GROUNDWATER BASICS AND DARCY’s LAW


Overview. Groundwater is a valuable resource of subsurface water that can flow freely and consequently can be pumped, commonly in prodigious quantities. Normally out of view, this water can be seen in wells and sometimes emerges as springs. Groundwater is the largest volume of accessible fresh water, totaling about 22% of Earth’s fresh water inventory, with the remainder mostly stored in remote icecaps. As discussed earlier, the quantity of fresh water present in familiar reservoirs such as rivers, lakes and the air, is quite small by comparison.


Because groundwater resides beneath our feet, out of the realm of ordinary observation, most humans have little understanding of this valuable resource, even though billions rely on it for their drinking water supply. This is unfortunate, because in many cases groundwater is effectively a non-renewable resource that must be managed wisely. Instead, in many parts of the world, groundwater pumping dwarfs the natural recharge rate, so the resource is steadily decreasing in quantity. In many such areas groundwater is also being degraded in quality, greatly diminishing its value. The largest use of groundwater is irrigation, so the diminishing quantity and quality of groundwater means that the global food supply is not sustainable.


This chapter will discuss the inventories and nature of subsurface waters, and then will cover the physics of groundwater behavior in some detail. After those topics are understood, attention will be directed to some real examples, as well as to the surprising variety of problems that are associated with the overexploitation of available groundwater supplies.


Water Profile


The best way to understand groundwater is to observe it directly, and a good approach would be to dig a hole. However, in most areas the hole would need to be rather deep, rendering the groundwater inaccessible by casual effort. Nevertheless, as discussed in chapter 2, many ancient people dug wells, which are simply holes that are deep enough to intersect groundwater, allowing convenient access by humans.


So, what vertical zones are encountered when digging a well? First, in most areas, one encounters a few inches or feet of soil, typically comprising a fine-grained matrix of mineral matter that may include dark organic material (Fig. 1). Soils have complex character that varies from place to place, and a whole field of science is dedicated to their description, but what is key here is that soils typically contain moisture that is essential to plants. This soil water resides along with air in the pores of the soil, typically in a state of suction (sub-atmospheric pressure) that can be measured with a tensiometer. Under normal conditions soil water does not flow, but it gradually dries out, until it is replenished up to the “field capacity” by intermittent rainfall. Abundant rain that exceeds this capacity of the soil to hold water percolates downward to replenish groundwater, or moves laterally away as subsurface runoff.



Figure 1. Sketch of the Water Profile, not to scale, showing the vertical succession of zones relevant to hydrology (REC)


Below the soil are additional layers, collectively called the “Intermediate zone”, that contain water plus air in pores. This zone may represent subsoil, colluvium, alluvium, or other unconsolidated clastic materials, or alternatively could be fractured bedrock. At sufficient depth a “capillary zone” is encountered, where water is drawn upward by surface forces, so it is also under a condition of tension. The vertical extent of this capillary zone can be as great as several feet, depending on the pore size of the matrix. All water within and above this zone is collectively included in the unsaturated zone, also called the vadose zone.


The base of the unsaturated zone is the water table, which is the level to which water will rise in a normal well under “unconfined” conditions (defined below). That is, if one looks down a well, the water surface that one sees far below defines the water table at that location (Fig. 1). In regions of sufficient rainfall, stream and lake surfaces occur where the water table intersects Earth’s surface. The water table is seldom horizontal, but tends to rise under hills and be low in valleys, forming what is oft called a “subdued replica of the topography” (Fig. 2).


Below the water table the pores no longer contain air. The pore pressure at the water table is one atmosphere, but below this level is the saturated zone, sometimes called the “phreatic zone”, where pore pressure increases downward at a rapid rate.


Groundwater resides within the saturated zone. Layers that contain useful quantities of groundwater that can move freely are called aquifers. Alternatively, if water quantity or mobility is low, the layer is called an aquitard. Of course, aquifers and aquitards can be interbedded, and a continuum exists between these types, ranging down to the theoretical limit of completely impermeable materials (aquicludes). The latter condition is effectively realized at great depths, for example in ductile zones. Fluid behavior is abnormal under such conditions, as discussed in a subsequent chapter.



Figure 2. Cross-sectional of a hill. The water table, whose surface is traditionally denoted with a small inverted triangle, varies with position, and separates the overlying vadose zone from groundwater in the underlying saturated zone. Groundwater flow lines are shown schematically (long arrows). Note the position of the water table at the stream bank, and also note the level of water in the two observation wells whose screens (openings in an otherwise solid pipe) are depicted by the short horizontal lines. Why do these water levels closely coincide with the height of the water table in only one well? After USGS.


Basic Definitions


Because the behavior, environment, and driving forces of groundwater are unfamiliar, some unfamiliar terms are used to describe them. For easy reference the most important terms and their typical units are defined here. Some of these terms refer to the water, some to the rock matrix, and some to both, and all have very specific meanings.


Water Properties


Density (ρ, kg/m3). Density is the mass per unit volume of a material. The density of cool, natural groundwater is very close to 1 g/cm3, or one metric ton per cubic meter (1000 kg/m3). Because the compressibility of water is low and temperature varies little in shallow rocks, the density of water in shallow aquifers is nearly constant. However, density variations are important in hydrothermal systems or where salinity is high.


