Tag Archives: hydraulic conductivity

Henry Darcy’s Law; a conceptual leap

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Artesian flow

Artesian flow

Henry Darcy’s pivotal experiments with sand-filled tubes (in 1856) established an empirical relationship between hydraulic gradient (that is basically an expression of the hydraulic potential energy available for flow) and discharge. A modern rewrite of the basic equation that he deduced from experiments, the eponymous Darcy’s Law, is:

Q = -KA(Δh/D) where                                                (1)

  • Q is discharge (that has dimensions L3/T),
  • K is a proportionality constant, subsequently called the hydraulic conductivity (L/T)
  • A the cross-section area of a flow tube (L2) (Q is also proportional to A), and
  • Δh/D the head difference between two locations along a flow path, at distance D. Note that hydraulic head h is the sum of the pressure head and elevation head.

The hydraulic conductivity (a term borrowed from electrical theory), has dimensional units of distance and time (commonly expressed as cm/s, feet/s). Thus, in mathematical terms, K is expressed as a velocity, also known as the Darcy velocity.

Darcy’s empirically derived law is pivotal to modern, quantitative hydrogeological modelling. The description of his experiments and derivation of the law were published in Note D, an appendix to a lengthy report on the Dijon Fountains (680 pages!): Les Fontaines Purlieus’ de la Ville de Dijon (The Public Fountains of the City of Dijon). While the appendix might seem like an afterthought, it was in fact the culmination of two decades of observation, testing, experimentation, and the creative ability to extend his ideas below the surface, literally and figuratively.

 

How did Darcy arrive at this point of discovery?

Henry Darcy (1803-1858) was a French engineer who rose to prominence in the 1830s, at least in the public’s eyes, as the designer and executor of a modern water supply system for the city of Dijon, completed about 1840. Several other European cities modelled their own water supply networks on his design. The primary water supply for the Dijon network was a well dug into a groundwater (artesian) spring; the pipe supply network extended 28km – all gravity fed.

Darcy was familiar with the hydraulic theory and practice of the time; the theory of hydraulics was well established but the general understanding of aquifer dynamics was limited. Some of the important ingredients that contributed to his thinking and intuition were:

  • Bernoulli’s (1738) mathematical expression for energy conservation during fluid flow; in other words, flow requires an energy gradient. Bernoulli’s equation can be written as:

            V2/g + z + P/ρ.g = a constant known as hydraulic potential         (2)

where V2/g is kinetic energy, z the elevation head, and P/ρ.g the pressure head. By ignoring the kinetic energy component, that is insignificant for most groundwater problems, the equation reduces to a statement where hydraulic potential is the sum of z + P/ρ.g. The value of this statement is that it allowed Darcy (and us) to tease apart the components of hydraulic head in real wells and experiments.

  • He had developed an expertise with the practical problem of pressure losses between the entry and exit points of pipes used to transport water; he surmised and calculated the effects of surface roughness on energy, and therefore pressure losses.
  • Another crucial discovery, based on his work with pipes, was that at very low flow rates in small-diameter pipes, the head loss (or head gradient) was proportional to flow rate, that he would later discover could be applied to aquifer flow.
  • He was familiar with the flow of water through natural and constructed sand filters that were used to clean river water and was aware that frictional energy losses also applied to this kind of flow.
  • He had measured well drawdown for various pumping rates and observed well recovery.
  • He had general knowledge of aquifer geology and aquifer recharge by precipitation.

Darcy’s conceptual leap was to equate the physical nature of these observations (in pipes, filters, and the behaviour of boreholes) with flow through porous aquifer media. The experiments he designed and performed bridged the gap between concept and empirical evidence.

 

Darcy’s experiments

Darcy began his experiments in 1855. They were based in part on his observations of flow through sand filters, but what he needed was a way to quantify head loss (hydraulic gradient).

A diagram showing the original experimental apparatus is reproduced here. Modern groundwater texts commonly redraw the apparatus configuration to be more reflective of aquifer flow – I have added a duplicate diagram.

