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Transport in Porous Media

, Volume 7, Issue 2, pp 163–185 | Cite as

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

  • Constantinos V. Chrysikopoulos
  • Peter K. Kitanidis
  • Paul V. Roberts
Article

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/L3

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

Dij

hydrodynamic dispersion coefficient,L2/t

D

dispersion coefficient tensor

Eij

constant

F

arbitrary global or local function

Hij

function of local coordinates

j

imaginary number unit:j=√−1

kd,kd0

dimensionless partition or distribution coefficients

kf

forward sorption rate coefficient,t−1

kr

reverse sorption rate coefficient,t−1

Kd

partition of distribution coefficient (liquid volume/solids mass),L3/M

li

characteristic linear dimension of a unit element,L

li

basic vectors which define a unit element

mm

liquid-phase local moments

Mm

continuous and discrete representation of liquid-phase global moments

ns

outer unit vector normal to∂V0

o

origin of a local coordinate system

O

order of magnitude, origin of global coordinate system

pm

solid-phase local moments

Pm

continuous and discrete representation of solid-phase global moments

qi

local Cartesian coordinates,L

q

local position vector within a unit element

qi

interface of a unit element

d3q

differential volume within a unit element

Qi

global Cartesian coordinates,L

Q

discrete position vector of a general point

Qn

discrete position vector locating the origin of the nth unit element

R,R0

retardation factors defined in Equations (8) and (85), respectively

s±i

faces of the unit element

ds

infinitesimal area on∂V0

S*

solid-phase or sorbed solute concentration (solute mass/liquid volume),M/L3

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{S} _{\text{n}}^*\)

total solid-phase solute mass within then unit element

t

time,t

U

average interstitial velocity,L/t

U

velocity vector

V0

domain of a unit element

∂Vo

external surface of a unit element

W

mass of solute injected,M

Zij

function of local coordinates

г

constant

δ()

Dirac delta function

δij

Kronecker delta

θ

porosity (liquid volume/aquifer volume),L3/L3

λ

spectrum coefficient

Μ

spectrum coefficients,L

ρ

bulk density of the solid matrix (solids mass/aquifer volume),M/L3

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

nth 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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Copyright information

© Kluwer Academic Publishers 1992

Authors and Affiliations

  • Constantinos V. Chrysikopoulos
    • 1
  • Peter K. Kitanidis
    • 1
  • Paul V. Roberts
    • 1
  1. 1.Department of Civil EngineeringStanford UniversityStanfordUSA

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