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The quinzhee wall, rated

Snow shelters · viscosity, settlement and warmth

A quinzhee starts as a heap: snow is shovelled into a pile and deliberately mixed as it goes, working out to about 250 kg/m³, then left alone for a day to sinter before anyone hollows it. The pile's compactive viscosity follows Kojima's exponential with η₀ = 8.5 × 10⁶ Pa·s and a density coefficient f = 0.021 per kg/m³. A layer partway down the pile carries about 600 Pa of overburden — roughly 30 cm of snow sitting above it. After the 24 h wait the shelter is dug out, leaving a wall 40 cm thick. Find the compactive viscosity of the mixed pile, the density that loaded layer settles to during the wait, and the R-value of the finished wall.

σLρtR

Every number in this problem is editable — change any value below and the whole chain recalculates.

Given
  • η₀ = 8,500,000 Pa·s — Zero-density viscosity intercept (Kojima)
  • f = 0.021 per kg/m³ — Density coefficient
  • ρ₀ = 250 kg/m³ — Density of the mixed pile
  • σ = 600 Pa — Overburden partway down
  • t = 24 h — Sinter wait before hollowing
  • L = 40 cm — Finished wall thickness
Determine
  1. (a)the compactive viscosity of the mixed pile
  2. (b)the density the loaded layer settles to in the 24 h wait
  3. (c)the R-value of the finished 40 cm wall
Step 1 of 3(a) · solve for Compactive viscosity

Kojima's exponential is the whole reason mixing works: every ln2/0.021 ≈ 33 kg/m³ of density MULTIPLIES the stiffness by two, so hauling loose 100 kg/m³ powder up to a mixed 250 kg/m³ pile buys viscosity by factors, not percent. At 250 kg/m³ the pile sits at 1.62 × 10⁹ Pa·s — the stiffening that makes a quinzhee carveable at all.

ηlogρsη0f
Rearranged for η
η=η0 efρs\eta = \eta_0 \, e^{f \rho_s}
Your values, in your units
η=(8,500,000 Pa⋅s)⋅e(0.021 m3/kg)⋅(250 kg/m3)\eta = \left( 8{,}500{,}000\ \text{Pa}{\cdot}\text{s} \right) \cdot e^{\left( 0.021\ \text{m}^{3}\text{/kg} \right) \cdot \left( 250\ \text{kg/m}^{3} \right)}
Answer
η=1,619,800,000 Pa⋅s\eta = 1{,}619{,}800{,}000\ \text{Pa}{\cdot}\text{s}

Carried onward at full precision, not this rounded figure.

Open the Compactive Viscosity of Snow (Kojima) solver →

Step 2 of 3(b) · solve for Density after settling

The viscosity from (a) is exactly what resists the settling here — the pile stiffens as it densifies, so under 600 Pa the whole 24 h wait adds only about 3 %, landing at 258 kg/m³. The overnight wait was never about density: it is buying sintered bond strength between the disturbed grains, and the density barely moves while that happens.

ρ0tσρs
Rearranged for ρs
ρs=ρ0 exp⁡ ⁣(σ tη)\rho_s = \rho_0 \, \exp\!\left( \frac{\sigma \, t}{\eta} \right)
1,619,800,000 Pa·scarried from step 1
Your values, in your units
ρs=(250 kg/m3)⋅exp⁡ ⁣((600 Pa)⋅(24 h)(1,619,810,000 Pa⋅s))\rho_s = \left( 250\ \text{kg/m}^{3} \right) \cdot \exp\!\left( \frac{\left( 600\ \text{Pa} \right) \cdot \left( 24\ \text{h} \right)}{\left( 1{,}619{,}810{,}000\ \text{Pa}{\cdot}\text{s} \right)} \right)
Answer
ρs=258.13 kg/m3\rho_s = 258.13\ \text{kg/m}^{3}

Carried onward at full precision, not this rounded figure.

Open the Snowpack Settlement (Viscous Compaction) solver →

Step 3 of 3(c) · solve for R-value of the snow wall

Sturm's fit puts conductivity at 258 kg/m³ around 0.093 W/(m·K), so 40 cm of wall rates RSI 4.31 — about R-24 imperial, warmer than most stick-built house walls. The same wall packed to 400 kg/m³ would rate far worse, because density buys strength at the direct price of warmth.

ventLRsinkρs
Rearranged for R
R=Lkeff(ρs)R = \frac{L}{k_{\mathrm{eff}}(\rho_s)}
258.13 kg/m³carried from step 2
Your values, in your units
R=(40 cm)keff ⁣((258.13 kg/m3))R = \frac{\left( 40\ \text{cm} \right)}{k_{\mathrm{eff}}\!\left(\left( 258.13\ \text{kg/m}^{3} \right)\right)}
Answer
R=4.3147 RSI (m2⋅K/W)R = 4.3147\ \text{RSI (m}^{2}{\cdot}\text{K/W)}

Carried onward at full precision, not this rounded figure.

Open the R-Value of a Snow Wall solver →

Answer

Therefore the mixed pile carves at a compactive viscosity of about 1.62 GPa·s, the loaded layer settles only to 258 kg/m³ over the day's wait — and the finished 40 cm wall rates RSI 4.31, about R-24 imperial, a snow heap outperforming most stick-built walls.

Why this order

The quinzhee's one trick is in step (a): undisturbed snowpack is layered and treacherous, but shovelling and mixing it breaks every grain's bonds and leaves a jumble of fractured crystals in contact, and fractured crystals sinter — necks of ice grow at every touch point while the pile sits. The exponential viscosity law says the mixed, densified pile is orders of magnitude stiffer than the powder it came from, and the 24-hour wait converts that stiffness into bonded strength. Step (b) closes the loop on itself: the very viscosity the mixing bought is what holds the pile up under its own weight, which is why the wait changes density by 3 % and strength by a great deal more.

Where people go wrong is reading step (c) as a safety rating. These pages compute material properties — a viscosity, a density, an R-value — and none of them ever says a shelter is safe: collapse in a snow shelter is burial, and nobody digs alone. The other trap is chasing density for strength: pack the wall to 400 kg/m³ and it hardens beautifully while the R-value collapses, because the still air in the pore space was doing the insulating all along. A quinzhee is a compromise struck at moderate density, and both ends of this chain are needed to see why.

Carried values move at full precision, not the rounded figure shown — chaining rounded numbers compounds error.