Rugged Energy Landscapes in Multiphase Porous Media Flow: A Discrete-Domain Description

Abstract:

Immiscible displacements in porous media involve a complex sequence of pore-scale events, from the smooth, reversible displacement of interfaces to abrupt interfacial reconfigurations and rapid pore invasion cascades. Discontinuous changes in pressure or saturation have been referred to as Haines jumps, and they emerge as a key mechanism to understand the origin of hysteresis in porous media flow. Hysteresis persists at the many-pore scale: when multiple cycles of drainage and imbibition of a porous sample are conducted, a dense hysteresis diagram emerges. The interpretation of hysteresis as a consequence of irreversible transitions and multistability is at the heart of early hysteresis models, and in recent experiments, and points to an inherently non-equilibrium behavior. For a given volume fraction of fluids occupying the pore space, many stable configurations are possible, due to the tortuous network of nonuniform pores and throats that compose the porous medium, and to complex wetting and capillary transitions. Multistability indicates that porous media systems exhibit rugged energy landscapes, where the system may remain pinned at local energy minima for long times.

We address the question of developing a zero-dimensional model that inherits the path-dependence and `'bursty'' behavior of immiscible displacements, and propose a discrete-domain model that captures the role of metastability and local equilibria in the origin of hysteresis. We describe the porous medium and fluid system as a discrete set of weakly connected, multistable compartments, charaterized by a unique free energy function. This description does not depend explicitly on past saturations, turning points, or drainage/imbibition labels. The system behaves hysteretically, and we rationalize its behavior as sweeping a complex metastability diagram, with dissipation arising from discrete switches among metastable branches. The hysteretic behavior of the pressure-saturation curve is controlled by the topography of the energy landscape, through the number of metastable regions of the compartments and characteristic height of the energy barriers separating the different basins. Our model opens the door to fully explore the interplay between hysteresis and fluctuations in multiphase displacements in porous media.