Joint quantification of uncertainty on spatial and non-spatial reservoir parameters Comparison between the Joint Modeling Method and Distance Kernel Method

The experimental design methodology is widely used to quantify uncertainty in the oil and gas industry. This technique is adapted for uncertainty quantification on non-spatial parameters, such as bubble point pressure, oil viscosity, and aquifer strength. However, it is not well adapted for the case of geostatistical (spatial) uncertainty, due to the discrete nature of many input parameters as well as the potential nonlinear response with respect to those parameters. One way to handle this type of uncertainty, derived from the experimental design theory, is called the joint modeling method (JMM). This method, originally proposed in a petroleum context by Zabalza (2000), incorporates both non-spatial and spatial parameters within an experimental design framework. The method consists of the construction of two models, a mean model which accounts for the non-spatial parameters and a dispersion model which accounts for the spatial uncertainty. Classical Monte-Carlo simulation is then applied to obtain the probability density and quantiles of the response of interest (for example the cumulative oil production). Another method to quantify spatial uncertainty is the distance kernel method (DKM) proposed recently by Scheidt and Caers (2007), which defines a realization-based model of uncertainty. Based on a distance measure between realizations, the methodology uses kernel methods to select a small subset of representative realizations which have the same characteristics as the entire set. Flow simulations are then run on the subset, allowing for an efficient and accurate quantification of uncertainty. In this work, we extend the DKM to address uncertainty in both spatial and non-spatial parameters, and propose it as an alternative to the joint JMM. Both methods are applied to a synthetic test case which has spatial uncertainty on the channel representation of the facies, and non-spatial uncertainties on the channel permeability, porosity, and connate water saturation. The results show that the DKM provides for a more accurate quantification of uncertainty with fewer reservoir simulations. Finally, we propose a third method which combines aspects of the DKM and the JMM. This third method again shows improvement in efficiency compared to the JMM alone.