# How to calculate the average velocity over the flowpath length?

I’m working on building a GIS based groundwater travel time distribution (TTD) model similar to the one developed by Schilling and Wolter (2007) http://lilt.ilstu.edu/ewpeter/GEO435/Readings/2012/Schilling_Kelly.pdf.

The conceptual model of cell TTD is illustrated below (Schilling and Wolter, 2007).

As you know Velocity = -k (dh/dl)/n

I have calculated the velocity using raster calculator. I used the permeability values from the soil map as a substitute for hydraulic conductivity, the slope from the DEM as substitute for hydraulic gradient (dh/dl) and porosity (n) one value only from literature. I calculated the distance till the nearest river using Euclidean Distance.

After getting the velocity and the distance, I can easily calculate the time using raster calculator. However, it is not straightforward like this. Schilling & Wolter (2007) described how to calculate the time as follows “The time needed for groundwater in each 5-m cell to travel from the cell to the stream network was then determined by averaging all the individual cell velocities over the ﬂowpath length from the cell to the stream”

So, in order to calculate the time, I have to calculate the average velocities over the flowpath length which I don’t know how to do. It would be much appreciated if you can help me in this regard.

• Your work appears inconsistent. If you truly calculated velocity (not just speed), you can see that the groundwater rarely flows in a straight line towards its discharge point. Thus the Euclidean distance has little relevance. You need to track the flow down flow lines instead and, at each cell, record the cumulative time and cumulative distance from the discharge point. That's what Schilling & Wolter are saying. (BTW, this is not an average velocity, it's an average speed.) You can adapt a costdistance or pathdistance calculation to do this. – whuber Aug 30 '13 at 15:28
• @whuber Many thanks for your help and detailed reply. I do agree with you the Euclidean distance was not the right choice. What about using the flow length to calculate the distance and using the flow accumulation as the weight raster with a null value for the river network (i.e. cells with flow accumulation values >=100000 for example)? It would be much appreciated if you can explain in more detail how can I use cost distance and path distance to calculate the travel time? – shiny Sep 2 '13 at 2:33