Transport With Bimolecular Reactions in a Fracture-Matrix System: Analytical Solutions With Applications to In Situ Chemical Oxidation

Abstract

We present analytical solutions for transport with bimolecular reactions in a single fracture embedded within an infinite rock matrix. The fracture and matrix are initially assumed to contain one aqueous species (B) at a uniform concentration. A second aqueous species (A) is injected into the fracture and reacts with B and an additional immobile species (N) in the rock matrix. Under these conditions, moving reaction fronts form and propagate along the fracture and into the rock matrix. We employ a composite similarity variable involving two space variables to derive analytical solutions for all species concentrations and the geometry of reaction fronts in the fracture and matrix. The behavior of the reaction-diffusion equations in the rock matrix is posed as a Stefan problem. For uniform advection in the fracture, our analytical solutions establish that the reaction fronts propagate as the square root of time in both the matrix and the fracture. Our analytical solutions agree very well with numerical simulations. We extend our analytical solutions to nonuniform flows in the fracture by invoking a travel-time transformation. We present applications of our analytical solutions to in situ chemical oxidation of dense nonaqueous phase liquids in fractured rock, wherein an oxidant (A, e.g., permanganate) is injected through fractures and consumed by bimolecular reactions with dissolved dense nonaqueous phase liquids (B, e.g., trichloroethylene) and natural organic matter (N) in the fracture and rock matrix. Our analytical solutions are also relevant to a broad class of reactive transport problems in fracture-matrix systems where moving reaction fronts occur.