{"id":1127,"date":"2026-05-12T21:46:19","date_gmt":"2026-05-12T21:46:19","guid":{"rendered":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/permeability\/"},"modified":"2026-05-13T03:08:26","modified_gmt":"2026-05-13T03:08:26","slug":"permeability","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/permeability\/","title":{"raw":"3.3 Permeability","rendered":"3.3 Permeability"},"content":{"raw":"<div id=\"permeability\" class=\"level1\">\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nPorosity is an important parameter in hydrogeology; however, it is not the only quantity that we need to understand to characterize groundwater systems. Different materials transmit fluids with varying degrees of difficulty. The ease with which a porous medium transmits fluid is called the <em>permeability<\/em>.\r\n\r\n<\/div>\r\n<div id=\"darcys-law\" class=\"level3\">\r\n<h3>Darcy\u2019s Law<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nHenry Darcy (1856) experimented with fluid flow through a pipe filled with sand (<a href=\"#Xa031578b577e0b9d17f6c9ceb7bd82e94c5ab4f\"><span data-custom-style=\"Hyperlink\">Figure\u00a014<\/span><\/a>a) as part of a larger project to design filtration systems for the public fountains of Dijon, France. He forced water through the pipe under a hydraulic head\u2014that is, he filled up a reservoir at a higher elevation than the pipe and allowed gravity to drive water through the sand\u2014then he measured:\r\n\r\n<\/div>\r\n<ul>\r\n \t<li>the rate of discharge through the pipe ([latex]Q[\/latex]); and<\/li>\r\n \t<li>the loss in head ([latex]dh[\/latex]) across some length of the pipe ([latex]dl[\/latex]).<\/li>\r\n<\/ul>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nHe concluded that discharge of water through the pipe is directly proportional to the energy loss and inversely proportional to the length of the path of flow (<a href=\"#Xa031578b577e0b9d17f6c9ceb7bd82e94c5ab4f\"><span data-custom-style=\"Hyperlink\">Figure\u00a014<\/span><\/a>b). Or, if we call the loss in head over a length the hydraulic gradient, he found that the discharge is directly proportional to the hydraulic gradient. This statement is expressed mathematically as Equation\u00a0<a href=\"#Xb28a07bc251c0d3f6ecf18f8974031da56ba18f\"><span data-custom-style=\"Hyperlink\">(22)<\/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]\\frac{Q}{A} \\propto \\frac{dh}{dl}[\/latex]<\/th>\r\n<th style=\"text-align: right;\"><span id=\"Xb28a07bc251c0d3f6ecf18f8974031da56ba18f\" class=\"anchor\">(22)<\/span><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n<div class=\"bc-figure figure\" data-custom-style=\"Captioned Figure\"><img style=\"display: block; margin-left: auto; margin-right: auto; ;width: 4.95556in; height: 5.22396in;\" src=\"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-content\/uploads\/sites\/42\/2026\/05\/file16-2.png\" alt=\"A diagram of a cylinder with a line graph and a diagram of a line graph Description automatically generated with medium confidence\" \/>\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=\"Xa031578b577e0b9d17f6c9ceb7bd82e94c5ab4f\" class=\"anchor\"><strong>Figure\u00a014\u00a0\u2011\u00a0<\/strong>a) Darcy tube; b) graph showing experimental results.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"what-does-this-mean\" class=\"level3\">\r\n<h3>What Does This Mean?<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nNotice that the data fall on a relatively straight line when graphed, meaning that the proportionality is linear. Whenever you have a relationship where one quantity is linearly proportional to another, the slope of the line allows you to set the two quantities equal to each other. In mathematical terms, this slope is a <em>constant of proportionality<\/em> or a <em>coefficient of proportionality<\/em>. In this specific equation, we call the constant the <em>hydraulic conductivity<\/em> ([latex]K[\/latex]). Inserting [latex]K[\/latex] and rearranging the equation to solve for <em>Q<\/em> gives us Darcy\u2019s law, as expressed by Equation\u00a0<a href=\"#Xede9d2642fb26d393fd30ecfb014f1bc1ce3ec8\"><span data-custom-style=\"Hyperlink\">(23)<\/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]Q = - KA\\frac{dh}{dl}[\/latex]<\/th>\r\n<th style=\"text-align: right;\"><span id=\"Xede9d2642fb26d393fd30ecfb014f1bc1ce3ec8\" class=\"anchor\">(23)<\/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]Q[\/latex]<\/th>\r\n<th style=\"text-align: center;\">=<\/th>\r\n<th>volumetric discharge (L<sup>3<\/sup>T<sup>-1<\/sup>)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\">[latex]K[\/latex]<\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>hydraulic conductivity (LT<sup>-1<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\">[latex]A[\/latex]<\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>area perpendicular to flow (L<sup>2<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\">[latex]dh[\/latex]<\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>difference in head (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\">[latex]dl[\/latex]<\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>distance over which <em>dh<\/em> occurs (L)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nIf we divide through by the cross\u2011sectional area, the volumetric discharge reduces to a 1\u2011dimensional discharge (i.e., velocity) called the <em>Darcian velocity<\/em> ([latex]q[\/latex]), as shown by Equation\u00a0<a href=\"#X1d8b6e7794a178121074700ced4156e31784a26\"><span data-custom-style=\"Hyperlink\">(24)<\/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]q = - K\\frac{dh}{dl}[\/latex]<\/th>\r\n<th style=\"text-align: right;\"><span id=\"X1d8b6e7794a178121074700ced4156e31784a26\" class=\"anchor\">(24)<\/span><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nDarcian velocity is also called the <em>specific discharge<\/em>. This value should not be considered to represent the actual velocity of groundwater flow as not all of the cross-sectional area perpendicular to flow is actually open and available for fluid flow. This is discussed in more detail in Section\u00a03.3.7.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nWhen we deal with aquifers, we often use a quantity called the transmissivity ([latex]T[\/latex]), which is defined as the hydraulic conductivity multiplied by the saturated thickness of the aquifer ([latex]b[\/latex]), as expressed by Equation\u00a0<a href=\"#X728c121492f6de6c2462cc0cbf7ffae1f487ba5\"><span data-custom-style=\"Hyperlink\">(25)<\/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]T = Kb[\/latex]<\/th>\r\n<th style=\"text-align: right;\"><span id=\"X728c121492f6de6c2462cc0cbf7ffae1f487ba5\" class=\"anchor\">(25)<\/span><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThe importance of this value will become evident in later sections of this book; for now, it is useful to be aware of the definition of the term.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nIn summary, hydraulic conductivity is a quantity that characterizes the ease with which water moves through a porous medium. Darcy\u2019s study indicated that different porous media (i.e., different types of sand with different grain size distributions) have different [latex]K[\/latex] values, indicating that each media has a distinct value of [latex]K[\/latex]. It is important to realize, however, that this relationship is for water specifically; if we are studying some other fluid\u2014petroleum, high\u2011salinity brine, molasses, and so on\u2014we need to use a different coefficient.