Darcy’s apparatus for determining the relationship between aquifer discharge and head loss, 1856. Figure 3 Plate 24, with some additional annotation

Darcy’s apparatus for determining the relationship between aquifer discharge and head loss, 1856. Figure 3 Plate 24, with some additional annotation

 

An alternative representation of Darcy’s experiment, shown as tube (pipe) flow in a sand aquifer. Instead of mercury manometers, the piezometers measure the water levels directly, relative to a datum. Each water level represents the total hydraulic head at the point of measurement in the aquifer. The distance D between piezometers allows the calculation of hydraulic gradient.

An alternative representation of Darcy’s experiment, shown as tube (pipe) flow in a sand aquifer. Instead of mercury manometers, the piezometers measure the water levels directly, relative to a datum. Each water level represents the total hydraulic head at the point of measurement in the aquifer. The distance D between piezometers allows the calculation of hydraulic gradient.

Darcy’s aquifer was represented by a vertical steel tube, sealed at both ends, with an air-bleed valve at the top. Two mercury manometers were used to measure pressures so that head values top and bottom of the tube could be calculated – the manometers measured atmospheric pressure ± the height of water. Water was added at the top of the tube and discharged from the base.

Darcy and his assistant performed several experimental runs. The tube was filled with water for each run. Sand was added from the top and allowed to settle on the bottom. The thickness of sand was varied systematically (this is D in the above expression), and for each sand thickness the flow rates were also varied. Flow of water into the tube was kept constant for each run; the discharge measured as volume per unit time (the vagaries of water supply at the time meant that keeping flow constant was a bit of a problem). Once steady state conditions had been established in each run, the manometer (pressure) levels were read.

The following graph shows the observed linear relationship between discharge Q and head loss (or head difference) based on Darcy’s data.

Darcy’s data for Set 1 experiments (there were two sets), replotted as Q versus head gradient for each thickness of sand. Modified from Brown, 2002, Fig. 6.

Darcy’s data for Set 1 experiments (there were two sets), replotted as Q versus head gradient for each thickness of sand. Modified from Brown, 2002, Fig. 6.

Darcy’s key observations were:

  1. Discharge is proportional to the head loss, or head difference along the flow path (Q ∝ Δh), and
  2. Discharge is inversely proportional to the distance that water travels between the two points of head measurement (Q ∝ 1/D) (in his experiments this is the thickness of sand).

He expressed the proportionalities as:            Q = KA (h1 + z1) – (h2 + z2)/D          (3)

where Q, K, A, and D as noted above, h is the pressure head, z is the elevation head at the two points of measurement (1 and 2) that are a distance D apart. This is Darcy’s Law. The form of the equation is basically the same as equation (1).

He noted that K varied depending on the sand used (particularly its packing that probably varied slightly for each experiment). We now know that K (hydraulic conductivity) not only varies with differences in the porous medium, but also with the nature of the liquid. Thus, if the sand is the same in two separate experiments, but water is used in one and oil in the other, then the values of K will be different. Later considerations by other workers would establish that K is also a function of dynamic viscosity.

 

And so…

Darcy’s Law tells us that, under steady state conditions, there must be a hydraulic (head) gradient for flow to occur in an aquifer; in essence it restates the fundamental physical principle that for mechanical work to be done (i.e., to move water from one location to another) there must be an energy potential.

The Law also provides a quantitative solution to determining the parameters for groundwater flow, provided we know something about the porous medium and the fluid itself (that is expressed as hydraulic conductivity). Of course, we can also determine K if we know Q and Δh, and we can measure K experimentally using a permeameter – an instrument that looks similar to Darcy’s original apparatus.

Darcy’s Law describes a flux, and as such is cast in the same mathematical form as Ohm’s Law (current is proportional to voltage), and Fick’s Law (molecular diffusion is proportional to a concentration gradient.

Darcy’s Law is a crucial component of fluid flow modelling, particularly for solving important questions about groundwater, for example the sustainability of aquifers to pumping.

 

Credits: The historical background to Darcy’s life, his scientific and social contributions were gleaned from Freeze (1994; Brown (2002- open access); and Simmons (2008 – PDF).

