Solute Mixing Under Unstable Two-Phase Flow in Heterogeneous Porous Media

Heterogeneous porous media saturated with two liquid phases represent a complex system that can be observed in many natural and engineering processes. The transport of passive solutes in this type of environment is at the centre of our research whose final aim is to quantify and mathematically describe the physical mechanisms that regulate the displacement of the solute, such as twisting and stretching. To analyse and quantify the dynamics of the solute mixing and dispersion, we rely on numerical simulations where a passive solute is transported by two fluids through a heterogeneous porous media, such as a reservoir or an aquifer. Based on the mutual miscibility of the fluids two main scenarios are identified, one where the fluids that transport the passive solute are miscible and one where they are immiscible. In both cases the passive solute can freely cross the interface between the two fluids. The setup for the numerical experiment is a three-dimensional flow and transport domain where permeability is represented by an indicator multi-Gaussian random field whose continuous parent fields are characterised by exponential covariance function. We prescribe the mean flow while periodic conditions are applied to the permeability on the lateral boundaries. The injection of the less viscous fluid into the domain saturated with a more viscous fluid happens along a control plane perpendicular to the mean flow direction. The consequent displacement of the more viscous fluid by a less viscous fluid leads to fingering instabilities. The flow fluctuations are governed by the unstable displacement of the two fluids and the spatial heterogeneity. To study the mixing of a passive solute in this flow, we consider an instantaneous solute injection over the control plane at time zero. For both scenarios, the solute dispersion is quantified in terms of the spatial moments of the solute distribution while mixing is measured through the scalar dissipation rate, dilution index, and the probability density function of concentration point values. Mixing metrics that show regular trends are fitted using power and exponential laws. Compared to the constant viscosity case, it is observed that the viscosity difference between the liquid phases enhances the mixing of the passive solute.