{"id":1116,"date":"2026-05-12T21:46:14","date_gmt":"2026-05-12T21:46:14","guid":{"rendered":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/fluid-energy\/"},"modified":"2026-05-13T03:07:41","modified_gmt":"2026-05-13T03:07:41","slug":"fluid-energy","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/fluid-energy\/","title":{"raw":"2.2 Fluid Energy","rendered":"2.2 Fluid Energy"},"content":{"raw":"<div id=\"fluid-energy\" class=\"level1\">\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nIn simple, everyday terms, we think of hydraulic head as an elevation. More specifically, hydraulic head is the elevation of water in a manometer in a pressurized water pipe (<a href=\"#X2e5db15e8fd9c6662e914995714d00e7791673e\"><span data-custom-style=\"Hyperlink\">Figure\u00a06<\/span><\/a>) or in a piezometer (<a href=\"#Xb6b77400bc96f88fcafbc832b988aa2b417b76b\"><span data-custom-style=\"Hyperlink\">Figure\u00a07<\/span><\/a>).\r\n\r\n<\/div>\r\n<ul>\r\n \t<li>A <em>manometer<\/em> is a vertical tube in a pressurized water pipe and is used to measure fluid pressure in the pipe.<\/li>\r\n \t<li>A <em>piezometer<\/em> is a vertical tube with an open or slotted interval\u2014usually called the screened interval or the well screen\u2014inserted into the ground and used to measure hydraulic head in an aquifer; it is basically a well that was constructed for the sole purpose of measuring groundwater levels.<\/li>\r\n<\/ul>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nThe water level in a piezometer is a manifestation of the total fluid energy in a groundwater system so measuring a water elevation in a well is a measurement of the energy present in the fluid at a certain point in a groundwater system\u2014that is, the interval of the piezometer that is open to the aquifer (<a href=\"#Xb6b77400bc96f88fcafbc832b988aa2b417b76b\"><span data-custom-style=\"Hyperlink\">Figure\u00a07<\/span><\/a>).\r\n\r\n<\/div>\r\n<div data-custom-style=\"Captioned Figure\">\r\n\r\n<img style=\"display: block; margin-left: auto; margin-right: auto; ;width: 3.84in; height: 2.34428in;\" src=\"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-content\/uploads\/sites\/42\/2026\/05\/file8-2.jpg\" alt=\"A diagram of a graph Description automatically generated\" \/>\r\n\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>\r\n<div class=\"figcaption-text\">\r\n\r\n<span id=\"X2e5db15e8fd9c6662e914995714d00e7791673e\" class=\"anchor\"><strong>Figure\u00a06\u00a0\u2011\u00a0<\/strong>Pipe of flowing water with manometers showing the loss of head along the flow path.<\/span>\r\n\r\n<\/div><\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Captioned Figure\">\r\n\r\n<img style=\"display: block; margin-left: auto; margin-right: auto; ;width: 3.64954in; height: 2.17443in;\" src=\"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-content\/uploads\/sites\/42\/2026\/05\/file9-2.jpg\" alt=\"\" \/>\r\n\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>\r\n<div class=\"figcaption-text\">\r\n\r\n<span id=\"Xb6b77400bc96f88fcafbc832b988aa2b417b76b\" class=\"anchor\"><strong>Figure\u00a07\u00a0-\u00a0<\/strong>Cross\u2011section of aquifer showing hydraulic heads in three piezometers. The line with an inverted triangle represents the depth to which water rises within each piezometer.<\/span>\r\n\r\n<\/div><\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nThe fluid energy in an aquifer varies throughout the aquifer and is different from one point to the next. This results in different water levels in piezometers located at different points throughout the aquifer, and these variations in water level represent variations in fluid energy that in turn drive the movement of water throughout the aquifer. As a result, an aquifer is a dynamic system of flowing water.\r\n\r\n<\/div>\r\n<div id=\"what-do-we-mean-by-energy\" class=\"level3\">\r\n<h3>What do we mean by <em>Energy<\/em>?<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nEverything in the universe has some amount of energy associated with it, and that energy is present in various forms. Variations in energy are what drive all natural processes. The key to understanding physical processes is in understanding the distribution of energy in a system. We differentiate between potential energy and kinetic energy.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\n<em>Potential energy<\/em> is energy stored in a piece of matter or at a point in a system. It is generally associated with position within a system (measured as an elevation or distance above a datum) or with the thermodynamics of the system (measured as pressure, temperature, or chemical potential).\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\n<em>Kinetic energy<\/em> is energy associated with motion and is generally measured as a velocity.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nAt every point in an aquifer, the fluid possesses some total amount of energy that is the <em>sum<\/em> of all the potential and kinetic energies in the fluid. The fluid energy at a point in an aquifer manifests itself as the water level in a piezometer. We can also say that the water level, or hydraulic head, represents the total energy in the aquifer at a given point, and we can use the various energy components of the hydraulic head\u2014elevation, pressure, velocity, and so on\u2014to understand the driving forces behind fluid motion in the subsurface.\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"the-bernoulli-equation\" class=\"level3\">\r\n<h3>The Bernoulli Equation<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nFluid energies\u2014and, subsequently, water levels\u2014vary from one point in an aquifer to the next. Let us consider the first and second laws of thermodynamics. The first law of thermodynamics states that energy is conserved in any system; that is, it cannot be created or destroyed, and any changes in energy must be accounted for in any system as shown by Equation\u00a0<a href=\"#Xe9be407ce20ceef189f44d3e8740924bb09e1d6\"><span data-custom-style=\"Hyperlink\">(3)<\/span><\/a>.\r\n\r\n<\/div>\r\n<table style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 8%;\" \/> <col style=\"width: 64%;\" \/> <col style=\"width: 11%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th style=\"text-align: center;\">energy added<em>\u2212<\/em> energy subtracted\u00a0= change in total energy<\/th>\r\n<th style=\"text-align: right;\"><span id=\"Xe9be407ce20ceef189f44d3e8740924bb09e1d6\" class=\"anchor\">(3)<\/span><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThis is the same as the conservation of mass, Equation\u00a0<a href=\"\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/the-hydrologic-cycle\/#Xa2cfc3a788a424ca34d0dd91adbeea74369a84a\"><span data-custom-style=\"Hyperlink\">(1)<\/span><\/a>, but stated for energy in a system. We could also express the first law in terms of the difference in energy at two points in a dynamic system as described by Equation\u00a0<a href=\"#Xbd9446ef147e84a634faebd8d490ca966e84d2c\"><span data-custom-style=\"Hyperlink\">(4)<\/span><\/a>.