{"id":401,"date":"2020-10-25T02:22:19","date_gmt":"2020-10-25T02:22:19","guid":{"rendered":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/?post_type=chapter&p=401"},"modified":"2020-12-28T19:48:24","modified_gmt":"2020-12-28T19:48:24","slug":"spatial-and-directional-variation-of-hydraulic-conductivity","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/chapter\/spatial-and-directional-variation-of-hydraulic-conductivity\/","title":{"raw":"5.4 Spatial and Directional Variation of Hydraulic Conductivity","rendered":"5.4 Spatial and Directional Variation of Hydraulic Conductivity"},"content":{"raw":"As discussed previously in this book, the transmission properties of earth materials are influenced by the processes that are active when the material is formed or deposited, as well as the changes in the material that occur during post-formation processes. Both sets of processes influence the direction and magnitude of hydraulic conductivity at each location throughout the material. In the general case, hydraulic conductivity varies with direction, and when aligned with the principal coordinate axes, it is represented as K<\/em>x<\/em><\/sub>, K<\/em>y<\/em><\/sub> and K<\/em>z<\/em><\/sub>.\r\n\r\nIf formation of the earth material results in uniform pore sizes that are well connected in all directions at a given location (e.g., K<\/em>x<\/em><\/sub> = K<\/em>y<\/em><\/sub> = K<\/em>z<\/em><\/sub>) the hydraulic conductivity at that location is isotropic<\/em>. In a material with isotropic hydraulic conductivity, water passes through a material with equal ease regardless of the direction that the water is forced through the material. That is, given the same gradient in any direction, the resulting flow rate is the same. Figure 33a provides a schematic of an isotropic porous medium where K<\/em> is the same in every direction at every location. In contrast, if the pore structure and connections resulted in a setting where hydraulic conductivity is lower in one direction than another, the porous medium is anisotropic<\/em>, for example, where K<\/em>x<\/em><\/sub> = K<\/em>y<\/em><\/sub>, but K<\/em>z<\/em><\/sub> has a smaller value as shown in Figure 33b. The medium shown in Figure 33b could be the result of compacting the medium shown in Figure 33a, or from a depositional process that oriented non-spherical grains in a preferred direction. Anisotropy can also involve different values of K<\/em> in all of the principal directions K<\/em>x<\/em><\/sub> \u2260 K<\/em>y<\/em><\/sub> \u2260 K<\/em>z<\/em><\/sub>. In most sediments and sedimentary rocks, the deposition process produces micro layering resulting in horizontal hydraulic conductivities being about the same, K<\/em>x<\/em><\/sub> = K<\/em>y<\/em><\/sub>, and vertical values, K<\/em>z<\/em><\/sub>, that are smaller. This results in isotropic conditions in the horizontal plane (map view) with anisotropic conditions in a cross-sectional representation as shown in Figure 33b.\r\n\r\n[caption id=\"attachment_402\" align=\"alignnone\" width=\"878\"]\"Schematics Figure 33 -<\/strong> Schematics of a representative volume of a porous medium with isotropic and anisotropic hydraulic conductivity: a) an isotropic porous medium where the magnitude and direction of K<\/em>x<\/em><\/sub>, K<\/em>y<\/em><\/sub> and K<\/em>z<\/em><\/sub> are equal; b) an anisotropic medium where the direction and magnitude of K<\/em>x<\/em><\/sub> equals K<\/em>y<\/em><\/sub>, while K<\/em>z<\/em><\/sub> has a smaller value. The material shown in (b) could be the result of compacting the material shown in (a).[\/caption]\r\n\r\nWhen hydraulic conductivity distributions are observed at the broader scale of a geologic formation (not at only one location), two general conditions can be found in four combinations as illustrated in Figure 34. The formation can be either homogeneous or heterogeneous<\/em>, and it can be either isotropic or anisotropic:\r\n
    \r\n \t
  • If all locations have the same value of K<\/em>x<\/em><\/sub>, K<\/em>y<\/em><\/sub>, K<\/em>z<\/em><\/sub>, then the hydraulic conductivity distribution is isotropic<\/em> and homogeneous<\/em> (Figure 34a).<\/li>\r\n \t
  • If the same anisotropic conditions are present at all locations within a formation, then the hydraulic conductivity distribution is anisotropic<\/em> and homogeneous<\/em> (Figure 34b).<\/li>\r\n \t
  • If the hydraulic conductivity is isotropic at all locations, but the isotropic values differ from location to location, then the hydraulic conductivity distribution is isotropic<\/em> and heterogeneous<\/em> (Figure 34c).<\/li>\r\n \t
