Summary
An inversion method for the integration of dynamic (pressure) data directly
into statistical moment equations (SMEs) is presented. The method is
demonstrated for incompressible flow in heterogeneous reservoirs. In addition
to information about the mean, variance, and correlation structure of the
permeability, few permeability measurements are assumed available. Moreover,
few measurements of the dependent variable are available. The first two
statistical moments of the dependent variable (pressure) are conditioned on all
available information directly. An iterative inversion scheme is used to
integrate the pressure data into the conditional statistical moment equations
(CSMEs). That is, the available information is used to condition, or improve
the estimates of, the first two moments of permeability, pressure, and velocity
directly. This is different from Monte Carlo (MC) -based geostatistical
inversion techniques, where conditioning on dynamic data is performed for one
realization of the permeability field at a time. In the MC approach, estimates
of the prediction uncertainty are obtained from statistical post-processing of
a large number of inversions, one per realization.
Several examples of flow in heterogeneous domains in a quarter-five-spot
setting are used to demonstrate the CSME-based method. We found that as the
number of pressure measurements increases, the conditional mean pressure
becomes more spatially variable, while the conditional pressure variance gets
smaller. Iteration of the CSME inversion loop is necessary only when the number
of pressure measurements is large. Use of the CSME simulator to assess the
value of information in terms of its impact on prediction uncertainty is also
presented.
Introduction
The properties of natural geologic formations (e.g., permeability) rarely
display uniformity or smoothness. Instead, they usually show significant
variability and complex patterns of correlation. The detailed spatial
distributions of reservoir properties, such as permeability, are needed to make
performance predictions using numerical reservoir simulation. Unfortunately,
only limited data are available for the construction of these detailed
reservoir-description models. Consequently, our incomplete knowledge
(uncertainty) about the property distributions in these highly complex natural
geologic systems means that significant uncertainty accompanies predictions of
reservoir flow performance.
To deal with the problem of characterizing reservoir properties that exhibit
such variability and complexity of spatial correlation patterns when only
limited data are available, a probabilistic framework is commonly used. In this
framework, the reservoir properties (e.g., permeability) are assumed to be a
random space function. As a result, flow-related properties such as pressure,
velocity, and saturations are random functions. We assume that the available
information about the permeability field includes a few measurements in
addition to the spatial correlation structure, which we take here as the
two-point covariance. This incomplete knowledge (uncertainty) about the
detailed spatial distribution of permeability is the only source of uncertainty
in our problem. Uncertainty about the detailed distribution of the permeability
field in the reservoir leads to uncertainty in the computed predictions of the
flow field (e.g., pressure).
© 2006. Society of Petroleum Engineers
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History
- Original manuscript received:
3 December 2004
- Revised manuscript received:
16 February 2006
- Manuscript approved:
8 March 2006
- Version of record:
20 June 2006
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