Dimension Reduction and Asymptotic Approximation of Reactive Transport in a Bulk Domain with a Branched Thin Fracture

Abstract

We study nonlinear reactive transport in a two-dimensional bulk domain containing a thin branched fracture composed of three narrow branches of thickness epsilon connected through a junction node of diameter O(epsilon). The microscopic model couples nonlinear parabolic reaction-diffusion equations in the bulk with an advection-diffusion equation in the fracture, where the longitudinal Peclet number is of order epsilon^{-1}, leading to advection-dominated transport along the branches. Nonlinear side-dependent flux conditions describe the coupling between the bulk and the fracture. As epsilon tends to zero, the fracture collapses to a one-dimensional graph, and we derive a recurrent structure of effective limit problems: a first-order hyperbolic problem on the graph satisfying the classical Kirchhoff transmission condition at the node, and reaction-diffusion equations in the bulk with nonlinear Robin conditions on the graph edges involving the graph solution. The node boundary conditions of the microscopic model do not affect these leading-order limits. To capture the influence of the node geometry, we construct node-layer and corner-layer correctors and determine subsequent terms of the asymptotic expansion. Their coefficients solve auxiliary boundary-value problems in unbounded domains with outlets at infinity and in corner-type geometries. We assemble a complete multiscale approximation combining bulk, branch, node-layer, and corner-layer contributions, and establish quantitative error estimates in appropriate energy norms. These estimates demonstrate the accuracy of the approximation relative to epsilon and depend explicitly on the corner angles of the limiting graph, reflecting the geometric complexity of the fracture network.