The transition toward deep decarbonisation relies heavily on sector coupling, proposing the use of existing natural gas networks to transport and store green hydrogen. However, hydrogen's low volumetric energy density poses a significant threat to the inherent thermal energy storage capacity of these infrastructures. This paper presents a fully dynamic, isothermal, one-dimensional transient partial differential equation model to quantify the thermodynamic, environmental, and economic impacts of hydrogen blending on the Portuguese high-pressure gas network. By simulating a 1278 km topology over a continuous 30-day horizon, this study investigates the linepack depletion paradox under two distinct injection scenarios: stochastic renewable curtailment and a dedicated continuous solar and wind power plant. Results indicate that blending at a nominal 70 bar inevitably induces a dynamic linepack deficit, reaching 10.85 GWh under continuous injection, critically limiting the flexibility buffer required by the wider energy system, while offering a negligible baseline CO₂ emission reduction of merely 0.05% to 0.50%. Furthermore, this study mathematically disproves the viability of elevating network baseline pressures to recover storage capacity. Boosting the operational pressure to 80 bar successfully restores the lost thermal inertia but triggers a severe environmental backfire, actively increasing CO₂ emissions by up to 0.44% due to the forced injection of compensatory fossil gas. Economically, this high-pressure recovery regime imposes a highly non-linear penalty on the grid, surging compressor electrical power demand by up to 36.18%. Consequently, the Levelised Cost of Hydrogen for curtailment escalates to an unfeasible value exceeding 12 €/kg.