---
title: "Laplace equation for 2D seepage"
subject: "Soil Mechanics"
chapter: 7
key: "semester_4/soil_mechanics:laplace-equation-two-dimensional-flow"
source: https://www.bctengineer.com/q/semester_4/soil_mechanics/laplace-equation-two-dimensional-flow
site: BCT Engineering
---

# Laplace equation for 2D seepage

Soil Mechanics · Chapter 7: Seepage through Soils

- Asked in: 2075 Baisakh, 4 marks
- Video: [Derivation of Laplace Equation for 2D Flow - Permeability of Soils and Seepage Analysis](https://www.youtube.com/watch?v=eF8bB3McqoY) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/soil_mechanics/laplace-equation-two-dimensional-flow
- Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-7

## Question

Derive the Laplace equation for two-dimensional flow in the soil.

## Short answer

For steady 2D flow in isotropic soil, continuity gives ∂v_x/∂x + ∂v_z/∂z = 0. Using Darcy’s law v_x = k ∂h/∂x, v_z = k ∂h/∂z, substituting yields ∂²h/∂x² + ∂²h/∂z² = 0, the Laplace equation.

## Answer

**Assumptions:** the soil is homogeneous, isotropic and saturated; water and soil grains are incompressible; flow is steady, laminar and obeys Darcy's law; flow takes place in the $x$–$z$ plane.

Consider an elemental soil block $dx \times dz$ with unit thickness. Let the discharge velocity components be $v_x$ and $v_z$.

```text
          v_z + (∂v_z/∂z)dz
              ↑
      +---------------+
 v_x →|    dx × dz    |→ v_x + (∂v_x/∂x)dx
      +---------------+
              ↑
             v_z
```

#### Continuity

Steady flow with incompressible water: inflow = outflow.

$$
v_x\,dz + v_z\,dx = \left(v_x + \frac{\partial v_x}{\partial x}dx\right)dz + \left(v_z + \frac{\partial v_z}{\partial z}dz\right)dx
$$

$$
\frac{\partial v_x}{\partial x} + \frac{\partial v_z}{\partial z} = 0 \quad (1)
$$

#### Darcy's law

$$
v_x = k_x\,i_x = k_x\frac{\partial h}{\partial x}, \qquad v_z = k_z\,i_z = k_z\frac{\partial h}{\partial z} \quad (2)
$$

(the sign depends on the direction chosen for $h$; it does not affect the result).

Substituting (2) in (1):

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

For isotropic soil $k_x = k_z$:

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

This is the **Laplace equation**. Its solutions represent two families of curves, equipotential lines $h = $ constant and flow lines, which form the flow net.

## Check yourself

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

1. What does continuity for steady flow of incompressible water give for the element?
   A. dv_x/dx + dv_z/dz = 0
   B. dv_x/dx - dv_z/dz = 0
   C. v_x + v_z = 0
   D. dv_x/dz + dv_z/dx = 0
   Answer: A. Inflow equals outflow, which reduces to the sum of velocity gradients being zero. (Key point: Continuity)

2. Which step combines with continuity to give the Laplace equation?
   A. Replacing v with the shear stress
   B. Substituting v_x = k_x dh/dx and v_z = k_z dh/dz into the continuity equation
   C. Using the Terzaghi consolidation equation
   D. Assuming zero head at the boundary
   Answer: B. Putting Darcy's law in continuity gives k_x h_xx + k_z h_zz = 0. (Key point: Darcy's law substitution)

3. What assumption reduces k_x d2h/dx2 + k_z d2h/dz2 = 0 to the Laplace equation?
   A. Flow is unsteady
   B. The soil is anisotropic
   C. k_x = k_z (isotropic soil)
   D. Water is compressible
   Answer: C. With k_x = k_z the permeability cancels, giving d2h/dx2 + d2h/dz2 = 0. (Key point: Isotropy)

4. What do the solutions of the Laplace equation represent?
   A. Only the flow lines
   B. Equipotential lines and flow lines forming the flow net
   C. Stress contours
   D. Settlement curves
   Answer: B. The two families of curves (h = constant and flow lines) form the flow net. (Key point: Meaning of solution)

## Related in Soil Mechanics

- [Seepage in anisotropic soil](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aseepage-anisotropic-soil-derivation) — Soil Mechanics, chapter 7
- [Seepage in inclined permeable layer](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Ainclined-permeable-layer-seepage) — Soil Mechanics, chapter 6
- [Sheet pile seepage quantity numerical](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Asheet-pile-seepage-quantity-numerical) — Soil Mechanics, chapter 7
- [Impervious dam seepage discharge](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aimpervious-dam-seepage-discharge-numerical) — Soil Mechanics, chapter 7
- [Constant head permeameter, seepage velocity](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aconstant-head-permeameter-seepage-velocity) — Soil Mechanics, chapter 6
- [Derive Terzaghi 1D consolidation theory](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aterzaghi-1d-consolidation-derivation) — Soil Mechanics, chapter 9, asked 2×

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