Specific weight (kg/m2-s2). The specific weight, a term used commonly in hydrology but infrequently in this text, is density times the gravitational constant "g".


Kinematic Viscosity (ν, m2/s). The kinematic viscosity of a fluid is equal to the dynamic viscosity divided by the density, and describes the resistance of a fluid to flow. For example, the kinematic viscosity of water (10-6 m2/s) is much lower than that of syrup or honey (10-3 to 10-2 m2/s). The viscosity of shallow groundwater is generally considered to be constant.


Rock (Matrix) Properties


Density (ρ, kg/m3). The rock density typically refers to the density of the dry mineral matrix, which is typically about 2500 kg/m3, but density may also refer to the saturated porous medium.


Porosity (φ, dimensionless). Porosity is the ratio of void (pore) volume to the total volume of a rock. This important parameter controls the amount of water a saturated medium can hold. The porosity of rocks varies greatly, from near zero to 30% or more.


Permeability (k, m2). Permeability describes the capacity of a medium to transmit fluid. Permeability is the most important property that controls subsurface fluid flow, and it varies over an enormous range in common rocks, from as little as 10-19 m2 in unfractured igneous rocks to 10-8 m2 in coarse gravel.


Hydraulic Conductivity (K, m/s). The hydraulic conductivity is the proportionality constant in Darcy’s Law. K has units of velocity, and it depends on the properties of both fluid and medium. Specifically, K is a lumped parameter that is equal to kg/ν, that is, the permeability (k) of the matrix times the gravitational constant (g) divided by the kinematic viscosity (ν) of the fluid. Typical values of K in good aquifers are about 10-6 to 10-3 m/s, and very low (less than 10-10) m/s in aquitards.


Hydraulic Diffusivity (D, m2/s). The hydraulic diffusivity is the proportionality constant in the hydraulic diffusion equation, and is the quotient of the transmissivity and storativity, defined below.


Governing Variables


Pressure (P, nt/m2). Pressure, the familiar quantity with units of force per unit area, is erroneously believed by many to drive groundwater flow. Instead, the relevant quantity is head, which includes the effects of pressure.


Head (h, m). The hydraulic head is the level to which water will rise in an unconfined (water table) aquifer. At deeper levels, or in a confined aquifer, the head is the level to which water will rise in a special pipe, called a piezometer, with one open end sunk to the level of interest in the aquifer, and the other end open to the atmosphere. Fluid flow is driven by lateral or vertical differences in head.


Rock Unit Character


Aquifer: A saturated formation whose porosity and permeability are sufficient to yield a useful quantity of groundwater.


Aquitard: A saturated formation that holds water but is not capable of transmitting it rapidly.


Unconfined aquifer: A saturated formation whose upper surface is the water table under normal, 1 bar atmospheric pressure. Also called a water table aquifer.


Confined aquifer: A saturated formation whose upper “confining” surface is an aquitard, which isolates the formation from normal surface conditions.


FLUID STATICS


A first step in understanding groundwater behavior is to consider the static (no flow) condition, that for example is realized in a swimming pool. Anyone who has made a shallow dive has experienced the rapid increase of pressure with depth, with significant ear discomfort occurring at depths of only 2 or 3m. This increase in pressure with depth is rapid because the density of water is more than 800x greater than that of air, and because pressure is the direct result of the weight of all overlying material. In fact, the mass of a 10-m deep column of water is equal to the mass of the entire atmosphere in a column of equal basal area.


In quantitative terms, the change of pressure (P) with height (z) is given by the hydrostatic equation, presented earlier in Chapter 2:


dP/dz = -ρg


If water density (ρ) is constant, this integrates to:


P = ρg(h - z)


where h is the head. The pressure at a particular point having an elevation z above some useful datum (e.g., sealevel) depends on that elevation as well as on h, which is the elevation of the water surface relative to the same datum. Note that h-z is the water depth, so that the pressure is proportional to this depth times the water density. At the water surface, where z equals h, the indicated pressure is zero; thus P is actually the gauge pressure, or water pressure, which neglects normal atmospheric pressure (Fig. 3, left).


Although the hydrostatic equation is straightforward, its consequences provide important insights concerning groundwater behavior. The quantity head (h) is a scalar with units of meters (or feet), and it represents the energy of position per unit weight of the water. Also, for static (no flow) conditions, the surfaces of constant pressure and constant density are horizontal, and so must coincide with gravitational equipotential surfaces.


Now that pressure variations with depth are understood in a pool of homogeneous fluid, a key remaining question is how does pressure vary with depth in a saturated porous medium? Conceptually, these two situations can be visualized by comparing a bucket of water with a bucket of marbles that is then filled with water (Fig. 3, right). Now imagine two ants submerged to the same depth in these buckets, and ask, what is the difference in the pressures on these ants? Interestingly, the answer is – None. The pore pressure depends only on the fluid depth, and the effect of the marbles is nil, because they support themselves. However, at the contact points between the marbles, the pressure is much higher, because these points must bear the weight of the marbles. Thus, two different pressures exist in the “porous medium” bucket, the hydrostatic pore pressure, which is the fluid (pore) pressure at the particular depth, and the greater rock pressure at that depth. This is the normal condition for the shallow groundwater environment, if the groundwater can communicate freely with its own water table, which is at atmospheric pressure. However, if the sediment is fine grained, communication of pore fluids with the atmospheric surface might be restricted, and the pore fluids will begin to carry the rock load, producing abnormally-high fluid pressures (AHPs, discussed in Ch. 9).



