Since 1996, super-critical CO$2$ has been injected at a rate of $\sim$0.85 Mt yr$^{-1}$ into a pristine, saline aquifer at the Sleipner carbon capture and storage project. A suite of time-lapse, three-dimensional seismic reflection surveys have been acquired over the injection site. This suite includes a pre-injection survey acquired in 1994 and seven post-injection surveys acquired between 1999 and 2010. Nine consistently bright reflections within the reservoir, mapped on all post-injection surveys, are interpreted to be thin layers of CO$2$ trapped beneath mudstone horizons. The areal extents of these CO$2$ layers are observed to either increase or remain constant with time. However, volume flux of CO$2$ into these layers has proven difficult to measure accurately. In addition, the complex planform of the shallowest layer, Layer 9, has proven challenging to explain using reservoir simulations. In this dissertation, the spatial distribution of CO$2$ in Layer~9 is measured in three dimensions using a combination of seismic reflection amplitudes and changes in two-way travel time between time-lapse seismic reflection surveys. The CO$2$ volume in this layer is shown to be growing at an increasing rate through time. To investigate CO$2$ flow within Layer~9, a numerical gravity current model that accounts for topographic gradients is developed. This vertically-integrated model is computationally efficient, allowing it to be inverted to find reservoir properties that minimise differences between measured and modelled CO$2$ distributions. The best-fitting reservoir permeability agrees with measured values from nearby wells. Rapid northward migration of CO$2$ in Layer~9 is explained by a high permeability channel, inferred from spectral decomposition of the seismic reflection surveys. This numerical model is found to be capable of forecasting CO$2$ flow by comparing models calibrated on early seismic reflection surveys to observed CO$2$ distributions from later surveys. Numerical and analytical models are then used to assess the effect of the proximity of an impermeable base on the flow of a buoyant fluid, motivated by the variable thickness of the uppermost reservoir. Spatial gradients in the confinement of the reservoir are found to direct the flow of CO$2$ when the current is of comparable thickness to the reservoir. Finally, CO$2$ volume in the second shallowest layer, Layer~8, is measured using structural analysis and numerical modelling. CO$2$ in Layer~8 is estimated to have reached the spill point of its structural trap by 2010. CO$2$ flux into the upper two layers is now $\sim$40\%oftotalCO$_2$ flux injected at the base of the reservoir, and is increasing with time. This estimate is supported by observations of decreasing areal growth rate of the lower layers. The uppermost layers are therefore expected to contribute significantly to the total reservoir storage capacity in the future. CO$2$ flow within Layer~9 beyond 2010 is forecast to be predominantly directed towards a topographic dome located $\sim$3 kmnorthoftheinjectionpoint.ThisdissertationshowsthatadvancesindeterminingthespatialdistributionandflowofCO$_2$ in the sub-surface can be made by a combination of careful seismic interpretation and numerical flow modelling.