{"id":1600,"date":"2026-02-14T04:22:11","date_gmt":"2026-02-14T04:22:11","guid":{"rendered":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/ch014-xhtml\/"},"modified":"2026-10-21T03:59:48","modified_gmt":"2026-10-21T03:59:48","slug":"development-of-the-cone-of-depression-under-transient-conditions","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/development-of-the-cone-of-depression-under-transient-conditions\/","title":{"raw":"3.1 Development of the Cone of Depression Under Transient Conditions","rendered":"3.1 Development of the Cone of Depression Under Transient Conditions"},"content":{"raw":"
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\r\n\r\nThe region surrounding a pumping well where water levels are drawn down by pumping is referred to as the cone of depression. Maps, cross sections, and 3D representations of the cone of depression can be plotted using head or drawdown values (Figure 2<\/a>). Drawdown is computed by taking the static head in a well (corrected for background influences on head) minus the measured head at a time since the pumping began (Figure 2<\/a>). Flow to the well is radial under isotropic and homogeneous conditions (i.e., where hydrogeologic properties are constant and the same at all points within the hydrogeologic system being investigated; Woessner & Poeter (2020)). Drawdown occurs in the pumping well, observation wells and any other wells within the cone of depression. Transient conditions exist when pumping induces continuing drawdown over time. Water levels in the pumping well and in nearby unpumped observation wells decrease rapidly when pumping starts, then the rate of drawdown slows as the test progresses (Figure 3<\/a>).\r\n\r\n<\/div>\r\n
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\r\n\r\nFigure 2 -<\/strong> Groundwater flow to a pumping well penetrating an isotropic and homogeneous confined aquifer.<\/span>\r\n\r\n<\/div><\/blockquote>\r\n<\/div>\r\n
\r\n
a) Cross\u2011sectional view of the response of water levels (vertical black arrows) in the pumping well and surrounding observation wells. The decline in water levels represented by a potentiometric surface is shown as a drawdown curve. The groundwater velocity in the confined aquifer is highest near the well (large blue vectors).<\/blockquote>\r\n<\/div>\r\n
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b) Map view of the drawdown distribution around a pumping well. Numbers represent values of drawdown in units of length. A and B, located at the red dots, are unpumped observation wells. The zone of measurable drawdown is referred to as the cone of depression, which is centered at the pumping well (black dot). The flow of water (blue arrows) to the well is radial and equal in all directions when the aquifer is isotropic and homogeneous. Velocity (black vectors) increases from the outside edge of the cone of depression where the hydraulic gradient is very small to the pumping well where the gradient is large.<\/blockquote>\r\n<\/div>\r\n
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\r\n\r\n\"\"\r\n\r\n<\/div>\r\nFigure 3<\/strong> \u2011<\/strong> Schematic of the changes in drawdown with time for a pumping well and the two observation wells shown in Figure 2<\/a><\/span> where Well A is closer to the pumping well than Well B. Drawdown values are plotted on arithmetic scales. Positive numbers represent potentiometric surface decline. Drawdown has dimensions of length (e.g., units of m, ft, and so on) and time may be in units of seconds, minutes, hours, or days depending on the length of the test and properties of the system tested. Conceptually, in a confined isotropic and homogeneous system under a constant pumping rate, the rate of drawdown is logarithmic with time, and drawdown is greatest at the pumping well. Drawdown at the observation wells begins sometime after pumping starts and is less in observation wells located farther from the pumping well.\r\n\r\n<\/div><\/blockquote>\r\n<\/div>\r\n
