Our research goal, to develop an analytic expression of transient fluid temperature in wellbores during flow initiation or shut-in well, has progressed very well.  As this is the last annual report for this project, we describe in the following paragraphs the work we have done over the first two years as well as that of the last year.

         A typical well will be shut-off many times over its producing life for workover and other purposes.  Shutting off a producing well or restarting a shut-off well often triggers transients in flow, pressure, and temperature in the wellbore fluid.  Although these are interdependent processes, momentum and mass transients dissipate quickly with minimal effects on heat transfer.  This assumption allows us to decouple heat transfer from the other two transport processes, and we derive the following analytic expression for fluid temperature, T_f, as a function of 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 well configuration.

         Applying our model to field data and results obtained from rigorous simulations, we noted that while the model's estimates are generally accurate, its performance needs improving.  Our analyses indicate that the assumption of a constant fluid temperature gradient, dT_f/dz, and hence L_R, in the model is the primary source of the problem. Although a weak function of time at late times, L_R varies significantly at early times. Due to changes in well configuration, the overall heat-transfer coefficient, U_to (and hence L_R), may depend on depth. Changes in heat-transfer coefficient of the tubing/casing annulus fluid with temperature may also cause U_to to be as function of time. As production continues, heat transfer from the wellbore causes a gradual rise in the temperature of the surrounding formation, in turn causing a slow decrease in heat exchange between the wellbore fluid and its surroundings. We also found that neglecting flow transients caused errors in temperature estimates.

         Over the last three years, the support from ACS-PRF allowed us to develope two approaches that remove the assumptions of constant L_R and flow rate. Backward Euler and Newton-Raphson iteration schemes replace the constant L_R assumption, leading to an implicit solution shown below, that 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. The following expression is used to update the formation temperature near the wellbore at the new time level.

         Variable earth-temperature scheme updates the formation temperature in the immediate vicinity of the wellbore. This scheme has slightly improved the accuracy of fluid temperature computations and bottomhole pressure predictions.

         In addition, we have started working on removing the assumptions of zero mass flux during a buildup and constant mass flux during a drawdown.  This will allow us the flexibility to mimic energy transport during afterflow and multirate tests. 

         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. Surrounding formation temperature is updated every timestep to account for changes in heat-transfer rate between the hot wellbore fluid and the cooler formation. Optional hybrid numerical differentiation routine removes the limitations imposed by the constant relaxation-parameter assumption used in previous models.

         A paper acknowledging ACS-PRF support and detailing some of these findings was presented at the 2006 international meeting of the Society of Petroleum Engineers.