Figure 3. Left. Pressure variations with depth in a deep pool of water. A diver at a depth of 20m experiences a water pressure of approximately 2 bars; because water is incompressible, the change of pressure with depth is nearly linear at 0.1 bar/m. Right. Same pool, but filled with huge boulders. If the diver could swim between the voids (pores), he would experience the same fluid (pore) pressure as he would at the same depth in open water, but if the boulders were to shift, he would be crushed! (REC)


Fluid Dynamics in Permeable Media


Fluid dynamics contrasts with fluid statics because the fluid is in motion. Fortunately, the dynamics of groundwater flow are much simpler than those of aeronautics, where high velocities and turbulent conditions prevail, and physical properties vary dramatically because air is compressible. In contrast, normal groundwater in the shallow crust has nearly constant density and viscosity, and its movement is dominated by viscous, laminar flow thru minute cracks and pores, that is slow enough that inertial effects are negligible.


Before embarking on a quantitative treatment of groundwater flow, it is useful to consider four “thought experiments” that will help eliminate many common misconceptions. The central question is, what drives groundwater flow in a permeable medium, and in particular, does the water flow down hill, down pressure, or down head? To determine the answer, consider 1) an artesian well; 2) a swimming pool; 3) a convective gyre, and 4) a complex metamorphic or magmatic system.


First, what is observationally remarkable about a flowing artesian well? The fluid flows straight uphill, out of the ground, without the need for pumping or other inducement. Clearly, groundwater need not flow downhill.


Second, consider a swimming pool. As just discussed, the pressure is much higher at the bottom than at the top. Nevertheless, the water does not jump out of the pool, as it must were it required to flow to lower pressure! Instead, the fluid is static, and the large vertical pressure gradient produces no motion.


Now consider a convective gyre, which we could construct by placing a hot vertical plate of metal in the pool. The fluid can be induced to flow in a circular gyre under such conditions. Next to the hot plate, the fluid is heated and expands and ascends, so it moves from high pressure to low pressure, congruent with popular conceptions. However, this movement must be compensated by a limb where cool fluid descends, that clearly must involve flow from low to high pressure! When this return flow moves laterally toward the plate, it moves from low temperature to high temperature; this lateral movement must likewise be compensated by an opposite limb that moves laterally away from the plate, now featuring flow from high temperature to low temperature.


Finally, a direct consequence of these thought experiments is that, in complex metamorphic or magmatic environments, aqueous fluids can move in any direction, irrespective of what way is uphill or downhill, or in which direction pressures or temperatures become higher or lower. Many geologists and engineers are hopelessly confused about this. However, shallow groundwater will always flow toward progressively lower values of head.


Darcy's Law and the Darcy tube


Shortly after the dreaded disease cholera was recognized to be contracted from polluted water, Henry Darcy, a sanitation engineer, developed a means to use sand filtration to purify drinking water supplies. The fundamental basis of groundwater theory, Darcy’s Law, was a consequence of his seminal 1856 publication, Public water supply for Dijon, France. Basically, Darcy filled a large vertical pipe with sand, equipped with two mercury manometers situated near its upper and lower ends. Though manometers are typically used to measure pressure, in his setup they were actually used to determine head. Darcy observed that the volumetric flow rate (Q, m3/s) of water flow through his column was proportional to the difference in head between his upper and lower manometers, that is:


Q ∝ (hu -hl)/L


where (hu -hl) is the difference (m) in the manometer readings, and L is the spacing (m) between them.


The best way to understand Darcy’s relationship, and the movement of fluids through permeable media in general, is to conduct experiments with a sand-packed column similar to Darcy’s apparatus. Hubert (1940) describes such an apparatus (Fig. 4) that has the important, additional capability of being tilted at an arbitrarily angle, rather than being fixed in a vertical orientation. Although some effort will be made here to describe some results, there is no substitute for constructing this simple device and observing how it behaves.


The apparatus depicted in Fig 4 is a ~ 1 m long, 2.5 cm diameter glass tube packed with medium-coarse sand. This tube contains two nipples that allow upper and lower manometers, constructed of ~6 mm dia. glass tubing, to be attached with a short length of flexible plastic tubing. The sand can be held in place with some small mats of steel or copper wool, and the ends are finally capped with two corks penetrated by another short piece of glass tubing, to which supply and drain lines constructed of plastic tubing can be attached.


Water can be passed through this apparatus from an upper reservoir. The flow rate can be adjusted with the aid of a clamp on the lower plastic tube (drain line). This convenient control, however, does not change the fact that the rate of water flow through the sand column between the manometers is governed only by quantities that can be directly read on those manometers. In other words, external conditions cannot affect the interior of the Darcy tube in a way that is not recorded by the bordering manometers.



