Understanding Geothermal Gradient and Calculation in Excel

The geothermal gradient is the rate of change in temperature with respect to increasing depth in Earth’s interior. The temperature of the Earth’s crust rises with depth due to the heat flow from the much hotter mantle. Away from tectonic plate boundaries, temperature rises in about 25-30°C per kilometer of depth near the surface in most of the world. However, in some cases, the temperature may drop with increasing depth, especially near the surface, a phenomenon known as inverse or negative geothermal gradient. The effects of weather, the Sun, and season only reach a depth of roughly 10-20 meters.

The geothermal gradient is expressed in degrees Celsius per kilometer (°C/km), Kelvin per kilometer (K/km), or milliKelvin per meter (mK/m). The geothermal gradient is influenced by atmospheric temperature, and the uppermost layers of the solid planet are at the temperature produced by the local weather, decaying to approximately the annual mean-average ground temperature at a shallow depth of about 10-20 meters depending on the type of ground, rock, etc.

Basic Theory

The Earth’s interior is heated primarily by the decay of radioactive elements and the residual heat from its
formation. The geothermal gradient is the change in temperature per unit depth below the Earth’s surface. It is
typically expressed in units of degrees Celsius per kilometer (°C/km) or degrees Fahrenheit per mile
(°F/mi).

The geothermal gradient can be influenced by factors such as local geology, tectonic activity, and regional
variations. In petroleum engineering, understanding the geothermal gradient is crucial for estimating subsurface
temperatures, which, in turn, impacts fluid properties and drilling operations.

Procedures for Calculation

The geothermal gradient (GG) can be calculated using the formula:

$GG = \frac{{T_{\text{bottom}} - T_{\text{surface}}}}{{\text{Depth}}}$

Where:

$GG$ is the geothermal gradient,
$T_{\text{bottom}}$ is the temperature at the bottom of the well or borehole,
$T_{\text{surface}}$ is the temperature at the Earth’s surface, and
$\text{Depth}$ is the depth of the well or borehole.

Excel Calculation

Let’s create an Excel table to calculate the geothermal gradient based on a scenario. Suppose we have the
following data:

Surface Temperature (°C)	Bottom Temperature (°C)	Depth (km)	Geothermal Gradient (°C/km)
20	150	3	= (150 – 20) / 3

Result

The calculated geothermal gradient is $\frac{{150 - 20}}{{3}} = 43.33 \, \text{°C/km}$ .

MATLAB Comparison

To compare the results, the same scenario can be solved in MATLAB using similar calculations. Here’s an example
MATLAB code:

% MATLAB code for geothermal gradient calculation
surface_temp = 20;  % in °C
bottom_temp = 150;  % in °C
depth = 3000;       % in meters

geothermal_gradient = (bottom_temp - surface_temp) / (depth / 1000);  % convert depth to km
disp(['Geothermal Gradient (MATLAB): ' num2str(geothermal_gradient) ' °C/km']);

Run the MATLAB code, and compare the result with the Excel calculation. The values should match.