Non-Linear Thermal Diffusion Around a Wellbore: The Case with Temperature Dependent Volumetric Heat Capacity

Problem description

This example is an extension of the linear thermal diffusion problem presented in Pure Thermal Diffusion Around a Wellbore. It uses the thermal single-phase flow solver to model a non-linear thermal diffusion problem around a wellbore where the volumetric heat capacity of the solid rock depends linearly on the temperature.

Input file

This benchmark example uses no external input file and everything required is contained within two GEOS xml files that are located at:

inputFiles/singlePhaseFlow/thermalCompressible_2d_base.xml

and

inputFiles/singlePhaseFlow/thermalCompressible_temperatureDependentVolumetricHeatCapacity_benchmark.xml

In this example, we focus on the Constitutive tag.

Constitutive

The reference value of the volumetric heat capacity of the medium around the wellbore and its derivative with respect to temperature are defined in the SolidInternalEnergy XML block:

    <SolidInternalEnergy
      name="rockInternalEnergy_nonLinear"
      referenceVolumetricHeatCapacity="4.56e6"
      dVolumetricHeatCapacity_dTemperature="1e6"
      referenceTemperature="0"
      referenceInternalEnergy="0"/>

Results and benchmark

A good agreement between the results obtained using GEOS and the reference results that are obtained by the classical finite difference method is shown in the figure below:

(Source code)

To go further

Feedback on this example

This concludes the example of a non-linear thermal diffusion problem around a wellbore due to temperature dependent volumetric heat capacity of rock. For any feedback on this example, please submit a GitHub issue on the project’s GitHub page.