{"id":1803,"date":"2026-02-14T04:27:45","date_gmt":"2026-02-14T04:27:45","guid":{"rendered":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/ch078-xhtml\/"},"modified":"2026-08-01T11:00:33","modified_gmt":"2026-08-01T11:00:33","slug":"test-methods-used-to-estimate-aquitard-properties","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/test-methods-used-to-estimate-aquitard-properties\/","title":{"raw":"16.2 Test Methods Used to Estimate Aquitard Properties","rendered":"16.2 Test Methods Used to Estimate Aquitard Properties"},"content":{"raw":"<div id=\"test-methods-used-to-estimate-aquitard-properties\" class=\"level1\">\r\n<div data-custom-style=\"First Paragraph\">\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nThis section focuses on the physical characterization of hydrogeologic properties of aquitards. In this book, we purposefully use the terms pumping test and hydraulic test instead of aquifer test. Conceptually, the pumping\u2011test techniques described in previous sections are applicable to both highly conductive and, in some settings, lower\u2011permeability units that may be considered aquitards. This section addresses some of the specific methods used to characterize the hydrogeologic properties of aquitards. Cherry and others (2006) and Bradbury and others (2006) also describe methods used to evaluate the integrity of aquitards as this condition relates to the potential for migration of contaminants from an impacted source through an aquitard.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nCherry and others (2006) suggest that characterization of aquitard hydraulic properties can be grouped into the application of external and internal methods. Internal methods include laboratory methods applied to samples and cores of the aquitard, performing hydraulic tests within a borehole penetrating the aquitard, and observing head responses to recharge or loading. External methods involve pumping a productive water\u2011bearing unit above or below an aquitard and then interpreting time\u2011drawdown data from observation wells in the principal water\u2011bearing unit and, in some cases, in the adjacent aquifer and\/or aquitard.\r\n\r\n<\/div>\r\n<div id=\"internal-methods\" class=\"level3\">\r\n<h3>Internal Methods<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nDirect internal investigation of an aquitard includes collecting and testing sample cores as well as installing wells and conducting pumping and\/or slug tests. Indirect internal methods include monitoring the response of the near surface unconsolidated aquitard material to recharge events and monitoring the aquitard\u2019s response to external loading.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nLaboratory Methods\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nResearchers have observed that hydraulic\u2011property values estimated from laboratory investigations of samples of uniform aquitards without significant secondary affects (developed macropores and\/or fracturing) may be equivalent to those derived from field\u2011scale methods (Cherry et al., 2006). However, laboratory\u2011scale samples of heterogeneous materials may poorly represent field\u2011scale properties, as the density and general interconnectedness of zones of different lithologies and secondary permeability features may be missing, overrepresented, or underrepresented at the laboratory scale. In some cases, laboratory samples are repacked and yield unrepresentative results. When any of these conditions occur, laboratory\u2011derived hydraulic conductivity values often underestimate field\u2011scale permeability (Haefner, 2000; Neuzil, 1986; van der Kamp, 2001).\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\n<span id=\"TextLinkToBox9\"><\/span><a href=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/box-9-laboratory-methods-used\/\">Box 9<\/a> illustrates some of the laboratory methods such as the falling\u2011head permeameter, triaxial cell, and consolidometer that are used to derive hydraulic conductivities of low\u2011permeability material.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nField Methods\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThe following subsections describe internal field methods for characterizing hydrogeologic properties of aquitards.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nPumping tests\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nConceptually, a pumping well and observation wells could be constructed in an aquitard. In most settings, the low yield potential and permeability of the unit would limit the pumping rate and the distance at which observation wells would respond to pumping. In addition, if more permeable units underlie or overlie the aquitard, they must be accounted for in the analysis. In some settings, pumping a single well at a low constant rate with measurements of drawdown and time may be most useful. This is discussed in Section 12. If a successful pumping test is conducted, drawdown responses in the aquitard would be analyzed using methods described in Sections 7 through 10.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nSlug Tests\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nSlug tests, as described in Section 14, determine internal horizontal hydraulic conductivities of screened portions of aquitards. Hydraulic\u2011conductivity values reflect the nature of the material within a few meters of the test interval. The local nature of the test may not capture conditions where secondary permeability or material heterogeneity dominate the formation. Slug testing does not provide direct measurement of the vertical hydraulic conductivity. Estimates of vertical hydraulic conductivity are often assumed to be an order of magnitude smaller than horizontal values in stratified rock or sediments. However, in some settings horizontal to vertical hydraulic conductivity ratios can be 1000:1 (Anderson et al., 2015). In contrast, aquitard vertical hydraulic conductivity may be greater than horizontal values in laterite or saprolite materials where remnant structures such as weathered quartz veins may be present (Cherry et al., 2004).\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nChanges in the near\u2011well hydraulic conductivity can occur when wells and piezometers are installed in unlithified aquitards (e.g., Cherry et al., 2006). Drilling can smear silt and clay along the borehole wall, reducing permeability. This skin effect artificially slows the rate of the head response causing hydraulic conductivity values to be underestimated. If the boring, well\u2011completion, and development processes enhance the permeability, values of hydraulic conductivity at the well will be overestimated. Perforated intervals should be developed prior to testing (e.g., Sterrett, 2007).\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nSlug\u2011test responses in aquitards are overdamped as discussed in Section 12. In aquitards, head changes during a slug test occur slowly and can take hours to days or months to fully recover. In some low\u2011permeability settings, the static water level after the drilling and completion of a piezometer or monitoring well may be below the fully recovered head. Often, collection of water\u2011level data as recovery occurs can be used as a slug\u2011out test and analyzed accordingly as explained in Section 14. Often in low\u2011permeability materials, monitoring wells with small diameters (smaller volumes to be filled) are used to observe head change. Aquitard well design must accommodate slugs or pneumatic methods, and water\u2011level monitoring tools.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nConstant\u2011Drawdown Tests\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nCharacterization of a low\u2011permeable unit can also be evaluated using a constant\u2011drawdown method. This method usually involves the head in the well to rise or fall and then at some point during the recovery water is continuously injected or extracted at varying flow rates to keep the head constant. Data collected are the well\u2011design factors and the inflow or outflow rate, <em>Q<\/em>, needed to keep the unrecovered head (<em>H<\/em>) constant (e.g., Mieussens &amp; Ducasse, 1977; Tavenas et al., 1990; Neville &amp; Markle, 2000) as illustrated in <a href=\"#Xf33ba11a54bffa5f40d1303b43652b0e04c574d\">Figure 141<\/a>.