Data‐Driven Discovery of Classical and Anomalous Transport Models From Sparse Breakthrough Curves in Heterogeneous Porous Media via Symbolic Regression

Abstract

Natural porous media are highly heterogeneous, with flow and transport controlled by pore‐space morphology and connectivity. Consequently, classical continuum models often fail to describe observed transport under strong heterogeneity. A key challenge is identifying physically interpretable transport models from sparse observational data. We address this using a physics‐informed, data‐driven machine‐learning framework combining symbolic regression with genetic programming system identification, a stochastic optimization approach that discovers candidate partial differential equations (PDEs) directly from data. We apply it to two classical experimental data sets of solute transport. For Scheidegger's (1959) breakthrough‐curve measurements in Berea sandstones, a long‐standing benchmark for anomalous transport, the framework identifies data‐consistent, physically interpretable models using only sparse outlet observations and physical constraints. These include fractional advection–dispersion equations with temporal and spatial fractional operators: the temporal derivative represents memory effects from solute trapping in dead‐end pores, whereas the spatial derivative captures nonlocal transport from preferential flow paths and conductivity heterogeneity. For the Levy and Berkowitz (2003, https://doi.org/10.1016/s0169‐7722(02)00204‐8 ) data, the framework recovers both classical and anomalous transport representations depending on medium heterogeneity, distinguishing Fickian from non‐Fickian regimes from sparse breakthrough curves alone. The identified models are validated against independent data at different injection rates. Both data sets are intentionally sparse (one and four breakthrough curves, respectively), so these results should be read as evidence of feasibility under severe data scarcity, not as a general demonstration across broad classes of porous media. They illustrate the potential of symbolic‐regression‐based PDE discovery as an interpretable, physically constrained framework for transport‐model identification in heterogeneous geologic media.