Figure. 4. The Darcy tube is simply a pipe packed with sand, configured with manometers so that the transmittal of water through the device can be studied. a) In the horizontal configuration, both pressure head (Ph) and hydraulic head (h) decrease in the direction of fluid flow, but elevation head (z) is constant. b) If the Darcy tube is tilted downward, both h and z decrease, but Ph increases in the direction of flow. c) If the Darcy tube is tilted upward, Ph and h both decrease, but z increases as the fluid flows uphill. The only parameter that consistently decreases is h, and the flow rate depends linearly on its gradient. See text. (REC)


Accurate sketches are shown for a steady flow through the sand column with three different orientations (tilts) of the Darcy tube- a) horizontal orientation; b) “downhill” tilt, and c) “uphill” tilt. Steady flow conditions require that the fluid level in the upper reservoir is kept constant, which can be realized if that reservoir is very large, or alternatively if water is added to that reservoir as fast as it flows out. After stable flow is achieved, the flow rate through the tube is easily measured from the time in seconds required to fill a 10 ml graduated cylinder from the lower outlet. The level of water in the two manometers must also be measured, using any convenient datum, such as the bench top, in a consistent manner. It is also informative, although unnecessary, to also measure the height of the center of the sand column, immediately below the two manometers, using the same datum.


Consider the horizontal configuration (Fig 4a), which depicts a steady flow of 10 ml per 300 seconds. The water levels in the upper and lower manometers, designated heads hu and hl, are recorded at heights of 80 and 70 cm above the bench top. The heights of the sand column immediately below the manometers, designated zu and zl, are identical at 45 cm in this case; these values are known as the “elevation head”. The two quantities hu-zu and hl-zl, here equal to 35 and 25 cm, clearly represent the height of water in the manometers above the sand column at the two places, and clearly, these “depths” must be directly proportional to the fluid pressure on the sand at those points; accordingly, these differences are called the “pressure heads” (Ph=P/ρg). Of course, the total head at each manometer position is the elevation head plus the pressure head, as can be seen by adding these quantities, all of which are measured in cm. Now consider how the flow behaves- in this case the flow is horizontal, but down pressure, and down head.


Next, consider the configuration shown in Fig 4b, where the Darcy tube is tilted downward, and a steady flow of 10 ml per 200 seconds is realized. Again, measurements of the water levels and base elevations of the upper and lower manometers will quantify the variables of interest. For this case the flow is downhill, but up pressure (low P to hi P), and down head.


Finally, consider the configuration shown in Fig 4c, where the Darcy tube is tilted uphill, and a steady flow of 10 ml per 600 seconds is realized. Again, make the relevant measurements. In this case the flow is uphill, strongly down pressure (hi P to low P), and down head.


While casual reflection of the above experiments will provide a qualitative explanation of flow behavior, a little more effort will quantify it. Plot the flow rates observed in the three experiments versus: a) the quantity zu - zl, which defines the effect of elevation; b) the quantity, hu-zu-hl+zl, which is proportional to the effect of pressure, and c) hu-hl, which defines the effect of head. In only one case will the three points define a straight line, and very importantly, that line is special because it passes though the origin (0,0) of the plot, where it defines the condition for no flow. The slope of the line is also significant- what does it mean?


The beginner is strongly encouraged to make these graphs before reading on. Better yet, he should conduct his own experiments with a Darcy tube, as this effort will reward him with very compelling and consistent results.


Darcy's Law: A Modern Treatment


Modern versions of Darcy’s important equation define flow in terms of the specific discharge (q), which is the ratio Q/A, where A is the cross sectional area of the column. Note that q is actually the volumetric flux, a vector with units of m3/m2-s. Because these units reduce to those of velocity (m/s), q is commonly also called the Darcy velocity. Thus, the modern version of Darcy’s law is:


q = -K ∂h/∂z


where K is the hydraulic conductivity, defined earlier, and ∂h/∂z is the hydraulic gradient (dimensionless, m/m) in the direction of flow. This is the single most useful equation in groundwater theory, and underlies almost all advanced treatments of any problem. In particular, this equation describes the fluid flux in one-dimension, that is, the volume of fluid that passes through a unit, cross-sectional area per unit of time.


For many situations it is necessary to use the three-dimensional form of Darcy’s Law, which is:


q = - K ∇h


where ∇ is the gradient operator, defined by:


∇ = i ∂/∂x + j ∂/∂y + k ∂/∂z


where i, j and k are unit vectors.


Equations with analogous proportionalities are extremely important in physics; for example Fick’s law of diffusion relates the flux of matter to the concentration gradient, while Fourier’s law relates the flux of heat to the temperature gradient.


Because the Darcy velocity is actually a flux, it is different in magnitude than the actual microscopic velocity (u) of the fluid. For fluid moving through a porous medium, these velocities differ by a factor equal to the effective (flow) porosity, specifically:


u = q/φ


Clearly, u is always greater than q in a permeable medium; that is, u > q.


Example: Alluvial Groundwater


The broad floodplain of the lower Missouri River is underlain by 24 to 36 m of alluvial sediments, that are dominated by fine soils, silt and clays at shallow depth, then by sands and silt, and finally by coarse gravel at depth. Groundwater pumped from these sediments supplies thousands of citizens in many municipalities including Kansas City, Independence, and Columbia Missouri. It is well known that the head in this aquifer responds rather quickly to changes in the water level (stage) of the Missouri River and also to rainfall events.


