# Hydraulic head

**Hydraulic head**is a specific measurement of water pressure ortotal energy perunit weight above ageodetic datum . It is usually measured as a water surface elevation, expressed in units of length, but represents the energy at the entrance (or bottom) of apiezometer . In an aquifer, it can be calculated from the depth to water in a piezometric well (a specializedwater well ), and given information of the piezometer's elevation and screen depth. Hydraulic head can similarly be measured in a column of water using a standpipe piezometer by measuring the height of the water surface in the tube relative to a common datum. In both cases, although the measurement is made at the water surface, the hydraulic head measurement represents the total energy at the entrance (base) of the piezometer. The hydraulic head can be used to determine a "hydraulic gradient" between two or more points.**"Head" in fluid dynamics**In

fluid dynamics ,**head**is the difference inelevation between two points in a column offluid , and the resultingpressure of the fluid at the lower point. Head is normally expressed in units of height (e.g.meters ), although some texts (incorrectly) refer to it in units ofpressure such as pascals (theSI unit of pressure). Head refers to the constant right hand side in the incompressible steady version ofBernoulli's equation .This is best understood by considering a

waterwheel : the head is the vertical distance from the top of the waterwheel to the free surface of the millpond.More generally, when considering a flow, one says that head is lost if

energy is dissipated, often throughturbulence ; equations such as theProny equation and theDarcy-Weisbach equation have been used to calculate the head loss due tofriction . In the context ofsteam train s, one talks of a "good head of steam", referring to the pressure in theboiler .The

**static head**of apump is the maximum height (pressure) it can deliver. The capacity of the pump can be read from its Q-H curve (flow vs. height).Head is used to describe the

energy inincompressible fluid s. Head has units of distance and equals the fluid's energy per unitweight . Head is useful in specifyingcentrifugal pump s because their pumping characteristics tend to be independent of the fluid's density.There are four types of head used to calculate the total head in and out of a pump:

*"Static head " is due togravitational force on a column of fluid.

*"Velocity head " is due to the motion of a fluid (kinetic energy ).

*"Resistance head " (or "friction head") is due to the frictional forces against a fluid's motion.

*"Pressure head " is due to other mechanical forces acting on a fluid.**Components of hydraulic head**The total hydraulic head is composed of "pressure head", "velocity head" and "elevation head". The pressure head is the equivalent gauge pressure of a column of water at the base of the piezometer, the velocity head is the

kinetic energy from the motion of water, and the elevation head is the relativepotential energy in terms of an elevation. This can be expressed as::$h\; =\; psi\; +\; h\_v\; +\; z\; ,$where:$h$ is the hydraulic head (Length in m or ft),:$h\_v$ is thevelocity head , measured by aPitot tube (Length in m or ft),:$psi$ is thepressure head , in terms of the elevation difference of the water column relative to the piezometer bottom (Length in m or ft), and:$z$ is the elevation at the piezometer bottom (Length in m or ft)In

groundwater studies, the velocity head is assumed to be zero and is ignored. This is because groundwater moves very slowly, and the kinetic energy loss is very low. Thus, for groundwater studies, hydraulic head can be defined simply as::$h\; =\; psi\; +\; z\; ,$In an example with a 400m deep piezometer, with an elevation of 1000m, and a depth to water of 100m: "z"=600m, "ψ"=300m, and "h"=900 m.

The pressure head can be expressed as::$psi\; =\; frac\{P\}\{gamma\}\; =\; frac\{P\}\{\; ho\; g\}$where:$P$ is the gauge pressure (Force per unit area, often Pa or psi),:$gamma$ is the

unit weight of water (Force per unit volume, typically N·m^{−3}or lbf/ft³),:$ho$ is thedensity of the water (Mass per unit volume, frequently kg·m^{−3}), and:$g$ is thegravitational acceleration (velocity change per unit time, often m·s^{−2})In faster moving water where the

