{"id":279,"date":"2022-01-13T23:17:39","date_gmt":"2022-01-13T23:17:39","guid":{"rendered":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/chapter\/exercise-3-solution\/"},"modified":"2022-01-18T04:13:53","modified_gmt":"2022-01-18T04:13:53","slug":"exercise-3-solution","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/chapter\/exercise-3-solution\/","title":{"raw":"Exercise 3 Solution","rendered":"Exercise 3 Solution"},"content":{"raw":"
\r\n

The cumulative land subsidence is due to the compaction of the three layers (Equation 13<\/a>):<\/p>\r\n

\u03b7<\/em>tot<\/em><\/sub> = \u03b7<\/em>aquifer1<\/em><\/sub> + \u03b7<\/em>aquitard<\/em><\/sub> + \u03b7<\/em>aquifer2<\/em><\/sub><\/p>\r\n

The compaction of each aquifer can be obtained by means of Equation 12<\/a>:<\/p>\r\n

\u03b7<\/em>aquifer1<\/em><\/sub> = t<\/em>aquifer1<\/em><\/sub> c<\/em>b<\/em>,aquifer<\/em>1<\/sub> \u2206\u03c3<\/em>z<\/em>,aquifer1<\/em><\/sub><\/p>\r\n

\u03b7<\/em>aquifer2<\/em><\/sub> = t<\/em>aquifer2<\/em><\/sub> c<\/em>b<\/em>,aquifer<\/em>2<\/sub> \u2206\u03c3<\/em>z<\/em>,aquifer2<\/em><\/sub><\/p>\r\n

The ultimate aquitard compaction amounts to (Equation 28<\/a>):<\/p>\r\n

\u03b7<\/em>aquitard<\/em><\/sub> = t<\/em>aquitard<\/em><\/sub> c<\/em>b<\/em>,aquitard<\/em><\/sub> 0.5 (\u2206\u03c3<\/em>z<\/em>,aquifer1<\/em><\/sub> + \u2206\u03c3<\/em>z<\/em>,aquifer2<\/em><\/sub>)<\/p>\r\n

The sum becomes:<\/p>\r\n

\u03b7<\/em>tot<\/em><\/sub> = (t<\/em>aquifer1<\/em><\/sub> c<\/em>b<\/em>,aquifer1<\/em><\/sub> \u2206\u03c3<\/em>z<\/em>,aquifer1<\/em><\/sub>) +\u00a0(t<\/em>aquitard<\/em><\/sub>\u00a0c<\/em>b<\/em>,aquitard<\/em><\/sub>\u00a00.5\u00a0(\u2206\u03c3<\/em>z<\/em>,aquifer1<\/em><\/sub>\u00a0+\u00a0\u2206\u03c3<\/em>z<\/em>,aquifer2<\/em><\/sub>)) +\u00a0(t<\/em>aquifer2<\/em><\/sub>\u00a0c<\/em>b<\/em>,aquifer2<\/em><\/sub>\u00a0\u2206\u03c3<\/em>z<\/em>,aquifer2<\/em><\/sub>)<\/p>\r\n

Taking into account the relationships between the compressibility of the three layers, c<\/em>b<\/em><\/sub>,aquifer<\/em><\/sub>1<\/em><\/sub> = 2 c<\/em>b<\/em><\/sub>,aquifer2<\/em><\/sub> and c<\/em>b<\/em><\/sub>,aquifer1<\/em><\/sub> = 0.1 c<\/em>b<\/em><\/sub>, aquitard<\/em><\/sub> we can write the expression in terms of the compressibility of aquifer 1:<\/p>\r\n

