An explainable physics-informed neural network prototype for synthetic proppant transport in a two-dimensional fracture domain

Abstract

This study examines a single-domain physics-informed neural network (PINN) prototype on a normalized two-dimensional rectangular domain representing a planar fracture cross section. The recovered implementation maps (x,y,t) to (u,v,p,Cp) and penalizes velocity divergence, pressure gradients, and a scalar advection–diffusion–settling residual together with supervised concentration error; it does not enforce the full Navier–Stokes momentum equations. The archived run uses all 5000 generated concentration labels for both training and evaluation and reports an in-sample R2=0.635. To provide independent verification, we add a seeded 3500/750/750 training/validation/test protocol on manufactured analytical u, v, p, and Cp fields with exact forcing. Matched data-only, reported-proxy, and full-residual PINNs use the same corrected 3–64–64–64–4 architecture. On the 750 held-out points, the full-residual PINN gives Cp root mean square error 1.71×10−4, mean absolute percentage error 0.610%, maximum absolute error 9.12×10−4, and R2=0.973. Its relative L2 errors are 0.0228, 0.606, 0.356, and 0.00834 for u, v, p, and Cp, respectively. The full residual improves held-out concentration, horizontal velocity, continuity, and scalar-transport consistency, whereas the data-only model remains better for v and p. SHapley Additive exPlanations, Integrated Gradients, residual maps, and the controlled comparison provide an inspectable synthetic benchmark without implying experimental or high-fidelity fracture validation.

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