Reynolds number exceeds 10, such as in ariver , the velocity head can be calculated usingBernoulli's principle s as::$h\_v\; =\; frac\{v^2\}\{2g\}$where $v$ is relativevelocity of the water (distance per unit time, often m·s^{−1}or ft·s^{−1})**Fresh water head**The pressure head is dependent on the

density of water, which can vary depending on both thetemperature andchemical composition (salinity , in particular). This means that the hydraulic head calculation is dependent on the density of the water within the piezometer. If one or more hydraulic head measurements are to be compared, they need to be standardized, usually to their "fresh water head", which can be calculated as::$h\_\{fw\}\; =\; psi\; frac\{\; ho\}\{\; ho\_\{fw\; +\; z$where:$h\_\{fw\}\; ,$ is the fresh water head (Length, measured in m or ft), and:$ho\_\{fw\}\; ,$ is thedensity of fresh water (Mass per unit volume, typically in kg·m^{−3})**Hydraulic gradient**The "hydraulic gradient" is a vector gradient between two or more hydraulic head measurements over the length of the flow path. It is also called the "

darcy slope ", since it determines the quantity of a "darcy flux ", or discharge. Adimensionless hydraulic gradient can be calculated between two piezometers as::$i\; =\; frac\{dh\}\{dl\}\; =\; frac\{h\_2\; -\; h\_1\}\{mathrm\{length$where:$i$ is the hydraulic gradient (dimensionless),:$dh$ is the difference between two hydraulic heads (Length, usually in m or ft), and:$dl$ is the flow path length between the two piezometers (Length, usually in m or ft)The hydraulic gradient can be expressed in vector notation, using the

del operator. This requires a hydraulic head field, which can only be practically obtained from a numerical model, such asMODFLOW . InCartesian coordinates , this can be expressed as::$abla\; h\; =\; left(\{frac\{partial\; h\}\{partial\; x,\{frac\{partial\; h\}\{partial\; y,\{frac\{partial\; h\}\{partial\; z\; ight)\; =\; \{frac\{partial\; h\}\{partial\; xmathbf\{i\}\; +\; \{frac\{partial\; h\}\{partial\; ymathbf\{j\}\; +\; \{frac\{partial\; h\}\{partial\; zmathbf\{k\}$This vector describes the direction of the groundwater flow, where negative values indicate flow along the dimension, and zero indicates "no flow". As with any other example in physics, energy must flow from high to low, which is why the flow is in the negative gradient. This vector can be used in conjunction withDarcy's law and atensor ofhydraulic conductivity to determine the flux of water in three dimensions.**Hydraulic head in groundwater**The distribution of hydraulic head through an

aquifer determines where groundwater will flow. In a hydrostatic example (first figure), where the hydraulic head is constant, there is no flow. However, if there is a difference in hydraulic head from the top to bottom due to draining from the bottom (second figure), the water will flow downward, due to the difference in head, also called the "hydraulic gradient".**Atmospheric pressure**Even though it is convention to use

gauge pressure in the calculation of hydraulic head, it is more correct to use total pressure (gauge pressure +atmospheric pressure ), since this is truly what drives groundwater flow. Often detailed observations ofbarometric pressure are not available at each well through time, so this is often disregarded (contributing to large errors at locations where hydraulic gradients are low or the angle between wells is acute.)The effects of changes in

atmospheric pressure upon water levels observed in wells has been known for many years. The effect is a direct one, an increase in atmospheric pressure is an increase in load on the water in the aquifer, which increases the depth to water (lowers the water level elevation). Pascal first qualitatively observed these effects in the 1600s, and they were more rigorously described by the soil physicistEdgar Buckingham (working for the USDA) using air flow models in 1907.**Analogs to other fields**Hydraulic head is a measure of energy, and has many analogs in

physics andchemistry , where the same mathematical principles and rules apply:

*"Hydraulic head" is analogous to:

**magnetic monopole

**electric charge

**heat (i.e.,temperature )

**concentration

*A continuous "field" of hydraulic heads is analogous to:

**anelectric field

**amagnetic field

*Similardifferential operator s can be applied to the fields, to find:

**thegradient , or the direction of flow

**thedivergence of flow

**the curl, or if the field is rotating**ee also***

Bernoulli's principle , from which we derive pressure head and velocity head

*fluid dynamics

*Pressure head

*Darcy's law

* Total Dynamic Head**References*** Bear, J. 1972. "Dynamics of Fluids in Porous Media", Dover. ISBN 0-486-65675-6.

* for other references which discuss hydraulic head in the context of hydrogeology, see that page's further reading section

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