Transport in Porous Media

, Volume 124, Issue 1, pp 221–236 | Cite as

Thermal Conduction in Deforming Isotropic and Anisotropic Granular Porous Media with Rough Grain Surface

  • Roohollah AskariEmail author
  • S. Hossein Hejazi
  • Muhammad Sahimi


Resistance to the heat flow in solid–solid contact areas plays a fundamental role in heat transfer in unconsolidated porous materials. In the present work, we study thermal conduction in granular porous media that undergo deformation due to an external compressing pressure. The media’s grains have rough surface, with the roughness profile following the statistics of self-affine fractals that have been shown to be abundant in natural porous media. We utilize a fractal contact model of rough surfaces in order to estimate the deformation of the contact areas, which is a function of roughness fractal parameters, the grains’ Young modulus, and the compressing pressure. For porous media saturated by a single fluid, the effects of various factors, such as the porosity, the grains’ overlap (consolidation), and shapes (circular vs. elliptical), are all studied. Increasing the compressing pressure enhances heat transfer due to deformation of the rough surface of the gains. The thermal conductivity of the medium is strongly affected by the porosity, when the grains’ conductivity is much larger than that of the fluid that saturates the pore space. Furthermore, we show that thermal anisotropy is a decreasing function of roughness deformation. In other words, granular media with rougher grains exhibit larger anisotropy as measured by the ratio of the directional thermal conductivities. Whereas in one type of granular media the anisotropy eventually vanishes at very high compressing pressure, it persists in a second model of anisotropic media that we study.


Roughness deformation Anisotropy Thermal conductivity Granular porous media Thermal contact resistance 

List of symbols

\( a \)

Contact area of a deforming rough surface

\( D \)

Fractal dimension

\( G \)

Fractal roughness parameter


Hurst exponent

\( K_{\text{e}} \)

Effective thermal conductivity

\( K_{\text{s}} \)

Matrix thermal conductivity

\( K_{\text{f}} \)

Fluid thermal conductivity

\( K_{\parallel } \)

Effective thermal conductivity parallel to the main axis of grains and the overlap direction

\( K_{ \bot } \)

Effective thermal conductivity perpendicular to the main axis of grains and the overlap direction

\( L \)

Sample’s length

\( P_{\text{e}} \)

Compressing pressure

\( S \)

Power spectrum of a rough surface


Young’s modulus

\( \delta \)

Height of the deformation

\( \nu \)

Scaling parameter

\( \omega \)



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Authors and Affiliations

  1. 1.Department of Geological and Mining Engineering and SciencesMichigan Technological UniversityHoughtonUSA
  2. 2.Department of Chemical and Petroleum EngineeringUniversity of CalgaryCalgaryCanada
  3. 3.Mork Family Department of Chemical Engineering and Materials ScienceUniversity of Southern CaliforniaLos AngelesUSA

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