An effective continuum model is derived to describe the nucleation and subsequent growth of a gas phase from a supersaturated, slightly compressible binary liquid in a porous medium, driven by solute diffusion. The evolution of the gas results from reducing the system pressure at a constant rate or withdrawing the liquid at a constant rate. The model addresses two stages before the onset of bulk gas flow, nucleation and gas-phase growth. Negligible gradients due to gravity or viscous forces are assumed, so the critical gas saturation signaling the onset of bulk gas flow is only a function of the nucleation fraction. Important quantities characterizing the process, such as the pore number fraction hosting activated sites, deviation from thermodynamic equilibrium, maximum supersaturation, and critical gas saturation, depend crucially on the nucleation characteristics of the medium. Heterogeneous nucleation models using preexisting gas trapped in hydrophobic cavities or in terms of a rate-dependent nucleation are used to investigate the nucleation behavior. Using a simpler analytical model the relevant quantities during nucleation can be expressed in terms of a simple combination of dimensionless parameters. The theory predicts that the maximum supersaturation is a weakly increasing function of rate, which in the region of typical experimental parameters can be approximated as a power law. It depends sensitively on the probability density function of nucleation cavity sizes. The final nucleation fraction (thus, the critical gas saturation) is predicted to be a power law of the decline rate. The theoretical exponents agree well with experimental data. The subsequent evolution of the gas phase and the approach to the critical gas saturation is also described.