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

  • Juvenal A. Letelier (a1) (a2), Nicolás Mujica (a3) and Jaime H. Ortega (a4)
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 $\langle \unicode[STIX]{x1D717}\rangle _{\unicode[STIX]{x1D70F}}\sim Ra^{\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D716})-1}$ .

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Corresponding author
Email address for correspondence: juvenal.letelier@ing.uchile.cl
References
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Adams, B. M., Kuehn, T. H., Bielicki, J. M., Randolph, J. B. & Saar, M. O. 2014 On the importance of the thermosiphon effect in CPG (CO2 plume geothermal) power systems. Energy 69, 409418.
Ahlers, G., Funfschilling, D. & Bodenschatz, E. 2009 Transitions in heat transport by turbulent convection at Rayleigh numbers up to 1015 . New J. Phys. 11, 123001.
Backhaus, S., Turitsyn, K. & Ecke, R. E. 2011 Convective instability and mass transport of diffusion layers in a Hele-Shaw geometry. Phys. Rev. Lett. 106, 104501.
Battistelli, A., Calore, C. & Pruess, K. 1997 The simulator TOUGH2/EWASG for modelling geothermal reservoirs with brines and non-condensible gas. Geothermics 26 (4), 437464.
Benson, S., Cook, P. et al. 2006 Underground geological storage. In Carbon Dioxide Capture and Storage. Special Report of the Intergovernmental Panel on Climate Change (ed. Metz, B., Davidson, O., de Coninck, H., Loos, M. & Meyer, L.), pp. 195276. Cambridge University Press.
Bizon, C., Werne, J., Predtechensky, A. A., Julien, K., McCormick, W. D., Swift, J. B. & Swinney, H. L. 1997 Plume dynamics in quasi-2D turbulent convection. Chaos 7, 107124.
Brown, D. 2000 A hot dry rock geothermal energy concept utilizing supercritical CO2 instead of water. In Proceedings of the Twenty-Fifth Workshop on Geothermal Reservoir Engineering, pp. 233238. Stanford University.
Cherkaoui, A. & Wilcock, W. 2001 Laboratory studies of high Rayleigh number circulation in an open-top Hele-Shaw cell: an analog to mid-ocean ridge hydrothermal systems. J. Geophys. Res. 106, 10983.
Cooper, C., Crews, J., Schumer, R., Breitmeyer, R., Voepel, H. & Decker, D. 2014 Experimental investigation of transient thermal convection in porous media. Trans. Porous Med. 104, 335347.
Croucher, A. & O’Sullivan, M. 2008 Application of the computer code TOUGH2 to the simulation of supercritical conditions in geothermal systems. Geothermics 37, 622634.
Davies, G. F. 1999 Dynamic Earth: Plates, Plumes and Mantle Convection. Cambridge University Press.
Elder, J. W. 1967 Steady free convection in a porous medium heated from below. J. Fluid Mech. 27, 2948.
Foster, T. D. 1965 Stability of a homogeneous fluid cooled uniformly from above. Phys. Fluids 8, 12491257.
Gondret, P. & Rabaud, M. 1997 Shear instability of two-fluid parallel flow in a Hele-Shaw cell. Phys. Fluids 9, 32673274.
Grossmann, S. & Lohse, D. 2011 Multiple scaling in the ultimate regime of thermal convection. Phys. Fluids 23, 045108.
Hewitt, D. R., Neufeld, J. A. & Lister, J. R. 2012 Ultimate regime of high Rayleigh number convection in a porous medium. Phys. Rev. Lett. 108, 224503.
Hidalgo, J. J., Fe, J., Cueto-Felgueroso, L. & Juanes, R. 2012 Scaling of convective mixing in porous media. Phys. Rev. Lett. 109, 264503.
Jenny, P., Lee, J. S., Meyer, D. W. & Tchelepi, H. A. 2014 Scale analysis of miscible density-driven convection in porous media. J. Fluid Mech. 749, 519541.
Jha, B., Cueto-Felgueroso, L. & Juanes, R. 2011 Quantifying mixing in viscously unstable porous media flows. Phys. Rev. E 84, 066312.
Jha, B., Cueto-Felgueroso, L. & Juanes, R. 2013 Synergetic fluid mixing from viscous fingering and alternating injection. Phys. Rev. Lett. 111, 144501.
Joseph, D., Huang, A. & Hu, H. 1996 Non-solenoidal velocity effects and Korteweg stresses in simple mixtures of incompressible liquids. Physica D 97, 104125.
