The dynamics of heat and mass transfer in porous media are of practical importance in several scientific and engineering applications. The characteristics of the transport inside the pore space are governed by the mechanisms that occur at the pore level. Dispersion and growth of the thermal and solutal transition zones are among the dynamic characteristics of a displacement process that is accompanied by heat and mass transfer. Recent advances in the computational power and high-resolution imaging techniques provide the opportunity to investigate these phenomena at the pore level. In this study, a direct pore-scale methodology is developed to model heat and mass transport in heterogeneous unconsolidated granular porous media during viscously stable and unstable flows. The pore-level heterogeneity of the porous domains is described by the standard deviation of their solid grains’ diameters. The velocity, temperature, and concentration fields are obtained by conducting numerical simulations of flow and transport on the discretized solution domains of several 2D and 3D geometries. The pore-level distributions of the temperature and concentration are used to calculate the length of the thermal and solutal transition zones. Then, the temporal scaling of the transition zone length is employed to identify the nature of the transport under different flow conditions and pore space geometrical attributes. By matching the effluent temperature and concentration profiles to the analytical solutions of the macroscale heat and mass transfer equations, the longitudinal components of the thermal and solutal dispersion tensors are determined and then correlated with the relevant governing parameters. The computed dispersion coefficients can be directly used in the macroscale reservoir simulators to accurately predict the performance of a solvent-based or thermal recovery process. The outcomes of this study find applications in the design and implementation of an efficient isothermal or non-isothermal miscible displacement in porous media.