\ \begin{array}{l}2{a}_{14}{a}_{22}{V}_{1}{V}_{2}-2{a}_{14}{a}_{23}{V}_{2}^{2}-2{a}_{14}{a}_{24}{V}_{2}^{2}+{a}_{21}^{2}{V}_{1}^{2}+2{a}_{21}{a}_{22}{V}_{1}^{2}+2{a}_{21}{a}_{23}{V}_{1}{V}_{2}+2{a}_{11}{a}_{24}{V}_{1}{V}_{2}+\\ {a}_{22}^{2}{V}_{1}^{2}+2{a}_{22}{a}_{23}{V}_{1}{V}_{2}+2{a}_{22}{a}_{24}{V}_{1}{V}_{2}+{a}_{23}^{2}{V}_{2}^{2}+2{a}_{23}{a}_{24}{V}_{2}^{2}+{a}_{24}^{2}{V}_{2}^{2}+4{a}_{31}{a}_{41}{V}_{1}^{2}+4{a}_{31}{a}_{42}{V}_{1}^{2}+\\ \begin{array}{l}4{a}_{31}{a}_{43}{V}_{1}{V}_{2}+4{a}_{31}{a}_{44}{V}_{1}{V}_{2}+4{a}_{32}{a}_{41}{V}_{1}^{2}+4{a}_{32}{a}_{42}{V}_{1}^{2}+4{a}_{32}{a}_{43}{V}_{1}{V}_{2}+4{a}_{32}{a}_{44}{V}_{1}{V}_{2}+\\ 4{a}_{33}{a}_{41}{V}_{1}{V}_{2}+4{a}_{33}{a}_{42}{V}_{1}{V}_{2}+4{a}_{33}{a}_{43}{V}_{2}^{2}+4{a}_{33}{a}_{44}{V}_{2}^{2}+4{a}_{34}{a}_{41}{V}_{1}{V}_{2}+4{a}_{34}{a}_{42}{V}_{1}{V}_{2}+\\ {4{a}_{34}{a}_{43}{V}_{2}^{2}+4{a}_{34}{a}_{44}{V}_{2}^{2})}^\frac{1}{2}\end{array}\end{array}\end{array}\end{array}$$

(63)

