Equivalent (effective) permeability for 2D flow through anisotropic soil For a soil with permeabilities kx and ky in x- and y-directions, determine the effective permeability k_eq applicable to two-dimensional flow analyses.

Introduction / Context:
Natural soils often exhibit directional permeability due to stratification or fabric, leading to anisotropic hydraulic conductivities. For two-dimensional seepage, an equivalent isotropic permeability is used after a coordinate transformation, preserving Laplace’s equation form.

Given Data / Assumptions:

• Anisotropic soil with conductivities kx and ky along orthogonal axes.
• Steady, incompressible, 2D flow conditions.
• Goal: find an equivalent permeability that makes transformed flow nets valid.

Concept / Approach:
By stretching coordinates x and y with factors proportional to sqrt(kx) and sqrt(ky), the anisotropic Laplace equation can be recast into an isotropic form. The effective permeability corresponding to this transformation equals the geometric mean of principal conductivities.
k_eq = (kx * ky)^(1/2)

Step-by-Step Solution:
1) Recognize anisotropy with principal values kx, ky.2) Apply coordinate transform: x' = x * (kx)^(1/2), y' = y * (ky)^(1/2).3) Under transform, governing equation becomes isotropic with k_eq = (kx ky)^(1/2).4) Use k_eq in discharge/head-loss computations for 2D flow.

Verification / Alternative check:
When kx = ky, the geometric mean reduces to kx, confirming consistency. For layered media, this result aligns with classical derivations of transformed flow nets.

Why Other Options Are Wrong:
Arithmetic sum or RMS do not arise from the governing equation; kx/ky is dimensionally inconsistent for permeability.

Common Pitfalls:
Confusing geometric mean (2D flow) with arithmetic/harmonic means used for series/parallel 1D layered systems; mixing axes after rotation.

Final Answer:
(kx ky)^(1/2)