{"id":1725,"date":"2026-02-14T04:25:12","date_gmt":"2026-02-14T04:25:12","guid":{"rendered":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/ch057-xhtml\/"},"modified":"2026-10-21T04:05:37","modified_gmt":"2026-10-21T04:05:37","slug":"special-considerations-when-using-drawdown-data-from-a-pumping-well","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/special-considerations-when-using-drawdown-data-from-a-pumping-well\/","title":{"raw":"12.1 Special Considerations When Using Drawdown Data from a Pumping Well","rendered":"12.1 Special Considerations When Using Drawdown Data from a Pumping Well"},"content":{"raw":"<div id=\"special-considerations-when-using-drawdown-data-from-a-pumping-well\" class=\"level1\">\r\n<div data-custom-style=\"First Paragraph\">\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nUnder ideal conditions, analytical solutions (Sections 7 through 10) and their simplifying assumptions can be applied to single\u2011well pumping\u2011test data to determine corresponding values of <em>T<\/em> and <em>S<\/em>. However, wells need to be fully penetrating and 100 % efficient. When using drawdown in the pumping well for the analysis, the pumping well radius is often substituted in the equations that require a radial distance to an observation well. The screen or perforated\u2011casing radius is used unless the well construction includes an envelope of gravel pack. When this occurs, the well radius is defined as the effective well radius and includes the radius of the perforated interval and gravel pack.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nHowever, when a single pumping well is the only source of drawdown data, it must be recognized that the recorded drawdown data are often influenced by partial penetration, well\u2011bore storage, and well loss, which are functions of the well design. Partial\u2011penetration and well\u2011loss effects result in lower water levels in the pumping well as compared to the adjacent aquifer. Well\u2011bore\u2011storage effects temporarily reduce the rate of early water level decline when large diameter wells are pumping at low rates. Each of these conditions are usually present to some degree and directly affect the predicted or reported drawdown at the pumping well. As with observation well test records, pumping\u2011well drawdowns can be impacted by well interference, barometric, tidal, and, in unconfined cases, recharge events and direct evapotranspiration.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nMost often partial penetration and well\u2011loss conditions result in greater drawdown measured in the pumping well than if they were not present. As a result, computed <em>T<\/em> and <em>S<\/em> values are lower than properties of the formation. Ideally, pumping\u2011well drawdown would be corrected before analyses were undertaken.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nDuring the early\u2011time period of some pumping tests\u2014especially in large\u2011diameter wells\u2014pumped water is derived not only from the aquifer but also from water stored in the well casing or annular space (and filter pack) surrounding the screened or perforated interval. This results in a reduction in the observed drawdown in the pumping well and is referred to as a wellbore storage effect. As time progresses, formational water makes up an increasing portion of the well discharge until wellbore storage effects are sufficiently small, and flow is supplied by the aquifer. The influence of wellbore storage on drawdown data is most easily observed using log\u2011log plots of time\u2011drawdown data. The early\u2011time data will appear as a straight line diverging from the Theis curve. Papadopulos and Cooper (1967) developed an analytical method and a set of type curves that can be used to analyze confined\u2011aquifer\u2011test data affected by wellbore storage. Moench (1997) generated an analytical solution for pumping unconfined aquifers that accounts for well bore storage. He also provides a set of type curves.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThe total observed pumping\u2011well drawdown can be visualized as having several components as shown in <a href=\"#Xbdfb767a3839ac96403047b0d9d15425042ebaf\">Figure 79<\/a> and explained by Equation\u00a0<a href=\"#X4c56c45c5337f35abff2d4912fe1dcfbb8ea397\">(79)<\/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]s_{T} = s_{F\\ } + s_{pp\\ } + s_{WL\\ } + s_{I\\ } + s_{O\\ }[\/latex]<\/th>\r\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"X4c56c45c5337f35abff2d4912fe1dcfbb8ea397\" class=\"anchor\">(79)<\/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: 3%;\" \/> <col style=\"width: 67%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\"><em>s<sub>T<\/sub><\/em><\/th>\r\n<th style=\"text-align: center;\">=<\/th>\r\n<th>total observed drawdown in the pumping well (L)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>s<sub>F<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>component of drawdown based solely on the formational properties (<em>T<\/em>, <em>S<\/em>) (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>s<sub>pp<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>component of drawdown resulting from partial penetration of the perforated interval (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>s<sub>WL<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>component of drawdown from linear and non\u2011linear well loss (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>s<sub>I<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>component of drawdown from well interference (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>s<sub>O<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>component of drawdown from other factors with a positive or negative sign (e.g., wellbore storage, barometric effects, tides, recharge) (L)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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: 3.20833in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file83-1.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=\"Xbdfb767a3839ac96403047b0d9d15425042ebaf\" class=\"anchor\"><strong>Figure 79 \u2011<\/strong> Schematic of a partially penetrating pumping well in a confined aquifer (sand pattern) showing components of the total drawdown (<em>s<sub>T<\/sub><\/em>) as represented by the red profile. The formational drawdown (<em>s<sub>F<\/sub><\/em>) is shown in the blue drawdown profile and is the drawdown if the well is 100 percent efficient and fully penetrating. Additional components contributing to the total drawdown observed in the pumping well may include the effects of partial penetration which results in longer flow paths to the well as indicate by the blue flow lines (<em>s<sub>pp<\/sub><\/em>), well loss resulting from the well construction, design, and pumping rate (<em>s<sub>WL<\/sub><\/em>), drawdown from wells pumping nearby which is called well interference (<em>s<sub>I<\/sub><\/em>), and other factors causing the water levels to rise or fall such as wellbore storage and barometric effects (<em>s<sub>0<\/sub><\/em>).<\/span>\r\n\r\n<\/div><\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nThe measured water levels in the pumping well will be similar to the theoretical drawdown shown in blue on <a href=\"#Xbdfb767a3839ac96403047b0d9d15425042ebaf\">Figure 79<\/a> if:\r\n\r\n<\/div>\r\n<ul>\r\n \t<li>the pumping well is fully penetrating and 100 percent efficient,<\/li>\r\n \t<li>wellbore storage has been depleted<\/li>\r\n \t<li>there are no head losses due to turbulent flow caused by high flow rates and\/or obstructions in the well, and<\/li>\r\n \t<li>there is no interference from nearby pumping wells or other hydraulic stresses on the unit being tested.<\/li>\r\n<\/ul>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nThe following sections describe how the factors that require special consideration influence drawdown data in the pumping well and present methods used to quantify the magnitude of their effect and to correct the drawdown data.\r\n\r\n<\/div>\r\n<div id=\"partial-penetration\" class=\"level3\">\r\n<h3>Partial Penetration<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nA partially penetrating pumping well produces vertical components of flow near the well in both confined and unconfined aquifers. The vertical flow component creates longer flow paths that result in the flow system expending more energy to deliver water to the well than if all flow is horizontal. Partial\u2011penetration effects can occur in an unconfined aquifer as pumping begins even if a well is fully penetrating when the saturated thickness adjacent to the pumping well is reduced significantly during the test.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nTodd and Mays (2005) provide methods to compute the increase in drawdown attributed to the head loss during groundwater flow to a partially penetrating well. They report that if the perforated interval of the well penetrates 85 percent or more of the aquifer then correction is not necessary, but it is generally accepted that correction is not necessary if the screened portion of a well penetrates 80 percent or more of the aquifer thickness.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nWhen considering the effects of partial penetration only, the total drawdown is composed of the formational drawdown, <em>s<sub>F<\/sub><\/em>, and the additional drawdown \u2206<em>s<sub>pp<\/sub><\/em> caused by partial penetration as shown in Equation\u00a0<a href=\"#X8a962088b2297c02a791059b7f3536a840dcdf0\">(80)<\/a> and <a href=\"#X4f513417674715fefa1e005a0db23beb5283a33\">Figure 80<\/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]s_{T} = s_{F} + \\mathrm{\\Delta}s_{pp}[\/latex]<\/th>\r\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"X8a962088b2297c02a791059b7f3536a840dcdf0\" class=\"anchor\">(80)<\/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: 3%;\" \/> <col style=\"width: 67%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\"><em>s<sub>T<\/sub><\/em><\/th>\r\n<th style=\"text-align: center;\">=<\/th>\r\n<th>total drawdown in pumping well including effects of partial penetration and formational drawdown (L)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>s<sub>F<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>formational drawdown (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\">\u2206<em>s<sub>pp<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>drawdown due to partial penetration (L)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Captioned Figure\">\r\n\r\n<img style=\"display: block; margin-left: auto; margin-right: auto; ;width: 4.99525in; height: 2.75in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file84-1.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=\"X4f513417674715fefa1e005a0db23beb5283a33\" class=\"anchor\"><strong>Figure 80 \u2011<\/strong> Total drawdown, <em>s<sub>T<\/sub><\/em>, associated with a partially penetrating pumping well when both formational drawdown, <em>s<sub>F<\/sub><\/em>, and drawdown related to the effects of partial penetration, \u2206<em>s<sub>pp<\/sub><\/em> occur.<\/span>\r\n\r\n<\/div><\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nHuisman (1972) and Todd and Mays (2005) present methods to compute the effects of partial penetration using steady\u2011state formulations, the variables shown in <a href=\"#X4f513417674715fefa1e005a0db23beb5283a33\">Figure 80<\/a> and <a href=\"#Xc11fd70da34645e52b4f599ce8e6d55bf3d2ed8\">Figure 81<\/a>, and a penetration factor <em>p<\/em> = <em>h<sub>s<\/sub><\/em>\/<em>b<\/em>, screen length (<em>h<sub>s<\/sub><\/em>) divided by the saturated aquifer thickness (<em>b<\/em>).\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: 2.27562in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file85-1.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=\"Xc11fd70da34645e52b4f599ce8e6d55bf3d2ed8\" class=\"anchor\"><strong>Figure 81 \u2011<\/strong> Schematic of a confined aquifer (sand pattern) and the perforated interval (horizontal black lines) for two production wells. <em>b<\/em> represents the initial saturated thickness of the aquifer. <em>h<sub>s<\/sub><\/em> is the length of the perforated interval and <em>r<sub>w<\/sub><\/em> is the radius of the well (after Todd &amp; Mays, 2005).<\/span>\r\n\r\n<\/div><\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nThe additional drawdown associated with partial penetration under steady\u2011state conditions in a confined aquifer as represented on the left\u2011hand side of <a href=\"#X4f513417674715fefa1e005a0db23beb5283a33\">Figure 80<\/a> is defined for p\u00a0&gt;\u00a00.2 by Todd &amp; Mays (2003) in Equation\u00a0<a href=\"#Xd44f87d47e7555f427f8859b4b3fed147f86070\">(81)<\/a>. When the well screen is centered in the formation as shown in the right\u2011hand portion of <a href=\"#Xc11fd70da34645e52b4f599ce8e6d55bf3d2ed8\">Figure 81<\/a>, then <em>p<\/em> = (<em>h<sub>s<\/sub><\/em>\/2)\/(<em>b<\/em>\/2<em>)<\/em> and this ratio is substituted into Equation\u00a0<a href=\"#Xd44f87d47e7555f427f8859b4b3fed147f86070\">(81)<\/a> that becomes Equation\u00a0<a href=\"#X45c3b3df8c2d80bbcad1b2caaf270c353b75fe3\">(82)<\/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]\\mathrm{\\Delta}s_{pp} = \\frac{Q_{p}}{(2\\pi T)}\\left( \\frac{1 - p}{p} \\right)\\ln\\left( \\frac{(1 - p)h_{s}}{r_{w}} \\right)[\/latex]<\/th>\r\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"Xd44f87d47e7555f427f8859b4b3fed147f86070\" class=\"anchor\">(81)<\/span><\/th>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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<td><\/td>\r\n<td style=\"text-align: center;\">[latex]\\mathrm{\\Delta}s_{pp} = \\frac{Q_{p}}{(2\\pi T)}\\left( \\frac{1 - p}{p} \\right)\\ln\\left( \\frac{(1 - p)h_{s}}{2r_{w}} \\right)[\/latex]<\/td>\r\n<td style=\"text-align: right; white-space: nowrap;\"><span id=\"X45c3b3df8c2d80bbcad1b2caaf270c353b75fe3\" class=\"anchor\">(82)<\/span><\/td>\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: 3%;\" \/> <col style=\"width: 67%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\">\u2206<em>s<sub>pp<\/sub><\/em><\/th>\r\n<th style=\"text-align: center;\">=<\/th>\r\n<th>additional drawdown resulting from partial penetration (L)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>Q<sub>p<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>partially penetrating well pumping rate (L<sup>3<\/sup>T<sup>\u20111<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>T<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>transmissivity (L<sup>2<\/sup>T<sup>\u20111<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>p<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>penetration factor, <em>h<sub>s<\/sub><\/em>\/<em>b<\/em> (dimensionless)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>h<sub>s<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>length of well screen (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>b<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>saturated