\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: 3.98389in; height: 4.57391in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file146-2.jpg\" alt=\"\" \/>\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=\"Xf33ba11a54bffa5f40d1303b43652b0e04c574d\" class=\"anchor\"><strong>Figure 141 \u2011\u00a0<\/strong>Slug\u2011out constant drawdown test set up for a monitoring well. The dashed blue line is the static water level. A slug of water is removed from the well (<em>H<sub>0<\/sub><\/em>) and at some time during the recovery the unrecovered head, <em>H<\/em>, in the well is stabilized by extracting volumes of water over time. The test time, <em>t<\/em>, starts when the variable discharge begins. <em>L<\/em> is the length of the perforated interval and <em>d<\/em> is the diameter.<\/span>\r\n\r\n<\/div><\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nTavenas and others (1990) developed a method to analyze the data using plots of <em>Q<\/em>(<em>t<\/em>) versus 1\/<em>t<\/em><sup>0.5<\/sup>. This relationship should be linear after 5 to 10 minutes from the test initiation. Using an arithmetic plot, a line is fitted to the linear portion of the data and then extrapolated to 1\/<em>t<\/em><sup>0.5<\/sup> = 0. The steady\u2011state flow, <em>Q<sub>infinity<\/sub><\/em>, is determined by the intersection with the y axis. Using the steady\u2011state flow rate, <em>K<sub>h<\/sub><\/em> is computed as shown in Equation\u00a0<a href=\"#Xf2e02eff9727819a74b4f4dcfeece82a6d2f73d\">(127)<\/a>.\r\n\r\n<\/div>\r\n<table class=\"gw-eq-table\" style=\"width: 100%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 10%;\" \/> <col style=\"width: 75%;\" \/> <col style=\"width: 15%;\" \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<th><\/th>\r\n<th style=\"text-align: center;\">[latex]K_{h} = \\ \\frac{Q_{infinity}}{F\\ H}[\/latex]<\/th>\r\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"Xf2e02eff9727819a74b4f4dcfeece82a6d2f73d\" class=\"anchor\">(127)<\/span><\/th>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nwhere:\r\n\r\n<\/div>\r\n<table style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 14%;\" \/> <col style=\"width: 2%;\" \/> <col style=\"width: 68%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\"><em>K<sub>h<\/sub><\/em><\/th>\r\n<th style=\"text-align: left;\">=<\/th>\r\n<th>horizontal hydraulic conductivity (LT<sup>\u20111<\/sup>)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>Q<sub>infinitity<\/sub><\/em><\/td>\r\n<td style=\"text-align: left;\">=<\/td>\r\n<td>Steady\u2011state flow rate from graph (L<sup>3<\/sup>T<sup>\u20111<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>F<\/em><\/td>\r\n<td style=\"text-align: left;\">=<\/td>\r\n<td>shape factor for the monitoring well (function of radius, screen length and other factors) (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>H<\/em><\/td>\r\n<td style=\"text-align: left;\">=<\/td>\r\n<td>unrecovered head as illustrated in <a href=\"#Xf33ba11a54bffa5f40d1303b43652b0e04c574d\">Figure 141<\/a> (L)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThe shape factor can be approximated as suggested by Hvorslev (1951) and shown in Equation\u00a0<a href=\"#X8e75ea4e1d347c05f6ca9b4cbea9d8d2ffa48db\">(128)<\/a>.\r\n\r\n<\/div>\r\n<table class=\"gw-eq-table\" style=\"width: 100%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 10%;\" \/> <col style=\"width: 75%;\" \/> <col style=\"width: 15%;\" \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<th><\/th>\r\n<th style=\"text-align: center;\">[latex]F\\ = \\ \\frac{2\\ \\pi\\ L}{\\ln\\left( \\frac{L}{d} + \\ \\left( 1 + \\left( \\frac{L}{d} \\right)^{2} \\right)^{.5} \\right)}[\/latex]<\/th>\r\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"X8e75ea4e1d347c05f6ca9b4cbea9d8d2ffa48db\" class=\"anchor\">(128)<\/span><\/th>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nwhere:\r\n\r\n<\/div>\r\n<table style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 14%;\" \/> <col style=\"width: 2%;\" \/> <col style=\"width: 68%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\"><em>L<\/em><\/th>\r\n<th style=\"text-align: left;\">=<\/th>\r\n<th>perforated length (L)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>d<\/em><\/td>\r\n<td style=\"text-align: left;\">=<\/td>\r\n<td>diameter of the perforated interval (L)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nTavenas and others (1990) provide additional discussion of parameters. The Jacob\u2011Lohman (1952) solution as outlined by Lohman (1972) may also be applicable for analyzing constant\u2011head tests. This method assumes a constant head from a flowing well where the change in discharge over time is recorded.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nWhen aquitard material lies at or near the surface, permeability has been investigated using other methods referenced by Cherry and others (2006). These include monitoring and analyzing head changes within the aquitard induced by\r\n\r\n<\/div>\r\n<ul>\r\n \t<li>Natural, precipitation\u2011induced recharge (e.g., Davis, 1972; Keller et al., 1989; Boldt\u2011Leppin &amp; Hendry, 2003),<\/li>\r\n \t<li>flow into a large, augured cavity (Keller et al., 1989), and<\/li>\r\n \t<li>construction loading and unloading (van der Kamp &amp; Maathuis, 1985).<\/li>\r\n<\/ul>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nAquitard properties can also be estimated based on the calibration of one\u2011dimensional and three\u2011dimensional numerical models given hydraulic test data from a site (e.g., Pavelko, 2004; Hart et al., 2005; Anderson et al., 2015).\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"external-methods\" class=\"level3\">\r\n<h3>External Methods<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nExternal methods include pumping adjacent water\u2011bearing units and observing the changes in water levels in the aquitard or in monitoring wells in the pumped aquifer. Examples of these methods are presented in the following subsections.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\n<strong>Pumping Test with An Observation Well in the Aquitar<\/strong>d\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nNeuman and Witherspoon (1972) develop a method to determine the diffusivity, <em>K\u2019<\/em>\/<em>S<sub>s<\/sub>\u2019<\/em> (aquitard horizontal hydraulic conductivity and specific storage), when a pumping test is conducted in the underlying or overlying aquifer and a monitoring well or piezometer is installed in the aquitard in concert with a monitoring well in the aquifer as shown in <a href=\"#Xbf4ebcb5bfaa915ba000ff4369fd5e93fb794b2\">Figure 142<\/a>. The aquitard well and aquifer\u2011observation well are at the same radial distance from the pumping well. When an aquifer monitoring well is not present, the drawdown in the pumped aquifer can be projected for the specified radial distance using a Theis analysis of the test data (Neuman &amp; Witherspoon, 1972).\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: 4.83096in; height: 4.5974in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file147-2.jpg\" alt=\"\" \/>\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=\"Xbf4ebcb5bfaa915ba000ff4369fd5e93fb794b2\" class=\"anchor\"><strong>Figure 142 \u2011<\/strong> Schematic of a pumping test conducted in an aquifer and observation wells in that unit (well B) as well as the overlying aquitard (well C). <em>z<\/em> is the distance from the top of the water bearing unit to the open interval of the observation well in the aquitard. Neuman and Witherspoon (1972) assumed the observation well opening was represented by a point (indicated by A). Rowe and Nadarajah (1993) accounted for the length of well screen where <em>z<\/em> is the distance from the boundary to the center of the well screen (dashed) (C). The radial distance to the observation well is <em>r<\/em>. <em>b\u2019<\/em> is the thickness of the aquitard unit. At a time after the pumping starts and when drawdown in the aquitard is observed, values of drawdown <em>s\u2019<\/em> in the aquitard and <em>s<\/em> in the unit being pumped (measured from the initial static water level elevation) are recorded.<\/span>\r\n\r\n<\/div><\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nThe method assumes units being pumped and aquitards are infinite in lateral extent and the monitoring wells are located within about 60 m of the pumping well. Detailed observations of drawdown should be collected at early test times. The usable early\u2011time response of drawdown in the main pumped water\u2011bearing unit and aquitard are conservatively constrained as indicated by Equation\u00a0<a href=\"#X21c12aae5a3aae64a718396d349dd0a421352b0\">(129)<\/a>.