The behavior of this aquifer is exemplified by a monitored observation well near Jefferson City (Fig. 5a). This well was selected for detailed study because long-term rainfall records and river levels are available for stations located nearby (Fig. 5a). The thoughtful reader of Observational Hydrology will appreciate how fortuitous it is to have so many different recording stations in one small area, and will recognize the effort and dedication required to operate and maintain these facilities for long periods of time. However, the detailed relationships between well head, river level, and rainfall appear to be complicated or even chaotic (Fig. 5b).



Figure. 5. a) A large alluvial aquifer with a depth of ~35m lies the beneath the ~3 km-wide Missouri River floodplain. Long-term observations have been made of the head in an observation well (small inset photo), of the stage of the Missouri River, and of rainfall at proximal facilities managed by USGS and NOAA; these data appear to be very complex in detail (b). c) Use of a flux equation based on Darcy’s Law provides an accurate simulation (red) of the seemingly complex head variations in the well (blue), using only the stage measured on Jan. 1, 1996, the measured river level and precipitation records, and constant values for parameters a, b, and c in the equation in the text. Modified after Criss and Criss (2012; Ground Water, 50, 571-577).


Because Darcy’s law is straightforward, it has great applicability to many problems, including this one. In the case of the alluvial aquifer, it is clear that any flow into to aquifer, or rainfall added, will increase the amount of water it holds, so its level will increase. An equation was crafted to express the daily change in head (∆h), which could be simplified to:


∆h = a(S-h) +bPeff +c


where a, b, and c are constants particular to the location, and Peff is precipitation corrected for evapotranspiration losses. Note that the first term on the right, which is much more important than the other terms, is simply Darcy’s law recast for this situation. That is, the lateral flux of fluid q will produce a change in head (∆h) because any inflowing (or outflowing) water must be stored in (or lost from) the porous floodplain sediment. This fluid flow will be proportional to a constant times the gradient in head, which must be proportional to S-h, that is to the difference between the river level S and the head h in the well.


Remarkably, this straightforward relationship can be used to simulate the head record to great accuracy (Fig. 5c). In particular, using only one head measurement (in 1996), the heads for the subsequent 15 years were calculated almost perfectly using only this equation and the records for river level and precipitation. In practice, each calculated daily change (∆h) was added to the head calculated for the previous day, to get the estimated head for the next day, and this was continued for 15 years. This exercise illustrates the power of Darcy’s law to explain datasets that appear, on face value, to be hopelessly complex.


Hydraulic Potential(Φ)


Darcy’s law is an empirical relationship, but in his famous book, Theory of Groundwater Motion, Hubbert (1940) provided its theoretical basis. He proved that the head is the energy per unit weight of the fluid, and defined a quantity called the hydraulic potential (Φ) that is the energy per unit mass. This potential is:


Φ = gh = gz + P/ρw


and is simply the gravitational constant times the head. Hubbert showed that this quantity is equal to the energy required to move a fluid element from place to place, which involves it being lifted, accelerated, and compressed or decompressed. For slow flow of an incompressible fluid element, the kinetic energy term is negligible, and the potential of the fluid element can be expressed in terms of its position, height, pressure and density, as written above.


The hydraulic potential is a scalar quantity that has a unique, continuously varying magnitude at every point along the path followed by a moving packet of groundwater. Importantly, the gradient of the hydraulic potential is the force per unit mass acting on the fluid element:


Force/unit mass = ∇Φ= g - ∇P/ρ


Force/unit weight = ∇h = 1 - ∇P/ρg


These relationships enabled Hubbert (1940, J. Geol. 48, p. 785-944) to rewrite Darcy's Law in a theoretical framework, specifically:



Here, the qm (kg/m2-s) is the volumetric flux qv times the fluid density. Note that the mass flux is directly proportional to the permeability and to the force per unit mass (in brackets), and inversely proportional to the fluid viscosity. These proportionalities make perfect sense.


Applying the above equation to a static fluid body provides other important insights into groundwater behavior. For no flow, qm equals zero, and this condition can be realized only if either the permeability is zero, or if the force acting on the fluid element is zero. In the latter case, the pressure gradient ∇P must be equal to ρg. However, according to equation 1, a single term of ∇P, specifically the vertical pressure gradient ∂P/∂z, is equal to ρg. This implies that the other two terms in ∇P, specifically the horizontal pressure gradients ∂P/∂x and ∂P/∂y, must be equal to zero if the fluid is not moving. A corollary is that differences in pressure along a horizontal plane, or a vertical pressure gradient that is different in magnitude than ρg, will require the fluid to flow.


GROUNDWATER: ADVANCED CONCEPTS


Fundamental Concepts. When motion is possible, objects spontaneously move to positions of lower energy, as we witness when we drop a stone. The same idea applies to “packets” of groundwater. Thus, groundwater flow can be deduced by considering how the head (h), which is directly proportional to the hydraulic potential Φ varies in space. It was shown above that Φ =gh, and that this potential represents the energy per unit mass of the fluid.


