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         The support from ACS-PRF has helped us to successfully pursue modeling transient heat transport in wellbores.  That, in turn, has allowed us to develop a robust analytic expression of transient fluid temperature in wellbores, to improve estimation of flowing fluid temperature for complex wellbores under steady-state conditions, to apply the fluid temperature dependence on flow rate to estimate production, and to develop applications of the temperature model to many systems, including geothermal wells. Our work has also led to vigorous research in this area by other workers and has generated research interests in a number of our students.  As this is the final report for the project, we will describe in the following paragraphs the work we have done since the initiation of this project as well as that of the last year. 

         For well testing, workover, and other purposes, most wells are shut-off and restarted many times over their producing lives.  Shutting off a producing well or restarting a shut-off well often triggers long-lasting transients in flow, pressure, and temperature in the wellbore fluid.  Although these transport processes are interdependent, momentum and mass transients dissipate quickly in these systems, causing the duration of their effects on heat transfer to be very short lived.  Assuming that that momentum and mass transients have ceased allows us to decouple heat transfer from the other two transport processes and help us derive the following analytic expression of transient fluid temperature, T_f, as a function of producing (or shut-in) time and well depth, in terms of flow rate, w, and fluid mass m,

where                                     

and                             

The parameters L_R and f depend on fluid and formation thermal properties, wellbore heat transfer coefficient, U_to, and configuration of the well.

         The expression for T_f was developed by assuming constant L_R and flow rate.   Over the last few years, we developed two approaches that partially remove these assumptions. The constant L_R assumption is replaced by combining backward Euler and Newton-Raphson iteration schemes. The solution is implicit in nature and provides an efficient algorithm for fast convergence and stability.  A method to account for change in heat transfer rate with time as production continues is also developed. We believe the proposed improvements in the model would accurately mimic afterflow during surface shut-in by computing velocity profile at each timestep and its consequent impact on temperature and density profiles in the wellbore.