# Generalized Taylor-Aris moment analysis of the transport of sorbing solutes through porous media with spatially-periodic retardation factor

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## Abstract

Taylor-Aris dispersion theory, as generalized by Brenner, is employed to investigate the macroscopic behavior of sorbing solute transport in a three-dimensional, hydraulically homogeneous porous medium under steady, unidirectional flow. The porous medium is considered to possess spatially periodic geochemical characteristics in all three directions, where the spatial periods define a rectangular parallelepiped or a unit-element. The spatially-variable geochemical parameters of the solid matrix are incorporated into the transport equation by a spatially-periodic distribution coefficient and consequently a spatially-periodic retardation factor. Expressions for the effective or large-time coefficients governing the macroscopic solute transport are derived for solute sorbing according to a linear equilibrium isotherm as well as for the case of a first-order kinetic sorption relationship. The results indicate that for the case of a chemical equilibrium sorption isotherm the longitudinal macrodispersion incorporates a second term that accounts for the eflect of averaging the distribution coefficient over the volume of a unit element. Furthermore, for the case of a kinetic sorption relation, the longitudinal macrodispersion expression includes a third term that accounts for the effect of the first-order sorption rate. Therefore, increased solute spreading is expected if the local chemical equilibrium assumption is not valid. The derived expressions of the apparent parameters governing the macroscopic solute transport under local equilibrium conditions agreed reasonably with the results of numerical computations using particle tracking techniques.

## Key words

Moment analysis sorbing solute transport spatially-periodic retardation## Nomenclature

*A*amplitude of oscillation of the retardation factor

**b**vector of wavenumbers:

**b**=(*b*_{ x },*b*_{ y },*b*_{ z })^{ T }- \(\begin{gathered} ` \hfill \\ b \hfill \\ \end{gathered}\)
normalized vector:\(b = (b_x /l_x , b_y /l_y , b_z /l_z )_{}^T\)

*C*liquid-phase solute concentration (solute mass/liquid volume),

*M*/*L*^{3}*C*^{*}solid-phase or sorbed solute concentration (solute mass/solids mass),

*M/M*- \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C} _n\)
total liquid-phase solute mass within the nth unit element,

*M**D*_{ij}hydrodynamic dispersion coefficient,

*L*^{2}/*t***D**dispersion coefficient tensor

*E*_{ij}constant

- F
arbitrary global or local function

*H*_{ij}function of local coordinates

*j*imaginary number unit:

*j*=√−1*k*_{d},*k*_{d}^{0}dimensionless partition or distribution coefficients

*k*_{f}forward sorption rate coefficient,

*t*^{−1}*k*_{r}reverse sorption rate coefficient,

*t*^{−1}*K*_{d}partition of distribution coefficient (liquid volume/solids mass),

*L*^{3}/*M**l*_{i}characteristic linear dimension of a unit element,

*L***l**_{i}basic vectors which define a unit element

**m**_{m}liquid-phase local moments

**M**_{m}continuous and discrete representation of liquid-phase global moments

**n**_{s}outer unit vector normal to

*∂V*_{0}*o*origin of a local coordinate system

*O*order of magnitude, origin of global coordinate system

**p**_{m}solid-phase local moments

**P**_{m}continuous and discrete representation of solid-phase global moments

*q*_{i}local Cartesian coordinates,

*L***q**local position vector within a unit element

*∂***q**_{i}interface of a unit element

*d*^{3}**q**differential volume within a unit element

*Q*_{i}global Cartesian coordinates,

*L***Q**discrete position vector of a general point

**Q**_{n}discrete position vector locating the origin of the nth unit element

*R*,*R*^{0}retardation factors defined in Equations (8) and (85), respectively

*s*_{±i}faces of the unit element

*ds*infinitesimal area on

*∂V*_{0}*S*^{*}solid-phase or sorbed solute concentration (solute mass/liquid volume),

*M*/*L*^{3}- \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{S} _{\text{n}}^*\)
total solid-phase solute mass within the

*n*unit element*t*time,

*t**U*average interstitial velocity,

*L/t*- U
velocity vector

*V*_{0}domain of a unit element

*∂V*_{o}external surface of a unit element

*W*mass of solute injected,

*M**Z*_{ij}function of local coordinates

*г*constant

*δ*()Dirac delta function

*δ*_{ij}Kronecker delta

- θ
porosity (liquid volume/aquifer volume),

*L*^{3}/*L*^{3}*λ*spectrum coefficient

*Μ*spectrum coefficients,

*L**ρ*bulk density of the solid matrix (solids mass/aquifer volume),

*M*/*L*^{3}- ⌆
summation

- Φ
function of local coordinates

**0**null vector

- ∃
an element of

- ∇
_{q} vector operator (del): ∇

_{q}=[∂/∂*q*_{ x },∂/∂*q*_{ y },∂/∂*q*_{ z }]^{ T }- \(\mathop {{\text{def}}}\limits_ =\)
equals by definition

- ∀
for all

- ∥
magnitude of a vector, Euclidean norm

- 〚 〛
jump in the value of a function across equivalent points on opposite faces of a unit element

## Subscripts

*i, j*direction of principal axes:

*i, j = x, y, z**m, n*integer summation indicies

**n****n**th unit element: {**n**} = {*n*_{ x },*n*_{ y },*n*_{ z }}*x, y, z*principal directions of a Cartesian coordinate system

## Superscripts

*T*transpose

- *
indicates the solid-phase

- ◊
effective global coefficient

- ⊲
indicates the value of a function minus its average over the volume of a unit element

- ‡
complex conjugate

- •
local equilibrium sorption

- • •
first-order reversible kinetic sorption (overbar) average over the volume of a unit element

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