Perturbative corrections for the scaling of heat transport in a Hele-Shaw geometry and its application to geological vertical fractures

Abstract

In this work, we investigate numerically the perturbative effects of cell aperture in heat transport and thermal dissipation rate for a vertical Hele-Shaw geometry, which is used as an analogue representation of a planar vertical fracture at the laboratory scale. To model the problem, we derive a two-dimensional set of equations valid for this geometry. For Hele-Shaw cells heated from below and above, with periodic boundary conditions in the horizontal direction, the model gives new nonlinear scalings for both the time-averaged Nusselt number $\langle Nu\rangle _{\unicode[STIX]{x1D70F}}$ and dimensionless mean thermal dissipation rate $\langle \unicode[STIX]{x1D717}\rangle _{\unicode[STIX]{x1D70F}}$ in the high-Rayleigh regime. We demonstrate that $\langle Nu\rangle _{\unicode[STIX]{x1D70F}}$ and $\langle \unicode[STIX]{x1D717}\rangle _{\unicode[STIX]{x1D70F}}$ depend upon the cell anisotropy ratio $\unicode[STIX]{x1D716}$, which measures the ratio between the cell gap and height. We show that $\langle Nu\rangle _{\unicode[STIX]{x1D70F}}$ values in the high-Rayleigh regime decrease when $\unicode[STIX]{x1D716}$ grows, supporting the field observations at the fracture scale. When $\unicode[STIX]{x1D716}\ll 1$, our results are in agreement with the scalings found using the Darcy model. The numerical results satisfy the theoretical relation $\langle Nu\rangle _{\unicode[STIX]{x1D70F}}=Ra\langle \unicode[STIX]{x1D717}\rangle _{\unicode[STIX]{x1D70F}}$, which is obtained from the model. This latter relation is valid for all values of Rayleigh number considered. The perturbative effects of cell aperture are observed only in the exponents of the scalings $\langle Nu\rangle _{\unicode[STIX]{x1D70F}}\sim Ra^{\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D716})}$ and $\langle \unicode[STIX]{x1D717}\rangle _{\unicode[STIX]{x1D70F}}\sim Ra^{\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D716})-1}$.

How non-Darcy effects influence scaling laws in Hele-Shaw convection experiments

“…In particular, for b * = 0.80 mm and 1.00 mm, corresponding to a decrease of resistance to the flow, there is an important decrease of F , while hypothetical two-dimensional simulations would give a constant F . The case of the reported strong reduction of F in our experiments may be interpreted as the combined action of spurious transversal solute fluxes in the Hele-Shaw cell, which produce the dispersion effects as discussed by Letelier et al (2019), and the transition towards to the three-dimensional flow regime. Numerical simulations can replicate a perfect two-dimensional situation and therefore will be used here to benchmark and analyse our data in the limit of small thicknesses.…”

“…To fully appreciate the important role of hydrodynamic dispersion on this type of experiment, we present in figure 4 a global view of all our measurements of F , which is the time average of the solute flux F(t) during the constant flux regime. We underline here that Letelier et al (2019) used the scalar dissipation rate rather than F , but as discussed by De Paoli et al 2019, for the present configuration, F is equivalent to the mean scalar dissipation rate. In figure 4(a), we present F as a function of Ra, which is varied both by changing H * and b * , see (1.1).…”

“…Subsequent studies investigated the influence of the geometry of the domain on the features of the flow. In particular, with the aid of perturbation techniques and numerical simulations, Letelier, Mujica & Ortega (2019) investigated the effect of the combined action of the Rayleigh-Darcy number and the anisotropy ratio = b * / √ 12H * . In accordance with the value of the parameter 2 Ra, they have been able to identify three flow configurations: (i) Darcy regime ( 2 Ra → 0), where the flow is two-dimensional and well described by Darcy simulations; (ii) Hele-Shaw regime ( 2 Ra 1), where the flow is still two-dimensional but influenced by gap-induced dispersion; and (iii) three-dimensional regime ( 2 Ra 1), when the effects of the third dimension become non-negligible.…”

Vigorous convection in porous media

“…(e.g. [37,55]). In fact, the first equality in (3.5) is often taken as the explicit definition of Nu, from which (3.4) can be derived.…”

“…Of course, it may be that correctly accounting for the role of dispersion in this limit helps to alleviate this issue, but nevertheless, the limit in which small plumes becomes of a comparable size to pores is clearly realizable. Promising recent experimental [48] and computational [55] studies have begun to examine these issues in the simpler context of a Hele-Shaw cell, where one can at least rigorously determine the asymptotic corrections associated with a non-infinitesimal gap thickness. Experiments of convection in real porous media that are far from the Darcy limit (i.e.…”