Analytical Calculations For Heat Transfer From Fractures
Authors: Wheeler J.A. 
Published: December 2007

Abstract
Successful thermal recovery processes require injection rates which in some reservoirs are attainable only by fracturing the formation. To increase our understanding of thermal processes in fractured systems, we have processes in fractured systems, we have developed three new analytical models which relate temperature as a function of time and position to injection conditions and reservoir position to injection conditions and reservoir characteristics. The models are easier to apply than numerical solutions, and unlike models previously available, approximate the effects of convective heat transfer in the pay sand and heat loss to the overburden.
The model for hot water injection into a vertical fracture accounts for convection of energy in the fracture, conduction and convection in the pay sand and conduction in the overburden. The calculation was designed primarily to aid analysis of temperature primarily to aid analysis of temperature observations made in the vicinity of a fracture for reservoir evaluation purposes.
The steam injection model takes into account convection and condensation in a vertical fracture and conduction in the formation. A special radial case is also presented for those interested in heat transfer from a very thin, permeable zone. Unlike any previous analytical model for steam injection, this model predicts the presence of an isothermal steam zone preceded by a nonisothermal hot water zone in the fracture. The model applies only to thick, impermeable formations, since it does not include heat losses to the overburden and underburden and convection of energy in the formation.
The "latent heat model" accounts for conduction and convection of energy in the pay sand and conduction in adjacent strata. Although a simplified block-like temperature distribution is assumed to exist in the fracture, this model provides a useful alternative to the steam model. The steam model does not apply to thin sands for which heat losses to adjacent strata are significant or to permeable sands for which convection in the formation is significant.
The report deals primarily with the derivations of the three models. Partial differential equations for each model are formulated in the text and solved in the appendices. The assumptions inherent in the models are examined critically. Calculations are included to check the validity of the models, to emphasize the potential importance of convection in the pay sand as a heat transfer mechanism, and to illustrate the application of the models to the problem of determining fracture orientation. problem of determining fracture orientation
Introduction
Economical injection rates are attainable in some cases only by fracturing the formation. The injection of steam or hot water at subfracture pressures may be uneconomical because of low formation permeability or high cold-oil viscosity. In either case, the injection pressure can be increased until the formation pressure can be increased until the formation is fractured and acceptable injection rates are achieved. This procedure radically affects the heat and mass transfer characteristics of the system.
Numerous mathematical models of heat transfer in nonfractured systems have been published, but relatively few studies have been published, but relatively few studies have been made of heat transfer in fractured systems. Spillette has reviewed models of the former type.