Methane-rich water moves through conduits beneath the seafloor whose surfaces are formed through precipitation reactions. To understand how such submarine fluid conduit and venting systems form and grow, we develop a detailed mathematical model for this reaction–advection system and we quantify the evolution of an ensemble of similar filaments. We show that this growth can be described by a superposition of advection and dispersion. We analyse analogue laboratory experiments of chemical-garden type to study the growth of a single filament undergoing a precipitation reaction with the surrounding environment. We apply these findings to geological fluid conduit and venting systems, showing that their irregular trajectories can lead to very effective spreading within the surrounding seabed, thus enhancing contact and exchanges of chemicals between the conduit and external fluids. We discuss how this methane venting leads to the formation of marine authigenic carbonate rocks, and for confirmation we analyse two field samples from the Gulf of Cadiz for composition and mineralogy of the precipitates. We note the implications of this work for hydrate melting and methane escape from the seabed. This article is protected by copyright. All rights reserved.