An Exploration of Hysteresis & Dynamic Effect in Porous Media Flow


This talk summarizes our 7 year long mathematical and modelling journey into understanding and explaining a wide range of phenomenon occurring due to hysteresis and dynamical capillarity effect (termed together as non-equilibrium effects) in multiphase flow through porous media. Hysteresis, referring to the history dependent disparity between imbibition and drainage characteristics of porous media flow processes, was first experimentally observed in 1930. Along with the dynamic capillarity effect, hysteresis is responsible for phenomena such as viscous fingering [Glass et al (1989)], formation of saturation overshoots and plateaus [DiCarlo (2004)]. But despite their demonstrable impact on flow behaviour, the non-equilibrium effects are generally disregarded in the standard equilibrium model due to their inherent complexity. Based on thermodynamic arguments, a comparatively simple play-type hysteresis model was proposed in [Beliaev & Hassanizadeh (2001)], incorporating also the dynamic capillarity effect in a closed form capillary pressure relationship. However, the model is incomplete in the sense that it fails to model some commonly occurring physical processes such as horizontal redistribution of water in homogeneous soil.

In our work, we propose a simple extension to the play-type hysteresis model that can be determined through experimentally obtained hysteresis curves [van Duijn & Mitra (2018)]. The model is mathematically well-posed [Mitra (2021)], thermodynamically consistent, and was used to cover all cases of the horizontal redistribution problem.

To explain the occurrence of saturation overshoots in water infiltration experiments we firstly looked at gravity driven flow of water through soil (modelled by the Richards equation). Under the standard infiltration setting, travelling waves were shown to exist [van Duijn et al (2018)]. They were then used to conclude that saturation overshoots occur when the dynamic effects are strong. Hysteresis plays a key role in determining the shape of the saturation profiles [Mitra & van Duijn (2018)]. The results were generalized to two-phase flows, including hysteresis in relative permeability [Mitra et al (2020)]. Entropy solutions for the corresponding hyperbolic Buckley-Leverett equation were derived. They differ significantly from the standard Oleinik entropy solutions, and can possess fundamentally different components such as stationary shocks [Mitra & van Duijn (2021)], stable and expanding plateaus.

Finally, the viscous fingering phenomenon was studied using higher dimensional travelling wave solutions [Mitra et al (2020)]. It was proved that in the presence of both hysteresis and dynamic capillarity effects, finger shaped solutions exist that propagate with constant speed.