Our plant pathologist thinks this may be slime flux, which can be relatively common on tree trunks but which can also emerge from other wood, such as woody roots. The odor would be coming from naturally-occurring yeasts and other microbes that are consuming/fermenting the ooze that contains some sap sugars.

There is no way to remedy it directly and the microbe population will wane once the oozing slows or stops, having run out of food. Why that area is staying so wet might be of more concern, at least with regards to plant health since few species can tolerate saturated root zones for very long.

Miri

Do you think if I add some more top soil or lime it will dry out that area?

Probably not, since it won't address why that area is collecting water (besides being slightly lower-grade, we presume, since we can't see the surrounding land level in the photo). There are many plant species tolerant of poor drainage, depending on how sunny or shady that area is in summer and whether deer browsing is a risk, but turfgrass and certain tree roots will not thrive in soggy conditions if they persist for too long. The tree's root system at least covers a wider area, so if most of its root zone is not this wet, the tree might be fine as-is, but the turfgrass may need replacing with other plants in the trouble spot if the situation persists.

Miri