Transport scaling in porous media convection

We present a theory to describe the Nusselt number, $\operatorname {\mathit {Nu}}$ , corresponding to the heat or mass flux, as a function of the Rayleigh–Darcy number, $\operatorname {\mathit {Ra}}$ , the ratio of buoyant driving force over diffusive dissipation, in convective porous media flows. First, we derive exact relationships within the system for the kinetic energy and the thermal dissipation rate. Second, by segregating the thermal dissipation rate into contributions from the boundary layer and the bulk, which is inspired by the ideas of the Grossmann and Lohse theory (J. Fluid Mech., vol. 407, 2000; Phys. Rev. Lett., vol. 86, 2001), we derive the scaling relation for $\operatorname {\mathit {Nu}}$ as a function of $\operatorname {\mathit {Ra}}$ and provide a robust theoretical explanation for the empirical relations proposed in previous studies. Specifically, by incorporating the length scale of the flow structure into the theory, we demonstrate why heat or mass transport differs between two-dimensional and three-dimensional porous media convection. Our model is in excellent agreement with the data obtained from numerical simulations, affirming its validity and predictive capabilities.