Math 2.0

Fluid mechanics Definition

Dupuit–Forchheimer assumption

Builds on Darcy's lawGroundwater flow equationPorous medium equation

Definition

Consider an unconfined aquifer of hydraulic conductivity K>0K > 0 on a horizontal impermeable base z=0z = 0, with water table at height z=h(x,y,t)>0z = h(x, y, t) > 0. The Dupuit–Forchheimer assumption states that:

  1. the hydraulic head is constant on each vertical line, so it equals h(x,y,t)h(x, y, t) at every depth;
  2. hence, by Darcy's law, the flow is horizontal with specific discharge −K∇h-K \nabla h, the same at every depth, where ∇=(∂x,∂y)\nabla = (\partial_x, \partial_y).

The discharge per unit width is then −Kh∇h-K h \nabla h. Conservation of mass gives the Boussinesq equation Sy∂h∂t=∇⋅(Kh∇h)+N.S_y \frac{\partial h}{\partial t} = \nabla \cdot (K h \nabla h) + N. Here Sy>0S_y > 0 is the specific yield and NN is the recharge rate.

Remarks

Example: steady flow in a strip 0≤x≤L0 \le x \le L between water levels h(0)=h1h(0) = h_1 and h(L)=h2h(L) = h_2, with no recharge. The discharge per unit width q=−Kh dh/dxq = -K h \, dh/dx is constant, so h2h^2 is linear in xx, and q=K(h12−h22)2L,h(x)2=h12−(h12−h22)xL.q = \frac{K (h_1^2 - h_2^2)}{2 L}, \qquad h(x)^2 = h_1^2 - (h_1^2 - h_2^2) \frac{x}{L}. This is Dupuit's formula; the discharge it gives is known to be exact for this geometry, although the water table shape is not.

Non-example: near a pumping well the water table drops steeply and the flow has a strong vertical part, so clause 1 fails. The assumption also ignores the seepage face that forms at the well.

The Boussinesq equation with N=0N = 0 and constant KK is ∂th=K2Sy∇2(h2)\partial_t h = \frac{K}{2 S_y} \nabla^2 (h^2), a porous medium equation with exponent 22. It is the nonlinear counterpart of the groundwater flow equation of a confined aquifer.

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  1. Euclidean space object
  2. Function concept
  3. Groundwater flow equation concept
  4. Porous medium equation concept
  5. Stokes flow concept
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  9. Differentiable function of several variables concept
  10. Partial derivative concept
  11. Vector field concept
  12. Gradient concept
  13. Divergence concept
  14. Darcy's law statement
  15. Dupuit–Forchheimer assumption definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Dupuit–Forchheimer_assumption
About this page
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MSC76S05, 86A05
DefinesDupuit–Forchheimer assumption
UsesDarcy's law, Groundwater flow equation, Porous medium equation
Changed3 November 2026

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