Application of Thermal Taylor Dispersion to Upscaling of Geothermal Processes in Heterogeneous Formations

Well-logging data show that geothermal formations typically feature layered heterogeneities. This imposes a challenge in numerical simulations, in particular in the upscaling of geothermal processes. The goal of our study is to develop an approach to (1) simplify the description of heterogeneous geothermal formations and (2) provide an accurate representation of convection/dispersion processes for simulating the up-scaled system.

In geothermal processes, transverse thermal conduction causes extra spreading of the cooling front: thermal Taylor dispersion. We derive a model from an energy balance for effective thermal diffusivity, α_eff, to represent this phenomenon in layered media. α_eff, accounting for transverse heat conduction, is much greater than the longitudinal thermal diffusivity, leading to a remarkably larger effective dispersion. A ratio of times is defined for longitudinal thermal convection and transverse thermal conduction, referred to as transverse thermal-conduction number N_TC. The value of N_TC is an indicator of thermal equilibrium in the vertical cross-section. Both N_TC and α_eff equations are verified by a match with numerical solutions for convection/conduction in a two-layer system. For N_TC > 5, the system behaves as a single layer with thermal diffusivity α_eff.

When N_TC > 5, a two-layer system can be combined and represented with α_eff and average properties of the two layers. We illustrate upscaling approach for simulation of geothermal processes in stratified formations, by grouping layers based on the condition of N_TC > 5 and the α_eff model. Specifically, N_TC is calculated for every adjacent two layers, and the paired layers with a maximum value of N_TC are grouped first. This procedure repeats on the grouped system until no adjacent layers meet the criterion N_TC > 5. The upscaled layer properties of the grouped system are used as new inputs in the numerical simulations. The effectiveness of the upscaling approach is validated by a good agreement in numerical solutions for thermal convection/dispersion using original and average layer properties, respectively (Figs. 1 and 2 in the Supplementary Data File). The upscaling approach is applied to well-log data of a geothermal reservoir in Copenhagen featuring many interspersed layers. After upscaling, the formation with 93 layers of thickness 1 – 3 meters is upscaled to 12 layers (Fig. 3 in the Supplementary Data File). The effective thermal diffusivity α_eff in the flow direction is about a factor of 10 times greater than original thermal diffusivity of the rock. Thus, α_eff should be used as simulation inputs for representing more accurately geothermal processes in the up-scaled system.