Darcy's law
|
|
Darcy's Law is a phenomologically derived constituative equation that describes the flow of a fluid through a porous medium (typically water through an aquifer). The law was formulated by Henry Darcy based on the results of 1855 and 1856 experiments on the flow of water through beds of sand. It, along with the conservation of mass, comprises the groundwater flow equation, which is one the basic building relationships of hydrogeology.
Darcy's Law (an expression of conservation of momentum) is a relationship determined experimentally by Henry Darcy, which has since been proved theoretically from simplifications made to the Navier-Stokes equations. It is analogous to Fourier's law in the field of heat conduction, Ohm's law in the field of electrical networks, or Fick's law in diffusion theory. This simple relationship relates the steady-state discharge rate through a porous medium to the hydraulic gradient (change in hydraulic head over a distance) and the hydraulic conductivity (k).
- <math>Q=-kA \frac{h_a - h_b}{L}<math>
It shows that the total discharge, Q (units of volume per time, e.g., m3/s) is proportional to the hydraulic conductivity, k, the cross-sectional area to flow, A, and the hydraulic gradient (the hydraulic head drop between two points a and b, divided by the distance between them, L). The negative sign is needed because water flows from high head to low head. Dividing both sides of the equation by the area and using more general notation leads to
- <math>q=-k \nabla h<math>
where q is the flux (discharge per unit area, with units of length per time, m/s) and <math>\nabla h<math> indicates the mathematical gradient of the hydraulic head. This value of flux is often referred to as the Darcy flux, it is not the velocity which the water traveling through the pores is experiencing. The pore velocity (v) is related to the Darcy flux (q) by the porosity (n). The flux is divided by porosity to account for the fact that only a fraction of the total aquifer volume is available for flow, this pore velocity would be the velocity a conservative tracer would experience if carried by water through the aquifer.
- <math>v=\frac{q}{n}<math>
Darcy's law is a simple mathematical statement which neatly summarizes several familiar properties that groundwater flowing in aquifers exhibits, including:
- if there is no hydraulic gradient (difference in hydraulic head over a distance), no flow occurs (this is hydrostatic conditions),
- if there is a hydraulic gradient, flow will occur from high head towards low head (opposite the direction of increasing gradient - hence the negative sign in Darcy's law),
- the greater the hydraulic gradient (through the same aquifer material), the greater the discharge, and
- the discharge of water will be often be different — through different aquifer materials (or even through the same material, in a different direction) — even if the same hydraulic gradient exists in both cases.
A graphical illustration of the use of the steady-state groundwater flow equation (based on Darcy's law and the conservation of mass) is in the construction of flownets, to quantify the amount of groundwater flowing under a dam.
Darcy's law is only valid for slow, viscous flow; fortunately, most groundwater flow cases fall in this category. Typically any flow with a Reynold's number less than one is clearly laminar, and it would be valid to apply Darcy's law. Experimental tests have shown that for flow regimes with values of Reynold's number up to 10 may still be Darcian. Reynold's number (a dimensionless parameter) for porous media flow is typically expressed as
- <math>Re = \frac{\rho v d_{30}}{\mu}<math>.
Where ρ is the density of water (units of mass per volume), v is the specific discharge (not the pore velocity — with units of length per time), d30 is a representative grain diameter for the porous media (often taken as the 30% passing size from a grain size analysis using sieves - with units of length), and μ is the viscosity of the fluid.
Additional forms of Darcy's Law
For very short time scales, a time derivative of flux may be added to Darcy's law, which results in valid solutions at very small times (in heat transfer, this is called the modified form of Fourier's law),
- <math>\tau \frac{\partial q}{\partial t}+q=-k \nabla h<math>,
where τ is a very small time constant which causes this equation to reduce to the normal form of Darcy's law at "normal" times (> nanoseconds). The main reason for doing this is that the regular groundwater flow equation (diffusion equation) leads to singularities at constant head boundaries at very small times. This form is more mathematically rigorous, but leads to a hyperbolic groundwater flow equation, which is more difficult to solve and is only useful at very small times, typically out of the realm of practical use.
Another extension to the traditional form of Darcy's Law is the Brinkman term, which is used to account for transitional flow between boundaries (introduced by Brinkman in 1947),
- <math>\beta \nabla^{2}q +q =-k \nabla h<math>,
where β is an effective viscosity term. This correction term accounts for flow through medium where the grains of the media are porous themselves, but is difficult to use, and is typically neglected.