Just as it does on the surface of the earth, water also flows underground, but only through pores in the soil and underlying geologic structure. Although these pores are very small and account for only a small portion of the underground volume, it is possible for water to move large distances underground, albeit very slowly. Other fluids, such as natural gas and oil, also may be moved out of underground locations by pumping from wells. The movement of contaminated fluid from a solid waste landfill into a potable water aquifer located beneath it is an example of unwanted underground flow. Thus the motion of fluids through porous rock, induced by pressure and gravity forces, can be of great practical importance.
It is possible to filter harmful particles from a fluid stream by passing it through a porous solid whose pores are too small to permit the passage of of the particles. In other cases, the large surface area of the microscopic pores may provide sites for chemical catalysis or adsorbtion of components of the fluid. Flow through filters and catalyst beds is an engineering application of flow through porous media.
Although the fluid passages through a porous solid are irregular in size and length, we might expect the average flow speed through them to be comparable to the flow through a circular tube of diameter D, as given by equation 6.42:
To generalize this relationship to a porous solid, where the pore size and length is variable, we introduce a proportionality constant k, called the permeability, that includes an averaging of the pore size, shape and length, to determine the volume flow rate per unit area of solid, , as:
Equation 6.65 is called Darcy's Law. The permeability k, which has the dimensions of length squared, is an approximate measure of the average of the square of the pore diameters. The unit of permeability, called the darcy, is .
The velocity of the fluid in a pore will be much higher than because the latter is based upon the volume of fluid passing through a unit area, only a small portion of which is composed of pores. Thus a fluid particle will move a distance L in a time that is much smaller than , but it is the quantity that is observable in porous flows and that is of most practical interest.
For incompressible flows, the velocity field must satisfy the condition of mass conservation, equation 3.17 :
Equations 6.65 and 6.66 together enable us to find the field of flow in a porous medium. The methods of solution for a Darcy flow are similar to those used in irrotational flows (see chapter 11 ).
The fraction of the volume of a solid that is occupied by the fluid is called the porosity. The porosity of a porous solid determines its capacity to store fluids, but does not explicitly affect the flow field .
Darcy's Law, equation 6.65, has the same form as Ohm's law for the electric current density in an electrical conductor having an electrical conductivity :
where is the electric potential, or voltage. The free electrons move through the conductor just like the viscous fluid in a porous solid, encountering resistance from the atoms of the electrical conductor. In electrical superconductors, where , a finite current can flow with no voltage drop (). In superfluid helium, below , the superfluid phase has zero viscosity and can flow through the normal phase at a finite velocity without a pressure drop, as in Darcy's law with . Both superconductivity and superfluidity are examples of quantum behavior exhibited on a macroscopic scale.