{"id":1615,"date":"2023-12-07T17:30:46","date_gmt":"2023-12-07T17:30:46","guid":{"rendered":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/?post_type=chapter&p=1615"},"modified":"2023-12-11T15:21:30","modified_gmt":"2023-12-11T15:21:30","slug":"exercise-3","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/chapter\/exercise-3\/","title":{"raw":"Exercise 3","rendered":"Exercise 3"},"content":{"raw":"Sandstone and limestone samples subjected to laboratory compression triaxial tests (Tests 1 through 4 in the tables below) undergo shear fracturing. The fractures are generated under the following critical stress states:\r\n

Sandstone<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\r\n

Test #<\/strong><\/p>\r\n<\/td>\r\n

\r\n

Principal stresses (MPa)<\/strong><\/p>\r\n<\/td>\r\n

\r\n

\u03c3<\/em>n<\/span><\/sub> (MPa)<\/span><\/strong><\/p>\r\n<\/td>\r\n

\r\n

\u03c4<\/em> (MPa)<\/span><\/strong><\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

1<\/p>\r\n<\/td>\r\n

\r\n

\u03c3<\/strong><\/em>3<\/span><\/sub> = 27.5, <\/span>\u03c3<\/strong><\/em>1<\/span><\/sub> = 116.25<\/span><\/p>\r\n<\/td>\r\n

\r\n

<\/p>\r\n<\/td>\r\n

\r\n

<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

2<\/p>\r\n<\/td>\r\n

\r\n

\u03c3<\/strong><\/em>3<\/span><\/sub> = 68.75, <\/span>\u03c3<\/strong><\/em>1<\/span><\/sub> = 217.5<\/span><\/p>\r\n<\/td>\r\n

\r\n

<\/p>\r\n<\/td>\r\n

\r\n

<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Limestone<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\r\n

Test #<\/strong><\/p>\r\n<\/td>\r\n

\r\n

Principal stresses (MPa)<\/strong><\/p>\r\n<\/td>\r\n

\r\n

\u03c3<\/em>n<\/span><\/sub> (MPa)<\/span><\/strong><\/p>\r\n<\/td>\r\n

\r\n

\u03c4<\/em> (MPa)<\/span><\/strong><\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

3<\/p>\r\n<\/td>\r\n

\r\n

\u03c3<\/strong><\/em>3<\/span><\/sub> = 7.5,\u00a0<\/span>\u03c3<\/strong><\/em>1<\/span><\/sub> = 192.5<\/span><\/p>\r\n<\/td>\r\n

\r\n

<\/p>\r\n<\/td>\r\n

\r\n

<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

4<\/p>\r\n<\/td>\r\n

\r\n

\u03c3<\/strong><\/em>3<\/span><\/sub> = 41.25,\u00a0<\/span>\u03c3<\/strong><\/em>1<\/span><\/sub> = 308.75<\/span><\/p>\r\n<\/td>\r\n

\r\n

<\/p>\r\n<\/td>\r\n

\r\n

<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

a)\u00a0 Construct the Mohr diagram for the sandstone and plot the principal stresses for tests 1 and 2; draw the Mohr circles and determine the failure envelope equation of the sandstone. In a separate diagram, follow the same procedure for the limestone, plotting the principal stresses of tests 3 and 4.<\/p>\r\n

b)\u00a0 What is the value of the slope (\u03c6<\/span><\/em>, <\/span>friction angle) of the sandstone failure envelope? What is the value of \u03c6 <\/span><\/em>for the limestone failure envelope?<\/p>\r\n

c)\u00a0 For each rock, what is the angle between the shear conjugate fractures (2\u03b8<\/span><\/em>)? <\/span>What is the angle between \u03c3<\/span>1<\/span> and the shear fractures (\u03b8<\/span><\/em>)?<\/p>\r\n

d)\u00a0 Determine the critical normal stress (\u03c3<\/span>n<\/span><\/sub>) <\/span>and shear stress (\u03c4<\/span><\/em>) at which the shear fractures were generated in each test and enter their values in columns \u03c3<\/span>n<\/span> and \u03c4 <\/span>in the two tables.<\/p>\r\n

e)\u00a0 Compare the critical Mohr circles and the failure envelopes of the two rocks. Which rock has the greater cohesion (C<\/i>)? Which rock would undergo brittle deformation under lower stresses?<\/p>\r\n

f)\u00a0 In the case where both rocks are present in the same geological context and are subjected to the same stress fields (same tectonic events), should we expect one of them to have a denser fracture network? Explain your answer. What should you expect regarding the connected fracture network in each rock?<\/p>\r\n

Click to return to where text linked to Exercise 3<\/strong><\/a><\/p>\r\n

Click for solution to Exercise 3<\/strong><\/a><\/p>\r\n ","rendered":"

Sandstone and limestone samples subjected to laboratory compression triaxial tests (Tests 1 through 4 in the tables below) undergo shear fracturing. The fractures are generated under the following critical stress states:<\/p>\n

