To understand, describe, simulate, and predict the dynamics of geophysical lava flows for hazard mitigation, this research work presents mathematical modelling and simulations of three-dimensional shallow free-surface flows of Herschel–Bulkley viscoplastic fluids down an inclined non-flat topography with varying basal conditions. Starting from a systematic perturbation analysis, lower-order models of the Navier–Stokes equations are derived by employing the multi-regime approach, which allows to model different flow regimes originating from the variation of the mean-slope and/or the basal conditions. In particular, the lubrication model, the shallow water equations, and the depth-averaged heat equation are derived for multi-regime viscoplastic flows applicable to a natural setting consisting of an irregular topography with perturbations in basal slipperiness.

Two flow regimes (Regime A, the basic/classical one, and Regime B, the enriched one with a corrective pressure term) corresponding to different balances between shear and pressure forces are defined and investigated. Flow models corresponding to these regimes are calculated as perturbations of the zeroth-order solutions. The classical reference models in the literature are recovered by considering their respective cases on a flat-inclined surface. The flow solutions of the two regimes are compared; the difference appears, in particular, in the vicinity of sharp changes of slopes. Nonetheless, both regime models are compared with experiments and are found to be in good agreement.

Moreover, the derived shallow water equations and the lubrication approximation are compared to test their validity limits in different flow regimes in terms of the inclination angle, aspect ratio, basal perturbations, and Reynolds numbers. In the limit of low Reynolds numbers, both models are observed to agree well with viscoplastic experiments on a flat topography. The significant difference is observed at early times of dam-break, where the lubrication approximation overestimates the speed of the front position compared to the experiments. Also, it is shown that the flow dynamics over a perturbed topography are well predicted by the shallow water equations compared to the lubrication model. Generally, the shallow water equations are observed to have a wider predictive limit (e.g., in terms of the slope angle and Reynolds numbers) than the lubrication approximation.

Furthermore, to investigate the effects of rheological parameters: power-law index (n), consistency index (K), and yield stress (τc), on flow height and velocity of Herschel– Bulkley fluids over different topographies, three practical examples of dam-break flow cases are studied: a dam-break on an inclined flat surface, a dam-break over a nonflat topography, and a dam-break over a wet bed. The effects of bed slopes and depth ratios between upstream and downstream fluid levels on flow dynamics are also analyzed. The numerical results are compared with experimental data from the literature and are found to be in good agreement. Results show that increasing any of the three rheological parameters decreases the fluid front position, peak height, and mean velocity, for both dry and wet bed conditions.

Lastly, by formulating an inverse problem, we can infer the rheological parameters (K, τc, n) from free-surface measurements based on experimental data of flows around an occlusion. The rheological identification problem is formulated to minimize an objective function that measures the discrepancy between the elevation hydrographs from the model output and experimental data. The inverse solver is tested on both synthetic and laboratory data. The set of rheological parameters inferred is compared with the values measured on a rheometer for the fluid used in the experiments. The results have shown that hydrograph measurements at the wetted solid-fluid interface contain information of the fluid flow, which can be used to retrieve the unknown rheology for hazard mitigation and/or aid prediction of the wetting dynamics.

Novel results include the derivation and validation of the shallow water models for 3D Herschel–Bulkley flows that account for basal elevation and basal slipperiness, a formal comparison of the lubrication and shallow water equations with experiments, numerical simulations of viscoplastic dam-break flows over a dry non-flat bed and over a wet bed, and the inference of rheological parameters for viscoplastic flows using elevation hydrographs around an occlusion.