Heat Transfer in Hydraulic Fracturing

Abstract

Summary An analytical heat transfer model is presented for the combined convection along a vertical fracture with conduction and convection in the reservoir. The model couples the energy (heal transfer) and fracture propagation equations resulting in a closed-form, dimensionless-temperature integral solution. Major model enhancements. include incorporation ofa finite film coefficient,time-dependent fracture temperature for calculating the instantaneous heat flux,energy storage in the fracture, andcoupled energy and fracture propagation equations. The introduction of a power-law Nusselt number to determine the convection-film coefficient is also new and unique. The influence of fluid leakoff and a finite convection-film coefficient as heat-reduction mechanisms is discussed. The effect of energy storage and a time-dependent fracture temperature on the heat transfer rate is also illustrated. Finally, simulation studies illustrating the effect of heat transfer on fracture propagation characteristics are presented. Introduction Heat transfer in hydraulic fracturing is an important consideration in high-temperature formations when the fracture-fluid rheology is temperature-dependent. The change in rheology with temperature and time in the fracture affects not only fracture propagation char-acteristics and proppant transport but also many other temperature dependent processes. A number of heat transfer models account for many of the complex heat-transfer mechanisms in the fracture and formation. The major difference between many of these analytical models is in coupling the energy and fracture propagation equations. These analytical models also do not include energy storage, a finite film coefficient, or the effect of a changing fracture-face temperature. The heat flux at the fracture face generally is calculated from the energy equation for a semi-infinite slab initially at a uniform temperature and subjected to a step change in temperature at the surface. This combined convection (fluid loss) and conduction heat flux is then coupled with the convective heat transfer in the fracture. The main difference in most of the analytical models comes from the various assumptions related to fluid leakoff, mass and energy conservation in the fracture, and heat flux. An analytical heat transfer model is presented that accounts for the combined transient heat conduction and convection from the fracture with a convection boundary condition at the fracture face. This model also, includes the additional mechanisms of a time-dependent fracture-face temperature and energy storage. The model is formulated for two-dimensional (2D) vertical hydraulic fracture geometries. The solution methodology, however, can be extended to pseudo-three-dimensional (3D) fracture models with the method of streamlines [modified one-dimensional (ID) fracture flow]. The constitutive relationships coupling the energy and fracture propagation equations are provided in detail for Geertsma-de Klerk (GdK) and Perkins-Kem-Nordgren (PKN) -type fracture geometries. The coupled system of equations results in a closed-form, analytical, integral dimensionless-temperature solution. A parametric study illustrates the effect of various heat transfer mechanisms and parameters on the dimensionless-temperature profile. Comparisons with other analytical and numerical models are also given. Finally, the effect of heat transfer on fracture propagation characteristics for GdK- and PKN-VN fracture geometries is illustrated. Dimensionless-Temperature Solution The solution methodology for solving the coupled continuity and energy-balance equations is based on an integral method for tracking an infinitesimal element of fluid in time and space. The equations of mass conservation, fracture propagation, and energy are then solved simultaneously. Therefore, unlike the analytical solutions of Whitsitt and Dysart, Which assume a given form for leakoff and the flow rate in the fracture, this model can easily be coupled to any 2D fracture model. The major heat transfer mechanisms and processes considered arecoupled energy and fracture propagation equations,energy and mass storage, andwall heat flux based on a finite convection-film coefficient and time-dependent temperature. The wall heat flux also accounts for fluid leakoff. Including these mechanisms results in a more realistic heat transfer model than the and a lytical models of Refs. 1 through 5 and many of the discrete models presented in Ref. 7. The resulting dimensionless fracture-temperature solution for the combined transient heat conduction and convection from the fracture with a convective boundary condition at the fracture face is ................(1a) or ................(1b) where ................(2) ................(3) ................(4) ................(5) ................(6) ................(7) and ..............(8) See Ref. 11 for a detailed summary of the model assumptions and solution methodology. The heat transfer mechanisms of convection in the fracture, fluid leakoff (convection) in the formation, and a finite convective-film coefficient at the fracture face are represented by, and, respectively. SPEPE November 1989 P. 423^