\u03b7<\/em>tot<\/em><\/sub> = (t<\/em>aquifer1<\/em><\/sub>\u00a0c<\/em>b<\/em>,aquifer1<\/em><\/sub>\u00a0\u0394\u03c3<\/em>z<\/em>,aquifer1<\/em><\/sub>) +\u00a0(t<\/em>aquitard<\/em><\/sub>\u00a010\u00a0c<\/em>b<\/em>,aquifer1<\/em><\/sub>\u00a00.5(\u2206\u03c3<\/em>z<\/em>,aquifer1<\/em><\/sub>\u00a0+\u00a0\u2206\u03c3<\/em>z<\/em>,aquifer2<\/em><\/sub>)) +\u00a0(t<\/em>aquifer2<\/em><\/sub>\u00a00.5\u00a0c<\/em>b<\/em>,aquifer1<\/em><\/sub>\u00a0\u2206\u03c3<\/em>z<\/em>,aquifer2<\/em><\/sub>)<\/p>\r\n

Substituting the known subsidence (0.1 m), the decreased pressure in aquifers 1 and 2 (20 m and 15 m, and converting those pressures from height of water to kg\/m3<\/sup>, so 20,000 kg\/m3<\/sup>and 15,000 kg\/m3<\/sup>), and their thicknesses (20 m and 40 m), as well as the 15 m aquitard thickness, results in:<\/p>\r\n

[latex]\\displaystyle 0.1\\ m=(20\\ \\textup{m}\\ c_{b,aquifer1}1.96\\ \\textup{bar})[\/latex] [latex]\\displaystyle +\\left ( 15\\ \\textup{m}\\ 10\\ c_{b,aquifer1}\\ 0.5\\left ( 20,000\\frac{\\textup{kg}}{\\textup{m}^{2}}+15,000\\frac{\\textup{kg}}{\\textup{m}^{2}} \\right ) \\right )[\/latex] [latex]\\displaystyle +\\left ( 40\\ \\textup{m}\\ 0.5\\ c_{b,aquifer1}1.47\\ \\textup{bar} \\right )[\/latex]<\/p>\r\n

There are 9.806710\u2011<\/sup>5<\/sup> bars per 1 kg\/m2<\/sup> , so:<\/p>\r\n

0.1 m = (20\u00a0m\u00a0c<\/em>b<\/em>,<\/sub>aquifer1<\/sub><\/em> 1.96\u00a0bar) +\u00a0(15\u00a0m\u00a010\u00a0c<\/em>b<\/em>,aquifer1<\/em><\/sub>\u00a00.5\u00a0(1.96bar\u00a0+\u00a01.47\u00a0bar)) +\u00a0(40\u00a0m\u00a00.5\u00a0cb<\/em>,aquifer1<\/em><\/sub>\u00a01.47bar)<\/p>\r\n

carrying out the multiplications produces:<\/p>\r\n

0.1 m = (39.2\u00a0m\u00a0bar\u00a0c<\/em>b<\/em>,aquifer1<\/em><\/sub>) +\u00a0(257.4\u00a0m\u00a0bar\u00a0c<\/em>b<\/em>,aquifer1<\/em><\/sub>) +\u00a0(29.4\u00a0m\u00a0bar\u00a0c<\/em>b<\/em>,aquifer1<\/em><\/sub>)<\/p>\r\n

0.1 m = 326.1 m bar c<\/em>b<\/em>,aquifer1<\/em><\/sub><\/p>\r\n

[latex]\\displaystyle c_{b,aquifer1}=\\frac{0.1\\ \\textup{m}}{326.1\\ \\textup{m}\\ \\textup{bar}}=3.1\\times 10^{-4}\\textup{bar}^{-1}[\/latex]<\/p>\r\n

and,<\/p>\r\n

c<\/em>b<\/em>,aquifer2<\/em><\/sub> = 0.5 c<\/em>b<\/em>,aquifer1<\/em><\/sub> = 1.5 \u00d7 10-4<\/sup> bar-1<\/sup><\/p>\r\n

and<\/p>\r\n

c<\/em>b<\/em>,aquitard<\/em><\/sub> = 10 c<\/em>b<\/em>,aquifer1<\/em><\/sub> = 3.1 \u00d7 10-3<\/sup> bar-1<\/sup>.<\/p>\r\n

