It's fragrant, produces purple-blue flowers in spring, and adapts well to sun or shade. I just described Creeping Charlie also known as ground ivy. A weed most gardeners are trying to eliminate from their lawns.
Digging is a difficult, long-term process and one where the creeping Charlie usually outlasts the gardener. As you can see, this weed creeps along the soil surface, rooting as it grows. Any little piece of stem that gets left behind will root and re-infest the area.
You may want to try the new chelated iron weed killers. This natural selective herbicide kills broadleaf weeds but not your grass.
Treating with a broadleaf weed killer when the plants are in full bloom or after a hard freeze will kill the plant. Just keep in mind there are lots of seeds in the soil waiting for a bit of space to sprout and grow.
Be sure to read and follow label directions carefully.