Reactive transport modelling (RTM) is essential for subsurface environmental management but is fundamentally constrained by traditional geochemical solvers. These solvers incur prohibitive computational costs and frequently suffer from numerical instabilities such as convergence failures, particularly in microbially-mediated kinetic reaction systems. While machine learning surrogates offer acceleration, they often lack physical consistency when dealing with stiff biogeochemical dynamics. Here we propose PHYGNET (Physically GNN Network), which maps microbial reaction networks into a directed graph by representing species and reactions as nodes and edges. It embeds Monod-type kinetics within a physics layer to enforce mass conservation and thermodynamic hierarchies, and incorporates a residual corrector for refinement. By successfully coupling with COMSOL, PHYGNET demonstrates the capability to execute full reactive transport simulations. Benchmark tests reveal that, in contrast to the severe super-linear time penalties faced by traditional solvers at engineering scales, PHYGNET maintains stable sub-linear scaling via tensor parallelism. At a scale of 10⁵ grid nodes, PHYGNET achieved an acceleration (up to 3524-fold) without numerical crashes. Furthermore, its escalating speedup ratio establishes a "small-sample training, ultra-large-scale inference" paradigm that effectively offsets initial data generation costs. Overall, PHYGNET provides an efficient and physically consistent framework for accelerating Monod-type microbial reactive transport simulations, offering a practical pathway for large-scale environmental applications.