Hydraulic head
Encyclopedia
Hydraulic head or piezometric head is a specific measurement of water pressure above a geodetic datum. It is usually measured as a water surface elevation, expressed in units of length, at the entrance (or bottom) of a piezometer
. In an aquifer, it can be calculated from the depth to water in a piezometric well (a specialized water 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. The hydraulic head can be used to determine a hydraulic gradient between two or more points.
, head is a concept that relates the energy in an incompressible fluid to the height of an equivalent static column of that fluid. From Bernoulli's Principle
, the total energy at a given point in a fluid is the energy associated with the movement of the fluid, plus energy from pressure in the fluid, plus energy from the height of the fluid relative to an arbitrary datum. Head is expressed in units of height such as meters or feet.
The static head of a pump
is the maximum height (pressure) it can deliver. The capability of the pump can be read from its Q-H curve (flow vs. height).
Head is equal to the fluid's energy per unit weight
. Head is useful in specifying centrifugal 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:
ing from an elevation
(in a vacuum
) will reach a speed
when arriving at elevation z=0, or when we rearrange it as a head: 
where
is the acceleration due to gravity
The term
is called the velocity head, expressed as a length measurement. In a flowing fluid, it represents the energy of the fluid due to its bulk motion.
The total hydraulic head of a fluid is composed of pressure head and elevation head. The pressure head is the equivalent gauge
pressure
of a column of water at the base of the piezometer, and the elevation head is the relative potential energy
in terms of an elevation. The head equation, a simplified form of the Bernoulli Principle for incompressible fluids, can be expressed as:
where
is the hydraulic head (Length
in m or ft), also known as the piezometric head.
is the pressure head
, in terms of the elevation difference of the water column relative to the piezometer bottom (Length
in m or ft), and
is the elevation at the piezometer bottom (Length
in m or ft)
In an example with a 400 m deep piezometer, with an elevation of 1000 m, and a depth to water of 100 m: z = 600 m, ψ = 300 m, and h = 900 m.
The pressure head can be expressed as:
where
is the gauge pressure (Force per unit area, often Pa or psi),
is the unit weight of water (Force per unit volume, typically N·m−3 or lbf
/ft³),
is the density
of the water (Mass per unit volume, frequently kg·m−3), and
is the gravitational acceleration
(velocity change per unit time, often m·s−2)
of water, which can vary depending on both the temperature
and chemical 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:
where
is the fresh water head (Length, measured in m or ft), and
is the density
of fresh water (Mass per unit volume, typically in kg·m−3)
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. A dimensionless hydraulic gradient can be calculated between two piezometers as:
where
is the hydraulic gradient (dimensionless),
is the difference between two hydraulic heads (Length, usually in m or ft), and
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 as MODFLOW
. In Cartesian coordinates, this can be expressed as:
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 with Darcy's law
and a tensor
of hydraulic conductivity
to determine the flux of water in three dimensions.
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.
), since this is truly what drives groundwater flow. Often detailed observations of barometric 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 17th century, and they were more rigorously described by the soil physicist
Edgar Buckingham (working for the USDA
) using air flow models in 1907.
; turbulence
dissipates even more energy for high Reynolds number flows. Head loss is divided into two main categories, "major losses" associated with energy loss per length of pipe, and "minor losses" associated with bends, fittings, valves, etc. The most common equation used to calculate major head losses is the Darcy–Weisbach equation. Older, more empirical approaches are the Hazen-Williams equation
and the Prony equation
.
For relatively short pipe systems, with a relatively large number of bends and fittings, minor losses can easily exceed major losses. In design, minor losses are usually estimated from tables using coefficients or a simpler and less accurate reduction of minor losses to equivalent length of pipe.
and chemistry
, where the same mathematical principles and rules apply:
Piezometer
A piezometer is either a device used to measure static liquid pressure in a system by measuring the height to which a column of the liquid rises against gravity, or a device which measures the pressure of groundwater at a specific point...
. In an aquifer, it can be calculated from the depth to water in a piezometric well (a specialized water well
Water well
A water well is an excavation or structure created in the ground by digging, driving, boring or drilling to access groundwater in underground aquifers. The well water is drawn by an electric submersible pump, a trash pump, a vertical turbine pump, a handpump or a mechanical pump...
), 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. The hydraulic head can be used to determine a hydraulic gradient between two or more points.
"Head" in fluid dynamics
In fluid dynamicsFluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
, head is a concept that relates the energy in an incompressible fluid to the height of an equivalent static column of that fluid. From Bernoulli's Principle
Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...
, the total energy at a given point in a fluid is the energy associated with the movement of the fluid, plus energy from pressure in the fluid, plus energy from the height of the fluid relative to an arbitrary datum. Head is expressed in units of height such as meters or feet.
The static head of a pump
Pump
A pump is a device used to move fluids, such as liquids, gases or slurries.A pump displaces a volume by physical or mechanical action. Pumps fall into three major groups: direct lift, displacement, and gravity pumps...
is the maximum height (pressure) it can deliver. The capability of the pump can be read from its Q-H curve (flow vs. height).