Sandstone<\/p>\n\n\n\n\n\n
\n

Test #<\/strong><\/p>\n<\/td>\n

\n

Principal stresses (MPa)<\/strong><\/p>\n<\/td>\n

\n

\u03c3<\/em>n<\/span><\/sub> (MPa)<\/span><\/strong><\/p>\n<\/td>\n

\n

\u03c4<\/em> (MPa)<\/span><\/strong><\/p>\n<\/td>\n<\/tr>\n

\n

1<\/p>\n<\/td>\n

\n

\u03c3<\/strong><\/em>3<\/span><\/sub> = 27.5, <\/span>\u03c3<\/strong><\/em>1<\/span><\/sub> = 116.25<\/span><\/p>\n<\/td>\n

\n

\n<\/td>\n

\n

\n<\/td>\n<\/tr>\n

\n

2<\/p>\n<\/td>\n

\n

\u03c3<\/strong><\/em>3<\/span><\/sub> = 68.75, <\/span>\u03c3<\/strong><\/em>1<\/span><\/sub> = 217.5<\/span><\/p>\n<\/td>\n

\n

\n<\/td>\n

\n

\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Limestone<\/p>\n\n\n\n\n\n
\n

Test #<\/strong><\/p>\n<\/td>\n

\n

Principal stresses (MPa)<\/strong><\/p>\n<\/td>\n

\n

\u03c3<\/em>n<\/span><\/sub> (MPa)<\/span><\/strong><\/p>\n<\/td>\n

\n

\u03c4<\/em> (MPa)<\/span><\/strong><\/p>\n<\/td>\n<\/tr>\n

\n

3<\/p>\n<\/td>\n

\n

\u03c3<\/strong><\/em>3<\/span><\/sub> = 7.5,\u00a0<\/span>\u03c3<\/strong><\/em>1<\/span><\/sub> = 192.5<\/span><\/p>\n<\/td>\n

\n

\n<\/td>\n

\n

\n<\/td>\n<\/tr>\n

\n

4<\/p>\n<\/td>\n

\n

\u03c3<\/strong><\/em>3<\/span><\/sub> = 41.25,\u00a0<\/span>\u03c3<\/strong><\/em>1<\/span><\/sub> = 308.75<\/span><\/p>\n<\/td>\n

\n

\n<\/td>\n

\n

\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

a)\u00a0 Construct the Mohr diagram for the sandstone and plot the principal stresses for tests 1 and 2; draw the Mohr circles and determine the failure envelope equation of the sandstone. In a separate diagram, follow the same procedure for the limestone, plotting the principal stresses of tests 3 and 4.<\/p>\n

b)\u00a0 What is the value of the slope (\u03c6<\/span><\/em>, <\/span>friction angle) of the sandstone failure envelope? What is the value of \u03c6 <\/span><\/em>for the limestone failure envelope?<\/p>\n

c)\u00a0 For each rock, what is the angle between the shear conjugate fractures (2\u03b8<\/span><\/em>)? <\/span>What is the angle between \u03c3<\/span>1<\/span> and the shear fractures (\u03b8<\/span><\/em>)?<\/p>\n

d)\u00a0 Determine the critical normal stress (\u03c3<\/span>n<\/span><\/sub>) <\/span>and shear stress (\u03c4<\/span><\/em>) at which the shear fractures were generated in each test and enter their values in columns \u03c3<\/span>n<\/span> and \u03c4 <\/span>in the two tables.<\/p>\n

e)\u00a0 Compare the critical Mohr circles and the failure envelopes of the two rocks. Which rock has the greater cohesion (C<\/i>)? Which rock would undergo brittle deformation under lower stresses?<\/p>\n

f)\u00a0 In the case where both rocks are present in the same geological context and are subjected to the same stress fields (same tectonic events), should we expect one of them to have a denser fracture network? Explain your answer. What should you expect regarding the connected fracture network in each rock?<\/p>\n

Click to return to where text linked to Exercise 3<\/strong><\/a><\/p>\n

Click for solution to Exercise 3<\/strong><\/a><\/p>\n

 <\/p>\n","protected":false},"author":6,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1615","chapter","type-chapter","status-publish","hentry"],"part":1403,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/wp-json\/pressbooks\/v2\/chapters\/1615","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":18,"href":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/wp-json\/pressbooks\/v2\/chapters\/1615\/revisions"}],"predecessor-version":[{"id":2106,"href":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/wp-json\/pressbooks\/v2\/chapters\/1615\/revisions\/2106"}],"part":[{"href":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/wp-json\/pressbooks\/v2\/parts\/1403"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/wp-json\/pressbooks\/v2\/chapters\/1615\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/wp-json\/wp\/v2\/media?parent=1615"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/wp-json\/pressbooks\/v2\/chapter-type?post=1615"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/wp-json\/wp\/v2\/contributor?post=1615"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/wp-json\/wp\/v2\/license?post=1615"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}