\r\n\r\n<\/div>\r\n<table style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 6%;\" \/> <col style=\"width: 75%;\" \/> <col style=\"width: 3%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th>[latex]{total\\ energy}_{(at\\ point\\ 1)} + {energy\\ added\\ or\\ lost}_{(betwen\\ point\\ 1\\ and\\ 2)} = {total\\ energy}_{(at\\ point\\ 2)}[\/latex]<\/th>\r\n<th style=\"text-align: right;\"><span id=\"Xbd9446ef147e84a634faebd8d490ca966e84d2c\" class=\"anchor\">(4)<\/span><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThe second law of thermodynamics states that isolated systems tend to move towards increasing entropy. In a dynamic system like an aquifer, water moves from a point of higher energy (i.e., lower entropy) to a point of lower energy (higher entropy); in other words, groundwater spontaneously moves in the direction of decreasing hydraulic head. It is important to recognize that groundwater does NOT (necessarily) flow downhill relative to the land surface\u2014it flows in the direction of decreasing hydraulic head.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThe <em>Bernoulli equation<\/em> shown in Equation\u00a0<a href=\"#Xc58887b3a2241092004ebba159f2318883cd7b5\"><span data-custom-style=\"Hyperlink\">(5)<\/span><\/a> describes the total energy of a fluid at all positions along a flow path in a closed system and is basically an expression of the first and second laws of thermodynamics.\r\n\r\n<\/div>\r\n<table style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 8%;\" \/> <col style=\"width: 64%;\" \/> <col style=\"width: 11%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th style=\"text-align: center;\">[latex]z_{1} + \\frac{p_{1}}{\\rho_{w}g} + \\frac{v_{1}^{2}}{2g} + H_{a} = z_{2} + \\frac{p_{2}}{\\rho_{w}g} + \\frac{v_{2}^{2}}{2g} + H_{L} + H_{E}[\/latex]<\/th>\r\n<th style=\"text-align: right;\"><span id=\"Xc58887b3a2241092004ebba159f2318883cd7b5\" class=\"anchor\">(5)<\/span><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nwhere (parameter dimensions are dark green font with mass as M, length as L, time as T):\r\n\r\n<\/div>\r\n<table style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 12%;\" \/> <col style=\"width: 5%;\" \/> <col style=\"width: 67%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\">[latex]z[\/latex]<\/th>\r\n<th style=\"text-align: center;\">[latex]=[\/latex]<\/th>\r\n<th>elevation (L)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\">[latex]p[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\r\n<td>pressure (ML<sup>-1<\/sup>T<sup>-2<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\">[latex]\\rho_{w}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\r\n<td>fluid density (ML<sup>-3<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\">[latex]g[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\r\n<td>gravitational acceleration (LT<sup>-2<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\">[latex]v[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\r\n<td>velocity (LT<sup>-1<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\">[latex]H_{a}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\r\n<td>heat energy added (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\">[latex]H_{L}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\r\n<td>mechanical energy lost (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\">[latex]H_{E}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\r\n<td>heat energy extracted (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\">subscripts<\/td>\r\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\r\n<td>represent two different positions along the flow path<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nIf we assume that no heat energy is added to or subtracted from the system, and that the only change is the loss of mechanical energy from one point to the next, we can rewrite the equation as Equation\u00a0<a href=\"#X07a408df356fbbe4c2fde8dcf9a7e50066c073e\"><span data-custom-style=\"Hyperlink\">(6)<\/span><\/a>.\r\n\r\n<\/div>\r\n<table style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 8%;\" \/> <col style=\"width: 64%;\" \/> <col style=\"width: 11%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th style=\"text-align: center;\">[latex]z_{1} + \\frac{p_{1}}{\\rho_{w}g} + \\frac{v_{1}^{2}}{2g} + I_{1} = z_{2} + \\frac{p_{2}}{\\rho_{w}g} + \\frac{v_{2}^{2}}{2g} + I_{2}[\/latex]<\/th>\r\n<th style=\"text-align: right;\"><span id=\"X07a408df356fbbe4c2fde8dcf9a7e50066c073e\" class=\"anchor\">(6)<\/span><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nWhere <em>I<\/em> is the internal energy at each point (i.e., the remainder of the potential energy not described by the other terms) and [latex]I_{2}\\ - \\ I_{1}[\/latex] is equal to [latex]H_{L}[\/latex] in Equation\u00a0<a href=\"#Xc58887b3a2241092004ebba159f2318883cd7b5\"><span data-custom-style=\"Hyperlink\">(5)<\/span><\/a>.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nWe can understand this equation by considering each individual term in the equation as representing a specific component of energy\u2014that is:\r\n\r\n<\/div>\r\n<ul>\r\n \t<li>elevation,<\/li>\r\n \t<li>pressure,<\/li>\r\n \t<li>kinetic (velocity), and<\/li>\r\n \t<li>other internal energies (thermal, chemical).<\/li>\r\n<\/ul>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nThe equation therefore describes the change in energy from one point along a flow path to the next. The interesting thing about it to us as hydrogeologists is that, since it expresses all the components of energy in the same units (i.e., length), we can use it to compare the relative magnitude of the individual components. When we do that, we see that for most groundwater situations the flow velocity is slow so <em>we can ignore the internal and the kinetic (velocity) components because they are so much smaller than the elevation and pressure components<\/em>. We can also use the equation to mathematically \u201cconvert\u201d one form of energy to another\u2014for example, we can calculate how much a certain change in elevation will result in a change in pressure.