Global Resource Uncertainty Using a Spatial/Multivariate Decomposition Approach
Authors: Ren W. , Leuangthong O. , Deutsch C.V. 
Published: March 2010

Abstract
Quantifying uncertainty in petroleum resources is important for development planning and decision making. Increasingly, geostatistical techniques are used to integrate diverse data sources and provide a defensible model of uncertainty. Petroleum resources are calculated from a combination of variables including thickness, porosity and saturation. Uncertainty in global petroleum resources are calculated stepwise:establish the local uncertainty in each variable using a conventional Gaussian geostatistical model;sample the local distributions with spatial correlation using a p-field based technique;modify the p-field samples to have the correct multivariate variability using the LU technique; and,assemble the distribution of uncertainty over any volume using the joint spatial/multivariate realizations.
The alternatives to this technique are a simplistic Monte Carlo simulation without spatial correlation or a more complex high resolution geostatistical model. Speed and mathematical consistency are the main advantages of the proposed technique. The theoretical basis of the spatial/multivariate decomposition approach is developed with the assumptions and implementation details. A synthetic example from a realistic case study is presented showing the global uncertainty in oil-in-place over arbitrarily large areas.

Introduction
Geostatistical techniques have been increasingly used for reservoir characterization for two main reasons:different data sources can be integrated to predict a reservoir property between wells; and,an assessment of uncertainty in the estimation can be obtained(1).
Quantifying global uncertainty in petroleum resources is important for reservoir development planning and decision making. There are two challenges that must be addressed:the scale of uncertainty ---local uncertainty must be scaled to global uncertainty; and,the multivariate relationship between the variables that go into resource calculations ---predictions of uncertainty must account for the correlation between variables.
Conventional geostatistical techniques predict local uncertainty at the scale of the data. Global uncertainty refers to the petroleum resource, or the oil-in-place (OIP), for an arbitrarily large area. The global uncertainty in OIP cannot be calculated by simply summing the local uncertainties. The spatial continuity of the variables must be considered in scaling local uncertainty to global uncertainty. If the variable is very discontinuous, then the uncertainty decreases quickly with scale. If the variable is continuous, then the uncertainty decreases slowly, but fewer data are needed to constrain the uncertainty. Simulation must be used to combine uncertainty reconciling these two notions.
OIP is calculated from several reservoir properties, such as net pay thickness, porosity and oil saturation. Treating the variables independently has a risk of underestimating the global resource uncertainty. High values average out with low values. The correlation between the multiple constituent variables of OIP must be calculated.
A spatial/multivariate decomposition approach is proposed for assessing the global uncertainty in OIP from local uncertainties. The joint spatial and multivariate correlations are taken into account. The key idea is to simulate a set of spatially correlated probability values (a ‘p-field’) and then simultaneously draw the variable of interest at multiple locations. The LU decomposition is used to account for the multivariate correlations at each location.