Summary
In reservoir evaluation problems, the reservoir properties are largely
unknown. To infer these properties from observations of the reservoir
production is referred to as history matching or production history
conditioning. Traditionally, this is done by repeated fluid-flow simulations,
where all the available production data are used simultaneously to arrive at a
set of history-matched reservoir models. In recent years, the amount of data
continuously collected from a reservoir under production has been on the
increase. Hence, the need for automatic, continuous model updating is apparent.
The ensemble Kalman filter has been shown to be suitable for this purpose.
However, large reservoir evaluation problems require upscaling reservoir
properties to perform the necessary number of fluid-flow simulations.
Traditional ensemble Kalman filtering is shown to give bias in the production
history conditioned reservoir representations. The loss in accuracy and
precision introduced by performing fluid-flow simulations on a coarser scale
should be accounted for, but this is rarely or never done. We introduce the
scale-corrected ensemble Kalman filter approach in order to quantify the loss
in accuracy and precision. A reference scale is defined and all uncertainty
quantifications are made relative to this scale, although the fluid flow
simulations are made on a coarser scale. The production history conditioned
reservoir representation will be accurate with realistic precision measures on
this reference scale. The methodology is demonstrated on a large case study
inspired by the characteristics of the Troll field in the North Sea.
Introduction
One of the objectives of reservoir evaluation is to find the optimal well
configuration and well-operating conditions for a given reservoir. Forecasts of
hydrocarbon production for a given recovery strategy can be used to determine
this. Quantification of the uncertainty both in the prediction of the reservoir
properties and in the forecast of the production properties should be an
integral part of the evaluation process.
The assessment of the uncertainty in the production forecasts requires
repeated fluid-flow simulations. This is done by using a reservoir production
simulator. The reservoir conditions needed as input to this simulator are in
practice largely unknown, however. Therefore, the uncertainty in the reservoir
properties must be described by a stochastic reservoir model, taking all the
available data into account. Prior to starting production, the available data
are static data. After the reservoir has been in production for a while, an
observed production history is also available. The observed production history
should be used to update the reservoir model, and thereby improve the
production forecasts. In petroleum-related literature this is referred to as
"history matching."
Traditionally, production history conditioning is performed through repeated
fluid-flow simulations, where the reservoir properties are tuned to the
production history, either manually or automatically by minimizing an objective
function involving the mismatch between simulated and observed production.
There are two problems with this methodology. The first problem is the
computational cost of repeated fluid-flow simulations, which severely restricts
the size of the reservoir models to which the production history conditioning
can be applied. The second problem is that the reservoir models are updated
using all the available production data simultaneously. This means that when
new production data become available, the entire production history
conditioning process must be repeated. In recent years, the use of permanent
sensors for monitoring dynamic production properties has increased, requiring
more frequent updating of the reservoir models.
Ideally, the observations should be included in the model sequentially as
they become available. This approach requires continuous or sequential
production history-conditioning techniques. The Kalman filter has been widely
used for this type of time series problem. However, the Kalman filter is most
appropriate when the number of variables in the model is low and the
observations are linearly related to the model. This is not the case in
spatio-temporal reservoir evaluation problems, where the number of model
parameters is typically very high, and the relation between the reservoir model
and the production observations, represented by a fluid-flow simulator, is
highly nonlinear.
Several extensions to the Kalman filter techniques have been suggested,
among these the ensemble Kalman filter, developed by Evensen (1994). The
ensemble Kalman filter is used to update both the reservoir properties and the
production properties. The computations are based on an ensemble of
realizations of the reservoir and production properties, from which relevant
statistics concerning the model uncertainty can be estimated. At times where
new observations become available, all ensemble members are updated to honor
these observations. Consequently, the realizations are always kept up to date
with the latest observations. The ensemble Kalman filter methodology has been
applied to numerous cases in various fields, such as weather forecasting
(Evensen 1994; Houtekamer and Mitchell 1998), ground water hydrology (Reichle
et al. 2002), and petroleum engineering (Nævdal et al. 2002, 2005; Gu and
Oliver 2005; Wen and Chen 2006; Haugen et al. 2006). For a review of recent
progress see Evensen (2007).
The ensemble Kalman filter is shown to perform well with an ensemble size of
around 100 members. In practice, however, the computational demands by
fluid-flow simulation on large reservoir models prohibit ensembles of this
size. This problem is typically overcome by performing fluid-flow simulations
on a coarser-scale representation of the reservoir variables. This upscaling is
known to introduce bias in the production forecasts, however, which should be
accounted for. For notational convenience, we will refer to this as
coarse-scale fluid flow simulation contrary to fine-scale fluid flow simulation
on the preferable fine-scale representation of the reservoir variables. Let us
emphasize that the same fluid-flow simulator is used; it is only the gridding
of the input variables which vary. In this paper we use the general ensemble
Kalman filtering framework of Evensen (1994), and extend it to correct for the
effect of using coarse-scale fluid flow simulators, using the approach of Omre
and Lødøen (2004). The basic idea of Omre and Lødøen is to use coarse-scale
fluid flow simulation to predict the results from fine-scale fluid-flow
simulation, and to assess the associated prediction uncertainty. The fine-scale
representation is termed the reference scale. This correction is feasible if
the coarse-scale fluid flow simulations capture the most important features of
the fine-scale fluid-flow simulations. We coin our approach scale-corrected
ensemble Kalman filter.
This paper proceeds as follows: We start by defining the notation and
describing the ensemble Kalman filter methodology. Then we motivate and present
our model extensions. We proceed by presenting a case study, which is inspired
by the characteristics of the Troll field in the North Sea. Further, we present
and discuss the results from our simulation studies, and finally we draw some
conclusions.
© 2008. Society of Petroleum Engineers
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History
- Original manuscript received:
9 May 2007
- Revised manuscript received:
5 October 2007
- Manuscript approved:
18 October 2007
- Version of record:
25 June 2008
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