*4.1. Circulation Rate*

Circulation rate is an indicator of the carrying capacity that can remove drilling cuttings. Therefore, it is a major factor in determining the rate of penetration (ROP) [19]. Figure 7a,b describe the TDR and fluid temperature in the annulus along with the whole well depth. The TDR is non-dimensionalized as Equation (20) for efficient comparison.

$$\text{TDR}\_{\text{D}} = \frac{\text{TDR}}{r\_{\text{c}}}.\tag{20}$$

**Figure 7.** The sensitivity analysis of thermal disturbance with various circulation rates: (**a**) TDR and (**b**) fluid temperature in the annulus.

The circulation rate for comparison was determined to be 20 L/s, 40 L/s, and 80 L/s, shown in solid blue, red, and gray, respectively. In Figure 7a, all three TDR curves have similar shapes regardless of circulation rate. The TDR curve shape forms the minimum point at a certain depth, and below the depth, the TDR gradually increases as the depth increases. Despite the similarity of the curve shape, the minimum TDR depth of each case depends on the circulation rate. The smaller the circulation rate, the deeper the minimum TDR depth. The annulus fluid temperature can be evidence for the different minimum TDR depths, shown in Figure 7b. Because the annulus fluid is in direct contact with the formation, heat transfer occurs between the annulus fluid and the formation.

The circulation rate indicates the amount of heat capacity; the increasing fluid circulation requires more heat to change the fluid temperature. Thus, as the circulation rate increases, the temperature gradient in the annulus becomes smaller. In Figure 7b, the curve with the faster circulation deviates farther away from the formation temperature distribution (dotted gray line). Each temperature curve demonstrates a different slope intersecting the formation temperature line at different depths. The intersect depth is observed to be around 1000 m for 80 L/s, 1,400 m for 40 L/s, and 2000 m for 20 L/s, respectively. The intersect depth corresponds to the minimum TDR depth. At this depth, the heat transfer and, thus, the thermal disturbance occur minimally due to the slightest temperature difference between the circulating mud and the surrounding formation.

As observed in the previous chapter, the TDR is maximum at the bottom-hole depth in all three cases, where the temperature difference is the largest in the whole depth. The dimensionless TDR is around 28 for 20 L/s, 30.5 for 40 L/s, and 32 for 80 L/s, respectively. However, as seen from the estimated values, the increasing rate of TDR slows down compared to the rising circulation rate. Additionally, the TDR is expected to have an upper limit due to the limit of the temperature difference between the annulus fluid and the formation because the fluid temperature in the annulus cannot be lower than the injecting temperature. In conclusion, the larger circulation rate increases the TDR, but its effect is limited above a certain circulation rate.
