Nonlinear Darcy–Stefan problem for radial Darcy flow in a porous medium during cold-fluid injection

About the authors:

Artem A. Tarasenko, Undergraduate Student, T. F. Gorbachev Kuzbass State Technical University, Branch in Novokuznetsk, Novokuznetsk, Russia

gamenk987@gmail.com

Evgeniya V. Prokopenko, Cand. Sci. (Phys.-Math.), Associate Professor, Head of the Department of Information Security, T. F. Gorbachev Kuzbass State Technical University, Kemerovo, Russia

pev.vtit@kuzstu.ru

Alyona S. Ashcheulova, Cand. Sci. (Phys.-Math.), Associate Professor, T. F. Gorbachev Kuzbass State Technical University, Kemerovo, Russia

asheulovaas@kuzstu.ru

Abstract:

A radially symmetric freezing problem for a saturated porous medium near an injection well during cold-fluid injection is examined. The relevance of the research is in the need to describe phase-change processes in the near-wellbore zone while accounting for radial flow and a moving boundary. The aim of the work is to develop and analyze a reduced Darcy–Stefan model for this setting. The problem is formulated in terms of the supercooling variable. A self-similar solution is obtained and expressed in terms of incomplete gamma functions. The front parameter is determined by a transcendental equation; the existence of a positive root is proved, and an explicit formula is derived for a particular case. A semi-implicit enthalpy scheme is used for numerical verification. An increase in the dimensionless filtration parameter β, which characterizes the ratio of filtration-induced convective transport to diffusive heat transfer, accelerates the freezing front; by the end of the simulation, the front-position error is lower for β > 1 than for β = 0.5. For β = 0.5, the maximum discrepancy is observed at the beginning of the simulation due to the influence of the inner boundary, the grid approximation, and the choice of the front-detection criterion.

References:

Borodin, S. L. (2015). Numerical solution of the Stefan’s problem. Tyumen State University Herald. Physical and Mathematical Modeling. Oil, Gas, Energy, 1(3), 164–175. https://doi.org/10.21684/2411-7978-2015-1-3-164-175 [In Russian]

Elkin, I. S. (2010). Process modeling of mass carry in a coal massif at humidification. Bulletin of the Kuzbass State Technical University, (1), 25–28. [In Russian]

Lee, K. Kh., Kazantsev, V. G., Tailakov, O. V., Lee, Kh. U., & Fomin, A. I. (2024). Coal formation damage accumulation model at coal mine degassing wells. Bulletin of the Kuzbass State Technical University, (1), 52–65. https://doi.org/10.26730/1999-4125-2024-1-52-65 [In Russian]

Musakaev, N. G., Borodin, S. L., & Romanyuk, S. N. (2014). Methods for solving the one-dimensional radial heat-transfer problem in frozen rocks surrounding a well. Bulletin of Tyumen State University. Physics and Mathematics. Informatics, (7), 19–26. [In Russian]

Musakaev, N. G., & Khasanov, M. K. (2014). Mathematical modeling of gas extraction from a gas-hydrate reservoir with account of the ice formation. Bulletin of Tyumen State University. Physics and Mathematics. Informatics, (7), 43–50. [In Russian]

Tailakov, O. V., Utkaev, E. A., Smyslov, A. I., & Kormin, A. N. (2014). Physical modelling of coal seams filtration properties fluctuation. Bulletin of the Kuzbass State Technical University, (6), 13–16. [In Russian]

Khyamyalaynen, V. A., Baev, M. A., & Shevtsov, A. G. (2017). About the influence of the strain-stress state of coal seam on the propped fracture permeability. Bulletin of the Kuzbass State Technical University, (6), 121–126. https://doi.org/10.26730/1999-4125-2017-6-121-126 [In Russian]

Shevchenko, L. A., Tkachenko, D. A., & Astrakov, S. N. (2016). Mathematical modeling gasdynamically state coal solid in zone borehole in the drilling process. Bulletin of the Kuzbass State Technical University, (1), 67–70. [In Russian]