\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"permeability-for-other-fluids\" class=\"level3\">\r\n<h3>Permeability for Other Fluids<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nThe ease with which a fluid moves through a porous medium is dependent on:\r\n\r\n<\/div>\r\n<ul>\r\n \t<li>properties of the media (size of pores and connectivity); and<\/li>\r\n \t<li>properties of the fluid (density and viscosity).<\/li>\r\n<\/ul>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nThe hydraulic conductivity, as measured and characterized by Darcy\u2019s law, is for water. This quantity not only characterizes the properties of the media, but it also contains the properties of the fluid moving through that media. Because of this, a hydraulic conductivity obtained for a sand by using water as the fluid in Darcy\u2018s apparatus cannot be used to calculate the flow rate of a different fluid\u2014for example, motor oil or mercury. This is not a big deal in most hydrogeology problems, since we generally deal with water in near\u2011surface environments, but it does concern us if we are dealing with multiple phases, or types of fluids\u2014for example, oil and water\u2014in an aquifer. To handle such multi\u2011phase flow problems we use a quantity called the <em>intrinsic permeability<\/em> ([latex]k).[\/latex] This value is a quantity that characterizes the ease with which any fluid flows through a porous medium, and only characterizes the porous medium itself. Equation\u00a0<a href=\"#X0cffb24992882ed7313af3ca8df2826ef77e52d\"><span data-custom-style=\"Hyperlink\">(26)<\/span><\/a> shows how intrinsic permeability can be used in Darcy\u2019s law.\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]Q = k\\frac{\\rho_{fluid}\\ g}{\\mu_{fluid}}[\/latex]<\/th>\r\n<th style=\"text-align: right;\"><span id=\"X0cffb24992882ed7313af3ca8df2826ef77e52d\" class=\"anchor\">(26)<\/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]k[\/latex]<\/th>\r\n<th style=\"text-align: center;\">=<\/th>\r\n<th>intrinsic permeability (L<sup>2<\/sup>)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\">[latex]\\rho_{fluid}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>density of the fluid (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;\">=<\/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]\\mu_{fluid}[\/latex]<\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>viscosity of the fluid (ML<sup>-1<\/sup>T<sup>-1<\/sup>)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThis equation requires that the user specify the properties of the fluid\u2014that is, density and viscosity. Otherwise, it is the same as Darcy\u2019s law. The hydraulic conductivity and the intrinsic permeability are related as shown by Equation\u00a0<a href=\"#X42f7bcc205e3e44bab32107c3d82f23f3d7acf1\"><span data-custom-style=\"Hyperlink\">(27)<\/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]k = K\\frac{\\mu_{fluid}}{\\rho_{fluid}\\ g}[\/latex]<\/th>\r\n<th style=\"text-align: right;\"><span id=\"X42f7bcc205e3e44bab32107c3d82f23f3d7acf1\" class=\"anchor\">(27)<\/span><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n<\/div>\r\n<div id=\"other-famous-equations-analogous-to-darcys-law\" class=\"level3\">\r\n<h3>Other Famous Equations Analogous to Darcy\u2019s Law<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nThe basic ideas conveyed by Darcy\u2019s law are:\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\na) discharge is driven by a gradient;\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nb) discharge is linearly proportional to a gradient; and\r\n\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>c) there is a constant of proportionality that characterizes the ease with which water flows through the system (a <em>conductivity<\/em> parameter).<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nThroughout the natural sciences we find numerous other physical processes that are mathematically identical to groundwater flow in that they have the same three properties. Examples including:\r\n\r\n<\/div>\r\n<ul>\r\n \t<li>heat flow (Fourier\u2019s Law);<\/li>\r\n \t<li>electrical flow (Ohm\u2019s Law); and<\/li>\r\n \t<li>diffusive flow of solutes (Fick\u2019s Law).<\/li>\r\n<\/ul>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nEach one of these equations has a discharge rate, a gradient or potential, and a constant of proportionality that represents conductivity. The importance of this is that we can use solutions from these other sciences and apply them to groundwater flow problems. This is discussed using a specific example in the section on Well Hydraulics (Section\u00a07).\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"the-limits-of-darcys-law\" class=\"level3\">\r\n<h3>The Limits of Darcy\u2019s Law<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nAlthough Darcy\u2019s law is regarded as a fundamental relationship in the Earth sciences, it has limitations, and there is a range of conditions over which it is valid.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nUpper limit\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nOne of the assumptions inherent in Darcy\u2019s law is that the flow is laminar. Darcy\u2019s law is not valid above the onset of turbulence in the system because the discharge is no longer linearly proportional to the gradient.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nLower limit\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThe lower limit of Darcy\u2019s law occurs at the <em>threshold gradient<\/em>, where fluid viscosity is too strong for the gradient to overcome the resistance to flow.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nApplicability\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nDarcy\u2019s law is still widely used because\r\n\r\n<\/div>\r\n<ol type=\"1\">\r\n \t<li>groundwater systems rarely become turbulent, and<\/li>\r\n \t<li>at gradients near the threshold gradient, flow is generally so small that it can be ignored.<\/li>\r\n<\/ol>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\n<em><strong>Thought Question 4<\/strong>: In what geologic situations might we be concerned about groundwater flow exceeding the limits of Darcy\u2019s law?<\/em>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"measuring-permeability\" class=\"level3\">\r\n<h3>Measuring Permeability<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nUnderstanding permeability is one thing; measuring it is another matter. Unfortunately, no one has developed a simple permeability meter with a digital display that you can point at a rock or sediment and instantly measure the permeability. Instead, we have a variety of ways of measuring permeability, both in the field and in the laboratory. These methods generally involve forcing fluid through a sample of the material, measuring the head loss and discharge rate, and calculating permeability from the measurements.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\n<em>Permeameters<\/em> or <em>Darcy tubes<\/em> are like the apparatus shown in in <a href=\"#Xa031578b577e0b9d17f6c9ceb7bd82e94c5ab4f\"><span data-custom-style=\"Hyperlink\">Figure\u00a014<\/span><\/a>a. The procedure involves filling the tube with a sample, running water through the sample under a constant head in the reservoir, reading the values of head loss (<em>dh<\/em>) and discharge (<em>Q<\/em>), then using those measurements and Darcy\u2019s law to calculate hydraulic conductivity. Permeameters are generally used for unconsolidated (i.e., loose) material like sand or gravel that can be packed into a column or minimally disturbed cores of the material. Some permeameters are designed to measure permeability in small cores taken from consolidated (i.e., solid) rock.