Figure 1: Artist's impression of a super-Earth.
The total mass of water on Earth forms a negligible fraction of the planet's total mass. Unlike Earth, water-rich super-Earths can have water mass fractions exceeding one percent. In the study by Thomas & Madhusudhan (2016), water-rich super-Earths are assumed to be comprised of Earth-like cores (i.e. 33 percent iron and 67 percent silicates) beneath heated water layers. At low temperatures and pressures, water exists as a liquid, vapour or solid (Ice Ih). At the high pressures and temperatures expected on these water-rich super-Earths, water can take on a number of alternate forms such as exotic high pressure ices (i.e. Ice V, VI, VII, X, etc), a supercritical fluid, or a superheated vapour.
Consider a water-rich super-Earth with 4 times the mass of Earth. Its internal structure is comprised of an Earth-like core beneath a layer of water constituting 5 percent of the planet's mass. If the pressure on the planet's surface is 100 bar, the increase in the planet's radius when the planet's surface temperature increases from 300 K to 1000 K is approximately 0.3 times the radius of Earth. The planet's radius is expected to increase by a larger amount with lower surface pressures and/or higher surface temperatures.
Surface pressure is an important factor in determining the radius of a water-rich super-Earth because the thermal expansion of water decreases under higher pressures. Take a water-rich super-Earth with 4 times the mass of Earth, a water mass fraction of 10 percent, and a surface temperature of 1000 K. If the planet's surface pressure is increased from 10 bar to 1000 bar, its water layer will be compressed significantly, and this is expected to decrease the planet's radius by a factor of two.
Figure 2: The radius of a water-rich super-Earth depends on its surface temperature and internal temperature profile (i.e. adiabatic or isothermal). A higher surface temperature leads to a larger planetary radius. This figure shows temperature-dependent internal structure models of water-rich super-Earths that are comprised of an Earth-like core under a water layer that makes up 30 percent of the planet's mass. Also, the surface pressure is assumed to be 100 bar. Thomas & Madhusudhan (2016)
Figure 3: The radius of a water-rich super-Earth depends a lot on its surface pressure. However, for pressures above 1000 bar the effect temperature has on the radius of a water-rich super-Earth becomes insignificant. This figure shows temperature-dependent internal structure models of water-rich super-Earths that are comprised of an Earth-like core under a water layer that makes up 30 percent of the planet's mass. Thomas & Madhusudhan (2016)
Nevertheless, the water content of a water-rich super-Earth does not have much of an effect on the planet's radius. For example, a water-rich super-Earth with 10 times the mass of Earth and a total water content of 50 percent, can increase its radius by 0.5 times the radius of Earth when its surface temperature is increased from 300 K to 1000 K. Given the same increase in surface temperature and the same planet, but now with a total water content of just 1 percent, the increase in the planet's radius is only slightly smaller, at 0.4 times the radius of Earth.
Thomas & Madhusudhan (2016), "In hot water: effects of temperature-dependent interiors on the radii of water-rich super-Earths", arXiv:1602.02758 [astro-ph.EP]