---
title: "Seepage in anisotropic soil"
subject: "Soil Mechanics"
chapter: 7
key: "semester_4/soil_mechanics:seepage-anisotropic-soil-derivation"
source: https://www.bctengineer.com/q/semester_4/soil_mechanics/seepage-anisotropic-soil-derivation
site: BCT Engineering
---

# Seepage in anisotropic soil

Soil Mechanics · Chapter 7: Seepage through Soils

- Asked in: 2073 Bhadra, 4 marks
- Video: [Seepage Through Anisotropic Soil | Lecture 20 | Geotechnical Engineering](https://www.youtube.com/watch?v=1tTaNLvW1xM) — GATE ACADEMY - CE | ME | CH by Umesh Dhande
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/soil_mechanics/seepage-anisotropic-soil-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-7

## Question

Derive the relationship for the seepage discharge through anisotropic soil.

## Short answer

Transform horizontal dimensions by √(kz/kx) to make the section Laplace‑governed; an equivalent isotropic permeability k'=√(kx kz). Draw a square flow net on the transformed section; seepage q = √(kx kz) h (Nf/Nd).

## Answer

In anisotropic soil the permeability differs in the two directions: $k_x$ (horizontal, usually larger) and $k_z$ (vertical). The flow net cannot be drawn directly with square fields, so the section is transformed.

#### Governing equation

Continuity for steady flow, $\dfrac{\partial v_x}{\partial x} + \dfrac{\partial v_z}{\partial z} = 0$, with Darcy's law $v_x = k_x\dfrac{\partial h}{\partial x}$ and $v_z = k_z\dfrac{\partial h}{\partial z}$ (sign neglected), gives

$$
k_x\frac{\partial^2 h}{\partial x^2} + k_z\frac{\partial^2 h}{\partial z^2} = 0
$$

#### Transformation

Put $X = x\sqrt{\dfrac{k_z}{k_x}}$. Then $\dfrac{\partial}{\partial x} = \sqrt{\dfrac{k_z}{k_x}}\dfrac{\partial}{\partial X}$ and the equation becomes

$$
k_z\frac{\partial^2 h}{\partial X^2} + k_z\frac{\partial^2 h}{\partial z^2} = 0 \;\Rightarrow\; \frac{\partial^2 h}{\partial X^2} + \frac{\partial^2 h}{\partial z^2} = 0
$$

which is the Laplace equation for an isotropic medium. So the section is redrawn with all horizontal dimensions multiplied by $\sqrt{k_z/k_x}$ (vertical dimensions unchanged), and the flow net with square fields is drawn on this transformed section.

#### Discharge

The flow across a vertical plane in the real section, through a vertical height $dz$:

$$
\Delta q = v_x\,dz = k_x\frac{\partial h}{\partial x}dz = k_x\sqrt{\frac{k_z}{k_x}}\frac{\partial h}{\partial X}dz = \sqrt{k_xk_z}\,\frac{\partial h}{\partial X}\,dz
$$

This is Darcy's law in the transformed section with the equivalent permeability

$$
k' = \sqrt{k_x\,k_z}
$$

For a flow net of square fields with total head loss $h$:

$$
\boxed{q = \sqrt{k_x k_z}\;h\,\frac{N_f}{N_d}}
$$

The same expression follows by considering the flow across a horizontal plane, so the result is independent of the direction.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the governing equation for anisotropic soil?
   A. k_x d2h/dx2 + k_z d2h/dz2 = 0
   B. d2h/dx2 = 0
   C. k_x dh/dx = k_z dh/dz
   D. d2h/dx2 + d2h/dz2 = k
   Answer: A. Continuity with Darcy's law in each direction gives this. (Key point: Governing equation)

2. Which transformation makes it Laplace's equation?
   A. X = x sqrt(k_z/k_x)
   B. X = x sqrt(k_x/k_z)
   C. X = x k_x/k_z
   D. Z = z k_z
   Answer: A. Scaling x by sqrt(k_z/k_x) makes both terms have coefficient k_z. (Key point: Transformation)

3. What is the equivalent permeability?
   A. (k_x + k_z)/2
   B. sqrt(k_x k_z)
   C. k_x k_z
   D. k_x/k_z
   Answer: B. The flux derivation yields k' = sqrt(k_x k_z). (Key point: Equivalent permeability)

4. The discharge formula is:
   A. q = sqrt(k_x k_z) h N_f/N_d
   B. q = k_x h N_f/N_d
   C. q = (k_x + k_z) h N_f/N_d
   D. q = k_z h N_d/N_f
   Answer: A. Use k' in the standard flow net formula on the transformed section. (Key point: Final discharge)

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