Heating of a homogeneous volume

• ThermoHydroMechanics/HeatingHomogeneousDomain/hex_THM.prj

This example is one of the mini-benchmarks of FE-Modelling Task Force (by Andrés Alcolea et. al.). The example is aimed to simulate the coupled THM processes in a fully water saturated unit cubic porous medium ($[0, 1]^3\,\text{m}^3$) with a linear homogeneous temperature increment from 20$^{\circ}$C to 30$^{\circ}$C in 100 days.

The gravity is not considered in all balance equations. Since the temperature is homogeneous, the specific heat capacity is set to zero and thermal conductivity can be any non-zero number. The liquid density is given as

$$\rho_L = \rho_0 \exp (\beta_L (p_L-p_0)+\alpha_L^T(T-T_{ref})) $$

with

• $\rho_0=1002.6$ kg/m$^3$ the initial liquid density,
• $\beta_L=4.5\cdot10^{-10}$ Pa the liquid compressibility,
• $\alpha_L^T=-2.0\cdot 10^{-4} \text{ K}^{-1}$ the liquid thermal expansivity,
• $T_{ref} = 273.15 \text{ K}$ the reference temperature.

While the liquid viscosity is defined as

$$\mu_L = \text{A}\exp(\text{B}/T)$$

with A=$2.1\cdot 10^{-6} \text{ Pa}\cdot\text{s}$, and B=1808.5 K.

The other material parameters are given below:

Property	Value	Unit
Young’s modulus	1	GPa
Poisson ratio	0.35	-
Solid thermal expansion	$3 \cdot 10^{-6}$	$\text{K}^{-1}$
Biot’s coefficient	0.96111	-
Porosity	0.1	-
Intrinsic permeability	$3.0 \cdot 10^{-20}$	m$^2$

Initially, the temperature is $20 ^{\circ}$C, the pore pressure is $2\cdot 10^6$ Pa, and all effective stress components are zero.

At the boundary surfaces, there is no heat or flow flux, and the normal displacement is fixed to zero.

As a CTest, only 5 time steps with a fixed time step size $1.728\cdot 10^4$ s are computed. If the example is run up to 50 time steps, corresponding to 100 days of simulation time, the variations of pressure and effective stress can be obtained as that are shown in the two figures below: