An efficient analytical approach for steady-state upscaling of relative permeability and capillary pressure

Upscaling in petroleum reservoir simulation captures the dynamic behavior of fine-scale models at the coarse-scale in a volume average sense. Traditional upscaling of immiscible two-phase flow can be implemented by solving the fluid flow equations at steady state in the capillary or viscous limit in fine scale, which requires modeling the single-phase flow numerically many times and is computationally demanding. In this study, an analytical approach is proposed, to replace the numerical simulation, for computing the relative permeability. The key idea is to utilize the perturbation expansion technique and Fourier analysis to derive an explicit expression of the equivalent permeability. The analytical expression accounts for spatial correlations and is more accurate than the simple averaging and renormalization methods. It matches well with the numerical method in all cases for the upscaling of the relative permeability and capillary pressure. The developed method is also validated by comparing the pressure and saturation results from the coarse-scale and fine-scale models. In the base case of SPE10 benchmark test, the numerical upscaling takes 436 s, whereas the analytical upscaling takes 13.6 s, which is about 30 times faster than the numerical method. Moreover, the analytical coefficients just need to be computed once for the whole space in a given geostatistical model, which naturally leads to improved efficiency and is clearly favorable for petroleum reservoir simulations.