{"id":1687,"date":"2026-02-14T04:24:13","date_gmt":"2026-02-14T04:24:13","guid":{"rendered":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/ch048-xhtml\/"},"modified":"2026-08-01T09:26:54","modified_gmt":"2026-08-01T09:26:54","slug":"formulating-equations-to-represent-the-delayed-yield-response","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/chapter\/formulating-equations-to-represent-the-delayed-yield-response\/","title":{"raw":"10.2 Formulating Equations to Represent the Delayed Yield Response","rendered":"10.2 Formulating Equations to Represent the Delayed Yield Response"},"content":{"raw":"
\r\n
\r\n
\r\n\r\nAs shown in Figure 58<\/a>, when some water table aquifers are pumped, the drawdown response begins by rapidly declining then slows, and may exhibit a temporary hiatus before continuing to decline. This response is attributed to three conditions: 1) at early times instantaneous release of stored water throughout the aquifer thickness occurs as water pressure declines, so the elastic storativity of the unit controls the rate of drawdown (S<\/em>=bSs<\/sub><\/em>); 2) at intermediate times the decreased pressure throughout the aquifer causes a vertical gradient that results in water draining from pores above the water table which has an affect similar to leakage from an overlying unit and thus slows the progression of water level decline; 3) at later times drawdown in the depressurized aquifer is controlled by the release of water stored in pores at the water table so specific yield controls the rate of drawdown (S<\/em>=Sy<\/sub><\/em>). The delayed yield response has been generally attributed to gravity\u2011drained water from the capillary fringe and the vadose zone reaching the water table after drawdown has started. The duration of the delayed yield phenomenon is thought to be controlled by vertical gradients created in the zone above the water table as the water table is lowered and the transmissivity of the aquifer (e.g., Domenico & Schwartz, 1998).\r\n\r\n<\/div>\r\n
\"\"\r\n
\r\n
\r\n\r\nFigure<\/strong> 58\u00a0\u2011\u00a0<\/strong>Conceptual model of an unconfined aquifer with delayed yield. As the water table begins to decline, water is released from storage in the saturated portion of the aquifer (S=Ss<\/sub>b<\/em>). A reduction in the rate of drawdown occurs as downward gradients near and above the water table increase and water moves downward from pores draining near the water table, within the migrating capillary fringe, and in the partially saturated pores of the vadose zone (e.g., Nwankwor at al., 1992). As time progresses, the cone of depression expands and the rate of drainage from the capillary fringe and vadose zone declines until drawdown follows that of the Theis model with storativity equal to specific yield (S<\/em>=Sy<\/sub><\/em>).<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
a) A cross\u2011sectional representation of an infinite unconfined aquifer that is pumped at a constant rate. The blue arrows represent groundwater flow paths. b<\/em> is the initial saturated thickness, while Kh<\/sub><\/em> and Kz<\/sub><\/em> are the horizontal and vertical hydraulic conductivities. In the initial stages of pumping the water table declines and a vertical gradient occurs in the zone above the water table. The capillary fringe water and associated vadose zone water (light blue and purple shaded regions above the water table) move vertically (dashed red arrows) recharging the water table. As pumping continues the rate of gravity drainage slows and drawdown becomes directly controlled by the Sy<\/sub><\/em> of the unconfined aquifer.<\/blockquote>\r\n<\/div>\r\n
\r\n
b) Conceptual arithmetic plots of the observed water level response to pumping when delayed yield occurs. The drawdown increases rapidly at first, then slows as the volume of water draining from the zone above the water table temporarily increases. As the availability of drainable water in the zone above the water table decreases, the rate of drawdown increases because it is controlled solely by drainage of pores at the water table. The double arrows indicate the period of delayed yield starting when drawdown data depart from the theoretical Theis curve defined by elastic storage, S<\/em> (black dashed curves) and ending when the data rejoin the Theis curve defined by drainage of pores at the water table, Sy<\/sub><\/em>, (black dot\u2011dashed lines).<\/blockquote>\r\n<\/div>\r\n
\r\n\r\nConceptually, as pumping begins most of the discharge is composed of stored water being released by compression of the aquifer matrix and expansion of the water, analogous to the response of a confined aquifer, S<\/em>=bSs<\/sub><\/em> (Figure 58<\/a> and Box 2). Work by Nwankwor and others (1992) shows that the contribution of gravity drainage from the vadose zone also occurs as pumping is initiated, however the volume reaching the water table is small. As the water table declines, vertical gradients above the water table increase with time over an expanding zone around the well, inducing more vertical flow from pores in the downwardly migrating capillary fringe and in the vadose zone above the fringe. Nwankwor and others (1992) refer to water in these pores as excess storage. The period of reduced drawdown is referred to as delayed yield. It is observed at intermediate times of the pumping test when the volume of inflow from above is sufficient to supplement the volume of water flowing from the formation to the well by temporally providing a second source of water to the aquifer. The initiation and duration of the delay are controlled by the volume and rate of drainage (excess storage) as well as the properties of the saturated portion of the aquifer (T).\r\n\r\n<\/div>\r\n
\r\n\r\nHistorically, some researchers attributed the mechanisms controlling delayed yield solely to processes occurring in the saturated portion of the aquifer (e.g., Neuman, 1975). Today it is known that processes in both the saturated and vadose zone are involved (Nwankwor et al., 1992; Lin et al., 2019).\r\n\r\n<\/div>\r\n
\r\n\r\nThe period of delayed yield can last from seconds to tens of hours (Todd & Mays, 2003). As pumping continues, the contribution of water from the zone above the water table decreases and release of water from storage is dominated by drainage of water from pores at the water table (S<\/em>=Sy<\/sub>),<\/em> thus heads begin to decline again after plateauing. Both the initial drawdown and the post\u2011delayed\u2011yield drawdown response mirror the Theis model in which pumped water is derived only from water released from aquifer storage (Figure 59<\/a>).\r\n\r\n<\/div>\r\n
\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
\r\n
\r\n
\r\n\r\nFigure 59\u00a0\u2011\u00a0<\/strong>Schematic of a log\u2011log plot of the response of an observation well during pumping of an unconfined aquifer. Early time data matches a Theis curve (red dashed curve) reflecting elastic storage (S<\/em>) before drainage is significant. The delayed yield portion of the curve departs from the Theis conditions during intermediate time as indicated by a double black arrow and vertical dotted black lines. As the test progresses drawdown increases and matches a late time Theis curve (black dashed curve) based on drainage of pores at the water table as controlled by specific yield (Sy<\/sub><\/em>).<\/span>\r\n\r\n<\/div><\/blockquote>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"
\n
\n
\n

