Identifiability and Conditioning of Flux-Parameter Inversion for the Buckley–Leverett Equation

We analyse inversion of the viscosity ratio M in the Buckley–Leverett equation, distinguishing structural identifiability, observation-operator conditioning, and numerical estimation error. The Riemann solution gives M=4s2−4s from the shock speed s, with relative condition number κ(M)=21+M/(1+M−1), which diverges as M↓0. The continuum physics objective admits exact solution–parameter pairs for every M>0, so identifying the true ratio requires observations. We examine numerical recovery using a viscous physics-informed neural network (PINN) with positive parameterisation and 30 paired repetitions per setting. At M=3, the median parameter error decreases from 514.25% without observations to 6.13% with 25 noisy observations. Additional data decrease the median field error, while median parameter errors are nonmonotone. Over M∈[0.25,10], κ decreases from 18.94 to 2.86, whereas the smooth-branch Cramér–Rao relative scale for 50 scattered observations with 5% Gaussian noise levels off near 7%. Shock crossings supply additional nonregular information. Finite-network physics losses remain small across the tested ratios, with different orderings on training and held-out points. Scalar controls assess finite-resolution viscous-model discrepancy; complementary PINN tests show poorer recovery as the front sharpens. Recovery accuracy therefore depends jointly on the observation operator, regularisation, and numerical estimator.