Stabilization of Bottom Hole Temperature With Finite Circulation Time and Fluid Flow

As an alternative to finite element or finite difference modelling, analytical solutions are derived by the method of Laplace transformation and numerical results obtained for several models of the bottom hole temperature stabilization. Included in the models are the finite circulation time, the thermal property contrast between the borehole mud and the surrounding formation, and the presence of radial or lateral fluid flow in the formation, all of which are found to have significant effects on the dissipation of the thermal disturbance induced by drilling.
The mud circulation is considered to have the effect of either maintaining the borehole mud at a constant temperature or supplying a constant amount of heat per unit length per unit time to the borehole. For small circulation times, the former reduces to the 'zero circulation' model in which the mud circulation creates an instantaneous temperature anomaly at the hole bottom; for small borehole radii, the latter reduces to the line source model and the traditional 'Homer plot'.
For typical drilling operations in which the bottom hole temperatures are measured several hours to several tens of hours after the hole is shut in, the new models generally predict higher equilibrium formation temperatures than does the Horner plot. However, predictions from the various models converge if the BHTs are taken after the hole has been shut in for a period which is greater than about five times the circulation period.