Kvernvold, O. & Tyvand, P. A. 1981 Dispersion effects on thermal convection in a Hele-Shaw cell. Intl J. Heat Mass Transfer 24, 887890.
Lele, S. K. 1992 Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 1642.
Letelier, J. A., Herrera, P., Mujica, N. & Ortega, J. H. 2016 Enhancement of synthetic schlieren image resolution using total variation optical flow: application to thermal experiments in a Hele-Shaw cell. Exp. Fluids 57 (2), 114.
López, D. L. & Smith, L. 1995 Fluid flow in fault zones: analysis of the interplay of convective circulation and topographically driven groundwater flow. Water Resour. Res. 31, 14891503.
Malkus, W. V. 1954 The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. A 225, 196212.
Murphy, H. D. 1979 Convective instabilities in vertical fractures and faults. J. Geophys. Res. 84, 61216130.
Neufeld, J. A., Hesse, M. A., Riaz, A., Hallworth, M. A., Tchelepi, H. A. & Huppert, H. E. 2010 Convective dissolution of carbon dioxide in saline aquifers. Geophys. Res. Lett. 37, L22404.
Nield, D. & Bejan, A. 2006 Convection in Porous Media, 3rd edn. Springer.
Nigon, B., Englert, A. & Pascal, C. 2015 Modeling to heat transport through fractures with emphasis to roughness and aperture variability. In EGU General Assembly Conf. Abstracts, vol. 17, p. 8821.
Oltean, C., Felder, C. H., Panfilov, M. & Bues, M. A. 2004 Transport with a very low density contrast in Hele-Shaw cell and porous medium: evolution of the mixing zone. Trans. Porous Med. 55, 339360.
Oltean, C., Golfier, C. & Bues, M. A. 2008 Experimental and numerical study of the validity of Hele-Shaw cell as analogue model for variable-density flow in homogeneous porous media. Adv. Water Resour. 31, 8295.
Otero, J., Dontcheva, L. A., Johnston, H., Worthing, R. A., Kurganov, A., Petrova, G. & Doering, C. R. 2004 High Rayleigh number convection in a fluid-saturated porous layer. J. Fluid Mech. 500, 263281.
Palm, E., Weber, J. E. & Kvernvold, O. 1972 On steady convection in a porous medium. J. Fluid Mech. 54, 153161.
Pramanik, S. & Mishra, M. 2015 Viscosity scaling of fingering instability in finite slices with Korteweg stress. Eur. Phys. Lett. 109, 64001.
Pruess, K.1991 TOUGH2: a general-purpose numerical simulator for multiphase nonisothermal flows. Tech. Rep. Lawrence Berkeley Laboratory, CA.
Randolph, J. B. & Saar, M. O. 2011 Coupling carbon dioxide sequestration with geothermal energy capture in naturally permeable, porous geologic formations: implications for CO2 sequestration. Energy Proc. 4, 22062213.
Riaz, A., Hesse, M., Tchelepi, H. A. & Orr, F. M. 2006 Onset of convection in a gravitationally unstable diffusive boundary layer in porous media. J. Fluid Mech. 548, 87111.
Ruyer-Quil, C. 2001 Inertial corrections to the Darcy law in a Hele-Shaw cell. C. R. Acad. Sci. Paris IIb 329, 337342.
Scheck-Wenderoth, M., Cacace, M., Petrovich, Y., Cherubini, Y., Noack, V., Onno, B., Sippel, J. & Bjorn, L. 2014 Models of heat transport in the Central European Basin System: effective mechanisms at different scales. Mar. Petrol. Geol. 55, 315331.
Scheidegger, A. E. 1974 The Physics of Flow through Porous Media. University of Toronto Press.
Schoofs, S., Spera, F. & Hansen, U. 1999 Chaotic thermohaline convection in low-porosity hydrothermal systems. Earth Planet. Sci. Lett. 174, 213229.
Sheu, L., Tam, L., Chen, J., Chen, H., Lin, K. & Kang, Y. 2008 Chaotic convection of viscoelastic fluids in porous media. Chaos, Solitons Fractals 37, 113124.
Waleffe, F., Boonkasane, A. & Smith, L. 2015 Heat transport by coherent Rayleigh–Bénard convection. Phys. Fluids 27, 051702.
Winters, K. B. & de la Fuente, A. 2012 Modelling rotating stratified flows at laboratory-scale using spectrally-based DNS. Ocean Model. 49–50, 4759.
Zhao, C., Hoobs, B. E. & Ord, A. 2008 Convective and Advective Heat Transfer in Geological Systems, chap. 9. Springer.
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