Mathematical theory of heat flow

This paper presents the mathematical theory of heat flow in the Earth’s crust, with emphasis on conductive heat transfer in homogeneous and isotropic media.
Starting from Fourier’s law, the fundamental differential equations governing onedimensional and three-dimensional heat flow are derived and extended to include temperature variation with time. The study incorporates the effects of physical parameters such as thermal conductivity, density, and specific heat, leading to the classical heat equation. Special attention is given to geothermal applications, including the interpretation of subsurface temperature distributions and their relation to geological structures. The influence of porosity and moisture content on thermal conductivity is also analysed. Furthermore, periodic temperature variations caused by solar radiation,
including diurnal and annual cycles, are mathematically modelled to evaluate temperature changes with depth. The presented theory provides a basis for understanding geothermal processes and their applications in geophysics, engineering geology, and subsurface exploration.