{"id":1664,"date":"2026-02-14T04:23:41","date_gmt":"2026-02-14T04:23:41","guid":{"rendered":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/ch042-xhtml\/"},"modified":"2026-08-01T09:02:30","modified_gmt":"2026-08-01T09:02:30","slug":"formulation-of-equations-to-address-leaky-confined-conditions","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulation-of-equations-to-address-leaky-confined-conditions\/","title":{"raw":"9.1 Formulation of Equations to Address Leaky Confined Conditions","rendered":"9.1 Formulation of Equations to Address Leaky Confined Conditions"},"content":{"raw":"
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\r\n\r\nCooper (1963), Jacob (1946), Hantush and Jacob (1954), and Hantush (1960) all contributed to developing analytical solutions that describe hydraulic responses to pumping in confined systems with leaky confining beds. As the confined unit is being pumped it is receiving water from leakage, thus not all Theis assumptions (Section 6.3) apply. Leaky confined conditions assume one or both of two settings.\r\n\r\n<\/div>\r\n
    \r\n \t
  1. Water enters the pumped confined unit from overlying or underlying units and release of water from storage in the confining beds is negligible. Overlying and underlying water\u2011bearing units separated by the confining units are sufficiently permeable and extensive such that heads in the unpumped overlying and underlying aquifers remain constant when pumping the principal aquifer.<\/li>\r\n \t
  2. Water enters the pumped confined unit from water released from storage within the confining units.<\/li>\r\n<\/ol>\r\n
    \r\n\r\nFigure 41<\/a> and Figure 42<\/a> present conceptual models of leaky confined settings with and without aquitard storage, respectively. The mathematics of these two settings are developed separately in Sections\u00a09.2 a d 9.3.\r\n\r\n<\/div>\r\n
    \"\"\r\n
    \r\n
    \r\n
    \r\n\r\nFigure 41 \u2011<\/strong> Conceptual model of a Hantush\u2011Jacob leaky confined system. a) As the production well is pumped, water starts to leak through the overlying aquitard. A difference in head, \u0394hv<\/sub><\/em>, (vertical blue double arrow) induces flow between the overlying unconfined system and the pumped confined aquifer. The overlying aquifer supplying water is extensive and permeable such that the static water level in that aquifer (water table in this example) does not change during pumping of the confined aquifer. The red arrows represent leakage from the overlying aquifer that flows through the confining unit. Leakance is defined by Darcy's Law (i.e., the product of the difference in head between the aquifers (\u0394hv<\/sub><\/em>) and the vertical hydraulic conductivity (K\u2019<\/em>) of the confining unit divided by the confining unit thickness (b\u2019<\/em>). The dashed vertical lines and cylinder outline the zone at a specified time where leakage enters the confined aquifer. b) Example of drawdown versus time at the pumping well and observation well. The water level changes begin to level out (stop declining) in this example.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
    \"\"\r\n
    \r\n
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    \r\n\r\nFigure 42 \u2011<\/strong> Conceptual model of the modified Hantush (1960) solution setting. A well is pumped in a confined unit where water is being derived from the release of storage in both the aquifer and the confining beds. a) In addition to water being released from storage in the confined water\u2011bearing unit, water from overlying aquitard 1 and\/or underlying aquitard 2 are flowing into the confined aquifer. The properties of aquitard 1 are vertical hydraulic conductivity, K\u2019<\/em>, storativity, S\u2019<\/em>, and thickness, b\u2019<\/em>. The properties of aquitard 2 are the vertical hydraulic conductivity, K\u201d<\/em>, storativity, S\u201d<\/em> and thickness b\u201d<\/em>. b) The drawdown in the pumping well and observation well are less than those observed if the pumped unit is totally confined. When leakage rates do not equal the test pumping rate, water levels will not stabilize, instead they will continue to decline.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"
    \n
    \n
    \n

    Cooper (1963), Jacob (1946), Hantush and Jacob (1954), and Hantush (1960) all contributed to developing analytical solutions that describe hydraulic responses to pumping in confined systems with leaky confining beds. As the confined unit is being pumped it is receiving water from leakage, thus not all Theis assumptions (Section 6.3) apply. Leaky confined conditions assume one or both of two settings.<\/p>\n<\/div>\n

      \n
    1. Water enters the pumped confined unit from overlying or underlying units and release of water from storage in the confining beds is negligible. Overlying and underlying water\u2011bearing units separated by the confining units are sufficiently permeable and extensive such that heads in the unpumped overlying and underlying aquifers remain constant when pumping the principal aquifer.<\/li>\n
    2. Water enters the pumped confined unit from water released from storage within the confining units.<\/li>\n<\/ol>\n
      \n

      Figure 41<\/a> and Figure 42<\/a> present conceptual models of leaky confined settings with and without aquitard storage, respectively. The mathematics of these two settings are developed separately in Sections\u00a09.2 a d 9.3.<\/p>\n<\/div>\n

      \"\"<\/p>\n
      \n
      \n
      \n

      Figure 41 \u2011<\/strong> Conceptual model of a Hantush\u2011Jacob leaky confined system. a) As the production well is pumped, water starts to leak through the overlying aquitard. A difference in head, \u0394hv<\/sub><\/em>, (vertical blue double arrow) induces flow between the overlying unconfined system and the pumped confined aquifer. The overlying aquifer supplying water is extensive and permeable such that the static water level in that aquifer (water table in this example) does not change during pumping of the confined aquifer. The red arrows represent leakage from the overlying aquifer that flows through the confining unit. Leakance is defined by Darcy’s Law (i.e., the product of the difference in head between the aquifers (\u0394hv<\/sub><\/em>) and the vertical hydraulic conductivity (K\u2019<\/em>) of the confining unit divided by the confining unit thickness (b\u2019<\/em>). The dashed vertical lines and cylinder outline the zone at a specified time where leakage enters the confined aquifer. b) Example of drawdown versus time at the pumping well and observation well. The water level changes begin to level out (stop declining) in this example.<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      \"\"<\/p>\n
      \n
      \n
      \n

      Figure 42 \u2011<\/strong> Conceptual model of the modified Hantush (1960) solution setting. A well is pumped in a confined unit where water is being derived from the release of storage in both the aquifer and the confining beds. a) In addition to water being released from storage in the confined water\u2011bearing unit, water from overlying aquitard 1 and\/or underlying aquitard 2 are flowing into the confined aquifer. The properties of aquitard 1 are vertical hydraulic conductivity, K\u2019<\/em>, storativity, S\u2019<\/em>, and thickness, b\u2019<\/em>. The properties of aquitard 2 are the vertical hydraulic conductivity, K\u201d<\/em>, storativity, S\u201d<\/em> and thickness b\u201d<\/em>. b) The drawdown in the pumping well and observation well are less than those observed if the pumped unit is totally confined. When leakage rates do not equal the test pumping rate, water levels will not stabilize, instead they will continue to decline.<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":8,"menu_order":35,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1664","chapter","type-chapter","status-publish","hentry"],"part":611,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1664","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":3,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1664\/revisions"}],"predecessor-version":[{"id":2976,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1664\/revisions\/2976"}],"part":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/parts\/611"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1664\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/media?parent=1664"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapter-type?post=1664"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/contributor?post=1664"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/license?post=1664"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}