$$\begin{array}{l}{\lambda }_{2}= \frac{1}{2}\left({a}_{11}{V}_{1}+{a}_{12}{V}_{1}+{a}_{13}{V}_{2}+{a}_{14}{V}_{2}+{a}_{21}{V}_{1}+{a}_{22}{V}_{1}+{a}_{23}{V}_{2}+{a}_{24}{V}_{2}\right){-\frac{1}{2}(a}_{11}^{2}{V}_{1}^{2}+\\ 2{a}_{11}{a}_{12}{V}_{1}^{2}+2{a}_{11}{a}_{13}{V}_{1}{V}_{2}+2{a}_{11}{a}_{14}{V}_{1}{V}_{2}-2{a}_{11}{a}_{21}{V}_{1}^{2}-2{a}_{11}{a}_{22}{V}_{1}^{2}-2{a}_{11}{a}_{23}{V}_{1}{V}_{2}-\\ \begin{array}{l}2{a}_{11}{a}_{24}{V}_{1}{V}_{2}+2{a}_{12}^{2}{V}_{1}^{2}+2{a}_{12}{a}_{13}{V}_{1}{V}_{2}+2{a}_{12}{a}_{14}{V}_{1}{V}_{2}-2{a}_{12}{a}_{21}{V}_{1}^{2}-2{a}_{12}{a}_{22}{V}_{1}^{2}-2{a}_{12}{a}_{23}{V}_{1}{V}_{2}-\\ 2{a}_{12}{a}_{24}{V}_{1}{V}_{2}+{a}_{13}^{2}{V}_{2}^{2}+2{a}_{13}{a}_{14}{V}_{2}^{2}-2{a}_{13}{a}_{21}{V}_{1}{V}_{2}-2{a}_{13}{a}_{22}{V}_{1}{V}_{2}-2{a}_{13}{a}_{23}{V}_{1}^{2}-2{a}_{13}{a}_{24}{V}_{2}^{2}+\\ \begin{array}{l}{a}_{14}^{2}{V}_{2}^{2}-2{a}_{14}{a}_{21}{V}_{1}{V}_{2}-2{a}_{14}{a}_{22}{V}_{1}{V}_{2}-2{a}_{14}{a}_{23}{V}_{2}^{2}-2{a}_{14}{a}_{24}{V}_{2}^{2}+{a}_{21}^{2}{V}_{1}^{2}+2{a}_{21}{a}_{22}{V}_{1}^{2}+\\ 2{a}_{21}{a}_{23}{V}_{1}{V}_{2}+2{a}_{11}{a}_{24}{V}_{1}{V}_{2}+{a}_{22}^{2}{V}_{1}^{2}+2{a}_{22}{a}_{23}{V}_{1}{V}_{2}+2{a}_{22}{a}_{24}{V}_{1}{V}_{2}+{a}_{23}^{2}{V}_{2}^{2}+2{a}_{23}{a}_{24}{V}_{2}^{2}+\\ \begin{array}{l}{a}_{24}^{2}{V}_{2}^{2}+4{a}_{31}{a}_{41}{V}_{1}^{2}+4{a}_{31}{a}_{42}{V}_{1}^{2}+4{a}_{31}{a}_{43}{V}_{1}{V}_{2}+4{a}_{31}{a}_{44}{V}_{1}{V}_{2}+4{a}_{32}{a}_{41}{V}_{1}^{2}+4{a}_{32}{a}_{42}{V}_{1}^{2}+\\ 4{a}_{32}{a}_{43}{V}_{1}{V}_{2}+4{a}_{32}{a}_{44}{V}_{1}{V}_{2}+4{a}_{33}{a}_{41}{V}_{1}{V}_{2}+4{a}_{33}{a}_{42}{V}_{1}{V}_{2}+4{a}_{33}{a}_{43}{V}_{2}^{2}+4{a}_{33}{a}_{44}{V}_{2}^{2}+\\ {4{a}_{34}{a}_{41}{V}_{1}{V}_{2}+4{a}_{34}{a}_{42}{V}_{1}{V}_{2}+4{a}_{34}{a}_{43}{V}_{2}^{2}+4{a}_{34}{a}_{44}{V}_{2}^{2})}^\frac{1}{2}\end{array}\end{array}\end{array}\end{array}$$

(64)

For the 3D hexagonal case,

$$\begin{array}{l}{\lambda }_{1}=\frac{1}{3\sqrt[3]{2}}\{[{-2{a}^{3}+6{a}^{2}b+(4\left(-{a}^{2}+2ab-{b}^{2}-3{c}^{2}-3{d}^{2}-3{e}^{2}\right)}^{3}+(-2{a}^{3}+6{a}^{2}b-6a{b}^{2}+\\ {{18a{c}^{2}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde)}^{2})}^\frac{1}{2}-6a{b}^{2}+18a{c}^{2}-9a{d}^{2}-\\ \begin{array}{l}-{9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde]}^\frac{1}{3}\}-[\sqrt[3]{2}\left(-{a}^{2}+2ab-{b}^{2}-3{c}^{2}-3{d}^{2}-3{e}^{2}\right)]/\\ {\{3[-2{a}^{3}+6{a}^{2}b+(4\left(-{a}^{2}+2ab-{b}^{2}-3{c}^{2}-3{d}^{2}-3{e}^{2}\right)}^{3}+(-2{a}^{3}+6{a}^{2}b-6a{b}^{2}+18a{c}^{2}-\\ \begin{array}{l}{{-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde)}^{2})}^\frac{1}{2}-6a{b}^{2}-+18a{c}^{2}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}\\ {18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde]}^\frac{1}{3}\}+\frac{2a+b}{3}\end{array}\end{array}\end{array}$$

