SPE10 upscaling — pressure-solve vs simple averaging

SPE10 model 2 (60×220×85 ≈ 1.12M cells) is deliberately brutal: permeability spans ~5 orders of magnitude. A full-resolution run took 8h32m wall on the HPC box — fine to run once, hopeless for parameter sweeps. Upscaling is the way out.

Two families of upscaling

Simple averages

Porosity upscales by arithmetic (volume-weighted) mean:

\bar\phi = \frac{1}{V}\sum_i \phi_i\,V_i

Permeability has the layered bounds — arithmetic (parallel flow) and harmonic (series flow):

K_\text{arith} = \frac{1}{V}\sum_i K_i V_i, \qquad K_\text{harm} = \left(\frac{1}{V}\sum_i \frac{V_i}{K_i}\right)^{-1}

On a 5-orders-of-magnitude field these bounds are far apart, so a single scalar average is unreliable.

Flow-based (what OPM does)

upscale_perm solves the single-phase pressure equation on each coarse block

\nabla\cdot\big(\mathbf{K}\,\nabla p\big) = 0

under periodic / fixed / linear boundary conditions, then back-computes the effective permeability tensor $\mathbf{K}_\text{eff}$ from the Darcy flux response. This is the only method that stays honest on SPE10’s contrast.

OPM `opm-upscaling` tools

binary	what it does
`upscale_perm`	single-phase → effective $\mathbf{K}$ tensor
`upscale_relperm`	two-phase steady state → upscaled $k_r(S)$
`upscale_avg`	arithmetic / geometric / harmonic means (fast, crude)

# effective permeability of a coarse block, periodic BCs
upscale_perm -bc p fine_block.grdecl