  • If at different locations within the formation, different sets of anisotropic conditions are present, then the hydraulic conductivity distribution is anisotropic<\/em> and heterogeneous<\/em> (Figure 34d).<\/li>\r\n<\/ul>\r\n[caption id=\"attachment_403\" align=\"alignnone\" width=\"1024\"]\"Figure Figure 34 -<\/strong> Examples of hydraulic conductivity distributions for a geologic formation enclosed by the blue box. When point-values of K<\/em>x<\/em><\/sub>, K<\/em>y<\/em><\/sub> and K<\/em>z<\/em><\/sub> are observed at many locations in a formation, the hydraulic conductivity distribution can be described in terms of homogeneity (same throughout) or heterogeneity (variable from location to location) and having isotropy (at a location the same in all directions) or anisotropy (at a location not the same in all directions). a) Isotropic and homogeneous conditions: the values of hydraulic conductivity in each coordinate direction are equal at all locations. b) Anisotropic and homogeneous conditions: one or more of the values of hydraulic conductivity in each coordinate direction are not equal, but this relationship is the same at all locations within the region. c) Isotropic and heterogeneous conditions: hydraulic conductivity is equal in all directions at a location, but values differ at other locations. d) Anisotropic and heterogeneous conditions: one of more of the directional values of hydraulic conductivity can differ at each location and they can all be different at each location.[\/caption]\r\n\r\nThe three-dimensional components of hydraulic conductivity can be used to derive the hydraulic conductivity, K<\/em>s<\/em><\/sub>, associated with a particular groundwater flow path (q<\/em>) constructed within the groundwater system. For example, if the spatial hydraulic conductivity conditions (K<\/em>x<\/em><\/sub>, K<\/em>y<\/em><\/sub>, K<\/em>z<\/em><\/sub>) are known, the hydraulic conductivity associated with a flow line at some angle to the coordinate system can be established using a hydraulic conductivity ellipsoid (Freeze and Cherry, 1979, Chapter 2). In two dimensions an ellipse can be constructed using the square root of K<\/em>max<\/em><\/sub> and K<\/em>min<\/em><\/sub> values as shown in Figure 35 and expressed in Equations 34 and 35.\r\n\r\n[caption id=\"attachment_407\" align=\"alignnone\" width=\"924\"]\"Figure Figure 35 -<\/strong> The determination of the hydraulic conductivity, K<\/em>s<\/em><\/sub>, associated with an arbitrary flow line q<\/em>s<\/em><\/sub> at angle \u03b8<\/em> to the x coordinate in an isotropic and anisotropic material. a) Specific discharge q<\/em>s<\/em><\/sub> in an arbitrary direction of flow in an isotropic material. b) Hydraulic conductivity ellipse for isotropic conditions; K<\/em>s<\/em><\/sub> is the same at all angles. c) Specific discharge q<\/em>s<\/em><\/sub> in an arbitrary direction of flow in an anisotropic material. d) Hydraulic conductivity ellipse for anisotropic conditions showing K<\/em>s<\/em><\/sub> varies with angle (after Freeze and Cherry, 1979).[\/caption]\r\n\r\n\r\n\r\n
    <\/td>\r\n[latex]\\displaystyle \\frac{1}{K_{s}}=\\frac{\\cos ^{2}\\theta }{K_{x}}+\\frac{\\sin ^{2}\\theta }{K_{z}}[\/latex]<\/td>\r\n(34)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nEquation 34 relates the hydraulic conductivity, K<\/em>s<\/em><\/sub>, in any angular direction, \u03b8<\/em>, to the components K<\/em>x<\/em><\/sub> and K<\/em>z<\/em><\/sub>. Equation 34 can be written in rectangular coordinates by setting x<\/em> = r<\/em> cos \u03b8<\/em>\u00a0 and z<\/em> = r<\/em> sin \u03b8<\/em>\u00a0 resulting in Equation 35.\r\n\r\n\r\n\r\n
    <\/td>\r\n[latex]\\displaystyle \\frac{r^2}{K_s}=\\frac{x^2}{K_x}+\\frac{z^2}{K_z}[\/latex]<\/td>\r\n(35)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nEquation 35 is the equation of an ellipse. Major axes are the square root of K<\/em>x<\/em><\/sub> and K<\/em>z<\/em><\/sub> as shown in Figure 35d (Freeze and Cherry, 1979). As indicated in Figure 35b and d, the hydraulic conductivity in any direction, K<\/em>s<\/em><\/sub>, in an isotropic or anisotropic medium can be determined graphically if K<\/em>x<\/em><\/sub> and K<\/em>z<\/em><\/sub> are known. Although the x<\/em>, y<\/em>, and z<\/em> axes are shown here in their classic orientation, the axes may be oriented in any direction within a formation as shown in Figure 36.\r\n\r\n[caption id=\"attachment_410\" align=\"alignnone\" width=\"364\"]\"Figure Figure 36 -<\/strong> The hydraulic conductivity ellipse may have any orientation in a subsurface material.[\/caption]","rendered":"

