I agree with Paul, I spent over 12 years in Minot ND, Grand Forks ND and Great Falls MT… Those places are crazy cold!
Snow is actually a very good insulation blanket. This is why Eskimos can survive in the Artic at -60, but the temp in the igloo is a balmy 33± deg. Even body heat in a properly designed snow cave will raise the inside temp to right about the freezing point. This is the latent heat concept, where the temp will not raise much above freezing because it takes a huge amount of energy to force the snow to change states into water. Anyway, I’ve digresses…
Also a consideration is mineral and soil competition of the earth itself in the affected region. …As you know, heat travels by conduction, convection and even radiation. Never the less, the temp of soil is normally around 55 deg F at 8’ to 10’ deep all year, regardless of outside temp. Some things do influence movement of heat in the earth. Highly compacted soil will conduct heat (or loss of heat) slightly better, than loose fill soil. This concept as applied to blown insulation in an attic is apparent. Mashed insulation just does not insulate as well, and also mashed soil follows in the same way to various degrees.
So if you had extremely compact soil, zero snow cover, and it was -40 for 6 straight weeks, you may freeze a pipe at 4 foot deep. But I’d carefully examine the point the pipe froze up at… is there a 3 foot deep bolder covering the area where the pipe broke… with its top exposed to the ambient air, and to what extent do the minerals in the highly compacted rock conduct heat away from the soil near the pipe.