Performance analysis and key findings
Analysis of how different risk aversion parameters (λ) affect the expected NPV and standard deviation of the selected well portfolio.
Plot showing how mean NPV decreases as λ increases
As risk aversion increases, the optimizer selects well portfolios with lower expected NPV but also lower risk.
Plot showing how standard deviation decreases as λ increases
The standard deviation of the portfolio NPV decreases with increasing λ, demonstrating effective risk management.
Plot showing the efficient frontier of mean NPV vs. standard deviation
The efficient frontier represents the optimal trade-off between return (mean NPV) and risk (standard deviation). Each point on the curve corresponds to a different value of the risk aversion parameter λ.
Analysis of how the low-rank approximation affects solution quality and computational efficiency.
Plot showing how approximation error decreases as rank increases
The relative error in utility decreases as the rank of the approximation increases, with diminishing returns at higher ranks.
Plot showing number of differently selected wells vs. rank
As the rank increases, the solutions from the approximated problem more closely match those from the full problem, with fewer differently selected wells.
Analysis of the computational efficiency gained through low-rank approximation techniques.
Plot showing computation time scaling with problem size
The computation time for the full QKP grows rapidly with problem size, while the low-rank approximation shows much better scaling.
Plot showing computational speedup vs. rank
Lower ranks provide greater computational speedup, with a trade-off in solution quality. The optimal rank depends on the specific problem and accuracy requirements.
Plot showing the trade-off between approximation error and computational speedup
This plot illustrates the trade-off between solution quality and computational efficiency. Points closer to the bottom-left corner represent better trade-offs, with lower error and higher speedup.
A detailed case study demonstrating the application of the QKP Well Placement Optimizer to a realistic scenario.
| Parameter | Value |
|---|---|
| Reservoir Grid | 50 × 50 cells |
| Potential Well Locations | 100 |
| Wells to Select | 10 |
| Risk Aversion (λ) | 0.0, 0.5, 1.0, 2.0, 5.0 |
| Variogram Model | Exponential, range = 10 |
| RSVD Ranks | 10, 20, 30, 50, Full |
Map showing optimal well placements for different risk aversion values
This visualization shows how the optimal well locations change with different risk aversion parameters. Risk-neutral portfolios (λ = 0) tend to cluster wells in high-value areas, while risk-averse portfolios (high λ) spread wells to diverse locations to reduce correlation.
Using a low-rank approximation with rank = 20 achieved a 10× computational speedup while maintaining solution quality within 2% of the optimal solution for most risk aversion parameters.