Head is equal to the fluid's energy per unit weight
Weight
In science and engineering, the weight of an object is the force on the object due to gravity. Its magnitude , often denoted by an italic letter W, is the product of the mass m of the object and the magnitude of the local gravitational acceleration g; thus:...
. Head is useful in specifying centrifugal pump
Centrifugal pump
A centrifugal pump is a rotodynamic pump that uses a rotating impeller to create flow by the addition of energy to a fluid. Centrifugal pumps are commonly used to move liquids through piping...
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:
- Velocity head is due to the bulk motion of a fluid (kinetic energyKinetic energyThe kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...
). - Elevation head is due to the fluid's weight, the gravitational force acting on a column of fluid.
- Pressure headPressure headPressure head is a term used in fluid mechanics to represent the internal energy of a fluid due to the pressure exerted on its container. It may also be called static pressure head or simply static head...
is due to the static pressureStatic pressureIn fluid mechanics the term static pressure has several uses:* In the design and operation of aircraft, static pressure is the air pressure in the aircraft’s static pressure system....
, the internal molecular motion of a fluid that exerts a force on its container. - Resistance head (or friction head or Head Loss) is due to the frictional forces acting against a fluid's motion by the container.
Components of hydraulic head
A mass free fallFree fall
Free fall is any motion of a body where gravity is the only force acting upon it, at least initially. These conditions produce an inertial trajectory so long as gravity remains the only force. Since this definition does not specify velocity, it also applies to objects initially moving upward...
ing from an elevation
(in a vacuumVacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...
) will reach a speed
Speed
In kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...
when arriving at elevation z=0, or when we rearrange it as a head: where
is the acceleration due to gravityThe term
Term (mathematics)
A term is a mathematical expression which may form a separable part of an equation, a series, or another expression.-Definition:In elementary mathematics, a term is either a single number or variable, or the product of several numbers or variables separated from another term by a + or - sign in an...
is called the velocity head, expressed as a length measurement. In a flowing fluid, it represents the energy of the fluid due to its bulk motion.The total hydraulic head of a fluid is composed of pressure head and elevation head. The pressure head is the equivalent gauge
Gauge (engineering)
In engineering, a gauge or gage, is used to make measurements. A wide variety of tools exist which serve such funtions, ranging from simple pieces of material against which sizes can be measured to complex pieces of machinery...
pressure
Pressure measurement
Many techniques have been developed for the measurement of pressure and vacuum. Instruments used to measure pressure are called pressure gauges or vacuum gauges....
of a column of water at the base of the piezometer, and the elevation head is the relative potential energy
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...
in terms of an elevation. The head equation, a simplified form of the Bernoulli Principle for incompressible fluids, can be expressed as:
where
is the hydraulic head (LengthLength
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...
in m or ft), also known as the piezometric head.
is the pressure headPressure head
Pressure head is a term used in fluid mechanics to represent the internal energy of a fluid due to the pressure exerted on its container. It may also be called static pressure head or simply static head...
, in terms of the elevation difference of the water column relative to the piezometer bottom (Length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...
in m or ft), and
is the elevation at the piezometer bottom (LengthLength
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...
in m or ft)
In an example with a 400 m deep piezometer, with an elevation of 1000 m, and a depth to water of 100 m: z = 600 m, ψ = 300 m, and h = 900 m.
The pressure head can be expressed as:
where
is the gauge pressure (Force per unit area, often Pa or psi), is the unit weight of water (Force per unit volume, typically N·m−3 or lbfPound-force
The pound force is a unit of force in some systems of measurement including English engineering units and British gravitational units.- Definitions :...
/ft³),
is the densityDensity
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
of the water (Mass per unit volume, frequently kg·m−3), and
is the gravitational accelerationGravitational acceleration
In physics, gravitational acceleration is the acceleration on an object caused by gravity. Neglecting friction such as air resistance, all small bodies accelerate in a gravitational field at the same rate relative to the center of mass....
(velocity change per unit time, often m·s−2)
Fresh water head
The pressure head is dependent on the densityDensity
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
of water, which can vary depending on both the temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
and chemical composition (salinity
Salinity
Salinity is the saltiness or dissolved salt content of a body of water. It is a general term used to describe the levels of different salts such as sodium chloride, magnesium and calcium sulfates, and bicarbonates...
, 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:
where
is the fresh water head (Length, measured in m or ft), and is the densityDensity
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
of fresh water (Mass per unit volume, typically in kg·m−3)
Hydraulic gradient
The hydraulic gradient is a vector gradientGradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
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. A dimensionless hydraulic gradient can be calculated between two piezometers as:
where
is the hydraulic gradient (dimensionless), is the difference between two hydraulic heads (Length, usually in m or ft), and 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
Del
In vector calculus, del is a vector differential operator, usually represented by the nabla symbol \nabla . When applied to a function defined on a one-dimensional domain, it denotes its standard derivative as defined in calculus...
operator
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
. This requires a hydraulic head field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, which can only be practically obtained from a numerical model, such as MODFLOW
MODFLOW
MODFLOW is the U.S. Geological Survey modular finite-difference flow model, which is a computer code that solves the groundwater flow equation. The program is used by hydrogeologists to simulate the flow of groundwater through aquifers...