thickness of aquifer (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>r<sub>w<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>radius of the pumping well (L)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nBased on the Thiem equation for unconfined aquifers, the drawdown resulting from partial penetration is defined by Todd and Mays (2005) as shown in Equation\u00a0<a href=\"#X8252c2b0870bbb6dd4f2c0dac351ac510e09580\">(85)<\/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]\\mathrm{\\Delta}s_{pp}2h_{w} = \\frac{Q_{p}}{(\\pi K)}\\left( \\frac{1 - p}{p} \\right)\\ln\\frac{\\left( (1 - p)h_{s} \\right)}{r_{w}}[\/latex]<\/th>\r\n<th style=\"text-align: right; white-space: nowrap;\">(83)<\/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: 3%;\" \/> <col style=\"width: 67%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\">\u2206<em>s<sub>pp<\/sub><\/em><\/th>\r\n<th style=\"text-align: center;\">=<\/th>\r\n<th>drawdown resulting from partial penetration (L)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>h<sub>w<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>saturated thickness of the unconfined aquifer at pre\u2011pumping conditions (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>Q<sub>p<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>partially penetrating well pumping rate (L<sup>3<\/sup>T<sup>\u20111<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>K<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>hydraulic conductivity (LT<sup>\u20111<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>p<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>penetration factor <em>p<\/em> = <em>h<sub>s<\/sub><\/em>\/<em>b<\/em> (dimensionless)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>h<sub>s<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>length of well screen (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>r<sub>w<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>radius of the pumping well (L)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThe total drawdown in an unconfined aquifer is defined as shown in Equation\u00a0<a href=\"#X68c3a4b5559185adf4d9c021f057fba901b4151\">(84)<\/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]s_{T} = \\sqrt{s_{F}^{2} + \\mathrm{\\Delta}s_{pp}2h_{w}}[\/latex]<\/th>\r\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"X68c3a4b5559185adf4d9c021f057fba901b4151\" class=\"anchor\">(84)<\/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: 7%;\" \/> <col style=\"width: 1%;\" \/> <col style=\"width: 37%;\" \/> <col style=\"width: 37%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\"><em>s<sub>T<\/sub><\/em><\/th>\r\n<th style=\"text-align: center;\">=<\/th>\r\n<th>total drawdown in pumping well including effects of partial penetration and formational drawdown (L)<\/th>\r\n<th><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>s<sub>F<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>formational drawdown (L)<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\">\u2206<em>s<sub>pp<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>drawdown from partial penetration (L)<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>h<sub>w<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>saturated thickness of the unconfined aquifer at a fully penetrating well (L)<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThe partial\u2011penetration conditions that cause the additional drawdown are conceptualized as being constant once pumping begins. The partial\u2011penetration correction can be applied to transient or steady\u2011state drawdown data.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\n<strong>Example<\/strong>\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nA well with a radius of 0.15 m finished in a confined unit is pumped at a continuous rate of 4000 m<sup>3<\/sup>\/d to near steady state and the total drawdown is 34 m. The screened interval is 10 m in length, and it is located just below the upper boundary of a 45 m thick aquifer. If the aquifer transmissivity is 940 m<sup>2<\/sup>\/d, how much of the drawdown is due to partial penetration?\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThe penetration factor is 10 m\/45 m = 0.22. Using Equation\u00a0<a href=\"#Xd44f87d47e7555f427f8859b4b3fed147f86070\">(81)<\/a>, \u2206<em>s<sub>pp<\/sub><\/em> is computed as shown here.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\r\n\r\n[latex]\\mathrm{\\Delta}s_{pp} = \\frac{Q_{p}}{(2\\pi T)}\\left( \\frac{1 - p}{p} \\right)\\ln\\frac{\\left( (1 - p)h_{s} \\right)}{r_{w}}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\r\n\r\n[latex]\\mathrm{\\Delta}s_{pp} = \\frac{4000\\frac{m^{3}}{d}}{2\\ (3.14)\\ 940\\frac{m^{2}}{d}}\\frac{1 - (0.22)}{(0.22)}\\ln\\frac{\\left( (1 - 0.22)10\\ m \\right)}{0.15\\ m} = 9.5\\ m[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nIf a total observed drawdown in the partially penetrating well bore is 26.01 m, then rearranging Equation\u00a0<a href=\"#X8a962088b2297c02a791059b7f3536a840dcdf0\">(80)<\/a> the formational drawdown would be defined as <em>s<sub>F<\/sub><\/em> = <em>s<sub>T\u00a0<\/sub><\/em>\u2011\u00a0\u2206<em>s<sub>pp<\/sub><\/em>, so <em>s<sub>F<\/sub><\/em>\u00a0=\u00a026.01\u00a0m\u00a0\u20119.5\u00a0m\u00a0=\u00a016.51\u00a0m. If time\u2011drawdown data were collected during a transient test, drawdown would be corrected by \u20119.5m.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nIn addition to the equations presented above, other authors have addressed methods to assess partial penetration effects with analytical solutions for steady\u2011state and transient conditions (e.g., Sternberg, 1973; Brons &amp; Marting, 1961; Bradbury &amp; Rothschild (1985); Hantush, 1966; Neuman, 1975; Kipp 1973). Most commercial hydraulic\u2011test\u2011analysis software packages incorporate methods to correct data for partial penetration once well construction data are entered.\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"well-loss-and-using-stepdrawdown-tests-to-assess-loss\" class=\"level3\">\r\n<h3>Well Loss and Using Step\u2011Drawdown Tests to Assess Loss<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nWell loss is the component of the total drawdown in a production well due to the loss of energy resulting from turbulent flow of water through the screened or slotted interval (or the damaged borehole wall), as well as flow inside the casing to the pump intake as shown in <a href=\"#X555319ba517abd21addc2fe61df17c3fefc162e\">Figure 82<\/a> and expressed by Equation\u00a0<a href=\"#X8252c2b0870bbb6dd4f2c0dac351ac510e09580\">(85)<\/a>.\r\n\r\n<\/div>\r\n<div class=\"bc-figure figure\" data-custom-style=\"Captioned Figure\">\r\n\r\n<img style=\"display: block; margin-left: auto; margin-right: auto; ;width: 5.83333in; height: 4.22235in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file86-1.jpg\" alt=\"\" \/>\r\n<div class=\"bc-figcaption figcaption\">\r\n<div data-custom-style=\"Image Caption\">\r\n<div class=\"figcaption-text\">\r\n\r\n<span id=\"X555319ba517abd21addc2fe61df17c3fefc162e\" class=\"anchor\"><strong>Figure 82 \u2011<\/strong> Schematic showing the components of well loss in a fully penetrating pumping well finished in an isotropic and homogeneous confined aquifer. The measured drawdown is greater than if the well was 100 percent efficient and only the formational drawdown occurred as shown with the theoretical drawdown curve (<em>s<sub>F<\/sub><\/em>). The well loss is conceptualized as having two drawdown components, linear well loss (<em>s<sub>LWL<\/sub><\/em>), and non\u2011linear well loss (<em>s<sub>NLWL<\/sub><\/em>). As water in the vicinity of the wellbore converges to the perforated interval, head loss occurs as it passes through the area around the well that was impacted by well construction (linear well loss). Head loss also occurs as water moves through the perforations and up the well bore to the pump (non\u2011linear well loss).<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\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]s_{WL} = s_{LWL\\ } + s_{NLWL}[\/latex]<\/th>\r\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"X8252c2b0870bbb6dd4f2c0dac351ac510e09580\" class=\"anchor\">(85)<\/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: 3%;\" \/> <col style=\"width: 67%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\"><em>s<sub>WL<\/sub><\/em><\/th>\r\n<th style=\"text-align: center;\">=<\/th>\r\n<th>total well loss (L)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>s<sub>LWL<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>linear well loss (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>s<sub>NLWL<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>non\u2011linear well loss (L)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nWhen additional drawdown in a pumping well occurs from well loss, the total drawdown in the pumping well is the drawdown occurring due to the transmission and storage properties of the formation, <em>s<sub>F<\/sub><\/em>, and the total well loss (<em>s<sub>WL<\/sub><\/em>) as expressed in Equation\u00a0<a href=\"#Xf25dd1f73ec3e48fab37761d38d00355caf580d\">(86)<\/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]s_{T} = s_{F} + s_{WL}[\/latex]<\/th>\r\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"Xf25dd1f73ec3e48fab37761d38d00355caf580d\" class=\"anchor\">(86)<\/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: 3%;\" \/> <col style=\"width: 67%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\"><em>s<sub>T<\/sub><\/em><\/th>\r\n<th style=\"text-align: center;\">=<\/th>\r\n<th>total drawdown (L)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>s<sub>F<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>formational drawdown (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>s<sub>WL<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>total well loss drawdown (L)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nHydraulic gradients near the well bore are often affected by formational damage caused by well construction. These include zones of lower hydraulic conductivity (introduction of well cuttings and drilling fluids) that are present adjacent to the well borehole called skin effects (e.g., Sterrett, 2007). As the converging flow enters this area, gradients increase and additional head loss (linear well loss [<em>s<sub>LWL<\/sub><\/em>]) occurs. These are termed linear losses because their magnitude is directly proportional to the well\u2011discharge rate. Sometimes the damage zone has less resistance to flow because of performed well development and\/or pumping that mobilize fines in the formation, drawing them into the well and enhancing formation conductivity near the well such that <em>s<sub>LWL<\/sub><\/em> can be negative (Sterrett, 2007). The second component of head loss occurs as flow passes through the perforated interval (screen openings or cut slots) and moves up the well bore to the pump. This reduction in the total head is referred to as non\u2011linear well loss (<em>s<sub>NLHL<\/sub><\/em>). These are termed non\u2011linear losses because their magnitude increases nonlinearly as well discharge increases. <a href=\"#X6cf199d11e682b59f7c2ab9cf87ddcd4a5cf49a\">Figure 83<\/a> illustrates the impact of these loses on drawdown.\r\n\r\n<\/div>\r\n<div class=\"bc-figure figure\" data-custom-style=\"Captioned Figure\">\r\n\r\n<img style=\"display: block; margin-left: auto; margin-right: auto; ;width: 5.83333in; height: 3.78136in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file87-1.jpg\" alt=\"\" \/>\r\n<div class=\"bc-figcaption figcaption\">\r\n<div data-custom-style=\"Image Caption\">\r\n<div class=\"figcaption-text\">\r\n\r\n<span id=\"X6cf199d11e682b59f7c2ab9cf87ddcd4a5cf49a\" class=\"anchor\"><strong>Figure 83<\/strong> \u2013 Schematic of the change in water levels as water flows to the fully penetrating well bore completed in an isotropic and homogeneous confined aquifer being pumped at a constant rate (sand pattern). The small diameter wells (imaginary) are included in the diagram to illustrate the effect of well loss. The blue profile shows the theoretical drawdown (<em>s<sub>F<\/sub><\/em>) derived when the well has zero well loss (100 percent efficient). The red profile is created by adding the effects of well loss (<em>s<sub>LWL<\/sub><\/em>+<em>s<sub>NLWL<\/sub><\/em>) (red line). Zones of formational, linear, and non\u2011linear well loss are illustrated at the base of the diagram.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nWell loss is a function of the flow rate, being larger at higher pumping rates as shown in Equation\u00a0<a href=\"#Xebdd0b49fd9709e043160d7c2166077bafb3b31\">(87)<\/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]s_{T} = B_{1}Q + B_{2}Q + CQ^{2}[\/latex]<\/th>\r\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"Xebdd0b49fd9709e043160d7c2166077bafb3b31\" class=\"anchor\">(87)<\/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: 3%;\" \/> <col style=\"width: 67%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\"><em>s<sub>T<\/sub><\/em><\/th>\r\n<th style=\"text-align: center;\">=<\/th>\r\n<th>total drawdown (L)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>B<sub>1<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>formational aquifer properties (TL<sup>\u20112<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>Q<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>well discharge (L<sup>3<\/sup>T<sup>\u20111<\/sup>)<\/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: center;\">=<\/td>\r\n<td>formational conditions for linear well loss (TL<sup>\u20112<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>C<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>coefficient representing a non\u2011linear well loss coefficient (T<sup>2<\/sup>L<sup>\u20115<\/sup>)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nIn Equation\u00a0<a href=\"#Xebdd0b49fd9709e043160d7c2166077bafb3b31\">(87)<\/a>, <em>B<sub>1<\/sub>Q<\/em>=<em>s<sub>F<\/sub><\/em>, <em>B<sub>2<\/sub>Q<\/em>=<em>s<sub>LWL<\/sub><\/em>, and <em>CQ<\/em><sup>2<\/sup>=<em>s<sub>NLWL<\/sub><\/em>. The values of <em>B<sub>1<\/sub><\/em> and <em>B<sub>2<\/sub><\/em> are not separable when time\u2011drawdown data are available for only the pumping well. Jacob (1947) developed a relationship between the value <em>BQ<\/em> representing the formational drawdown and the non\u2011linear or turbulent well loss component, <em>CQ<\/em><sup>2<\/sup> (Equation\u00a0<a href=\"#Xd330de5326d90a18103ee2f560e6bd0a8494c17\">(88)<\/a>). <em>B<\/em> in this relationship can be conceptualized as <em>B<sub>1<\/sub><\/em>+<em>B<sub>2<\/sub><\/em>. For a confined aquifer, <em>B<\/em> can be calculated using the Cooper\u2011Jacob transient approximation in which the linear well loss component is formally included. That is, <em>B<\/em> equals [latex]\\frac{1}{4\\pi T}\\ \\ln\\left\\{ \\frac{Tt}{{1.78r}^{2}S} \\right\\},[\/latex] as shown in Equation\u00a0<a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/computing-t-and-s-from-hydraulic-test-data-using-the-theis-method\/#X14f949bdb63b8c17d62a6dd05551d2a372677e2\">(31)<\/a> of Section\u00a08.3.2. Rorabaugh (1953) suggests that the commonly used squared term on <em>C<\/em> in the non\u2011linear well loss term can be replaced with a variable that is near two but can be less or greater than two. Rorabaugh (1953) provides additional information. Traditionally, a value of two is assumed.