Thermal effects

Drilling mud is usually cooler than the geological formations in the subsurface. Because of such difference, drilling mud usually lowers the temperature of the rock near the wellbore. The process is time dependent and temperature $T$ variations in time and space (in the absence of fluid flow) can be modeled using the heat diffusivity equation:

$$\frac{\partial T}{\partial t} = D_T \nabla^2 T$$ (6.22)

where heat diffusivity is $D_T = k_T/(\rho c_p)$ proportional to the heat conductivity $k_T$, and inversely proportional to the rock mass density $\rho$ and the heat capacity $c_p$. The operator $\nabla^2=(\partial^2 / \partial x^2 + \partial^2 / \partial y^2 + \partial^2 / \partial z^2 )$ indicates variations in space. The heat diffusivity equation and the equations of thermo-elasticity (Section ) permit solving the changes of strains and stresses around the wellbore due to time-dependent changes of temperature (Example in Fig. 6.25).

Figure 6.25: Example of solution of heat diffusivity and thermoelastic solution around a wellbore. [make your own]

At steady-state conditions, the change in hoop stress $\Delta \sigma_{\theta \theta}$ around any point in the wellbore due to a change in temperature $\Delta T$ is:

$$\Delta \sigma_{\theta \theta} = \frac{\alpha_L E \Delta T}{1-\nu}$$ (6.23)

where $\alpha_L$ is the linear thermal expansion coefficient. Cooling leads to hoop stress relaxation and possibly tensile effective stress, while heating leads to increased compression i the tangential direction.