Non-local dispersion and the reassessment of Richardson's t3-scaling law

The Richardson-scaling law states that the mean square separation of a fluid particle pair grows according to t3 within the inertial range and at intermediate times. The theories predicting this scaling regime assume that the pair separation is within the inertial range and that the dispersion is local, which means that only eddies at the scale of the separation contribute. These assumptions ignore the structural organization of the turbulent flow into large-scale shear layers, where the intense small-scale motions are bounded by the large-scale energetic motions. Therefore, the large scales contribute to the velocity difference across the small-scale structures. It is shown that, indeed, the pair dispersion inside these layers is highly non-local and approaches Taylor dispersion in a way that is fundamentally different from the Richardson-scaling law. Also, the layer's contribution to the overall mean square separation remains significant as the Reynolds number increases. This calls into question the validity of the theoretical assumptions. Moreover, a literature survey reveals that, so far, t3 scaling is not observed for initial separations within the inertial range. We propose that the intermediate pair dispersion regime is a transition region that connects the initial Batchelor- with the final Taylor-dispersion regime. Such a simple interpretation is shown to be consistent with observations and is able to explain why t3 scaling is found only for one specific initial separation outside the inertial range. Moreover, the model incorporates the observed non-local contribution to the dispersion, because it requires only small-time-scale properties and large-scale properties.

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

<p>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.</p><p>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<sub>eff</sub>, to represent this phenomenon in layered media. α<sub>eff</sub>, 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<sub>TC</sub>. The value of N<sub>TC</sub> is an indicator of thermal equilibrium in the vertical cross-section. Both N<sub>TC</sub> and α<sub>eff</sub> equations are verified by a match with numerical solutions for convection/conduction in a two-layer system. For N<sub>TC</sub> > 5, the system behaves as a single layer with thermal diffusivity α<sub>eff</sub>.</p><p>When N<sub>TC</sub> > 5, a two-layer system can be combined and represented with α<sub>eff</sub> 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<sub>TC</sub> > 5 and the α<sub>eff</sub> model. Specifically, N<sub>TC</sub> is calculated for every adjacent two layers, and the paired layers with a maximum value of N<sub>TC</sub> are grouped first. This procedure repeats on the grouped system until no adjacent layers meet the criterion N<sub>TC</sub> > 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<sub>eff</sub> in the flow direction is about a factor of 10 times greater than original thermal diffusivity of the rock. Thus, α<sub>eff</sub> should be used as simulation inputs for representing more accurately geothermal processes in the up-scaled system.</p><p> </p><p> </p>