Okay. Then they add more and it will boil quickly. I guess the question boiles (huehue) down to how much water you can turn into stream per amount of lava or the inverse, how much lava you can cool down per amount of water.
The phase change from liquid water to stream will, by the way, not just contribute significantly but be by far the majority of energy needed. Simply heating water up, ignoring the phase change and changes of the heat capacity, with the same energy as it takes to go from liquid to gas (2257 kJ/kg) would result in a temperature rise of… dT = 2257 kJ/kg / 4.2 kJ/(kg*K) = 537 K