Return to Exercise 3<\/a><\/p>\r\n\r\n<\/div>","rendered":"

\n

The cumulative land subsidence is due to the compaction of the three layers (Equation 13<\/a>):<\/p>\n

\u03b7<\/em>tot<\/em><\/sub> = \u03b7<\/em>aquifer1<\/em><\/sub> + \u03b7<\/em>aquitard<\/em><\/sub> + \u03b7<\/em>aquifer2<\/em><\/sub><\/p>\n

The compaction of each aquifer can be obtained by means of Equation 12<\/a>:<\/p>\n

\u03b7<\/em>aquifer1<\/em><\/sub> = t<\/em>aquifer1<\/em><\/sub> c<\/em>b<\/em>,aquifer<\/em>1<\/sub> \u2206\u03c3<\/em>z<\/em>,aquifer1<\/em><\/sub><\/p>\n

\u03b7<\/em>aquifer2<\/em><\/sub> = t<\/em>aquifer2<\/em><\/sub> c<\/em>b<\/em>,aquifer<\/em>2<\/sub> \u2206\u03c3<\/em>z<\/em>,aquifer2<\/em><\/sub><\/p>\n

The ultimate aquitard compaction amounts to (Equation 28<\/a>):<\/p>\n

\u03b7<\/em>aquitard<\/em><\/sub> = t<\/em>aquitard<\/em><\/sub> c<\/em>b<\/em>,aquitard<\/em><\/sub> 0.5 (\u2206\u03c3<\/em>z<\/em>,aquifer1<\/em><\/sub> + \u2206\u03c3<\/em>z<\/em>,aquifer2<\/em><\/sub>)<\/p>\n

The sum becomes:<\/p>\n

\u03b7<\/em>tot<\/em><\/sub> = (t<\/em>aquifer1<\/em><\/sub> c<\/em>b<\/em>,aquifer1<\/em><\/sub> \u2206\u03c3<\/em>z<\/em>,aquifer1<\/em><\/sub>) +\u00a0(t<\/em>aquitard<\/em><\/sub>\u00a0c<\/em>b<\/em>,aquitard<\/em><\/sub>\u00a00.5\u00a0(\u2206\u03c3<\/em>z<\/em>,aquifer1<\/em><\/sub>\u00a0+\u00a0\u2206\u03c3<\/em>z<\/em>,aquifer2<\/em><\/sub>)) +\u00a0(t<\/em>aquifer2<\/em><\/sub>\u00a0c<\/em>b<\/em>,aquifer2<\/em><\/sub>\u00a0\u2206\u03c3<\/em>z<\/em>,aquifer2<\/em><\/sub>)<\/p>\n

Taking into account the relationships between the compressibility of the three layers, c<\/em>b<\/em><\/sub>,aquifer<\/em><\/sub>1<\/em><\/sub> = 2 c<\/em>b<\/em><\/sub>,aquifer2<\/em><\/sub> and c<\/em>b<\/em><\/sub>,aquifer1<\/em><\/sub> = 0.1 c<\/em>b<\/em><\/sub>, aquitard<\/em><\/sub> we can write the expression in terms of the compressibility of aquifer 1:<\/p>\n

\u03b7<\/em>tot<\/em><\/sub> = (t<\/em>aquifer1<\/em><\/sub>\u00a0c<\/em>b<\/em>,aquifer1<\/em><\/sub>\u00a0\u0394\u03c3<\/em>z<\/em>,aquifer1<\/em><\/sub>) +\u00a0(t<\/em>aquitard<\/em><\/sub>\u00a010\u00a0c<\/em>b<\/em>,aquifer1<\/em><\/sub>\u00a00.5(\u2206\u03c3<\/em>z<\/em>,aquifer1<\/em><\/sub>\u00a0+\u00a0\u2206\u03c3<\/em>z<\/em>,aquifer2<\/em><\/sub>)) +\u00a0(t<\/em>aquifer2<\/em><\/sub>\u00a00.5\u00a0c<\/em>b<\/em>,aquifer1<\/em><\/sub>\u00a0\u2206\u03c3<\/em>z<\/em>,aquifer2<\/em><\/sub>)<\/p>\n