\r\n\r\nWater levels in the pumping well begin to recover (rise) after pumping stops. With time, water levels begin to recover in observation wells within the cone of depression, with recovery occurring first in wells nearest to the pumping well. Once the pump is shut off, water within the cone of depression continues to flow toward the well under the pumping\u2011created hydraulic gradients. As water is no longer being extracted from the pumping well, groundwater from the surrounding aquifer region replenishes aquifer storage, and water levels begin to recover. Recovery is rapid at first, then slows as hydraulic gradients within the cone of depression decrease (Figure 4<\/a>). Similar patterns are observed in observation wells.\r\n\r\n<\/div>\r\n
\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
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\r\n\r\nFigure 4 -<\/strong> Illustration of drawdown and recovery for a pumping well discharging at a constant rate. The dashed blue line represents the corrected static water level. The red line is the drawdown during pumping. The brown vertical line represents the time when the pump is turned off. The blue line shows the recovering water levels after the pumping has ended. The vertical blue arrow shows the residual or unrecovered drawdown. The dot\u2011dashed red line is the projected drawdown that would continue to occur if the pump was not turned off.<\/span>\r\n\r\n<\/div><\/blockquote>\r\n<\/div>\r\n
\r\n\r\nThe background water level is rarely static because it is subject to the influence of other features connected to and within the groundwater system (e.g., pumping of nearby wells; changing water levels in bodies of surface water such as a stream, lake, or ocean; recharge from precipitation). Consequently, water levels need to be recorded for some time prior to the start of pumping and for some time after the end of pumping, to infer the trend and correct the static water level for rises and falls of the background water levels during the test. Corrections for variation in the background water level may need to be applied when analyzing the test as discussed in Section\u00a05.\r\n\r\n<\/div>\r\n
\r\n\r\nWater pumped from a well in a confined aquifer is released from storage as the head declines due to compaction of the water\u2011bearing unit and expansion of stored water as pressure decreases (Woessner & Poeter, 2020). The unit is not physically dewatered, rather the drawdown cone forms in the potentiometric surface and the hydrogeologic unit remains confined. During recovery, the observed decrease in drawdown is caused by water reentering storage as the increased pressure expands the water\u2011bearing unit and compresses the water within the unit. The recovery is rapid at first and slows with time.\r\n\r\n<\/div>\r\n
\r\n\r\nWhen an unconfined unit is pumped, water levels decline as water flows to the wellbore and a cone of depression develops in the water table (Figure 4<\/a>). In contrast to a confined system, a portion of the aquifer is physically dewatered as water drains by gravity from saturated pores and the water table elevation declines over time. Water released from storage is a combination of water released from elastic storage as in the confined system (compression of the matrix and expansion of the water), and water that drains from saturated pores as the water level declines. The drawdown is rapid at first then slows with time. In some settings at early times after pumping begins, drainage of water from the capillary fringe and vadose zone recharges the aquifer temporarily slowing the decline of the water table elevation. As pumping continues, drawdown resumes its earlier logarithmic rate of change. Specifics of pumping unconfined aquifers are discussed in Section 10.\r\n\r\n<\/div>\r\n
\r\n\r\nPumping reduces the saturated thickness of an unconfined aquifer at and near the well as the cone of depression expands over time. Under isotropic and homogeneous conditions and with a fully penetrating production well, groundwater flow is radial to the well and mostly horizontal (Figure 5<\/a>). Near the well, however, where pores have drained, flow parallels the sloping water table. In this region, vertical flow occurs (Figure 5<\/a>a).\r\n\r\n<\/div>\r\n
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\r\n\r\nFigure 5 -<\/strong> Schematic of the cone of depression in the water table of an unconfined aquifer. The aquifer is assumed to be isotropic and homogeneous, and the wells fully penetrate the aquifer.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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a) Schematic of a cone of depression showing drawdown data at surrounding observation wells and the pumping well. The blue\u2011gray block represents a fully saturated portion of the unconfined aquifer. The light brown shaded area represents the portion of the aquifer dewatered during pumping. Flow (represented by arrows) to the well is radial. Near the pumping well a vertical flow component is present as flow parallels the curved water table.<\/blockquote>\r\n<\/div>\r\n
