The physical problem of advection, dispersion, and reaction of a suite of chemicals is formulated mathematically as a set of coupled time-dependent nonlinear partial differential equations. Partially based on the Gear scheme for time discretization, the algorithm developed in this paper decouples equations into separate linearized problems. Then, using finite element approximation, these resulting linear advection-dispersion equations are further transformed, via a re-assembling of variational formulation, into an elliptic Stokes-like problem, which can be solved by an existing Stokes solver previously employed when solving the flow problem to produce a velocity background. The methodology is implemented in the simulation of transport and transformations of an electron donor, an electron acceptor, and active biomass. This application provides insights into biofilm evolution and pore clogging, and demonstrates mesh self-adjustment in the process of biofilm growth.