where $\mathbf{k}_{pT}$ is the phenomenological coefficient of thermo-osmosis and $\mathbf{k}_{Tp}$ the phenomenological coefficient of thermo-filtration.
It can be shown that $\mathbf{k}_{Tp}=T*\mathbf{k}_{pT}$ (Zhou et al. 1998).
/home/buchwalj/.local/lib/python3.10/site-packages/vtuIO.py:147: PerformanceWarning: DataFrame is highly fragmented. This is usually the result of calling `frame.insert` many times, which has poor performance. Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
An analytical solution was provided by Zhou et al. 1998 and can be obtained via [github](https://github.com/joergbuchwald/thermo-osmosis_analytical_solution).
For this example we used $\mathbf{k}_{pT}=2.7e-10\, m^2/(s K)$ and a fully saturated material. More details on model parameters can be found in the corresponding project files.
The Thermo-Richards (TR) model uses a correction to account for mechanical effects in the mass-balance equation. See Buchwald et al. 2021 for further details.
The differences between the analytical solution and OGS is assumed to come from the neglectance of the advective heat-flux in the analytical solution.
## References
[1] Zhou, Y., Rajapakse, R. K. N. D., & Graham, J. (1998). A coupled thermoporoelastic model with thermo-osmosis and thermal-filtration, International Journal of Solids and Structures, 35(34-35), 4659-4683.
[2] Buchwald, J., Kaiser, S., Kolditz, O., & Nagel, T. (2021). Improved predictions of thermal fluid pressurization in hydro-thermal models based on consistent incorporation of thermo-mechanical effects in anisotropic porous media. International Journal of Heat and Mass Transfer, 172, 121127.