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Storage of CO2 in subsurface porous media may mitigate climate change and contribute to enhanced oil recovery. Naturally fractured aquifers or petroleum reservoirs are abundant but have some challenges. Only the matrix blocks (which are disconnected by fractures) contribute significantly to storage space and injected fluids must displace pre-existing wetting fluid held in place by capillary forces. Luckily, high fluid density differences can contribute to such displacement and even enable enhanced oil recovery before converting a producing oil reservoir to a storage site. A model is described for displacement of wetting fluid (brine or oil) by non-wetting fluid (CO2 or gas) from a matrix block due to gravity drainage in an interplay between gravity- capillarity and compressibility. The amount CO2 that can be stored in the blocks depends on the matrix saturations obtained and the densities. A low CO2 pressure results in high density difference with brine allowing more volume of CO2 to invade the blocks, while a higher pressure reduces the density difference but increases the CO2 density. The optimal conditions to store CO2 are determined as function of matrix block parameters (height, permeability, etc.) and pressure conditions. We also describe ideas for analytical solutions describing co-current and counter-current displacement accounting for the mentioned key mechanisms (capillary forces, gravity and compressibility) although these were not completed before the conference.