 

Other posts in the Groundwater Series

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The Architecture of Connected Holes; A Different Way to Look at the Liquid Earth

“My water well taps into an underground river” and other myths

Coastal aquifers; groundwater at sea

Groundwater contamination; messing around with aquifers

Landslide! How groundwater affects the stability of slopes

GRACE meets LANDSAT; Eyes in the sky monitoring long-terms changes in water resources

A misspent youth serves to illustrate groundwater flow

Contrails, analogies, and visualizing groundwater flow

Springs and seeps

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Mineralogy of sandstones: Porosity and permeability

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well sorted sandstone with about 20% porosity. Each grain has a dusting of diagenetic clays

Porosity and permeability – the flow of water and other geofluids

This is part of the How To…series  on the Mineralogy of sandstones

Nearly all geological processes require the presence of water in one form or another. Most sedimentation occurs in water (aeolian deposits are the obvious exception). Sediment burial and compaction involve the expulsion of water. Diagenesis would not take place in the absence of water; hydrocarbons would not migrate to traps and minerals would not be concentrated in ore bodies. Aqueous fluids under pressure reduce cohesion and friction promoting rock deformation.  Metamorphism would be painfully slow, even by geological standards if it were not for the transfer of mass in hot aqueous fluids.

All these processes require not only the presence of water, but its continual movement or flow. Below Earth’s surface, the residence and flow of aqueous fluids requires two fundamental rock-sediment properties:
– voids, commonly in the form of intergranular pores and fractures, and
– connectivity among the voids.
The first of these is referred to as porosity; the second as permeability.

There are two main types of porosity: intergranular porosity that characterizes sands, gravels and mud, and fracture porosity in hard rock. Fracture porosity forms during brittle failure of hard rock or cooling of lava flows. Fracture networks that are connected can provide pathways for fluid flow even when the host rock is impervious (e.g. granite, basalt, indurated sandstone). Highly productive aquifers are not uncommon in fractured bedrock.

Fracture porosity in a columnar jointed lava flow, British Columbia

Intergranular porosity is the void space between detrital grain contacts and is expressed as a percentage of the total sediment-rock volume. It is a dimensionless number (i.e. it has no units of measure). All sediments begin life with some porosity.  Well sorted beach, river and dune sands have initial porosities ranging from 30% – 40%, muds as high as 70%. These values represent the total void space, namely the large pores plus lots of microporosity in tiny nooks and crannies between grains and crystals. Hydrogeologists have found it useful to define effective porosity as that which permits easy movement of fluid. This excludes microporosity where surface tension forces inhibit flow. Effective porosity is always less than total porosity. Follow this link to a simple experiment designed to measure porosity.

As sediment is buried, the grains settle (i.e. they become more closely packed) as they begin to compact.  The reduction in porosity by mechanical compaction continues during sediment burial, in concert with the precipitation of cements (chemical diagenesis).  This is particularly evident during the compaction of mud. The high initial porosity of mud is due to micro-pores between clay particles that have dimensions measured in microns. Compaction compresses the clays and drives off the interstitial water. Compaction (porosity-depth) curves for mud, like the example shown below, typically show a loss of porosity that at shallow depths is almost exponential, becoming approximately linear at depths where shale forms; total porosities in shale are extremely low.

 

Shale porosity - depth (compaction) curve

The conduits for fluid flow (water, oil, gas) from one pore space to another are the narrow connections adjacent to grain contacts. These connections are commonly referred to as pore throats. Pore throats are susceptible to blockage during sediment compaction (lithic sandstones are prone to this) and by cementation, particularly clay cements.

 

Schematic of sandstone burial sequence with compaction and loss of porosity

Diagram of pore-filling cements and occlusion of porosity

Porosity can also be enhanced during burial diagenesis. The primary mechanism for formation of secondary porosity is the dissolution, or partial dissolution of framework grains like feldspar and carbonate bioclasts. Many of these secondary pores are larger than the associated intergranular pore spaces; this is an important diagnostic clue to their identification. Likewise, carbonate and clay cements may be prone to dissolution, resulting in enhanced post-depositional porosity.