\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"hydraulic-head-and-hydraulic-potential\" class=\"level3\">\r\n<h3>Hydraulic Head and Hydraulic Potential<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nIf we go with the assumption that we can ignore velocity and internal energy components when dealing with groundwater, we can drop all that out of the equation and express the fluid energy as the sum of the elevation and pressure components. That sum is what we call <em>hydraulic head<\/em>; in physical terms, it is the fluid energy per unit weight, and in mathematical terms, it is described by Equation\u00a0<a href=\"#X37fb933696958c1be29f36fb7fe03c3100b61bf\"><span data-custom-style=\"Hyperlink\">(7)<\/span><\/a>.\r\n\r\n<\/div>\r\n<table style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 8%;\" \/> <col style=\"width: 64%;\" \/> <col style=\"width: 11%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th style=\"text-align: center;\">[latex]h = z + \\frac{p}{\\rho_{w}g}[\/latex]<\/th>\r\n<th style=\"text-align: right;\"><span id=\"X37fb933696958c1be29f36fb7fe03c3100b61bf\" class=\"anchor\">(7)<\/span><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nwhere (parameter dimensions are dark green font with mass as M, length as L, time as T):\r\n\r\n<\/div>\r\n<table style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 10%;\" \/> <col style=\"width: 5%;\" \/> <col style=\"width: 68%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\">[latex]h[\/latex]<\/th>\r\n<th style=\"text-align: center;\">[latex]=[\/latex]<\/th>\r\n<th>hydraulic head (L)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\">[latex]z[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\r\n<td>elevation (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\">[latex]p[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\r\n<td>pressure (ML<sup>-1<\/sup>T<sup>-2<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\">[latex]\\rho_{w}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\r\n<td>fluid density (ML<sup>-3<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\">[latex]g[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\r\n<td>gravitational acceleration (LT<sup>-2<\/sup>)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nIf we multiply both sides of the equation by gravitational acceleration ([latex]g[\/latex]), we get a quantity called <em>hydraulic potential<\/em> ([latex]\\Phi[\/latex]; i.e., the Greek letter <em>phi<\/em>), which is the fluid energy per unit mass, as shown in Equation\u00a0<a href=\"#X9f656c84f65f537f1fd6da9dd4114c1ffd9f87b\"><span data-custom-style=\"Hyperlink\">(8)<\/span><\/a> and Equation\u00a0<a href=\"#X56a9b16d2635df4bbdb5403ef041e9750713872\"><span data-custom-style=\"Hyperlink\">(9)<\/span><\/a>.\r\n\r\n<\/div>\r\n<table style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 8%;\" \/> <col style=\"width: 64%;\" \/> <col style=\"width: 11%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th style=\"text-align: center;\">[latex]\\Phi = gz + \\frac{p}{\\rho_{w}}[\/latex]<\/th>\r\n<th style=\"text-align: right;\"><span id=\"X9f656c84f65f537f1fd6da9dd4114c1ffd9f87b\" class=\"anchor\">(8)<\/span><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nsuch that\r\n\r\n<\/div>\r\n<table style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 8%;\" \/> <col style=\"width: 64%;\" \/> <col style=\"width: 11%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th style=\"text-align: center;\">[latex]\\Phi = gh[\/latex]<\/th>\r\n<th style=\"text-align: right;\"><span id=\"X56a9b16d2635df4bbdb5403ef041e9750713872\" class=\"anchor\">(9)<\/span><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThe hydraulic potential is simply a way of expressing the same fluid energy so that it is independent of gravity (in case you would want to compare a typical aquifer on Earth with, say, the liquid methane aquifers on Titan). We do not deal with hydraulic potential any further in this book; hydraulic head is sufficient for the problems that are addressed in this book.\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"physical-description-of-the-components-of-head\" class=\"level3\">\r\n<h3>Physical Description of the Components of Head<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nFrom Equation\u00a0<a href=\"#X37fb933696958c1be29f36fb7fe03c3100b61bf\"><span data-custom-style=\"Hyperlink\">(7)<\/span><\/a>, we can see that hydraulic head is the sum of the elevation component (or <em>elevation head<\/em>) and the pressure component (or <em>pressure head<\/em>). The physical meaning of pressure head and elevation head are defined and described in <a href=\"#X0951004273bbcb92279dbe6dd69577840509d57\"><span data-custom-style=\"Hyperlink\">Figure\u00a08<\/span><\/a>. This figure shows an idealized cross\u2011section of a piezometer with a screened interval at the bottom. <a href=\"#X0951004273bbcb92279dbe6dd69577840509d57\"><span data-custom-style=\"Hyperlink\">Figure\u00a08<\/span><\/a>a shows the depths and elevations measured in the field. <a href=\"#X0951004273bbcb92279dbe6dd69577840509d57\"><span data-custom-style=\"Hyperlink\">Figure\u00a08<\/span><\/a>b shows the pressure, elevation, and total heads relative to a datum. In this case, the datum is mean sea level.\r\n\r\n<\/div>\r\n<div class=\"bc-figure figure\" data-custom-style=\"Captioned Figure\"><img style=\"display: block; margin-left: auto; margin-right: auto; ;width: 3.8in; height: 2.2in;\" src=\"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-content\/uploads\/sites\/42\/2026\/05\/file10-2.png\" alt=\"A diagram of a diagram of a ground surface Description automatically generated\" \/>\r\n<div class=\"bc-figcaption figcaption\">\r\n<div data-custom-style=\"Image Caption\">\r\n<div class=\"figcaption-text\">\r\n\r\n<span id=\"X0951004273bbcb92279dbe6dd69577840509d57\" class=\"anchor\"><strong>Figure\u00a08\u00a0\u2011\u00a0<\/strong>Hypothetical well showing depths, elevations, and heads.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nFrom <a href=\"#X0951004273bbcb92279dbe6dd69577840509d57\"><span data-custom-style=\"Hyperlink\">Figure\u00a08<\/span><\/a>b, we see that the elevation head is the height of the screened interval above the datum. The datum is arbitrarily chosen\u2014if we wanted, we could choose the middle of the screened interval as the datum, and the elevation head would be zero. However, it is important to realize that hydraulic head is only useful to us when we are looking at multiple wells in the same aquifer, and we need to have a constant datum for all those wells in order to make the comparison meaningful. Sea level is usually chosen, but it is not necessary that we choose that as the datum.