Relative impacts of permeability heterogeneity and viscosity contrast on solute mixing

Solute transport in porous media is affected by several factors. The heterogeneous structure of the permeability field is a key factor controlling the spreading and mixing behaviors of a solute cloud. On the other hand, other factors such as the viscosity contrast between the dissolved solute and the ambient fluid can also play an important role. Although both these mixing mechanisms (field heterogeneity and viscosity contrast) have been acknowledged and studied, more investigations are needed in order to better characterize the effect of the variation of both the degree of viscous fingering and the level of disorder of the porous medium. This work aims to explore the impact of field heterogeneity and viscosity contrast on the transport behavior of an inert solute in a two-dimensional flow field. To achieve this, we performed high-resolution numerical simulations based on the spectral method to solve coupled flow and transport equations for a given range of viscosity contrast and log-permeability variance. We analyze the degree and rate of mixing, contour length of the solute cloud, spatial statistics of the concentration field, and arrival times at a control plane to characterize spreading and mixing in the domain. Through the use of numerical simulations, we provide a quantitative separation of the impacts of fingering and heterogeneity and we parametrize the concentration probability distribution function. We find that the interplay among viscous fingering, high-permeability channeling, and low-permeability stagnation at small scales create important features in the spreading and mixing characteristics. In particular, our results indicate that at early times viscosity contrast has a more significant impact on mixing than permeability heterogeneity, and the effect of viscosity contrast on early and late arrival times at a control plane is enhanced by increasing levels of permeability heterogeneity; on the other hand, heterogeneity reduces the peak concentration at a control plane and causes larger solute cloud spreading when the solute is more viscous than the ambient fluid compared to when the solute is less viscous. Moreover, we find that the concentration cumulative distribution function of the solute cloud can be described as a beta distribution for the range of viscosity contrast and permeability heterogeneity considered.