{"id":1660,"date":"2023-12-07T21:01:16","date_gmt":"2023-12-07T21:01:16","guid":{"rendered":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/?post_type=chapter&p=1660"},"modified":"2023-12-11T21:25:46","modified_gmt":"2023-12-11T21:25:46","slug":"solution-exercise-3","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/chapter\/solution-exercise-3\/","title":{"raw":"Solution Exercise 3\u200c","rendered":"Solution Exercise 3\u200c"},"content":{"raw":"

a) The first Mohr diagram below shows the Mohr circles of tests 1 and 2 (sandstone tests) and a straight line drawn to intercept each circle at a single point. The same procedure is repeated on the second Mohr diagram for tests 3 and 4 (limestone tests). These lines are the failure envelopes for the sandstone and limestone, and the points represent the critical stresses at which shear fractures were formed.<\/p>\r\n

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

\"\"<\/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

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

\r\n

40.5<\/span><\/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

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

\r\n

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

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

\"\"<\/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

\u03c3n<\/span><\/sub> (MPa)<\/span><\/strong><\/p>\r\n<\/td>\r\n

\r\n

\u03c4 (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, <\/span>\u03c3<\/strong><\/em>1<\/span><\/sub> = 192.5<\/span><\/p>\r\n<\/td>\r\n

\r\n

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

\r\n

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

\r\n

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

\r\n

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

b) The slopes, \u03c6 <\/span><\/em>friction angle, of the sandstone and limestone failure envelopes are manually measured as 25\u00b0 <\/span>and 32\u00b0<\/span>, respectively.<\/p>\r\n

c) The measured 2\u03b8 <\/span>values between the shear conjugate fractures are 65\u00b0 <\/span>and 58\u00b0 <\/span>for the sandstone and limestone, respectively. The angle between the \u03c3<\/span>1<\/span><\/sub> and the shear fractures is equal to \u03b8<\/span><\/em>, so these are 32.5\u00b0 <\/span>and 29\u00b0 <\/span>for the sandstone and limestone, respectively.<\/p>\r\n

d) The critical normal stress (\u03c3<\/span>n<\/span><\/sub>) and shear stress (\u03c4<\/span><\/em>) that generated the shear fractures in<\/p>\r\n

e) each of the four tests are indicated along the shear and normal stress axes in the Mohr diagrams; the values are also shown in the corresponding tables.<\/p>\r\n

f) The cohesion of each rock corresponds to the value at which each failure envelope intercepts the shear stress axis. The cohesion of the limestone is 47 MPa and is greater than that of the sandstone which is 15 MPa. Consequently, the sandstone can be fractured under lower stress.<\/p>\r\n

g) In the case that both rocks occur in the same geological context and are subjected to the same stress fields, one should expect that the sandstone would have a denser fracture network because milder stress conditions would fracture the sandstone but not the limestone. One possible consequence is that a more densely connected fracture network is expected for the sandstone.<\/p>\r\n

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

Return to Exercise 3<\/strong><\/span><\/a><\/p>","rendered":"

a) The first Mohr diagram below shows the Mohr circles of tests 1 and 2 (sandstone tests) and a straight line drawn to intercept each circle at a single point. The same procedure is repeated on the second Mohr diagram for tests 3 and 4 (limestone tests). These lines are the failure envelopes for the sandstone and limestone, and the points represent the critical stresses at which shear fractures were formed.<\/p>\n

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

\"\"<\/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

52.5<\/span><\/p>\n<\/td>\n

\n

40.5<\/span><\/p>\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

111.25<\/span><\/p>\n<\/td>\n

\n

70<\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

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

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

\n

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

\n

\u03c3n<\/span><\/sub> (MPa)<\/span><\/strong><\/p>\n<\/td>\n

\n

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

\n

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

\n

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

\n

51.25<\/span><\/p>\n<\/td>\n

\n

79.25<\/span><\/p>\n<\/td>\n<\/tr>\n

\n

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

\n

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

\n

106.25<\/span><\/p>\n<\/td>\n

\n

113<\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

b) The slopes, \u03c6 <\/span><\/em>friction angle, of the sandstone and limestone failure envelopes are manually measured as 25\u00b0 <\/span>and 32\u00b0<\/span>, respectively.<\/p>\n

c) The measured 2\u03b8 <\/span>values between the shear conjugate fractures are 65\u00b0 <\/span>and 58\u00b0 <\/span>for the sandstone and limestone, respectively. The angle between the \u03c3<\/span>1<\/span><\/sub> and the shear fractures is equal to \u03b8<\/span><\/em>, so these are 32.5\u00b0 <\/span>and 29\u00b0 <\/span>for the sandstone and limestone, respectively.<\/p>\n

d) The critical normal stress (\u03c3<\/span>n<\/span><\/sub>) and shear stress (\u03c4<\/span><\/em>) that generated the shear fractures in<\/p>\n

e) each of the four tests are indicated along the shear and normal stress axes in the Mohr diagrams; the values are also shown in the corresponding tables.<\/p>\n

f) The cohesion of each rock corresponds to the value at which each failure envelope intercepts the shear stress axis. The cohesion of the limestone is 47 MPa and is greater than that of the sandstone which is 15 MPa. Consequently, the sandstone can be fractured under lower stress.<\/p>\n

g) In the case that both rocks occur in the same geological context and are subjected to the same stress fields, one should expect that the sandstone would have a denser fracture network because milder stress conditions would fracture the sandstone but not the limestone. One possible consequence is that a more densely connected fracture network is expected for the sandstone.<\/p>\n

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

Return to Exercise 3<\/strong><\/span><\/a><\/p>\n","protected":false},"author":6,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1660","chapter","type-chapter","status-publish","hentry"],"part":1650,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/wp-json\/pressbooks\/v2\/chapters\/1660","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":10,"href":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/wp-json\/pressbooks\/v2\/chapters\/1660\/revisions"}],"predecessor-version":[{"id":2179,"href":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/wp-json\/pressbooks\/v2\/chapters\/1660\/revisions\/2179"}],"part":[{"href":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/wp-json\/pressbooks\/v2\/parts\/1650"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/wp-json\/pressbooks\/v2\/chapters\/1660\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/wp-json\/wp\/v2\/media?parent=1660"}],"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=1660"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/wp-json\/wp\/v2\/contributor?post=1660"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/structural-geology-applied-to-fractured-aquifer-characterization\/wp-json\/wp\/v2\/license?post=1660"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}