{"id":278,"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-2-solution\/"},"modified":"2022-01-18T02:14:49","modified_gmt":"2022-01-18T02:14:49","slug":"exercise-2-solution","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/chapter\/exercise-2-solution\/","title":{"raw":"Exercise 2 Solution","rendered":"Exercise 2 Solution"},"content":{"raw":"
\r\n

Question 1: land subsidence after 1 month<\/p>\r\n

Because of the small hydraulic conductivity, it can reasonably be assumed that pressure depletion has not yet significantly propagated into the aquitard. Therefore, similarly to Equation 13<\/a>, the total subsidence \u03b7<\/em>tot<\/em><\/sub> is the sum of compaction of the phreatic aquifer and the confined aquifer only: \u03b7<\/em>tot<\/em><\/sub> = \u03b7<\/em>p<\/em><\/sub> + \u03b7<\/em>c<\/em><\/sub>.<\/p>\r\n

The compaction of the phreatic aquifer amounts to \u03b7<\/em>p<\/em><\/sub> \u2245 t<\/em>p<\/em><\/sub> c<\/em>b<\/em><\/sub>,<\/em><\/sub>p<\/em><\/sub> \u0394\u03c3<\/em>z,p<\/em><\/sub> with t<\/em>p<\/em><\/sub> the time\u2011averaged aquifer thickness below the water table (Equation 10<\/a>). Denoting the porosity with \u03d5<\/em> and the water saturation with \u03b8<\/em>w<\/em><\/sub>,<\/em><\/sub> the effective stress change is given by Equation 7<\/a>.<\/p>\r\n

\u2206\u03c3<\/em>z<\/em>,p<\/em><\/sub> = \u03b3<\/em>w<\/em><\/sub> \u2206z<\/em>p<\/em><\/sub> (1 \u2212 \u03d5<\/em> + \u03b8<\/em>w<\/em><\/sub>) = 1000 kg\/m3<\/sup> 5 m (1 \u2212 0.35 + 0.10)<\/p>\r\n

= 1000 kg\/m3<\/sup> 3.75 m = [latex]3570\\ \\textup{kg}\/\\textup{m}^{2}\\ \\frac{9.8067\\times 10^{-5}\\textup{bar}}{\\textup{kg}\/\\textup{m}^{2}}[\/latex] = 0.37 bar<\/p>\r\n

Therefore:<\/p>\r\n

\u03b7<\/em>p<\/em><\/sub> = t<\/em>p<\/em><\/sub> c<\/em>b<\/em>,p<\/em><\/sub> \u2206\u03c3<\/em>z<\/em>,p<\/em><\/sub> = [latex]\\frac{1}{2}(25+20)\\textup{m}\\frac{1\\times 10^{-4}}{\\textup{bar}}0.37\\textup{bar}[\/latex] = 0.00083 m = 0.83 mm.<\/p>\r\n

The compaction of the confined aquifer amounts to:<\/p>\r\n\r\n<\/div>\r\n

[latex]\\\\eta _{c}=b_{c}\\ c_{b,c}\\ \\gamma _{w}\\ \\left ( \\Delta z_{c}-\\Delta z_{p}(\\phi -\\theta _{w}) \\right )[\/latex]<\/p>\r\n

[latex]\\displaystyle \\eta _{c}=50\\ \\textup{m}\\frac{2\\times 10^{-5}}{\\textup{bar}} \\left ( \\left ( 1000\\frac{\\textup{kg}}{\\textup{m}^{3}}\\left ( 25\\ \\textup{m}-5\\ \\textup{m}(0.35-0.1) \\right ) \\right )\\frac{9.8067\\times 10^{-5}\\textup{bar}}{\\frac{\\textup{kg}}{\\textup{m}^{2}}} \\right )[\/latex]<\/p>\r\n

