# Physical Processes in Geology

Lab 2: Introduction to Thermal Modeling
Due Date:
The objective of this lab is to apply basic equations of conductive heat transfer
to geologic problems. This is practice for next week, when we will use this
technique on real data.
Background:
Temperature diffuses in the same way chemicals diffuse or landscapes diffuse; these are all
described by a simple differential equation:

6T 2 K 82T

70? (922
There are 2D and 3D versions of this equation, but we will only consider temperature in
1D (depth). The constant K: is called the thermal diffusivity, which describes how rapidly
temperature diffuses in a material. It is a function of a material’s thermal conductivity,
density, and heat capacity. Notice that if the second derivative with respect to 2 on the right
hand side is large (i.e. if there is a sharp peak or trough in temperature), the left hand time
derivative will also be large, so the temperature will change rapidly at that location. The
effect of this is that temperature spikes will tend to be smoothed over time, and the larger
the spike, the faster the temperature change.
Exact solutions to the thermal diffusion equation involve the error function, but we will
use a numerical approximation scheme commonly employed for differential equations, called
the finite difference method. The finite difference equation we will use for this lab exercise
is:

T2105) 2 T2¢(t _ 1) + A[T2i+1(t _ 1) + TZi-1(t_ 1) _ 2TZ1 (t _ 1”

where A = [565-532
Although this looks imposing, it is quite simple. It says that the temperature at time step t
and spot 2,- (T z, (t)) is the temperature in the same spot at the previous time step (T z, (t – 1))
plus an adjustment factor that is an approximation to the second derivative.
To solve for the entire behavior of the system, we need initial and boundary conditions,
such as the temperature distribution at time zero.
We could optimize the speed of the finite difference method by programming it in Matlab
or another programming language, but it is more illuminating (and requires no programming
knowledge) to do the calculations in Microsoft Excel. In the Excel template you can download
from ANGEL, the rows (from top to bottom) are the times, and the columns (from left to
right) are the depths.