\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]s_{T} = BQ + CQ^{2}[\/latex]<\/th>\r\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"Xd330de5326d90a18103ee2f560e6bd0a8494c17\" class=\"anchor\">(88)<\/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: 3%;\" \/> <col style=\"width: 67%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\"><em>s<sub>T<\/sub><\/em><\/th>\r\n<th style=\"text-align: center;\">=<\/th>\r\n<th>total drawdown in the pumping well (L)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>B<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>represents formational aquifer properties (TL<sup>\u20112<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>Q<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>well discharge (L<sup>3<\/sup>T<sup>\u20111<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>C<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>coefficient representing non\u2011linear well loss (T<sup>2<\/sup>L<sup>\u20115<\/sup>)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nWhen <em>T<\/em> and <em>S<\/em> are known, Equation\u00a0<a href=\"#Xd330de5326d90a18103ee2f560e6bd0a8494c17\">(88)<\/a> is used to correct pumping well drawdown for non\u2011linear well loss by computing <em>CQ<\/em><sup>2<\/sup> and subtracting it from the observed drawdown, <em>s<sub>F<\/sub><\/em> = <em>s<sub>T<\/sub><\/em>\u2011<em>CQ<\/em><sup>2<\/sup>. Step\u2011drawdown tests are used to estimate the well loss coefficient <em>C<\/em>.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nStep\u2011Drawdown Test and Methods to Compute Well Loss Coefficient and Estimate T\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nA step\u2011drawdown test (often referred to as a step test) is conducted by pumping a well at various rates and recording time\u2011drawdown data in the well. These tests can be used to investigate the drawdown response to pumping when flow rates increase and decrease. Step\u2011drawdown testing is used to characterize the well loss that occurs at various pumping rates. This methodology is most often used to quantify the well\u2011loss coefficient, <em>C<\/em>. The coefficient is used to compute well\u2011loss components of the total drawdown at a prescribed pumping rate and to correct drawdown versus time data to reflect drawdown attributed only to formational properties and conditions.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nDuring a step\u2011drawdown test, the pumping rate is increased from an initially low constant rate through a series of sequentially higher constant pumping rates (steps). Typically, each step is of equal duration, lasting from approximately 30 minutes to 2 hours (Kruseman &amp; de Ridder, 2000). Each step should be of sufficient duration to allow the dissipation of wellbore storage effects. Any number of steps can be conducted; however, at least three are recommended. The steps are usually increased sequentially but can be started and stopped allowing full recovery (<a href=\"#Xa4966914aec2d7df0d610c66bfb6cca1c46c9a2\">Figure 84<\/a>). The step test is usually designed so that the stepped pumping rates include or bracket the desired pumping rate for a production well. This allows computation of the well\u2011loss coefficient at that planned pumping rate. Step tests are also used to evaluate well performance including selecting pumping rates with lower well loss and identifying whether well conditions have changed over time (Sterrett, 2007).\r\n\r\n<\/div>\r\n<div class=\"bc-figure figure\" data-custom-style=\"Captioned Figure\">\r\n\r\n<img style=\"display: block; margin-left: auto; margin-right: auto; ;width: 5.77301in; height: 1.78827in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file88-1.jpg\" alt=\"\" \/>\r\n<div class=\"bc-figcaption figcaption\">\r\n<div data-custom-style=\"Image Caption\">\r\n<div class=\"figcaption-text\">\r\n\r\n<span id=\"Xa4966914aec2d7df0d610c66bfb6cca1c46c9a2\" class=\"anchor\"><strong>Figure 84 \u2011<\/strong> Step\u2011drawdown tests are most often conducted as one continuous test where the pumping rate is held constant for a period and then increased by a value of \u0394<em>Q<\/em> as shown in the left\u2011hand portion of the graph. The length of time each step is pumped, and the \u0394<em>Q<\/em> can vary. Time\u2011drawdown data are collected for each step. A second approach, shown in the middle and right\u2011hand side of the diagram, illustrates that step test data sets can also be derived when a well is pumped at a constant rate, then the pumping is stopped, and the well is allowed to recover. Repeating this method for various pumping rates creates a step test data set.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nStep\u2011drawdown test results are plotted on a semi\u2011log scale as drawdown versus time. The change in drawdown between each step is designated as \u2206<em>s<sub>step<\/sub><\/em> and the change in discharge between each step is defined as \u2206<em>Q<\/em>. <a href=\"#Xf66551b43812d9d65d38f9b5b412934bc23f683\">Figure 85<\/a> shows data from a continuous step test where each step reached steady\u2011state drawdown or near steady\u2011state. <a href=\"#X617b490325a2b46eba157de35ae9b20fb2f0b70\">Figure 86<\/a> shows a transient step test where each step lasted 30 minutes or more and drawdown was still declining as the next step started. In this case, \u2206<em>s<sub>step <\/sub><\/em>is the difference between the straight\u2011line projection of the drawdown of the previous step and the drawdown after 30 minutes of pumping at the new rate.\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: 3.75in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file89-1.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=\"Xf66551b43812d9d65d38f9b5b412934bc23f683\" class=\"anchor\"><strong>Figure 85 \u2011<\/strong> Step\u2011drawdown test where sequential constant pumping rates were used, and the drawdown reached steady\u2011state at each step. Dashed lines represent an extension of the observed drawdown trend. Values of \u2206<em>Q<\/em> and \u0394<em>s<sub>step<\/sub><\/em> are shown. Total pumping rates and drawdown for an individual step are obtained by adding the change in pumping rate and additional drawdown to those of the previous steps.<\/span>\r\n\r\n<\/div><\/blockquote>\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: 3.4in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file90-1.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=\"X617b490325a2b46eba157de35ae9b20fb2f0b70\" class=\"anchor\"><strong>Figure 86 \u2011<\/strong> Transient step\u2011drawdown test with sequential constant rates. Pumping of each step equaled or exceeded 30 minutes. Dashed lines represent extensions of the observed drawdown trends. Using a 30\u2011minute interval from the start of each step, values of \u0394<em>Q<\/em> and \u0394<em>s<sub>step<\/sub><\/em> are shown. Total pumping rates and drawdown for each step (after 30 minutes of pumping) are obtained by adding the changes in pumping rates and drawdown up to that point (i.e., \u03a3\u0394<em>Q<\/em> and \u03a3\u0394<em>s<sub>step<\/sub><\/em>).<\/span>\r\n\r\n<\/div><\/blockquote>\r\n<\/div>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nAnalysis of the steady\u2011state step\u2011drawdown test shown in <a href=\"#Xf66551b43812d9d65d38f9b5b412934bc23f683\">Figure 85<\/a> is described next. To estimate the turbulent flow coefficient, <em>C<\/em>, and the formational coefficient, <em>B,<\/em> of Equation\u00a0<a href=\"#Xd330de5326d90a18103ee2f560e6bd0a8494c17\">(88)<\/a>, an arithmetic plot of the total discharge for each step and the total drawdown divided by the total step discharge for each step (<em>s<\/em>\/<em>Q<sub>T<\/sub><\/em>) is plotted (<a href=\"#Xc95c6006dd77ca75de680a92eac5902c47b5830\">Figure 87<\/a>). <em>Q<sub>T<\/sub><\/em> is the cumulative pumping rate for a step, e.g., <em>Q<sub>T<\/sub><\/em> for step 2 is \u2206<em>Q<sub>1<\/sub><\/em>+\u2206<em>Q<sub>2<\/sub><\/em>, and for step 3 <em>Q<sub>T<\/sub><\/em> = \u2206<em>Q<sub>1<\/sub><\/em>+\u2206<em>Q<sub>2<\/sub><\/em> +\u2206<em>Q<sub>3<\/sub><\/em><sub>.<\/sub> This approach is referred to as the Hantush\u2011Bierschenk method (Hantush, 1964; Bierschenk, 1963).\r\n\r\n<\/div>\r\n<div class=\"bc-figure figure\" data-custom-style=\"Captioned Figure\">\r\n\r\n<img style=\"display: block; margin-left: auto; margin-right: auto; ;width: 5.83333in; height: 3.23418in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file91-1.jpg\" alt=\"\" \/>\r\n<div class=\"bc-figcaption figcaption\">\r\n<div data-custom-style=\"Image Caption\">\r\n<div class=\"figcaption-text\">\r\n\r\n<span id=\"Xc95c6006dd77ca75de680a92eac5902c47b5830\" class=\"anchor\"><strong>Figure 87<\/strong> \u2011 Example of a Hantush\u2011Biershenk plot used to analyze steady\u2011state step\u2011drawdown data to determine the turbulent well loss coefficient, <em>C<\/em>, and the value of <em>B<\/em>. a) Calculations of values of <em>Q<sub>T<\/sub><\/em>, <em>s,<\/em> and <em>s<\/em>\/<em>Q<sub>T<\/sub><\/em> for each step. b) Plot of <em>s<\/em>\/<em>Q<sub>T<\/sub><\/em> for each step versus the total pumping rate (<em>Q<sub>T<\/sub><\/em>) at the step. The slope of a fitted straight line (dashed line) is the value <em>C<\/em> and the intercept of the fitted line on the y\u2011axis is the value <em>B<\/em>.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThe well\u2011loss coefficient is derived from the slope of the fitted line shown in <a href=\"#Xc95c6006dd77ca75de680a92eac5902c47b5830\">Figure 87<\/a>. Given the scatter of the data in this example, <em>C<\/em> values will vary depending on the location of the fitted line. The well loss, <em>s<sub>WL<\/sub><\/em>, from turbulent conditions is then calculated at a specified discharge as <em>CQ<\/em><sup>2<\/sup>. The turbulent well\u2011loss coefficient described by Jacob (1950) is assumed to be a constant. For example, if the well producing the data shown in <a href=\"#Xc95c6006dd77ca75de680a92eac5902c47b5830\">Figure 87<\/a> was being pumped at 2,500 m<sup>3<\/sup>\/d, the turbulent well loss would be (0.0000017 d<sup>2<\/sup>\/m<sup>5<\/sup>) (2,500 m<sup>3<\/sup>\/d)<sup>2<\/sup> = 10.6 m.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThe coefficient <em>B<\/em> of Equation\u00a0<a href=\"#Xebdd0b49fd9709e043160d7c2166077bafb3b31\">(87)<\/a> is related to the formational properties and represents some linear well loss as discussed in previous sections. It is derived from the y\u2011axis intercept of the fitted straight line. If other time\u2011drawdown data collected from an observation well during a constant discharge pumping test are available, an estimate of <em>B<sub>1<\/sub><\/em> of Equation\u00a0<a href=\"#Xebdd0b49fd9709e043160d7c2166077bafb3b31\">(87)<\/a> can be derived using standard curve\u2011matching methods because the observation well is not pumped so it is 100 percent efficient.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nWithout an independent method to estimate <em>B<sub>1<\/sub><\/em><sub>,<\/sub> the value <em>B<\/em> derived from step test analyses is assumed to represent an estimate of the formational drawdown, <em>s<sub>F<\/sub><\/em> = <em>BQ<\/em>. In that case, aquifer properties will be underestimated because the drawdown used to estimate <em>B<\/em> contains some undefined well loss, and thus is greater than the drawdown for only formational conditions, <em>B<sub>1<\/sub><\/em>. When steady\u2011state conditions occur, the Thiem Equation can be applied to estimate <em>T<\/em> for confined conditions (Equations <a href=\"#X27238fe01a452c90be5fd7e1c8384bdae064a31\">(89)<\/a>, <a href=\"#X2dd9ea833c3d603d08fab40f7b0126bdc1b7dfb\">(90)<\/a>, <a href=\"#X698554e4a902ff85113373ee696aa6b62958b18\">(91)<\/a>, and <a href=\"#X6be352f4755f50e25d03419032a4ea5d36361d3\">(92)<\/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]s_{T} = \\ \\frac{Q}{2\\pi T}\\ln\\left( \\frac{r_{0}}{r_{w}} \\right) + CQ^{2}[\/latex]<\/th>\r\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"X27238fe01a452c90be5fd7e1c8384bdae064a31\" class=\"anchor\">(89)<\/span><\/th>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td style=\"text-align: center;\">[latex]s_{F} = s_{T} - CQ^{2}[\/latex]<\/td>\r\n<td style=\"text-align: right; white-space: nowrap;\"><span id=\"X2dd9ea833c3d603d08fab40f7b0126bdc1b7dfb\" class=\"anchor\">(90)<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td style=\"text-align: center;\">[latex]s_{F} = BQ = \\ \\frac{1}{2\\pi T}\\ln\\left( \\frac{r_{0}}{r_{w}} \\right)Q[\/latex]<\/td>\r\n<td style=\"text-align: right; white-space: nowrap;\"><span id=\"X698554e4a902ff85113373ee696aa6b62958b18\" class=\"anchor\">(91)<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td style=\"text-align: center;\">[latex]B = \\ \\frac{1}{2\\pi T}\\ln\\left( \\frac{r_{0}}{r_{w}} \\right)[\/latex]<\/td>\r\n<td style=\"text-align: right; white-space: nowrap;\"><span id=\"X6be352f4755f50e25d03419032a4ea5d36361d3\" class=\"anchor\">(92)<\/span><\/td>\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: 3%;\" \/> <col style=\"width: 67%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\"><em>B<\/em><\/th>\r\n<th style=\"text-align: center;\">=<\/th>\r\n<th>coefficient for a confined aquifer under steady\u2011state conditions (TL<sup>\u20112<\/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: center;\">=<\/td>\r\n<td>transmissivity (<em>Kb<\/em>) (L<sup>2<\/sup>T<sup>\u20111<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>Q<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>constant pumping rate (L<sup>3<\/sup>T<sup>\u20111<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>s<sub>T<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>static water level minus the head in the pumping well (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>s<sub>F<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\"><\/td>\r\n<td>drawdown attributed to formational properties (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>r<sub>W<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>radius of the pumping well (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>r<sub>0<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>radial distance where drawdown equals 0 (estimated) (L)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThe pumping\u2011well radius is derived from the well\u2011construction information. However, when a gravel pack is present the well radius is often replaced by the effective well radius; that includes the screen radius and radius of the surrounding gravel pack. The effective radius is then used as r<em><sub>w<\/sub><\/em> in the Thiem and Theis single\u2011well pumping equations.