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nWe see from <a href=\"#X0951004273bbcb92279dbe6dd69577840509d57\"><span data-custom-style=\"Hyperlink\">Figure\u00a08<\/span><\/a>b that the pressure head is the length of the column of water in the well above the screened interval. Keep in mind that <em>pressure<\/em> and <em>pressure head<\/em> are two different things. The <em>pressure<\/em> at the screened interval is the force per unit area of the column of fluid above that point, while the <em>pressure head<\/em> is the pressure divided by the product of the density of the fluid and gravitational acceleration and is manifest as the length of the column of water above the screen. If we know the density of the fluid and the length of the column of water in the well, we can calculate the pressure at the well screen.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nFinally, we see that the total head is just the sum of the other two heads or, more generally, the height of the water level in the well above the datum. This brings us back to the concept that the water level in a well is a direct measure of the fluid energy (<a href=\"\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/exercise-1\/#exercise-1\"><span id=\"extext01\" class=\"anchor\"><span data-custom-style=\"Hyperlink\">Exercise\u00a01<\/span><\/span><\/a>\u2ba7).\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"distribution-of-head-i.e.-fluid-energy-in-a-3d-aquifer\" class=\"level3\">\r\n<h3>Distribution of Head (i.e., fluid energy) in a 3\u2011D Aquifer<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nThere are three types of mathematical quantities:\r\n\r\n<\/div>\r\n<ul>\r\n \t<li><em>scalar<\/em> quantities have magnitude only (e.g., temperature);<\/li>\r\n \t<li><em>vector<\/em> quantities have both magnitude and direction (e.g., velocity); and,<\/li>\r\n \t<li><em>tensor<\/em> quantities have magnitude that varies with respect to direction (e.g., stress).<\/li>\r\n<\/ul>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nHead is a scalar quantity; it is measured at a point, and it has a single magnitude that doesn\u2019t vary with respect to direction. Therefore, when we talk about hydraulic head from a theoretical standpoint, we are talking about the fluid energy at an infinitesimal point in an aquifer. An aquifer, however, exists in three dimensions and contains an infinite number of points, and the head within an aquifer varies from point to point.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThe trends in that variation are what control the directions and magnitudes of groundwater flow in an aquifer. We call this variation the <em>hydraulic gradient<\/em>. The hydraulic gradient is defined as the change in hydraulic head over a change in length, and it is directly analogous to other physical gradients\u2014for example, topographic slope, thermal gradients, concentration gradients, and so on.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nSince hydraulic heads exist in three dimensions, hydraulic gradients are not necessarily horizontal. In reality, most groundwater flow is generally in a horizontal direction, and it is often a realistic assumption to ignore fluid movement in the vertical direction. However, there are many situations where vertical gradients\u2014and, subsequently, vertical flow\u2014are significant (<a href=\"\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/exercise-2\/#exercise-2\"><span data-custom-style=\"Hyperlink\">Exe<\/span><span id=\"extext02\" class=\"anchor\"><span data-custom-style=\"Hyperlink\">rcise\u00a02<\/span><\/span><\/a>\u2ba7).\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"why-do-we-care-about-this\" class=\"level3\">\r\n<h3>Why Do We Care About This?<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nThe important thing to realize here is that when we install a piezometer in an aquifer and measure a water level in that piezometer, we are measuring the fluid energy at a point (in three dimensions) that is located at the well screen. <a href=\"#X1b7fe8ac9e06d80ac02b7a581905adf4ca2479d\"><span data-custom-style=\"Hyperlink\">Figure\u00a09<\/span><\/a> shows three wells; wells b and c are completed at approximately the same depth, while well a is completed in a much shallower interval. The two deep wells indicate a gradient pointing to the right of the page; however, the water level in the shallow well\u2014located between the other two\u2014is much lower than either of the other two wells. This indicates that there is some sort of vertical gradient in the system. It could also indicate that the wells are completed in different geologic units that are not well connected. This is discussed further in later sections of this book (<a href=\"\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/exercise-3\/#exercise-3\"><span data-custom-style=\"Hyperlink\">Exerc<\/span><span id=\"extext03\" class=\"anchor\"><span data-custom-style=\"Hyperlink\">ise\u00a03<\/span><\/span><\/a>\u2ba7).\r\n\r\n<\/div>\r\n<div data-custom-style=\"Captioned Figure\">\r\n\r\n<img style=\"display: block; margin-left: auto; margin-right: auto; ;width: 4.69026in; height: 2.73256in;\" src=\"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-content\/uploads\/sites\/42\/2026\/05\/file11-2.jpg\" alt=\"\" \/>\r\n\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>\r\n<div class=\"figcaption-text\">\r\n\r\n<span id=\"X1b7fe8ac9e06d80ac02b7a581905adf4ca2479d\" class=\"anchor\"><strong>Figure\u00a09\u00a0\u2011\u00a0<\/strong>Cross\u2011section showing differences in head and their separation that can be used to calculate vertical and horizontal gradients. The large difference in water level of the shallow well (a) which is between wells (b) and (c) suggests there is a vertical gradient in the system, which could reflect that the wells are located in a zone of hydrodynamically-driven upward flow and\/or that the wells are completed in different geologic units that are not well connected.<\/span>\r\n\r\n<\/div><\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nMore information about hydraulic head and understanding the relationship between head and groundwater flow can be found in <a href=\"https:\/\/gw-project.org\/books\/conceptual-and-visual-understanding-of-hydraulic-head-and-groundwater-flow\/\"><span data-custom-style=\"Hyperlink\"><em><u>Conceptual and Visual Understanding of Hydraulic Head and Groundwater Flow<\/u><\/em><\/span><\/a>\u2197 by Cohen and Cherry (2020), which is available from The Groundwater Project as a free download or can be read online.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div id=\"fluid-energy\" class=\"level1\">\n<div data-custom-style=\"First Paragraph\">\n<p>In simple, everyday terms, we think of hydraulic head as an elevation. More specifically, hydraulic head is the elevation of water in a manometer in a pressurized water pipe (<a href=\"#X2e5db15e8fd9c6662e914995714d00e7791673e\"><span data-custom-style=\"Hyperlink\">Figure\u00a06<\/span><\/a>) or in a piezometer (<a href=\"#Xb6b77400bc96f88fcafbc832b988aa2b417b76b\"><span data-custom-style=\"Hyperlink\">Figure\u00a07<\/span><\/a>).