 

Secondary porosity caused by the dissolution of feldspar

Burial depths and temperatures where formation of secondary porosity is encountered commonly coincide with chemical reactions involving the break-down of organic matter. By-products of these reactions include carbon dioxide (and carbonic acid) and organic acids like acetic acid. There is a fundamental shift in pH and chemical equilibria, particularly for carbonates, and this promotes dissolution.

Secondary porosity can also form during subaerial exposure of rock and by bioturbation. However, the secondary porosity seen in most ancient sandstones is the product of  burial diagenesis.

Permeability measures the ease with which a fluid flows through sediment or rock. The flow of fluid from one part of a rock to another, or from an aquifer to a bore hole, depends on the connections among pores and fractures. It is possible for a rock or sediment to have high porosity but low permeability if the intergranular or intercrystal connectivity is low – mud and shale are prime examples. In coarse-grained sediments that are devoid of clay, there is a good correlation between porosity and permeability.  This relationship does not apply where there are significant amounts of clay.

Permeability can be expressed in two ways. Henry Darcy’s pivotal experiments with sand-filled tubes (in 1856) established an empirical relationship between hydraulic gradient (that basically is an expression of hydraulic potential energy) and discharge. The proportionality constant in this relationship is called the hydraulic conductivity (K) (a label borrowed from electrical theory), that has units of distance and time (cm/s, feet/s). In mathematical terms, hydraulic conductivity is expressed as a velocity, also known as the Darcy velocity. Hydraulic conductivity is the standard expression of permeability in groundwater studies. Its value depends not only on the connectivity of pores but also on the dynamic viscosity and density of the fluid (viscosity measures the resistance to flow – crude oil is more viscous than water). Thus, for any porous medium the value of K will be different for water and oil, a factor that is important in groundwater remediation.

The hydrocarbon industry deals with fluids of highly variable viscosity (water, oil, gas) and has opted for a standard expression of intrinsic permeability (k) that depends only on the porous medium. The unit is the Darcy that mathematically reduces to units of area (ft2, m2). It is basically a measure of pore size (the oil industry commonly uses the term millidarcy). Frequently used conversions to Darcys are:

1 m2 = 1.013 x 1012 Darcy

1 Darcy = 9.87 x 10-13 m2

Hydraulic conductivity (K) and intrinsic permeability (k) are related by fluid density and dynamic viscosity such that:

k (m2) = K (m/s) x (1.023 x 10-7 m.s) (the time components cancel)

Typical permeability values for unconsolidated sediment and some rock equivalents are shown in the table below.

Table listing typical values of permeability, expressed as hydraulic condictivity and in Darcys

As you can see, the permeability of shale is extremely low. This is the reason why shale beds make good seals to hydrocarbon reservoirs, and aquitards to confined aquifers. Fluid flow in shales and well-cemented sandstones or limestones can be enhanced by hydraulic fracturing. This process (fracing) is front and centre of shale oil production (notwithstanding all the pros and cons of this industrial process). But that is a story for another time.

 

Here are three excellent texts that detail the theoretical aspects of the above:

P.A. Allen and J.R. Allen, Basin Analysis: Principles and Applications. Blackwell 2005

C.W. Fetter.  Applied Hydrogeology, 2001. PrenticeHall

P.A.Domenico and F.W.Schwartz Physical and Chemical Hydrogeology,1998 John Wiley & Sons

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The Architecture of Connected Holes; A Different Way to Look at the Liquid Earth

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2nd in the Series on Groundwater

lake taupo

We commonly differentiate the solid earth in terms of its architecture, whether it is the foundations of great mountain ranges, or the solidified magmas that underpinned ancient volcanoes.  All rocks, whether layered sedimentary rocks or massive intrusive granites, have unique characteristics that define their physical, chemical and biological make up – their architecture.

WE can also think of groundwater in terms of its own architecture.  The productivity of an aquifer depends first and foremost on its porosity and permeability.  We can use these two fundamental properties to define the architecture of earth materials.

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