*3.2. Identification of Thermal Disturbance*

We identified thermal disturbance by circulating mud from the radial temperature distribution in Figure 5. The solid red and blue curves are the radial temperature at the bottom-hole depth (i.e., 4900 m) and 2000 m, respectively. The dotted gray lines are thermally undisturbed temperature (i.e., formation temperature before drilling circulation) for each depth. In Figure 5a, the temperature difference between the solid curve and the dotted line indicates how much cooling occurs in the formation by cold circulating mud. As the formation radius is much longer than the wellbore radius, the inconsistency of radial temperature varies according to the radial distance from the wellbore. The nearwellbore formation is cooled by the circulating mud. The formation temperature far from the wellbore does not change, that is, the thermal disturbance does not reach the outer boundary of the formation at this moment. The radial temperature change at the bottom hole appears to be larger than that at a depth of 2000 m.

**Figure 5.** Radial temperature distributions of different depths: (**a**) radial temperature distribution and (**b**) dimensionless radial temperature distribution.

The radial distance and temperature were non-dimensionalized for clarity, following Equations (18) and (19), respectively. The profiles of the two curves in Figure 5b are almost identical. However, the required radial distance for converging to the undisturbed temperature (i.e., the dimensionless temperature is one) is different for each depth. When the depth is 2000 m, the curve converges to the undisturbed temperature when the radial distance is approaching 10. Meanwhile, the red curve at the bottom-hole depth does not reach even when the dimensionless radial distance is 15. The result in Figure 5b concludes that the thermal disturbance in the formation by circulating mud (i.e., TDR) is more extensive at the bottom hole than at 2000 m.

*rD* <sup>=</sup> *<sup>r</sup> rc* , (18)

$$T\_D = \frac{T - T\_\varepsilon}{T\_\varepsilon - T\_\varepsilon}.\tag{19}$$

Figure 6 describes the thermal disturbance throughout the whole depth graphically. In this simulation, the outer reservoir boundary is 30 m, approximately 300 times the wellbore radius, which is sufficient length to describe thermal disturbance occurring over a few days. In the color map over the entire reservoir boundary (Figure 6a), the thermal disturbance is localized in the near-wellbore formation. Therefore, we represent Figure 6b as an enlarged version over *rD* ≤ 30 to visualize thermal disturbance effectively. Cooling and heating effects are determined by the temperature difference between the circulating mud and

the formation. In the upper section of the depth up to 1000 m, the formation temperature near the wellbore increases because the formation temperature in this depth range is lower than the injection temperature. As the depth rises from 1000 m, the cooling effect in the near-wellbore formation begins to appear. Thus, the thermal disturbance by the circulating inducing cooling increases with deeper depth, and its degree reaches the maximum at the bottom hole. From these observations, TDR is inferred to be increasing as the depth approaches the bottom-hole depth.

**Figure 6.** Temperature profiles: (**a**) a whole radial range and (**b**) an enlarged version.