\r\n\r\n<\/div>\r\n<table class=\"gw-eq-table\" style=\"width: 100%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 10%;\" \/> <col style=\"width: 75%;\" \/> <col style=\"width: 15%;\" \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<th><\/th>\r\n<th style=\"text-align: center;\">[latex]t\\ \\leq \\ 0.1\\ \\frac{S_{s}'\\ {b'}^{2}}{K\u2019}[\/latex]<\/th>\r\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"X21c12aae5a3aae64a718396d349dd0a421352b0\" class=\"anchor\">(129)<\/span><\/th>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nwhere:\r\n\r\n<\/div>\r\n<table style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 14%;\" \/> <col style=\"width: 2%;\" \/> <col style=\"width: 68%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\"><em>t<\/em><\/th>\r\n<th style=\"text-align: left;\">=<\/th>\r\n<th>the time since the beginning of the pumping (T)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>S<sub>s<\/sub>\u2019<\/em><\/td>\r\n<td style=\"text-align: left;\">=<\/td>\r\n<td>specific storage of the aquitard (L<sup>\u20111<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>b\u2019<\/em><\/td>\r\n<td style=\"text-align: left;\">=<\/td>\r\n<td>thickness of the aquitard(L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>K\u2019<\/em><\/td>\r\n<td style=\"text-align: left;\">=<\/td>\r\n<td>hydraulic conductivity of the aquitard (isotropic) (LT<sup>\u20111<\/sup>)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThe time computed by Equation\u00a0<a href=\"#X21c12aae5a3aae64a718396d349dd0a421352b0\">(129)<\/a> is considered overly conservative (Neuman &amp; Witherspoon, 1969). They suggest that the early\u2011time limit can be determined by observing when the straight\u2011line portion of a log\u2011log time\u2011drawdown plot for an observation well in the aquitard begins to depart from a linear slope.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nAt a radial distance (<em>r<\/em>) and a selected time (<em>t<\/em>), drawdown (<em>s<\/em>) is measured in a monitoring well in the water\u2011bearing unit and drawdown (<em>s\u2019<\/em>) is measured in a monitoring well or piezometer located in the aquitard. Both wells are located at the same radial distance from the pumping well. Neuman and Witherspoon (1972) assumed that the piezometer in the aquitard is represented as a point (the midpoint of the open interval) and head changes are instantaneous. They developed several curves that relate the drawdown ratio to parameters, <em>t<sub>D<\/sub><\/em><sub>,<\/sub> and <em>t<sub>D<\/sub>\u2019<\/em> (<a href=\"#Xbf4ebcb5bfaa915ba000ff4369fd5e93fb794b2\">Figure 142<\/a>). As stated previously, additional curves for matching can be generated from the appropriate analytical solution.\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: 4.5in; height: 5.47in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file148-2.jpg\" alt=\"\" \/>\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=\"Xb56b533c306c47a0428677f18c4168aa2b7f749\" class=\"anchor\"><strong>Figure 143 \u2011<\/strong> Neuman and Witherspoon (1972) ratio\u2011method plots showing the relationship between the drawdown in the pumped unit and an observation well (point source) in an aquitard (<em>s\u2019<\/em>\/<em>s<\/em>), and factors <em>t<sub>D<\/sub><\/em> and <em>t<sub>D<\/sub>\u2019<\/em>. <em>t<sub>D<\/sub><\/em> values between 10<sup>2<\/sup> and 10<sup>10<\/sup> are very close together and can be interpolated (modified from Neuman &amp; Witherspoon, 1972; Rowe &amp; Nadarajah, 1993).<\/span>\r\n\r\n<\/div><\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nParameters are defined as shown in Equations <a href=\"#Xecc6de8ff30942a9e7e3427bac2830b44bb3489\">(130)<\/a> and <a href=\"#Xf1a7baf90d3581f718f63f7f25184ca8e22f1e6\">(131)<\/a>.\r\n\r\n<\/div>\r\n<table class=\"gw-eq-table\" style=\"width: 100%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 10%;\" \/> <col style=\"width: 75%;\" \/> <col style=\"width: 15%;\" \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<th><\/th>\r\n<th style=\"text-align: center;\">[latex]t_{D} = \\frac{Tt}{Sr^{2}}[\/latex]<\/th>\r\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"Xecc6de8ff30942a9e7e3427bac2830b44bb3489\" class=\"anchor\">(130)<\/span><\/th>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nwhere:\r\n\r\n<\/div>\r\n<table style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 14%;\" \/> <col style=\"width: 2%;\" \/> <col style=\"width: 68%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\"><em>T<\/em><\/th>\r\n<th style=\"text-align: left;\">=<\/th>\r\n<th>pumped unit transmissivity (L<sup>2<\/sup>T<sup>\u20111<\/sup>)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>t<\/em><\/td>\r\n<td style=\"text-align: left;\">=<\/td>\r\n<td>time since the pumping began (T)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>S<\/em><\/td>\r\n<td style=\"text-align: left;\">=<\/td>\r\n<td>storativity of the pumped unit (dimensionless)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>r<\/em><\/td>\r\n<td style=\"text-align: left;\">=<\/td>\r\n<td>radial distance to the aquitard piezometer (L)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nand\r\n\r\n<\/div>\r\n<table class=\"gw-eq-table\" style=\"width: 100%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 10%;\" \/> <col style=\"width: 75%;\" \/> <col style=\"width: 15%;\" \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<th><\/th>\r\n<th style=\"text-align: center;\">[latex]t_{D}' = \\frac{K't}{S_{s}z^{2}}[\/latex]<\/th>\r\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"Xf1a7baf90d3581f718f63f7f25184ca8e22f1e6\" class=\"anchor\">(131)<\/span><\/th>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nwhere:\r\n\r\n<\/div>\r\n<table style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 14%;\" \/> <col style=\"width: 2%;\" \/> <col style=\"width: 68%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\"><em>K\u2019<\/em><\/th>\r\n<th style=\"text-align: left;\">=<\/th>\r\n<th>aquitard vertical hydraulic conductivity (LT<sup>\u20111<\/sup>)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>t<\/em><\/td>\r\n<td style=\"text-align: left;\">=<\/td>\r\n<td>time since pumping began (T)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>S<sub>s<\/sub>\u2019<\/em><\/td>\r\n<td style=\"text-align: left;\">=<\/td>\r\n<td>specific storage of aquitard (L<sup>\u20111<\/sup>))<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>z<\/em><\/td>\r\n<td style=\"text-align: left;\">=<\/td>\r\n<td>vertical distance from the top of the aquifer to the aquitard piezometer (L)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nWhen <em>t<sub>D<\/sub><\/em> is less than 100, estimates of <em>t<sub>D<\/sub>\u2019<\/em> should use the relationships shown in <a href=\"#Xb56b533c306c47a0428677f18c4168aa2b7f749\">Figure 143<\/a>. If <em>t<sub>D<\/sub><\/em> is greater than 100, multiple curves plot close together and estimated values of <em>t<sub>D<\/sub>\u2019<\/em> are often determined by using the curve labeled 10<sup>2<\/sup>. This will yield acceptable results with a small error (e.g., Neuman &amp; Witherspoon, 1972; Rowe &amp; Nadarajah, 1993).\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThe ratio method requires that the estimates of <em>T<\/em> and <em>S<\/em> be computed for the water\u2011bearing unit being pumped using early\u2011time data when leakage rates and volumes are low. Often a Theis analysis can be applied.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nIdeally, a standard monitoring well is installed in the pumping unit at the same radial distance from the pumping well as the monitoring well in the confining unit. Again, at early\u2011time, the ratio of drawdown in the aquitard well and aquifer\u2011monitoring well, <em>s\u2019<\/em>\/<em>s<\/em>, is computed. The relationships of <em>s\u2019<\/em>\/<em>s<\/em> and the computed <em>t<sub>D<\/sub><\/em> value are used to determine a value of <em>t<sub>D<\/sub>\u2019<\/em> by reading from the x axis of <a href=\"#Xb56b533c306c47a0428677f18c4168aa2b7f749\">Figure 143<\/a>. The unknown variable is <em>K\u2019<\/em>\/<em>S<sub>s<\/sub>\u2019<\/em>. A value of <em>S<sub>s<\/sub>\u2019<\/em> is derived from consolidation testing as described in Box 9.3 or the literature as explained in Box 2. Once <em>S<sub>s<\/sub>\u2019<\/em> is estimated, a value of <em>K\u2019<\/em> is calculated. Neuman and Witherspoon (1972) reported values of <em>K\u2019<\/em>\/<em>S<sub>s<\/sub>\u2019<\/em> as representing the portion of the aquitard defined by the distance <em>z<\/em> (e.g., the portion of the aquitard within the distance <em>z<\/em> from the aquifer unit which may overlie or underlie the aquitard).