Summary
The determination of optimal well settings is very demanding computationally
because the simulation model must be run many times during the course of the
optimization. For this reason, reduced-order modeling procedures, which are a
family of techniques that enable highly efficient simulations, may be very
useful for optimization problems. In this paper, we describe a recently
developed reduced-order modeling (ROM) technique that has been used in other
application areas, the trajectory piecewise linearization (TPWL) procedure, and
incorporate it in production-optimization computations. The TPWL methodology
represents solutions encountered during the optimization runs in terms of
Taylor-series expansions around previously simulated states. This requires a
small number of preprocessing (training) simulations using the full
(high-fidelity) model, during which pressure and saturation states and Jacobian
matrices are saved. These states and matrices are then projected into a
low-dimensional space using proper orthogonal decomposition (POD). Simulations
in this reduced space can be performed very efficiently; in this work, we
observe runtime speedups of a factor of 450. Overall speedups are, however,
less because of the preprocessing overhead.
We assess the TPWL representation for simulations of waterflood in a
heterogeneous 3D model containing more than 20,000 gridblocks and six wells.
The high degree of accuracy of the TPWL model is first demonstrated for several
testing simulations in which producer- and injector-well settings differ from
those used in the training runs. The TPWL representations are then used in
optimizations involving the determination of optimal bottomhole pressures
(BHPs) for a reservoir model with four production wells and two injection
wells. A gradient-based algorithm is applied for the optimizations. In the
first case, the BHPs of the producers and injectors are optimized at six
different times (36 control variables) and in the second case at 15 different
times (90 control variables). Results for optimized net present value (NPV)
using TPWL are shown to be in consistently close agreement with those computed
using high-fidelity simulations. Most significantly, when the optimal well
settings obtained using the TPWL procedure are applied in high-fidelity models,
the resulting NPVs are within approximately 0.5% of the values determined using
the high-fidelity simulations. Our overall conclusion is that the TPWL
representation may be quite useful in production-optimization problems.
© 2009. Society of Petroleum Engineers
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History
- Original manuscript received:
21 October 2008
- Meeting paper published:
2 February 2009
- Revised manuscript received:
18 June 2009
- Manuscript approved:
27 June 2009
- Published online:
3 December 2009
- Version of record:
17 June 2010
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