Estimate the depth below the ground at which the annual...

Estimate the depth below the ground at which the annual temperature variation will be $10 \%$ of that at the surface for
(i) dry soil, $k=1.0 \mathrm{~W} / \mathrm{m} \mathrm{K}$.
(ii) wet soil, $k=2.0 \mathrm{~W} / \mathrm{m} \mathrm{K}$.
Repeat for the diurnal temperature variation.

Key Concepts

Thermal Diffusivity

Thermal diffusivity is a material property defined as ? = k/(?c), where k is the thermal conductivity, ? is the density, and c is the specific heat capacity. It measures how quickly a material can conduct thermal energy relative to its ability to store it. In the context of soil temperature variations, thermal diffusivity controls the rate at which temperature oscillations at the surface propagate into the ground, affecting both amplitude attenuation and phase lag.

Penetration (Damping) Depth

Penetration depth, also known as damping depth (or skin depth), is the characteristic distance over which periodic temperature variations decay exponentially within a medium. It is given by the expression ? = ?(2?/?), where ? is the thermal diffusivity and ? is the angular frequency of the temperature oscillation (? = 2?/T, with T being the period). This concept is crucial in estimating the depth at which the amplitude of temperature variations reduces to a specified fraction of its surface value.

Periodic Heat Conduction

Periodic heat conduction describes the process by which cyclical temperature changes at the surface, such as diurnal and annual variations, diffuse into the subsurface. The governing heat conduction equation with periodic boundary conditions results in solutions that exhibit both attenuation in amplitude and a phase lag with increasing depth. This approach is used to model how the temperature signal diminishes (e.g., to 10% of its surface amplitude) as one moves deeper into the ground.

Soil Thermal Properties

Soil thermal properties, particularly thermal conductivity, density, and specific heat capacity, differ between conditions such as dry and wet soils. These differences influence the thermal diffusivity and, consequently, the penetration depth of temperature variations. In problems like the one given, contrasting values of thermal conductivity for dry and wet soils illustrate how moisture content can significantly affect thermal behavior in the subsurface.

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