Residual Saturation under Pressure-Controlled Drainage

Here, pressure-controlled drainage is formulated as bond percolation with trapping on the pore-network graph, establishing a direct connection between percolation theory and pressure--saturation relations. In two dimensions, the deviation of the residual saturation from its non-vanishing thermodynamic limit obeys a finite-size scaling law with exponent $\delta \approx 0.25$, independent of microscopic details of the lattice. In three dimensions, finite-size corrections decay more rapidly ($\delta \approx 0.75$), while the asymptotic residual saturation remains finite and depends on coordination number. This extends the standard invasion-percolation picture beyond the breakthrough state, where the invading cluster is fractal and the invaded-phase saturation vanishes in the infinite-size limit.