(65)

$$\begin{array}{l}{\lambda }_{2}=-\frac{1}{6\sqrt[3]{2}}\left(1-i\sqrt{3}\right)\{[-2{a}^{3}+6{a}^{2}b+(4{\left(-{a}^{2}+2ab-{b}^{2}-3{c}^{2}-3{d}^{2}-3{e}^{2}\right)}^{3}+\\ (-2{a}^{3}+6{a}^{2}b-6a{b}^{2}+18a{c}^{2}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde{)}^{2}{)}^\frac{1}{2}\\ \begin{array}{l}-{6a{b}^{2}+18a{c}^{2}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde]}^\frac{1}{3}\}+[\left(1+i\sqrt{3}\right)(-{a}^{2}+\\ 2ab-{b}^{2}-3{c}^{2}-3{d}^{2}-3{e}^{2})]/\{3\sqrt[3]{4}[-2{a}^{3}+6{a}^{2}b+(4{(-{a}^{2}+2ab-{b}^{2}-3{c}^{2}-3{d}^{2}-3{e}^{2})}^{3}+\\ \begin{array}{l}{\left({-2{a}^{3}+6{a}^{2}b-6a{b}^{2}+18a{c}^{2}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde)}^{2}\right)}^\frac{1}{2}-\\ {6a{b}^{2}+18a{c}^{2}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde]}^\frac{1}{3}\}+\frac{2a+b}{3}\end{array}\end{array}\end{array}$$

(66)

$$\begin{array}{l}{\lambda }_{3}=-\frac{1}{6\sqrt[3]{2}}(1+i\sqrt{3})\{[-2{a}^{3}+6{a}^{2}b+(4{\left(-{a}^{2}+2ab-{b}^{2}-3{c}^{2}-3{d}^{2}-3{e}^{2}\right)}^{3}+\\ {\left({-2{a}^{3}+6{a}^{2}b-6a{b}^{2}+18a{c}^{2}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde)}^{2}\right)}^\frac{1}{2}\\ \begin{array}{l}-{6a{b}^{2}+18a{c}^{2}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde]}^\frac{1}{3}\}+[(-{a}^{2}+\\ 2ab-{b}^{2}-3{c}^{2}-3{d}^{2}-3{e}^{2})]/\{3\sqrt[3]{4}[-2{a}^{3}+6{a}^{2}b+{(4\left(-{a}^{2}+2ab-{b}^{2}-3{c}^{2}-3{d}^{2}-3{e}^{2}\right)}^{3}+\\ \begin{array}{l}{\left(-2{a}^{3}+6{a}^{2}b-6a{b}^{2}+18a{c}^{2}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde\right)}^{2}{)}^\frac{1}{2}-\\ -{6a{b}^{2}+18a{c}^{2}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde]}^\frac{1}{3}\}+\frac{2a+b}{3}\end{array}\end{array}\end{array}$$

(67)

For the 3D tetragonal case,

$$\begin{array}{l}{\lambda }_{1}=\frac{1}{3\sqrt[3]{2}}{\{[-2{a}^{3}+6{a}^{2}b+(4\left(-{a}^{2}+2ab-{b}^{2}-3{c}^{2}-3{d}^{2}-3{e}^{2}\right)}^{3}+(-2{a}^{3}+6{a}^{2}b-6a{b}^{2}+\\ 18a{c}^{2}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde{)}^{2}{)}^\frac{1}{2}-6a{b}^{2}+18a{c}^{2}-9a{d}^{2}\\ \begin{array}{l}{9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde]}^\frac{1}{3}\}-[\sqrt[3]{2}\left(-{a}^{2}+2ab-{b}^{2}-3{c}^{2}-3{d}^{2}-3{e}^{2}\right)]/\\ \{3[-2{a}^{3}+6{a}^{2}b+{(4\left(-{a}^{2}+2ab-{b}^{2}-3{c}^{2}-3{d}^{2}-3{e}^{2}\right)}^{3}+(-2{a}^{3}+6{a}^{2}b-6a{b}^{2}+18a{c}^{2}-\\ \begin{array}{l}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde{)}^{2}{)}^\frac{1}{2}-6a{b}^{2}+18a{c}^{2}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-\\ {18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde]}^\frac{1}{3}\}+\frac{2a+b}{3}\end{array}\end{array}\end{array}$$

