Near-critical extrema in the properties of water may influence flow patterns in hydrothermal systems, but singularities in equations of state for H2O at its critical point have inhibited quantitative modeling. Posing governing equations in terms of pressure (P) and enthalpy (H) avoids these singularities and facilitates computation. Numerical simulations with a P-H based model show little near-critical enhancement in heat transfer for systems in which flow is driven by fixed pressure drops. However, in density-driven systems, near-critical variations in fluid properties can enhance convective heat transfer by a factor of 102 or more (''superconvection'') if permeability is sufficiently high. Near-critical two-phase processes (''heat pipes'') are at least equally effective at dissipating thermal energy. The restriction to high-permeability environments within a fairly narrow P-H window suggests that superconvection may be quite rare in natural systems.