[latex]\\displaystyle \\eta _{c}=50\\ \\textup{m}\\frac{2\\times 10^{-5}}{\\textup{bar}} \\left ( 1000\\frac{\\textup{kg}}{\\textup{m}^{3}}\\left ( 23.75\\ \\textup{m} \\right ) \\frac{9.8067\\times 10^{-5}\\textup{bar}}{\\frac{\\textup{kg}}{\\textup{m}^{2}}} \\right )[\/latex]<\/p>\r\n

[latex]\\displaystyle \\eta _{c}=50\\ \\textup{m}\\frac{2\\times 10^{-5}}{\\textup{bar}}(2.33\\ \\textup{bar})[\/latex]<\/p>\r\n\r\n

\r\n

\u03b7<\/em>c<\/em><\/sub> = 0.00233 m = 2.33 mm<\/p>\r\n

Hence:\u00a0 \u03b7<\/em>tot<\/em><\/sub> = 0.83 + 2.33 = 3.16 mm<\/p>\r\n

Question 2: land subsidence after 10 years<\/p>\r\n

In this case, aquitard compaction contributes to land subsidence \u03b7<\/em>tot<\/em><\/sub> = \u03b7<\/em>p<\/em><\/sub> + \u03b7<\/em>aqt<\/em><\/sub> + \u03b7<\/em>c<\/em><\/sub>. After a period of 10 years, it can be assumed that the pressure within the aquitard has reached an equilibrated distribution, with the ultimate aquitard compaction equal to (Equation 28<\/a>):<\/p>\r\n

[latex]\\eta _{aqt}=\\frac{1}{2}(\\Delta \\sigma _{z,p}+\\Delta \\sigma _{z,c})t_{aqt}c_{b,aqt}[\/latex] [latex]= 0.5(0.37\\ \\textup{bar+2.33\\ \\textup{bar}})\\ 20\\ \\textup{m}\\frac{1\\times 10^{-3}}{\\textup{bar}}[\/latex] = 0.027 m = 27 mm<\/p>\r\n

The cumulative land subsidence is: \u03b7<\/em>tot<\/em><\/sub> = 0.83 + 27.0 + 2.33 = 30.16 mm.<\/p>\r\n

Return to Exercise 2<\/a><\/p>\r\n\r\n<\/div>","rendered":"

\n

Question 1: land subsidence after 1 month<\/p>\n

Because of the small hydraulic conductivity, it can reasonably be assumed that pressure depletion has not yet significantly propagated into the aquitard. Therefore, similarly to Equation 13<\/a>, the total subsidence \u03b7<\/em>tot<\/em><\/sub> is the sum of compaction of the phreatic aquifer and the confined aquifer only: \u03b7<\/em>tot<\/em><\/sub> = \u03b7<\/em>p<\/em><\/sub> + \u03b7<\/em>c<\/em><\/sub>.<\/p>\n

The compaction of the phreatic aquifer amounts to \u03b7<\/em>p<\/em><\/sub> \u2245 t<\/em>p<\/em><\/sub> c<\/em>b<\/em><\/sub>,<\/em><\/sub>p<\/em><\/sub> \u0394\u03c3<\/em>z,p<\/em><\/sub> with t<\/em>p<\/em><\/sub> the time\u2011averaged aquifer thickness below the water table (Equation 10<\/a>). Denoting the porosity with \u03d5<\/em> and the water saturation with \u03b8<\/em>w<\/em><\/sub>,<\/em><\/sub> the effective stress change is given by Equation 7<\/a>.<\/p>\n

\u2206\u03c3<\/em>z<\/em>,p<\/em><\/sub> = \u03b3<\/em>w<\/em><\/sub> \u2206z<\/em>p<\/em><\/sub> (1 \u2212 \u03d5<\/em> + \u03b8<\/em>w<\/em><\/sub>) = 1000 kg\/m3<\/sup> 5 m (1 \u2212 0.35 + 0.10)<\/p>\n

