Assigning accumulated volumes in pipeline network

I'm working on finding a way to automate the application of volumes to line segments through a pipeline network. I'm open to any way of achieving this, be it through the creation of a geometric network or something else.

I've worked with generating a geometric network and using volume as a weight. This allowed for the tracing of upstream accumulation at specific points such as the sink or various network junction. This only partially helps me. I need the volume through each segment. I was told that the Trace Geometric Network processing tool might do the trick, but either didn't understand the results it gave me or it simply didn't solve my problem.

I saw some where in the forums about the Attribute Assistant Methods that are a part of ArcGIS for Local Governments and started trying to explore the tools that are a part of that to see if that could help me. I do not currently have access to the package and thus can't play with it myself to see what can work.