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nWhen steady\u2011state unconfined conditions are present, the total and formational drawdowns are usually approximated assuming the saturated thickness only changes a small amount during pumping. This allows application of the equations for confined aquifers, Equations <a href=\"#X27238fe01a452c90be5fd7e1c8384bdae064a31\">(89)<\/a> through <a href=\"#X6be352f4755f50e25d03419032a4ea5d36361d3\">(92)<\/a>.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nIn Equation\u00a0<a href=\"#X6be352f4755f50e25d03419032a4ea5d36361d3\">(92)<\/a>, the value of <em>B<\/em> is derived from the plot shown in <a href=\"#Xc95c6006dd77ca75de680a92eac5902c47b5830\">Figure 87<\/a>. The radius of the well is obtained from field measurements and well construction records. Once the value of <em>B<\/em> is graphically determined, values of <em>T<\/em> and <em>K=T\/b<\/em> can be computed using Equation\u00a0<a href=\"#X6be352f4755f50e25d03419032a4ea5d36361d3\">(92)<\/a>. However, the solutions depend on an estimate of the radial distance from the pumping well where drawdown is zero, <em>r<sub>0<\/sub><\/em>. An arbitrary value of 500\u00a0m is often assumed for steady\u2011state confined conditions because it is sufficiently large and using another sufficiently large value makes little difference in the result. When pumping rates are high, assumed distances can be increased by 1.5 to 2 times. Estimates of <em>T<\/em> change with different radial\u2011distance values (zero drawdown). However, because the radial distance is divided by the well radius and the natural logarithm is applied to the ratio, final estimates vary slightly when larger or smaller distances are used.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nResults of a transient step\u2011drawdown test such as the one shown in <a href=\"#X617b490325a2b46eba157de35ae9b20fb2f0b70\">Figure 86<\/a> can be analyzed using the same approach as described in <a href=\"#Xc95c6006dd77ca75de680a92eac5902c47b5830\">Figure 87<\/a>. <a href=\"#X40565ebc17ce2c272b1c757459ced8c9a0350b2\">Figure 88<\/a> is a plot of <em>s<\/em>\/<em>Q<sub>T<\/sub><\/em> versus <em>Q<sub>T<\/sub><\/em> for the transient test. Values of <em>s<\/em>\/<em>Q<sub>T<\/sub><\/em> for each step are computed using drawdown values measured at the end of the designated constant time, in this case the first 30 minutes.\r\n\r\n<\/div>\r\n<div class=\"bc-figure figure\" data-custom-style=\"Captioned Figure\">\r\n\r\n<img style=\"display: block; margin-left: auto; margin-right: auto; ;width: 5.83333in; height: 3.20002in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file92-1.jpg\" alt=\"\" \/>\r\n<div class=\"bc-figcaption figcaption\">\r\n<div data-custom-style=\"Image Caption\">\r\n<div class=\"figcaption-text\">\r\n\r\n<span id=\"X40565ebc17ce2c272b1c757459ced8c9a0350b2\" class=\"anchor\"><strong>Figure 88 \u2011<\/strong> A Hantush\u2011Biershenk plot used to analyze transient step\u2011drawdown data to determine the turbulent well loss coefficient, <em>C<\/em>, and the value of <em>B<\/em> in a confined system. a) Calculations of values of <em>Q<sub>T<\/sub><\/em>, <em>s<\/em> and <em>s<\/em>\/<em>Q<sub>T<\/sub><\/em> for each step using drawdown at the end of the 30\u2011minute interval since the step began. b) Plot of <em>s<\/em>\/<em>Q<sub>T<\/sub><\/em> for each step versus the total pumping rate (<em>Q<sub>T<\/sub><\/em>) at the step. The slope of a fitted straight line (dashed line) is the value <em>C<\/em> and the intercept of the fitted line on the y\u2011axis is the value <em>B<\/em>.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nAgain, using the relationship shown in Equation\u00a0<a href=\"#Xd330de5326d90a18103ee2f560e6bd0a8494c17\">(88)<\/a> shows <em>BQ<\/em> = <em>s<sub>F<\/sub><\/em>. Estimates of <em>T<\/em> will be higher if only <em>B<sub>1<\/sub><\/em> is used. Defining <em>B<\/em> using the Cooper\u2011Jacob approximation for a confined aquifer for <em>u<\/em>&lt;0.01 yields Equation\u00a0<a href=\"#X9f0dad1748e85a4741d22e1d1e0b1b77ea419f9\">(93)<\/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]B = \\frac{1}{4\\pi T}\\ 2.30\\ log\\left( 2.25\\ \\frac{Tt}{r_{w}^{2}S} \\right)[\/latex]<\/th>\r\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"X9f0dad1748e85a4741d22e1d1e0b1b77ea419f9\" class=\"anchor\">(93)<\/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: 3%;\" \/> <col style=\"width: 67%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\"><em>B<\/em><\/th>\r\n<th style=\"text-align: center;\">=<\/th>\r\n<th>formational coefficient for transient conditions (TL<sup>\u20112<\/sup>)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\">T<\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>transmissivity (L<sup>2<\/sup>T<sup>\u20111<\/sup>)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>t<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>constant time step length corresponding to the reported drawdown value (T)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>r<sub>W<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>radius of the pumping well (L)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\"><em>S<\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>storativity (dimensionless)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nTransient estimates of <em>\u0392<\/em> require an estimate of <em>S<\/em>. This is often derived by estimating the specific storage (<em>S<sub>s<\/sub><\/em>) of the aquifer material and computing a value of <em>S =S<sub>s<\/sub><\/em> (as discussed in Box 2). In addition, the estimate of <em>B<\/em> is time dependent. Thus, it is not used to estimate formational drawdown at other than early times (length of a step). The computed <em>T<\/em> values and estimates of <em>S<\/em> can be used to compute drawdowns at other times using appropriate analytical solutions (confined aquifer). If sufficient time\u2011drawdown data are collected in the first step of the step test these data can be corrected for non\u2011linear well loss and then used to curve match and compute additional estimates of <em>T<\/em> and <em>S<\/em>.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nJacob (1950) developed another method to estimate <em>C<\/em> using at least three consecutive steps from a step\u2011drawdown test. Most often this method is used for confined transient conditions and the change in drawdown is compared for equal time intervals since the step began, drawdown after 30 minutes or 1 hour for example (<a href=\"#X617b490325a2b46eba157de35ae9b20fb2f0b70\">Figure 86<\/a>). The turbulent well\u2011loss coefficients bracketed by two pumping rates are computed as shown in Equations <a href=\"#X370d1f00c7221b6f6777ba7c3c2125ee61606ba\">(94)<\/a>, <a href=\"#Xa6a68181081049d189f60dae7d14d137a551b0a\">(95)<\/a>, and <a href=\"#X9c3633e4f712c7dba92f15634229b15fe2b7763\">(96)<\/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]C_{1} = \\frac{\\left( \\frac{\\mathrm{\\Delta}s_{step2}}{\\mathrm{\\Delta}Q_{2}} \\right) - \\left( \\frac{\\mathrm{\\Delta}s_{step1}}{\\mathrm{\\Delta}Q_{1}} \\right)}{\\mathrm{\\Delta}Q_{1} + \\mathrm{\\Delta}Q_{2}}[\/latex]<\/th>\r\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"X370d1f00c7221b6f6777ba7c3c2125ee61606ba\" class=\"anchor\">(94)<\/span><\/th>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td style=\"text-align: center;\">[latex]C_{2} = \\frac{\\left( \\frac{\\mathrm{\\Delta}s_{step3}}{\\mathrm{\\Delta}Q_{3}} \\right) - \\left( \\frac{\\mathrm{\\Delta}s_{step2}}{\\mathrm{\\Delta}Q_{2}} \\right)}{\\mathrm{\\Delta}Q_{2} + \\mathrm{\\Delta}Q_{3}}[\/latex]<\/td>\r\n<td style=\"text-align: right; white-space: nowrap;\"><span id=\"Xa6a68181081049d189f60dae7d14d137a551b0a\" class=\"anchor\">(95)<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td style=\"text-align: center;\">[latex]C_{3} = \\frac{\\left( \\frac{\\mathrm{\\Delta}s_{step4}}{\\mathrm{\\Delta}Q_{4}} \\right) - \\left( \\frac{\\mathrm{\\Delta}s_{step3}}{\\mathrm{\\Delta}Q_{3}} \\right)}{\\mathrm{\\Delta}Q_{3} + \\mathrm{\\Delta}Q_{4}}[\/latex]<\/td>\r\n<td style=\"text-align: right; white-space: nowrap;\"><span id=\"X9c3633e4f712c7dba92f15634229b15fe2b7763\" class=\"anchor\">(96)<\/span><\/td>\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: 24%;\" \/> <col style=\"width: 2%;\" \/> <col style=\"width: 58%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: right;\"><em>C<sub>1<\/sub><\/em>, <em>C<sub>2<\/sub><\/em>, <em>C<sub>3<\/sub><\/em><\/th>\r\n<th style=\"text-align: center;\">=<\/th>\r\n<th>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nwell loss turbulence coefficient for steps 1\u20112, 2\u20113 and 3\u20114\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\n(T<sup>2<\/sup>L<sup>\u20115<\/sup>)\r\n\r\n<\/div><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: right;\">\u2206<em>s<sub>step1<\/sub><\/em>, \u2206<em>s<sub>step2<\/sub><\/em>, \u2206<em>s<sub>step3<\/sub><\/em>, \u2206<em>s<sub>step4<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>\r\n<div data-custom-style=\"Body Text\">\r\n\r\ndrawdown for each step after a constant pumping\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nperiod (L)\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: right;\">\u2206<em>Q<sub>1<\/sub><\/em>,\u00a0\u2206<em>Q<sub>2<\/sub><\/em>, \u2206<em>Q<sub>3<\/sub><\/em>, \u2206<em>Q<sub>4<\/sub><\/em><\/td>\r\n<td style=\"text-align: center;\">=<\/td>\r\n<td>changes in pumping rates between steps (L<sup>3<\/sup>T<sup>\u20111<\/sup>)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\n<em>C<\/em> can be computed based on the response of two sequential steps using the transient step\u2011drawdown data shown in <a href=\"#X617b490325a2b46eba157de35ae9b20fb2f0b70\">Figure 86<\/a>. Some researchers discourage using this approach because in the original formulation of <em>C<\/em> it is assumed to be constant (Jacob, 1950). However, often when this method is used most <em>C<\/em> values show variability. When values of <em>C<\/em> are similar, they can be averaged or a value more representative of the pumping rate selected is used to compute well loss.\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\n<strong>Example<\/strong>\r\n\r\n<\/div>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nUsing Jacob\u2019s method (1950) compute <em>C<\/em> from the transient data set shown in <a href=\"#X617b490325a2b46eba157de35ae9b20fb2f0b70\">Figure 86<\/a>, the time interval for analysis of each step is 30 minutes from the beginning of the step.\r\n\r\n<\/div>\r\n<table style=\"width: 66%;\" data-custom-style=\"Table\"><colgroup> <col style=\"width: 66%;\" \/> <\/colgroup>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\">[latex]C_{1} = \\frac{\\left( \\frac{1\\ m}{500\\frac{m^{3}}{d}} \\right) - \\left( \\frac{1.4\\ m}{600\\frac{m^{3}}{d}} \\right)}{600\\frac{m^{3}}{d} + 500\\frac{m^{3}}{d}} = - 3.0x10^{- 7}\\frac{m^{5}}{d^{2}}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]C_{2} = \\frac{\\left( \\frac{5.2\\ m}{1030\\frac{m^{3}}{d}} \\right) - \\left( \\frac{1.0\\ m}{500\\frac{m^{3}}{d}} \\right)}{500\\frac{m^{3}}{d} + 1030\\frac{m^{3}}{d}} = 2.0x10^{- 6}\\frac{m^{5}}{d^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]C_{2} = \\frac{\\left( \\frac{4.2\\ m}{480\\frac{m^{3}}{d}} \\right) - \\left( \\frac{5.2\\ m}{1030\\frac{m^{3}}{d}} \\right)}{1030\\frac{m^{3}}{d} + 480\\frac{m^{3}}{d}} = 2.4x10^{- 6}\\frac{m^{5}}{d^{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-custom-style=\"Body Text\">\r\n\r\nThe coefficients are used to compute the additional drawdown caused by non\u2011linear well loss. The negative value computed between the first two steps suggests the well was developing when pumped at 1,100 m<sup>3<\/sup>\/d in the second step. If a step\u2011drawdown test is completed and then followed by a constant discharge pumping test using the single well, the computed well loss (additional drawdown) for that pumping rate would be subtracted from the observed drawdown data before time\u2011drawdown data were analyzed to estimate <em>T<\/em> and <em>S<\/em>.\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"well-interference-1\" class=\"level3\">\r\n<h3>Well Interference<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nIt is best if other nearby wells are not operating during a single pumping test. However, if the operating schedules of surrounding wells in the area are known, well interference can be computed using standard analytical solutions or they can be observed during non\u2011pumping test periods. This involves monitoring water levels prior to and after pumping to see if adjacent pumping wells are impacting the water levels at the test well. If the effects of pumping in the surrounding wells are observed in the time\u2011drawdown data of the test well, they can be removed as described in Section 11.1. Observed drawdown may be affected during both drawdown and recovery.\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"other-conditions-that-effect-pumping-well-drawdown\" class=\"level3\">\r\n<h3>Other Conditions that Effect Pumping Well Drawdown<\/h3>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nOther factors may influence water levels during a single\u2011well test\r\n\r\n<\/div>\r\n<ul>\r\n \t<li>wellbore storage effects can occur, and are more pronounced for larger\u2011diameter wells and lower pumping rates;<\/li>\r\n \t<li>confined systems can be affected by barometric pressure changes, tidal loading, and direct loading; and<\/li>\r\n \t<li>unconfined systems may be impacted by tidal loading, evapotranspiration cycles, river\u2011stage changes, and short\u2011 or long\u2011term recharge events.<\/li>\r\n<\/ul>\r\n<div data-custom-style=\"First Paragraph\">\r\n\r\nWellbore\u2011storage effects were discussed in Section 12.1. Additional details on the effects of changing barometric pressure as well as both tidal and direct loading on observed water levels are discussed in Section 5. Water\u2011level changes in the pumping well due to these factors will be small in most cases. These factors will have a greater effect on late\u2011time data when drawdown changes due to formational conditions are also small. If time\u2011drawdown data need to be corrected for these perturbations, Kruseman and de Ridder (2000) and Sterrett (2007) provide a list of corrections. Procedures for determining production\u2011well efficiency are described in <span id=\"TextLinkToBox4\" class=\"anchor\"><a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/box-4-production-well-efficiency\/\">Box 4<\/a><\/span>.