{"id":892,"date":"2020-12-09T05:01:55","date_gmt":"2020-12-09T05:01:55","guid":{"rendered":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/?post_type=chapter&p=892"},"modified":"2020-12-30T01:46:02","modified_gmt":"2020-12-30T01:46:02","slug":"solution-to-exercise-5","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/chapter\/solution-to-exercise-5\/","title":{"raw":"Solution to Exercise 5","rendered":"Solution to Exercise 5"},"content":{"raw":"5) A factory is disposing of hot waste water by injecting it into a 1000 m deep well that penetrates a confined aquifer. The sandstone aquifer contains water at 35 \u00b0C, a similar temperature as the waste water. A regulator wanted the company to model how far the contaminated water would travel in the aquifer over a 10-year period. Hydraulic conductivity values were sparse. They required the company to core part of the 50 m thick confined aquifer and determine a representative value of hydraulic conductivity. A portion of the core was placed in a constant head permeameter as illustrated in the accompanying figure.\r\n\r\n\"\"\r\n\r\n5a) If an average of 34.0 ml\/min was collected at the outlet, what is K in cm\/s?<\/em>\r\n\r\nHydraulic conductivity can be determined by rearranging Equation 15 of this book\r\n\r\n\r\n\r\n
[latex]\\displaystyle Q=-K\\frac{\\Delta h}{\\Delta L}\\ A[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nwhere:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Q<\/em><\/td>\r\n=<\/td>\r\nvolumetric flow rate (L3<\/sup>\/T)<\/td>\r\n<\/tr>\r\n
K<\/em><\/td>\r\n=<\/td>\r\nhydraulic conductivity, is the proportionality constant reflecting the ease with which water flows through a material (L\/T)<\/td>\r\n<\/tr>\r\n
\u0394h<\/em><\/td>\r\n=<\/td>\r\ndifference in hydraulic head between two measuring points as defined for Equation 14 (L)<\/td>\r\n<\/tr>\r\n
\u0394L<\/em><\/td>\r\n=<\/td>\r\nlength along the flow path between locations where hydraulic heads are measured (L)<\/td>\r\n<\/tr>\r\n
[latex]\\displaystyle \\frac{\\Delta h}{\\Delta L}[\/latex]<\/td>\r\n=<\/td>\r\ngradient of hydraulic head (dimensionless)<\/td>\r\n<\/tr>\r\n
A<\/em><\/td>\r\n=<\/td>\r\ncross-sectional area of flow perpendicular to the direction of flow (L2<\/sup>)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nRearranging:\r\n\r\n\r\n\r\n\r\n
[latex]\\displaystyle K=-\\frac{Q\\Delta L}{A\\Delta h}[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]\\displaystyle K=-\\frac{\\frac{34\\ {\\textup{cm}}^3}{\\textup{min}}\\ 30\\ \\textup{cm}\\ \\frac{1\\ \\textup{min}}{60\\ \\textup{s}}}{3.14\\ \\left(3\\ \\textup{cm}\\right)^2(48.5\\ \\textup{cm}-60\\ \\textup{cm})}=\\frac{\\frac{17\\ {\\textup{cm}}^4}{\\textup{s}}}{325\\ {\\textup{cm}}^{3\\ }}=0.052\\ \\frac{\\textup{cm}}{\\textup{s}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n5b) If the laboratory experiment was completed using 15 \u00b0C water, what is the intrinsic permeability of the sandstone in cm2<\/sup>?<\/em>\r\n\r\nIf the test was conducted at 15 \u00b0C, the intrinsic permeability, k<\/em>, of the sandstone in cm2<\/sup> can be calculated using Equation 31 of this book, using the relationships for the physical properties of water and temperature shown as Figure 28<\/a> of this book.\r\n\r\n\r\n\r\n
[latex]\\displaystyle K=\\frac{Cd^{2}g\\rho }{\\mu }=\\frac{kg\\rho }{\\mu }[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nwhere\r\n\r\n\r\n\r\n\r\n\r\n\r\n
k<\/em><\/td>\r\n=<\/td>\r\nintrinsic permeability (L2<\/sup>)<\/td>\r\n<\/tr>\r\n
\u03c1<\/em><\/td>\r\n=<\/td>\r\nfluid density (M\/L3<\/sup>)<\/td>\r\n<\/tr>\r\n
g<\/em><\/td>\r\n=<\/td>\r\ngravitational constant (acceleration of gravity) (L\/T2<\/sup>)<\/td>\r\n<\/tr>\r\n
\u03bc<\/em><\/td>\r\n=<\/td>\r\ndynamic viscosity (M\/LT)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nrearranging:\r\n\r\n\r\n\r\n
[latex]\\displaystyle k=\\frac{K\\mu }{g\\rho }=\\frac{0.052\\ \\frac{\\textup{cm}}{\\textup{s}}\\ 0.011\\ \\frac{\\textup{gm}\\cdot \\textup{cm}}{\\textup{s}}}{980\\ \\frac{\\textup{cm}}{\\textup{s}^{2}}\\ 0.998\\ \\frac{\\textup{g}}{\\textup{cm}^{2}}}=5.8\\times 10^{-7}\\ \\textup{cm}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n5c) What is the hydraulic conductivity of the sandstone if the water temperature is 35 \u00b0C (in cm\/s)?<\/em>\r\n\r\nIf the water temperature in the sandstone is 35 \u00b0C, Equation 31 of this book can be used to calculate, K<\/em>, of the sandstone at that temperature in cm\/s, using the relationships for the physical properties of water and temperature shown as Figure 28<\/a> of this book.\r\n\r\n\r\n\r\n
[latex]\\displaystyle K=\\frac{5.8\\times 10^{-7}\\ {\\textup{cm}}^{2}\\ 980\\ \\frac{\\textup{cm}}{\\textup{s}^{2}}\\ 0.994\\ \\frac{\\textup{g}}{\\textup{cm}^{3}}}{0.007\\ \\frac{\\textup{g}\\cdot \\textup{cm}}{\\textup{s}}}=\\frac{0.08\\ \\textup{cm}}{\\textup{s}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Return to Exercise 5<\/a><\/p>","rendered":"