\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nRowe and Nadarajah (1993) modified Neuman and Witherspoon\u2019s (1972) ratio method by accounting for the perforated length of an aquitard\u2011monitoring well. This contrasts with the point\u2011location assumption of Neuman and Witherspoon (1972) as shown in <a href=\"#Xbf4ebcb5bfaa915ba000ff4369fd5e93fb794b2\">Figure 142<\/a>. They note that the ratio method used the assumption that the piezometer had an immediate response to pressure change in the aquifer. Rowe and Nadarajah (1993) addressed the effects of using finite\u2011length perforated intervals in aquitard wells and presented graphics with correction factors for aquitard\u2011monitoring well\u2011drawdown data. The measurement location is represented as the midpoint of the screened interval. They also examined the effect of aquitard\u2011monitoring wells located near the top of the pumping\u2011unit boundary as expressed in Equation\u00a0<a href=\"#Xf67dc390446fc18c6c5cda27f9276e669d0129d\">(132)<\/a>. Rowe and Nadarajah (1993) provide more details in their publication.\r\n\r\n<\/div>\r\n<table class=\"gw-eq-table\" style=\"width: 100%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 10%;\" \/> <col style=\"width: 75%;\" \/> <col style=\"width: 15%;\" \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<th><\/th>\r\n<th style=\"text-align: center;\">[latex]K' = \\frac{t_{D}'\\ S_{s}'\\ z^{2}}{t}\\frac{B_{2}^{2}}{B_{1}}[\/latex]<\/th>\r\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"Xf67dc390446fc18c6c5cda27f9276e669d0129d\" class=\"anchor\">(132)<\/span><\/th>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nwhere:\r\n\r\n<\/div>\r\n<table style=\"width: 85%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 14%;\" \/> <col style=\"width: 2%;\" \/> <col style=\"width: 68%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\"><em>t\u2019<sub>D<\/sub><\/em><\/th>\r\n<th style=\"text-align: left;\">=<\/th>\r\n<th><em>K\u2019t<\/em>\/(<em>Ss\u2019z<\/em><sup>2<\/sup>) from Figure 1 in Rowe and Nadarajah (1993), which is the same as <a href=\"#Xbf4ebcb5bfaa915ba000ff4369fd5e93fb794b2\">Figure 142<\/a> in this book (L<sup>2<\/sup>)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>S<sub>s<\/sub>\u2019<\/em><\/td>\r\n<td style=\"text-align: left;\">=<\/td>\r\n<td>specific storage of aquitard (L<sup>\u20111<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>z<\/em><\/td>\r\n<td style=\"text-align: left;\">=<\/td>\r\n<td>distance from the aquifer to the midpoint of the piezometer screen in the aquitard (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>t<\/em><\/td>\r\n<td style=\"text-align: left;\">=<\/td>\r\n<td>elapsed time since pumping began (T)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>B<sub>1<\/sub><\/em><\/td>\r\n<td style=\"text-align: left;\">=<\/td>\r\n<td>correction factor for time (Figure 5 in Rowe and Nadarajah, 1993) (dimensionless)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>B<sub>2<\/sub><\/em><\/td>\r\n<td style=\"text-align: left;\">=<\/td>\r\n<td>correction factors for time (Figure 8 in Rowe and Nadarajah, 1993) (dimensionless)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nFigures for Rowe and Nadarajah (1993) correction factors are reproduced in <span id=\"TextLinkToBox10\" class=\"anchor\"><a href=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/box-10-reproduction-of-figures-from-rowe-and-nadarajah-1993-correction-factors\/\">Box 10<\/a><\/span> for Groundwater Project readers who do not have access to commercial journals.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nIn some settings, a number of monitoring wells are placed in an aquitard and the underlying or overlying aquifer is pumped to qualitatively determine if the aquitard is homogeneous or contains heterogeneities such as high\u2011permeability zones or fractures. Grisak and Cherry (1975) conducted a sequence of pumping tests with wells in an aquifer overlain by a surficial clay\u2011rich aquitard. Multi\u2011level monitoring wells were placed in the aquitard. By comparing the rates and magnitudes of responses they interpreted that some wells were finished in blocks of unfractured matrix and others intersected fractures in the aquitard. They then used Hantush analyses of the pumping tests to derive broad\u2011scale <em>K\u2019<\/em> values for the aquitards and finite\u2011element modeling to adjust estimates of <em>K\u2019<\/em> so model results matched observed observation\u2011well water levels.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nPumping Tests with Observation Wells in Adjacent Productive Units\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nPumping tests conducted in productive water\u2011bearing units confined by aquitards yield information on the hydraulic properties of aquitards. If water\u2011level responses to pumping mirror the Thesis solution, then the aquitards are acting as impermeable units. In these settings, leakage rates through the aquitard and water released from aquitard storage are sufficiently small and the aquitard is considered impermeable (aquiclude).\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nIn contrast, if aquifer drawdowns reflect the addition of water via the aquitard, then pumping\u2011test analyses can provide information on the vertical hydraulic conductivity of the aquitard and, in some cases, the storage properties of the adjacent confining unit.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nWhen pumping\u2011test results match the Hantush\u2011Jacob or Hantush models for leaky aquifers the general properties of the associated aquitards can be determined as illustrated in <a href=\"#Xc9d4da791bc66eb9cfcf027f8516c8d09dd5a51\">Figure 144<\/a>, which repeats the images of <a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/hantush-equation-for-a-leaky-confined-system-with-water-released-from-confining-bed-storage\/#X6b6c2ec720b85923f014beffc01d41c6be08513\">Figure 49<\/a> for the readers\u2019 convenience in recalling the groundwater\u2011system settings. Analysis methods are discussed in Section 9 and are not restated here.\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: 5.83333in; height: 4.64722in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file53-2.jpg\" alt=\"\" \/>\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=\"Xc9d4da791bc66eb9cfcf027f8516c8d09dd5a51\" class=\"anchor\"><strong>Figure 144 \u2011<\/strong> Standard pumping\u2011test methods that yield hydrogeologic properties of the confining units that are associated with a confined water\u2011bearing unit.<\/span>\r\n\r\n<\/div><\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>a) Drawdown in a confined aquifer that receives water from storage in one or two confining beds when the beds overlying and underlying the confining beds are not permeable.<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>b) Drawdown in a confined aquifer that receives water from storage in overlying and\/or underlying confining beds, as well as leakage of water through the confining bed from a highly transmissive unit such as an unconfined water\u2011table aquifer.<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"Block Text\">\r\n<blockquote>c) Drawdown in a confined aquifer receiving water via direct flow through a confining bed that has negligible storage from a highly transmissive aquifer above and\/or below the confining bed.<\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nAquitards generally restrict the movement of groundwater and contaminants. When building conceptual models, developing water budgets, mapping flow paths, assessing resident times, and constructing three\u2011dimensional numerical models, it is as important to establish the properties of aquitards as well as those of aquifers. This section provides basic methods used to characterize aquitards. Additional methods can be found in the literature and the reader is directed to these when assessing more complex settings (e.g., Cherry et al., 2006; Bradbury et al., 2006).\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div id=\"test-methods-used-to-estimate-aquitard-properties\" class=\"level1\">\n<div data-custom-style=\"First Paragraph\">\n<div data-custom-style=\"First Paragraph\">\n<p>This section focuses on the physical characterization of hydrogeologic properties of aquitards. In this book, we purposefully use the terms pumping test and hydraulic test instead of aquifer test. Conceptually, the pumping\u2011test techniques described in previous sections are applicable to both highly conductive and, in some settings, lower\u2011permeability units that may be considered aquitards. This section addresses some of the specific methods used to characterize the hydrogeologic properties of aquitards. Cherry and others (2006) and Bradbury and others (2006) also describe methods used to evaluate the integrity of aquitards as this condition relates to the potential for migration of contaminants from an impacted source through an aquitard.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Cherry and others (2006) suggest that characterization of aquitard hydraulic properties can be grouped into the application of external and internal methods. Internal methods include laboratory methods applied to samples and cores of the aquitard, performing hydraulic tests within a borehole penetrating the aquitard, and observing head responses to recharge or loading. External methods involve pumping a productive water\u2011bearing unit above or below an aquitard and then interpreting time\u2011drawdown data from observation wells in the principal water\u2011bearing unit and, in some cases, in the adjacent aquifer and\/or aquitard.<\/p>\n<\/div>\n<div id=\"internal-methods\" class=\"level3\">\n<h3>Internal Methods<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>Direct internal investigation of an aquitard includes collecting and testing sample cores as well as installing wells and conducting pumping and\/or slug tests. Indirect internal methods include monitoring the response of the near surface unconsolidated aquitard material to recharge events and monitoring the aquitard\u2019s response to external loading.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Laboratory Methods<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Researchers have observed that hydraulic\u2011property values estimated from laboratory investigations of samples of uniform aquitards without significant secondary affects (developed macropores and\/or fracturing) may be equivalent to those derived from field\u2011scale methods (Cherry et al., 2006). However, laboratory\u2011scale samples of heterogeneous materials may poorly represent field\u2011scale properties, as the density and general interconnectedness of zones of different lithologies and secondary permeability features may be missing, overrepresented, or underrepresented at the laboratory scale. In some cases, laboratory samples are repacked and yield unrepresentative results. When any of these conditions occur, laboratory\u2011derived hydraulic conductivity values often underestimate field\u2011scale permeability (Haefner, 2000; Neuzil, 1986; van der Kamp, 2001).<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p><span id=\"TextLinkToBox9\"><\/span><a href=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/box-9-laboratory-methods-used\/\">Box 9<\/a> illustrates some of the laboratory methods such as the falling\u2011head permeameter, triaxial cell, and consolidometer that are used to derive hydraulic conductivities of low\u2011permeability material.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Field Methods<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>The following subsections describe internal field methods for characterizing hydrogeologic properties of aquitards.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Pumping tests<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Conceptually, a pumping well and observation wells could be constructed in an aquitard. In most settings, the low yield potential and permeability of the unit would limit the pumping rate and the distance at which observation wells would respond to pumping. In addition, if more permeable units underlie or overlie the aquitard, they must be accounted for in the analysis. In some settings, pumping a single well at a low constant rate with measurements of drawdown and time may be most useful. This is discussed in Section 12. If a successful pumping test is conducted, drawdown responses in the aquitard would be analyzed using methods described in Sections 7 through 10.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Slug Tests<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Slug tests, as described in Section 14, determine internal horizontal hydraulic conductivities of screened portions of aquitards. Hydraulic\u2011conductivity values reflect the nature of the material within a few meters of the test interval. The local nature of the test may not capture conditions where secondary permeability or material heterogeneity dominate the formation. Slug testing does not provide direct measurement of the vertical hydraulic conductivity. Estimates of vertical hydraulic conductivity are often assumed to be an order of magnitude smaller than horizontal values in stratified rock or sediments. However, in some settings horizontal to vertical hydraulic conductivity ratios can be 1000:1 (Anderson et al., 2015). In contrast, aquitard vertical hydraulic conductivity may be greater than horizontal values in laterite or saprolite materials where remnant structures such as weathered quartz veins may be present (Cherry et al., 2004).<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Changes in the near\u2011well hydraulic conductivity can occur when wells and piezometers are installed in unlithified aquitards (e.g., Cherry et al., 2006). Drilling can smear silt and clay along the borehole wall, reducing permeability. This skin effect artificially slows the rate of the head response causing hydraulic conductivity values to be underestimated. If the boring, well\u2011completion, and development processes enhance the permeability, values of hydraulic conductivity at the well will be overestimated. Perforated intervals should be developed prior to testing (e.g., Sterrett, 2007).<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Slug\u2011test responses in aquitards are overdamped as discussed in Section 12. In aquitards, head changes during a slug test occur slowly and can take hours to days or months to fully recover. In some low\u2011permeability settings, the static water level after the drilling and completion of a piezometer or monitoring well may be below the fully recovered head. Often, collection of water\u2011level data as recovery occurs can be used as a slug\u2011out test and analyzed accordingly as explained in Section 14. Often in low\u2011permeability materials, monitoring wells with small diameters (smaller volumes to be filled) are used to observe head change. Aquitard well design must accommodate slugs or pneumatic methods, and water\u2011level monitoring tools.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Constant\u2011Drawdown Tests<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Characterization of a low\u2011permeable unit can also be evaluated using a constant\u2011drawdown method. This method usually involves the head in the well to rise or fall and then at some point during the recovery water is continuously injected or extracted at varying flow rates to keep the head constant. Data collected are the well\u2011design factors and the inflow or outflow rate, <em>Q<\/em>, needed to keep the unrecovered head (<em>H<\/em>) constant (e.g., Mieussens &amp; Ducasse, 1977; Tavenas et al., 1990; Neville &amp; Markle, 2000) as illustrated in <a href=\"#Xf33ba11a54bffa5f40d1303b43652b0e04c574d\">Figure 141<\/a>.<\/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: 3.98389in; height: 4.57391in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file146-2.jpg\" alt=\"\" \/><\/p>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote>\n<div class=\"figcaption-text\">\n<p><span id=\"Xf33ba11a54bffa5f40d1303b43652b0e04c574d\" class=\"anchor\"><strong>Figure 141 \u2011\u00a0<\/strong>Slug\u2011out constant drawdown test set up for a monitoring well. The dashed blue line is the static water level. A slug of water is removed from the well (<em>H<sub>0<\/sub><\/em>) and at some time during the recovery the unrecovered head, <em>H<\/em>, in the well is stabilized by extracting volumes of water over time. The test time, <em>t<\/em>, starts when the variable discharge begins. <em>L<\/em> is the length of the perforated interval and <em>d<\/em> is the diameter.