A probabilistic framework for dynamic data integration (history matching) has become accepted practice. The idea is to build an ensemble of reservoir models, all of which being consistent with the geologic scenario and also honoring all available (static and dynamic) information. In addition to data assimilation, the probabilistic framework provides an assessment of the prediction uncertainty due to incomplete knowledge of the reservoir description. Methods based on Monte Carlo Simulation (MCS) are widely used. This is driven by the generality and simplicity of MCS. As a black-box approach, only pre/post-processing of conventional flow simulations is 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 an alternative direct approach for model calibration and uncertainty quantification. Specifically, we describe a Statistical Moment Equations (SME) framework for both the forward and inverse problems. In the SME method, the equations governing the statistical moments of the quantities of interest (e.g., pressure, saturation) are derived and solved directly. Immiscible two-phase flow problems are investigated, where we assume that in addition to statistical information (and a few measurements) about the permeability field, measurements of pressure, saturation, and flow rate are available at a few locations and several times. For the forward problem, the flow (pressure and total-velocity) equations are solved on regular grid, while a streamline-based strategy is used to solve the transport moments. A Kriging-based inversion algorithm, in which the statistical moments of permeability are conditioned directly based on available dynamic data, is used. We analyze the behaviors of the saturation moments and their evolution as they are conditioned on measurements, in both space and time. Moreover, we compare the SME inversion scheme with Kalman filter approaches (an MCS inversion approach) for dynamic data integration, from a mathematical framework perspective.

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