An Analytical Solution for Wellbore Heat Transmission in Layered Formations (includes associated papers 23410 and 23411 )

Abstract

Summary. This paper presents a new analytical solution for wellbore heat transmission. Previous treatments of the wellbore heat-transfer problem are improved in several aspects:nonhomogeneous formations are approximated as layered formations with different physical properties;closed-form analytical solutions are obtained in both real and Laplace space;anda more accurate formula is provided for the transient heat-conduction function, f(tD). Introduction Heat is transferred to or from the wellbore when there is a difference in temperature between the surrounding formation and the injected (or produced) fluid. To evaluate the feasibility of a thermal recovery project, it is necessary to estimate the heat losses or gains of the flowing fluid in wellbores, the changes in temperature with time and depth, and the heat-transfer conditions between wellbore and formation. A quantitative description of heat exchange between a wellbore and surrounding formations often is also required when one attempts to estimate formation temperatures from wellbore measurements. Studies of wellbore heat transmission during hot or cold fluid injection have appeared in the literature since the 1950's. The techniques available for dealing with wellbore heat transmission include analytical and numerical methods. Lessem et al. and Squier et al. derived and solved similar systems of differential equations describing the temperature behavior of gas and hot-water injection wells. They neglected wellbore thermal resistance and made the following assumptions. 1.No conductive heat transfer occurs in the vertical direction of either the flowing fluid or the formation. 2. The mass flow rate of gas or water is constant throughout the injection or production system. 3. The volumetric heat capacities of fluids and formation are constant. 4. The formation is homogeneous and isotropic with constant thermal conductivity. 5. The fluid temperature is the same as the formation temperature on the wellbore surface. Subsequent work introduced another assumption, that vertical heat transfer in the wellbore was considered steady state. The classic study by Ramey on wellbore heat transmission improved Moss and White's approach to incorporate an overall heat-transfer coefficient. Ramey presented an approximate solution for the temperatures of fluids, tubing, and casing as a function of time and depth in a well used for hot-fluid injection. Satter suggested a similar method for analyzing wellbore heat loss when condensing steam flow is considered and provided a sample procedure for a given set of reservoir properties. Ramey and Willhite gave an expression for the overall heat-transfer coefficient for any well completion and the early-time values of the transient heat-conduction function. Durrant and Thambynayagam's more recent work provided an iterative procedure for the wellbore heat-transmission problem during flow of steam/water mixtures that includes vertical heat conduction. The numerical models by Farouq Ali and Wooley were more comprehensive than the analytical models. They include both horizontal and vertical heat conduction in the formation and can deal with different well operation conditions. The numerical methods, however, are often too complicated for field applications or for reservoir simulation studies because many of the required wellbore and formation heat-transfer properties are rarely known precisely. The mathematical model for wellbore heat transmission presented in this paper adopts assumptions similar to those of Lessem et al. The maim differences are that we introduce an overall heat-transfer coefficient to consider the wellbore heat resistance and that we treat the surrounding earth as consisting of an arbitrary number of layers with different thermal and physical properties and arbitrary initial temperature distributions(Fig. 1). An analytical solution has been obtained in both real and Laplace space for prediction of wellbore heat transmission. The numerical results calculated from the analytical solutions are compared with Ramey's long-time approximation. illustrative applications are given for predicting wellbore heat trans-mission for engineering designs or reservoir simulation studies in petroleum and geothermal reservoir development.