<\/span><\/p>\n<\/div>\n<\/blockquote>\n<\/div>\n<div data-custom-style=\"First Paragraph\">\n<p>Tavenas and others (1990) developed a method to analyze the data using plots of <em>Q<\/em>(<em>t<\/em>) versus 1\/<em>t<\/em><sup>0.5<\/sup>. This relationship should be linear after 5 to 10 minutes from the test initiation. Using an arithmetic plot, a line is fitted to the linear portion of the data and then extrapolated to 1\/<em>t<\/em><sup>0.5<\/sup> = 0. The steady\u2011state flow, <em>Q<sub>infinity<\/sub><\/em>, is determined by the intersection with the y axis. Using the steady\u2011state flow rate, <em>K<sub>h<\/sub><\/em> is computed as shown in Equation\u00a0<a href=\"#Xf2e02eff9727819a74b4f4dcfeece82a6d2f73d\">(127)<\/a>.<\/p>\n<\/div>\n<table class=\"gw-eq-table\" style=\"width: 100%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 10%;\" \/>\n<col style=\"width: 75%;\" \/>\n<col style=\"width: 15%;\" \/><\/colgroup>\n<tbody>\n<tr>\n<th><\/th>\n<th style=\"text-align: center;\">[latex]K_{h} = \\ \\frac{Q_{infinity}}{F\\ H}[\/latex]<\/th>\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"Xf2e02eff9727819a74b4f4dcfeece82a6d2f73d\" class=\"anchor\">(127)<\/span><\/th>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>where:<\/p>\n<\/div>\n<table style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 14%;\" \/>\n<col style=\"width: 2%;\" \/>\n<col style=\"width: 68%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\"><em>K<sub>h<\/sub><\/em><\/th>\n<th style=\"text-align: left;\">=<\/th>\n<th>horizontal hydraulic conductivity (LT<sup>\u20111<\/sup>)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><em>Q<sub>infinitity<\/sub><\/em><\/td>\n<td style=\"text-align: left;\">=<\/td>\n<td>Steady\u2011state flow rate from graph (L<sup>3<\/sup>T<sup>\u20111<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>F<\/em><\/td>\n<td style=\"text-align: left;\">=<\/td>\n<td>shape factor for the monitoring well (function of radius, screen length and other factors) (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>H<\/em><\/td>\n<td style=\"text-align: left;\">=<\/td>\n<td>unrecovered head as illustrated in <a href=\"#Xf33ba11a54bffa5f40d1303b43652b0e04c574d\">Figure 141<\/a> (L)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>The shape factor can be approximated as suggested by Hvorslev (1951) and shown in Equation\u00a0<a href=\"#X8e75ea4e1d347c05f6ca9b4cbea9d8d2ffa48db\">(128)<\/a>.<\/p>\n<\/div>\n<table class=\"gw-eq-table\" style=\"width: 100%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 10%;\" \/>\n<col style=\"width: 75%;\" \/>\n<col style=\"width: 15%;\" \/><\/colgroup>\n<tbody>\n<tr>\n<th><\/th>\n<th style=\"text-align: center;\">[latex]F\\ = \\ \\frac{2\\ \\pi\\ L}{\\ln\\left( \\frac{L}{d} + \\ \\left( 1 + \\left( \\frac{L}{d} \\right)^{2} \\right)^{.5} \\right)}[\/latex]<\/th>\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"X8e75ea4e1d347c05f6ca9b4cbea9d8d2ffa48db\" class=\"anchor\">(128)<\/span><\/th>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>where:<\/p>\n<\/div>\n<table style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 14%;\" \/>\n<col style=\"width: 2%;\" \/>\n<col style=\"width: 68%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\"><em>L<\/em><\/th>\n<th style=\"text-align: left;\">=<\/th>\n<th>perforated length (L)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><em>d<\/em><\/td>\n<td style=\"text-align: left;\">=<\/td>\n<td>diameter of the perforated interval (L)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>Tavenas and others (1990) provide additional discussion of parameters. The Jacob\u2011Lohman (1952) solution as outlined by Lohman (1972) may also be applicable for analyzing constant\u2011head tests. This method assumes a constant head from a flowing well where the change in discharge over time is recorded.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>When aquitard material lies at or near the surface, permeability has been investigated using other methods referenced by Cherry and others (2006). These include monitoring and analyzing head changes within the aquitard induced by<\/p>\n<\/div>\n<ul>\n<li>Natural, precipitation\u2011induced recharge (e.g., Davis, 1972; Keller et al., 1989; Boldt\u2011Leppin &amp; Hendry, 2003),<\/li>\n<li>flow into a large, augured cavity (Keller et al., 1989), and<\/li>\n<li>construction loading and unloading (van der Kamp &amp; Maathuis, 1985).<\/li>\n<\/ul>\n<div data-custom-style=\"First Paragraph\">\n<p>Aquitard properties can also be estimated based on the calibration of one\u2011dimensional and three\u2011dimensional numerical models given hydraulic test data from a site (e.g., Pavelko, 2004; Hart et al., 2005; Anderson et al., 2015).<\/p>\n<\/div>\n<\/div>\n<div id=\"external-methods\" class=\"level3\">\n<h3>External Methods<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>External methods include pumping adjacent water\u2011bearing units and observing the changes in water levels in the aquitard or in monitoring wells in the pumped aquifer. Examples of these methods are presented in the following subsections.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p><strong>Pumping Test with An Observation Well in the Aquitar<\/strong>d<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Neuman and Witherspoon (1972) develop a method to determine the diffusivity, <em>K\u2019<\/em>\/<em>S<sub>s<\/sub>\u2019<\/em> (aquitard horizontal hydraulic conductivity and specific storage), when a pumping test is conducted in the underlying or overlying aquifer and a monitoring well or piezometer is installed in the aquitard in concert with a monitoring well in the aquifer as shown in <a href=\"#Xbf4ebcb5bfaa915ba000ff4369fd5e93fb794b2\">Figure 142<\/a>. The aquitard well and aquifer\u2011observation well are at the same radial distance from the pumping well. When an aquifer monitoring well is not present, the drawdown in the pumped aquifer can be projected for the specified radial distance using a Theis analysis of the test data (Neuman &amp; Witherspoon, 1972).<\/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: 4.83096in; height: 4.5974in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file147-2.jpg\" alt=\"\" \/><\/p>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote>\n<div class=\"figcaption-text\">\n<p><span id=\"Xbf4ebcb5bfaa915ba000ff4369fd5e93fb794b2\" class=\"anchor\"><strong>Figure 142 \u2011<\/strong> Schematic of a pumping test conducted in an aquifer and observation wells in that unit (well B) as well as the overlying aquitard (well C). <em>z<\/em> is the distance from the top of the water bearing unit to the open interval of the observation well in the aquitard. Neuman and Witherspoon (1972) assumed the observation well opening was represented by a point (indicated by A). Rowe and Nadarajah (1993) accounted for the length of well screen where <em>z<\/em> is the distance from the boundary to the center of the well screen (dashed) (C). The radial distance to the observation well is <em>r<\/em>. <em>b\u2019<\/em> is the thickness of the aquitard unit. At a time after the pumping starts and when drawdown in the aquitard is observed, values of drawdown <em>s\u2019<\/em> in the aquitard and <em>s<\/em> in the unit being pumped (measured from the initial static water level elevation) are recorded.<\/span><\/p>\n<\/div>\n<\/blockquote>\n<\/div>\n<div data-custom-style=\"First Paragraph\">\n<p>The method assumes units being pumped and aquitards are infinite in lateral extent and the monitoring wells are located within about 60 m of the pumping well. Detailed observations of drawdown should be collected at early test times. The usable early\u2011time response of drawdown in the main pumped water\u2011bearing unit and aquitard are conservatively constrained as indicated by Equation\u00a0<a href=\"#X21c12aae5a3aae64a718396d349dd0a421352b0\">(129)<\/a>.