It is useful to consider the static (no flow) condition in a swimming pool, that may contain only water, or may also contain large cobbles (Fig. 3). In either of these cases the head is represented by the elevation of the water surface, and it does not vary with lateral position. In fact, this quantity does not vary with vertical position either, so the head is constant and all fluid packets in the pool have the same energy per unit mass. Thus, there is no inducement for the fluid to move, laterally or vertically, so the fluid is static. Because h is constant, the gradient in head (∇h) is zero, so flow is zero.


In contrast, if the fluid is in motion, the situation is dynamic and the head does vary from place to place. Suppose that the flow is purely horizontal; as we will see, this is possible only in a horizontal, confined aquifer of constant thickness. In this case, planes of constant head, analogous to equipotential surfaces, will be a succession of parallel vertical planes, and if the medium is isotropic, as hereafter assumed unless otherwise stated, the fluid will flow in the down-gradient direction that is perpendicular to them. This flow can visualized as “flowlines”, constructed perpendicular to the equipotential surfaces.


In real situations, the configuration of equipotential planes is often more complex and in general the planes will be curved. Nevertheless, no two equipotential planes can ever intersect but instead they must be subparallel, otherwise a single fluid element would have more than one energy. Because the groundwater flowlines are constructed perpendicular to these planes, they must also be curved, and the situation must be visualized in 3-D. Here, some simple rules are helpful. Fluid flowlines cannot intersect- otherwise you would have a fluid packet with two velocities. Also, Darcy’s law, q= -K∇h, still applies at every point in space. In the isotropic case, where K is a simple scalar, the fluid flux (Darcy velocity) is everywhere perpendicular to the equipotential planes, and it’s magnitude is inversely proportional to their spacing.


If the hydraulic conductivity K is a constant, the equipotential surfaces will be evenly spaced in a confined aquifer of constant thickness. If K varies laterally, but remains isotropic at any place, then if K is smaller in some part of the aquifer layer, the hydraulic gradient will have to be steeper there, which means that the equipotential planes will become more closely spaced in that zone. This is because q= -K∇h, and because q must be constant- that is, the fluid flux must be the same for every cross-sectional plane in the aquifer; otherwise excess fluid would accumulate in some region, which is impossible because the fluid is basically incompressible. What happens instead is that the product K∇h remains constant, sot that q remains constant, so if K is smaller in some zone, ∇h is proportionately larger, and vice-versa. If values of K are anisotropic, the effect on equipotential surfaces and flowlines is more complex, as discussed under “permeability anisotropy”, below.


Flow Net. A flow net is a conceptual network of intersecting equipotential lines and fluid flowlines, defined on some plane of interest. If K is isotropic as usually assumed, then these two sets of lines are perpendicular.


Flowlines, also called streamlines, represent the instantaneous flow directions. For steady flow these flowlines are the actual path followed by the fluid “packets”. For transient flow, however, the actual fluid paths, called path lines, are not coincident with the flowlines.


A few rules are useful for the construction of flow nets. The equipotential surfaces will be perpendicular to an impermeable boundary, so the flowlines will be parallel to it (Fig 6 left). The opposite relationships hold near a constant head boundary, such as near a lake or river (Fig. 6 middle). Finally, these families will have an angular relationship to a water table boundary, with the flowlines being directed downgradient and into the subsurface (Fig. 6 right). A more complex model for a natural system exhibits several of these concepts (Fig. 7).



Figure 6. Conceptual flow nets near different types of boundaries. Dashed red lines show the family of equipotential surfaces; perpendicular arrows are the flowlines. After Freeze and Cherry.



Figure 7. Model flow net for a vertical section of the surficial aquifer in coastal Maryland. The equipotential surfaces are indicated by the dashed contour lines, while the flowlines are shown by the orthogonal blue arrows. Note that rainfall is recharged on the small hills on the left, where it infiltrates downward, and then moves toward regions of lower head near the ocean. The freshwater can emerge, sometimes as springs, beneath the ocean surface. From USGS.


Potentiometric surface. The potentiometric surface is a map of the hydraulic head measured on a planar surface of interest. The x-y position is the spatial (map) coordinate, but contours on the map represent the head rather than the topographic elevation. Though straightforward in concept, confusion can arise because this is a 2-D, planar slice through a succession of curved, 3-D equipotential surfaces that fill all space. In contrast, a topographic surface is truly two dimensional, however irregular- such a surface can be likened to a wrinkled sheet of paper.


A water table map is a special type of potentiometric surface that is easily understood and widely used (Fig. 8). However, this map represents the head only on the curved plane of the water table, and need not indicate the head at greater depth. Only if the equipotential planes are perfectly vertical can a unique value of head be assigned to every x-y position that is valid at all aquifer depths. Vertical equipotential planes are a useful approximation in many but not all circumstances; if they are not vertical, a unique equipotential map can be drawn at any desired elevation.



Fig. 8. Water table map of the southern Sacramento Valley from Spring 2015. Chronic cones of depressions below sea level form on the east side that is dominated by urban development but relies on groundwater for the majority of their water supply. West side is mostly flood furrow agricultural that in part contributes to annual recharge. Data from CA Department of Water Resources; base map USGS.


