Analysis of the heat transfer in subsurface porous media with considering Robin-type boundaries and arbitrary surface temperature variations

Abstract

Heat occurs everywhere in the environment and has been used as a natural groundwater tracer in many hydrogeological studies. Several studies have developed various mathematical models to analyze the heat transfer in subsurface porous media. However, most of these models have two limitations: (1) the media have semi-infinite thicknesses, and (2) the surface temperature variations follow specific temporal patterns. As a result, the models are qualified for only some specific situations. To handle these geothermal problems effectively, it is essential to develop a general model. This study therefore develops a heat transfer model for simulating spatiotemporal distributions of the subsurface temperature (i.e., geotemperature) in a finite-thickness porous medium. This model comprises a heat conduction-advection equation that is subject to the condition of arbitrary surface temperature variations and a Robin-type condition for the bottom convective boundary. The solution of this model is obtained via the Laplace transform and Duhamel's principle and validated by a finite-difference solution. In a case study, we consider saturated sandy soils with the thermal properties of heat capacity cρ = 2.96×106 J/m3/ °C and thermal conductivity λ = 2.2 W/m/ °C for analyzing the geotemperature via the present solution. By way of the solution, the results show that the model can reduce to a pure-conduction model at the thermal Peclet number Pe≤10−4 and to a pure-advection model at Pe≥102. The Robin boundary condition can become the Dirichlet boundary condition at the dimensionless heat transfer coefficient Hf ≥ 103 and the Neumann boundary condition at Hf ≤ 10−2. The responses of the geotemperature to the changes in the thermal parameters are inspected in a sensitivity analysis. The present solution agrees better with the field data from the Sendai Plain in Japan than the existing solutions subject to the limitation of semi-infinite-thickness porous media on predicting geotemperature-depth profiles. As stated above, the developed solution can solve relatively comprehensive geothermal problems and greatly benefits the studies on the heat transport in subsurface porous media.