<\/p>\n<\/div>\n<table class=\"gw-eq-table\" style=\"width: 100%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 10%;\" \/>\n<col style=\"width: 75%;\" \/>\n<col style=\"width: 15%;\" \/><\/colgroup>\n<tbody>\n<tr>\n<th><\/th>\n<th style=\"text-align: center;\">[latex]t\\ \\leq \\ 0.1\\ \\frac{S_{s}'\\ {b'}^{2}}{K\u2019}[\/latex]<\/th>\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"X21c12aae5a3aae64a718396d349dd0a421352b0\" class=\"anchor\">(129)<\/span><\/th>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>where:<\/p>\n<\/div>\n<table style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 14%;\" \/>\n<col style=\"width: 2%;\" \/>\n<col style=\"width: 68%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\"><em>t<\/em><\/th>\n<th style=\"text-align: left;\">=<\/th>\n<th>the time since the beginning of the pumping (T)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><em>S<sub>s<\/sub>\u2019<\/em><\/td>\n<td style=\"text-align: left;\">=<\/td>\n<td>specific storage of the aquitard (L<sup>\u20111<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>b\u2019<\/em><\/td>\n<td style=\"text-align: left;\">=<\/td>\n<td>thickness of the aquitard(L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>K\u2019<\/em><\/td>\n<td style=\"text-align: left;\">=<\/td>\n<td>hydraulic conductivity of the aquitard (isotropic) (LT<sup>\u20111<\/sup>)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>The time computed by Equation\u00a0<a href=\"#X21c12aae5a3aae64a718396d349dd0a421352b0\">(129)<\/a> is considered overly conservative (Neuman &amp; Witherspoon, 1969). They suggest that the early\u2011time limit can be determined by observing when the straight\u2011line portion of a log\u2011log time\u2011drawdown plot for an observation well in the aquitard begins to depart from a linear slope.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>At a radial distance (<em>r<\/em>) and a selected time (<em>t<\/em>), drawdown (<em>s<\/em>) is measured in a monitoring well in the water\u2011bearing unit and drawdown (<em>s\u2019<\/em>) is measured in a monitoring well or piezometer located in the aquitard. Both wells are located at the same radial distance from the pumping well. Neuman and Witherspoon (1972) assumed that the piezometer in the aquitard is represented as a point (the midpoint of the open interval) and head changes are instantaneous. They developed several curves that relate the drawdown ratio to parameters, <em>t<sub>D<\/sub><\/em><sub>,<\/sub> and <em>t<sub>D<\/sub>\u2019<\/em> (<a href=\"#Xbf4ebcb5bfaa915ba000ff4369fd5e93fb794b2\">Figure 142<\/a>). As stated previously, additional curves for matching can be generated from the appropriate analytical solution.<\/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: 4.5in; height: 5.47in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file148-2.jpg\" alt=\"\" \/><\/p>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote>\n<div class=\"figcaption-text\">\n<p><span id=\"Xb56b533c306c47a0428677f18c4168aa2b7f749\" class=\"anchor\"><strong>Figure 143 \u2011<\/strong> Neuman and Witherspoon (1972) ratio\u2011method plots showing the relationship between the drawdown in the pumped unit and an observation well (point source) in an aquitard (<em>s\u2019<\/em>\/<em>s<\/em>), and factors <em>t<sub>D<\/sub><\/em> and <em>t<sub>D<\/sub>\u2019<\/em>. <em>t<sub>D<\/sub><\/em> values between 10<sup>2<\/sup> and 10<sup>10<\/sup> are very close together and can be interpolated (modified from Neuman &amp; Witherspoon, 1972; Rowe &amp; Nadarajah, 1993).<\/span><\/p>\n<\/div>\n<\/blockquote>\n<\/div>\n<div data-custom-style=\"First Paragraph\">\n<p>Parameters are defined as shown in Equations <a href=\"#Xecc6de8ff30942a9e7e3427bac2830b44bb3489\">(130)<\/a> and <a href=\"#Xf1a7baf90d3581f718f63f7f25184ca8e22f1e6\">(131)<\/a>.<\/p>\n<\/div>\n<table class=\"gw-eq-table\" style=\"width: 100%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 10%;\" \/>\n<col style=\"width: 75%;\" \/>\n<col style=\"width: 15%;\" \/><\/colgroup>\n<tbody>\n<tr>\n<th><\/th>\n<th style=\"text-align: center;\">[latex]t_{D} = \\frac{Tt}{Sr^{2}}[\/latex]<\/th>\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"Xecc6de8ff30942a9e7e3427bac2830b44bb3489\" class=\"anchor\">(130)<\/span><\/th>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>where:<\/p>\n<\/div>\n<table style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 14%;\" \/>\n<col style=\"width: 2%;\" \/>\n<col style=\"width: 68%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\"><em>T<\/em><\/th>\n<th style=\"text-align: left;\">=<\/th>\n<th>pumped unit transmissivity (L<sup>2<\/sup>T<sup>\u20111<\/sup>)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><em>t<\/em><\/td>\n<td style=\"text-align: left;\">=<\/td>\n<td>time since the pumping began (T)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>S<\/em><\/td>\n<td style=\"text-align: left;\">=<\/td>\n<td>storativity of the pumped unit (dimensionless)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>r<\/em><\/td>\n<td style=\"text-align: left;\">=<\/td>\n<td>radial distance to the aquitard piezometer (L)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>and<\/p>\n<\/div>\n<table class=\"gw-eq-table\" style=\"width: 100%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 10%;\" \/>\n<col style=\"width: 75%;\" \/>\n<col style=\"width: 15%;\" \/><\/colgroup>\n<tbody>\n<tr>\n<th><\/th>\n<th style=\"text-align: center;\">[latex]t_{D}' = \\frac{K't}{S_{s}z^{2}}[\/latex]<\/th>\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"Xf1a7baf90d3581f718f63f7f25184ca8e22f1e6\" class=\"anchor\">(131)<\/span><\/th>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>where:<\/p>\n<\/div>\n<table style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 14%;\" \/>\n<col style=\"width: 2%;\" \/>\n<col style=\"width: 68%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\"><em>K\u2019<\/em><\/th>\n<th style=\"text-align: left;\">=<\/th>\n<th>aquitard vertical hydraulic conductivity (LT<sup>\u20111<\/sup>)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><em>t<\/em><\/td>\n<td style=\"text-align: left;\">=<\/td>\n<td>time since pumping began (T)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>S<sub>s<\/sub>\u2019<\/em><\/td>\n<td style=\"text-align: left;\">=<\/td>\n<td>specific storage of aquitard (L<sup>\u20111<\/sup>))<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>z<\/em><\/td>\n<td style=\"text-align: left;\">=<\/td>\n<td>vertical distance from the top of the aquifer to the aquitard piezometer (L)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>When <em>t<sub>D<\/sub><\/em> is less than 100, estimates of <em>t<sub>D<\/sub>\u2019<\/em> should use the relationships shown in <a href=\"#Xb56b533c306c47a0428677f18c4168aa2b7f749\">Figure 143<\/a>. If <em>t<sub>D<\/sub><\/em> is greater than 100, multiple curves plot close together and estimated values of <em>t<sub>D<\/sub>\u2019<\/em> are often determined by using the curve labeled 10<sup>2<\/sup>. This will yield acceptable results with a small error (e.g., Neuman &amp; Witherspoon, 1972; Rowe &amp; Nadarajah, 1993).<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>The ratio method requires that the estimates of <em>T<\/em> and <em>S<\/em> be computed for the water\u2011bearing unit being pumped using early\u2011time data when leakage rates and volumes are low. Often a Theis analysis can be applied.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Ideally, a standard monitoring well is installed in the pumping unit at the same radial distance from the pumping well as the monitoring well in the confining unit. Again, at early\u2011time, the ratio of drawdown in the aquitard well and aquifer\u2011monitoring well, <em>s\u2019<\/em>\/<em>s<\/em>, is computed. The relationships of <em>s\u2019<\/em>\/<em>s<\/em> and the computed <em>t<sub>D<\/sub><\/em> value are used to determine a value of <em>t<sub>D<\/sub>\u2019<\/em> by reading from the x axis of <a href=\"#Xb56b533c306c47a0428677f18c4168aa2b7f749\">Figure 143<\/a>. The unknown variable is <em>K\u2019<\/em>\/<em>S<sub>s<\/sub>\u2019<\/em>. A value of <em>S<sub>s<\/sub>\u2019<\/em> is derived from consolidation testing as described in Box 9.3 or the literature as explained in Box 2. Once <em>S<sub>s<\/sub>\u2019<\/em> is estimated, a value of <em>K\u2019<\/em> is calculated. Neuman and Witherspoon (1972) reported values of <em>K\u2019<\/em>\/<em>S<sub>s<\/sub>\u2019<\/em> as representing the portion of the aquitard defined by the distance <em>z<\/em> (e.g., the portion of the aquitard within the distance <em>z<\/em> from the aquifer unit which may overlie or underlie the aquitard).