Dynamic Data Integration and Quantification of Prediction Uncertainty Using Statistical-Moment Equations

Abstract

Summary The use of a probabilistic framework for dynamic data integration (history matching) has become accepted practice. In this framework, one constructs an ensemble of reservoir models, in which each realization honors the available (static and dynamic) information. The variations in the flow performance across the ensemble provide an assessment of the prediction uncertainty due to incomplete knowledge of the reservoir properties (e.g., permeability distribution). Methods based on Monte Carlo simulation (MCS) are widely used because of the generality and simplicity of MCS. As a black-box approach, only pre- and post-processing of conventional flow simulations are needed. To achieve reasonable accuracy in estimating the statistical moments of flow-performance predictions, however, large numbers of realizations are usually necessary. Here, we use a different, and direct, approach for model calibration and uncertainty quantification. Specifically, we describe a statistical-moment-equations (SMEs) framework for both the forward and inverse problems associated with immiscible two-phase flow. In the SME method, the equations governing the statistical moments of the quantities of interest (e.g., pressure and saturation) are derived and solved directly. We assume that statistical information and a few measurements are available for the permeability field. As for the dynamic properties, we assume that measurements of pressure, saturation, and flow rate are available at a few locations and at several times. For the forward problem, the flow (pressure and total-velocity) SMEs are solved on a regular grid, while a streamline-based strategy is used to solve the transport SMEs. We use a kriging-based inversion algorithm, in which the first two statistical moments of permeability are conditioned directly using the available dynamic data. We analyze the behaviors of the saturation moments and their evolution as they are conditioned on measurements, in both space and time. Moreover, we discuss the relationship between the widely used MCS-based Kalman-filter approach and our SME inversion scheme.