{"id":1394,"date":"2026-05-12T21:47:44","date_gmt":"2026-05-12T21:47:44","guid":{"rendered":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/solution-exercise-6\/"},"modified":"2026-05-13T03:38:03","modified_gmt":"2026-05-13T03:38:03","slug":"solution-exercise-6","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/chapter\/solution-exercise-6\/","title":{"raw":"Solution Exercise 6","rendered":"Solution Exercise 6"},"content":{"raw":"
\r\n
    \r\n \t
  1. The items and their dimensions are defined in the table of Exercise 5. Their SI units are as follows.<\/li>\r\n<\/ol>\r\n<\/colgroup>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    <\/th>\r\n[latex]Q = k\\frac{\\rho_{fluid}g}{\\mu_{fluid}}A\\frac{dh}{dl}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    [latex]Q[\/latex]<\/td>\r\n=<\/td>\r\nvolumetric discharge (m3<\/sup> sec-1<\/sup>)<\/td>\r\n<\/tr>\r\n
    [latex]k[\/latex]<\/td>\r\n=<\/td>\r\nintrinsic permeability (m2<\/sup>)<\/td>\r\n<\/tr>\r\n
    [latex]\\rho_{fluid}[\/latex]<\/td>\r\n=<\/td>\r\ndensity of the fluid (kg m-3<\/sup>)<\/td>\r\n<\/tr>\r\n
    [latex]g[\/latex]<\/td>\r\n=<\/td>\r\ngravitational acceleration (m sec-2<\/sup>)<\/td>\r\n<\/tr>\r\n
    A<\/em><\/td>\r\n=<\/td>\r\ncross\u2011sectional area perpendicular to flow (m2<\/sup>)<\/td>\r\n<\/tr>\r\n
    dh\/dl<\/em><\/td>\r\n=<\/td>\r\nhydraulic gradient ([latex]:[\/latex])<\/td>\r\n<\/tr>\r\n
    [latex]\\mu_{fluid}[\/latex]<\/td>\r\n=<\/td>\r\nviscosity of the fluid (kg m-1<\/sup> sec-1<\/sup>)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
      \r\n \t
    1. Variables are [latex]Q,\\ A,\\frac{dh}{dl},\\ k,\\ \\rho_{fluid},\\ \\mu_{fluid}[\/latex]. Constants are [latex]g[\/latex].<\/li>\r\n<\/ol>\r\n
      \r\n
      3. Discharge is directly proportional to permeability, fluid density, gravitational acceleration, area, and hydraulic gradient. Discharge is inversely proportional to viscosity.<\/blockquote>\r\n<\/div>\r\n
      \r\n
      4. There are no exponents or logarithms in the equation; therefore, all proportionality relationships are linear.<\/blockquote>\r\n<\/div>\r\n
      \r\n\r\nThought Question 19<\/strong>: Explain why discharge is directly proportional to density but inversely proportional to viscosity.<\/em>\r\n\r\n<\/div>\r\n
      \r\n\r\nReturn to Exercise 6<\/strong><\/span><\/a>\u2ba5\r\n\r\n<\/div>\r\n
      \r\n\r\nReturn to where text linked to Exercise\u00a06<\/strong><\/span><\/a>\u2ba5\r\n\r\n<\/div>\r\n<\/div>","rendered":"
      \n
        \n
      1. The items and their dimensions are defined in the table of Exercise 5. Their SI units are as follows.<\/li>\n<\/ol>\n\n\n\n\n\n<\/colgroup>\n\n\n\n\n\n\n\n\n\n\n
        <\/th>\n[latex]Q = k\\frac{\\rho_{fluid}g}{\\mu_{fluid}}A\\frac{dh}{dl}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
        [latex]Q[\/latex]<\/td>\n=<\/td>\nvolumetric discharge (m3<\/sup> sec-1<\/sup>)<\/td>\n<\/tr>\n
        [latex]k[\/latex]<\/td>\n=<\/td>\nintrinsic permeability (m2<\/sup>)<\/td>\n<\/tr>\n
        [latex]\\rho_{fluid}[\/latex]<\/td>\n=<\/td>\ndensity of the fluid (kg m-3<\/sup>)<\/td>\n<\/tr>\n
        [latex]g[\/latex]<\/td>\n=<\/td>\ngravitational acceleration (m sec-2<\/sup>)<\/td>\n<\/tr>\n
        A<\/em><\/td>\n=<\/td>\ncross\u2011sectional area perpendicular to flow (m2<\/sup>)<\/td>\n<\/tr>\n
        dh\/dl<\/em><\/td>\n=<\/td>\nhydraulic gradient ([latex]:[\/latex])<\/td>\n<\/tr>\n
        [latex]\\mu_{fluid}[\/latex]<\/td>\n=<\/td>\nviscosity of the fluid (kg m-1<\/sup> sec-1<\/sup>)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
          \n
        1. Variables are [latex]Q,\\ A,\\frac{dh}{dl},\\ k,\\ \\rho_{fluid},\\ \\mu_{fluid}[\/latex]. Constants are [latex]g[\/latex].<\/li>\n<\/ol>\n
          \n

          3. Discharge is directly proportional to permeability, fluid density, gravitational acceleration, area, and hydraulic gradient. Discharge is inversely proportional to viscosity.<\/p><\/blockquote>\n<\/div>\n

          \n

          4. There are no exponents or logarithms in the equation; therefore, all proportionality relationships are linear.<\/p><\/blockquote>\n<\/div>\n

          \n

          Thought Question 19<\/strong>: Explain why discharge is directly proportional to density but inversely proportional to viscosity.<\/em><\/p>\n<\/div>\n

          \n

          Return to Exercise 6<\/strong><\/span><\/a>\u2ba5<\/p>\n<\/div>\n

          \n

          Return to where text linked to Exercise\u00a06<\/strong><\/span><\/a>\u2ba5<\/p>\n<\/div>\n<\/div>\n","protected":false},"author":8,"menu_order":117,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1394","chapter","type-chapter","status-publish","hentry"],"part":1531,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters\/1394","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\/1394\/revisions"}],"predecessor-version":[{"id":1824,"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters\/1394\/revisions\/1824"}],"part":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/parts\/1531"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/basic-hydrogeology-an-introduction-to-the-fundamentals-of-groundwater-science\/wp-json\/pressbooks\/v2\/chapters\/1394\/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=1394"}],"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=1394"},{"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=1394"},{"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=1394"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}