    As discussed previously in this book, the transmission properties of earth materials are influenced by the processes that are active when the material is formed or deposited, as well as the changes in the material that occur during post-formation processes. Both sets of processes influence the direction and magnitude of hydraulic conductivity at each location throughout the material. In the general case, hydraulic conductivity varies with direction, and when aligned with the principal coordinate axes, it is represented as K<\/em>x<\/em><\/sub>, K<\/em>y<\/em><\/sub> and K<\/em>z<\/em><\/sub>.<\/p>\n

    If formation of the earth material results in uniform pore sizes that are well connected in all directions at a given location (e.g., K<\/em>x<\/em><\/sub> = K<\/em>y<\/em><\/sub> = K<\/em>z<\/em><\/sub>) the hydraulic conductivity at that location is isotropic<\/em>. In a material with isotropic hydraulic conductivity, water passes through a material with equal ease regardless of the direction that the water is forced through the material. That is, given the same gradient in any direction, the resulting flow rate is the same. Figure 33a provides a schematic of an isotropic porous medium where K<\/em> is the same in every direction at every location. In contrast, if the pore structure and connections resulted in a setting where hydraulic conductivity is lower in one direction than another, the porous medium is anisotropic<\/em>, for example, where K<\/em>x<\/em><\/sub> = K<\/em>y<\/em><\/sub>, but K<\/em>z<\/em><\/sub> has a smaller value as shown in Figure 33b. The medium shown in Figure 33b could be the result of compacting the medium shown in Figure 33a, or from a depositional process that oriented non-spherical grains in a preferred direction. Anisotropy can also involve different values of K<\/em> in all of the principal directions K<\/em>x<\/em><\/sub> \u2260 K<\/em>y<\/em><\/sub> \u2260 K<\/em>z<\/em><\/sub>. In most sediments and sedimentary rocks, the deposition process produces micro layering resulting in horizontal hydraulic conductivities being about the same, K<\/em>x<\/em><\/sub> = K<\/em>y<\/em><\/sub>, and vertical values, K<\/em>z<\/em><\/sub>, that are smaller. This results in isotropic conditions in the horizontal plane (map view) with anisotropic conditions in a cross-sectional representation as shown in Figure 33b.<\/p>\n

    \"Schematics
    Figure 33 –<\/strong> Schematics of a representative volume of a porous medium with isotropic and anisotropic hydraulic conductivity: a) an isotropic porous medium where the magnitude and direction of K<\/em>x<\/em><\/sub>, K<\/em>y<\/em><\/sub> and K<\/em>z<\/em><\/sub> are equal; b) an anisotropic medium where the direction and magnitude of K<\/em>x<\/em><\/sub> equals K<\/em>y<\/em><\/sub>, while K<\/em>z<\/em><\/sub> has a smaller value. The material shown in (b) could be the result of compacting the material shown in (a).<\/figcaption><\/figure>\n

    When hydraulic conductivity distributions are observed at the broader scale of a geologic formation (not at only one location), two general conditions can be found in four combinations as illustrated in Figure 34. The formation can be either homogeneous or heterogeneous<\/em>, and it can be either isotropic or anisotropic:<\/p>\n

      \n
    • If all locations have the same value of K<\/em>x<\/em><\/sub>, K<\/em>y<\/em><\/sub>, K<\/em>z<\/em><\/sub>, then the hydraulic conductivity distribution is isotropic<\/em> and homogeneous<\/em> (Figure 34a).<\/li>\n
    • If the same anisotropic conditions are present at all locations within a formation, then the hydraulic conductivity distribution is anisotropic<\/em> and homogeneous<\/em> (Figure 34b).<\/li>\n
    • If the hydraulic conductivity is isotropic at all locations, but the isotropic values differ from location to location, then the hydraulic conductivity distribution is isotropic<\/em> and heterogeneous<\/em> (Figure 34c).<\/li>\n
    • If at different locations within the formation, different sets of anisotropic conditions are present, then the hydraulic conductivity distribution is anisotropic<\/em> and heterogeneous<\/em> (Figure 34d).<\/li>\n<\/ul>\n
      \"Figure
      Figure 34 –<\/strong> Examples of hydraulic conductivity distributions for a geologic formation enclosed by the blue box. When point-values of K<\/em>x<\/em><\/sub>, K<\/em>y<\/em><\/sub> and K<\/em>z<\/em><\/sub> are observed at many locations in a formation, the hydraulic conductivity distribution can be described in terms of homogeneity (same throughout) or heterogeneity (variable from location to location) and having isotropy (at a location the same in all directions) or anisotropy (at a location not the same in all directions). a) Isotropic and homogeneous conditions: the values of hydraulic conductivity in each coordinate direction are equal at all locations. b) Anisotropic and homogeneous conditions: one or more of the values of hydraulic conductivity in each coordinate direction are not equal, but this relationship is the same at all locations within the region. c) Isotropic and heterogeneous conditions: hydraulic conductivity is equal in all directions at a location, but values differ at other locations. d) Anisotropic and heterogeneous conditions: one of more of the directional values of hydraulic conductivity can differ at each location and they can all be different at each location.<\/figcaption><\/figure>\n

