Effects of hydrodynamic dispersion on plume lengths for instantaneous bimolecular reactions

In this article a new two-dimensional analytical model is presented, which explores the effects of longitudinal (α_L) and transverse (α_T) dispersivity on the transient and steady state length of a plume undergoing an instantaneous bimolecular reaction. A solute Bis injected into a steady, uniform flow field in a homogeneous porous medium initially filled with a solute A. Mixing of the solutes occurs only through longitudinal and transversal dispersion and reaction between the solutes is assumed to occur where both solutes are simultaneously present. Explicit steady state solutions (in the limit t → ∞) are presented, which describe the distributions of reactants and products in x-y space in the form of the modified Bessel function of zero order and second kind. When α_T ≤ α_L ≤ 10 α_T, i.e., limits typically used in practical engineering scenarios, the steady state length of a contaminant plume is obtained as a simple algebraic expression including α_T; the influence of α_L upon plume length being negligible. This phenomenon has been observed experimentally and in numerical simulations. The analysis provided in this paper proves that this behavior follows directly from the governing equations. In addition, the effects of hydrodynamic dispersion on the transient plume development are investigated. The assumption that α_L is only important for the transient development of the plume is verified through the use of a numerical solution.