{"id":234,"date":"2022-01-13T23:17:24","date_gmt":"2022-01-13T23:17:24","guid":{"rendered":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/chapter\/pumping-from-a-water-table-aquifer\/"},"modified":"2022-01-17T00:03:27","modified_gmt":"2022-01-17T00:03:27","slug":"pumping-from-a-water-table-aquifer","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/chapter\/pumping-from-a-water-table-aquifer\/","title":{"raw":"2.2 Pumping from a Water Table Aquifer","rendered":"2.2 Pumping from a Water Table Aquifer"},"content":{"raw":"
\r\n

For the sake of simplicity, assume a water table aquifer is horizontal and the piezometric decline due to pumping over a given time interval is \u0394z<\/em>.<\/em> Let \u03b8<\/em>w<\/em><\/sub> be the moisture content (i.e., the fraction of the total porous medium volume occupied by water, moisture content is equal to porosity in a fully saturated medium) within the unsaturated zone above A, and, between the phreatic surfaces (labeled as the piezometric levels A and B in Figure\u00a013) after the piezometric surface has declined from A to B.<\/a><\/p>\r\n

\"Sketch<\/p>\r\n

Figure\u00a0<\/strong>13<\/strong>\u00a0<\/strong>\u2011\u00a0<\/strong>Sketch of a pumped water table aquifer.<\/p>\r\n

As a result of lowering the piezometric level, there will be an increase of effective stress due to drainage of water from the zone between A and B because that zone is no longer under the influence of pore water pressure (Archimedes\u2019 upward buoyant force). At any point location between B and C, the geostatic stress, \u03c3<\/em>c<\/em><\/sub>, is decreased by the quantity \u03b3<\/em>\u0394z<\/em>(\u03d5<\/em>\u00a0\u2011\u00a0\u03b8<\/em>w<\/em><\/sub>), with \u03b3<\/em> being the specific weight of water with dimensions ML-2<\/sup>T-2<\/sup>, and the quantity \u03b3<\/em>\u0394z<\/em> being p<\/em>. Therefore, the effective stress, \u03c3<\/em>z<\/em><\/sub>, is increased by (Equation\u00a08),<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n\u0394\u03c3<\/em>z<\/em><\/sub> = \u03b3<\/em>\u0394z<\/em>(1 - \u03d5<\/em> +\u03b8<\/em>w<\/em><\/sub>)<\/td>\r\n(8)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

that is to say, it is increased by the difference in Archimedes\u2019 force exerted upon the solid grains before and after pumping. The layers underlying the water table aquifer, where p <\/span><\/em>remains constant, experience a \u03c3<\/em>z<\/em><\/sub> reduction equal to the \u03c3<\/em>c<\/em><\/sub> reduction (that is, \u03b3<\/em>\u0394z<\/em>(\u03d5<\/em>\u00a0\u2011\u00a0\u03b8<\/em>w<\/em><\/sub>)) with a resulting small rebound. As the magnitude of the rebound is small, it is discarded in the following calculation. Let\u2019s refer to the mid\u2011point between B and C in Figure\u00a013. The stress \u03c3<\/em>z<\/em><\/sub>0<\/sub> before withdrawal is as shown in Equation\u00a09.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n\u03c3<\/em>z<\/em>0<\/sub> = (1 - \u03d5<\/em>)[\u03b3<\/em>\u2032(d<\/em> + \u0394z<\/em> + s<\/em>0<\/sub>\/2) \u2212 \u03b3(<\/em>\u0394z<\/em> + s<\/em>0<\/sub>\/2)] + \u03b3\u03b8<\/em>w<\/em><\/sub>d<\/em><\/td>\r\n(9)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nwhere:\r\n\r\n\r\n\r\n
\u03b3<\/em>\u2032<\/td>\r\n=<\/td>\r\nspecific weight of the solid grains (ML-<\/sup>2<\/sup>T-2<\/sup>)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