(68)

$$\begin{array}{l}{\lambda }_{2}=-\frac{1}{6\sqrt[3]{2}}(1-i\sqrt{3}){\{[-2{a}^{3}+6{a}^{2}b+(4\left(-{a}^{2}+2ab-{b}^{2}-3{c}^{2}-3{d}^{2}-3{e}^{2}\right)}^{3}+\\ {\left(-2{a}^{3}+6{a}^{2}b-6a{b}^{2}+18a{c}^{2}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde\right)}^{2}{)}^\frac{1}{2}-\\ \begin{array}{l}{6a{b}^{2}+18a{c}^{2}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde]}^\frac{1}{3}\}+[(1+i\sqrt{3})(-{a}^{2}+\\ {2ab-{b}^{2}-3{c}^{2}-3{d}^{2}-3{e}^{2})]/\{3\sqrt[3]{4}[-2{a}^{3}+6{a}^{2}b+(4\left(-{a}^{2}+2ab-{b}^{2}-3{c}^{2}-3{d}^{2}-3{e}^{2}\right)}^{3}+\\ \begin{array}{l}{{\left(-2{a}^{3}+6{a}^{2}b-6a{b}^{2}+18a{c}^{2}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde\right)}^{2})}^\frac{1}{2}-\\ {6a{b}^{2}+18a{c}^{2}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde]}^\frac{1}{3}\}+\frac{2a+b}{3}\end{array}\end{array}\end{array}$$

(69)

$$\begin{array}{l}{\lambda }_{3}=-\frac{1}{6\sqrt[3]{2}}(1+i\sqrt{3})\{[-2{a}^{3}+6{a}^{2}b+({\left(-{a}^{2}+2ab-{b}^{2}-3{c}^{2}-3{d}^{2}-3{e}^{2}\right)}^{3}+\\ {\left(-2{a}^{3}+6{a}^{2}b-6a{b}^{2}+18a{c}^{2}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde\right)}^{2}{)}^\frac{1}{2}-\\ \begin{array}{l}{6a{b}^{2}+18a{c}^{2}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde]}^\frac{1}{3}\}+[\left(1-i\sqrt{3}\right)(-{a}^{2}+\\ 2ab-{b}^{2}-3{c}^{2}-3{d}^{2}-3{e}^{2})]/\{3\sqrt[3]{4}[-2{a}^{3}+6{a}^{2}b+(4{\left(-{a}^{2}+2ab-{b}^{2}-3{c}^{2}-3{d}^{2}-3{e}^{2}\right)}^{3}+\\ \begin{array}{l}{{(-2{a}^{3}+6{a}^{2}b-6a{b}^{2}+18a{c}^{2}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde)}^{2})}^\frac{1}{2}-\\ {6a{b}^{2}+18a{c}^{2}-9a{d}^{2}-9a{e}^{2}+2{b}^{3}-18b{c}^{2}+9b{d}^{2}+9b{e}^{2}+54cde]}^\frac{1}{3}\}+\frac{2a+b}{3}\end{array}\end{array}\end{array}$$

(70)