Substituting the known subsidence (0.1 m), the decreased pressure in aquifers 1 and 2 (20 m and 15 m, and converting those pressures from height of water to kg\/m3<\/sup>, so 20,000 kg\/m3<\/sup>and 15,000 kg\/m3<\/sup>), and their thicknesses (20 m and 40 m), as well as the 15 m aquitard thickness, results in:<\/p>\n

\"\displaystyle 0.1\ m=(20\ \textup{m}\ c_{b,aquifer1}1.96\ \textup{bar})\" \"\displaystyle +\left ( 15\ \textup{m}\ 10\ c_{b,aquifer1}\ 0.5\left ( 20,000\frac{\textup{kg}}{\textup{m}^{2}}+15,000\frac{\textup{kg}}{\textup{m}^{2}} \right ) \right )\" \"\displaystyle +\left ( 40\ \textup{m}\ 0.5\ c_{b,aquifer1}1.47\ \textup{bar} \right )\"<\/p>\n

There are 9.806710\u2011<\/sup>5<\/sup> bars per 1 kg\/m2<\/sup> , so:<\/p>\n

0.1 m = (20\u00a0m\u00a0c<\/em>b<\/em>,<\/sub>aquifer1<\/sub><\/em> 1.96\u00a0bar) +\u00a0(15\u00a0m\u00a010\u00a0c<\/em>b<\/em>,aquifer1<\/em><\/sub>\u00a00.5\u00a0(1.96bar\u00a0+\u00a01.47\u00a0bar)) +\u00a0(40\u00a0m\u00a00.5\u00a0cb<\/em>,aquifer1<\/em><\/sub>\u00a01.47bar)<\/p>\n

carrying out the multiplications produces:<\/p>\n

0.1 m = (39.2\u00a0m\u00a0bar\u00a0c<\/em>b<\/em>,aquifer1<\/em><\/sub>) +\u00a0(257.4\u00a0m\u00a0bar\u00a0c<\/em>b<\/em>,aquifer1<\/em><\/sub>) +\u00a0(29.4\u00a0m\u00a0bar\u00a0c<\/em>b<\/em>,aquifer1<\/em><\/sub>)<\/p>\n

0.1 m = 326.1 m bar c<\/em>b<\/em>,aquifer1<\/em><\/sub><\/p>\n

\"\displaystyle c_{b,aquifer1}=\frac{0.1\ \textup{m}}{326.1\ \textup{m}\ \textup{bar}}=3.1\times 10^{-4}\textup{bar}^{-1}\"<\/p>\n

and,<\/p>\n

c<\/em>b<\/em>,aquifer2<\/em><\/sub> = 0.5 c<\/em>b<\/em>,aquifer1<\/em><\/sub> = 1.5 \u00d7 10-4<\/sup> bar-1<\/sup><\/p>\n

and<\/p>\n

c<\/em>b<\/em>,aquitard<\/em><\/sub> = 10 c<\/em>b<\/em>,aquifer1<\/em><\/sub> = 3.1 \u00d7 10-3<\/sup> bar-1<\/sup>.<\/p>\n

Return to Exercise 3<\/a><\/p>\n<\/div>\n","protected":false},"author":1,"menu_order":34,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-279","chapter","type-chapter","status-publish","hentry"],"part":185,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/pressbooks\/v2\/chapters\/279","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":9,"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/pressbooks\/v2\/chapters\/279\/revisions"}],"predecessor-version":[{"id":482,"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/pressbooks\/v2\/chapters\/279\/revisions\/482"}],"part":[{"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/pressbooks\/v2\/parts\/185"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/pressbooks\/v2\/chapters\/279\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/wp\/v2\/media?parent=279"}],"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=279"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/wp\/v2\/contributor?post=279"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/wp\/v2\/license?post=279"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}