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b) A map view of the cone of depression plotted as drawdown. The flow (black arrows) is radial. Numbers represent values of drawdown in units of length. Gradient increases with proximity to the pumping well (drawdown contours are closer together). The velocity (blue vectors) increases as groundwater moves from the edge of the cone to the center. The red dot represents an observation well.<\/blockquote>\r\n<\/div>\r\n
\r\n\r\nUnder ideal conditions, the rate of drawdown decreases with time. This is because the volume of confined aquifer material in the cylinder below the cone of depression forming in the potentiometric surface increases with time by the square of the radius of the cone of depression. Water is withdrawn at the same rate over time, but the cylindrical volume contributing water grows as the cone of depression expands. Thus, less drawdown is required to yield the same amount of water from storage. The volume of the cylinder of aquifer material below the cone of depression is h<\/em>\u03c0r<\/em>2<\/sup> where h<\/em> is the thickness of the confined aquifer and where r1<\/sub><\/em> is the radius of the cone at Time 1 and r2<\/sub><\/em> is the radius at Time 2 as shown in Figure 6<\/a>.\r\n\r\n<\/div>\r\n
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\r\n\r\nFigure 6 -<\/strong> As the duration of pumping increases in this isotropic and homogeneous confined aquifer the rate of water\u2011level decline decreases (Figure 3<\/a><\/span>) due to the increasing volume of aquifer contributing water to the pumped well. A rapid decline in head occurs in the early period of pumping. At Time 1 the volume of water that has been pumped from the well was derived from the release of water stored in the aquifer as controlled by the storativity (S<\/em>) of the aquifer within the dark blue cylinder of radius r1<\/sub><\/em> in the confined unit. S<\/em> is a dimensionless number reflecting the volume of water produced for one unit of drawdown per one unit of area. Thus, the volume released from the aquifer after a period of pumping is the product of three entities: S, the average drawdown over the area of the cone of depression, and the surface area defined by the perimeter of the cone (i.e., Volume<\/em>=S<\/em>(AverageDrawdown)(Area). A drawdown cone is also shown at Time 2 after pumping twice as long at the same constant rate. Because the pumping is constant and for the same duration, the same volume of water is released from storage between Time 1 and Time 2 throughout the light blue cylinder of radius r2<\/sub><\/em> and this zone has the same storativity (S<\/em>). The drawdowns between Time 1 and Tine 2 are smaller than between Time 0 and Time 1 because the same volume of water is yielded from a larger area.\r\n\r\n<\/div><\/blockquote>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"
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The region surrounding a pumping well where water levels are drawn down by pumping is referred to as the cone of depression. Maps, cross sections, and 3D representations of the cone of depression can be plotted using head or drawdown values (Figure 2<\/a>). Drawdown is computed by taking the static head in a well (corrected for background influences on head) minus the measured head at a time since the pumping began (Figure 2<\/a>). Flow to the well is radial under isotropic and homogeneous conditions (i.e., where hydrogeologic properties are constant and the same at all points within the hydrogeologic system being investigated; Woessner & Poeter (2020)). Drawdown occurs in the pumping well, observation wells and any other wells within the cone of depression. Transient conditions exist when pumping induces continuing drawdown over time. Water levels in the pumping well and in nearby unpumped observation wells decrease rapidly when pumping starts, then the rate of drawdown slows as the test progresses (Figure 3<\/a>).<\/p>\n<\/div>\n