To obtain Equation\u00a09 we used Equation\u00a03<\/a> where \u03c3<\/em>c<\/em><\/sub> is equal to the geostatic weight of a soil column with height h<\/em>\u00a0=\u00a0d<\/em>\u00a0+\u00a0\u0394z<\/em>\u00a0+\u00a0s<\/em>0<\/sub>\/2<\/em>, <\/em>that is, \u03c3<\/em>c<\/em><\/sub>\u00a0=\u00a0\u03b3\u03b8<\/em>w<\/em><\/sub>d<\/em>\u00a0<\/em>+\u00a0\u03b3<\/em>\u2032h<\/em>(1 \u2212 \u03d5<\/em>)\u00a0+\u00a0\u03b3\u03d5<\/em>(h<\/em> \u2212 d<\/em>), and p<\/em>\u00a0=\u00a0\u03b3<\/em>(h<\/em> \u2212 d<\/em>). <\/em>We thus locate the \u03c3<\/em>z<\/em><\/sub>0 <\/sub>point in Figure\u00a012a and, by making use of Equation\u00a08, compute the subsidence at a given depth according to Equation\u00a04<\/a> as follows.<\/p>\r\n

[latex]\\displaystyle \\eta =\\frac{\\Delta e}{1+e_{0}}\\left ( s_{0}+\\frac{\\Delta z}{2} \\right )[\/latex]<\/p>\r\n

It is easier to visualize the relationship between compaction and void ratio using an abstract version of Figure\u00a02 in which all solids are grouped with no pore space and all pore space occupies the remainder of the volume, with example values assigned and calculations carried out, as explained in <\/a>Box\u00a0<\/span><\/a>2<\/span><\/a>. Box\u00a02 also provides a worked example of calculating the change in effective stress for a decline in the piezometric level of an unconfined aquifer as shown in Figure\u00a013.<\/p>\r\n

If s<\/em>0<\/sub> is large, we can divide s<\/em>0<\/sub> into a number of sublayers and implement the above calculation for the mid\u2011point of each sublayer (\u0394\u03c3<\/em>z<\/em><\/sub> is the same for each sublayer while \u03c3<\/em>z<\/em><\/sub>0<\/sub> changes).<\/p>\r\n

In summary, the compaction of a phreatic aquifer is shown in Equation\u00a010.<\/a><\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\eta =\\left ( s_{0}+\\frac{\\Delta z}{2} \\right )c_{b}\\Delta \\sigma _{z}[\/latex]<\/td>\r\n(10)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

In Equation\u00a010, c<\/em>b<\/em><\/sub> is the uniaxial vertical soil compressibility, and \u0394\u03c3<\/em>z<\/em><\/sub> is the change in the effective intergranular stress (Equation 8).<\/p>\r\n\r\n<\/div>","rendered":"

\n

For the sake of simplicity, assume a water table aquifer is horizontal and the piezometric decline due to pumping over a given time interval is \u0394z<\/em>.<\/em> Let \u03b8<\/em>w<\/em><\/sub> be the moisture content (i.e., the fraction of the total porous medium volume occupied by water, moisture content is equal to porosity in a fully saturated medium) within the unsaturated zone above A, and, between the phreatic surfaces (labeled as the piezometric levels A and B in Figure\u00a013) after the piezometric surface has declined from A to B.<\/a><\/p>\n

\"Sketch<\/p>\n

Figure\u00a0<\/strong>13<\/strong>\u00a0<\/strong>\u2011\u00a0<\/strong>Sketch of a pumped water table aquifer.<\/p>\n

As a result of lowering the piezometric level, there will be an increase of effective stress due to drainage of water from the zone between A and B because that zone is no longer under the influence of pore water pressure (Archimedes\u2019 upward buoyant force). At any point location between B and C, the geostatic stress, \u03c3<\/em>c<\/em><\/sub>, is decreased by the quantity \u03b3<\/em>\u0394z<\/em>(\u03d5<\/em>\u00a0\u2011\u00a0\u03b8<\/em>w<\/em><\/sub>), with \u03b3<\/em> being the specific weight of water with dimensions ML-2<\/sup>T-2<\/sup>, and the quantity \u03b3<\/em>\u0394z<\/em> being p<\/em>. Therefore, the effective stress, \u03c3<\/em>z<\/em><\/sub>, is increased by (Equation\u00a08),<\/a><\/p>\n\n\n\n
<\/td>\n\u0394\u03c3<\/em>z<\/em><\/sub> = \u03b3<\/em>\u0394z<\/em>(1 – \u03d5<\/em> +\u03b8<\/em>w<\/em><\/sub>)<\/td>\n(8)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

