This study presents a conceptual Dual-Permeability Non-Equilibrium (DPNE) model that accounts for both physical and chemical non-equilibria to describe the reactive solute transport through a porous medium. A semi-analytical solution of the DPNE model is derived in the Laplace domain, which is then numerically inverted to obtain concentrations in different domains at different times and depths. The derived semi-analytical solution is validated using experimental data and existing analytical (for simple problems) and numerical solutions. The Global Sensitivity Analysis (GSA) is performed to identify model parameters with an impact on nonreactive tracer breakthrough curves (BTC) for a selected data set. A sensitivity-based calibration of the DPNE model’s parameters is carried out to simulate a BTC with multiple inflection points. The DPNE model can better describe the experimental datasets than the model based on the dual-porosity (MIM) concept with first-order mass transfer. Since the DPNE model considers detailed sorption and mass transfer dynamics, it is suitable for process-based investigations.