Uncertainty propagation and sensitivity analysis of three-phase flow in porous media using polynomial chaos expansion

Abstract

In this study, polynomial chaos expansion (PCE) will be utilized to investigate propagation and evolution of uncertainty from uncertain model parameters to model outputs. Unknown PCE coefficients are calculated using probabilistic collocation method (PCM), which is a non-intrusive technique. The proposed approach is evaluated on a 2D reservoir under three-phase flow condition. By use of PCEs, dependent model outputs, such as well bottomhole pressure and production rates, are represented as functions of uncertain model parameters, including parameters defining the capillary pressure (Pc) and relative permeability (kr) curves. Accuracy, compatibility and efficiency of the proposed approach are compared with Monte Carlo (MC) method, which is considered as the reference case. Mean values, standard deviations and histograms for well bottomhole pressure and production rates estimated from the PCM-based PCE are closely matched to those obtained from MC method. This discloses the fact that the proposed approach can successfully restore the simulation model to represent propagation of uncertainty from uncertain model parameters toward model outputs and to perform sensitivity analysis (SA). Unlike the MC approach which requires an extremely large number of realizations to obtain accurate results, the PCM-based PCE requires to perform model evaluations only at the selected collocation points, thus the computational cost is significantly reduced. Subsequently, using the proposed PCE as the surrogate model of the reservoir, Sobol's sensitivity indices are calculated analytically. Results of SA reveal that those parameters defining Pc curves have negligible effect on the model outputs compared with those parameters defining kr curves, and thus may be ignored during history matching process. Moreover, it is observed that the effects caused by interactions between uncertain kr and Pc parameters have negligible contribution compared to the individual effect of each parameter.