that is to say, it is increased by the difference in Archimedes\u2019 force exerted upon the solid grains before and after pumping. The layers underlying the water table aquifer, where p <\/span><\/em>remains constant, experience a \u03c3<\/em>z<\/em><\/sub> reduction equal to the \u03c3<\/em>c<\/em><\/sub> reduction (that is, \u03b3<\/em>\u0394z<\/em>(\u03d5<\/em>\u00a0\u2011\u00a0\u03b8<\/em>w<\/em><\/sub>)) with a resulting small rebound. As the magnitude of the rebound is small, it is discarded in the following calculation. Let\u2019s refer to the mid\u2011point between B and C in Figure\u00a013. The stress \u03c3<\/em>z<\/em><\/sub>0<\/sub> before withdrawal is as shown in Equation\u00a09.<\/a><\/p>\n\n\n\n
<\/td>\n\u03c3<\/em>z<\/em>0<\/sub> = (1 – \u03d5<\/em>)[\u03b3<\/em>\u2032(d<\/em> + \u0394z<\/em> + s<\/em>0<\/sub>\/2) \u2212 \u03b3(<\/em>\u0394z<\/em> + s<\/em>0<\/sub>\/2)] + \u03b3\u03b8<\/em>w<\/em><\/sub>d<\/em><\/td>\n(9)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n
\u03b3<\/em>\u2032<\/td>\n=<\/td>\nspecific weight of the solid grains (ML–<\/sup>2<\/sup>T-2<\/sup>)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

To obtain Equation\u00a09 we used Equation\u00a03<\/a> where \u03c3<\/em>c<\/em><\/sub> is equal to the geostatic weight of a soil column with height h<\/em>\u00a0=\u00a0d<\/em>\u00a0+\u00a0\u0394z<\/em>\u00a0+\u00a0s<\/em>0<\/sub>\/2<\/em>, <\/em>that is, \u03c3<\/em>c<\/em><\/sub>\u00a0=\u00a0\u03b3\u03b8<\/em>w<\/em><\/sub>d<\/em>\u00a0<\/em>+\u00a0\u03b3<\/em>\u2032h<\/em>(1 \u2212 \u03d5<\/em>)\u00a0+\u00a0\u03b3\u03d5<\/em>(h<\/em> \u2212 d<\/em>), and p<\/em>\u00a0=\u00a0\u03b3<\/em>(h<\/em> \u2212 d<\/em>). <\/em>We thus locate the \u03c3<\/em>z<\/em><\/sub>0 <\/sub>point in Figure\u00a012a and, by making use of Equation\u00a08, compute the subsidence at a given depth according to Equation\u00a04<\/a> as follows.<\/p>\n

\"\displaystyle \eta =\frac{\Delta e}{1+e_{0}}\left ( s_{0}+\frac{\Delta z}{2} \right )\"<\/p>\n

It is easier to visualize the relationship between compaction and void ratio using an abstract version of Figure\u00a02 in which all solids are grouped with no pore space and all pore space occupies the remainder of the volume, with example values assigned and calculations carried out, as explained in <\/a>Box\u00a0<\/span><\/a>2<\/span><\/a>. Box\u00a02 also provides a worked example of calculating the change in effective stress for a decline in the piezometric level of an unconfined aquifer as shown in Figure\u00a013.<\/p>\n

If s<\/em>0<\/sub> is large, we can divide s<\/em>0<\/sub> into a number of sublayers and implement the above calculation for the mid\u2011point of each sublayer (\u0394\u03c3<\/em>z<\/em><\/sub> is the same for each sublayer while \u03c3<\/em>z<\/em><\/sub>0<\/sub> changes).<\/p>\n

In summary, the compaction of a phreatic aquifer is shown in Equation\u00a010.<\/a><\/p>\n\n\n\n
<\/td>\n\"\displaystyle \eta =\left ( s_{0}+\frac{\Delta z}{2} \right )c_{b}\Delta \sigma _{z}\"<\/td>\n(10)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

In Equation\u00a010, c<\/em>b<\/em><\/sub> is the uniaxial vertical soil compressibility, and \u0394\u03c3<\/em>z<\/em><\/sub> is the change in the effective intergranular stress (Equation 8).<\/p>\n<\/div>\n","protected":false},"author":1,"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-234","chapter","type-chapter","status-publish","hentry"],"part":121,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/pressbooks\/v2\/chapters\/234","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":11,"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/pressbooks\/v2\/chapters\/234\/revisions"}],"predecessor-version":[{"id":450,"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/pressbooks\/v2\/chapters\/234\/revisions\/450"}],"part":[{"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/pressbooks\/v2\/parts\/121"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/pressbooks\/v2\/chapters\/234\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/wp\/v2\/media?parent=234"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/pressbooks\/v2\/chapter-type?post=234"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/wp\/v2\/contributor?post=234"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/wp\/v2\/license?post=234"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}