Porosity, Permeability and Darcy’s Law

Porosity

Groundwater occurs in the void spaces of earth materials: soils, unconsolidated sediments, and rock.

Even rock that was formed as a solid mass will fracture as it is brought towards the surface. These fractures provide space for the storage and movement of groundwater.

Porosity is defined as the percentage of a volume of rock (the all purpose term hydrologists use for ‘earth materials’) that is empty space.

Effective porosity: void spaces that are too small to admit water molecules are of little interest to hydrologists. The amount of voidspace available for fluid flow is the effective porosity.

Even large voidspaces that are interconnected by small pore throats are unavailable for fluid flow.

Fortunately, studies have shown that even fine clays have pore throats that are larger than water molecules, so that, at least in sediments, effective porosity is equal to total porosity.

Total porosity can be computed from measurements of density:

p bulk is the density of the bulk aquifer material

p pd is the density of the particles that make up the aquifer material

For most rock and soil, the particle density is about 2.65 g/cm3, roughly the density of quartz and clay minerals.

Factors affecting porosity

Grain size: In and of itself, grain size has no effect on porosity. Well rounded sediments that are packed into the same arrangement generally have porosities from 26% to 48% depending on the packing.

A room full of bowling balls and a room full of BBs would have the same porosities if the spheres were packed the same way.

Sorting: Well sorted sediments generally have higher porosities than poorly sorted sediments for the simple reason that if a sediment is a range of particle sizes then the smaller particles may fill in the voids between the larger particles.

Sorting is measured as a ratio of the larger to smaller particle sizes in the sediment. This measure is called a uniformity coefficient.

d60 the grain size below which 60% of the sediment is finer

d10 the grain size below which 10% of the sediment is finer
 
 

Grain shape: Irregularly shaped particles tend not to pack as neatly as rounded particles, resulting in higher proportions of voidspace.

Clay and organic content: Organic particles tend to be irregularly shaped and can increase voidspace. Clay particles tend to electrostatically repell one-another along the surface of the particles. This results in a relatively large proportion of voidspace.

Primary vs secondary porosity

Primary porosity is the porosity that exists between individual grains in the rock.

Secondary porosity is the porosity that results from fracturing, dissolution, and separation of the rock after its formation.

Specific yield

Knowing the porosity of a rock will not tell you how much water can be removed from that rock (sediment, soil, etc..).

Water is sticky. Water molecules are polar - meaning that they carry a net positivee and negative charge across the molecule - so they like to attract each other, and they like to attract to other charged materials, including most surfaces.

We see the sticky nature of water as surface tension and as the tendency for water to coat surfaces with a thin film.

If you saturate a volume of sediment and then allow the water to drain out of the sediment under gravity, all of the sediments will be coated with a thin film of pendular water (water left ‘hanging’ on the grains).

Specific yield is thus a measure of how much water can drain away from the rock under gravity vs how much water the rock actually holds (the total voidspace).

Specific retention, contrariwise, is the amount of water that the rock retains as a surface film on the sediments.

The sum of specific yield and specific retention is the total porosity.

What effects specific yield?

Grain surface area - as grainsize decreases, total surface area increases, leading to smaller specific yields at smaller grain sizes.

The highest specific yields come from coarse sands and fine gravels. At larger gravel sizes specific yield decreases (why???).

The lowest specific yields come from clays, which have small particles and voidspaces with large surface areas. Clays can have porosities of 50% combined with specific yields of only 3%.

Permeability

Permeability describes of how easily water is able to move through rock. Permeability is related to the connectedness of the void spaces and to the grain size of the rock.

Obviously, a rock could be extremely porous, but if each pore was isolated from the others, the rock would be impermeable and thus make a lousy aquifer. Often volcanic rocks will have many vessicles, but the vessicles will be isolated, rendering the rock impermeable.

Grain size affects permeability in a mannor similar to the way that it effects specific yield.

The thin film of water that clings to the surface of particles is tenaceous. If voidspaces are large, then additional water can easily move past the water coated particle surfaces.

However, if voidspaces are small, then the surface film of water can actually choke the movement of additional water through the small spaces.

This explains why clays are so impermeable, even though their porosities can be as high as 50%. The voidspaces in clay are small and clogged with pendular water.

So from the above discussion, we can see why sands make the best aquifers and clays make the worst (clays are not referred to as aquifers at all, rather they are called aquicludes).

Hydraulic Conductivity

Hydraulic conductivity is a measure of how easily a particular fluid will pass through a particular earth material.

It is calculated using a formula widely known as Darcy’s Law (the E=MC2 of hydrology).

Darcy was a French engineer who made some experiments in the mid-nineteeth century on how fluids move through sediments. What Darcy did was to measure the rate of flow of water draining by gravity through pipes filled with different types of materials. The relationships that he discovered now bear his name.

What did Darcy discover?

First, flow is directly proportional to the area of the pipe (no surprise there - the bigger the pipe, the more water will flow).

Flow is also proportional to the difference in height of the water from when it begins flowing through the pipe to when it comes out of the pipe. We call the height of water in an aquifer (measured from some defined datum) the hydraulic head.

Obviously, it is the difference in hydraulic head from the start of the pipe to the end of the pipe that is generating the pressure pushing the water through the pipe. We call this difference in height = pressure the hydraulic gradient.

Finally, flow is inversely proportional to the length of the pipe. The more sediment that the water has to flow through, the more it will be impeded.

So, Darcy made his measurements and wrote the following formula:

ha?hb is more generally expressed as dh, and L can be thought of as dL as well.

dh is negative because the change in elevation is from high to low.

We can rearrange this equation to solve for K:

The coefficient K is called the Hydraulic Conductivity and is particular to specific combinations of fluid and earth material. The greater the value of K, the higher will be the rate of flow of a fluid through a material.

Hydraulic conductivity is similar to permeability. The difference is that permeability is a property that is specific to the sediment, while hydraulic conductivity describes the behavior of a specific fluid and a specific sediment.

The units of hydraulic conductivity can be obtained from the above equation - L/T. However, commonly hydraulic conductivity is reported as cubic units per time per area. One can see that this conversion is accomplished easily by multiplying L/T X L2/L2.

In lab today, we will experimentally determine the hydraulic conductivities of three different sediments using a simple version of Darcy’s original experiment.

The apparatus used to measure hydraulic conductivity is called a permeameter.