V-GT-02 — steady compressible Darcy outflow (P² profile)

Tags: gas_transport, darcy, pressure_closure, steady

References:

• gas-reservoir P² solution (Muskat-form compressible Darcy, standard porous-media texts)
• technical reference Ch. 9 (wood-pyrolysis overpressure magnitudes)

Problem statement

The canonical porous-flow exact solution (gas-reservoir form): isothermal ideal gas, uniform volumetric generation ġ (zero-order gas-producing reaction at clamped T), sealed back at z = 0, Dirichlet pressure P∞ at z = L. The steady mass balance d/dz[(κM/μRT)·P·dP/dz] = −ġ integrates to

P²(z) = P∞² + (μRT·ġ/(κM))·(L² − z²), exit mass flux = ġ·L (exact)

Nonlinear in P; exercises the algebraic ideal-gas pressure closure, face mobility, Darcy upwinding, and the DirichletPressureBC outflow — all at once. The existing Darcy test only checks v = −(κ/μ)∇P pointwise.

Setup notes (the four pitfalls the survey flagged):

• P is NOT persisted in PyrolysisSolution: reconstructed per cell from sol.ξ/sol.T via porosity + idealgaspressure with the same material.
• Initial gas is seeded with backgroundgasconcentration so P(z,0) = P∞ (starting from ξ_gas = 0 would start from vacuum).
• μ in the exact profile is the code's own Sutherland value effectivegasviscosity(T_iso), not a hand-picked constant.
• λ = 0 on every component kills the Fickian term: pure Darcy outflow.
• ρintrinsic = 1e5 (φ = 0.99, verification-only value): solid→gas conversion frees pore volume at dφ/dt = ġ/ρintrinsic, and the gas stored in that new volume steals (P̄M/RT)/ρintrinsic of the exit flux — a mesh-independent bias (~3.6e-4 at ρintrinsic = 2000) that would floor the convergence study; at 1e5 it is ~7e-6, below discretization error. Permeability mixes on the renormalized solid basis, so κ_eff = κ exactly regardless of φ.

Timescales: pressure equilibrates in ≪ 0.1 s (gas inventory / production ≈ 0.07 s); compared at t = 1 s ≈ steady, with an explicit steadiness QoI.

Quantities of interest (n = 320)

Convergence

Observed order 0.8798 (L2), expected 1.0.

Solver configuration