“Connection Gas” is distinct increase of gas above a normal background gas level when bottom up occurs after a connection. If you see a small high gas peak in short period of time and when you calculate back you will see that it comes from every connection, you can suspect that this is the Connection Gas.
You may have heard about gas show, connection gas and trip gas. What are differences of those term?
Gas show: It is gas that appears in drilling mud and it indicates the presence of gas zones or soluble gas in oil.
Connection gas: It is relatively small amount of gas that enters well when mud pump is stopped because ECD (equivalent circulating density) decreases when stop pumping. Connection gas indicates that mud weight in hole is less than formation pressure.
The difference of gas show and connection gas:If there is connection gas, gas shown in a mud logger screen increases and decreases sharply like small gas peak; however, gas show demonstrates the smoother curve than connection gas. Lag stroke, lag time are used to find where the location of connection gas or gas show comes from. Especially, when the connection gas presents, lag time of gas peak will be relative to the time when pumps are off.
Trip gas: It is gas entering to well bore when tripping out of hole. There are several factors as balled-up bit, fast pulling out rate, bigger diameter BHA, etc can create swab effect that can reduce hydrostatic pressure while tripping out of hole. If too much reduction in hydrostatic pressure is present, formation gas can influx into the well bore, called “Trip gas”.
Reference book: 
Well Control Books
