Application of the heat equation to the study of underground temperature

TL;DR

This paper extends the one-dimensional heat equation to include rainwater flow, analyzing its impact on underground temperature propagation and providing practical tools for geophysical applications and climate studies.

Contribution

It introduces an analytical and numerical approach to model underground temperatures with water flow, highlighting effects neglected in standard textbooks.

Findings

Rainwater circulation influences subsurface temperature propagation.

Soil dampens surface temperature variations, affecting insulation.

Finite difference Python code models layered geology and temperature dynamics.

Abstract

Modeling underground temperatures provides a practical application of the one-dimensional heat equation. In this work, the one-dimensional heat equation in surface soil is extended to include heat carried by the vertical flow of rainwater through the soil. Analytical solutions, with and without water flow, illustrate the influence of rainwater circulation on the sub-surface propagation of seasonal temperature variations, an important effect that is generally neglected in textbooks. The surface temperature variations are damped by the soil, and this effect was used by the Troglodytae in Egypt or the Petra in South Jordan to insulate against extreme temperatures. For a realistic case of horizontally layered geology, a finite difference Python code was developed for the same purpose. Subsurface temperatures were also measured over a full year at depths up to 1.8 m and used to estimate the thermal skin depth and thermal wavelength. This study provides students with a practical example of how a textbook physics problem can be modified to extract information of contemporary importance in geophysics and global warming.