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div id=\"special-considerations-when-using-drawdown-data-from-a-pumping-well\" class=\"level1\">\n<div data-custom-style=\"First Paragraph\">\n<div data-custom-style=\"First Paragraph\">\n<p>Under ideal conditions, analytical solutions (Sections 7 through 10) and their simplifying assumptions can be applied to single\u2011well pumping\u2011test data to determine corresponding values of <em>T<\/em> and <em>S<\/em>. However, wells need to be fully penetrating and 100 % efficient. When using drawdown in the pumping well for the analysis, the pumping well radius is often substituted in the equations that require a radial distance to an observation well. The screen or perforated\u2011casing radius is used unless the well construction includes an envelope of gravel pack. When this occurs, the well radius is defined as the effective well radius and includes the radius of the perforated interval and gravel pack.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>However, when a single pumping well is the only source of drawdown data, it must be recognized that the recorded drawdown data are often influenced by partial penetration, well\u2011bore storage, and well loss, which are functions of the well design. Partial\u2011penetration and well\u2011loss effects result in lower water levels in the pumping well as compared to the adjacent aquifer. Well\u2011bore\u2011storage effects temporarily reduce the rate of early water level decline when large diameter wells are pumping at low rates. Each of these conditions are usually present to some degree and directly affect the predicted or reported drawdown at the pumping well. As with observation well test records, pumping\u2011well drawdowns can be impacted by well interference, barometric, tidal, and, in unconfined cases, recharge events and direct evapotranspiration.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Most often partial penetration and well\u2011loss conditions result in greater drawdown measured in the pumping well than if they were not present. As a result, computed <em>T<\/em> and <em>S<\/em> values are lower than properties of the formation. Ideally, pumping\u2011well drawdown would be corrected before analyses were undertaken.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>During the early\u2011time period of some pumping tests\u2014especially in large\u2011diameter wells\u2014pumped water is derived not only from the aquifer but also from water stored in the well casing or annular space (and filter pack) surrounding the screened or perforated interval. This results in a reduction in the observed drawdown in the pumping well and is referred to as a wellbore storage effect. As time progresses, formational water makes up an increasing portion of the well discharge until wellbore storage effects are sufficiently small, and flow is supplied by the aquifer. The influence of wellbore storage on drawdown data is most easily observed using log\u2011log plots of time\u2011drawdown data. The early\u2011time data will appear as a straight line diverging from the Theis curve. Papadopulos and Cooper (1967) developed an analytical method and a set of type curves that can be used to analyze confined\u2011aquifer\u2011test data affected by wellbore storage. Moench (1997) generated an analytical solution for pumping unconfined aquifers that accounts for well bore storage. He also provides a set of type curves.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>The total observed pumping\u2011well drawdown can be visualized as having several components as shown in <a href=\"#Xbdfb767a3839ac96403047b0d9d15425042ebaf\">Figure 79<\/a> and explained by Equation\u00a0<a href=\"#X4c56c45c5337f35abff2d4912fe1dcfbb8ea397\">(79)<\/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]s_{T} = s_{F\\ } + s_{pp\\ } + s_{WL\\ } + s_{I\\ } + s_{O\\ }[\/latex]<\/th>\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"X4c56c45c5337f35abff2d4912fe1dcfbb8ea397\" class=\"anchor\">(79)<\/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: 3%;\" \/>\n<col style=\"width: 67%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\"><em>s<sub>T<\/sub><\/em><\/th>\n<th style=\"text-align: center;\">=<\/th>\n<th>total observed drawdown in the pumping well (L)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><em>s<sub>F<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>component of drawdown based solely on the formational properties (<em>T<\/em>, <em>S<\/em>) (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>s<sub>pp<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>component of drawdown resulting from partial penetration of the perforated interval (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>s<sub>WL<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>component of drawdown from linear and non\u2011linear well loss (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>s<sub>I<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>component of drawdown from well interference (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>s<sub>O<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>component of drawdown from other factors with a positive or negative sign (e.g., wellbore storage, barometric effects, tides, recharge) (L)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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: 3.20833in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file83-1.jpg\" alt=\"\" \/><\/p>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote>\n<div class=\"figcaption-text\">\n<p><span id=\"Xbdfb767a3839ac96403047b0d9d15425042ebaf\" class=\"anchor\"><strong>Figure 79 \u2011<\/strong> Schematic of a partially penetrating pumping well in a confined aquifer (sand pattern) showing components of the total drawdown (<em>s<sub>T<\/sub><\/em>) as represented by the red profile. The formational drawdown (<em>s<sub>F<\/sub><\/em>) is shown in the blue drawdown profile and is the drawdown if the well is 100 percent efficient and fully penetrating. Additional components contributing to the total drawdown observed in the pumping well may include the effects of partial penetration which results in longer flow paths to the well as indicate by the blue flow lines (<em>s<sub>pp<\/sub><\/em>), well loss resulting from the well construction, design, and pumping rate (<em>s<sub>WL<\/sub><\/em>), drawdown from wells pumping nearby which is called well interference (<em>s<sub>I<\/sub><\/em>), and other factors causing the water levels to rise or fall such as wellbore storage and barometric effects (<em>s<sub>0<\/sub><\/em>).<\/span><\/p>\n<\/div>\n<\/blockquote>\n<\/div>\n<div data-custom-style=\"First Paragraph\">\n<p>The measured water levels in the pumping well will be similar to the theoretical drawdown shown in blue on <a href=\"#Xbdfb767a3839ac96403047b0d9d15425042ebaf\">Figure 79<\/a> if:<\/p>\n<\/div>\n<ul>\n<li>the pumping well is fully penetrating and 100 percent efficient,<\/li>\n<li>wellbore storage has been depleted<\/li>\n<li>there are no head losses due to turbulent flow caused by high flow rates and\/or obstructions in the well, and<\/li>\n<li>there is no interference from nearby pumping wells or other hydraulic stresses on the unit being tested.<\/li>\n<\/ul>\n<div data-custom-style=\"First Paragraph\">\n<p>The following sections describe how the factors that require special consideration influence drawdown data in the pumping well and present methods used to quantify the magnitude of their effect and to correct the drawdown data.<\/p>\n<\/div>\n<div id=\"partial-penetration\" class=\"level3\">\n<h3>Partial Penetration<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>A partially penetrating pumping well produces vertical components of flow near the well in both confined and unconfined aquifers. The vertical flow component creates longer flow paths that result in the flow system expending more energy to deliver water to the well than if all flow is horizontal. Partial\u2011penetration effects can occur in an unconfined aquifer as pumping begins even if a well is fully penetrating when the saturated thickness adjacent to the pumping well is reduced significantly during the test.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Todd and Mays (2005) provide methods to compute the increase in drawdown attributed to the head loss during groundwater flow to a partially penetrating well. They report that if the perforated interval of the well penetrates 85 percent or more of the aquifer then correction is not necessary, but it is generally accepted that correction is not necessary if the screened portion of a well penetrates 80 percent or more of the aquifer thickness.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>When considering the effects of partial penetration only, the total drawdown is composed of the formational drawdown, <em>s<sub>F<\/sub><\/em>, and the additional drawdown \u2206<em>s<sub>pp<\/sub><\/em> caused by partial penetration as shown in Equation\u00a0<a href=\"#X8a962088b2297c02a791059b7f3536a840dcdf0\">(80)<\/a> and <a href=\"#X4f513417674715fefa1e005a0db23beb5283a33\">Figure 80<\/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]s_{T} = s_{F} + \\mathrm{\\Delta}s_{pp}[\/latex]<\/th>\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"X8a962088b2297c02a791059b7f3536a840dcdf0\" class=\"anchor\">(80)<\/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: 3%;\" \/>\n<col style=\"width: 67%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\"><em>s<sub>T<\/sub><\/em><\/th>\n<th style=\"text-align: center;\">=<\/th>\n<th>total drawdown in pumping well including effects of partial penetration and formational drawdown (L)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><em>s<sub>F<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>formational drawdown (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\">\u2206<em>s<sub>pp<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>drawdown due to partial penetration (L)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Captioned Figure\">\n<p><img decoding=\"async\" style=\"display: block; margin-left: auto; margin-right: auto; ;width: 4.99525in; height: 2.75in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file84-1.jpg\" alt=\"\" \/><\/p>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote>\n<div class=\"figcaption-text\">\n<p><span id=\"X4f513417674715fefa1e005a0db23beb5283a33\" class=\"anchor\"><strong>Figure 80 \u2011<\/strong> Total drawdown, <em>s<sub>T<\/sub><\/em>, associated with a partially penetrating pumping well when both formational drawdown, <em>s<sub>F<\/sub><\/em>, and drawdown related to the effects of partial penetration, \u2206<em>s<sub>pp<\/sub><\/em> occur.<\/span><\/p>\n<\/div>\n<\/blockquote>\n<\/div>\n<div data-custom-style=\"First Paragraph\">\n<p>Huisman (1972) and Todd and Mays (2005) present methods to compute the effects of partial penetration using steady\u2011state formulations, the variables shown in <a href=\"#X4f513417674715fefa1e005a0db23beb5283a33\">Figure 80<\/a> and <a href=\"#Xc11fd70da34645e52b4f599ce8e6d55bf3d2ed8\">Figure 81<\/a>, and a penetration factor <em>p<\/em> = <em>h<sub>s<\/sub><\/em>\/<em>b<\/em>, screen length (<em>h<sub>s<\/sub><\/em>) divided by the saturated aquifer thickness (<em>b<\/em>).<\/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: 2.27562in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file85-1.jpg\" alt=\"\" \/><\/p>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote>\n<div class=\"figcaption-text\">\n<p><span id=\"Xc11fd70da34645e52b4f599ce8e6d55bf3d2ed8\" class=\"anchor\"><strong>Figure 81 \u2011<\/strong> Schematic of a confined aquifer (sand pattern) and the perforated interval (horizontal black lines) for two production wells. <em>b<\/em> represents the initial saturated thickness of the aquifer. <em>h<sub>s<\/sub><\/em> is the length of the perforated interval and <em>r<sub>w<\/sub><\/em> is the radius of the well (after Todd &amp; Mays, 2005).<\/span><\/p>\n<\/div>\n<\/blockquote>\n<\/div>\n<div data-custom-style=\"First Paragraph\">\n<p>The additional drawdown associated with partial penetration under steady\u2011state conditions in a confined aquifer as represented on the left\u2011hand side of <a href=\"#X4f513417674715fefa1e005a0db23beb5283a33\">Figure 80<\/a> is defined for p\u00a0&gt;\u00a00.2 by Todd &amp; Mays (2003) in Equation\u00a0<a href=\"#Xd44f87d47e7555f427f8859b4b3fed147f86070\">(81)<\/a>. When the well screen is centered in the formation as shown in the right\u2011hand portion of <a href=\"#Xc11fd70da34645e52b4f599ce8e6d55bf3d2ed8\">Figure 81<\/a>, then <em>p<\/em> = (<em>h<sub>s<\/sub><\/em>\/2)\/(<em>b<\/em>\/2<em>)<\/em> and this ratio is substituted into Equation\u00a0<a href=\"#Xd44f87d47e7555f427f8859b4b3fed147f86070\">(81)<\/a> that becomes Equation\u00a0<a href=\"#X45c3b3df8c2d80bbcad1b2caaf270c353b75fe3\">(82)<\/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]\\mathrm{\\Delta}s_{pp} = \\frac{Q_{p}}{(2\\pi T)}\\left( \\frac{1 - p}{p} \\right)\\ln\\left( \\frac{(1 - p)h_{s}}{r_{w}} \\right)[\/latex]<\/th>\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"Xd44f87d47e7555f427f8859b4b3fed147f86070\" class=\"anchor\">(81)<\/span><\/th>\n<\/tr>\n<\/tbody>\n<\/table>\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<td><\/td>\n<td style=\"text-align: center;\">[latex]\\mathrm{\\Delta}s_{pp} = \\frac{Q_{p}}{(2\\pi T)}\\left( \\frac{1 - p}{p} \\right)\\ln\\left( \\frac{(1 - p)h_{s}}{2r_{w}} \\right)[\/latex]<\/td>\n<td style=\"text-align: right; white-space: nowrap;\"><span id=\"X45c3b3df8c2d80bbcad1b2caaf270c353b75fe3\" class=\"anchor\">(82)<\/span><\/td>\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: 3%;\" \/>\n<col style=\"width: 67%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\">\u2206<em>s<sub>pp<\/sub><\/em><\/th>\n<th style=\"text-align: center;\">=<\/th>\n<th>additional drawdown resulting from partial penetration (L)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><em>Q<sub>p<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>partially penetrating well pumping rate (L<sup>3<\/sup>T<sup>\u20111<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>T<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>transmissivity (L<sup>2<\/sup>T<sup>\u20111<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>p<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>penetration factor, <em>h<sub>s<\/sub><\/em>\/<em>b<\/em> (dimensionless)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>h<sub>s<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>length of well screen (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>b<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>saturated thickness of aquifer (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>r<sub>w<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>radius of the pumping well (L)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>Based on the Thiem equation for unconfined aquifers, the drawdown resulting from partial penetration is defined by Todd and Mays (2005) as shown in Equation\u00a0<a href=\"#X8252c2b0870bbb6dd4f2c0dac351ac510e09580\">(85)<\/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]\\mathrm{\\Delta}s_{pp}2h_{w} = \\frac{Q_{p}}{(\\pi K)}\\left( \\frac{1 - p}{p} \\right)\\ln\\frac{\\left( (1 - p)h_{s} \\right)}{r_{w}}[\/latex]<\/th>\n<th