\n

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

\n
\n
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Figure 2 –<\/strong> Groundwater flow to a pumping well penetrating an isotropic and homogeneous confined aquifer.<\/span><\/p>\n<\/div>\n<\/blockquote>\n<\/div>\n

\n

a) Cross\u2011sectional view of the response of water levels (vertical black arrows) in the pumping well and surrounding observation wells. The decline in water levels represented by a potentiometric surface is shown as a drawdown curve. The groundwater velocity in the confined aquifer is highest near the well (large blue vectors).<\/p><\/blockquote>\n<\/div>\n

\n

b) Map view of the drawdown distribution around a pumping well. Numbers represent values of drawdown in units of length. A and B, located at the red dots, are unpumped observation wells. The zone of measurable drawdown is referred to as the cone of depression, which is centered at the pumping well (black dot). The flow of water (blue arrows) to the well is radial and equal in all directions when the aquifer is isotropic and homogeneous. Velocity (black vectors) increases from the outside edge of the cone of depression where the hydraulic gradient is very small to the pumping well where the gradient is large.<\/p><\/blockquote>\n<\/div>\n

\n
\n
\n
\n

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

Figure 3<\/strong> \u2011<\/strong> Schematic of the changes in drawdown with time for a pumping well and the two observation wells shown in Figure 2<\/a><\/span> where Well A is closer to the pumping well than Well B. Drawdown values are plotted on arithmetic scales. Positive numbers represent potentiometric surface decline. Drawdown has dimensions of length (e.g., units of m, ft, and so on) and time may be in units of seconds, minutes, hours, or days depending on the length of the test and properties of the system tested. Conceptually, in a confined isotropic and homogeneous system under a constant pumping rate, the rate of drawdown is logarithmic with time, and drawdown is greatest at the pumping well. Drawdown at the observation wells begins sometime after pumping starts and is less in observation wells located farther from the pumping well.<\/p>\n<\/div>\n<\/blockquote>\n<\/div>\n

\n

Water levels in the pumping well begin to recover (rise) after pumping stops. With time, water levels begin to recover in observation wells within the cone of depression, with recovery occurring first in wells nearest to the pumping well. Once the pump is shut off, water within the cone of depression continues to flow toward the well under the pumping\u2011created hydraulic gradients. As water is no longer being extracted from the pumping well, groundwater from the surrounding aquifer region replenishes aquifer storage, and water levels begin to recover. Recovery is rapid at first, then slows as hydraulic gradients within the cone of depression decrease (Figure 4<\/a>). Similar patterns are observed in observation wells.<\/p>\n<\/div>\n

\n

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

\n
\n
\n

Figure 4 –<\/strong> Illustration of drawdown and recovery for a pumping well discharging at a constant rate. The dashed blue line represents the corrected static water level. The red line is the drawdown during pumping. The brown vertical line represents the time when the pump is turned off. The blue line shows the recovering water levels after the pumping has ended. The vertical blue arrow shows the residual or unrecovered drawdown. The dot\u2011dashed red line is the projected drawdown that would continue to occur if the pump was not turned off.<\/span><\/p>\n<\/div>\n<\/blockquote>\n<\/div>\n

\n

The background water level is rarely static because it is subject to the influence of other features connected to and within the groundwater system (e.g., pumping of nearby wells; changing water levels in bodies of surface water such as a stream, lake, or ocean; recharge from precipitation). Consequently, water levels need to be recorded for some time prior to the start of pumping and for some time after the end of pumping, to infer the trend and correct the static water level for rises and falls of the background water levels during the test. Corrections for variation in the background water level may need to be applied when analyzing the test as discussed in Section\u00a05.<\/p>\n<\/div>\n

\n

Water pumped from a well in a confined aquifer is released from storage as the head declines due to compaction of the water\u2011bearing unit and expansion of stored water as pressure decreases (Woessner & Poeter, 2020). The unit is not physically dewatered, rather the drawdown cone forms in the potentiometric surface and the hydrogeologic unit remains confined. During recovery, the observed decrease in drawdown is caused by water reentering storage as the increased pressure expands the water\u2011bearing unit and compresses the water within the unit. The recovery is rapid at first and slows with time.<\/p>\n<\/div>\n

\n

When an unconfined unit is pumped, water levels decline as water flows to the wellbore and a cone of depression develops in the water table (Figure 4<\/a>). In contrast to a confined system, a portion of the aquifer is physically dewatered as water drains by gravity from saturated pores and the water table elevation declines over time. Water released from storage is a combination of water released from elastic storage as in the confined system (compression of the matrix and expansion of the water), and water that drains from saturated pores as the water level declines. The drawdown is rapid at first then slows with time. In some settings at early times after pumping begins, drainage of water from the capillary fringe and vadose zone recharges the aquifer temporarily slowing the decline of the water table elevation. As pumping continues, drawdown resumes its earlier logarithmic rate of change. Specifics of pumping unconfined aquifers are discussed in Section 10.<\/p>\n<\/div>\n

