2x4 walls with 2x6 insulation?

compressing insulation doesnt work too well, the air space designed in it is part of the r value, they do make high density insulation with a higher r value for the same thickness..sealing any draft coming through the wall would be better time spent ...

As downhill said "sealing any draft coming through the wall would be better time spent" is the top priority with that board sheathing and possible a poor air barrier on the outside.

High density 5-15 fiberglass or r-15 Roxul would be my choice, with pieces of house wrap cut to fit above and below that 2x4 that runs on the diagonal. Seal that house wrap in place and you have your air barrier.

Bud

Just wanted to add, when you fill those cavities with batt insulation, be sure to cut out for the diagonal so it is a perfect fit.

Ayuh,.... The only way I've found that can get ya the insulation R-factor of a 2x6 wall in a 2x4 wall is closed cell spray foam, installed by a pro-spray foam installer,...

+1

SPF works quite well here.

I might fill that cavity with some rigid foam to limit any over expansion of foam inside the gaps in the sheathing from leaking to the outside.

Now would be a good time to build the wall out with 2 X 2's.
Also make sure to add a nailing plate where that drain runs through the bottom plate.

Extruded polystyrene insulation (the blue or pink stuff) has an R value of 5.0 per inch. With 3.5 inch studs, that would give you an R value of 17.5, not counting the wood sheathing on the exterior and the drywall inside.

You could cut the extruded polystyrene to fit your stud cavities, and then hold it in place around the perimeter of each stud cavity with expanding foam (like the Great Stuff product).

One of the benefits of using extruded polystyrene insulation is that it doesn't require a vapour barrier. It's impermeable to air, and so it doesn't need a vapour barrier to prevent air seepage through it.

The R value of wood is 1.0 per inch.