A Second-Order Damage Tensor Is Sufficient for Damage-Induced Permeability; Scalar Crack Density Is Not

Reservoir simulators close the damage-permeability coupling with a state variable, yet it has never been tested whether that variable carries enough information, because laboratory experiments cannot hold damage fixed while the stress path varies. We build a numerical reference that can: phase-field fracture under crack-length control, so the softening branch is traversed, coupled to FFT-Galerkin homogenization of Darcy flow on the same periodic cell. Across five loading paths and 210 damage states on one microstructure, permeability at matched scalar crack density spreads by up to 43.5 percent, and the spread grows with damage. Matching the full second-order damage tensor instead collapses the spread to 3.0 percent, with no residual floor: load-path memory enters permeability only through the tensor. The exponential damage law in common use, granted its own best-fit exponent, still errs by 57 percent in the median, underpredicting at moderate damage and overpredicting beyond tr D = 0.3.