How to patch hole in interior chimney wall

brickholemasonry

I was recently removing plaster from the interior wall of my chimney and discovered a huge hole that had been patched by some drywall.

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The chimney is still used to vent my furnace and water heater, but there are no active fireplaces anymore. I would like to patch this hole with brick. My currently plan is to take out the top and bottom bricks so that the hole becomes rectangular and then fill it with new bricks. Something like this:

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Is this the correct way to approach this problem? Is there a simpler way to solve this while still filling the hole with brick? Are there tools and techniques for getting grout into the bricks like this?

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