<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Rowe and Nadarajah (1993) modified Neuman and Witherspoon\u2019s (1972) ratio method by accounting for the perforated length of an aquitard\u2011monitoring well. This contrasts with the point\u2011location assumption of Neuman and Witherspoon (1972) as shown in <a href=\"#Xbf4ebcb5bfaa915ba000ff4369fd5e93fb794b2\">Figure 142<\/a>. They note that the ratio method used the assumption that the piezometer had an immediate response to pressure change in the aquifer. Rowe and Nadarajah (1993) addressed the effects of using finite\u2011length perforated intervals in aquitard wells and presented graphics with correction factors for aquitard\u2011monitoring well\u2011drawdown data. The measurement location is represented as the midpoint of the screened interval. They also examined the effect of aquitard\u2011monitoring wells located near the top of the pumping\u2011unit boundary as expressed in Equation\u00a0<a href=\"#Xf67dc390446fc18c6c5cda27f9276e669d0129d\">(132)<\/a>. Rowe and Nadarajah (1993) provide more details in their publication.<\/p>\n<\/div>\n<table class=\"gw-eq-table\" style=\"width: 100%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 10%;\" \/>\n<col style=\"width: 75%;\" \/>\n<col style=\"width: 15%;\" \/><\/colgroup>\n<tbody>\n<tr>\n<th><\/th>\n<th style=\"text-align: center;\">[latex]K' = \\frac{t_{D}'\\ S_{s}'\\ z^{2}}{t}\\frac{B_{2}^{2}}{B_{1}}[\/latex]<\/th>\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"Xf67dc390446fc18c6c5cda27f9276e669d0129d\" class=\"anchor\">(132)<\/span><\/th>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>where:<\/p>\n<\/div>\n<table style=\"width: 85%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 14%;\" \/>\n<col style=\"width: 2%;\" \/>\n<col style=\"width: 68%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\"><em>t\u2019<sub>D<\/sub><\/em><\/th>\n<th style=\"text-align: left;\">=<\/th>\n<th><em>K\u2019t<\/em>\/(<em>Ss\u2019z<\/em><sup>2<\/sup>) from Figure 1 in Rowe and Nadarajah (1993), which is the same as <a href=\"#Xbf4ebcb5bfaa915ba000ff4369fd5e93fb794b2\">Figure 142<\/a> in this book (L<sup>2<\/sup>)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><em>S<sub>s<\/sub>\u2019<\/em><\/td>\n<td style=\"text-align: left;\">=<\/td>\n<td>specific storage of aquitard (L<sup>\u20111<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>z<\/em><\/td>\n<td style=\"text-align: left;\">=<\/td>\n<td>distance from the aquifer to the midpoint of the piezometer screen in the aquitard (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>t<\/em><\/td>\n<td style=\"text-align: left;\">=<\/td>\n<td>elapsed time since pumping began (T)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>B<sub>1<\/sub><\/em><\/td>\n<td style=\"text-align: left;\">=<\/td>\n<td>correction factor for time (Figure 5 in Rowe and Nadarajah, 1993) (dimensionless)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>B<sub>2<\/sub><\/em><\/td>\n<td style=\"text-align: left;\">=<\/td>\n<td>correction factors for time (Figure 8 in Rowe and Nadarajah, 1993) (dimensionless)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>Figures for Rowe and Nadarajah (1993) correction factors are reproduced in <span id=\"TextLinkToBox10\" class=\"anchor\"><a href=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/box-10-reproduction-of-figures-from-rowe-and-nadarajah-1993-correction-factors\/\">Box 10<\/a><\/span> for Groundwater Project readers who do not have access to commercial journals.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>In some settings, a number of monitoring wells are placed in an aquitard and the underlying or overlying aquifer is pumped to qualitatively determine if the aquitard is homogeneous or contains heterogeneities such as high\u2011permeability zones or fractures. Grisak and Cherry (1975) conducted a sequence of pumping tests with wells in an aquifer overlain by a surficial clay\u2011rich aquitard. Multi\u2011level monitoring wells were placed in the aquitard. By comparing the rates and magnitudes of responses they interpreted that some wells were finished in blocks of unfractured matrix and others intersected fractures in the aquitard. They then used Hantush analyses of the pumping tests to derive broad\u2011scale <em>K\u2019<\/em> values for the aquitards and finite\u2011element modeling to adjust estimates of <em>K\u2019<\/em> so model results matched observed observation\u2011well water levels.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Pumping Tests with Observation Wells in Adjacent Productive Units<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Pumping tests conducted in productive water\u2011bearing units confined by aquitards yield information on the hydraulic properties of aquitards. If water\u2011level responses to pumping mirror the Thesis solution, then the aquitards are acting as impermeable units. In these settings, leakage rates through the aquitard and water released from aquitard storage are sufficiently small and the aquitard is considered impermeable (aquiclude).<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>In contrast, if aquifer drawdowns reflect the addition of water via the aquitard, then pumping\u2011test analyses can provide information on the vertical hydraulic conductivity of the aquitard and, in some cases, the storage properties of the adjacent confining unit.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>When pumping\u2011test results match the Hantush\u2011Jacob or Hantush models for leaky aquifers the general properties of the associated aquitards can be determined as illustrated in <a href=\"#Xc9d4da791bc66eb9cfcf027f8516c8d09dd5a51\">Figure 144<\/a>, which repeats the images of <a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/hantush-equation-for-a-leaky-confined-system-with-water-released-from-confining-bed-storage\/#X6b6c2ec720b85923f014beffc01d41c6be08513\">Figure 49<\/a> for the readers\u2019 convenience in recalling the groundwater\u2011system settings. Analysis methods are discussed in Section 9 and are not restated here.<\/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: 5.83333in; height: 4.64722in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file53-2.jpg\" alt=\"\" \/><\/p>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote>\n<div class=\"figcaption-text\">\n<p><span id=\"Xc9d4da791bc66eb9cfcf027f8516c8d09dd5a51\" class=\"anchor\"><strong>Figure 144 \u2011<\/strong> Standard pumping\u2011test methods that yield hydrogeologic properties of the confining units that are associated with a confined water\u2011bearing unit.<\/span><\/p>\n<\/div>\n<\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>a) Drawdown in a confined aquifer that receives water from storage in one or two confining beds when the beds overlying and underlying the confining beds are not permeable.<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>b) Drawdown in a confined aquifer that receives water from storage in overlying and\/or underlying confining beds, as well as leakage of water through the confining bed from a highly transmissive unit such as an unconfined water\u2011table aquifer.<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote><p>c) Drawdown in a confined aquifer receiving water via direct flow through a confining bed that has negligible storage from a highly transmissive aquifer above and\/or below the confining bed.<\/p><\/blockquote>\n<\/div>\n<div data-custom-style=\"First Paragraph\">\n<p>Aquitards generally restrict the movement of groundwater and contaminants. When building conceptual models, developing water budgets, mapping flow paths, assessing resident times, and constructing three\u2011dimensional numerical models, it is as important to establish the properties of aquitards as well as those of aquifers. This section provides basic methods used to characterize aquitards. Additional methods can be found in the literature and the reader is directed to these when assessing more complex settings (e.g., Cherry et al., 2006; Bradbury et al., 2006).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":8,"menu_order":71,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1803","chapter","type-chapter","status-publish","hentry"],"part":595,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1803","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":9,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1803\/revisions"}],"predecessor-version":[{"id":3768,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1803\/revisions\/3768"}],"part":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/parts\/595"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1803\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/media?parent=1803"}],"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=1803"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/contributor?post=1803"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/license?post=1803"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}