In this work, we introduce a novel neural operator, the Solute Transport Operator Network (STONet), to efficiently model contaminant transport in micro-cracked porous media. STONet's model architecture is specifically designed for this problem and uniquely integrates an enriched DeepONet structure with a transformer-based multi-head attention mechanism, enhancing performance without incurring additional computational overhead compared to existing neural operators. The model combines different networks to encode heterogeneous properties effectively and predict the rate of change of the concentration field to accurately model the transport process. The training data is obtained using finite element (FEM) simulations by random sampling of micro-fracture distributions and applied pressure boundary conditions, which capture diverse scenarios of fracture densities, orientations, apertures, lengths, and balance of pressure-driven to density-driven flow. Our numerical experiments demonstrate that, once trained, STONet achieves accurate predictions, with relative errors typically below 1% compared with FEM simulations while reducing runtime by approximately two orders of magnitude. This type of computational efficiency facilitates building digital twins for rapid assessment of subsurface contamination risks and optimization of environmental remediation strategies. The data and code for the paper will be published at https://github.com/ehsanhaghighat/STONet.