<\/p>\n<\/div>\n<ul>\n<li>A <em>manometer<\/em> is a vertical tube in a pressurized water pipe and is used to measure fluid pressure in the pipe.<\/li>\n<li>A <em>piezometer<\/em> is a vertical tube with an open or slotted interval\u2014usually called the screened interval or the well screen\u2014inserted into the ground and used to measure hydraulic head in an aquifer; it is basically a well that was constructed for the sole purpose of measuring groundwater levels.<\/li>\n<\/ul>\n<div data-custom-style=\"First Paragraph\">\n<p>The water level in a piezometer is a manifestation of the total fluid energy in a groundwater system so measuring a water elevation in a well is a measurement of the energy present in the fluid at a certain point in a groundwater system\u2014that is, the interval of the piezometer that is open to the aquifer (<a href=\"#Xb6b77400bc96f88fcafbc832b988aa2b417b76b\"><span data-custom-style=\"Hyperlink\">Figure\u00a07<\/span><\/a>).<\/p>\n<\/div>\n<div data-custom-style=\"Captioned Figure\">\n<p><img decoding=\"async\" style=\"display: block; margin-left: auto; margin-right: auto; ;width: 3.84in; height: 2.34428in;\" src=\"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-content\/uploads\/sites\/42\/2026\/05\/file8-2.jpg\" alt=\"A diagram of a graph Description automatically generated\" \/><\/p>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote>\n<div class=\"figcaption-text\">\n<p><span id=\"X2e5db15e8fd9c6662e914995714d00e7791673e\" class=\"anchor\"><strong>Figure\u00a06\u00a0\u2011\u00a0<\/strong>Pipe of flowing water with manometers showing the loss of head along the flow path.<\/span><\/p>\n<\/div>\n<\/blockquote>\n<\/div>\n<div data-custom-style=\"Captioned Figure\">\n<p><img decoding=\"async\" style=\"display: block; margin-left: auto; margin-right: auto; ;width: 3.64954in; height: 2.17443in;\" src=\"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-content\/uploads\/sites\/42\/2026\/05\/file9-2.jpg\" alt=\"\" \/><\/p>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote>\n<div class=\"figcaption-text\">\n<p><span id=\"Xb6b77400bc96f88fcafbc832b988aa2b417b76b\" class=\"anchor\"><strong>Figure\u00a07\u00a0&#8211;\u00a0<\/strong>Cross\u2011section of aquifer showing hydraulic heads in three piezometers. The line with an inverted triangle represents the depth to which water rises within each piezometer.<\/span><\/p>\n<\/div>\n<\/blockquote>\n<\/div>\n<div data-custom-style=\"First Paragraph\">\n<p>The fluid energy in an aquifer varies throughout the aquifer and is different from one point to the next. This results in different water levels in piezometers located at different points throughout the aquifer, and these variations in water level represent variations in fluid energy that in turn drive the movement of water throughout the aquifer. As a result, an aquifer is a dynamic system of flowing water.<\/p>\n<\/div>\n<div id=\"what-do-we-mean-by-energy\" class=\"level3\">\n<h3>What do we mean by <em>Energy<\/em>?<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>Everything in the universe has some amount of energy associated with it, and that energy is present in various forms. Variations in energy are what drive all natural processes. The key to understanding physical processes is in understanding the distribution of energy in a system. We differentiate between potential energy and kinetic energy.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p><em>Potential energy<\/em> is energy stored in a piece of matter or at a point in a system. It is generally associated with position within a system (measured as an elevation or distance above a datum) or with the thermodynamics of the system (measured as pressure, temperature, or chemical potential).<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p><em>Kinetic energy<\/em> is energy associated with motion and is generally measured as a velocity.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>At every point in an aquifer, the fluid possesses some total amount of energy that is the <em>sum<\/em> of all the potential and kinetic energies in the fluid. The fluid energy at a point in an aquifer manifests itself as the water level in a piezometer. We can also say that the water level, or hydraulic head, represents the total energy in the aquifer at a given point, and we can use the various energy components of the hydraulic head\u2014elevation, pressure, velocity, and so on\u2014to understand the driving forces behind fluid motion in the subsurface.<\/p>\n<\/div>\n<\/div>\n<div id=\"the-bernoulli-equation\" class=\"level3\">\n<h3>The Bernoulli Equation<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>Fluid energies\u2014and, subsequently, water levels\u2014vary from one point in an aquifer to the next. Let us consider the first and second laws of thermodynamics. The first law of thermodynamics states that energy is conserved in any system; that is, it cannot be created or destroyed, and any changes in energy must be accounted for in any system as shown by Equation\u00a0<a href=\"#Xe9be407ce20ceef189f44d3e8740924bb09e1d6\"><span data-custom-style=\"Hyperlink\">(3)<\/span><\/a>.<\/p>\n<\/div>\n<table style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 8%;\" \/>\n<col style=\"width: 64%;\" \/>\n<col style=\"width: 11%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th><\/th>\n<th style=\"text-align: center;\">energy added<em>\u2212<\/em> energy subtracted\u00a0= change in total energy<\/th>\n<th style=\"text-align: right;\"><span id=\"Xe9be407ce20ceef189f44d3e8740924bb09e1d6\" class=\"anchor\">(3)<\/span><\/th>\n<\/tr>\n<\/thead>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>This is the same as the conservation of mass, Equation\u00a0<a href=\"\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/the-hydrologic-cycle\/#Xa2cfc3a788a424ca34d0dd91adbeea74369a84a\"><span data-custom-style=\"Hyperlink\">(1)<\/span><\/a>, but stated for energy in a system. We could also express the first law in terms of the difference in energy at two points in a dynamic system as described by Equation\u00a0<a href=\"#Xbd9446ef147e84a634faebd8d490ca966e84d2c\"><span data-custom-style=\"Hyperlink\">(4)<\/span><\/a>.