style=\"text-align: right; white-space: nowrap;\">(83)<\/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: 3%;\" \/>\n<col style=\"width: 67%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\">\u2206<em>s<sub>pp<\/sub><\/em><\/th>\n<th style=\"text-align: center;\">=<\/th>\n<th>drawdown resulting from partial penetration (L)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><em>h<sub>w<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>saturated thickness of the unconfined aquifer at pre\u2011pumping conditions (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>Q<sub>p<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>partially penetrating well pumping rate (L<sup>3<\/sup>T<sup>\u20111<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>K<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>hydraulic conductivity (LT<sup>\u20111<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>p<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>penetration factor <em>p<\/em> = <em>h<sub>s<\/sub><\/em>\/<em>b<\/em> (dimensionless)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>h<sub>s<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>length of well screen (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>r<sub>w<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>radius of the pumping well (L)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>The total drawdown in an unconfined aquifer is defined as shown in Equation\u00a0<a href=\"#X68c3a4b5559185adf4d9c021f057fba901b4151\">(84)<\/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]s_{T} = \\sqrt{s_{F}^{2} + \\mathrm{\\Delta}s_{pp}2h_{w}}[\/latex]<\/th>\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"X68c3a4b5559185adf4d9c021f057fba901b4151\" class=\"anchor\">(84)<\/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: 7%;\" \/>\n<col style=\"width: 1%;\" \/>\n<col style=\"width: 37%;\" \/>\n<col style=\"width: 37%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\"><em>s<sub>T<\/sub><\/em><\/th>\n<th style=\"text-align: center;\">=<\/th>\n<th>total drawdown in pumping well including effects of partial penetration and formational drawdown (L)<\/th>\n<th><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><em>s<sub>F<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>formational drawdown (L)<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\">\u2206<em>s<sub>pp<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>drawdown from partial penetration (L)<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>h<sub>w<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>saturated thickness of the unconfined aquifer at a fully penetrating well (L)<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>The partial\u2011penetration conditions that cause the additional drawdown are conceptualized as being constant once pumping begins. The partial\u2011penetration correction can be applied to transient or steady\u2011state drawdown data.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p><strong>Example<\/strong><\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>A well with a radius of 0.15 m finished in a confined unit is pumped at a continuous rate of 4000 m<sup>3<\/sup>\/d to near steady state and the total drawdown is 34 m. The screened interval is 10 m in length, and it is located just below the upper boundary of a 45 m thick aquifer. If the aquifer transmissivity is 940 m<sup>2<\/sup>\/d, how much of the drawdown is due to partial penetration?<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>The penetration factor is 10 m\/45 m = 0.22. Using Equation\u00a0<a href=\"#Xd44f87d47e7555f427f8859b4b3fed147f86070\">(81)<\/a>, \u2206<em>s<sub>pp<\/sub><\/em> is computed as shown here.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\n<p>[latex]\\mathrm{\\Delta}s_{pp} = \\frac{Q_{p}}{(2\\pi T)}\\left( \\frac{1 - p}{p} \\right)\\ln\\frac{\\left( (1 - p)h_{s} \\right)}{r_{w}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<div class=\"display-math\" data-custom-style=\"Display Equation\">\n<p>[latex]\\mathrm{\\Delta}s_{pp} = \\frac{4000\\frac{m^{3}}{d}}{2\\ (3.14)\\ 940\\frac{m^{2}}{d}}\\frac{1 - (0.22)}{(0.22)}\\ln\\frac{\\left( (1 - 0.22)10\\ m \\right)}{0.15\\ m} = 9.5\\ m[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>If a total observed drawdown in the partially penetrating well bore is 26.01 m, then rearranging Equation\u00a0<a href=\"#X8a962088b2297c02a791059b7f3536a840dcdf0\">(80)<\/a> the formational drawdown would be defined as <em>s<sub>F<\/sub><\/em> = <em>s<sub>T\u00a0<\/sub><\/em>\u2011\u00a0\u2206<em>s<sub>pp<\/sub><\/em>, so <em>s<sub>F<\/sub><\/em>\u00a0=\u00a026.01\u00a0m\u00a0\u20119.5\u00a0m\u00a0=\u00a016.51\u00a0m. If time\u2011drawdown data were collected during a transient test, drawdown would be corrected by \u20119.5m.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>In addition to the equations presented above, other authors have addressed methods to assess partial penetration effects with analytical solutions for steady\u2011state and transient conditions (e.g., Sternberg, 1973; Brons &amp; Marting, 1961; Bradbury &amp; Rothschild (1985); Hantush, 1966; Neuman, 1975; Kipp 1973). Most commercial hydraulic\u2011test\u2011analysis software packages incorporate methods to correct data for partial penetration once well construction data are entered.<\/p>\n<\/div>\n<\/div>\n<div id=\"well-loss-and-using-stepdrawdown-tests-to-assess-loss\" class=\"level3\">\n<h3>Well Loss and Using Step\u2011Drawdown Tests to Assess Loss<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>Well loss is the component of the total drawdown in a production well due to the loss of energy resulting from turbulent flow of water through the screened or slotted interval (or the damaged borehole wall), as well as flow inside the casing to the pump intake as shown in <a href=\"#X555319ba517abd21addc2fe61df17c3fefc162e\">Figure 82<\/a> and expressed by Equation\u00a0<a href=\"#X8252c2b0870bbb6dd4f2c0dac351ac510e09580\">(85)<\/a>.<\/p>\n<\/div>\n<div class=\"bc-figure figure\" data-custom-style=\"Captioned Figure\">\n<p><img decoding=\"async\" style=\"display: block; margin-left: auto; margin-right: auto; ;width: 5.83333in; height: 4.22235in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file86-1.jpg\" alt=\"\" \/><\/p>\n<div class=\"bc-figcaption figcaption\">\n<div data-custom-style=\"Image Caption\">\n<div class=\"figcaption-text\">\n<p><span id=\"X555319ba517abd21addc2fe61df17c3fefc162e\" class=\"anchor\"><strong>Figure 82 \u2011<\/strong> Schematic showing the components of well loss in a fully penetrating pumping well finished in an isotropic and homogeneous confined aquifer. The measured drawdown is greater than if the well was 100 percent efficient and only the formational drawdown occurred as shown with the theoretical drawdown curve (<em>s<sub>F<\/sub><\/em>). The well loss is conceptualized as having two drawdown components, linear well loss (<em>s<sub>LWL<\/sub><\/em>), and non\u2011linear well loss (<em>s<sub>NLWL<\/sub><\/em>). As water in the vicinity of the wellbore converges to the perforated interval, head loss occurs as it passes through the area around the well that was impacted by well construction (linear well loss). Head loss also occurs as water moves through the perforations and up the well bore to the pump (non\u2011linear well loss).<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\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]s_{WL} = s_{LWL\\ } + s_{NLWL}[\/latex]<\/th>\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"X8252c2b0870bbb6dd4f2c0dac351ac510e09580\" class=\"anchor\">(85)<\/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: 3%;\" \/>\n<col style=\"width: 67%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\"><em>s<sub>WL<\/sub><\/em><\/th>\n<th style=\"text-align: center;\">=<\/th>\n<th>total well loss (L)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><em>s<sub>LWL<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>linear well loss (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>s<sub>NLWL<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>non\u2011linear well loss (L)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>When additional drawdown in a pumping well occurs from well loss, the total drawdown in the pumping well is the drawdown occurring due to the transmission and storage properties of the formation, <em>s<sub>F<\/sub><\/em>, and the total well loss (<em>s<sub>WL<\/sub><\/em>) as expressed in Equation\u00a0<a href=\"#Xf25dd1f73ec3e48fab37761d38d00355caf580d\">(86)<\/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]s_{T} = s_{F} + s_{WL}[\/latex]<\/th>\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"Xf25dd1f73ec3e48fab37761d38d00355caf580d\" class=\"anchor\">(86)<\/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: 3%;\" \/>\n<col style=\"width: 67%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\"><em>s<sub>T<\/sub><\/em><\/th>\n<th style=\"text-align: center;\">=<\/th>\n<th>total drawdown (L)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><em>s<sub>F<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>formational drawdown (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>s<sub>WL<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>total well loss drawdown (L)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>Hydraulic gradients near the well bore are often affected by formational damage caused by well construction. These include zones of lower hydraulic conductivity (introduction of well cuttings and drilling fluids) that are present adjacent to the well borehole called skin effects (e.g., Sterrett, 2007). As the converging flow enters this area, gradients increase and additional head loss (linear well loss [<em>s<sub>LWL<\/sub><\/em>]) occurs. These are termed linear losses because their magnitude is directly proportional to the well\u2011discharge rate. Sometimes the damage zone has less resistance to flow because of performed well development and\/or pumping that mobilize fines in the formation, drawing them into the well and enhancing formation conductivity near the well such that <em>s<sub>LWL<\/sub><\/em> can be negative (Sterrett, 2007). The second component of head loss occurs as flow passes through the perforated interval (screen openings or cut slots) and moves up the well bore to the pump. This reduction in the total head is referred to as non\u2011linear well loss (<em>s<sub>NLHL<\/sub><\/em>). These are termed non\u2011linear losses because their magnitude increases nonlinearly as well discharge increases. <a href=\"#X6cf199d11e682b59f7c2ab9cf87ddcd4a5cf49a\">Figure 83<\/a> illustrates the impact of these loses on drawdown.<\/p>\n<\/div>\n<div class=\"bc-figure figure\" data-custom-style=\"Captioned Figure\">\n<p><img decoding=\"async\" style=\"display: block; margin-left: auto; margin-right: auto; ;width: 5.83333in; height: 3.78136in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file87-1.jpg\" alt=\"\" \/><\/p>\n<div class=\"bc-figcaption figcaption\">\n<div data-custom-style=\"Image Caption\">\n<div class=\"figcaption-text\">\n<p><span id=\"X6cf199d11e682b59f7c2ab9cf87ddcd4a5cf49a\" class=\"anchor\"><strong>Figure 83<\/strong> \u2013 Schematic of the change in water levels as water flows to the fully penetrating well bore completed in an isotropic and homogeneous confined aquifer being pumped at a constant rate (sand pattern). The small diameter wells (imaginary) are included in the diagram to illustrate the effect of well loss. The blue profile shows the theoretical drawdown (<em>s<sub>F<\/sub><\/em>) derived when the well has zero well loss (100 percent efficient). The red profile is created by adding the effects of well loss (<em>s<sub>LWL<\/sub><\/em>+<em>s<sub>NLWL<\/sub><\/em>) (red line). Zones of formational, linear, and non\u2011linear well loss are illustrated at the base of the diagram.<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Well loss is a function of the flow rate, being larger at higher pumping rates as shown in Equation\u00a0<a href=\"#Xebdd0b49fd9709e043160d7c2166077bafb3b31\">(87)<\/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]s_{T} = B_{1}Q + B_{2}Q + CQ^{2}[\/latex]<\/th>\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"Xebdd0b49fd9709e043160d7c2166077bafb3b31\" class=\"anchor\">(87)<\/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: 3%;\" \/>\n<col style=\"width: 67%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\"><em>s<sub>T<\/sub><\/em><\/th>\n<th style=\"text-align: center;\">=<\/th>\n<th>total drawdown (L)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><em>B<sub>1<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>formational aquifer properties (TL<sup>\u20112<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>Q<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>well discharge (L<sup>3<\/sup>T<sup>\u20111<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>B<sub>2<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>formational conditions for linear well loss (TL<sup>\u20112<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>C<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>coefficient representing a non\u2011linear well loss coefficient (T<sup>2<\/sup>L<sup>\u20115<\/sup>)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>In Equation\u00a0<a href=\"#Xebdd0b49fd9709e043160d7c2166077bafb3b31\">(87)<\/a>, <em>B<sub>1<\/sub>Q<\/em>=<em>s<sub>F<\/sub><\/em>, <em>B<sub>2<\/sub>Q<\/em>=<em>s<sub>LWL<\/sub><\/em>, and <em>CQ<\/em><sup>2<\/sup>=<em>s<sub>NLWL<\/sub><\/em>. The values of <em>B<sub>1<\/sub><\/em> and <em>B<sub>2<\/sub><\/em> are not separable when time\u2011drawdown data are available for only the pumping well. Jacob (1947) developed a relationship between the value <em>BQ<\/em> representing the formational drawdown and the non\u2011linear or turbulent well loss component, <em>CQ<\/em><sup>2<\/sup> (Equation\u00a0<a href=\"#Xd330de5326d90a18103ee2f560e6bd0a8494c17\">(88)<\/a>). <em>B<\/em> in this relationship can be conceptualized as <em>B<sub>1<\/sub><\/em>+<em>B<sub>2<\/sub><\/em>. For a confined aquifer, <em>B<\/em> can be calculated using the Cooper\u2011Jacob transient approximation in which the linear well loss component is formally included. That is, <em>B<\/em> equals [latex]\\frac{1}{4\\pi T}\\ \\ln\\left\\{ \\frac{Tt}{{1.78r}^{2}S} \\right\\},[\/latex] as shown in Equation\u00a0<a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/computing-t-and-s-from-hydraulic-test-data-using-the-theis-method\/#X14f949bdb63b8c17d62a6dd05551d2a372677e2\">(31)<\/a> of Section\u00a08.3.2. Rorabaugh (1953) suggests that the commonly used squared term on <em>C<\/em> in the non\u2011linear well loss term can be replaced with a variable that is near two but can be less or greater than two. Rorabaugh (1953) provides additional information. Traditionally, a value of two is assumed.