# Experimental and Numerical Study of the Onset of Transient Natural Convection in a Fractured Porous Medium

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## Abstract

In this study, analysis of transient natural convection in a porous medium with a vertical fracture across it in a rectangular enclosure was performed experimentally and numerically. A number of thermocouples in distinctive distances in the model were used to measure temperature distribution and confirm the numerical simulations. A detailed CFD simulation was carried out, which was based on the finite volume method to extend our experimental observations. The onset of transient natural convection in the fractured porous media was observed by experimental setup and successfully simulated by numerical analysis. A modified Rayleigh number for the constant surface temperature at the bottom boundary condition of the model was calculated in this study (i.e., use of \(K_\mathrm{avg}\) instead of *K*), which was near to the theoretical value of linear stability analysis. Moreover, the critical time of the onset convection in the fractured model was predicted. The average Nusselt number was estimated to be about 3.5 for this fractured porous model.

## Keywords

Transient natural convection Fractured porous media Rayleigh number Critical length and time scales CFD## List of Symbols

- AAPD
Average absolute percent deviation, \((\hbox {AAPD}=\frac{100}{\hbox {number of data}}\sum |\frac{(\text {Value}_{\mathrm{simulated}} -\text {Value}_{\mathrm{experimental}})}{\text {Value}_{\mathrm{experimental}}}|)\)

*b*Fracture thickness (m)

*c*Specific heat capacity (J/kg K)

*d*Packed sphere mean diameter (m)

*D*Free molecular diffusion coefficient in a viscous fluid \((\hbox {m}^{2}/\hbox {s})\)

- \(D_\mathrm{m}\)
Effective molecular diffusion in porous medium \((\hbox {m}^{2}/\hbox {s})\)

*g*Acceleration gravity \((\hbox {m}/\hbox {s}^{2})\)

*k*Thermal conductivity (W/m K)

*K*Permeability \((\hbox {m}^{2})\)

*l*Boundary layer growth thickness (m)

*L*Wide of porous medium (m)

*Le*Lewis number

- \(\alpha _\mathrm{m}\)
Effective thermal diffusivity in porous medium \((\hbox {m}^{2}/\hbox {s})\)

*Nu*Nusselt number

*p*Pressure (Pa)

*Pr*Prandtl number

*Ra*Rayleigh number

- REV
Representative elementary volume

*Sc*Schmidt number

- \(t_\mathrm{c}\)
Critical time of natural convection (s)

- \(T_\mathrm{f}\)
Hot temperature of the lower wall (K)

- \(T_\mathrm{o}\)
Initial temperature of the fluid (K)

*U*Velocity (m/s)

- \(U_{0}\)
Overall uncertainty \((^{\circ }\hbox {C})\)

- \(U_\mathrm{D}\)
Uncertainty associated with data acquisition system \((^{\circ }\hbox {C})\)

- \(U_\mathrm{T}\)
Uncertainty associated with thermocouple \((^{\circ }\hbox {C})\)

- \(\gamma \)
Fracture inclination from vertical direction (Degree)

- \(\phi \)
Porosity

- \(\widetilde{a_\mathrm{c} }\)
Dimensionless wave number

- \(\beta \)
Thermal volume expansion coefficient (1/K)

- \(\mu \)
Liquid viscosity (kg/m s)

- \(\nu \)
Kinematic viscosity \((\hbox {m}^{2}/\hbox {s})\)

- \(\rho \)
Density \((\hbox {kg}/\hbox {m}^{3})\)

- \(\Delta T\)
Temperature difference \((T_{\mathrm{H}}-\hbox {Tc})\) (K)

- \(\lambda _\mathrm{C}\)
Critical wavelength for convection cell (m)

- \(_{\sigma }\)
Heat capacity ratio

## Subscripts

- f
Fluid

- s
Solid part of porous medium

- o
Initial condition

- c
Critical

- m
Porous medium

## Notes

### Acknowledgements

The authors would like to gratefully thank Iranian South Oilfields Company (NISOC) for their technical help and financial support by Grant Number of 904. Also, the authors are grateful from Mr. Ehsan Farrokhi from South Pars Gas Complex (SPGC) for his technical help.

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