A detailed description of coupled heat and momentum transfer in a one-dimensional porous medium saturated by a supercritical fluid subject to a heat flux at one of its boundaries is reported in this study. The derivation is performed with an asymptotic analysis of the linearized macroscopic mass, momentum, and energy equations, assuming a van der Waals supercritical fluid and the mean-field theory, considering Darcy's law for momentum transfer. Three regimes of heat transfer separated by two crossovers are highlighted. The first regime, far enough from the critical point (CP), corresponds to the classical piston effect (PE), as in a plain fluid. While nearing the CP, a first crossover is found, giving rise to a second regime in which the PE in the bulk fluid is supplemented by a temperature gradient that results from a pressure gradient buildup. Below this crossover, the critical speeding up by thermocompression of the bulk stops and a porous saturation of the characteristic heat transfer timescale is reached. Closer to the CP, a second crossover is identified that corresponds to the presence of a viscous pressure drop in the whole domain. Below this second crossover, the PE is faded away and, in this third regime, heat transfer returns to a classical diffusive process. The two crossovers are characterized by a single parameter that is the ratio between the acoustic time (i.e., the sound wave travel time through a pore) and the pore-scale viscous diffusion time, namely, the acoustic pore Reynolds number.