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\n<em>Gas permeameters<\/em> are like regular permeameters except they use a gas (usually nitrogen) as the fluid instead of water. These can be portable for field use or set up in a laboratory. Gas permeameters require additional mathematical adjustments to account for the compressibility of air.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\n<em>Well\/piezometer tests<\/em>, also known as pumping tests or hydraulic tests, are tests performed in the field on wells or piezometers. These tests generally involve pumping water at a known rate while simultaneously monitoring water levels in the pumping well and, if possible, any nearby wells. Wells can also be tested by displacing a known volume of water from the well and monitoring the recovery of the water levels over time; this is known as a slug test. More information on pumping and slug testing can be found in <a href=\"https:\/\/gw-project.org\/books\/an-introduction-to-hydraulic-testing-in-hydrogeology-pumping-slug-and-packer-methods\/\"><span data-custom-style=\"Hyperlink\"><u><em>An Introduction to Hydraulic Testing in Hydrogeology: Basic Pumping, Slug, and Packer Method<\/em>s<\/u>\u2197<\/span><\/a> by Woessner and others (2023), which is available as a free download from The Groundwater Project. Well testing is discussed in more detail in Section\u00a08.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nNumerous researchers have developed empirical equations to calculate permeability\u2014that is, equations expressing relationships based on experimental data, as opposed to equations derived from first principles. The empirical equations are based on observed correlations between permeability and properties of the rock, primarily grain size distributions, average grain size, or <em>effective<\/em> grain size. In general, larger grain size is correlated with higher permeability; however, the relationship is not that simple. As indicated by <a href=\"\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/porosity\/#X02d64f01094029903069370a435dbdfd18ce577\"><span data-custom-style=\"Hyperlink\">Figure\u00a013<\/span><\/a>c and <a href=\"\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/porosity\/#X02d64f01094029903069370a435dbdfd18ce577\"><span data-custom-style=\"Hyperlink\">Figure\u00a013<\/span><\/a>d, the distribution of grain sizes can be just as significant as overall size, with well\u2011sorted sediments\u2014that is, very little variation in grain size\u2014correlated with greater permeability.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nPermeability can also be estimated using inverse methods, which involve using data collected in the field\u2014for example, water levels in wells, groundwater flow rates from discharge to springs and streams, and dimensions of the aquifer\u2014to estimate, or \u201cback out\u201d, permeability values using numerical models and automated model calibration techniques.\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fluid-velocity\" class=\"level3\">\r\n<h3>Fluid Velocity<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nWe have stated that dividing the Darcy equation by the cross\u2011sectional area reduces the volumetric discharge ([latex]Q[\/latex]) down to a one\u2011dimensional discharge ([latex]q[\/latex]) that we call the specific discharge or Darcian velocity. In reality, however, groundwater is not flowing through the entire cross\u2011sectional area; it is only flowing through the pore spaces, which make up a percentage of the total area. We must therefore modify the specific discharge by dividing it by the effective porosity to get a value called the average linear velocity ([latex]v[\/latex]), which is a more accurate characterization of the groundwater velocity. Calculation of specific discharge is shown by Equation\u00a0<a href=\"#Xcc725fa3d2c9dd21d195d523315b1349c755865\"><span data-custom-style=\"Hyperlink\">(28)<\/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]v = \\frac{q}{\\varphi_{eff}}[\/latex]<\/th>\r\n<th style=\"text-align: right;\"><span id=\"Xcc725fa3d2c9dd21d195d523315b1349c755865\" class=\"anchor\">(28)<\/span><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nIt is common for practicing hydrogeologists to use estimates of hydraulic conductivity, effective porosity, and hydraulic gradient to calculate an average linear velocity and then use that velocity to develop estimates about travel times of contaminants in the aquifers that they study. These calculated travel times are then used to develop plans for remediation (i.e., cleaning up) of groundwater contamination at sites that have been impacted by leaking tanks or accidental releases of contaminants. Several decades of experience in assessing and remediating groundwater contamination has shown that travel time estimates based on the average linear velocity are almost always much longer than observed travel times, which means that contaminants generally move much faster than the average linear velocity estimates. <span id=\"extext05\" class=\"anchor\"><a href=\"\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/exercise-5\/#exercise-5\"><span data-custom-style=\"Hyperlink\">Exercise\u00a05<\/span><\/a>\u2ba7\u00a0provides an opportunity to make calculations related to groundwater velocity.<\/span>\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThese observed discrepancies are likely associated with the fact that the true velocity of the groundwater varies from point to point in an aquifer depending on the size of the pores, the tortuosity of the flow paths, and constriction at the pore throats. If one were able to measure flow velocities at many points in an ideal aquifer, one would see very fast velocities at a few locations, very slow to even stagnant water in other locations, with a wide range of velocities in between. Plotting a frequency histogram of these velocities would likely show us a normal distribution centered around the average linear velocity calculated by Equation <a href=\"#Xcc725fa3d2c9dd21d195d523315b1349c755865\"><span data-custom-style=\"Hyperlink\">(28)<\/span><\/a>. However, given the observation that contaminants arrive sooner than expected, we can conclude it is common for contaminants to move along paths with higher velocities. Thus, we need to be cautious about using the average linear velocity to estimate travel times. A discussion of alternative methods for developing meaningful estimates of groundwater velocity is beyond the scope of this book. Readers are encouraged to review <a href=\"https:\/\/gw-project.org\/books\/groundwater-velocity\/\"><span data-custom-style=\"Hyperlink\"><em><u>Groundwater Velocity<\/u><\/em><\/span><\/a>\u2197 by Devlin (2020), available from The Groundwater Project as a free download or can be read online.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\n<em><strong>Thought Question 5<\/strong>: How is it that the variations in actual groundwater velocity can result in faster than expected arrival of contaminants at a potential receptor (i.e., a downgradient water well) versus estimates derived from calculating the average linear velocity?<\/em>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"permeability-distributions-in-aquifers\" class=\"level3\">\r\n<h3>Permeability Distributions in Aquifers<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nAs mentioned, the definitions of scalar, vector, and tensor quantities in mathematics are:\r\n\r\n<\/div>\r\n<ul>\r\n \t<li>scalar quantities have magnitude only;<\/li>\r\n \t<li>vector quantities have magnitude and direction; and<\/li>\r\n \t<li>tensor quantities have magnitude that varies with respect to direction.