. In Cartesian coordinates, this can be expressed as:
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 with Darcy's law
Darcy's law
Darcy's law is a phenomenologically derived constitutive equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on the results of experiments on the flow of water through beds of sand...
and a tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
of hydraulic conductivity
Hydraulic conductivity
Hydraulic conductivity, symbolically represented as K, is a property of vascular plants, soil or rock, that describes the ease with which water can move through pore spaces or fractures. It depends on the intrinsic permeability of the material and on the degree of saturation...
to determine the flux of water in three dimensions.
Hydraulic head in groundwater
The distribution of hydraulic head through an aquifer
Aquifer
An aquifer is a wet underground layer of water-bearing permeable rock or unconsolidated materials from which groundwater can be usefully extracted using a water well. The study of water flow in aquifers and the characterization of aquifers is called hydrogeology...
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 pressureAtmospheric pressure
Atmospheric pressure is the force per unit area exerted into a surface by the weight of air above that surface in the atmosphere of Earth . In most circumstances atmospheric pressure is closely approximated by the hydrostatic pressure caused by the weight of air above the measurement point...
), since this is truly what drives groundwater flow. Often detailed observations of barometric pressure are not available at each well
Water well
A water well is an excavation or structure created in the ground by digging, driving, boring or drilling to access groundwater in underground aquifers. The well water is drawn by an electric submersible pump, a trash pump, a vertical turbine pump, a handpump or a mechanical pump...
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
Atmospheric pressure
Atmospheric pressure is the force per unit area exerted into a surface by the weight of air above that surface in the atmosphere of Earth . In most circumstances atmospheric pressure is closely approximated by the hydrostatic pressure caused by the weight of air above the measurement point...
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
Blaise Pascal
Blaise Pascal , was a French mathematician, physicist, inventor, writer and Catholic philosopher. He was a child prodigy who was educated by his father, a tax collector in Rouen...
first qualitatively observed these effects in the 17th century, and they were more rigorously described by the soil physicist
Soil physics
Soil physics is the study of soil physical properties and processes. It is applied to management and prediction under natural and managed ecosystems. Soil physics deals with the dynamics of physical soil components and their phases as solids, liquids, and gases. It draws on the principles of...
Edgar Buckingham (working for the USDA
United States Department of Agriculture
The United States Department of Agriculture is the United States federal executive department responsible for developing and executing U.S. federal government policy on farming, agriculture, and food...
) using air flow models in 1907.
Head loss
In any real moving fluid, energy is dissipated due to frictionFriction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...
; turbulence
Turbulence
In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time...
dissipates even more energy for high Reynolds number flows. Head loss is divided into two main categories, "major losses" associated with energy loss per length of pipe, and "minor losses" associated with bends, fittings, valves, etc. The most common equation used to calculate major head losses is the Darcy–Weisbach equation. Older, more empirical approaches are the Hazen-Williams equation
Hazen-Williams equation
The Hazen–Williams equation is an empirical formula which relates the flow of water in a pipe with the physical properties of the pipe and the pressure drop caused by friction. It is used in the design of water pipe systems such as fire sprinkler systems, water supply networks, and irrigation systems...
and the Prony equation
Prony equation
The Prony equation is a historically important equation in hydraulics, used to calculate the head loss due to friction within a given run of pipe...
.
For relatively short pipe systems, with a relatively large number of bends and fittings, minor losses can easily exceed major losses. In design, minor losses are usually estimated from tables using coefficients or a simpler and less accurate reduction of minor losses to equivalent length of pipe.
Analogs to other fields
Hydraulic head is a measure of energy, and has many analogs in physicsPhysics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and chemistry
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
, where the same mathematical principles and rules apply:
- Hydraulic head is analogous to:
- magnetic monopoleMagnetic monopoleA magnetic monopole is a hypothetical particle in particle physics that is a magnet with only one magnetic pole . In more technical terms, a magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring...
- electric chargeElectric chargeElectric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
- heatHeatIn physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
(i.e., temperatureTemperatureTemperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
) - concentrationConcentrationIn chemistry, concentration is defined as the abundance of a constituent divided by the total volume of a mixture. Four types can be distinguished: mass concentration, molar concentration, number concentration, and volume concentration...
- magnetic monopole
- A continuous "fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
" of hydraulic heads is analogous to:- an electric fieldElectric fieldIn physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
- a magnetic fieldMagnetic fieldA magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
- an electric field
- Similar differential operatorDifferential operatorIn mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
s can be applied to the fields, to find:- the gradientGradientIn vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
, or the direction of flow - the divergenceDivergenceIn vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
of flow - the curl, or if the field is rotating
- the gradient
See also
- Borda–Carnot equationBorda–Carnot equationIn fluid dynamics the Borda–Carnot equation is an empirical description of the mechanical energy losses of the fluid due to a flow expansion. It describes how the total head reduces due to the losses. This in contrast with Bernoulli's principle for dissipationless flow , where the total head is a...
- Total dynamic head