\n

Pumping reduces the saturated thickness of an unconfined aquifer at and near the well as the cone of depression expands over time. Under isotropic and homogeneous conditions and with a fully penetrating production well, groundwater flow is radial to the well and mostly horizontal (Figure 5<\/a>). Near the well, however, where pores have drained, flow parallels the sloping water table. In this region, vertical flow occurs (Figure 5<\/a>a).<\/p>\n<\/div>\n

\n

\"\"<\/p>\n

\n
\n
\n

Figure 5 –<\/strong> Schematic of the cone of depression in the water table of an unconfined aquifer. The aquifer is assumed to be isotropic and homogeneous, and the wells fully penetrate the aquifer.<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

\n

a) Schematic of a cone of depression showing drawdown data at surrounding observation wells and the pumping well. The blue\u2011gray block represents a fully saturated portion of the unconfined aquifer. The light brown shaded area represents the portion of the aquifer dewatered during pumping. Flow (represented by arrows) to the well is radial. Near the pumping well a vertical flow component is present as flow parallels the curved water table.<\/p><\/blockquote>\n<\/div>\n

\n

b) A map view of the cone of depression plotted as drawdown. The flow (black arrows) is radial. Numbers represent values of drawdown in units of length. Gradient increases with proximity to the pumping well (drawdown contours are closer together). The velocity (blue vectors) increases as groundwater moves from the edge of the cone to the center. The red dot represents an observation well.<\/p><\/blockquote>\n<\/div>\n

\n

Under ideal conditions, the rate of drawdown decreases with time. This is because the volume of confined aquifer material in the cylinder below the cone of depression forming in the potentiometric surface increases with time by the square of the radius of the cone of depression. Water is withdrawn at the same rate over time, but the cylindrical volume contributing water grows as the cone of depression expands. Thus, less drawdown is required to yield the same amount of water from storage. The volume of the cylinder of aquifer material below the cone of depression is h<\/em>\u03c0r<\/em>2<\/sup> where h<\/em> is the thickness of the confined aquifer and where r1<\/sub><\/em> is the radius of the cone at Time 1 and r2<\/sub><\/em> is the radius at Time 2 as shown in Figure 6<\/a>.<\/p>\n<\/div>\n

\n

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

\n
\n
\n

Figure 6 –<\/strong> As the duration of pumping increases in this isotropic and homogeneous confined aquifer the rate of water\u2011level decline decreases (Figure 3<\/a><\/span>) due to the increasing volume of aquifer contributing water to the pumped well. A rapid decline in head occurs in the early period of pumping. At Time 1 the volume of water that has been pumped from the well was derived from the release of water stored in the aquifer as controlled by the storativity (S<\/em>) of the aquifer within the dark blue cylinder of radius r1<\/sub><\/em> in the confined unit. S<\/em> is a dimensionless number reflecting the volume of water produced for one unit of drawdown per one unit of area. Thus, the volume released from the aquifer after a period of pumping is the product of three entities: S, the average drawdown over the area of the cone of depression, and the surface area defined by the perimeter of the cone (i.e., Volume<\/em>=S<\/em>(AverageDrawdown)(Area). A drawdown cone is also shown at Time 2 after pumping twice as long at the same constant rate. Because the pumping is constant and for the same duration, the same volume of water is released from storage between Time 1 and Time 2 throughout the light blue cylinder of radius r2<\/sub><\/em> and this zone has the same storativity (S<\/em>). The drawdowns between Time 1 and Tine 2 are smaller than between Time 0 and Time 1 because the same volume of water is yielded from a larger area.<\/p>\n<\/div>\n<\/blockquote>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":8,"menu_order":7,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1600","chapter","type-chapter","status-publish","hentry"],"part":513,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1600","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\/1600\/revisions"}],"predecessor-version":[{"id":4182,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1600\/revisions\/4182"}],"part":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/parts\/513"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1600\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/media?parent=1600"}],"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=1600"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/contributor?post=1600"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/license?post=1600"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}