<\/li>\r\n<\/ul>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nPermeability is a tensor quantity. Not only does it vary from place to place, but it also varies with respect to direction. We can use two vectors to graphically represent a permeability tensor and to describe the variations in permeability throughout a system.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nWhen we talk about variations of permeability in an aquifer, we use the terms heterogeneity and anisotropy. <em>Heterogeneity<\/em> is variation in a parameter with respect to <em>location<\/em> within a system. <em>Anisotropy<\/em> is variation in a parameter varies with respect to <em>direction<\/em>.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\n<a href=\"#Xa109fe8b5fde3400548453167085dae4bec0de9\"><span data-custom-style=\"Hyperlink\">Figure\u00a015<\/span><\/a> graphically shows the relationship between these two terms. It is important not to use these terms interchangeably\u2014each has a specific meaning.\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: 3.83869in; height: 2.42708in;\" src=\"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-content\/uploads\/sites\/42\/2026\/05\/file17-2.jpg\" alt=\"A diagram of different types of arrows 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=\"Xa109fe8b5fde3400548453167085dae4bec0de9\" class=\"anchor\"><strong>Figure\u00a015\u00a0\u2011\u00a0<\/strong>Graphical representation of the relationship between heterogeneity and anisotropy.<\/span>\r\n\r\n<\/div><\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nIt should be noted that <em>all<\/em> aquifers and groundwater systems have some degree of heterogeneity with respect to permeability. It is essentially impossible to determine the actual distribution of permeabilities in an aquifer, so it is common for practicing hydrogeologists to assume that the system can be approximated as a homogeneous system with a single permeability or a heterogeneous system with a small number of homogenous permeability zones. Anisotropy in aquifer systems is generally associated with the natural layering of geologic formations such that permeability in the vertical direction is generally less than that in the horizontal direction. Anisotropy is also associated with systematic fractures (e.g., fault zones, joint sets) and is also associated with certain depositional environments (e.g., fluvial). More information on the impact of fractures on groundwater flow can be found in <a href=\"https:\/\/gw-project.org\/books\/fractures-and-faults-in-sandstone-and-sandstone-shale-mudstone-sequences-and-their-impact-on-groundwater\/\"><span data-custom-style=\"Hyperlink\"><em><u>Fractures and Faults in Sandstone and Sandstone\u2011Shale\/Mudstone Sequences and Their Impact on Groundwater<\/u><\/em><\/span><\/a>\u2197 by Aydin and others (2023), section 5.6 of <a href=\"https:\/\/gw-project.org\/books\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/\"><span data-custom-style=\"Hyperlink\"><em>Hydrogeologic Properties of Earth Materials and Principles of Groundwater Flow<\/em><\/span><\/a>\u2197 by Woessner and Poeter (2020), and <a href=\"https:\/\/gw-project.org\/books\/structural-geology-applied-to-fractured-aquifer-characterization\/\"><span data-custom-style=\"Hyperlink\"><em><u>Structural Geology Applied to Fractured Aquifer Characterization<\/u><\/em><\/span><\/a>\u2197 by Fernandes and others (2023). Each book is available as a free download from The Groundwater Project.\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"statistical-distribution\" class=\"level3\">\r\n<h3>Statistical Distribution<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nPermeability is a quantity that varies over 19 orders of magnitude! When we are studying an aquifer and we measure many permeabilities in that aquifer, our measurements typically cover several orders of magnitude. When multiple measured aquifer permeability values are available it is common to take an average of those values and use that average to perform calculations or develop models. Since permeability generally ranges across several orders of magnitude, we must be careful when averaging permeabilities to get a single value to characterize an aquifer.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nIn mathematics there are several ways of calculating a mean (i.e., average) for a set of data. The relevant methods for us are listed here.\r\n\r\n<\/div>\r\n<ul>\r\n \t<li><em>Arithmetic mean:<\/em> Add up the values and divide by the number of measurements.<\/li>\r\n<\/ul>\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]\\frac{\\sum_{i = 1}^{n}\\left( k_{i} \\right)}{n}[\/latex]<\/th>\r\n<th style=\"text-align: right;\">(29)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n<ul>\r\n \t<li><em>Geometric mean:<\/em> Add up the logs of each value and divide by number of measurements and raise ten to that power.<\/li>\r\n<\/ul>\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>[latex]10^{\\left( \\frac{\\sum_{i = 1}^{n}\\left( \\log\\left( k_{i} \\right) \\right)}{n} \\right)}[\/latex]<\/th>\r\n<th style=\"text-align: right;\">(30)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n<ul>\r\n \t<li><em>Harmonic mean:<\/em> Divide the number of measurements by the sum of the reciprocal of each measurement.<\/li>\r\n<\/ul>\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>[latex]\\frac{n}{\\sum_{i = 1}^{n}\\left( \\frac{1}{k_{i}} \\right)}[\/latex]<\/th>\r\n<th style=\"text-align: right;\">(31)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nWe are generally used to dealing with quantities that follow normal distributions, in which the values are distributed around an arithmetic mean value. Examples of normally distributed quantities include the heights of people in a population, the grades of students in a large class, and variations in air temperatures over time at a single location. Permeability is a log\u2011normally distributed value (<a href=\"#X0b7a36742abcc21f647ea9f2dcc4c161732ed37\"><span data-custom-style=\"Hyperlink\">Figure\u00a016<\/span><\/a>). A log\u2011normal distribution means that the logarithm of permeability is distributed around a mean, and if we calculate an arithmetic mean for log-normally distributed data we are likely over\u2011estimating the value that could serve as a representative value for the entire formation. The implication is that we should not use the arithmetic mean of multiple measured permeability values to characterize the permeability of an aquifer; instead, it is more appropriate to use a geometric mean\u2014that is, average the logarithms rather than the actual values.\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: 4.2294in; height: 3.58441in;\" src=\"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-content\/uploads\/sites\/42\/2026\/05\/file18-2.png\" alt=\"A diagram of normal distribution 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=\"X0b7a36742abcc21f647ea9f2dcc4c161732ed37\" class=\"anchor\"><strong>Figure\u00a016\u00a0\u2011\u00a0<\/strong>Normal versus log\u2011normal distribution.<\/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\nA more detailed discussion of the properties of porous media and the relationships between those properties and groundwater flow can be found in <a href=\"https:\/\/gw-project.org\/books\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/\"><span data-custom-style=\"Hyperlink\"><em>Hydrogeologic Properties of Earth Materials and Principles of Groundwater Flow<\/em><\/span><\/a>\u2197 by Woessner and Poeter (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=\"permeability\" class=\"level1\">\n<div data-custom-style=\"First Paragraph\">\n<p>Porosity is an important parameter in hydrogeology; however, it is not the only quantity that we need to understand to characterize groundwater systems. Different materials transmit fluids with varying degrees of difficulty. The ease with which a porous medium transmits fluid is called the <em>permeability<\/em>.