Uncertainty Quantification of Two-phase Flow in Heterogeneous Reservoirs Using a Streamline-based Pdf Formulation

Oil recovery forecast relies on predictions of saturation fields from reservoir simulations. These predictions, in return, are dependent on the quality of the input data available, such as the reservoir permeability and porosity fields. However, because of inherent uncertainties in these input data, due to sparse and possibly inaccurate measurements, a stochastic framework is usually adopted to describe their variability. Probability distributions are then assigned to the input fields. In this stochastic context, predicting the saturation fields consists in determining the saturation probability distributions. For nonlinear immiscible two-phase flow in highly heterogeneous porous media, we propose an efficient distribution method to estimate the saturation cumulative distribution function (CDF) and probability density function (PDF). While high heterogeneity is traditionally regarded as an obstacle to designing stochastic methods, this distribution method exploits the physics of highly heterogeneous porous media. The method identifies hidden smooth random fields (RFs) that explain the saturation distribution, together with the nonlinear mapping from these hidden RFs to the saturation. We show that the statistics and distributions of these hidden RFs can be estimated very efficiently and simply, leading to a method that is computationally inexpensive compared to full Monte Carlo (MC) simulations. We provide numerical evidence of the agreement between MC-based and distribution-based saturation CDF, PDF and moments, in high heterogeneous cases such as high variance lognormal permeability fields, anisotropic layered fields and channelized systems. Finally, we show how the saturation CDF can be used for risk assessment by relying not only on the mean and standard deviation of the saturation, but primarily on saturation quantiles, obtained from probability of exceedance.