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In a statement the environmental group said Abigail Smith and Nick Hanafin started climbing the Majestic Centre this morning and were expected to take all day.
They were attempting to get to OMV's offices on the 20th floor of the building to "deliver a message".
The protest was against the arrival of a self-propelled drill rig, to be used by OMV to drill exploratory wells off the Taranaki Coast and in the Great South Basin, Greenpeace said.
The oil company has applied for a marine consent to operate in the southern ocean, and earlier this year started a maintenance project at the Pohokura platform off the coast of Taranaki.
Climate campaigner Amanda Larsson said oil companies like OMV had known for 30 years that burning fossil fuels was driving climate change.
"We've seen governments including those of the UK, Wales and Scotland declare national climate emergencies, and councils all over New Zealand including Wellington, Auckland, Christchurch, and Dunedin doing the same.
"We're living through a climate emergency. If we don't halve global carbon emissions in just a decade, we'll be locked into extreme weather and heating that will threaten all life on Earth, including our own," she said.