For the 3D orthorhombic case,

$$\begin{array}{l}{\lambda }_{1}=\frac{1}{3\sqrt[3]{2}}\{[2{a}^{3}-3{a}^{2}b-3{a}^{2}c+(4{\left(-{a}^{2}+ab+ac-{b}^{2}+bc-{c}^{2}-3{d}^{2}-3{e}^{2}-3{f}^{2}\right)}^{3}+\\ (2{a}^{3}-3{a}^{2}b-3{a}^{2}c-3a{b}^{2}+12abc-3a{c}^{2}+9a{d}^{2}-18a{e}^{2}+9a{f}^{2}+2{b}^{3}-3{b}^{2}c-3b{c}^{2}+\\ \begin{array}{l}{{9b{d}^{2}+9b{e}^{2}-18b{f}^{2}+2{c}^{2}-18c{d}^{2}+9c{e}^{2}+9c{f}^{2}+54def)}^{2})}^\frac{1}{2}-6a{b}^{2}+9a{c}^{2}-9a{d}^{2}+2{b}^{3}-\\ {9b{c}^{2}+9b{d}^{2}+54def]}^\frac{1}{3}\}-[\sqrt[3]{2}(-{a}^{2}+2ab+ac-{b}^{2}+bc-{c}^{2}-3{d}^{2}-3{e}^{2}-3{f}^{2})]/\{3[2{a}^{3}-\\ \begin{array}{c}3{a}^{2}b-3{a}^{2}c+(4{\left(-{a}^{2}+ab+ac-{b}^{2}+bc-{c}^{2}-3{d}^{2}-3{e}^{2}-3{f}^{2}\right)}^{3}+(2{a}^{3}-3{a}^{2}b-3{a}^{2}c-\\ \begin{array}{l}3a{b}^{2}+12abc-3a{c}^{2}+9a{d}^{2}-18a{e}^{2}+9a{f}^{2}+2{b}^{3}-3{b}^{2}\mathrm{c}-3b{c}^{2}+9b{d}^{2}+9b{e}^{2}-18b{f}^{2}+\\ \begin{array}{l}{{2{c}^{2}-18c{d}^{2}+9c{e}^{2}+9c{f}^{2}+54def)}^{2})}^\frac{1}{2}-6a{b}^{2}+9a{c}^{2}-9a{d}^{2}+2{b}^{3}-9b{c}^{2}+9b{d}^{2}+\\ 54\mathrm{def}{]}^\frac{1}{3}\}+\frac{a+b+c}{3}\end{array}\end{array}\end{array}\end{array}\end{array}$$

(71)

$$\begin{array}{l}{\lambda }_{2}=-\frac{1}{6\sqrt[3]{2}}(1-i\sqrt{3})\{[2{a}^{3}-3{a}^{2}b-3{a}^{2}c+(4(-{a}^{2}+ab+ac-{b}^{2}+bc-{c}^{2}-3{d}^{2}-3{e}^{2}-\\ {3{f}^{2})}^{3}+(2{a}^{3}-3{a}^{2}b-3{a}^{2}c-3a{b}^{2}+12abc-3a{c}^{2}+9a{d}^{2}-18a{e}^{2}+9a{f}^{2}+2{b}^{3}-3{b}^{2}c-\\ \begin{array}{l}{{3b{c}^{2}+9b{d}^{2}+9b{e}^{2}-18b{f}^{2}+2{c}^{2}-18c{d}^{2}+9c{e}^{2}+9c{f}^{2}+54def)}^{2})}^\frac{1}{2}-6a{b}^{2}+9a{c}^{2}-\\ {9a{d}^{2}+2{b}^{3}-9b{c}^{2}+9b{d}^{2}+54def]}^\frac{1}{3}\}+[(1+i\sqrt{3})(-{a}^{2}+2ab+ac-{b}^{2}+bc-{c}^{2}-3{d}^{2}-\\ \begin{array}{l}3{e}^{2}-3{f}^{2})]/\{3\sqrt[3]{4}[2{a}^{3}-3{a}^{2}b-3{a}^{2}c+(4(-{a}^{2}+a\mathrm{b}+ac-{b}^{2}+bc-{c}^{2}-3{d}^{2}-3{e}^{2}\\ {3{f}^{2})}^{3}+(2{a}^{3}-3{a}^{2}b-3{a}^{2}c-3a{b}^{2}+12abc-3a{c}^{2}+9a{d}^{2}-18a{e}^{2}+9a{f}^{2}+2{b}^{3}-3{b}^{2}c-\\ \begin{array}{l}{{3b{c}^{2}+9b{d}^{2}+9b{e}^{2}-18b{f}^{2}+2{c}^{2}-18c{d}^{2}+9c{e}^{2}+9c{f}^{2}+54def)}^{2})}^\frac{1}{2}-6a{b}^{2}+9a{c}^{2}-\\ {9a{d}^{2}+2{b}^{3}-9b{c}^{2}+9b{d}^{2}+54def]}^\frac{1}{3}\}+\frac{a+b+c}{3}\end{array}\end{array}\end{array}\end{array}$$