Confined aquifers do not have a water table, because no 1 atm surface is present that separates water-filled and air-filled pores. A potentiometric map can nevertheless be drawn (Fig. 9); this imaginary surface will represent the levels to which water will rise in a dense network of piezometers, which are non-pumping observation wells that are cased to the aquifer. Simple interpretations of such maps are based on the assumption of vertical equipotential planes and constant aquifer thickness.



Figure 9. Map of the virgin potential surface of the Dakota aquifer, a confined aquifer hosted by the Cretaceous Dakota sandstone that is overlain by Cretaceous marine shales. This remarkable map by Darton (1909) shows that he correctly envisioned that the groundwater flow extended hundreds of miles, across the entire state of South Dakota, with recharge in the Black Hills to the west.


If highly detailed information is needed, one standard but expensive method is to install a system of nested piezometers, with each “nest’ representing a cluster of individual piezometers at a single site that are cased to different depths. In this manner, at each x-y position the head can be measured at several depths so vertical head gradients can be determined; the horizontal gradients can be assessed by comparing one nest to others.


Basic Definitions: In what follows, a few key properties of aquifers will repeatedly arise. For easy reference, these are described here.


Transmissivity (T) is a measure of water to move in an aquifer. This term has mks units of m2/s, and specifically describes the rate of flow of water thru a unit-wide, typically vertical strip of aquifer under a unit hydraulic gradient. Transmissivity is equal to the product K·m, where K is the hydraulic conductivity and m is the aquifer thickness. Good aquifers have T ≥ 0.015 m2/s.


Storativity (S) is a dimensionless measure of the quantity of pumpable water stored in an aquifer. Specifically, S represents the volume water released per unit area per unit head drop. Several different parameters contribute to the total value of S, most importantly the specific yield (Sy) and specific storage (Ss), described below, and aquifer thickness m. In particular,


For an unconfined aquifer: S = Sy + m·Ss ~ Sy; typical values are 0.01 to 0.30

For a confined aquifer: S = m·Ss ; typical values are 0.005 to 0.00005


Specific yield (Sy) is the storativity of an unconfined aquifer, and is the volume of water drained per unit area when the head is dropped by a unit amount. The specific retention (Sr) is the water retained during the same process, and the porosity φ is: φ = Sy + Sr + the unconnected porosity. In typical clastic sediments, Sy ranges from 0.01 to 0.30, but it increases with grain size, being high in gravels and low in clays, which have high Sr.

Specific storage (Ss) has units of 1/length, and represents the volume water released from storage per unit volume of the aquifer per unit head drop. This quantity depends on several factors, including the compressibility of the aquifer and the compressibility of the water itself.


Hydraulic Diffusivity (D) is the Transmissivity:Storativity ratio T/S, and has mks units of m2/s. For a confined aquifer, D = K/Ss

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Aquifer Types


Unconfined Aquifer: Also known as a water table aquifer, unconfined aquifers have the water table as their upper surface; this surface is in contact with an air phase at 1 atm pressure.


Perched Aquifer: An unconfined aquifer above the main water table; generally situated above a lens of low-permeability material.


Confined Aquifer: An aquifer that is significantly isolated from the conditions at the Earth’s surface, due to overlying strata with low permeability. Also known as an artesian aquifer, the head in confined aquifers rises above the local water table.


Flowing Artesian aquifer: A confined aquifer whose head rises above the land surface, so that water actively flows out of the collar of a well drilled into it.


Permeability and Porosity


If Hubbert’s equation is compared to the conventional version of Darcy’s Law, qv = -K∇h, one can determine the relationship between the hydraulic conductivity (K) and the permeability (k) and kinematic viscosity ν=μ/ρ:


K = kg/ν


where g is the gravitational acceleration. This relationship clearly shows that K is a lumped parameter, whose value depends on a rock property (k) and a fluid property (ν). However, we are generally interest in the flow of cool shallow groundwater on Earth, so both ρ, g, and ν can be considered to be constants, respectively 0.01 cm2/s and 980 cm/s2. Thus, under normal conditions K depends primarily on the permeability. Of course, if the flow of a different fluid such as petroleum or natural gas were under consideration, different values of ρ and ν would apply. Also, g would be different on different planetary bodies, such as Titan or Mars.


The above equation provides a simple recipe for converting permeability into hydraulic conductivity, or vice-versa, for cool groundwaters, as these parameters differ only by a factor of g/ν. For example, to determine K (cm/s) from permeability (cm2), the factor is about 980/.011; if K is desired in units of cm/d, an additional factor of 86,400 sec/day is included.


Permeability and porosity are commonly used interchangeably, and although they are proportional, they are very different quantities. For starters, they have different units, so they cannot be directly compared any more than a $5 bill can be compared to a 5 gallon jug. Lack of attention to units, including the use of equations that contain “constants”, particularly hidden constants with absurd or inconsistent units, is a major problem in hydrology.


