ANALYTICAL SOLUTIONS OF HEAT TRANSPORT IN FRACTURED RESERVOIRS WITH A GENERALIZED MULTIRATE MEMORY FUNCTION

Numerous analytical solutions have been developed for modeling thermal perturbations to underground formations caused by deep-well injection of fluids. Each solution has been derived for a specific boundary value problem and a simplified flow network with one set of parallel fractures. In this paper, new generalized solutions G^*{x,s} are developed using (existing) global transfer functions G₀^*{x,s} and a new memory function g^*(s), where x and s are the space and Laplace variable. The memory function represents the solutions of conductive heat exchange between fractures and matrix blocks and between fractured aquifers and unfractured aquitards. The memory function is developed to account for multirate exchange induced by different shapes, sizes, properties, and volumetric fractions of matrix blocks bounded by multiple sets of orthogonal fractures with different spacing. The global transfer functions represent the fundamental solutions to convective, convective-conductive, and convective-dispersive heat transport in fractures (or aquifers) without exchange, and are available for various (1-D linear, 1-D radial, 2-D dipole, and single-well injection-withdrawal) flow fields. The new solutions with exchange are developed using G^*{x,s}=B^*(s)G₀^*{x,s[1+θg^*(s)]}, thereby greatly simplifying solution development in a novel way, where θ and B^*(s) are a fracture-matrix scaling factor and the inlet boundary condition function. The new solutions are applied to several example problems, showing that heat transport in fractured aquifers is significantly impacted by (1) thermal dispersion in fractures that is rarely considered, (2) multirate heat exchange with a wide range of size and anisotropy of rectangular matrix blocks, and (3) heat exchange between aquifers and aquitards.