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L. A. M. Rocha gratefully acknowledges funding from the Fundação para a Ciência e Tecnologia (FCT), Portugal (grant SFRH/BD/130401/2017). C. Pimentel acknowledges funding from Juan de la Cierva‐Formación (grant FJC2018‐035820‐I) from the Spanish Ministry of Science. J. H. E. Cartwright and C. Ignacio Sainz‐Díaz acknowledge the financial support of the Spanish MINCINN projects FIS2016‐77692‐C2‐2‐P and PCIN‐2017‐098. I. Sánchez‐Almazo thanks the crew of the Cornide de Saavedra vessel, as well as the members of the Tasyo project, funded by the Spanish Marine Science and Technology Program, for allowing her to participate in the 2000–2001 Anastasya cruises. The authors acknowledge the contribution of the COST Action chemobrionics, CA17120., 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 analyze analog 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 analyze 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., Ciencia e Tecnologia, Fundação para a Ciência e Tecnologia, MINCINN PCIN‐2017‐098, European Cooperation in Science and Technology CA17120, Fundação para a Ciência e a Tecnologia FJC2018‐035820‐I, SFRH/BD/130401/2017, Ministerio de Ciencia e Innovación