When rocks deform, most of the deformation transfers into changes of porosity. As a matter of fact, the solid phase can also deform and change volume. In that case the equation that links stresses and deformations needs to be corrected according to Biot's effective stress:
where is the Biot coefficient and is the identity tensor. In the case of linear elastic and isotropic porous media, the Biot coefficient is:
where is the drained bulk modulus of the porous solid and is the unjacketed bulk modulus. The unjacketed bulk modulus is equal to the mineral bulk modulus when all porosity is connected and the rock has a mono-mineral composition. The values of the Biot coefficient range from the value of porosity to 1. Most rocks and sediments subjected to large depths have a Biot coefficient that ranges from 0.4 to 0.95, decreasing in value as rocks get stiffer. In anisotropic media, the poroelasticity correction factor becomes a tensor . The corrections for poroelasticity become significant in tight rocks with high stiffness, and low and unconnected porosity.
Changes of temperature in solids change the equilibrium distance between molecules, and therefore induce strains. Imagine a solid heated up, but not allowed to dilate in vertical direction (Figure 3.28). The result is an increase of stress in vertical direction rather than a deformation in vertical direction.
The coefficient of thermal dilation quantifies strains as a function of changes of temperature at constant pressure , and it is defined as
Typical linear thermal dilation coefficients of rock range from 5 to 10 1/C. You may compare this range to the linear thermal dilation coefficient of steel 1/C and water 1/C.
Under unconstrained (displacement) conditions, a negative change in temperature causes shrinkage and a positive change in temperature causes dilation. The elastic equations extended to consider thermal changes make explicit that stresses can be changed as a result of a change in temperature and/or as a result of a change of volumetric strains.
Eq. 3.44 does not include the effects of pore pressure. The coupled thermo-poro-elastic equations are covered in the Advanced Geomechanics course.
Thermal(-induced) stresses can cause reductions in fracture gradient when drilling with relatively cold drilling mud. Thermal(-induced) stresses can also cause enhanced fractured reactivation when injecting cold fluids in a hot reservoir, such as in deep geothermal energy recovery.
EXAMPLE 3.7: Derive an expression of the thermal swelling stress (in vertical direction) for the example shown in Figure 3.28.
Let us assume the axis 3 in the vertical direction, then , , . Then, for , a simplification of Eq. 3.44 results in,
Solving for and pluging it the equation results in