Although many analytical wellbore heat loss models have been presented, none has addressed the problems associated with (i) directional wells and (ii) 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.