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


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.


Multi-species transport Finite domain Analytical solution Integral transform 

List of Symbols

b0j, b1j

Auxiliary coefficients


Dimensional reference solute concentration


Dimensional solute concentration of the jth species


Dimensional inlet boundary concentration of the jth species

cj (x, t)

Dimensionless solute concentration of the jth species


Dimensionless inlet boundary concentration of the jth species

C1j, C2j

Auxiliary coefficients


Dispersion coefficient


Filter function


Integral coefficient

Gj (X)

Dimensional initial concentration of the jth species

gj (x)

Dimensionless initial concentration of the jth species

H1, H2


i, j, k



Domain length




Constant used in algebraic substitution


Peclet number


Constant used in algebraic substitution


Retardation coefficient for the jth species

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

Integral coefficient


Dimensional time


Dimensionless time

Tj (x, t)

Unknown function


Constant pore water velocity


Auxiliary coefficient


Dimensional spatial coordinate


Dimensionless spatial coordinate

Greek Symbols




Damkholer number


Kronecker delta

θj(x, t)

Unknown function in purely diffusive equation

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

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


First-order decay constant for the jth species







\({\phi_j (x)}\)

Auxiliary function



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

Normalized eigenfunction


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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