<p>The Laplace equation, (d^2 Ψ)/(dx^2 )+(d^2 Ψ)/(dy^2 )+(d^2 Ψ)/(dz^2 )=0, is just a steady state 3D flow equation. It's a black box conservation of hydraulic potential. Diffusion doesn't come into it. The Diffusion equation (assuming homogeneous isotropic conditions) is (∂^2 Ψ)/(∂x^2)+(∂^2Ψ)/(∂y^2)+(∂^2 Ψ)/(∂z^2)= S_s/K  ∂h/∂t. This discretizes the time element, and is very much dependent upon the diffusive properties of the aquifer. Note that S_s/K is the reciprocal of the 'hydraulic diffusivity', hence the name of the equation.  </p>

CC BY-SA 3.0
2016-08-24T11:11:12.043Z
Can I infer from your answer that diffusion equation is more realistic as compared to Laplace when it comes to groundwater movement in aquifer?
Ather Cheema
CC BY-SA 3.0
2016-08-24T11:22:14.193Z
As I see it, the Diffusion equation is a more specific and useful development of the more general and idealized Laplace equation. In fact there is a whole swag of equations, all based upon Laplace,. In the case of groundwater - solute transport, for example, we eventually end up with a more complex equation with sources, sinks, leakance and non isotropic factors.
Gordon Stanger
CC BY-SA 3.0
2016-08-24T03:42:21.943Z
2016-08-24T03:42:21.943Z
Gordon Stanger
<p>What is the difference in using Laplace equation (say only in 1D) to simulate water flow in a porous media with that of using diffusion equation for same purpose?</p>

<hydrology><hydrogeology><fluid-dynamics>
comparing use of Laplace equation with diffusion equation to simulate flow of fluid/water through porous media?
Ather Cheema
# comparing use of Laplace equation with diffusion equation to simulate flow of fluid/water through porous media?

(Asked by: username_0 on 2016-08-23)

What is the difference in using Laplace equation (say only in 1D) to simulate water flow in a porous media with that of using diffusion equation for same purpose?

**Tags:** hydrology, hydrogeology, fluid-dynamics


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## 1 answers

### Answer by username_1 on 2016-08-24. Score: 2

The Laplace equation, (d^2 Ψ)/(dx^2 )+(d^2 Ψ)/(dy^2 )+(d^2 Ψ)/(dz^2 )=0, is just a steady state 3D flow equation. It's a black box conservation of hydraulic potential. Diffusion doesn't come into it. The Diffusion equation (assuming homogeneous isotropic conditions) is (∂^2 Ψ)/(∂x^2)+(∂^2Ψ)/(∂y^2)+(∂^2 Ψ)/(∂z^2)= S\_s/K ∂h/∂t. This discretizes the time element, and is very much dependent upon the diffusive properties of the aquifer. Note that S\_s/K is the reciprocal of the 'hydraulic diffusivity', hence the name of the equation.

> **Comments:**
> **username_0**: Can I infer from your answer that diffusion equation is more realistic as compared to Laplace when it comes to groundwater movement in aquifer?  
> **username_1**: As I see it, the Diffusion equation is a more specific and useful development of the more general and idealized Laplace equation. In fact there is a whole swag of equations, all based upon Laplace,. In the case of groundwater - solute transport, for example, we eventually end up with a more complex equation with sources, sinks, leakance and non isotropic factors.  

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