• Journal of the Korean Mathematical Society
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• v.46 no.5
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• pp.895-905
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• 2009

Let $B^{n+1}$ be the unit ball in the complex vector space $\mathbb{C}^{n+1}$ with the standard Hermitian metric. Let ${\Sigma}^n={\partial}B^{n+1}=S^{2n+1}$ be the boundary sphere with the induced CR structure. Let f : ${\Sigma}^n{\hookrightarrow}{\Sigma}^N$ be a local CR immersion. If N < 3n - 1, the asymptotic vectors of the CR second fundamental form of f at each point form a subspace of the CR(horizontal) tangent space of ${\Sigma}^n$ of codimension at most 1. We study the higher order derivatives of this relation, and we show that a linearly full local CR immersion f : ${\Sigma}^n{\hookrightarrow}{\Sigma}^N$, N $\leq$ 3n-2, can only occur when N = n, 2n, or 2n + 1. As a consequence, it gives an extension of the classification of the rational proper holomorphic maps from $B^{n+1}$ to $B^{2n+2}$ by Hamada to the classification of the rational proper holomorphic maps from $B^{n+1}$ to $B^{3n+1}$.

• The Journal of Engineering Geology
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• v.28 no.2
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• pp.217-246
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• 2018

During the selection and characterization of target formations in the Small-scale Offshore $CO_2$ Storage Demonstration Project in the Pohang Basin, we have carefully investigated the possibility of induced earthquakes and leakage of $CO_2$ during the injection, and have designed the storage processes to minimize these effects. However, people in Pohang city have a great concern on $CO_2$-injection-intrigued seismicity, since they have greatly suffered from the 5.4 magnitude earthquake on Nov. 15, 2017. The research team of the project performed an extensive self-investigation on the safety issues, especially on the possible $CO_2$ leakage from the target formation and induced earthquakes. The target formation is 10 km apart from the epicenter of the Pohang earthquake and the depth is also quite shallow, only 750 to 800 m from the sea bottom. The project performed a pilot injection in the target formation from Jan. 12 to Mar. 12, 2017, which implies that there are no direct correlation of the Pohang earthquake on Nov. 15, 2017. In addition, the $CO_2$ injection of the storage project does not fracture rock formations, instead, the supercritical $CO_2$ fluid replaces formation water in the pore space gradually. The self-investigation results show that there is almost no chance for the injection to induce significant earthquakes unless injection lasts for a very long time to build a very high pore pressure, which can be easily monitored. The amount of injected $CO_2$ in the project was around 100 metric-tonne that is irrelevant to the Pohang earthquake. The investigation result on long-term safety also shows that the induced earthquakes or the reactivation of existing faults can be prevented successfully when the injection pressure is controlled not to demage cap-rock formation nor exceed Coulomb stresses of existing faults. The project has been performing extensive studies on critical stress for fracturing neighboring formations, reactivation stress of existing faults, well-completion processes to minimize possible leakage, transport/leakage monitoring of injected $CO_2$, and operation procedures for ensuring the storage safety. These extensive studies showed that there will be little chance in $CO_2$ leakage that affects human life. In conclusion, the Small-scale Offshore $CO_2$ Storage Demonstration Project in the Pohang Basin would not cause any induced earthquakes nor signifiant $CO_2$ leakage that people can sense. The research team will give every effort to secure the safety of the storage site.