Statistical moment equations for forward and inverse modeling of multiphase flow in porous media
- A probabilistic framework for dynamic data integration (history matching) has become accepted practice in reservoir engineering applications. The idea is to build an ensemble of reservoir models consistent with the geologic scenario, where each member of the ensemble honors all the 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 for MCS. To achieve reasonable accuracy in flow performance predictions; however, large numbers of realizations are usually necessary. Without an adequate number of realizations, it is not possible to quantify, or analyze, the uncertainty space. Here, we develop a Statistical Moment Equations (SME) approach for both the forward (flow-performance predictions) and inverse (conditioning on dynamic data) problems. In the SME method, the equations governing the statistical moments of the quantities of interest (pressure, saturation, travel-time, total- and water-production rates) are derived and solved directly. We study nonlinear, immiscible, two-phase flow problems, 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 for a few locations at different times. For the forward problem, the flow (pressure and total-velocity) equations are solved on a regular grid, while a streamline-based strategy is used to solve the nonlinear transport moments. A Kriging-based inversion algorithm, in which the statistical moments of the permeability are conditioned directly, is developed. The computational efficiency of the inversion scheme is discussed. We also analyze the behaviors of the statistical moments of pressure and saturation as a function of the available measurements in both space and time. We also compare the SME inversion scheme with the Kalman filter approach for dynamic data integration. The nature of the prediction uncertainty space and how it changes in the presence of dynamic measurements, such as saturation, are investigated. The analysis helps explain the success of the SME Kriging based inversion scheme in updating the statistical moments of the permeability field in the presence of measurements at different locations and different acquisition times.
|Type of resource
|electronic; electronic resource; remote
|1 online resource.
|Stanford University, Department of Energy Resources Engineering
|Li, Liyong, 1962-
|Li, Liyong, 1962-
|Statement of responsibility
|Submitted to the Department of Energy Resources Engineering.
|Ph. D. Stanford University 2010
- © 2010 by Pipat Likanapaisal
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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