Quantification of Uncertainty for Numerical Simulations
with Confidence Intervals
We present a prediction and uncertainty assessment methodology for numerical
simulation. The methodology allows prediction of confidence intervals.
It has been developed jointly with a number of colleagues. It is a work
in progress in the sense that not all components of the methodology
are complete. The methodology, at its present level of development,
will be illustrated in two specific cases: the flow of oil in petroleum
reservoirs (with prediction of production rates) and an analysis of
solution errors for the simulation of shock wave interactions.
The formalism assesses uncertainty and yields confidence intervals associated
with its prediction. In the terminology of verification and validation,
these predictions can be verified as exact within a framework for statistical
inference, but they are not validated as being descriptive of a physical
situation. In fact the present illustrative examples are simplified
and are not intended to represent an experimental or engineering system.
The methodology combines new developments in the traditional areas of
oil reservoir upscaling and history matching with a new theory for numerical
solution errors and with Bayesian inference. For the shock wave simulations,
the new result is an error analysis for simple shock wave interactions.
The significance of our methods, in the petroleum reservoir context,
is their ability to predict the risk, or uncertainty associated with
production rate forecasts, and not just the production rates themselves.
The latter feature of this method, which is not standard, is useful
for evaluation of decision alternatives. For shock wave interactions,
the significance of the methodology will be to contribute to verification
and validation of simulation codes.