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 owing 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.
© 2011. Society of Petroleum Engineers
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History
- Original manuscript received:
22 November 2008
- Meeting paper published:
2 February 2009
- Revised manuscript received:
20 June 2011
- Manuscript approved:
24 June 2011
- Published online:
19 December 2011
- Version of record:
13 March 2012
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