<\/p>\n<\/div>\n<div id=\"darcys-law\" class=\"level3\">\n<h3>Darcy\u2019s Law<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>Henry Darcy (1856) experimented with fluid flow through a pipe filled with sand (<a href=\"#Xa031578b577e0b9d17f6c9ceb7bd82e94c5ab4f\"><span data-custom-style=\"Hyperlink\">Figure\u00a014<\/span><\/a>a) as part of a larger project to design filtration systems for the public fountains of Dijon, France. He forced water through the pipe under a hydraulic head\u2014that is, he filled up a reservoir at a higher elevation than the pipe and allowed gravity to drive water through the sand\u2014then he measured:<\/p>\n<\/div>\n<ul>\n<li>the rate of discharge through the pipe ([latex]Q[\/latex]); and<\/li>\n<li>the loss in head ([latex]dh[\/latex]) across some length of the pipe ([latex]dl[\/latex]).<\/li>\n<\/ul>\n<div data-custom-style=\"First Paragraph\">\n<p>He concluded that discharge of water through the pipe is directly proportional to the energy loss and inversely proportional to the length of the path of flow (<a href=\"#Xa031578b577e0b9d17f6c9ceb7bd82e94c5ab4f\"><span data-custom-style=\"Hyperlink\">Figure\u00a014<\/span><\/a>b). Or, if we call the loss in head over a length the hydraulic gradient, he found that the discharge is directly proportional to the hydraulic gradient. This statement is expressed mathematically as Equation\u00a0<a href=\"#Xb28a07bc251c0d3f6ecf18f8974031da56ba18f\"><span data-custom-style=\"Hyperlink\">(22)<\/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]\\frac{Q}{A} \\propto \\frac{dh}{dl}[\/latex]<\/th>\n<th style=\"text-align: right;\"><span id=\"Xb28a07bc251c0d3f6ecf18f8974031da56ba18f\" class=\"anchor\">(22)<\/span><\/th>\n<\/tr>\n<\/thead>\n<\/table>\n<div class=\"bc-figure figure\" data-custom-style=\"Captioned Figure\"><img decoding=\"async\" style=\"display: block; margin-left: auto; margin-right: auto; ;width: 4.95556in; height: 5.22396in;\" src=\"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-content\/uploads\/sites\/42\/2026\/05\/file16-2.png\" alt=\"A diagram of a cylinder with a line graph and a diagram of a line graph Description automatically generated with medium confidence\" \/><\/p>\n<div class=\"bc-figcaption figcaption\">\n<div data-custom-style=\"Image Caption\">\n<div class=\"figcaption-text\">\n<p><span id=\"Xa031578b577e0b9d17f6c9ceb7bd82e94c5ab4f\" class=\"anchor\"><strong>Figure\u00a014\u00a0\u2011\u00a0<\/strong>a) Darcy tube; b) graph showing experimental results.<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"what-does-this-mean\" class=\"level3\">\n<h3>What Does This Mean?<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>Notice that the data fall on a relatively straight line when graphed, meaning that the proportionality is linear. Whenever you have a relationship where one quantity is linearly proportional to another, the slope of the line allows you to set the two quantities equal to each other. In mathematical terms, this slope is a <em>constant of proportionality<\/em> or a <em>coefficient of proportionality<\/em>. In this specific equation, we call the constant the <em>hydraulic conductivity<\/em> ([latex]K[\/latex]). Inserting [latex]K[\/latex] and rearranging the equation to solve for <em>Q<\/em> gives us Darcy\u2019s law, as expressed by Equation\u00a0<a href=\"#Xede9d2642fb26d393fd30ecfb014f1bc1ce3ec8\"><span data-custom-style=\"Hyperlink\">(23)<\/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]Q = - KA\\frac{dh}{dl}[\/latex]<\/th>\n<th style=\"text-align: right;\"><span id=\"Xede9d2642fb26d393fd30ecfb014f1bc1ce3ec8\" class=\"anchor\">(23)<\/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]Q[\/latex]<\/th>\n<th style=\"text-align: center;\">=<\/th>\n<th>volumetric discharge (L<sup>3<\/sup>T<sup>-1<\/sup>)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">[latex]K[\/latex]<\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>hydraulic conductivity (LT<sup>-1<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\">[latex]A[\/latex]<\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>area perpendicular to flow (L<sup>2<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\">[latex]dh[\/latex]<\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>difference in head (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\">[latex]dl[\/latex]<\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>distance over which <em>dh<\/em> occurs (L)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>If we divide through by the cross\u2011sectional area, the volumetric discharge reduces to a 1\u2011dimensional discharge (i.e., velocity) called the <em>Darcian velocity<\/em> ([latex]q[\/latex]), as shown by Equation\u00a0<a href=\"#X1d8b6e7794a178121074700ced4156e31784a26\"><span data-custom-style=\"Hyperlink\">(24)<\/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]q = - K\\frac{dh}{dl}[\/latex]<\/th>\n<th style=\"text-align: right;\"><span id=\"X1d8b6e7794a178121074700ced4156e31784a26\" class=\"anchor\">(24)<\/span><\/th>\n<\/tr>\n<\/thead>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>Darcian velocity is also called the <em>specific discharge<\/em>. This value should not be considered to represent the actual velocity of groundwater flow as not all of the cross-sectional area perpendicular to flow is actually open and available for fluid flow. This is discussed in more detail in Section\u00a03.3.7.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>When we deal with aquifers, we often use a quantity called the transmissivity ([latex]T[\/latex]), which is defined as the hydraulic conductivity multiplied by the saturated thickness of the aquifer ([latex]b[\/latex]), as expressed by Equation\u00a0<a href=\"#X728c121492f6de6c2462cc0cbf7ffae1f487ba5\"><span data-custom-style=\"Hyperlink\">(25)<\/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]T = Kb[\/latex]<\/th>\n<th style=\"text-align: right;\"><span id=\"X728c121492f6de6c2462cc0cbf7ffae1f487ba5\" class=\"anchor\">(25)<\/span><\/th>\n<\/tr>\n<\/thead>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>The importance of this value will become evident in later sections of this book; for now, it is useful to be aware of the definition of the term.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>In summary, hydraulic conductivity is a quantity that characterizes the ease with which water moves through a porous medium. Darcy\u2019s study indicated that different porous media (i.e., different types of sand with different grain size distributions) have different [latex]K[\/latex] values, indicating that each media has a distinct value of [latex]K[\/latex]. It is important to realize, however, that this relationship is for water specifically; if we are studying some other fluid\u2014petroleum, high\u2011salinity brine, molasses, and so on\u2014we need to use a different coefficient.<\/p>\n<\/div>\n<\/div>\n<div id=\"permeability-for-other-fluids\" class=\"level3\">\n<h3>Permeability for Other Fluids<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>The ease with which a fluid moves through a porous medium is dependent on:<\/p>\n<\/div>\n<ul>\n<li>properties of the media (size of pores and connectivity); and<\/li>\n<li>properties of the fluid (density and viscosity).