(72)

$$\begin{array}{l}{\lambda }_{3}=-\frac{1}{6\sqrt[3]{2}}(1+i\sqrt{3})\{[2{a}^{3}-3{a}^{2}b-3{a}^{2}c+(4(-{a}^{2}+ab+ac-{b}^{2}+bc-{c}^{2}-3{d}^{2}-3{e}^{2}-\\ {3{f}^{2})}^{3}+(2{a}^{3}-3{a}^{2}b-3{a}^{2}c-3a{b}^{2}+12abc-3a{c}^{2}+9a{d}^{2}-18a{e}^{2}+9a{f}^{2}+2{b}^{3}-3{b}^{2}c-\\ \begin{array}{l}{{3b{c}^{2}+9b{d}^{2}+9b{e}^{2}-18b{f}^{2}+2{c}^{2}-18c{d}^{2}+9c{e}^{2}+9c{f}^{2}+54def)}^{2})}^\frac{1}{2}-6a{b}^{2}+9a{c}^{2}-\\ {9a{d}^{2}+2{b}^{3}-9b{c}^{2}+9b{d}^{2}+54def]}^\frac{1}{3}\}+[(1-i\sqrt{3})(-{a}^{2}+2ab+ac-{b}^{2}+bc-{c}^{2}-3{d}^{2}-\\ \begin{array}{l}3{e}^{2}-3{f}^{2})]/\{3\sqrt[3]{4}[2{a}^{3}-3{a}^{2}b-3{a}^{2}c+(4(-{a}^{2}+ab+ac-{b}^{2}+bc-{c}^{2}-3{d}^{2}-3{e}^{2}-\\ {3{f}^{2})}^{3}+(2{a}^{3}-3{a}^{2}b-3{a}^{2}c-3a{b}^{2}+12abc-3a{c}^{2}+9a{d}^{2}-18a{e}^{2}+9a{f}^{2}+2{b}^{3}-3{b}^{2}c-\\ \begin{array}{l}3b{c}^{2}+9b{d}^{2}+9b{e}^{2}-18b{f}^{2}+2{c}^{2}-18c{d}^{2}+9c{e}^{2}+9c{f}^{2}+54def{)}^{2}{)}^\frac{1}{2}-6a{b}^{2}+9a{c}^{2}-\\ {9a{d}^{2}+2{b}^{3}-9b{c}^{2}+9b{d}^{2}+54def]}^\frac{1}{3}\}+\frac{a+b+c}{3}\end{array}\end{array}\end{array}\end{array}$$

(73)

Appendix 2: Validation Process

Following the validation process in section ‘Expressions and validations’, the first step involves calculating D_ij = ${a}_{ijkl}\frac{V_k{V}_l}{v}$ directly. Mark the result as D_ij1.