Porosity (φ): Porosity (φ) is conceptually simple, being the dimensionless ratio of the void volume Vvoid to the total volume Vt of a material:


φ = Vvoid/Vt


For simple packing of spheres of uniform size, φ = 47.9%, whereas for closest packing, φ = 26%. Lower values of φ can be obtained if grain size is not uniform, so that small grains can fit within the largest spaces. In a simple medium, porosity is the factor that relates the actual microscopic velocity of the fluid (u), in the net direction of flow, to the Darcy velocity (qv):


u = qv


Porosity varies considerably among different rock types (Fig 10). Very low values (<0.1%) are common in unfractured crystalline rocks, but values can be quite high in well-sorted sand, uncompacted mud, or karstic limestone. Porosity decreases with depth, as compaction and dewatering occur, and secondary minerals are precipitated in void spaces. Under very deep conditions, rocks can flow plastically and porosity becomes almost nil.


Permeability (k): Permeability describes the ability of a medium to transmit flow under a hydraulic gradient. In particular, k depends on both the size and the interconnectivity of the pore (void) spaces. Thus, media with high porosity commonly have high permeability, but this need not be true. For example, a styrofoam cup has high porosity but very low permeability. Nevertheless, for a medium constituted of uniform spheres, k is proportional to the square of the grain diameter. Similarly, for parallel fractures with spacing L, it can be shown that k=a3/12L, where a is the fracture aperture.


A key consideration is that permeability varies by a huge factor, >1013, in common geologic materials (Fig. 11). Very low values are common in clays, shales, and unfractured crystalline rocks, whereas high values are typical of gravels, coarse sands, and karstic limestone. Like porosity, permeability tends to decrease with depth, but even more sharply. Also, in active geologic environments, k is a dynamic parameter due to fracturing, dissolution, cementation, and other processes. Finally, values of k depend on the scale of observation, particularly if it is primarily due to widely spaced fractures. Thus, values of k measured on small lab samples can be many orders of magnitude less than the most permeable parts of a drill core, which can be significantly less than the regional value for the particular rock unit.



Figure 10 (a) Porosity ranges for common rock types and hypothetical cubic packing structures. FCC (Face-centered cubic); BCC (Body centered cubic). (b) Permeability ranges for common rock types. Bottom scale, units of cm2; top scale, units of darcys (d).


Permeability Anisotropy: The permeability can be anisotropic, for example if controlled by fractures with preferred orientation(s); this condition also occurs in layered stratigraphic sequences, particularly if low permeability shale layers are present. In some instances, it is useful to consider this effect in terms of an anisotropy ratio, namely the dimensionless ratio kx/kz, which commonly ranges from 1 to 10 within a typical sedimentary layer, but can be 106 or more in a stratigraphic sequence. Thus, flow normally tends to be layer or fracture parallel, and unlike the simple form of Darcy’s law, such flow need not be orthogonal to the gradient in hydraulic head. In this case, both the permeability and, of course, the hydraulic conductivity, are actually tensors rather than simple scalars.


First consider a simple case for layered strata (Fig 12). It can be shown that the hydraulic conductivity in the direction parallel to the strata (defined as the x direction) is the simple weighted average of the layer-parallel conductivities of the individual layers, but weighted by layer thickness. In contrast, the average conductivity in the orthogonal direction is a type of harmonic mean, the least familiar of the three Pythagorean means (Fig. 13). An interesting consequence is that flow in the x direction is dominated by the most permeable units, whereas flow in the vertical direction is dominated by the units with the least permeability. This makes sense- if one of the layers were an impervious plastic sheet, however thin, vertical flow would be impossible, yet such a layer would have negligible impact on the layer-parallel flow.


Now consider the effect of anisotroptic permeability, or hydraulic conductivity, on flow nets. Because the direction of fluid flow is not be simply perpendicular to the equipotential surfaces, the flow net will not be rectilinear. Formally, this case is treated by recognizing that, rendering the mathematics becomes more complicated because k and K must now be treated as tensors, so computational methods are mostly used to numerically evaluate such problems. Nevertheless, measurements that constrain the distribution of k and K in such situations are few. The thoughtful hydrogeologist, however, will recognize that for real anisotropic situations, the flow in a stratigraphic package will tend to become parallel to the bedding, or in fractured medium, become strongly oriented in the direction of the major fractures.



Figure 11. Permeability parallel to rock layers (Kx) is normally much greater than in the orthogonal direction (Kz). The simple mean, weighed by layer thickness m, describes the former case, while the harmonic mean describes the latter.


Topographic Flow


Figure 12. Conceptual flow net showing equipotential surfaces and flowlines beneath a hill, with piezometers added to illustrate the variations in head. Note that under the hilltop, the head decreases with piezometer depth, so water percolates downward. In contrast, under valleys, the head at deep levels is greater than the head at sallow levels, so groundwater ascends. After Fetter (1988) and Hubbert (1940).




Figure 13. (Top). A “groundwater outcrop” at Cedar Bog, Ohio, which is actually a fen. Unconfined groundwater ascends in this area, feeding numerous springs and seeps that coalesce in a broad swampy area that hosts many grasses and sedges including Pleistocene relict flora. (Bottom). Plastic piezometer tube at Cedar Bog. Position of fingers shows the elevation of water in the open-ended tube- note that this level, which indicates the head at the bottom of the plastic tube, is higher than the head at the surface, which is represented by the water level in the stream. Photos by Criss.