      The three-dimensional components of hydraulic conductivity can be used to derive the hydraulic conductivity, K<\/em>s<\/em><\/sub>, associated with a particular groundwater flow path (q<\/em>) constructed within the groundwater system. For example, if the spatial hydraulic conductivity conditions (K<\/em>x<\/em><\/sub>, K<\/em>y<\/em><\/sub>, K<\/em>z<\/em><\/sub>) are known, the hydraulic conductivity associated with a flow line at some angle to the coordinate system can be established using a hydraulic conductivity ellipsoid (Freeze and Cherry, 1979, Chapter 2). In two dimensions an ellipse can be constructed using the square root of K<\/em>max<\/em><\/sub> and K<\/em>min<\/em><\/sub> values as shown in Figure 35 and expressed in Equations 34 and 35.<\/p>\n

      \"Figure
      Figure 35 –<\/strong> The determination of the hydraulic conductivity, K<\/em>s<\/em><\/sub>, associated with an arbitrary flow line q<\/em>s<\/em><\/sub> at angle \u03b8<\/em> to the x coordinate in an isotropic and anisotropic material. a) Specific discharge q<\/em>s<\/em><\/sub> in an arbitrary direction of flow in an isotropic material. b) Hydraulic conductivity ellipse for isotropic conditions; K<\/em>s<\/em><\/sub> is the same at all angles. c) Specific discharge q<\/em>s<\/em><\/sub> in an arbitrary direction of flow in an anisotropic material. d) Hydraulic conductivity ellipse for anisotropic conditions showing K<\/em>s<\/em><\/sub> varies with angle (after Freeze and Cherry, 1979).<\/figcaption><\/figure>\n\n\n\n
      <\/td>\n\"\displaystyle \frac{1}{K_{s}}=\frac{\cos ^{2}\theta }{K_{x}}+\frac{\sin ^{2}\theta }{K_{z}}\"<\/td>\n(34)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Equation 34 relates the hydraulic conductivity, K<\/em>s<\/em><\/sub>, in any angular direction, \u03b8<\/em>, to the components K<\/em>x<\/em><\/sub> and K<\/em>z<\/em><\/sub>. Equation 34 can be written in rectangular coordinates by setting x<\/em> = r<\/em> cos \u03b8<\/em>\u00a0 and z<\/em> = r<\/em> sin \u03b8<\/em>\u00a0 resulting in Equation 35.<\/p>\n\n\n\n
      <\/td>\n\"\displaystyle \frac{r^2}{K_s}=\frac{x^2}{K_x}+\frac{z^2}{K_z}\"<\/td>\n(35)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Equation 35 is the equation of an ellipse. Major axes are the square root of K<\/em>x<\/em><\/sub> and K<\/em>z<\/em><\/sub> as shown in Figure 35d (Freeze and Cherry, 1979). As indicated in Figure 35b and d, the hydraulic conductivity in any direction, K<\/em>s<\/em><\/sub>, in an isotropic or anisotropic medium can be determined graphically if K<\/em>x<\/em><\/sub> and K<\/em>z<\/em><\/sub> are known. Although the x<\/em>, y<\/em>, and z<\/em> axes are shown here in their classic orientation, the axes may be oriented in any direction within a formation as shown in Figure 36.<\/p>\n

      \"Figure
      Figure 36 –<\/strong> The hydraulic conductivity ellipse may have any orientation in a subsurface material.<\/figcaption><\/figure>\n","protected":false},"author":1,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-401","chapter","type-chapter","status-publish","hentry"],"part":103,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/401","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":18,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/401\/revisions"}],"predecessor-version":[{"id":1142,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/401\/revisions\/1142"}],"part":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/parts\/103"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/401\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/wp\/v2\/media?parent=401"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapter-type?post=401"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/wp\/v2\/contributor?post=401"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/wp\/v2\/license?post=401"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}