A three-dimensional finite-element model for hydraulic fracturing has been developed that accounts for local thermal non-equilibrium between the injected fluid and host rock. The model also accounts for fluid flow and heat transfer within the fracture, heat conduction through the solid rock, deformation of the rock, and propagation of the fracture. Fluid flow through the fractures is modeled using the lubrication equation, and is fully coupled to the thermoelastic mechanical model through the pressure exerted by the fluid on the fracture walls, as well as by ensuring compatibility of fracture volumetric strains. Fractures are discretely modeled using triangular surfaces in an unstructured three-dimensional mesh. The growth of fractures is modeled using linear elastic fracture mechanics (LEFM), with the onset and direction of growth based on stress intensity factors that are computed for unstructured triangle-tetrahedral meshes. The model has been verified against analytical solutions available in the literature for penny-shaped (3D) fractures. A radial hydraulic fracture from a horizontal well is simulated to investigate the effects of the thermal non-equilibrium between the fracturing fluid and the host rock. For the case of very low matrix rock permeability, results show very little influence of thermal effects on the creation of hydraulic fractures.
In hydraulic fracturing, the fracturing fluid is commonly not in thermal equilibrium with the solid medium, leading to heat transfer between the fluid inside the fracture and the host rock, through the fractures surfaces. Natural and man-made examples are the creation of several-kilometers-long magma-driven dikes (Spence and Turcotte, 1985) and hydraulic fracturing of gas shale (Tran et al., 2013), respectively.
Hydraulic fracturing has attracted a great deal of attention over the past decades. Many analytical and semi-analytical solutions have been developed to quantify hydraulic fracturing variables of interest, such as injection pressure, fracture aperture and fracture length (cf. Adachi et al., 2007). These semi-analytical solutions provide the foundation for hydraulic fracturing design (e.g. Cleary, 1980; Cleary et al., 1988). These solutions are constructed by combining the equations for laminar flow through the fracture with elastic deformation of the adjacent rock. Fluid flow through the fracture is commonly modelled using lubrication theory (Batchelor, 1967) for an incompressible Newtonian fluid obeying the cubic law, while fracture aperture is calculated using linear elasticity in conjunction with Linear Elastic Fracture Mechanics (LEFM) for the computation of the mode I stress intensity factor at the fracture tip (Geertsma and de Klerk, 1969; Spence and Sharp, 1985). The lubrication equation is derivable from the general Navier-Stokes equation for the flow between two parallel plates (Zimmerman and Bodvarsson, 1996).
Heat conduction between the fluid inside fracture and surrounding rock solid has been of particular interest in many cases including magma-driven fractures (Spence and Turcotte, 1985), hydraulic fracturing of wells (Wang and Papamichos, 1999), and hydraulic fracturing of gas shale (Tran et al., 2013; Enayatpour and Patzek, 2013). Rock temperature at the surfaces of the hydraulic fracture is often considered constant and equal to the temperature of the injected fluid (Tran et al., 2013; Abousleiman et al., 2014). However, such an assumption does not satisfy conservation of energy, and does not account the fact that heat exchange between the fracturing fluid and the rock gradually causes the fracturing fluid to thermally equilibrate with the matrix rock. Consequently, an unrealistically large effect due to thermal non-equilibrium is predicted by such approaches.