Upscaling of Modeling of Thermal Dispersion in Stratified Geothermal Formations

Upscaling of geothermal properties is necessary given the computational cost of numerical simulations. Nevertheless, accurate upscaling of thermo-physical properties of layers combined in simulation grid blocks has been a long-standing challenge. In stratified porous media, non-uniform velocity between layers combined with transverse thermal conduction across layers causes spreading of the thermal front: thermal Taylor dispersion. Neither effect of heterogeneity is accounted for in conventional upscaling. Based on thermal Taylor dispersion, we develop a new upscaling technique for simulation of geothermal processes in stratified formations. In particular, we derive a model for effective longitudinal thermal diffusivity in the direction of flow, α_eff, to represent this phenomenon in two-layer media. α_eff, accounting for differences in velocity and transverse thermal conduction, is much greater than the thermal diffusivity of the rock itself, leading to a remarkably larger effective dispersion. We define a dimensionless number, N_TC, a ratio of times for longitudinal convection to transverse conduction, as an indicator transverse thermal equilibration of the system during cold-water injection. Both N_TC and α_eff equations are verified by a match to numerical solutions for convection/conduction in two-layer systems. We find that for N_TC > 5, thermal dispersion in the system behaves as a single layer with α_eff This suggests a two-layer medium satisfying N_TC > 5 can be combined into a single layer with an effective longitudinal thermal diffusivity α_eff. Compared with conventional approaches by averaging, the α_eff model provides more accurate upscaling of thermal diffusivity and thus more-accurate prediction of cooling-front breakthrough. In stratified geothermal reservoirs with a sequence of layers, upscaling can be conducted in stages, e.g. combining two layers satisfying the N_TC criterion in each stage. The application of the new technique to upscaling geothermal well-log data will be presented in a companion paper.