As shown in Figure 58<\/a>, when some water table aquifers are pumped, the drawdown response begins by rapidly declining then slows, and may exhibit a temporary hiatus before continuing to decline. This response is attributed to three conditions: 1) at early times instantaneous release of stored water throughout the aquifer thickness occurs as water pressure declines, so the elastic storativity of the unit controls the rate of drawdown (S<\/em>=bSs<\/sub><\/em>); 2) at intermediate times the decreased pressure throughout the aquifer causes a vertical gradient that results in water draining from pores above the water table which has an affect similar to leakage from an overlying unit and thus slows the progression of water level decline; 3) at later times drawdown in the depressurized aquifer is controlled by the release of water stored in pores at the water table so specific yield controls the rate of drawdown (S<\/em>=Sy<\/sub><\/em>). The delayed yield response has been generally attributed to gravity\u2011drained water from the capillary fringe and the vadose zone reaching the water table after drawdown has started. The duration of the delayed yield phenomenon is thought to be controlled by vertical gradients created in the zone above the water table as the water table is lowered and the transmissivity of the aquifer (e.g., Domenico & Schwartz, 1998).<\/p>\n<\/div>\n

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

Figure<\/strong> 58\u00a0\u2011\u00a0<\/strong>Conceptual model of an unconfined aquifer with delayed yield. As the water table begins to decline, water is released from storage in the saturated portion of the aquifer (S=Ss<\/sub>b<\/em>). A reduction in the rate of drawdown occurs as downward gradients near and above the water table increase and water moves downward from pores draining near the water table, within the migrating capillary fringe, and in the partially saturated pores of the vadose zone (e.g., Nwankwor at al., 1992). As time progresses, the cone of depression expands and the rate of drainage from the capillary fringe and vadose zone declines until drawdown follows that of the Theis model with storativity equal to specific yield (S<\/em>=Sy<\/sub><\/em>).<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n

