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

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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.

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Abbreviations

b0j , b1j :

Auxiliary coefficients

C 0 :

Dimensional reference solute concentration

C j :

Dimensional solute concentration of the jth species

C 0j :

Dimensional inlet boundary concentration of the jth species

c j (x, t):

Dimensionless solute concentration of the jth species

c 0j :

Dimensionless inlet boundary concentration of the jth species

C1j , C2j :

Auxiliary coefficients

D :

Dispersion coefficient

F j :

Filter function

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

Integral coefficient

G j (X):

Dimensional initial concentration of the jth species

g j (x):

Dimensionless initial concentration of the jth species

H1, H2:

Coefficients

i, j, k:

Indices

L :

Domain length

N i :

Norm

p :

Constant used in algebraic substitution

Pe :

Peclet number

q j :

Constant used in algebraic substitution

R j :

Retardation coefficient for the jth species

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

Integral coefficient

T :

Dimensional time

t :

Dimensionless time

T j (x, t):

Unknown function

U :

Constant pore water velocity

w j :

Auxiliary coefficient

X :

Dimensional spatial coordinate

x :

Dimensionless spatial coordinate

β i :

Eigenvalue

γ j :

Damkholer number

δ ik :

Kronecker delta

θ j (x, t):

Unknown function in purely diffusive equation

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

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

λ j :

First-order decay constant for the jth species

Λ j :

Coefficient

μ i :

Eigenvalue

τ :

Eigenvalue

\({\phi_j (x)}\) :

Auxiliary function

ψ i (x):

Eigenfunction

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

Normalized eigenfunction

Ω j (x):

Auxiliary function

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Authors and Affiliations

  1. Radioactive Waste Division, Brazilian Nuclear Energy Commission (DIREJ/DRS/CNEN), Rio de Janeiro, Brazil

    Jesús S. Pérez Guerrero

  2. U.S. Salinity Laboratory, USDA-ARS, Riverside, CA, USA

    Todd H. Skaggs

  3. Department of Mechanical Engineering, LTTC/COPPE, Federal University of Rio de Janeiro, UFRJ, Rio de Janeiro, Brazil

    M. Th. van Genuchten

Authors
  1. Jesús S. Pérez Guerrero
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  2. Todd H. Skaggs
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  3. M. Th. van Genuchten
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Correspondence to Todd H. Skaggs.

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Pérez Guerrero, J.S., Skaggs, T.H. & van Genuchten, M.T. Analytical Solution for Multi-Species Contaminant Transport Subject to Sequential First-Order Decay Reactions in Finite Media. Transp Porous Med 80, 373–387 (2009). https://doi.org/10.1007/s11242-009-9368-3

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