## 4 Results
### Effect of Leakage Rate (Cases 1, 2, and 3/Bubble Size = 20 mm, Leakage Area = 20,000 m)
(**a**) Center plane in x direction (x = 1000 m); (**b**) Center plane in y direction (y = 500 m); (**c**) xy plane near leakage area (z = -180 m)  
Figure 9: Contour maps of \(\Delta\)pCO\({}_{2}\) in the constant z planes of the mesoscale domain at 30 days after the start of leakage for case 3 (leakage rate = 100,000 t/y). Effect of Initial Diameter of CO\({}_{2}\) Bubble (Cases 5, 4, and 3/Leakage Rate = 100,000 t/y, Leakage Area = 20,000 m)  
Contour maps of some variables on the center plane in the x direction (x = 1000 m) of the small-scale domain at 30 days after the start of leakage are shown in Figure 11. As seen in the distribution of void rate and cell-averaged number densities of CO\({}_{2}\) bubbles, as the initial diameter of the bubbles increases, the rising distance of the bubbles increases, which results from the larger buoyant force of the relatively large volume of the bubble. Because the leakage rates are the same for these cases, the number of bubbles is high within the short band when the initial diameter is 5 mm. Comparing the distributions of DCO\({}_{2}\) or \(\Delta\)DCO\({}_{2}\) among the cases, one can find that the distribution when the diameter is 10 mm is quite different from those of 5 and 20 mm. The reason for this phenomenon is that the mass transfer rate of a CO\({}_{2}\) bubble is largely affected by the shape of the bubble. The total mass transfer per unit volume decreases as the diameter of a bubble decreases. However, if the diameter exceeds a certain value and changes to the spherical cap, there is a sudden jump, and it decreases as the diameter increases. The threshold is approximately 18 mm. Therefore, the mass transfer rates when the diameters are 5 and 20 mm are similar, but the rate becomes smaller when the diameter is 10 mm, as discussed by [PERSON] et al. (2009). The lower dissolution of the bubbles results in a low DCO\({}_{2}\) and \(\Delta\)DCO\({}_{2}\) when the diameter is 10 mm. Figure 10: Time change of contour maps of \(\Delta\)pCO\({}_{2}\) on the planes, where the full-3D small-scale domain is located, of the mesoscale domain for case 3 (leakage rate = 100,000 t/y). (a) yz plane (x = 81,000 m); (b) xz plane (y = 10,500 m); (c) xy plane (z = 180 m)  
Figure 12 shows the contour maps of \(\Delta\)pCO\({}_{2}\) on the center planes of the x, y, and z directions of the small-scale domain. The maximum \(\Delta\)pCO\({}_{2}\) for cases 5 and 3 are similar but do not exceed 500 ppm,  
Figure 11: Contour maps of (**a**) void rate, (**b**) number density of undissolved CO\({}_{2}\) bubbles, (**c**) DCO\({}_{2}\), and (**d**) \(\Delta\)DCO\({}_{2}\) on the center plane in the x direction (x = 1000 m) of the small-scale domain at 30 days after the start of leakage with the same leakage rates of 100,000 t/y and leakage area of 20,000 m\({}^{2}\) but different initial diameter of bubble. which is also clearly seen in Figure 13, where those on the constant x, y, and z planes of the mesoscale domain are illustrated. Figure 12: Contour maps of \(\Delta\)pCO\({}_{2}\) on the center planes of the x, y, and z directions of the small-scale domain at 30 days after the start of leakage with the same leakage rates of 100,000 t/y and leakage area of 20,000 m\({}^{2}\) but different initial diameters of the bubble. (**a**) Center plane in x direction (x = 1000 m); (**b**) Center plane in y direction (y = 500 m); (**c**) xy plane near leakage area (z = \(-\)180 m).