Define a 2D triclinic dispersivity tensor in accordance with the Eq. (26) matrix:

$${a}_{ijkl}=\left|\begin{array}{cccc}13& 14& 15& 16\\ {}9& 10& 11& 12\\ {}5& 6& 7& 8\\ {}1& 2& 3& 4\end{array}\right|$$

(74)

And the velocity tensor:

$${V}_k{V}_l=\left|\begin{array}{c}1\\ {}\begin{array}{c}1\\ {}1\\ {}1\end{array}\end{array}\right|$$

(75)

$$\Rightarrow {D}_{ij1}=\left|\begin{array}{cc}58& 26\\ {}10& 42\end{array}\right|$$

(76)

Rewriting to:

$${D}_{ij1}=\left|\begin{array}{cc}68& 0\\ {}0& 32\end{array}\right|$$

(77)

The second step involves calculating D_ij from the derived analytical expressions—the matrices of Eqs. (63) and (64). Mark the result as D_ij2.

a ₁₁ = 16, a₁₂ = 15, a₁₃ = 14, a₁₄ = 13, a₂₁ = 12, a₂₂ = 11, a₂₃ = 10, a₂₄ = 9, a₃₁ = 8, a₃₂ = 7, a₃₃ = 6, a₃₄ = 5, a₄₁ = 4, a₄₂ = 3, a₄₃ = 2, a₄₄ = 1, v_k = 1, v_l = 1 (the matrices of Eqs. 74 and 75).

Put a_ijkl, v_k, v_l values back into D₁₁ and D₂₂:

$${\displaystyle \begin{array}{c}{D}_{11}=\left\lceil 50+\frac{\sqrt{1296}}{2}\right\rceil =68\\ {}{D}_{22}=\left\lceil 50-\frac{\sqrt{1296}}{2}\right\rceil =32\end{array}}$$

(78)

$$\Rightarrow {D}_{ij2}=\left|\begin{array}{cc}68& 0\\ {}0& 32\end{array}\right|$$

(79)

Lastly, subtract D_ij1 from D_ij2:

$${D}_{ij1}-{D}_{ij2}=\left|\begin{array}{cc}68-68& 0-0\\ {}0-0& 32-32\end{array}\right|=\left|\begin{array}{cc}0& 0\\ {}0& 0\end{array}\right|$$

(80)

Since it equals 0, the calculation is correct.

Run another round of tests.

Define a 2D triclinic dispersivity tensor in accordance with the Eq. (26) matrix:

$${a}_{ijkl}=\left|\begin{array}{cccc}23& 5& 25& 26\\ {}19& 20& 31& 22\\ {}12& 15& 17& 9\\ {}3& 1& 7& 6\end{array}\right|$$

(81)

Velocity tensor:

$${V}_k{V}_l=\left|\begin{array}{c}2\\ {}\begin{array}{c}2\\ {}3\\ {}3\end{array}\end{array}\right|$$

(82)

$$\Rightarrow {D}_{ij1}=\left|\begin{array}{cc}303& 0\\ {}0& 143\end{array}\right|$$

(83)

Calculate D_ij2: a₁₁ = 23, a₁₂ = 5, a₁₃ = 25, a₁₄ = 26, a₂₁ = 19, a₂₂ = 20, a₂₃ = 31, a₂₄ = 22, a₃₁ = 12, a₃₂ = 15 a₃₃ = 17, a₃₄ = 9, a₄₁ = 3, a₄₂ = 1, a₄₃ = 7, a₄₄ = 6, v_k = 2, v_l = 3 (the matrices of Eqs. 81 and 82). The result is:

$${\displaystyle \begin{array}{c}{D}_{11}=223+\frac{\sqrt{25600}}{2}=303\\ {}{D}_{22}=223-\frac{\sqrt{25600}}{2}=143\end{array}}$$

(84)

$$\Rightarrow {D}_{ij2}=\left|\begin{array}{cc}303& 0\\ {}0& 143\end{array}\right|$$

(85)

Subtract D_ij1 from D_ij2:

$${D}_{ij1}-{D}_{ij2}=\left|\begin{array}{cc}0& 0\\ {}0& 0\end{array}\right|$$

(86)

The result equals 0.