Temperature Analysis in Hydraulic Fracturing

Abstract

The authors present a thermal analysis of hydraulic fracturing based on variation methods. Their purpose is to provide a theoretical method for determining fracturing fluid temperature as a function of time and location during fracture growth. They first develop an expression of the variational principle for the general problem of convective heat transfer in a porous solid. Its accuracy is confirmed by comparisons with exact relations for specific cases. It is then used to develop a partial-differential equation for fluid temperature as a function of time and location. In this development, they treat fracture dimensions and leakoff distribution as known functions. The differential equation is solved by the method of characteristics. An alternative method of successive approximations is also presented. This solution can be combined iteratively with a fracture propagation analysis to find self-consistent results for fracture dimensions, leakoff, and temperature. The authors consider results obtained this way with those obtained with the two-dimensional (2D) Lagrangian analysis. Results are presented as profiles of temperature vs. dimensionless fracture length. When these profiles are normalized in terms of reservoir temperature and wellbore temperature, they change little with time of treatment, fracturing conditions, or reservoir properties. A simple profile with two straight-line segmentsmore » is a good approximation for most fracturing treatments. This approximate profile is very useful for field operations. It provides a way to estimate fracturing temperatures rapidly during a treatment.« less