<\/li>\n<\/ul>\n<div data-custom-style=\"First Paragraph\">\n<p>The hydraulic conductivity, as measured and characterized by Darcy\u2019s law, is for water. This quantity not only characterizes the properties of the media, but it also contains the properties of the fluid moving through that media. Because of this, a hydraulic conductivity obtained for a sand by using water as the fluid in Darcy\u2018s apparatus cannot be used to calculate the flow rate of a different fluid\u2014for example, motor oil or mercury. This is not a big deal in most hydrogeology problems, since we generally deal with water in near\u2011surface environments, but it does concern us if we are dealing with multiple phases, or types of fluids\u2014for example, oil and water\u2014in an aquifer. To handle such multi\u2011phase flow problems we use a quantity called the <em>intrinsic permeability<\/em> ([latex]k).[\/latex] This value is a quantity that characterizes the ease with which any fluid flows through a porous medium, and only characterizes the porous medium itself. Equation\u00a0<a href=\"#X0cffb24992882ed7313af3ca8df2826ef77e52d\"><span data-custom-style=\"Hyperlink\">(26)<\/span><\/a> shows how intrinsic permeability can be used in Darcy\u2019s law.<\/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]Q = k\\frac{\\rho_{fluid}\\ g}{\\mu_{fluid}}[\/latex]<\/th>\n<th style=\"text-align: right;\"><span id=\"X0cffb24992882ed7313af3ca8df2826ef77e52d\" class=\"anchor\">(26)<\/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]k[\/latex]<\/th>\n<th style=\"text-align: center;\">=<\/th>\n<th>intrinsic permeability (L<sup>2<\/sup>)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">[latex]\\rho_{fluid}[\/latex]<\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>density of the fluid (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;\">=<\/td>\n<td>gravitational acceleration (LT<sup>-2<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\">[latex]\\mu_{fluid}[\/latex]<\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>viscosity of the fluid (ML<sup>-1<\/sup>T<sup>-1<\/sup>)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>This equation requires that the user specify the properties of the fluid\u2014that is, density and viscosity. Otherwise, it is the same as Darcy\u2019s law. The hydraulic conductivity and the intrinsic permeability are related as shown by Equation\u00a0<a href=\"#X42f7bcc205e3e44bab32107c3d82f23f3d7acf1\"><span data-custom-style=\"Hyperlink\">(27)<\/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]k = K\\frac{\\mu_{fluid}}{\\rho_{fluid}\\ g}[\/latex]<\/th>\n<th style=\"text-align: right;\"><span id=\"X42f7bcc205e3e44bab32107c3d82f23f3d7acf1\" class=\"anchor\">(27)<\/span><\/th>\n<\/tr>\n<\/thead>\n<\/table>\n<\/div>\n<div id=\"other-famous-equations-analogous-to-darcys-law\" class=\"level3\">\n<h3>Other Famous Equations Analogous to Darcy\u2019s Law<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>The basic ideas conveyed by Darcy\u2019s law are:<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>a) discharge is driven by a gradient;<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>b) discharge is linearly proportional to a gradient; and<\/p>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>c) there is a constant of proportionality that characterizes the ease with which water flows through the system (a <em>conductivity<\/em> parameter).<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"First Paragraph\">\n<p>Throughout the natural sciences we find numerous other physical processes that are mathematically identical to groundwater flow in that they have the same three properties. Examples including:<\/p>\n<\/div>\n<ul>\n<li>heat flow (Fourier\u2019s Law);<\/li>\n<li>electrical flow (Ohm\u2019s Law); and<\/li>\n<li>diffusive flow of solutes (Fick\u2019s Law).<\/li>\n<\/ul>\n<div data-custom-style=\"First Paragraph\">\n<p>Each one of these equations has a discharge rate, a gradient or potential, and a constant of proportionality that represents conductivity. The importance of this is that we can use solutions from these other sciences and apply them to groundwater flow problems. This is discussed using a specific example in the section on Well Hydraulics (Section\u00a07).<\/p>\n<\/div>\n<\/div>\n<div id=\"the-limits-of-darcys-law\" class=\"level3\">\n<h3>The Limits of Darcy\u2019s Law<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>Although Darcy\u2019s law is regarded as a fundamental relationship in the Earth sciences, it has limitations, and there is a range of conditions over which it is valid.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Upper limit<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>One of the assumptions inherent in Darcy\u2019s law is that the flow is laminar. Darcy\u2019s law is not valid above the onset of turbulence in the system because the discharge is no longer linearly proportional to the gradient.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Lower limit<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>The lower limit of Darcy\u2019s law occurs at the <em>threshold gradient<\/em>, where fluid viscosity is too strong for the gradient to overcome the resistance to flow.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Applicability<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Darcy\u2019s law is still widely used because<\/p>\n<\/div>\n<ol type=\"1\">\n<li>groundwater systems rarely become turbulent, and<\/li>\n<li>at gradients near the threshold gradient, flow is generally so small that it can be ignored.<\/li>\n<\/ol>\n<div data-custom-style=\"First Paragraph\">\n<p><em><strong>Thought Question 4<\/strong>: In what geologic situations might we be concerned about groundwater flow exceeding the limits of Darcy\u2019s law?<\/em><\/p>\n<\/div>\n<\/div>\n<div id=\"measuring-permeability\" class=\"level3\">\n<h3>Measuring Permeability<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>Understanding permeability is one thing; measuring it is another matter. Unfortunately, no one has developed a simple permeability meter with a digital display that you can point at a rock or sediment and instantly measure the permeability. Instead, we have a variety of ways of measuring permeability, both in the field and in the laboratory. These methods generally involve forcing fluid through a sample of the material, measuring the head loss and discharge rate, and calculating permeability from the measurements.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p><em>Permeameters<\/em> or <em>Darcy tubes<\/em> are like the apparatus shown in in <a href=\"#Xa031578b577e0b9d17f6c9ceb7bd82e94c5ab4f\"><span data-custom-style=\"Hyperlink\">Figure\u00a014<\/span><\/a>a. The procedure involves filling the tube with a sample, running water through the sample under a constant head in the reservoir, reading the values of head loss (<em>dh<\/em>) and discharge (<em>Q<\/em>), then using those measurements and Darcy\u2019s law to calculate hydraulic conductivity. Permeameters are generally used for unconsolidated (i.e., loose) material like sand or gravel that can be packed into a column or minimally disturbed cores of the material. Some permeameters are designed to measure permeability in small cores taken from consolidated (i.e., solid) rock.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p><em>Gas permeameters<\/em> are like regular permeameters except they use a gas (usually nitrogen) as the fluid instead of water. These can be portable for field use or set up in a laboratory. Gas permeameters require additional mathematical adjustments to account for the compressibility of air.