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The hoies on drainpipe go down. The idea is that the water accumulates in the drain rock surrounding the pipe. The pipe admits water through the bottom holes and conducts it longitudinally down the trench. The holes go down in order to keep rocks and dirt from getting into the pipe. For a normal curtain drain, you would embed the pipe in 1' of drain rock, then cover that with geotextile fabric or filter fabric, then backfill. Use "washed drain rock" or septic system rock (all graded to a uniform size -- unlike filling a hole or building a road you do NOT want the fines mixed in). The place where you buy the rock will know what you mean when you say "drain rock". In fact, you could design the whole works like it was a septic drain field working in revers. But as F14 says, if the backfill freezes, the gravel and drainpipe below it will not help. The barn will still flood. Better to regrade around the barn. Gutters would help a lot, but perhaps you are avoiding them due to snow sliding off the roof. One drain pipe option that might work would be to put the drain pipe below the frost line and backfill with gravel all the way to the surface. The resulting porous fill would not be able to accumulate enough water to freeze up and become impermeable. Kind of the same theory as the subbase below a floating slab, or the roadbase they put below the highway. You would have to keep plant matter from growing a soil layer on top of the gravel. This would be a lot of gravel and a tad expensive...
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