Computational Fluid Dynamics (CFD) is used in this study to assess the influence of temperature-dependent oil viscosity and density on the flow of oil up the well, and therefore the amount of insulation required. CFD is a difficult tool to apply to flows where the grid aspect ratio is as high as it needs to be to accommodate the full length of an oil well with a realistic number of grid points. Each model was therefore intensive in terms of computational effort and time. This study shows that by allowing oil viscosity and density to vary with temperature in a 2150 m vertical well with no insulation, the production output is significantly affected. The drop in production output is approximately 3% when oil viscosity varies with temperature, but when coupled with temperature-dependent density the loss in production increases to 22%. Ten CFD models, each with a different value of insulation heat transmission coefficient lying in the range 0.35 Wm^-2K^-1 to 16900 Wm^-2K^-1, are used to establish the temperature drop between riser inlet and outlet. The results obtained allow an operator to select an appropriate insulation based on the allowable temperature drop up the well, assuming all other properties are equal. The completion fluid region is situated outside the oil flow, tubing and insulation. The fluid is stationary which suggests that natural convection currents are present. Seven CFD models with annulus heights ranging from 1 m to 64 m are used to detect these currents, and assess the effectiveness of water as an insulating completion fluid. This thesis establishes that the natural convection currents do not split into multiple cells, but remain mono-cellular when the Grashof number is approximately 1x10⁸ and the Prandtl number is 2.3. This work also shows that heat loss due to natural convection from the completion fluid is an important contributory factor to the overall heat loss from a well, dependent on the well height.