In the inviscid and incompressible fluid flow regime, surface tension effects on the behavior of two initially spherical bubbles with same size rising axisymmetrically in an infinite and initially stationary liquid are investigated numerically with the VOF method. The numerical experiments are performed for two bubbles with two different bubble distances. The ratio of gas density to liquid density is 0.001, which is close to the case of air bubbles rising in water. In the case of Dis = 2.5, where Dis is defined as the ratio of the distance between the bubble centroids to the radius of the bubble, it is found from numerical experiments that there exist four critical Weber numbers We1, We2, We3 and We4, which are in between 10 and 100, 3 and 4, 1.5 and 1.8, and 0.2 and 0.3, respectively. In the case of Dis = 2.3, similar phenomena also appear but the corresponding four critical Weber numbers are lower than those in the case of Dis = 2.5. The mechanism of the above phenomena is analyzed theoretically and numerically.