<\/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]s_{T} = BQ + CQ^{2}[\/latex]<\/th>\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"Xd330de5326d90a18103ee2f560e6bd0a8494c17\" class=\"anchor\">(88)<\/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: 3%;\" \/>\n<col style=\"width: 67%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\"><em>s<sub>T<\/sub><\/em><\/th>\n<th style=\"text-align: center;\">=<\/th>\n<th>total drawdown in the pumping well (L)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><em>B<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>represents formational aquifer properties (TL<sup>\u20112<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>Q<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>well discharge (L<sup>3<\/sup>T<sup>\u20111<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>C<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>coefficient representing non\u2011linear well loss (T<sup>2<\/sup>L<sup>\u20115<\/sup>)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>When <em>T<\/em> and <em>S<\/em> are known, Equation\u00a0<a href=\"#Xd330de5326d90a18103ee2f560e6bd0a8494c17\">(88)<\/a> is used to correct pumping well drawdown for non\u2011linear well loss by computing <em>CQ<\/em><sup>2<\/sup> and subtracting it from the observed drawdown, <em>s<sub>F<\/sub><\/em> = <em>s<sub>T<\/sub><\/em>\u2011<em>CQ<\/em><sup>2<\/sup>. Step\u2011drawdown tests are used to estimate the well loss coefficient <em>C<\/em>.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Step\u2011Drawdown Test and Methods to Compute Well Loss Coefficient and Estimate T<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>A step\u2011drawdown test (often referred to as a step test) is conducted by pumping a well at various rates and recording time\u2011drawdown data in the well. These tests can be used to investigate the drawdown response to pumping when flow rates increase and decrease. Step\u2011drawdown testing is used to characterize the well loss that occurs at various pumping rates. This methodology is most often used to quantify the well\u2011loss coefficient, <em>C<\/em>. The coefficient is used to compute well\u2011loss components of the total drawdown at a prescribed pumping rate and to correct drawdown versus time data to reflect drawdown attributed only to formational properties and conditions.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>During a step\u2011drawdown test, the pumping rate is increased from an initially low constant rate through a series of sequentially higher constant pumping rates (steps). Typically, each step is of equal duration, lasting from approximately 30 minutes to 2 hours (Kruseman &amp; de Ridder, 2000). Each step should be of sufficient duration to allow the dissipation of wellbore storage effects. Any number of steps can be conducted; however, at least three are recommended. The steps are usually increased sequentially but can be started and stopped allowing full recovery (<a href=\"#Xa4966914aec2d7df0d610c66bfb6cca1c46c9a2\">Figure 84<\/a>). The step test is usually designed so that the stepped pumping rates include or bracket the desired pumping rate for a production well. This allows computation of the well\u2011loss coefficient at that planned pumping rate. Step tests are also used to evaluate well performance including selecting pumping rates with lower well loss and identifying whether well conditions have changed over time (Sterrett, 2007).<\/p>\n<\/div>\n<div class=\"bc-figure figure\" data-custom-style=\"Captioned Figure\">\n<p><img decoding=\"async\" style=\"display: block; margin-left: auto; margin-right: auto; ;width: 5.77301in; height: 1.78827in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file88-1.jpg\" alt=\"\" \/><\/p>\n<div class=\"bc-figcaption figcaption\">\n<div data-custom-style=\"Image Caption\">\n<div class=\"figcaption-text\">\n<p><span id=\"Xa4966914aec2d7df0d610c66bfb6cca1c46c9a2\" class=\"anchor\"><strong>Figure 84 \u2011<\/strong> Step\u2011drawdown tests are most often conducted as one continuous test where the pumping rate is held constant for a period and then increased by a value of \u0394<em>Q<\/em> as shown in the left\u2011hand portion of the graph. The length of time each step is pumped, and the \u0394<em>Q<\/em> can vary. Time\u2011drawdown data are collected for each step. A second approach, shown in the middle and right\u2011hand side of the diagram, illustrates that step test data sets can also be derived when a well is pumped at a constant rate, then the pumping is stopped, and the well is allowed to recover. Repeating this method for various pumping rates creates a step test data set.<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Step\u2011drawdown test results are plotted on a semi\u2011log scale as drawdown versus time. The change in drawdown between each step is designated as \u2206<em>s<sub>step<\/sub><\/em> and the change in discharge between each step is defined as \u2206<em>Q<\/em>. <a href=\"#Xf66551b43812d9d65d38f9b5b412934bc23f683\">Figure 85<\/a> shows data from a continuous step test where each step reached steady\u2011state drawdown or near steady\u2011state. <a href=\"#X617b490325a2b46eba157de35ae9b20fb2f0b70\">Figure 86<\/a> shows a transient step test where each step lasted 30 minutes or more and drawdown was still declining as the next step started. In this case, \u2206<em>s<sub>step <\/sub><\/em>is the difference between the straight\u2011line projection of the drawdown of the previous step and the drawdown after 30 minutes of pumping at the new rate.<\/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: 3.75in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file89-1.jpg\" alt=\"\" \/><\/p>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote>\n<div class=\"figcaption-text\">\n<p><span id=\"Xf66551b43812d9d65d38f9b5b412934bc23f683\" class=\"anchor\"><strong>Figure 85 \u2011<\/strong> Step\u2011drawdown test where sequential constant pumping rates were used, and the drawdown reached steady\u2011state at each step. Dashed lines represent an extension of the observed drawdown trend. Values of \u2206<em>Q<\/em> and \u0394<em>s<sub>step<\/sub><\/em> are shown. Total pumping rates and drawdown for an individual step are obtained by adding the change in pumping rate and additional drawdown to those of the previous steps.<\/span><\/p>\n<\/div>\n<\/blockquote>\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: 3.4in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file90-1.jpg\" alt=\"\" \/><\/p>\n<\/div>\n<div data-custom-style=\"Block Text\">\n<blockquote>\n<div class=\"figcaption-text\">\n<p><span id=\"X617b490325a2b46eba157de35ae9b20fb2f0b70\" class=\"anchor\"><strong>Figure 86 \u2011<\/strong> Transient step\u2011drawdown test with sequential constant rates. Pumping of each step equaled or exceeded 30 minutes. Dashed lines represent extensions of the observed drawdown trends. Using a 30\u2011minute interval from the start of each step, values of \u0394<em>Q<\/em> and \u0394<em>s<sub>step<\/sub><\/em> are shown. Total pumping rates and drawdown for each step (after 30 minutes of pumping) are obtained by adding the changes in pumping rates and drawdown up to that point (i.e., \u03a3\u0394<em>Q<\/em> and \u03a3\u0394<em>s<sub>step<\/sub><\/em>).<\/span><\/p>\n<\/div>\n<\/blockquote>\n<\/div>\n<div data-custom-style=\"First Paragraph\">\n<p>Analysis of the steady\u2011state step\u2011drawdown test shown in <a href=\"#Xf66551b43812d9d65d38f9b5b412934bc23f683\">Figure 85<\/a> is described next. To estimate the turbulent flow coefficient, <em>C<\/em>, and the formational coefficient, <em>B,<\/em> of Equation\u00a0<a href=\"#Xd330de5326d90a18103ee2f560e6bd0a8494c17\">(88)<\/a>, an arithmetic plot of the total discharge for each step and the total drawdown divided by the total step discharge for each step (<em>s<\/em>\/<em>Q<sub>T<\/sub><\/em>) is plotted (<a href=\"#Xc95c6006dd77ca75de680a92eac5902c47b5830\">Figure 87<\/a>). <em>Q<sub>T<\/sub><\/em> is the cumulative pumping rate for a step, e.g., <em>Q<sub>T<\/sub><\/em> for step 2 is \u2206<em>Q<sub>1<\/sub><\/em>+\u2206<em>Q<sub>2<\/sub><\/em>, and for step 3 <em>Q<sub>T<\/sub><\/em> = \u2206<em>Q<sub>1<\/sub><\/em>+\u2206<em>Q<sub>2<\/sub><\/em> +\u2206<em>Q<sub>3<\/sub><\/em><sub>.<\/sub> This approach is referred to as the Hantush\u2011Bierschenk method (Hantush, 1964; Bierschenk, 1963).<\/p>\n<\/div>\n<div class=\"bc-figure figure\" data-custom-style=\"Captioned Figure\">\n<p><img decoding=\"async\" style=\"display: block; margin-left: auto; margin-right: auto; ;width: 5.83333in; height: 3.23418in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file91-1.jpg\" alt=\"\" \/><\/p>\n<div class=\"bc-figcaption figcaption\">\n<div data-custom-style=\"Image Caption\">\n<div class=\"figcaption-text\">\n<p><span id=\"Xc95c6006dd77ca75de680a92eac5902c47b5830\" class=\"anchor\"><strong>Figure 87<\/strong> \u2011 Example of a Hantush\u2011Biershenk plot used to analyze steady\u2011state step\u2011drawdown data to determine the turbulent well loss coefficient, <em>C<\/em>, and the value of <em>B<\/em>. a) Calculations of values of <em>Q<sub>T<\/sub><\/em>, <em>s,<\/em> and <em>s<\/em>\/<em>Q<sub>T<\/sub><\/em> for each step. b) Plot of <em>s<\/em>\/<em>Q<sub>T<\/sub><\/em> for each step versus the total pumping rate (<em>Q<sub>T<\/sub><\/em>) at the step. The slope of a fitted straight line (dashed line) is the value <em>C<\/em> and the intercept of the fitted line on the y\u2011axis is the value <em>B<\/em>.<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>The well\u2011loss coefficient is derived from the slope of the fitted line shown in <a href=\"#Xc95c6006dd77ca75de680a92eac5902c47b5830\">Figure 87<\/a>. Given the scatter of the data in this example, <em>C<\/em> values will vary depending on the location of the fitted line. The well loss, <em>s<sub>WL<\/sub><\/em>, from turbulent conditions is then calculated at a specified discharge as <em>CQ<\/em><sup>2<\/sup>. The turbulent well\u2011loss coefficient described by Jacob (1950) is assumed to be a constant. For example, if the well producing the data shown in <a href=\"#Xc95c6006dd77ca75de680a92eac5902c47b5830\">Figure 87<\/a> was being pumped at 2,500 m<sup>3<\/sup>\/d, the turbulent well loss would be (0.0000017 d<sup>2<\/sup>\/m<sup>5<\/sup>) (2,500 m<sup>3<\/sup>\/d)<sup>2<\/sup> = 10.6 m.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>The coefficient <em>B<\/em> of Equation\u00a0<a href=\"#Xebdd0b49fd9709e043160d7c2166077bafb3b31\">(87)<\/a> is related to the formational properties and represents some linear well loss as discussed in previous sections. It is derived from the y\u2011axis intercept of the fitted straight line. If other time\u2011drawdown data collected from an observation well during a constant discharge pumping test are available, an estimate of <em>B<sub>1<\/sub><\/em> of Equation\u00a0<a href=\"#Xebdd0b49fd9709e043160d7c2166077bafb3b31\">(87)<\/a> can be derived using standard curve\u2011matching methods because the observation well is not pumped so it is 100 percent efficient.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Without an independent method to estimate <em>B<sub>1<\/sub><\/em><sub>,<\/sub> the value <em>B<\/em> derived from step test analyses is assumed to represent an estimate of the formational drawdown, <em>s<sub>F<\/sub><\/em> = <em>BQ<\/em>. In that case, aquifer properties will be underestimated because the drawdown used to estimate <em>B<\/em> contains some undefined well loss, and thus is greater than the drawdown for only formational conditions, <em>B<sub>1<\/sub><\/em>. When steady\u2011state conditions occur, the Thiem Equation can be applied to estimate <em>T<\/em> for confined conditions (Equations <a href=\"#X27238fe01a452c90be5fd7e1c8384bdae064a31\">(89)<\/a>, <a href=\"#X2dd9ea833c3d603d08fab40f7b0126bdc1b7dfb\">(90)<\/a>, <a href=\"#X698554e4a902ff85113373ee696aa6b62958b18\">(91)<\/a>, and <a href=\"#X6be352f4755f50e25d03419032a4ea5d36361d3\">(92)<\/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]s_{T} = \\ \\frac{Q}{2\\pi T}\\ln\\left( \\frac{r_{0}}{r_{w}} \\right) + CQ^{2}[\/latex]<\/th>\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"X27238fe01a452c90be5fd7e1c8384bdae064a31\" class=\"anchor\">(89)<\/span><\/th>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: center;\">[latex]s_{F} = s_{T} - CQ^{2}[\/latex]<\/td>\n<td style=\"text-align: right; white-space: nowrap;\"><span id=\"X2dd9ea833c3d603d08fab40f7b0126bdc1b7dfb\" class=\"anchor\">(90)<\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: center;\">[latex]s_{F} = BQ = \\ \\frac{1}{2\\pi T}\\ln\\left( \\frac{r_{0}}{r_{w}} \\right)Q[\/latex]<\/td>\n<td style=\"text-align: right; white-space: nowrap;\"><span id=\"X698554e4a902ff85113373ee696aa6b62958b18\" class=\"anchor\">(91)<\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: center;\">[latex]B = \\ \\frac{1}{2\\pi T}\\ln\\left( \\frac{r_{0}}{r_{w}} \\right)[\/latex]<\/td>\n<td style=\"text-align: right; white-space: nowrap;\"><span id=\"X6be352f4755f50e25d03419032a4ea5d36361d3\" class=\"anchor\">(92)<\/span><\/td>\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: 3%;\" \/>\n<col style=\"width: 67%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\"><em>B<\/em><\/th>\n<th style=\"text-align: center;\">=<\/th>\n<th>coefficient for a confined aquifer under steady\u2011state conditions (TL<sup>\u20112<\/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: center;\">=<\/td>\n<td>transmissivity (<em>Kb<\/em>) (L<sup>2<\/sup>T<sup>\u20111<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>Q<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>constant pumping rate (L<sup>3<\/sup>T<sup>\u20111<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>s<sub>T<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>static water level minus the head in the pumping well (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>s<sub>F<\/sub><\/em><\/td>\n<td style=\"text-align: center;\"><\/td>\n<td>drawdown attributed to formational properties (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>r<sub>W<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>radius of the pumping well (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>r<sub>0<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>radial distance where drawdown equals 0 (estimated) (L)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>The pumping\u2011well radius is derived from the well\u2011construction information. However, when a gravel pack is present the well radius is often replaced by the effective well radius; that includes the screen radius and radius of the surrounding gravel pack. The effective radius is then used as r<em><sub>w<\/sub><\/em> in the Thiem and Theis single\u2011well pumping equations.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>When steady\u2011state unconfined conditions are present, the total and formational drawdowns are usually approximated assuming the saturated thickness only changes a small amount during pumping. This allows application of the equations for confined aquifers, Equations <a href=\"#X27238fe01a452c90be5fd7e1c8384bdae064a31\">(89)<\/a> through <a href=\"#X6be352f4755f50e25d03419032a4ea5d36361d3\">(92)<\/a>.