Kenmore Water heater, Model # 153...

Kenmore Water heater, Model # 153.339462, Pilot lights, burner burns for approx. 10 - 15 minutes then both go out. after which it is more difficult to light the pilot, and it goes out again in less time that it did the first time. it does not appear to be a thermocouple issue, but the burner unit appears to have two, one is a copper colored tube, the other is silver in color. I'm thinking either the thermostat or perhaps a secondary thermocouple that checks the actual burner for flame. I need ideas please