Hello,

Thank you for contacting Ask Extension. Could you please send a picture of the trunk and the 'sap' that is coming out, to be able to diagnose the situation.

Thank you for contacting Ask Extension. If you had the tree cut down, and had the stump ground down, the roots are pretty much dead and cannot produce sap, only the tree's vascular system can do that. 

You do not state why you had the birch taken down. But when the tree was ground down it seems that you have a condition called Slime Flux. Slime flux is typically caused by bacteria that enter the tree through wounds or cracks in the trunk. When these bacteria multiply, they can cause a dark, wet mark in the wood, that could have transferred to the mulch that you have around the ground trunk. As the slime flux dries, a light gray to white encrustation will remain on the mulch. Many trees develop slime flux including elm, oak, and mulberry, and to a lesser extent maple, birch, butternut, redbud, sycamore, and walnut. There is no cure or fix for this slime flux. But you can use a dilute solution of bleach to reduce the smell. 

I have included several publications for your reference:

https://www.purdue.edu/fnr/extension/slime-flux-of-trees/

https://extension.psu.edu/discovering-bacterial-wetwood

If you need further help, please contact us back.