<\/p>\n<\/div>\n<table style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 6%;\" \/>\n<col style=\"width: 75%;\" \/>\n<col style=\"width: 3%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th><\/th>\n<th>[latex]{total\\ energy}_{(at\\ point\\ 1)} + {energy\\ added\\ or\\ lost}_{(betwen\\ point\\ 1\\ and\\ 2)} = {total\\ energy}_{(at\\ point\\ 2)}[\/latex]<\/th>\n<th style=\"text-align: right;\"><span id=\"Xbd9446ef147e84a634faebd8d490ca966e84d2c\" class=\"anchor\">(4)<\/span><\/th>\n<\/tr>\n<\/thead>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>The second law of thermodynamics states that isolated systems tend to move towards increasing entropy. In a dynamic system like an aquifer, water moves from a point of higher energy (i.e., lower entropy) to a point of lower energy (higher entropy); in other words, groundwater spontaneously moves in the direction of decreasing hydraulic head. It is important to recognize that groundwater does NOT (necessarily) flow downhill relative to the land surface\u2014it flows in the direction of decreasing hydraulic head.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>The <em>Bernoulli equation<\/em> shown in Equation\u00a0<a href=\"#Xc58887b3a2241092004ebba159f2318883cd7b5\"><span data-custom-style=\"Hyperlink\">(5)<\/span><\/a> describes the total energy of a fluid at all positions along a flow path in a closed system and is basically an expression of the first and second laws of thermodynamics.<\/p>\n<\/div>\n<table style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 8%;\" \/>\n<col style=\"width: 64%;\" \/>\n<col style=\"width: 11%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th><\/th>\n<th style=\"text-align: center;\">[latex]z_{1} + \\frac{p_{1}}{\\rho_{w}g} + \\frac{v_{1}^{2}}{2g} + H_{a} = z_{2} + \\frac{p_{2}}{\\rho_{w}g} + \\frac{v_{2}^{2}}{2g} + H_{L} + H_{E}[\/latex]<\/th>\n<th style=\"text-align: right;\"><span id=\"Xc58887b3a2241092004ebba159f2318883cd7b5\" class=\"anchor\">(5)<\/span><\/th>\n<\/tr>\n<\/thead>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>where (parameter dimensions are dark green font with mass as M, length as L, time as T):<\/p>\n<\/div>\n<table style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 12%;\" \/>\n<col style=\"width: 5%;\" \/>\n<col style=\"width: 67%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\">[latex]z[\/latex]<\/th>\n<th style=\"text-align: center;\">[latex]=[\/latex]<\/th>\n<th>elevation (L)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">[latex]p[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\n<td>pressure (ML<sup>-1<\/sup>T<sup>-2<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\">[latex]\\rho_{w}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\n<td>fluid density (ML<sup>-3<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\">[latex]g[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\n<td>gravitational acceleration (LT<sup>-2<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\">[latex]v[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\n<td>velocity (LT<sup>-1<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\">[latex]H_{a}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\n<td>heat energy added (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\">[latex]H_{L}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\n<td>mechanical energy lost (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\">[latex]H_{E}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\n<td>heat energy extracted (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\">subscripts<\/td>\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\n<td>represent two different positions along the flow path<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>If we assume that no heat energy is added to or subtracted from the system, and that the only change is the loss of mechanical energy from one point to the next, we can rewrite the equation as Equation\u00a0<a href=\"#X07a408df356fbbe4c2fde8dcf9a7e50066c073e\"><span data-custom-style=\"Hyperlink\">(6)<\/span><\/a>.<\/p>\n<\/div>\n<table style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 8%;\" \/>\n<col style=\"width: 64%;\" \/>\n<col style=\"width: 11%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th><\/th>\n<th style=\"text-align: center;\">[latex]z_{1} + \\frac{p_{1}}{\\rho_{w}g} + \\frac{v_{1}^{2}}{2g} + I_{1} = z_{2} + \\frac{p_{2}}{\\rho_{w}g} + \\frac{v_{2}^{2}}{2g} + I_{2}[\/latex]<\/th>\n<th style=\"text-align: right;\"><span id=\"X07a408df356fbbe4c2fde8dcf9a7e50066c073e\" class=\"anchor\">(6)<\/span><\/th>\n<\/tr>\n<\/thead>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>Where <em>I<\/em> is the internal energy at each point (i.e., the remainder of the potential energy not described by the other terms) and [latex]I_{2}\\ - \\ I_{1}[\/latex] is equal to [latex]H_{L}[\/latex] in Equation\u00a0<a href=\"#Xc58887b3a2241092004ebba159f2318883cd7b5\"><span data-custom-style=\"Hyperlink\">(5)<\/span><\/a>.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>We can understand this equation by considering each individual term in the equation as representing a specific component of energy\u2014that is:<\/p>\n<\/div>\n<ul>\n<li>elevation,<\/li>\n<li>pressure,<\/li>\n<li>kinetic (velocity), and<\/li>\n<li>other internal energies (thermal, chemical).<\/li>\n<\/ul>\n<div data-custom-style=\"First Paragraph\">\n<p>The equation therefore describes the change in energy from one point along a flow path to the next. The interesting thing about it to us as hydrogeologists is that, since it expresses all the components of energy in the same units (i.e., length), we can use it to compare the relative magnitude of the individual components. When we do that, we see that for most groundwater situations the flow velocity is slow so <em>we can ignore the internal and the kinetic (velocity) components because they are so much smaller than the elevation and pressure components<\/em>. We can also use the equation to mathematically \u201cconvert\u201d one form of energy to another\u2014for example, we can calculate how much a certain change in elevation will result in a change in pressure.<\/p>\n<\/div>\n<\/div>\n<div id=\"hydraulic-head-and-hydraulic-potential\" class=\"level3\">\n<h3>Hydraulic Head and Hydraulic Potential<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>If we go with the assumption that we can ignore velocity and internal energy components when dealing with groundwater, we can drop all that out of the equation and express the fluid energy as the sum of the elevation and pressure components. That sum is what we call <em>hydraulic head<\/em>; in physical terms, it is the fluid energy per unit weight, and in mathematical terms, it is described by Equation\u00a0<a href=\"#X37fb933696958c1be29f36fb7fe03c3100b61bf\"><span data-custom-style=\"Hyperlink\">(7)<\/span><\/a>.