Shiryaev, S. N. (2019). Multistage degassing of coal-methane seams in the interaction of gas filtration processes and deformation of rock mass. Bulletin of the Kuzbass State Technical University, (5), 64–73. https://doi.org/10.26730/1999-4125-2019-5-64-73 [In Russian]

Shishlyaev, V. V., & Pimenov, V. P. (2024). Assessment of filtration properties of a coal seam and the length of a hydraulic fracturing crack based on the results of quantitative interpretation of injection falloff tests. Bulletin of the Kuzbass State Technical University, (2), 35–44. https://doi.org/10.26730/1999-4125-2024-2-35-44 [In Russian]

Alexiades, V., & Solomon, A. D. (1993). Mathematical Modeling of Melting and Freezing Processes. Hemisphere Publishing Corporation.

Alzoubi, M. A., Xu, M., Hassani, F. P., Poncet, S., & Sasmito, A. P. (2020). Artificial ground freezing: A review of thermal and hydraulic aspects. Tunnelling and Underground Space Technology, 104, 103534. https://doi.org/10.1016/j.tust.2020.103534

Basu, B., & Date, A. W. (1988). Numerical modelling of melting and solidification problems — A review. Sadhana, 13(3), 169–213. https://doi.org/10.1007/BF02812200

Crank, J. (1984). Free and Moving Boundary Problems. Clarendon Press.

Gerasimov, T. (2022, December 20). Benchmark test for the heat conduction equation with phase change: 3D axisymmetric problem of soil freezing around a BHE. OpenGeoSys Documentation. Retrieved March 28, 2026, from https://www.opengeosys.org/stable/docs/benchmarks/heatconduction/heatconduction-soil-freezing/Heat_c...

Li, T., Zhou, Y., Shi, X., Hu, X., & Zhou, G. (2018). Analytical solution for the soil freezing process induced by an infinite line sink. International Journal of Thermal Sciences, 127, 232–241. https://doi.org/10.1016/j.ijthermalsci.2018.01.013

Rees, D. A. S., & Bassom, A. P. (2010). Radial injection of a hot fluid into a cold porous medium: The effects of local thermal nonequilibrium. Computational Thermal Sciences, 2(3), 221–230. https://doi.org/10.1615/ComputThermalScien.v2.i3.20

Rodrigues, J. F., & Urbano, J. M. (1999). On a Darcy–Stefan problem arising in freezing and thawing of saturated porous media. Continuum Mechanics and Thermodynamics, 11(3), 181–191. https://doi.org/10.1007/s001610050110

Rubinstein, L. I. (1971). The Stefan Problem. American Mathematical Society.

Sazhenkov, S. A. (2004). Application of the method of kinetic equation to the Darcy–Stefan problem. Conservation Laws Preprint Server, 2004-006.

Voller, V. R., Cross, M., & Markatos, N. C. (1987). An enthalpy method for convection/diffusion phase change. International Journal for Numerical Methods in Engineering, 24(1), 271–284. https://doi.org/10.1002/nme.1620240119

Voller, V. R., & Prakash, C. (1987). A fixed grid numerical modelling methodology for convection-diffusion mushy region phase-change problems. International Journal of Heat and Mass Transfer, 30(8), 1709–1719. https://doi.org/10.1016/0017-9310(87)90317-6

Voller, V. R., Swaminathan, C. R., & Thomas, B. G. (1990). Fixed grid techniques for phase change problems: A review. International Journal for Numerical Methods in Engineering, 30(4), 875–898. https://doi.org/10.1002/nme.1620300419

Xu, M., Akhtar, S., Zueter, A. F., Auger, V., Alzoubi, M. A., & Sasmito, A. P. (2020). Development of analytical solution for a two-phase Stefan problem in artificial ground freezing using singular perturbation theory. Journal of Heat Transfer, 142(12), 122401. https://doi.org/10.1115/1.4048137

Xu, M., Akhtar, S., Zueter, A., Alzoubi, M. A., Sushama, L., & Sasmito, A. P. (2021). Asymptotic analysis of a two-phase Stefan problem in annulus: Application to outward solidification in phase change materials. Applied Mathematics and Computation, 408, 126343. https://doi.org/10.1016/j.amc.2021.126343