Direct numerical simulations (DNS) indicate that at large values of the
Rayleigh number ($Ra$) convection in porous media self-organizes into
narrowly-spaced columnar flows, with more complex spatiotemporal features being
confined to boundary layers near the top and bottom walls. In this
investigation of high-$Ra$ porous media convection in a minimal flow unit, two
reduced modeling strategies are proposed that exploit these specific flow
characteristics. Both approaches utilize the idea of decomposition since the
flow exhibits different dynamics in different regions of the domain:
small-scale cellular motions generally are localized within the thermal and
vorticity boundary layers near the upper and lower walls, while in the
interior, the flow exhibits persistent large-scale structures and only a few
low (horizontal) wavenumber Fourier modes are active. Accordingly, in the first
strategy, the domain is decomposed into two near-wall regions and one interior
region. Our results confirm that suppressing the interior high-wavenumber modes
has negligible impact on the essential structural features and transport
properties of the flow. In the second strategy, a hybrid reduced model is
constructed by using Galerkin projection onto a fully \emph{a priori}
eigenbasis drawn from energy stability and upper bound theory, thereby
extending the model reduction strategy developed by Chini \emph{et al.}
(\emph{Physica~D}, vol. 240, 2011, pp. 241--248) to large $Ra$. The results
indicate that the near-wall upper-bound eigenmodes can economically represent
the small-scale rolls within the exquisitely-thin thermal boundary layers.
Relative to DNS, the hybrid algorithm enables over an order-of-magnitude
increase in computational efficiency with only a modest loss of accuracy.