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>In Equation\u00a0<a href=\"#X6be352f4755f50e25d03419032a4ea5d36361d3\">(92)<\/a>, the value of <em>B<\/em> is derived from the plot shown in <a href=\"#Xc95c6006dd77ca75de680a92eac5902c47b5830\">Figure 87<\/a>. The radius of the well is obtained from field measurements and well construction records. Once the value of <em>B<\/em> is graphically determined, values of <em>T<\/em> and <em>K=T\/b<\/em> can be computed using Equation\u00a0<a href=\"#X6be352f4755f50e25d03419032a4ea5d36361d3\">(92)<\/a>. However, the solutions depend on an estimate of the radial distance from the pumping well where drawdown is zero, <em>r<sub>0<\/sub><\/em>. An arbitrary value of 500\u00a0m is often assumed for steady\u2011state confined conditions because it is sufficiently large and using another sufficiently large value makes little difference in the result. When pumping rates are high, assumed distances can be increased by 1.5 to 2 times. Estimates of <em>T<\/em> change with different radial\u2011distance values (zero drawdown). However, because the radial distance is divided by the well radius and the natural logarithm is applied to the ratio, final estimates vary slightly when larger or smaller distances are used.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Results of a transient step\u2011drawdown test such as the one shown in <a href=\"#X617b490325a2b46eba157de35ae9b20fb2f0b70\">Figure 86<\/a> can be analyzed using the same approach as described in <a href=\"#Xc95c6006dd77ca75de680a92eac5902c47b5830\">Figure 87<\/a>. <a href=\"#X40565ebc17ce2c272b1c757459ced8c9a0350b2\">Figure 88<\/a> is a plot of <em>s<\/em>\/<em>Q<sub>T<\/sub><\/em> versus <em>Q<sub>T<\/sub><\/em> for the transient test. Values of <em>s<\/em>\/<em>Q<sub>T<\/sub><\/em> for each step are computed using drawdown values measured at the end of the designated constant time, in this case the first 30 minutes.<\/p>\n<\/div>\n<div class=\"bc-figure figure\" data-custom-style=\"Captioned Figure\">\n<p><img decoding=\"async\" style=\"display: block; margin-left: auto; margin-right: auto; ;width: 5.83333in; height: 3.20002in;\" src=\"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-content\/uploads\/sites\/40\/2026\/08\/file92-1.jpg\" alt=\"\" \/><\/p>\n<div class=\"bc-figcaption figcaption\">\n<div data-custom-style=\"Image Caption\">\n<div class=\"figcaption-text\">\n<p><span id=\"X40565ebc17ce2c272b1c757459ced8c9a0350b2\" class=\"anchor\"><strong>Figure 88 \u2011<\/strong> A Hantush\u2011Biershenk plot used to analyze transient step\u2011drawdown data to determine the turbulent well loss coefficient, <em>C<\/em>, and the value of <em>B<\/em> in a confined system. a) Calculations of values of <em>Q<sub>T<\/sub><\/em>, <em>s<\/em> and <em>s<\/em>\/<em>Q<sub>T<\/sub><\/em> for each step using drawdown at the end of the 30\u2011minute interval since the step began. b) Plot of <em>s<\/em>\/<em>Q<sub>T<\/sub><\/em> for each step versus the total pumping rate (<em>Q<sub>T<\/sub><\/em>) at the step. The slope of a fitted straight line (dashed line) is the value <em>C<\/em> and the intercept of the fitted line on the y\u2011axis is the value <em>B<\/em>.<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Again, using the relationship shown in Equation\u00a0<a href=\"#Xd330de5326d90a18103ee2f560e6bd0a8494c17\">(88)<\/a> shows <em>BQ<\/em> = <em>s<sub>F<\/sub><\/em>. Estimates of <em>T<\/em> will be higher if only <em>B<sub>1<\/sub><\/em> is used. Defining <em>B<\/em> using the Cooper\u2011Jacob approximation for a confined aquifer for <em>u<\/em>&lt;0.01 yields Equation\u00a0<a href=\"#X9f0dad1748e85a4741d22e1d1e0b1b77ea419f9\">(93)<\/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]B = \\frac{1}{4\\pi T}\\ 2.30\\ log\\left( 2.25\\ \\frac{Tt}{r_{w}^{2}S} \\right)[\/latex]<\/th>\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"X9f0dad1748e85a4741d22e1d1e0b1b77ea419f9\" class=\"anchor\">(93)<\/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: 3%;\" \/>\n<col style=\"width: 67%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\"><em>B<\/em><\/th>\n<th style=\"text-align: center;\">=<\/th>\n<th>formational coefficient for transient conditions (TL<sup>\u20112<\/sup>)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">T<\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>transmissivity (L<sup>2<\/sup>T<sup>\u20111<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>t<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>constant time step length corresponding to the reported drawdown value (T)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>r<sub>W<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>radius of the pumping well (L)<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><em>S<\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>storativity (dimensionless)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>Transient estimates of <em>\u0392<\/em> require an estimate of <em>S<\/em>. This is often derived by estimating the specific storage (<em>S<sub>s<\/sub><\/em>) of the aquifer material and computing a value of <em>S =S<sub>s<\/sub><\/em> (as discussed in Box 2). In addition, the estimate of <em>B<\/em> is time dependent. Thus, it is not used to estimate formational drawdown at other than early times (length of a step). The computed <em>T<\/em> values and estimates of <em>S<\/em> can be used to compute drawdowns at other times using appropriate analytical solutions (confined aquifer). If sufficient time\u2011drawdown data are collected in the first step of the step test these data can be corrected for non\u2011linear well loss and then used to curve match and compute additional estimates of <em>T<\/em> and <em>S<\/em>.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Jacob (1950) developed another method to estimate <em>C<\/em> using at least three consecutive steps from a step\u2011drawdown test. Most often this method is used for confined transient conditions and the change in drawdown is compared for equal time intervals since the step began, drawdown after 30 minutes or 1 hour for example (<a href=\"#X617b490325a2b46eba157de35ae9b20fb2f0b70\">Figure 86<\/a>). The turbulent well\u2011loss coefficients bracketed by two pumping rates are computed as shown in Equations <a href=\"#X370d1f00c7221b6f6777ba7c3c2125ee61606ba\">(94)<\/a>, <a href=\"#Xa6a68181081049d189f60dae7d14d137a551b0a\">(95)<\/a>, and <a href=\"#X9c3633e4f712c7dba92f15634229b15fe2b7763\">(96)<\/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]C_{1} = \\frac{\\left( \\frac{\\mathrm{\\Delta}s_{step2}}{\\mathrm{\\Delta}Q_{2}} \\right) - \\left( \\frac{\\mathrm{\\Delta}s_{step1}}{\\mathrm{\\Delta}Q_{1}} \\right)}{\\mathrm{\\Delta}Q_{1} + \\mathrm{\\Delta}Q_{2}}[\/latex]<\/th>\n<th style=\"text-align: right; white-space: nowrap;\"><span id=\"X370d1f00c7221b6f6777ba7c3c2125ee61606ba\" class=\"anchor\">(94)<\/span><\/th>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: center;\">[latex]C_{2} = \\frac{\\left( \\frac{\\mathrm{\\Delta}s_{step3}}{\\mathrm{\\Delta}Q_{3}} \\right) - \\left( \\frac{\\mathrm{\\Delta}s_{step2}}{\\mathrm{\\Delta}Q_{2}} \\right)}{\\mathrm{\\Delta}Q_{2} + \\mathrm{\\Delta}Q_{3}}[\/latex]<\/td>\n<td style=\"text-align: right; white-space: nowrap;\"><span id=\"Xa6a68181081049d189f60dae7d14d137a551b0a\" class=\"anchor\">(95)<\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: center;\">[latex]C_{3} = \\frac{\\left( \\frac{\\mathrm{\\Delta}s_{step4}}{\\mathrm{\\Delta}Q_{4}} \\right) - \\left( \\frac{\\mathrm{\\Delta}s_{step3}}{\\mathrm{\\Delta}Q_{3}} \\right)}{\\mathrm{\\Delta}Q_{3} + \\mathrm{\\Delta}Q_{4}}[\/latex]<\/td>\n<td style=\"text-align: right; white-space: nowrap;\"><span id=\"X9c3633e4f712c7dba92f15634229b15fe2b7763\" class=\"anchor\">(96)<\/span><\/td>\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: 24%;\" \/>\n<col style=\"width: 2%;\" \/>\n<col style=\"width: 58%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: right;\"><em>C<sub>1<\/sub><\/em>, <em>C<sub>2<\/sub><\/em>, <em>C<sub>3<\/sub><\/em><\/th>\n<th style=\"text-align: center;\">=<\/th>\n<th>\n<div data-custom-style=\"Body Text\">\n<p>well loss turbulence coefficient for steps 1\u20112, 2\u20113 and 3\u20114<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>(T<sup>2<\/sup>L<sup>\u20115<\/sup>)<\/p>\n<\/div>\n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">\u2206<em>s<sub>step1<\/sub><\/em>, \u2206<em>s<sub>step2<\/sub><\/em>, \u2206<em>s<sub>step3<\/sub><\/em>, \u2206<em>s<sub>step4<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>\n<div data-custom-style=\"Body Text\">\n<p>drawdown for each step after a constant pumping<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>period (L)<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\">\u2206<em>Q<sub>1<\/sub><\/em>,\u00a0\u2206<em>Q<sub>2<\/sub><\/em>, \u2206<em>Q<sub>3<\/sub><\/em>, \u2206<em>Q<sub>4<\/sub><\/em><\/td>\n<td style=\"text-align: center;\">=<\/td>\n<td>changes in pumping rates between steps (L<sup>3<\/sup>T<sup>\u20111<\/sup>)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p><em>C<\/em> can be computed based on the response of two sequential steps using the transient step\u2011drawdown data shown in <a href=\"#X617b490325a2b46eba157de35ae9b20fb2f0b70\">Figure 86<\/a>. Some researchers discourage using this approach because in the original formulation of <em>C<\/em> it is assumed to be constant (Jacob, 1950). However, often when this method is used most <em>C<\/em> values show variability. When values of <em>C<\/em> are similar, they can be averaged or a value more representative of the pumping rate selected is used to compute well loss.<\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p><strong>Example<\/strong><\/p>\n<\/div>\n<div data-custom-style=\"Body Text\">\n<p>Using Jacob\u2019s method (1950) compute <em>C<\/em> from the transient data set shown in <a href=\"#X617b490325a2b46eba157de35ae9b20fb2f0b70\">Figure 86<\/a>, the time interval for analysis of each step is 30 minutes from the beginning of the step.<\/p>\n<\/div>\n<table style=\"width: 66%;\" data-custom-style=\"Table\">\n<colgroup>\n<col style=\"width: 66%;\" \/> <\/colgroup>\n<thead>\n<tr>\n<th style=\"text-align: center;\">[latex]C_{1} = \\frac{\\left( \\frac{1\\ m}{500\\frac{m^{3}}{d}} \\right) - \\left( \\frac{1.4\\ m}{600\\frac{m^{3}}{d}} \\right)}{600\\frac{m^{3}}{d} + 500\\frac{m^{3}}{d}} = - 3.0x10^{- 7}\\frac{m^{5}}{d^{2}}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">[latex]C_{2} = \\frac{\\left( \\frac{5.2\\ m}{1030\\frac{m^{3}}{d}} \\right) - \\left( \\frac{1.0\\ m}{500\\frac{m^{3}}{d}} \\right)}{500\\frac{m^{3}}{d} + 1030\\frac{m^{3}}{d}} = 2.0x10^{- 6}\\frac{m^{5}}{d^{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">[latex]C_{2} = \\frac{\\left( \\frac{4.2\\ m}{480\\frac{m^{3}}{d}} \\right) - \\left( \\frac{5.2\\ m}{1030\\frac{m^{3}}{d}} \\right)}{1030\\frac{m^{3}}{d} + 480\\frac{m^{3}}{d}} = 2.4x10^{- 6}\\frac{m^{5}}{d^{2}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-custom-style=\"Body Text\">\n<p>The coefficients are used to compute the additional drawdown caused by non\u2011linear well loss. The negative value computed between the first two steps suggests the well was developing when pumped at 1,100 m<sup>3<\/sup>\/d in the second step. If a step\u2011drawdown test is completed and then followed by a constant discharge pumping test using the single well, the computed well loss (additional drawdown) for that pumping rate would be subtracted from the observed drawdown data before time\u2011drawdown data were analyzed to estimate <em>T<\/em> and <em>S<\/em>.<\/p>\n<\/div>\n<\/div>\n<div id=\"well-interference-1\" class=\"level3\">\n<h3>Well Interference<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>It is best if other nearby wells are not operating during a single pumping test. However, if the operating schedules of surrounding wells in the area are known, well interference can be computed using standard analytical solutions or they can be observed during non\u2011pumping test periods. This involves monitoring water levels prior to and after pumping to see if adjacent pumping wells are impacting the water levels at the test well. If the effects of pumping in the surrounding wells are observed in the time\u2011drawdown data of the test well, they can be removed as described in Section 11.1. Observed drawdown may be affected during both drawdown and recovery.<\/p>\n<\/div>\n<\/div>\n<div id=\"other-conditions-that-effect-pumping-well-drawdown\" class=\"level3\">\n<h3>Other Conditions that Effect Pumping Well Drawdown<\/h3>\n<div data-custom-style=\"First Paragraph\">\n<p>Other factors may influence water levels during a single\u2011well test<\/p>\n<\/div>\n<ul>\n<li>wellbore storage effects can occur, and are more pronounced for larger\u2011diameter wells and lower pumping rates;<\/li>\n<li>confined systems can be affected by barometric pressure changes, tidal loading, and direct loading; and<\/li>\n<li>unconfined systems may be impacted by tidal loading, evapotranspiration cycles, river\u2011stage changes, and short\u2011 or long\u2011term recharge events.<\/li>\n<\/ul>\n<div data-custom-style=\"First Paragraph\">\n<p>Wellbore\u2011storage effects were discussed in Section 12.1. Additional details on the effects of changing barometric pressure as well as both tidal and direct loading on observed water levels are discussed in Section 5. Water\u2011level changes in the pumping well due to these factors will be small in most cases. These factors will have a greater effect on late\u2011time data when drawdown changes due to formational conditions are also small. If time\u2011drawdown data need to be corrected for these perturbations, Kruseman and de Ridder (2000) and Sterrett (2007) provide a list of corrections. Procedures for determining production\u2011well efficiency are described in <span id=\"TextLinkToBox4\" class=\"anchor\"><a href=\"\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/box-4-production-well-efficiency\/\">Box 4<\/a><\/span>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":8,"menu_order":50,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1725","chapter","type-chapter","status-publish","hentry"],"part":581,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1725","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":7,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1725\/revisions"}],"predecessor-version":[{"id":4184,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1725\/revisions\/4184"}],"part":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/parts\/581"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1725\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/media?parent=1725"}],"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=1725"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/contributor?post=1725"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/license?post=1725"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}