Modelling geothermal reservoirs is challenging due to the large domain and wide range of length- and time-scales of interest. Attempting to represent all scales using a fixed computational mesh can be very computationally expensive. Application of dynamic mesh optimisation in other fields of computational fluid dynamics has revolutionised the accuracy and cost of numerical simulations. Here we present a new approach for modelling geothermal reservoirs based on unstructured meshes with dynamic mesh optimisation. The resolution of the mesh varies during a simulation, to minimize an error metric for solution fields of interest such as temperature and pressure. Efficient application of dynamic mesh optimisation in complex subsurface reservoirs requires a new approach to represent geologic heterogeneity and we use parametric spline surfaces to represent key geological features such as faults and lithology boundaries. The resulting 3D surface-based models are mesh free; a mesh is created only when required for numerical computations. Dynamic mesh optimisation preserves the surfaces and hence geologic heterogeneity. The governing equations are discretised using a double control volume finite element method that ensures heat and mass are conserved and provides robust solutions on distorted meshes. We apply the new method to a series of test cases that model sedimentary geothermal reservoirs. We demonstrate that dynamic mesh optimisation yields significant performance gains, reducing run times by up to 8 times whilst capturing flow and heat transport with the same accuracy as fixed meshes.