Average permeability of layered soil

Treat each zone of the soil as layers in series or parallel. Let the left block (zone 1) have permeability $k_1$, length $L_1$ and height $H = H_1 + H_2$. The right block (zone 2) has length $L_2$ and is made of $k_2$ (upper, $H_1$) over $k_3$ (lower, $H_2$).

  <------ L1 ------><------ L2 ------>
  +-----------------+-----------------+ ^
  |                 |   k2       H1   | |
  |       k1        +-----------------+ H
  |                 |   k3       H2   | |
  +-----------------+-----------------+ v

1. Horizontal flow (left to right)

Inside zone 2 the flow is along the layers, so the two layers act in parallel:

$$k_{h2} = \frac{k_2H_1 + k_3H_2}{H_1 + H_2}$$

Zone 1 and zone 2 are met one after the other, so they act in series (same flow, lengths add):

$$k_{avg,H} = \frac{L_1 + L_2}{\dfrac{L_1}{k_1} + \dfrac{L_2}{k_{h2}}}$$

2. Vertical flow (top to bottom)

Inside zone 2 the flow crosses the layers, so they act in series:

$$k_{v2} = \frac{H_1 + H_2}{\dfrac{H_1}{k_2} + \dfrac{H_2}{k_3}}$$

Zone 1 and zone 2 now carry flow side by side under the same head (parallel). Weighting by the widths:

$$k_{avg,V} = \frac{k_1 L_1 + k_{v2} L_2}{L_1 + L_2}$$

Substituting $k_{v2}$ gives the full expression:

$$k_{avg,V} = \frac{k_1L_1 + \dfrac{(H_1+H_2)\,L_2}{\dfrac{H_1}{k_2}+\dfrac{H_2}{k_3}}}{L_1+L_2}$$