<\/p>\n<\/div>\n<table style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 8%;\" \/>\n<col style=\"width: 64%;\" \/>\n<col style=\"width: 11%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th><\/th>\n<th style=\"text-align: center;\">[latex]h = z + \\frac{p}{\\rho_{w}g}[\/latex]<\/th>\n<th style=\"text-align: right;\"><span id=\"X37fb933696958c1be29f36fb7fe03c3100b61bf\" class=\"anchor\">(7)<\/span><\/th>\n<\/tr>\n<\/thead>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>where (parameter dimensions are dark green font with mass as M, length as L, time as T):<\/p>\n<\/div>\n<table style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 10%;\" \/>\n<col style=\"width: 5%;\" \/>\n<col style=\"width: 68%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\">[latex]h[\/latex]<\/th>\n<th style=\"text-align: center;\">[latex]=[\/latex]<\/th>\n<th>hydraulic head (L)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">[latex]z[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\n<td>elevation (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\">[latex]p[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\n<td>pressure (ML<sup>-1<\/sup>T<sup>-2<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\">[latex]\\rho_{w}[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\n<td>fluid density (ML<sup>-3<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\">[latex]g[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]=[\/latex]<\/td>\n<td>gravitational acceleration (LT<sup>-2<\/sup>)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>If we multiply both sides of the equation by gravitational acceleration ([latex]g[\/latex]), we get a quantity called <em>hydraulic potential<\/em> ([latex]\\Phi[\/latex]; i.e., the Greek letter <em>phi<\/em>), which is the fluid energy per unit mass, as shown in Equation\u00a0<a href=\"#X9f656c84f65f537f1fd6da9dd4114c1ffd9f87b\"><span data-custom-style=\"Hyperlink\">(8)<\/span><\/a> and Equation\u00a0<a href=\"#X56a9b16d2635df4bbdb5403ef041e9750713872\"><span data-custom-style=\"Hyperlink\">(9)<\/span><\/a>.<\/p>\n<\/div>\n<table style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 8%;\" \/>\n<col style=\"width: 64%;\" \/>\n<col style=\"width: 11%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th><\/th>\n<th style=\"text-align: center;\">[latex]\\Phi = gz + \\frac{p}{\\rho_{w}}[\/latex]<\/th>\n<th style=\"text-align: right;\"><span id=\"X9f656c84f65f537f1fd6da9dd4114c1ffd9f87b\" class=\"anchor\">(8)<\/span><\/th>\n<\/tr>\n<\/thead>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>such that<\/p>\n<\/div>\n<table style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 8%;\" \/>\n<col style=\"width: 64%;\" \/>\n<col style=\"width: 11%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th><\/th>\n<th style=\"text-align: center;\">[latex]\\Phi = gh[\/latex]<\/th>\n<th style=\"text-align: right;\"><span id=\"X56a9b16d2635df4bbdb5403ef041e9750713872\" class=\"anchor\">(9)<\/span><\/th>\n<\/tr>\n<\/thead>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>The hydraulic potential is simply a way of expressing the same fluid energy so that it is independent of gravity (in case you would want to compare a typical aquifer on Earth with, say, the liquid methane aquifers on Titan). We do not deal with hydraulic potential any further in this book; hydraulic head is sufficient for the problems that are addressed in this book.<\/p>\n<\/div>\n<\/div>\n<div id=\"physical-description-of-the-components-of-head\" class=\"level3\">\n<h3>Physical Description of the Components of Head<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>From Equation\u00a0<a href=\"#X37fb933696958c1be29f36fb7fe03c3100b61bf\"><span data-custom-style=\"Hyperlink\">(7)<\/span><\/a>, we can see that hydraulic head is the sum of the elevation component (or <em>elevation head<\/em>) and the pressure component (or <em>pressure head<\/em>). The physical meaning of pressure head and elevation head are defined and described in <a href=\"#X0951004273bbcb92279dbe6dd69577840509d57\"><span data-custom-style=\"Hyperlink\">Figure\u00a08<\/span><\/a>. This figure shows an idealized cross\u2011section of a piezometer with a screened interval at the bottom. <a href=\"#X0951004273bbcb92279dbe6dd69577840509d57\"><span data-custom-style=\"Hyperlink\">Figure\u00a08<\/span><\/a>a shows the depths and elevations measured in the field. <a href=\"#X0951004273bbcb92279dbe6dd69577840509d57\"><span data-custom-style=\"Hyperlink\">Figure\u00a08<\/span><\/a>b shows the pressure, elevation, and total heads relative to a datum. In this case, the datum is mean sea level.<\/p>\n<\/div>\n<div class=\"bc-figure figure\" data-custom-style=\"Captioned Figure\"><img decoding=\"async\" style=\"display: block; margin-left: auto; margin-right: auto; ;width: 3.8in; height: 2.2in;\" src=\"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-content\/uploads\/sites\/42\/2026\/05\/file10-2.png\" alt=\"A diagram of a diagram of a ground surface Description automatically generated\" \/><\/p>\n<div class=\"bc-figcaption figcaption\">\n<div data-custom-style=\"Image Caption\">\n<div class=\"figcaption-text\">\n<p><span id=\"X0951004273bbcb92279dbe6dd69577840509d57\" class=\"anchor\"><strong>Figure\u00a08\u00a0\u2011\u00a0<\/strong>Hypothetical well showing depths, elevations, and heads.<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>From <a href=\"#X0951004273bbcb92279dbe6dd69577840509d57\"><span data-custom-style=\"Hyperlink\">Figure\u00a08<\/span><\/a>b, we see that the elevation head is the height of the screened interval above the datum. The datum is arbitrarily chosen\u2014if we wanted, we could choose the middle of the screened interval as the datum, and the elevation head would be zero. However, it is important to realize that hydraulic head is only useful to us when we are looking at multiple wells in the same aquifer, and we need to have a constant datum for all those wells in order to make the comparison meaningful. Sea level is usually chosen, but it is not necessary that we choose that as the datum.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>We see from <a href=\"#X0951004273bbcb92279dbe6dd69577840509d57\"><span data-custom-style=\"Hyperlink\">Figure\u00a08<\/span><\/a>b that the pressure head is the length of the column of water in the well above the screened interval. Keep in mind that <em>pressure<\/em> and <em>pressure head<\/em> are two different things. The <em>pressure<\/em> at the screened interval is the force per unit area of the column of fluid above that point, while the <em>pressure head<\/em> is the pressure divided by the product of the density of the fluid and gravitational acceleration and is manifest as the length of the column of water above the screen. If we know the density of the fluid and the length of the column of water in the well, we can calculate the pressure at the well screen.