How to convert difference in elevation into distance?
When you're running up a slope, you're running slower. Down the hill you may run faster, if your knees allow it. But how much?
In German, there is the notion of "Leistungskilometer", which is translated to "effective kilometers", "technical kilometers", "power kilometers" or "energy expenditure kilometers", depending on the source. The definition is that 100m difference in elevation are equivalent to 1 flat kilometer, at least for calculating hiking times.
I find that for running this is not true. Using a factor of 10 for the difference in elevation gives me virtual flat paces that are way beyond my abilities. Say you're running 6 km up a 500m hill in 45 minutes. This is a pace of 7:30; with a factor 10 the flat equivalent would 6 + 10x0.5 = 11 kilometers, giving a pace of 4:05.
I think a pace of 5:18 (which is using a factor of 5) would be more realistic.
Are there common conventions how to account for slopes when running?
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The following calculator leverages more recent studies on determining the relationship between incline gradient, velocity, and calories in addition to in the field experiment data: Calcuator.
The best equivalent I know of is Naismith's rule.
"Allow 1 hour for every 3 miles (5 km) forward, plus 1 hour for every 2000 feet (600 metres) of ascent."
Secondly, when it comes to run it will come down to how good you personally are at running up hills. You may need to calculate the factor yourself.
Remember that Garmin devices display sea-level distance, not slope distance.
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