# Heat Transfer in Sandstones at Low Temperature

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## Abstract

This paper addresses experimental and modeling investigations of heat transfer in sandstones subject to low-temperature conditions. At low temperature, pore liquid (e.g., water) would freeze; thus, heat is transferred not only in the form of specific heat but also in the form of latent heat. Moreover, the melting point is not constant; it depends on the pore size. Considering these characteristics, a governing equation of heat transfer with phase transition is established using the equivalent heat-capacity method. To calculate the equivalent heat capacity, the relation between ice content and temperature is assessed by the pore-size distribution curve. Heating tests (from 77 to 293 K) of sandstone samples in three saturation conditions (water-saturated, oil-saturated, and dry) are conducted and simulated using the model established. The results reveal that the temperature sensitivity of the heat capacity of dry sandstone is more pronounced in the low-temperature regime than in the high-temperature regime. The thermal conductivity of dry sandstone increases with temperature in the low-temperature regime. This is different with the case of the high-temperature regime at which the thermal conductivity decreases with temperature. The temperature evolution curve for the water-saturated sample features a plateau regime, that is, the temperature remains quasi-constant with time. The analysis demonstrates that the position and length of this temperature plateau are governed by the pore-size distribution.

## Keywords

Freezing–thawing Sandstone Convective boundary Heat capacity Thermal conductivity Pore-size distribution## List of Symbols

*A*_{s}Convective area

- \(a,{\text{ }}b,{\text{ }}{a_j},{\text{ }}{b_j}\)
Coefficients describing the dependence of heat capacity on temperature;

*j*can be i (ice), w (water), s (solid) or o (oil)- Bi
Biot number

*c*Specific heat

*D*Spreading coefficient

*e*Thickness of pre-melting liquid film

*h*Convection coefficient

*k*Thermal conductivity

*L*Latent heat

*L*_{c}Characteristic length

*m, n*Coefficients describing the dependence of thermal conductivity on temperature

*p*Mercury pressure

*r*Pore radius

*r*_{i}Smallest pore-access radius invaded by ice crystals

- \({r_a}\)
Given radius

*R*Correlation coefficient

*S*_{w},*S*_{i}Molar entropy of water (w) and ice (i)

*t*Time

*T*Temperature

*T*_{i}Initial temperature

*T*_{∞}Environment temperature

*T*_{s}Temperatures of the solid surface

*T*_{m}Melting point at atmospheric pressure

*V*Volume

- \({\bar {V}_{\text{i}}},{\text{ }}{\bar {V}_{\text{w}}}\)
Molar volume of ice (i) and water (w)

*α*Contact angle of ice–water interface

*ρ*Density

- \({\rho _{\text{i}}},{\rho _{\text{w}}},{\rho _{\text{s}}}\)
Density of ice (i), water (w) and solid (s)

- \(\nabla\)
Gradient operator

*θ*_{i}Volumetric fraction of ice

- \(\phi\)
Porosity

- \({\gamma _{{\text{iw}}}},{\gamma _{{\text{si}}}},{\gamma _{{\text{sw}}}}\)
Interface stress of ice–water (iw), solid–ice (si) and solid–water (sw)

- \(\xi\)
Range of intermolecular forces

- \({\sigma _{{\text{Hg}}}}\)
Interfacial tension of mercury

## Notes

### Acknowledgements

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (Grant no. 51809275) and the Science Foundation of China University of Petroleum, Beijing (2462018BJC002).

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