Monotone and Bounded Neural Operators in Conservative Drift-Flux Models for Soluble Methane Influx in Drilling

Abstract

Drilling annulus transients involving gas influx remain difficult to model and interpret because surface observables are shaped by compressibility, slip, temperature variation, and the partitioning of methane between free and dissolved inventories in synthetic or oil-based drilling fluids. While mechanistic models provide important physical structure, their operational use is often constrained by uncertain closure relations and by numerical fragility when regimes change rapidly. This paper develops a physics--machine-learning framework that targets a specific gap: learning closure operators for annular multiphase transport and dissolution while guaranteeing physical admissibility of the coupled simulator. The approach embeds monotone and bounded neural operators into a conservative drift-flux annular model with an explicit dissolved-gas inventory. Neural components represent uncertain mappings such as drift velocity, effective friction, swelling, and mass-transfer intensity, but are constructed to satisfy invariance constraints that enforce nonnegativity of phase masses, holdup bounds, and thermodynamic consistency of dissolution. Training uses a hybrid objective combining sparse surface measurements with weak physics residual penalties and inequality barrier terms that enforce admissibility across the training distribution and under extrapolation. A stability analysis shows how monotonicity and Lipschitz constraints on learned operators yield discrete-time invariance of the admissible state set under implicit time integration. Numerical experiments spanning circulating and near-static regimes demonstrate that the learned-closure simulator reduces bias in pressure and pit-gain predictions under closure uncertainty while avoiding unphysical states that can arise in unconstrained learning. The resulting model class is designed as a reliable computational kernel for real-time monitoring and uncertainty propagation in soluble kick environments.