{"id":340,"date":"2020-10-24T11:53:58","date_gmt":"2020-10-24T11:53:58","guid":{"rendered":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/?post_type=chapter&p=340"},"modified":"2020-12-29T16:43:07","modified_gmt":"2020-12-29T16:43:07","slug":"hydraulic-conductivity","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/chapter\/hydraulic-conductivity\/","title":{"raw":"4.4 Hydraulic Conductivity","rendered":"4.4 Hydraulic Conductivity"},"content":{"raw":"The hydraulic conductivity<\/em> proportionality constant, K<\/em>, can be conceptualized as the relative ease of fluid passage through a porous material. It has direction and magnitude and is represented as a vector; however, the first part of this discussion presents it as a scalar value. Rearranging Darcy\u2019s law to solve for hydraulic conductivity generates Equation 25.\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle K=-Q\\frac{\\Delta L}{\\Delta h\\ A}[\/latex]<\/td>\r\n(25)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn this configuration, it becomes clear that the units of K<\/em> are L\/T because Q<\/em> units are (L3<\/sup>\/T), A<\/em> units (L2<\/sup>), h<\/em> units are (L), and L<\/em> units are (L). The units are presented as Equation 26.\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\frac{\\textup{L}}{\\textup{T}} : \\frac{\\textup{L}^{3}}{\\textup{T}} \\frac{\\textup{L}}{\\textup{L} \\textup{L}^{2}}[\/latex]<\/td>\r\n(26)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThus, the constant of proportionality, K<\/em>, has units of velocity (e.g., meters\/seconds, meters\/day). However, K<\/em> is not a velocity, rather it represents the transmission properties of the porous material. If water easily passes through a porous material it is described as having a high hydraulic conductivity; if water is poorly transmitted through a material it has a low hydraulic conductivity. These conditions are also referred to as permeable or of low permeability, respectively.\r\n

Intrinsic Permeability<\/h1>\r\nFreeze and Cherry (1979) describe the results of hydraulic conductivity experiments used to explore the relationship between physical properties of the porous media and the fluid. A number of columns filled with different sizes of glass beads were set up as Darcy columns and changes in specific discharge were observed. Based on these observations, the relationship of the specific discharge to measurable characteristics of the porous media and were noted as shown in Equations 27, 28, and 29.\r\n\r\n\r\n\r\n
<\/td>\r\nq<\/em> \u221d d<\/em>2<\/sup><\/td>\r\n(27)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nwhere:\r\n\r\n\r\n\r\n\r\n
q<\/em><\/td>\r\n=<\/td>\r\nspecific discharge (L\/T)<\/td>\r\n<\/tr>\r\n
d<\/em><\/td>\r\n=<\/td>\r\ndiameter of uniform glass beads comprising the porous medium (L)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n
<\/td>\r\nq<\/em> \u221d \u03c1 g<\/em><\/td>\r\n(28)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nwhere:\r\n\r\n\r\n\r\n\r\n
\u03c1<\/em><\/td>\r\n=<\/td>\r\nfluid density (M\/L3<\/sup>)<\/td>\r\n<\/tr>\r\n
g<\/em><\/td>\r\n=<\/td>\r\ngravitational constant (acceleration of gravity) (L\/T2<\/sup>)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle q\\propto \\frac{1}{\\mu }[\/latex]<\/td>\r\n(29)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nwhere:\r\n\r\n\r\n\r\n
\u03bc<\/em><\/td>\r\n=<\/td>\r\ndynamic viscosity (M\/(LT))<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhen considered together with Darcy\u2019s original observation that q<\/em> \u221d \u2212dh<\/em>\/dl<\/em>, these three relationships lead to a definition of hydraulic conductivity that includes physical characteristics of the porous media and the influence of fluid properties (Equation 30).\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\frac{Q}{A}=-\\frac{\\left ( \\frac{Cd^{2}g\\rho }{\\mu } \\right )dh}{dl}[\/latex]<\/td>\r\n(30)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nwhere:\r\n\r\n\r\n\r\n\r\n
Cd<\/em>2<\/sup><\/td>\r\n=<\/td>\r\nintrinsic permeability<\/em> (k<\/em>) of the porous medium (L2<\/sup>)<\/td>\r\n<\/tr>\r\n
C<\/em><\/td>\r\n=<\/td>\r\na constant of proportionality representing effects of particle shape, tortuosity<\/em>, and pore size distribution of the porous medium independent of fluid properties (dimensionless)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nOnce the properties of the fluid are known, the hydraulic conductivity can be calculated from the intrinsic permeability as shown in Equation 31.<\/a>\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle K=\\frac{Cd^{2}g\\rho }{\\mu }=\\frac{kg\\rho }{\\mu }[\/latex]<\/td>\r\n(31)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFor sediments and rocks, intrinsic permeability<\/em> (k<\/em>) incorporates the influence of all the media properties that affect flow, not only the mean grain diameter as was the case for the uniform glass spheres. It has units of L2<\/sup>. The intrinsic permeability represents the magnitude of variation in the diameters of the interconnected pores as well as the amount of branching and reconnecting of the pore pathways over a linear travel path, referred as the degree of tortuosity. Tortuosity<\/em> is a measure of actual distance traveled divided by the shortest distance between two locations. In general, the larger the diameter of the pores and the more efficiently they are interconnected (less tortuosity), the larger the intrinsic permeability. In contrast, a porous material with small diameter pores and many circuitous interconnected pathways (high tortuosity) would have a lower intrinsic permeability. Intrinsic permeability can also be computed if the hydraulic conductivity and fluid properties are known by rearranging Equation 31.\r\n

Fluid Properties<\/h1>\r\nThe specific weight and dynamic viscosity of the fluid also influence the hydraulic conductivity. The larger the specific weight of a fluid ( \u03b3<\/em> = g\u03c1<\/em>) and the lower the dynamic viscosity (\u03bc<\/em>), the higher the hydraulic conductivity. The specific weight<\/em>, \u03b3<\/em>, of a fluid is its density times the gravitational constant, \u03c1g<\/em>, and the dynamic viscosity<\/em>, \u03bc<\/em>, is the ratio of the shearing stress on a plane to the rate at which fluid velocity changes across the plane (internal resistance to flow). The influence of fluid properties on the value of hydraulic conductivity can be illustrated by visualizing how the flow rate would be affected if two Darcy sand-filled columns under the same hydraulic gradients were set up such that water flowed through one and molasses flowed through the other (Freeze and Cherry, 1979). It is assumed that the sand in each column has the same the structure and number of interconnected pores. Clearly the flow rate of the molasses would be slower than that of water. This is because the viscosity of molasses is typically more than a thousand times higher than that of water, while the specific weight of molasses is only about 1.5 times higher than water.\r\n\r\nFor groundwater systems, changes in density and viscosity caused by temperature need to be considered when computing hydraulic conductivities. Dynamic viscosity and density of water as a function of the water temperature is shown in Figure 28. Temperature has a more significant impact on viscosity than density.<\/a>\r\n\r\n[caption id=\"attachment_353\" align=\"alignnone\" width=\"1024\"]\"Graph Figure 28 -<\/strong> Dynamic viscosity and density of water as a function of water temperature. It is useful to note that 1 millipascal-second is equivalent to 0.01 gram\/centimeter-second (data from IAPWS, 2008).[\/caption]\r\n\r\nFor example: If a sand has an intrinsic permeability, k<\/em>, of 1 \u00d7 10-7<\/sup> cm2<\/sup>, and the water moving through the sand has a temperature of 10 \u00b0C, then (from Figure 28):\r\n