42771-B9
Modeling Transient Heat Transport in Wells
Our research goal, to develop an analytic expression of transient fluid temperature in wellbores during flow initiation or shut-in, has progressed well. As this is the last annual report for this project, we describe in the following the work we have done previously, as well as that of the last year.
A typical well will be shut-off many times over its producing life. Shutting off a well or restarting a shut-off well triggers transients in flow, pressure, and temperature in the wellbore fluid. Although these transport processes are interdependent, momentum and mass transients dissipate quickly, making their effects on heat transfer minimal. In this project we used this assumption to decouple heat transfer from the other two transport processes and developed the governing differential equation for fluid temperature,
Using appropriate initial and boundary conditions we then derived the following analytic expression of transient fluid temperature, Tf, as a function of time and well depth, in terms of flow rate, w, and fluid mass m,
where
The parameters LR and f depend on fluid and formation thermal properties, wellbore heat transfer coefficient, Uto, and configuration of the well.
A large part of the effort in this project went to developing a solution algorithm for the analytic expression and the numeric model for transient fluid temperature estimation. We used excel spread-sheet with visual basic macros for the analytic solution platform because of its popularity in the petroleum industry. It was also useful to us for examining the influence of various parameters and as a teaching tool for undergraduate students.
We were gratified to see that the analytic transient temperature expression we developed during this project agreed well with field data and results obtained from rigorous simulations. Extensive sensitivity analyses we performed showed us, however, that the model has short comings. Our analyses indicate that the assumption of a constant fluid temperature gradient, dTf/dz, (and hence LR) in the model is the primary source of the problem. Although a weak function of time at late times, LR varies significantly at early times, especially during a drawdown. Due to changes in well configuration, the overall heat-transfer coefficient, Uto (and hence LR), may depend on depth. Changes in heat-transfer coefficient of the tubing/casing annulus fluid with temperature may also cause Uto to be as function of time. As production continues, heat transfer from the wellbore causes a gradual rise in the temperature of the surrounding, in turn causing a decrease in heat exchange. We also found that neglecting flow transients caused errors in temperature estimates.
During the project period, the support from ACS-PRF allowed us to develop two approaches that partially remove the assumptions of constant LR and flow rate. The constant LR assumption is replaced by combining backward Euler and Newton-Raphson implicit iteration scheme. The final expression for fluid temperature is given by
We also developed the following expression to update the formation temperature near the wellbore at successive time intervals, thus accounting for change in heat transfer rate with time,
In addition, we have started working on removing the assumptions of constant mass flux during buildups and drawdowns. I received support from ACS Summer Research Fellowship program to work with Dr. Spindler, an assistant professor of Mathematics at the Bemidji State University. He was able to non-dimensionalize the governing equation, and identified and interpreted important emergent parameters. He also applied several perturbation techniques and tried to solve the equation more directly using orthogonal solutions methods and variation of parameters. Dr. Spindler is continuing his efforts and is planning to write a proposal for further support of his work.
We are happy with the results of our efforts in this project. We presented a paper detailing some of these findings at the international meeting of the Society of Petroleum Engineers (SPE) in 2006 and published them in SPE Production & Operations, a refereed journal; in 2007 (both acknowledged ACS-PRF support).
We are continuing to work on this project to develop simple implementations of the proposed improvements to the model. I plan to request a six-month extension of the project to support a student to perform various computations and sensitivity analyses.
