A three-dimensional numerical tool for the microscale simulation of smoldering in fixed beds of solid fuels is presented. The description is based on the local equations and accounts the local couplings of the transport and reaction mechanisms. The chemical model includes devolatilization and cracking of the kerogen, calcination of the carbonates contained in a mineral matrix and oxidation of the carbon char left by the pyrolysis. An extensive survey of the functioning regimes exhibits features that have to be taken into account in the operation of a reactor and in its macroscopic modeling. Three dimensionless numbers are shown to control the phenomenology, which embody the effects of the constituent properties and of the operating conditions. One of them, , provides an a priori criterion for the validity of a local equilibrium hypothesis and for the applicability of standard homogenized formulations. The numerical observations comply when is small with the expectations from a simple homogenized description, including quantitative predictions of the mean temperature profile, of the consumption of the various reactants and of the relative positions of the reaction fronts. Conversely, local equilibrium is not satisfied when is large and these approaches fail in several respects. The simple upscaled transport equations are unable to predict the evolution of some of the locally average state variable. Furthermore, strong local deviations of the state variables from their local averages, combined with the nonlinearity of the kinetic laws, cause the overall reaction rates to differ from those deduced from the mean values. Nevertheless, a successful heuristic model for the spread of the hot and potentially reactive region can be stated, which provides an avenue for further studies.