Summary
Dynamic optimization of waterflooding using optimal control theory has
significant potential to increase ultimate recovery, as has been shown in
various studies. However, optimal control strategies often lack robustness to
geological uncertainties. We present an approach to reduce the effect of
geological uncertainties in the field-development phase known as robust
optimization (RO). RO uses a set of realizations that reflect the range of
possible geological structures honoring the statistics of the geological
uncertainties. In our study, we used 100 realizations of a 3D reservoir in a
fluvial depositional environment with known main-flow direction. We optimized
the rates of the eight injection and four production wells over the life of the
reservoir, with the objective to maximize the average net present value (NPV).
We used a gradient-based optimization method in which the gradients are
obtained with an adjoint formulation. We compared the results of the RO
procedure to two alternative approaches: a nominal-optimization (NO) and a
reactive-control approach. In the reactive approach, each production well is
shut in when production is no longer profitable. The NO procedure is based on a
single realization. In our study, the NO procedure is performed on each of the
100 realizations in the set individually, resulting in 100 different
NO-production strategies. The control strategies were applied to each
realization, from which the average NPVs, the standard deviation, the
cumulative-distribution functions, and the probability-density functions were
determined. The RO results displayed a much smaller variance than the
alternatives, indicating an increased robustness to geological uncertainty.
Moreover, the RO procedure significantly improved the expected NPV compared to
the alternative methods (on average 9.5% higher than using reactive-control and
5.9% higher than the average of the NO strategies).
Introduction
In this paper, we consider the secondary-recovery phase of a petroleum
reservoir using waterflooding. In this case, a number of injection and
production wells are drilled to preserve a steady reservoir pressure and sweep
the reservoir. The use of smart wells expands the possibilities to manipulate
and control fluid-flow paths through the oil reservoir. The ability to
manipulate (to some degree) the progression of the oil/water front provides the
possibility to search for a control strategy that will result in maximization
of ultimate oil recovery.
Dynamic optimization of waterflooding using optimal control theory has
significant potential to increase ultimate recovery by delaying water
breakthrough and increasing sweep, as has been shown in various studies
(Brouwer and Jansen 2004). However, optimal control strategies often lack
robustness to geological uncertainties. By discarding these uncertainties, the
sensitivity to a possibly large system/model mismatch is not taken into account
within the optimization procedure. As a result, the optimal control strategy
may cease to be optimal or may even result in very poor performance.
Dealing with uncertainty is a topic encountered in many fields related to
modeling and control. It can essentially be divided into two different
strategies, which are not mutually exclusive: reducing the uncertainty itself
using measurements [i.e., history matching (Landa and Horne 1997, Li et al.
2003)] and reducing the sensitivity to the uncertainty. In this paper, we
consider a situation in which no production data are assumed to be available,
which rules out any history-matching approach to reduce the geological
uncertainty. Our study forms part of a larger research project to enable
closed-loop, model-based reservoir management (Jansen et al. 2005).
A suggested approach from the process industry, to optimization problems
that suffer from vast uncertainty and limited measurement information, is the
use of a so-called RO technique (Srinivasan et al. 2003, Terwiesch et al. 1998,
Ruppen et al. 1995). In RO, the optimization procedure is performed over a set
of realizations, actively accounting for the influence of the uncertainty. The
implementation of multiple realizations within the optimization process has
been addressed by Yeten et al. (2002). However, their study deviates in the way
the realizations are incorporated in the objective function, in the
optimization method, and in the number of realizations. The goal of our paper
is to present an RO procedure on the basis of a set of 100 realizations of a 3D
oil/water reservoir, which leads to a control strategy that accounts explicitly
for geological uncertainty.
© 2009. Society of Petroleum Engineers
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History
- Original manuscript received:
22 June 2006
- Meeting paper published:
24 September 2006
- Revised manuscript received:
14 March 2008
- Manuscript approved:
27 August 2008
- Published online:
16 March 2009
- Version of record:
1 March 2009
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