Permeability Averaging

Which Average?

Upscaling replaces a stack of fine cells with one coarse value. For most properties an arithmetic average is fine, but permeability is directional: the right average depends on how the flow meets the heterogeneity. Get it wrong and the coarse model conducts far too much or far too little.

Along and Across

In a layered rock, flow along the layers runs through them in parallel, so the high-permeability layers carry it and the effective permeability is the arithmetic mean. Flow across the layers passes through them in series, so the tightest layer throttles it and the effective permeability is the harmonic mean, dominated by the smallest value. The two can differ by orders of magnitude.

The Geometric Middle

The geometric mean sits between them and is a reasonable estimate for a randomly heterogeneous rock with no strong layering. The ordering arithmetic, then geometric, then harmonic always holds, and the gap widens with the permeability contrast. Knowing the flow direction relative to the layering is what picks the right one.