Numerical modeling of unsteady-state wellbore heat transmission

This paper proposes a fully-implicit thermal model, which couples the wellbore and the surrounding formation. The authors use this integrated model to simulate and predict the unsteady-state heat transmission and temperature distribution in the wellbore and the surrounding formation under variable operational constraints. This model employs refined grid discretization for both the wellbore and the surrounding formation domains. Detailed mass, energy and momentum balance analysis is given for each grid block, which is solved simultaneously at every time step. The model captures the near-wellbore boundary effects using geometric spacing. The resulting set of equations is solved by the Unsymmetric MultiFrontal solver (UMFPACK). We compared and validated the results of the numerical model, against both a conventional Ramey's approach and a rigorous analytical solution introduced by Hagoort. The results of the comparison show that this model is able to predict the temperature distribution in the wellbore and the surrounding formation from early to late time. Contrasted with Ramey's approach, this model yields more accurate results at any time scale, particularly for early time values when the wellbore temperature profile is determined by the wellbore inlet fluid frontier. Two case studies are discussed to test the feasibility of this model under variable operating conditions. The results of this procedure indicate that this model is applicable to both variable-temperature and multi-rate injection cases. The temperature profiles in the wellbore and the surrounding formation are presented with temperature contours, displaying how the temperature changes from near the wellbore to deep in the formation. Moreover, a superposition time is defined and integrated with Ramey's approach (as an alternative to the simulation model) in order to produce a quick and relatively accurate prediction of the wellbore temperature profile. This approach has wide application in cyclic steam-injection and geothermal wells under variable operating conditions.