A semi-analytical solution for heat transport in rock with parallel fractures and a heat source in both fracture and matrix

In recent decades, numerous analytical solutions have been developed to quantify temperature perturbations in fractured rock having mobile and immobile fluid phases. The assumption of one-dimensional heat conduction in the matrix, or neglecting heat dispersion and storage in fractures, however, are typical simplifications adopted to overcome the difficulties in mathematically representing the problem. In this study, we propose a two-dimensional semi-analytical solution framework based on a Green's function approach for a flexible heat source definition, including source dimensions, energy delivery strength and duration, and the presence of a heat source in the matrix and/or fracture. The solution fully accounts for heat conduction, advection, dispersion, and transient heat exchange between the fracture fluid and rock matrix in a system of parallel fractures. The solution having a strip heat source extending from a fracture into the matrix indicates that one-dimensional heat conduction in the matrix underestimates and overestimates temperature responses at early and later times, respectively. When cold-water is injected following the heating period, the fracture temperature near the heat source grows more slowly with increasing fracture aperture sizes; although, the fracture temperature in a larger fracture is retained for a longer distance in the downstream direction comparing to the scenario with a small aperture size. The fracture temperature growth is enhanced once the overlapped heating area between the adjacent parallel fractures has been developed. The transient temperature analyses imply that the spatial temperature variation is strongly associated with heat delivery strength. The early time temperature anomalies are closely related to the heat source configurations, and the later time temperatures in the domain are mainly determined by the total energy being delivered into the domain.