Onset of convection in a vertical slab of saturated porous media between two impermeable conducting blocks

Abstract

The onset of natural convection in a vertically oriented, finite thin slab of saturated porous material is considered. The slab is embedded between two impermeable conducting blocks of finite dimension. A vertical temperature difference is imposed between the upper and lower horizontal surfaces of the slab and blocks, and a linear temperature distribution is imposed on the outer vertical surfaces of the blocks. This configuration is used to model convection in a saturated, fractured rock zone like that associated with faulting. A linear stability analysis is developed for both convection in the slab and conduction in the block. The objective of the study is to obtain the critical Rayleigh number and mode of convection in the slab. When the block and the slab widths are both small compared to the other two dimensions one finds a large number of tall, narrow, three-dimensional cells. In contrast, a block of relatively large width promotes the formation of longer wavelength, weakly three-dimensional cells in the slab at a much lower Rayleigh number. The difference is related to the character of the temperature distribution in the solid block.