\n

a) A cross\u2011sectional representation of an infinite unconfined aquifer that is pumped at a constant rate. The blue arrows represent groundwater flow paths. b<\/em> is the initial saturated thickness, while Kh<\/sub><\/em> and Kz<\/sub><\/em> are the horizontal and vertical hydraulic conductivities. In the initial stages of pumping the water table declines and a vertical gradient occurs in the zone above the water table. The capillary fringe water and associated vadose zone water (light blue and purple shaded regions above the water table) move vertically (dashed red arrows) recharging the water table. As pumping continues the rate of gravity drainage slows and drawdown becomes directly controlled by the Sy<\/sub><\/em> of the unconfined aquifer.<\/p><\/blockquote>\n<\/div>\n

\n

b) Conceptual arithmetic plots of the observed water level response to pumping when delayed yield occurs. The drawdown increases rapidly at first, then slows as the volume of water draining from the zone above the water table temporarily increases. As the availability of drainable water in the zone above the water table decreases, the rate of drawdown increases because it is controlled solely by drainage of pores at the water table. The double arrows indicate the period of delayed yield starting when drawdown data depart from the theoretical Theis curve defined by elastic storage, S<\/em> (black dashed curves) and ending when the data rejoin the Theis curve defined by drainage of pores at the water table, Sy<\/sub><\/em>, (black dot\u2011dashed lines).<\/p><\/blockquote>\n<\/div>\n

\n

Conceptually, as pumping begins most of the discharge is composed of stored water being released by compression of the aquifer matrix and expansion of the water, analogous to the response of a confined aquifer, S<\/em>=bSs<\/sub><\/em> (Figure 58<\/a> and Box 2). Work by Nwankwor and others (1992) shows that the contribution of gravity drainage from the vadose zone also occurs as pumping is initiated, however the volume reaching the water table is small. As the water table declines, vertical gradients above the water table increase with time over an expanding zone around the well, inducing more vertical flow from pores in the downwardly migrating capillary fringe and in the vadose zone above the fringe. Nwankwor and others (1992) refer to water in these pores as excess storage. The period of reduced drawdown is referred to as delayed yield. It is observed at intermediate times of the pumping test when the volume of inflow from above is sufficient to supplement the volume of water flowing from the formation to the well by temporally providing a second source of water to the aquifer. The initiation and duration of the delay are controlled by the volume and rate of drainage (excess storage) as well as the properties of the saturated portion of the aquifer (T).<\/p>\n<\/div>\n

\n

Historically, some researchers attributed the mechanisms controlling delayed yield solely to processes occurring in the saturated portion of the aquifer (e.g., Neuman, 1975). Today it is known that processes in both the saturated and vadose zone are involved (Nwankwor et al., 1992; Lin et al., 2019).<\/p>\n<\/div>\n

\n

The period of delayed yield can last from seconds to tens of hours (Todd & Mays, 2003). As pumping continues, the contribution of water from the zone above the water table decreases and release of water from storage is dominated by drainage of water from pores at the water table (S<\/em>=Sy<\/sub>),<\/em> thus heads begin to decline again after plateauing. Both the initial drawdown and the post\u2011delayed\u2011yield drawdown response mirror the Theis model in which pumped water is derived only from water released from aquifer storage (Figure 59<\/a>).<\/p>\n<\/div>\n

\n

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

\n
\n
\n

Figure 59\u00a0\u2011\u00a0<\/strong>Schematic of a log\u2011log plot of the response of an observation well during pumping of an unconfined aquifer. Early time data matches a Theis curve (red dashed curve) reflecting elastic storage (S<\/em>) before drainage is significant. The delayed yield portion of the curve departs from the Theis conditions during intermediate time as indicated by a double black arrow and vertical dotted black lines. As the test progresses drawdown increases and matches a late time Theis curve (black dashed curve) based on drainage of pores at the water table as controlled by specific yield (Sy<\/sub><\/em>).<\/span><\/p>\n<\/div>\n<\/blockquote>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":8,"menu_order":41,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1687","chapter","type-chapter","status-publish","hentry"],"part":576,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1687","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":3,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1687\/revisions"}],"predecessor-version":[{"id":3295,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1687\/revisions\/3295"}],"part":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/parts\/576"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/1687\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/media?parent=1687"}],"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=1687"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/contributor?post=1687"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/an-introduction-to-hydraulic-testing-in-hydrogeology\/wp-json\/wp\/v2\/license?post=1687"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}