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p><em>Well\/piezometer tests<\/em>, also known as pumping tests or hydraulic tests, are tests performed in the field on wells or piezometers. These tests generally involve pumping water at a known rate while simultaneously monitoring water levels in the pumping well and, if possible, any nearby wells. Wells can also be tested by displacing a known volume of water from the well and monitoring the recovery of the water levels over time; this is known as a slug test. More information on pumping and slug testing can be found in <a href=\"https:\/\/gw-project.org\/books\/an-introduction-to-hydraulic-testing-in-hydrogeology-pumping-slug-and-packer-methods\/\"><span data-custom-style=\"Hyperlink\"><u><em>An Introduction to Hydraulic Testing in Hydrogeology: Basic Pumping, Slug, and Packer Method<\/em>s<\/u>\u2197<\/span><\/a> by Woessner and others (2023), which is available as a free download from The Groundwater Project. Well testing is discussed in more detail in Section\u00a08.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Numerous researchers have developed empirical equations to calculate permeability\u2014that is, equations expressing relationships based on experimental data, as opposed to equations derived from first principles. The empirical equations are based on observed correlations between permeability and properties of the rock, primarily grain size distributions, average grain size, or <em>effective<\/em> grain size. In general, larger grain size is correlated with higher permeability; however, the relationship is not that simple. As indicated by <a href=\"\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/porosity\/#X02d64f01094029903069370a435dbdfd18ce577\"><span data-custom-style=\"Hyperlink\">Figure\u00a013<\/span><\/a>c and <a href=\"\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/porosity\/#X02d64f01094029903069370a435dbdfd18ce577\"><span data-custom-style=\"Hyperlink\">Figure\u00a013<\/span><\/a>d, the distribution of grain sizes can be just as significant as overall size, with well\u2011sorted sediments\u2014that is, very little variation in grain size\u2014correlated with greater permeability.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Permeability can also be estimated using inverse methods, which involve using data collected in the field\u2014for example, water levels in wells, groundwater flow rates from discharge to springs and streams, and dimensions of the aquifer\u2014to estimate, or \u201cback out\u201d, permeability values using numerical models and automated model calibration techniques.<\/p>\n<\/div>\n<\/div>\n<div id=\"fluid-velocity\" class=\"level3\">\n<h3>Fluid Velocity<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>We have stated that dividing the Darcy equation by the cross\u2011sectional area reduces the volumetric discharge ([latex]Q[\/latex]) down to a one\u2011dimensional discharge ([latex]q[\/latex]) that we call the specific discharge or Darcian velocity. In reality, however, groundwater is not flowing through the entire cross\u2011sectional area; it is only flowing through the pore spaces, which make up a percentage of the total area. We must therefore modify the specific discharge by dividing it by the effective porosity to get a value called the average linear velocity ([latex]v[\/latex]), which is a more accurate characterization of the groundwater velocity. Calculation of specific discharge is shown by Equation\u00a0<a href=\"#Xcc725fa3d2c9dd21d195d523315b1349c755865\"><span data-custom-style=\"Hyperlink\">(28)<\/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]v = \\frac{q}{\\varphi_{eff}}[\/latex]<\/th>\n<th style=\"text-align: right;\"><span id=\"Xcc725fa3d2c9dd21d195d523315b1349c755865\" class=\"anchor\">(28)<\/span><\/th>\n<\/tr>\n<\/thead>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>It is common for practicing hydrogeologists to use estimates of hydraulic conductivity, effective porosity, and hydraulic gradient to calculate an average linear velocity and then use that velocity to develop estimates about travel times of contaminants in the aquifers that they study. These calculated travel times are then used to develop plans for remediation (i.e., cleaning up) of groundwater contamination at sites that have been impacted by leaking tanks or accidental releases of contaminants. Several decades of experience in assessing and remediating groundwater contamination has shown that travel time estimates based on the average linear velocity are almost always much longer than observed travel times, which means that contaminants generally move much faster than the average linear velocity estimates. <span id=\"extext05\" class=\"anchor\"><a href=\"\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/exercise-5\/#exercise-5\"><span data-custom-style=\"Hyperlink\">Exercise\u00a05<\/span><\/a>\u2ba7\u00a0provides an opportunity to make calculations related to groundwater velocity.<\/span><\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>These observed discrepancies are likely associated with the fact that the true velocity of the groundwater varies from point to point in an aquifer depending on the size of the pores, the tortuosity of the flow paths, and constriction at the pore throats. If one were able to measure flow velocities at many points in an ideal aquifer, one would see very fast velocities at a few locations, very slow to even stagnant water in other locations, with a wide range of velocities in between. Plotting a frequency histogram of these velocities would likely show us a normal distribution centered around the average linear velocity calculated by Equation <a href=\"#Xcc725fa3d2c9dd21d195d523315b1349c755865\"><span data-custom-style=\"Hyperlink\">(28)<\/span><\/a>. However, given the observation that contaminants arrive sooner than expected, we can conclude it is common for contaminants to move along paths with higher velocities. Thus, we need to be cautious about using the average linear velocity to estimate travel times. A discussion of alternative methods for developing meaningful estimates of groundwater velocity is beyond the scope of this book. Readers are encouraged to review <a href=\"https:\/\/gw-project.org\/books\/groundwater-velocity\/\"><span data-custom-style=\"Hyperlink\"><em><u>Groundwater Velocity<\/u><\/em><\/span><\/a>\u2197 by Devlin (2020), available from The Groundwater Project as a free download or can be read online.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p><em><strong>Thought Question 5<\/strong>: How is it that the variations in actual groundwater velocity can result in faster than expected arrival of contaminants at a potential receptor (i.e., a downgradient water well) versus estimates derived from calculating the average linear velocity?<\/em><\/p>\n<\/div>\n<\/div>\n<div id=\"permeability-distributions-in-aquifers\" class=\"level3\">\n<h3>Permeability Distributions in Aquifers<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>As mentioned, the definitions of scalar, vector, and tensor quantities in mathematics are:<\/p>\n<\/div>\n<ul>\n<li>scalar quantities have magnitude only;<\/li>\n<li>vector quantities have magnitude and direction; and<\/li>\n<li>tensor quantities have magnitude that varies with respect to direction.<\/li>\n<\/ul>\n<div data-custom-style=\"First Paragraph\">\n<p>Permeability is a tensor quantity. Not only does it vary from place to place, but it also varies with respect to direction. We can use two vectors to graphically represent a permeability tensor and to describe the variations in permeability throughout a system.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>When we talk about variations of permeability in an aquifer, we use the terms heterogeneity and anisotropy. <em>Heterogeneity<\/em> is variation in a parameter with respect to <em>location<\/em> within a system. <em>Anisotropy<\/em> is variation in a parameter varies with respect to <em>direction<\/em>.