 Layered porous permeability

ID: layered-porous-permeability

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Codex  0 2026-10-14

Consider planar horizontal layers with positive scalar intrinsic intrinsic permeabilities

k_{i}

, thicknesses

b_{i}

, and fractions

f_{i} = b_{i} / \sum_{j} b_{j}

. Assume each layer is uniform and isotropic, fully saturated by the same Newtonian fluid of constant mass density and dynamic viscosity, with perfect hydraulic contact and steady incompressible flow. Use boundaries that impose pure flow normal to the layers or a common gradient of hydraulic head along them; additional interface resistance and finite-edge effects are excluded.

For vertical flow normal to the layers, every layer carries the same normal Darcy flux by mass conservation. The hydraulic head drops add. Writing

B = \sum_{i} b_{i}

and taking positive

Δ h_{i}

as the drop along the flow direction gives

q = \frac{ρ g}{μ} k_{i} \frac{Δ h _{i}}{b _{i}}, Δ h = \sum_{i} Δ h_{i} = \frac{μ q}{ρ g} \sum_{i} \frac{b _{i}}{k _{i}}, k_{⊥} = \frac{B}{\sum _{i} b _{i} / k _{i}} = (\sum_{i} \frac{f _{i}}{k _{i}})^{- 1} .

(1)

This is the thickness-weighted harmonic mean. It is hydraulic head, rather than fluid pressure alone, that accounts correctly for gravity in the vertical calculation. A thin layer of very small permeability of a porous medium can control the whole discharge because all flow must pass through it.

For horizontal flow parallel to the layers, each layer experiences the same gradient of hydraulic head. For width

W

, each area is

A_{i} = W b_{i}

, and the individual volume fluxes add:

Q = \sum_{i} Q_{i} = - \frac{ρ g}{μ} \frac{d h}{d x} \sum_{i} k_{i} A_{i}, \frac{Q}{A} = - \frac{ρ g}{μ} \frac{d h}{d x} \sum_{i} f_{i} k_{i}, k_{∥} = \sum_{i} f_{i} k_{i} .

(2)

This is the thickness-weighted arithmetic mean. A highly permeable layer can carry most of the horizontal flow because it supplies a continuous route from inlet to outlet. Woessner and Poeter derive both cases for hydraulic conductivity; identical fluid properties allow the common factor

ρ g / μ

to cancel.^[ref]

For two equally thick layers, take

k_{1} = k_{0}

and

k_{2} = 9 k_{0}

, where

k_{0} = 1 0^{- 12} m^{2}

. The values 1 and 9 are therefore permeabilities in units of

k_{0}

, rather than dimensionless intrinsic intrinsic permeabilities. Then

k_{⊥} = \frac{2}{1/ k _{0} + 1/ ( 9 k _{0} )} = 1.8 k_{0}, k_{∥} = \frac{k _{0} + 9 k _{0}}{2} = 5 k_{0} .

(3)

Across the layers, the less permeable half takes

90%

of the total hydraulic head drop and the more permeable half only

10%

. Along the layers, the more permeable half carries

90%

of the total volume flux. At the same magnitude of imposed gradient of hydraulic head, horizontal Darcy flux is

5/1.8 = 25/9 ≃ 2.78

times the vertical Darcy flux.

The arrangement thus creates physical anisotropy from locally isotropic ingredients. In coordinates with

x, y

along the layers and

z

normal to them, the effective permeability of a porous medium tensor is

k_{eff} = diag (5, 5, 1.8) k_{0}

. A driving gradient of hydraulic head with both horizontal and vertical components generally produces a discharge in a different direction. This distinction matters when deciding whether a geological bed conducts aquifer flow along itself or acts as a barrier to flow across it.

Figure 1.
The same two layers give a harmonic mean across them and an arithmetic mean along them
. **Details**.

Regenerate the diagram by running python ../layered-porous-permeability.py from _media; it writes its PNG into the current directory.

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