The Rayleigh-Bénard convection of a nanoliquid-saturated porous medium confined in a very shallow enclosure is investigated theoretically using the modified Buongiorno - Brinkman model. In the study, the chosen nanoliquid-saturated porous medium is assumed to be made up of water well dispersed with copper(Cu) nanoparticles of five different shapes saturating in a 30% reinforced polycarbonate glass fiber(GF) porous material of high porosity and its effective thermophysical properties are calculated using the phenomenological laws or mixture theory. Two kinds of boundary conditions, viz., stress-free and rigid, are employed and the analytical solution is obtained in both cases. On the other hand, Rayleigh-Bénard convection in a very shallow domain of height 5mm and width 5cm filled with water-liquid and bounded by the rigid boundaries is simulated. The simulation results are then compared with the analytical results in the case of rigid boundaries. We found that the analytical results are in good agreement with those of the simulation results and this validates results of the present study. Linear and weakly non-linear stability analyses are performed to find the onset and the heat transport of the system. The effects of various parameters on the onset and heat transport of the system are depicted graphically and the physical explanation is provided for all observed results in the study. We found that the addition of dilute concentration of nanoparticles advances the onset and thereby enhances the heat transport in the system. Among five different shapes of copper nanoparticles, maximum and minimum heat transports are observed in the cases of blade and spherical shaped nanoparticles, respectively. The porous medium parameters: Brinkman number and porous parameter, show a stabilizing effect in the system. The existence of subcritical motions is also predicted for the system. The results of the Khanafer-Vafai-Lightstone(KVL) single-phase model, nanoliquid, base liquid and base liquid-saturated porous medium are obtained as limiting cases of the present study. Since nanoparticles and porous medium, respectively, show a destabilizing and stabilizing nature of influence in the system, the present work has possible applications in both heat removal and heat retainment systems.