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GATE 2019 CE (CE1) – Question 12

Geotechnical Engineering · Permeability and seepage: flow nets, uplift pressure, piping, capillarity · 1 mark · Multiple choice

For two-dimensional flow through isotropic soil in the x and z directions, let h denote hydraulic head. Which is Laplace’s continuity equation?

  1. $\frac{\partial h}{\partial x}+\frac{\partial h}{\partial z}=0$
  2. $\frac{\partial h}{\partial x}+\frac{\partial h}{\partial x}\frac{\partial h}{\partial z}+\frac{\partial h}{\partial z}=0$
  3. $\frac{\partial^2h}{\partial x^2}+\frac{\partial^2h}{\partial z^2}=0$
  4. $\frac{\partial^2h}{\partial x^2}+\frac{\partial^2h}{\partial x\partial z}+\frac{\partial^2h}{\partial z^2}=0$

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Correct answer: (C) $\frac{\partial^2h}{\partial x^2}+\frac{\partial^2h}{\partial z^2}=0$

Explanation

Darcy velocities are $v_x=-k\partial h/\partial x$ and $v_z=-k\partial h/\partial z$. Steady incompressible continuity gives $\partial v_x/\partial x+\partial v_z/\partial z=0$. For constant isotropic k, this becomes $$\frac{\partial^2h}{\partial x^2}+\frac{\partial^2h}{\partial z^2}=0.$$ Answer **C**.