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Transport in Porous Media

, Volume 80, Issue 2, pp 373–387 | Cite as

Analytical Solution for Multi-Species Contaminant Transport Subject to Sequential First-Order Decay Reactions in Finite Media

  • Jesús S. Pérez Guerrero
  • Todd H. SkaggsEmail author
  • M. Th. van Genuchten
Article

Abstract

Transport equations governing the movement of multiple solutes undergoing sequential first-order decay reactions have relevance in analyzing a variety of subsurface contaminant transport problems. In this study, a one-dimensional analytical solution for multi-species transport is obtained for finite porous media and constant boundary conditions. The solution permits different retardation factors for the various species. The solution procedure involves a classic algebraic substitution that transforms the advection-dispersion partial differential equation for each species into an equation that is purely diffusive. The new system of partial differential equations is solved analytically using the Classic Integral Transform Technique (CITT). Results for a classic test case involving a three-species nitrification chain are shown to agree with previously reported literature values. Because the new solution was obtained for a finite domain, it should be especially useful for testing numerical solution procedures.

Keywords

Multi-species transport Finite domain Analytical solution Integral transform 

List of Symbols

b0j, b1j

Auxiliary coefficients

C0

Dimensional reference solute concentration

Cj

Dimensional solute concentration of the jth species

C0j

Dimensional inlet boundary concentration of the jth species

cj (x, t)

Dimensionless solute concentration of the jth species

c0j

Dimensionless inlet boundary concentration of the jth species

C1j, C2j

Auxiliary coefficients

D

Dispersion coefficient

Fj

Filter function

\({\bar{{f}}_{ji}}\)

Integral coefficient

Gj (X)

Dimensional initial concentration of the jth species

gj (x)

Dimensionless initial concentration of the jth species

H1, H2

Coefficients

i, j, k

Indices

L

Domain length

Ni

Norm

p

Constant used in algebraic substitution

Pe

Peclet number

qj

Constant used in algebraic substitution

Rj

Retardation coefficient for the jth species

\({\bar{{S}}_{ji} (t)}\)

Integral coefficient

T

Dimensional time

t

Dimensionless time

Tj (x, t)

Unknown function

U

Constant pore water velocity

wj

Auxiliary coefficient

X

Dimensional spatial coordinate

x

Dimensionless spatial coordinate

Greek Symbols

βi

Eigenvalue

γj

Damkholer number

δik

Kronecker delta

θj(x, t)

Unknown function in purely diffusive equation

\({\bar{{\theta}}_{ji} (t)}\)

Integral transform of the function θ j (x, t)

λj

First-order decay constant for the jth species

Λj

Coefficient

μi

Eigenvalue

τ

Eigenvalue

\({\phi_j (x)}\)

Auxiliary function

ψi(x)

Eigenfunction

\({\tilde{\psi}_i (x)}\)

Normalized eigenfunction

Ωj(x)

Auxiliary function

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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Jesús S. Pérez Guerrero
    • 1
  • Todd H. Skaggs
    • 2
    Email author
  • M. Th. van Genuchten
    • 3
  1. 1.Radioactive Waste DivisionBrazilian Nuclear Energy Commission (DIREJ/DRS/CNEN)Rio de JaneiroBrazil
  2. 2.U.S. Salinity LaboratoryUSDA-ARSRiversideUSA
  3. 3.Department of Mechanical Engineering, LTTC/COPPEFederal University of Rio de Janeiro, UFRJRio de JaneiroBrazil

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