= 1000 kg\/m3<\/sup> 3.75 m = \"3570\ \textup{kg}/\textup{m}^{2}\ \frac{9.8067\times 10^{-5}\textup{bar}}{\textup{kg}/\textup{m}^{2}}\" = 0.37 bar<\/p>\n

Therefore:<\/p>\n

\u03b7<\/em>p<\/em><\/sub> = t<\/em>p<\/em><\/sub> c<\/em>b<\/em>,p<\/em><\/sub> \u2206\u03c3<\/em>z<\/em>,p<\/em><\/sub> = \"\frac{1}{2}(25+20)\textup{m}\frac{1\times 10^{-4}}{\textup{bar}}0.37\textup{bar}\" = 0.00083 m = 0.83 mm.<\/p>\n

The compaction of the confined aquifer amounts to:<\/p>\n<\/div>\n

\"\\eta _{c}=b_{c}\ c_{b,c}\ \gamma _{w}\ \left ( \Delta z_{c}-\Delta z_{p}(\phi -\theta _{w}) \right )\"<\/p>\n

\"\displaystyle \eta _{c}=50\ \textup{m}\frac{2\times 10^{-5}}{\textup{bar}} \left ( \left ( 1000\frac{\textup{kg}}{\textup{m}^{3}}\left ( 25\ \textup{m}-5\ \textup{m}(0.35-0.1) \right ) \right )\frac{9.8067\times 10^{-5}\textup{bar}}{\frac{\textup{kg}}{\textup{m}^{2}}} \right )\"<\/p>\n

\"\displaystyle \eta _{c}=50\ \textup{m}\frac{2\times 10^{-5}}{\textup{bar}} \left ( 1000\frac{\textup{kg}}{\textup{m}^{3}}\left ( 23.75\ \textup{m} \right ) \frac{9.8067\times 10^{-5}\textup{bar}}{\frac{\textup{kg}}{\textup{m}^{2}}} \right )\"<\/p>\n

\"\displaystyle \eta _{c}=50\ \textup{m}\frac{2\times 10^{-5}}{\textup{bar}}(2.33\ \textup{bar})\"<\/p>\n

\n

\u03b7<\/em>c<\/em><\/sub> = 0.00233 m = 2.33 mm<\/p>\n

Hence:\u00a0 \u03b7<\/em>tot<\/em><\/sub> = 0.83 + 2.33 = 3.16 mm<\/p>\n

Question 2: land subsidence after 10 years<\/p>\n

In this case, aquitard compaction contributes to land subsidence \u03b7<\/em>tot<\/em><\/sub> = \u03b7<\/em>p<\/em><\/sub> + \u03b7<\/em>aqt<\/em><\/sub> + \u03b7<\/em>c<\/em><\/sub>. After a period of 10 years, it can be assumed that the pressure within the aquitard has reached an equilibrated distribution, with the ultimate aquitard compaction equal to (Equation 28<\/a>):<\/p>\n

\"\eta _{aqt}=\frac{1}{2}(\Delta \sigma _{z,p}+\Delta \sigma _{z,c})t_{aqt}c_{b,aqt}\" \"= 0.5(0.37\ \textup{bar+2.33\ \textup{bar}})\ 20\ \textup{m}\frac{1\times 10^{-3}}{\textup{bar}}\" = 0.027 m = 27 mm<\/p>\n

The cumulative land subsidence is: \u03b7<\/em>tot<\/em><\/sub> = 0.83 + 27.0 + 2.33 = 30.16 mm.<\/p>\n

Return to Exercise 2<\/a><\/p>\n<\/div>\n","protected":false},"author":1,"menu_order":33,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-278","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\/278","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":6,"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/pressbooks\/v2\/chapters\/278\/revisions"}],"predecessor-version":[{"id":475,"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/pressbooks\/v2\/chapters\/278\/revisions\/475"}],"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\/278\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/wp\/v2\/media?parent=278"}],"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=278"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/wp\/v2\/contributor?post=278"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/land-subsidence-and-its-mitigation\/wp-json\/wp\/v2\/license?post=278"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}