We present a thermal analysis of hydraulic fracturing based on variational methods. Our purpose is to provide a theoretical method for determining fracturing fluid temperature as a function of time and location during fracture growth. We 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 partial-differential equation for fluid temperature as a function of time and location. In this development, we 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. We 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 segments is a good properties. A simple profile with two straight-line segments 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.
Hydraulic fracturing technology includes an abundance of theoretical work on the problem of fracture propagation. Theories have evolved from the simplest 2D analyses to modern three-dimensional (3D) computer models. In this evolution, much attention has been given to the problems of fracture mechanics, rock deformation, fluid leakoff, proppant transport, and fluid rheology. The equally important problem of fluid temperature distributions in the fracture has received little attention.
Temperature considerations are especially important in modern fracturing operations where gelled fluids are used almost exclusively. These non-Newtonian gels have viscosity and sand-carrying characteristics that are very temperature-sensitive. At critical temperatures, they begin to decompose and lose virtually all their sand-carrying capabilities. Extending these decomposition limits to higher temperatures has been an area of intensive research recently. Exploiting these efforts requires a confident knowledge of fluid temperatures in the fracture during the treatment.
Thus, the problem of fracture temperature is a practically important one. Nevertheless, theoretical contributions in this area have been few in number. Best known is the important paper of Whitsitt and Dysart, who used energy- and mass-balance principles to derive a relationship for temperature in a propagating crack as a function of location and time. Although this was a pioneering work, it has certain shortcomings. It relies on a complicated Laplace transform that is not strictly applicable. It overlooks some essential features of the problem, and its final result is not very adaptable to finding self-consistent values for crack dimensions and temperatures.
Additional contributions have been made by Wheeler, by Sinclair, and by Poulsen and Lee. Wheeler considered the problem of heat transfer by steam injected into an existing problem of heat transfer by steam injected into an existing fracture. Because the fracture was treated as static, the results say little about temperature changes in a propagating fracture. Sinclair applied Wheeler's results to a growing fracture but treated fracture propagation independently of heat transfer. Because the two are strongly coupled, this approach is not likely to produce realistic results. Poulsen and Lee suggested some modifications to Whitsitt and Dysart's method.
Variational methods provide a much more natural approach to the fracture temperature problem. Such methods have already been developed for heat conduction and convection in mixed solid/fluid systems. It remains only to extend these general methods to the problem of crack growth with leakoff. problem of crack growth with leakoff. This is the approach we have used. Our purpose is to provide a detailed analysis of fluid temperature as a function of time and location during fracture growth with leakoff. As a starting point. we assume that the time history of crack growth has been point. we assume that the time history of crack growth has been determined by a method that neglects temperature effects. Thus, crack dimensions and leakoff are treated as known functions in the thermal analysis.
A second analysis of crack growth gives a better approximation by including temperature effects found from the thermal analysis. This process is repeated until convergence is obtained. The final results for crack dimensions and temperature distribution are thus self-consistent.
The thermal analysis developed this way is general enough to be used with almost any crack propagation theory. We include in this paper an example of its use with the 2D Lagrangian analysis. We assume that wellbore temperature at the fracture entrance is known from an independent analysis, such as that given by Ramey.
We compare results obtained from our variational approach with exact solutions for specific cases to verify the general accuracy of the method. The accuracy has also been improved by appropriate adjustments of constants.
The variational approach leads to a general partial-differential equation for temperature in the fracture as a function of time and position. We present two methods of solution: one based on position. We present two methods of solution: one based on successive approximations and the other on the method of characteristics. The latter is used in our examples on the basis of Lagrangian propagation analysis.
General variational principles for a porous solid have already been developed. We wish to apply these principles to the problem of deriving temperature distributions in a propagating fracture.
We treat the problem of heat flow away from the fracture face by considering a porous half-space. We take the y axis normal to the plane boundary, y=O, as in Fig. 1. The positive direction is toward the porous medium. We assume that fluid leakoff can be represented by a uniform volumetric fluid velocity, v, parallel to the y axis.