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p><a href=\"#Xa109fe8b5fde3400548453167085dae4bec0de9\"><span data-custom-style=\"Hyperlink\">Figure\u00a015<\/span><\/a> graphically shows the relationship between these two terms. It is important not to use these terms interchangeably\u2014each has a specific meaning.<\/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: 3.83869in; height: 2.42708in;\" src=\"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-content\/uploads\/sites\/42\/2026\/05\/file17-2.jpg\" alt=\"A diagram of different types of arrows Description automatically generated\" \/><\/p>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote>\n<div class=\"figcaption-text\">\n<p><span id=\"Xa109fe8b5fde3400548453167085dae4bec0de9\" class=\"anchor\"><strong>Figure\u00a015\u00a0\u2011\u00a0<\/strong>Graphical representation of the relationship between heterogeneity and anisotropy.<\/span><\/p>\n<\/div>\n<\/blockquote>\n<\/div>\n<div data-custom-style=\"First Paragraph\">\n<p>It should be noted that <em>all<\/em> aquifers and groundwater systems have some degree of heterogeneity with respect to permeability. It is essentially impossible to determine the actual distribution of permeabilities in an aquifer, so it is common for practicing hydrogeologists to assume that the system can be approximated as a homogeneous system with a single permeability or a heterogeneous system with a small number of homogenous permeability zones. Anisotropy in aquifer systems is generally associated with the natural layering of geologic formations such that permeability in the vertical direction is generally less than that in the horizontal direction. Anisotropy is also associated with systematic fractures (e.g., fault zones, joint sets) and is also associated with certain depositional environments (e.g., fluvial). More information on the impact of fractures on groundwater flow can be found in <a href=\"https:\/\/gw-project.org\/books\/fractures-and-faults-in-sandstone-and-sandstone-shale-mudstone-sequences-and-their-impact-on-groundwater\/\"><span data-custom-style=\"Hyperlink\"><em><u>Fractures and Faults in Sandstone and Sandstone\u2011Shale\/Mudstone Sequences and Their Impact on Groundwater<\/u><\/em><\/span><\/a>\u2197 by Aydin and others (2023), section 5.6 of <a href=\"https:\/\/gw-project.org\/books\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/\"><span data-custom-style=\"Hyperlink\"><em>Hydrogeologic Properties of Earth Materials and Principles of Groundwater Flow<\/em><\/span><\/a>\u2197 by Woessner and Poeter (2020), and <a href=\"https:\/\/gw-project.org\/books\/structural-geology-applied-to-fractured-aquifer-characterization\/\"><span data-custom-style=\"Hyperlink\"><em><u>Structural Geology Applied to Fractured Aquifer Characterization<\/u><\/em><\/span><\/a>\u2197 by Fernandes and others (2023). Each book is available as a free download from The Groundwater Project.<\/p>\n<\/div>\n<\/div>\n<div id=\"statistical-distribution\" class=\"level3\">\n<h3>Statistical Distribution<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>Permeability is a quantity that varies over 19 orders of magnitude! When we are studying an aquifer and we measure many permeabilities in that aquifer, our measurements typically cover several orders of magnitude. When multiple measured aquifer permeability values are available it is common to take an average of those values and use that average to perform calculations or develop models. Since permeability generally ranges across several orders of magnitude, we must be careful when averaging permeabilities to get a single value to characterize an aquifer.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>In mathematics there are several ways of calculating a mean (i.e., average) for a set of data. The relevant methods for us are listed here.<\/p>\n<\/div>\n<ul>\n<li><em>Arithmetic mean:<\/em> Add up the values and divide by the number of measurements.<\/li>\n<\/ul>\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]\\frac{\\sum_{i = 1}^{n}\\left( k_{i} \\right)}{n}[\/latex]<\/th>\n<th style=\"text-align: right;\">(29)<\/th>\n<\/tr>\n<\/thead>\n<\/table>\n<ul>\n<li><em>Geometric mean:<\/em> Add up the logs of each value and divide by number of measurements and raise ten to that power.<\/li>\n<\/ul>\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>[latex]10^{\\left( \\frac{\\sum_{i = 1}^{n}\\left( \\log\\left( k_{i} \\right) \\right)}{n} \\right)}[\/latex]<\/th>\n<th style=\"text-align: right;\">(30)<\/th>\n<\/tr>\n<\/thead>\n<\/table>\n<ul>\n<li><em>Harmonic mean:<\/em> Divide the number of measurements by the sum of the reciprocal of each measurement.<\/li>\n<\/ul>\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>[latex]\\frac{n}{\\sum_{i = 1}^{n}\\left( \\frac{1}{k_{i}} \\right)}[\/latex]<\/th>\n<th style=\"text-align: right;\">(31)<\/th>\n<\/tr>\n<\/thead>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>We are generally used to dealing with quantities that follow normal distributions, in which the values are distributed around an arithmetic mean value. Examples of normally distributed quantities include the heights of people in a population, the grades of students in a large class, and variations in air temperatures over time at a single location. Permeability is a log\u2011normally distributed value (<a href=\"#X0b7a36742abcc21f647ea9f2dcc4c161732ed37\"><span data-custom-style=\"Hyperlink\">Figure\u00a016<\/span><\/a>). A log\u2011normal distribution means that the logarithm of permeability is distributed around a mean, and if we calculate an arithmetic mean for log-normally distributed data we are likely over\u2011estimating the value that could serve as a representative value for the entire formation. The implication is that we should not use the arithmetic mean of multiple measured permeability values to characterize the permeability of an aquifer; instead, it is more appropriate to use a geometric mean\u2014that is, average the logarithms rather than the actual values.<\/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: 4.2294in; height: 3.58441in;\" src=\"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-content\/uploads\/sites\/42\/2026\/05\/file18-2.png\" alt=\"A diagram of normal distribution 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=\"X0b7a36742abcc21f647ea9f2dcc4c161732ed37\" class=\"anchor\"><strong>Figure\u00a016\u00a0\u2011\u00a0<\/strong>Normal versus log\u2011normal distribution.<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>A more detailed discussion of the properties of porous media and the relationships between those properties and groundwater flow can be found in <a href=\"https:\/\/gw-project.org\/books\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/\"><span data-custom-style=\"Hyperlink\"><em>Hydrogeologic Properties of Earth Materials and Principles of Groundwater Flow<\/em><\/span><\/a>\u2197 by Woessner and Poeter (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":9,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1127","chapter","type-chapter","status-publish","hentry"],"part":1473,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters\/1127","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\/1127\/revisions"}],"predecessor-version":[{"id":1714,"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters\/1127\/revisions\/1714"}],"part":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/parts\/1473"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters\/1127\/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=1127"}],"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=1127"},{"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=1127"},{"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=1127"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}