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Laplace equation for 2D seepage

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.

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

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 xx–zz plane.

Consider an elemental soil block dx×dzdx \times dz with unit thickness. Let the discharge velocity components be vxv_x and vzv_z.

          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.

vx dz+vz dx=(vx+∂vx∂xdx)dz+(vz+∂vz∂zdz)dxv_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 ∂vx∂x+∂vz∂z=0(1)\frac{\partial v_x}{\partial x} + \frac{\partial v_z}{\partial z} = 0 \quad (1)

Darcy's law

vx=kx ix=kx∂h∂x,vz=kz iz=kz∂h∂z(2)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 hh; it does not affect the result).

Substituting (2) in (1):

kx∂2h∂x2+kz∂2h∂z2=0k_x\frac{\partial^2 h}{\partial x^2} + k_z\frac{\partial^2 h}{\partial z^2} = 0

For isotropic soil kx=kzk_x = k_z:

∂2h∂x2+∂2h∂z2=0\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=h = constant and flow lines, which form the flow net.

✓ Answer checked by Claude Opus 5.5 (high effort)

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