Although many analytical wellbore heat loss models have been presented, none has addressed the problems associated with (i) directional wells and (ij) changing injection conditions. Today, more and more wells are drilled directionally. Moreover, the injection rate, pressure, temperature and steam quality seldom remain constant throughout the injection period.This paper presents a semi-analytical model which can determine heat losses from directional wells (deviated or horizontal wells) under changing injection conditions.To model directional wells, the well is divided into a number of segments whose inclination angle can be varied with depth. The energy and pressure equations are coupled and solved iteratively for each segment. The changes in injection variables (rate, pressure, temperature and steam quality) are approximated by step functions and the method of superposition is used. A new empirical expression is used to determine the transient heat loss to the formation. This simple expression is valid for all times and gives results closer to the exact solution than the more commonly used long time asymptotic solution.The sample calculations in this paper illustrate the differences in heat losses between vertical and directional wells and demonstrate that the assumption of constant wellhead conditions may give unsatisfactory results in cases where the injection conditions vary.

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This work studies the depth of thermal penetration in the fast-transient process of heat transport, with emphasis on the rapid evolution of heat affected zone that reflects the combined behavior of thermalization and relaxation during the short-time transient. Employing the dual-phase-lag model, the heat balance integral is derived that contains both the phase lag of the heat flux vector (reflecting the relaxation behavior) and the phase lag of the temperature gradient (reflecting the thermalization behavior). The early-time responses where both phase lags are strong functions of temperature are modeled in detail, which emphasize the ways in which the classical diffusion behavior is retrieved as the thermalization and relaxation behaviors gradually diminish in the time-history. It shows that the classical diffusion model assuming Fourier’s law does not provide a conservative estimate for the response of temperature in the thermal process zone, which may result in unexpected early-time failure of conductors if not properly controlled. From a microscopic point of view, these unexpected factors are interpreted in terms of the microscopic parameters, including the coupling factor in phonon-electron interactions (for metallic structure) and the umklapp and normal relaxation times in phonon scattering (for semiconductors and insulators.