Investigation of saturation dependency of oil relative permeability during WAG process through linear and non-linear PCA

We characterize three-phase relative permeability data sets available in the literature in terms of basic descriptive statistics, bivariate correlation, as well as linear (PCA), nonlinear (NLPCA) and hierarchical principal component analyses (h-NLPCA). These studies are viewed in the context of the assessment of three-phase oil relative permeabilities for water alternating gas injection (WAG) protocols, where a proper (qualitative and quantitative) analysis of the dependence of observed three-phase oil relative permeability data on fluid saturations is of critical relevance for practical applications. Here, we focus on the characterization of the dependence of three-phase oil relative permeability on an identifiable set of Principal Components. We analyze the relationship between observed core scale three-phase oil relative permeability and input variables which are typically employed in the application of existing effective (pseudo-empirical) models. Input variables include saturations of fluids, saturations ending points, as well as two-phase relative permeabilities obtained from oil-water and oil-gas environments. The use of available prior information about saturation ending points is also discussed in the framework of Constrained Principal Component Analysis (CPCA). Our results show that: (i) the degree of nonlinearity displayed by the relationship between the input variables and three-phase oil relative permeability is in contrast with the fundamental assumptions underlying existing empirical models; (ii) a sigmoid-based empirical model can effectively characterize three-phase oil relative permeability as a function of fluid saturations, saturation ending points and oil relative permeability data collected under two-phase conditions.