In this first photo, you can see that the drain sits well above the low area where water is puddling.  This can be especially problematic on membrane roofs with chemical welds because sometimes the seams fail.  The smallest hole or tear can become a big problem when standing water is present.

In this photo, you can see that the drip edge is almost completely detached from the roof. The membrane seems to be pulling it up, leaving the roof vulnerable to wind and rain. This needs to be addressed immediately.

This photo shows that this section, along with much of the roof, is completely covered in standing water. In some places, it is more than 2 inches. This can be very problematic for EPDM roofing systems. If even the smallest puncture is beneath the water, it will eventually try to drain through. In the winter, this could turn to ice, causing excessive expansion and contraction of the membrane and the seams.

Vent boots and other penetrations can be especially vulnerable to a roof holding water, especially on EPDM membranes. They are tricky to install and adhere with a brush and specialized glue vs. TPO. Heat-welded to a mechanically or fully attached system typically.

This particular membrane hole has been going on for a while. A repair was attempted at some point but has long since begun to leak and cause the insulation board underneath to deteriorate. This will eventually cause damage to the eaves and the building beneath.

We were able to spot several areas that are likely to be leaking into the eaves, damaging soffits and facia board. So far, the damage does not seem to be extensive.

In the photos above, you can see a tapered system that we installed recently in Maryville, Tennessee. It is designed to move the water to a guttering system on one side of the building. Crickets are then added to channel water directly into the scupper drains. The pitch allows the water to move off the building efficiently well the welded seams and boots ensure a sound roof with permanent bonds. This is my ideal recommendation for your roof problems.