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Finally, we see that the total head is just the sum of the other two heads or, more generally, the height of the water level in the well above the datum. This brings us back to the concept that the water level in a well is a direct measure of the fluid energy (<a href=\"\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/exercise-1\/#exercise-1\"><span id=\"extext01\" class=\"anchor\"><span data-custom-style=\"Hyperlink\">Exercise\u00a01<\/span><\/span><\/a>\u2ba7).<\/p>\n<\/div>\n<\/div>\n<div id=\"distribution-of-head-i.e.-fluid-energy-in-a-3d-aquifer\" class=\"level3\">\n<h3>Distribution of Head (i.e., fluid energy) in a 3\u2011D Aquifer<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>There are three types of mathematical quantities:<\/p>\n<\/div>\n<ul>\n<li><em>scalar<\/em> quantities have magnitude only (e.g., temperature);<\/li>\n<li><em>vector<\/em> quantities have both magnitude and direction (e.g., velocity); and,<\/li>\n<li><em>tensor<\/em> quantities have magnitude that varies with respect to direction (e.g., stress).<\/li>\n<\/ul>\n<div data-custom-style=\"First Paragraph\">\n<p>Head is a scalar quantity; it is measured at a point, and it has a single magnitude that doesn\u2019t vary with respect to direction. Therefore, when we talk about hydraulic head from a theoretical standpoint, we are talking about the fluid energy at an infinitesimal point in an aquifer. An aquifer, however, exists in three dimensions and contains an infinite number of points, and the head within an aquifer varies from point to point.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>The trends in that variation are what control the directions and magnitudes of groundwater flow in an aquifer. We call this variation the <em>hydraulic gradient<\/em>. The hydraulic gradient is defined as the change in hydraulic head over a change in length, and it is directly analogous to other physical gradients\u2014for example, topographic slope, thermal gradients, concentration gradients, and so on.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Since hydraulic heads exist in three dimensions, hydraulic gradients are not necessarily horizontal. In reality, most groundwater flow is generally in a horizontal direction, and it is often a realistic assumption to ignore fluid movement in the vertical direction. However, there are many situations where vertical gradients\u2014and, subsequently, vertical flow\u2014are significant (<a href=\"\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/exercise-2\/#exercise-2\"><span data-custom-style=\"Hyperlink\">Exe<\/span><span id=\"extext02\" class=\"anchor\"><span data-custom-style=\"Hyperlink\">rcise\u00a02<\/span><\/span><\/a>\u2ba7).<\/p>\n<\/div>\n<\/div>\n<div id=\"why-do-we-care-about-this\" class=\"level3\">\n<h3>Why Do We Care About This?<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>The important thing to realize here is that when we install a piezometer in an aquifer and measure a water level in that piezometer, we are measuring the fluid energy at a point (in three dimensions) that is located at the well screen. <a href=\"#X1b7fe8ac9e06d80ac02b7a581905adf4ca2479d\"><span data-custom-style=\"Hyperlink\">Figure\u00a09<\/span><\/a> shows three wells; wells b and c are completed at approximately the same depth, while well a is completed in a much shallower interval. The two deep wells indicate a gradient pointing to the right of the page; however, the water level in the shallow well\u2014located between the other two\u2014is much lower than either of the other two wells. This indicates that there is some sort of vertical gradient in the system. It could also indicate that the wells are completed in different geologic units that are not well connected. This is discussed further in later sections of this book (<a href=\"\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/exercise-3\/#exercise-3\"><span data-custom-style=\"Hyperlink\">Exerc<\/span><span id=\"extext03\" class=\"anchor\"><span data-custom-style=\"Hyperlink\">ise\u00a03<\/span><\/span><\/a>\u2ba7).<\/p>\n<\/div>\n<div data-custom-style=\"Captioned Figure\">\n<p><img decoding=\"async\" style=\"display: block; margin-left: auto; margin-right: auto; ;width: 4.69026in; height: 2.73256in;\" src=\"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-content\/uploads\/sites\/42\/2026\/05\/file11-2.jpg\" alt=\"\" \/><\/p>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote>\n<div class=\"figcaption-text\">\n<p><span id=\"X1b7fe8ac9e06d80ac02b7a581905adf4ca2479d\" class=\"anchor\"><strong>Figure\u00a09\u00a0\u2011\u00a0<\/strong>Cross\u2011section showing differences in head and their separation that can be used to calculate vertical and horizontal gradients. The large difference in water level of the shallow well (a) which is between wells (b) and (c) suggests there is a vertical gradient in the system, which could reflect that the wells are located in a zone of hydrodynamically-driven upward flow and\/or that the wells are completed in different geologic units that are not well connected.<\/span><\/p>\n<\/div>\n<\/blockquote>\n<\/div>\n<div data-custom-style=\"First Paragraph\">\n<p>More information about hydraulic head and understanding the relationship between head and groundwater flow can be found in <a href=\"https:\/\/gw-project.org\/books\/conceptual-and-visual-understanding-of-hydraulic-head-and-groundwater-flow\/\"><span data-custom-style=\"Hyperlink\"><em><u>Conceptual and Visual Understanding of Hydraulic Head and Groundwater Flow<\/u><\/em><\/span><\/a>\u2197 by Cohen and Cherry (2020), which is available from The Groundwater Project as a free download or can be read online.<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":8,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1116","chapter","type-chapter","status-publish","hentry"],"part":1470,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters\/1116","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":2,"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters\/1116\/revisions"}],"predecessor-version":[{"id":1711,"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters\/1116\/revisions\/1711"}],"part":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/parts\/1470"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters\/1116\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/wp\/v2\/media?parent=1116"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapter-type?post=1116"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/wp\/v2\/contributor?post=1116"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/wp\/v2\/license?post=1116"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}