Current methods of monitoring subsurface CO$_2$, such as repeat seismic surveys, are episodic and require highly skilled personnel to acquire the data. Simulations based on simplified models have previously shown that muon radiography could be automated to continuously monitor CO$_2$ injection and migration, in addition to reducing the overall cost of monitoring. In this paper, we present a simulation of the monitoring of CO$_2$ plume evolution in a geological reservoir using muon radiography. The stratigraphy in the vicinity of a nominal test facility is modelled using geological data, and a numerical fluid flow model is used to describe the time evolution of the CO$_2$ plume. A planar detection region with a surface area of 1000 m$^2$ is considered, at a vertical depth of 776 m below the seabed. We find that one year of constant CO$_2$ injection leads to changes in the column density of $lesssim 1%$, and that the CO$_2$ plume is already resolvable with an exposure time of less than 50 days.
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