Upscaled Dynamic Relative Permeability for Unstable CO₂ Flow in Stratified Porous Media

In geologic porous media, layering is ubiquitous on all length scales ranging from sub-mm compositional laminations to km-thick formations. The current study presents a general framework for estimating the two-phase viscous limit dynamic relative permeability (k_r) for non-communicating stratified sedimentary rocks. The approach is a two-stage upscaling taking into account the influence of nested heterogeneity at intra- and inter-layer scales. Sub-layer heterogeneities, promoting microflow instabilities in individual layers, are modelled by tuning input relative permeability parameters based on the viscosity contrast between fluids. Macroscopic flow instability at the domain scale is tackled by a new analytical solution for the saturation distribution in stratified media considering the complexities of frontal advance theory. For each layer, three flow stages are considered, each with its unique time-dependent behaviour. The overall solution is presented as a set of equations derived by overlapping of the solutions in different layers. The upscaled dynamic k_r is figured out by history matching. The practicality of the method is demonstrated in application to measurements collected from the Otway International Research Facility, Victoria, Australia. The results show the significant impact of layer separation on the upscaled endpoint saturations in addition to required modifications on the current k_r formulas. A comparison with conventionally modelled steady-state sweep highlights the inadequacy of steady-state upscaling for viscous limit flows. The applicability of the method to estimate k_r from the unsteady-state lab experiments conducted on heterogeneous cores is also discussed.