Taylor-Bénard convection of water and water-based nanoliquids confined in three different types of high porosity rectangular enclosures, viz., shallow, square and tall, is studied analytically using both infinitesimal and finite amplitude stability analyses. We make use of the modified-Buongiorno-Brinkman model(MBBM) for the governing equations concerning nanoliquid-saturated porous enclosures bounded by rigid-rigid boundaries and obtain analytical results. Among three types of enclosures, maximum and minimum heat transfers are observed in tall and shallow enclosures respectively. Water well dispersed with a dilute concentration of single-walled carbon nanotubes(SWCNTs) is considered as a working medium. The water-SWCNTs is able to flow in the porous medium because the medium is loosely-packed with porosity in the range 0.5 ≤ ϕ ≤ 1. In addition to this, the maximum volume fraction of nanoparticles considered in the system is 6% and thus this does not alter the fluidity of the system. We found from the study that the presence of low concentration(volume fraction-0.06) of SWCNTs in a water-saturated porous medium effectively improves the heat transport of the system due to its high thermal conductivity and large surface area. Due to the presence of a porous medium, however, the onset of convection gets delayed and heat transport in nanoliquids gets substantially reduced in a Bénard-Brinkman configuration resulting from the weak thermal conductivity of the porous medium. Thus the porous medium acts as the heat storage system. Also, in a rotating frame of reference the heat transport gets reduced and rotation serves as an external mechanism of regulating heat transport in the system. The nonlinear dynamics of the system is studied using the 6-mode Lorenz model. Chaotic motion in the system is studied using the maximum Lyapunov exponent(MLE). The Hofp-bifurcation point of the system along with the MLE is used to investigate periodic, nearly periodic and mildly chaotic behaviors of the system.