5) A factory is disposing of hot waste water by injecting it into a 1000 m deep well that penetrates a confined aquifer. The sandstone aquifer contains water at 35 \u00b0C, a similar temperature as the waste water. A regulator wanted the company to model how far the contaminated water would travel in the aquifer over a 10-year period. Hydraulic conductivity values were sparse. They required the company to core part of the 50 m thick confined aquifer and determine a representative value of hydraulic conductivity. A portion of the core was placed in a constant head permeameter as illustrated in the accompanying figure.<\/p>\n

\"\"<\/p>\n

5a) If an average of 34.0 ml\/min was collected at the outlet, what is K in cm\/s?<\/em><\/p>\n

Hydraulic conductivity can be determined by rearranging Equation 15 of this book<\/p>\n\n\n\n
\"\displaystyle Q=-K\frac{\Delta h}{\Delta L}\ A\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n\n\n\n\n\n
Q<\/em><\/td>\n=<\/td>\nvolumetric flow rate (L3<\/sup>\/T)<\/td>\n<\/tr>\n
K<\/em><\/td>\n=<\/td>\nhydraulic conductivity, is the proportionality constant reflecting the ease with which water flows through a material (L\/T)<\/td>\n<\/tr>\n
\u0394h<\/em><\/td>\n=<\/td>\ndifference in hydraulic head between two measuring points as defined for Equation 14 (L)<\/td>\n<\/tr>\n
\u0394L<\/em><\/td>\n=<\/td>\nlength along the flow path between locations where hydraulic heads are measured (L)<\/td>\n<\/tr>\n
\"\displaystyle \frac{\Delta h}{\Delta L}\"<\/td>\n=<\/td>\ngradient of hydraulic head (dimensionless)<\/td>\n<\/tr>\n
A<\/em><\/td>\n=<\/td>\ncross-sectional area of flow perpendicular to the direction of flow (L2<\/sup>)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Rearranging:<\/p>\n\n\n\n\n
\"\displaystyle K=-\frac{Q\Delta L}{A\Delta h}\"<\/td>\n<\/tr>\n
\"\displaystyle K=-\frac{\frac{34\ {\textup{cm}}^3}{\textup{min}}\ 30\ \textup{cm}\ \frac{1\ \textup{min}}{60\ \textup{s}}}{3.14\ \left(3\ \textup{cm}\right)^2(48.5\ \textup{cm}-60\ \textup{cm})}=\frac{\frac{17\ {\textup{cm}}^4}{\textup{s}}}{325\ {\textup{cm}}^{3\ }}=0.052\ \frac{\textup{cm}}{\textup{s}}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

5b) If the laboratory experiment was completed using 15 \u00b0C water, what is the intrinsic permeability of the sandstone in cm2<\/sup>?<\/em><\/p>\n

If the test was conducted at 15 \u00b0C, the intrinsic permeability, k<\/em>, of the sandstone in cm2<\/sup> can be calculated using Equation 31 of this book, using the relationships for the physical properties of water and temperature shown as Figure 28<\/a> of this book.<\/p>\n\n\n\n
\"\displaystyle K=\frac{Cd^{2}g\rho }{\mu }=\frac{kg\rho }{\mu }\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where<\/p>\n\n\n\n\n\n\n
k<\/em><\/td>\n=<\/td>\nintrinsic permeability (L2<\/sup>)<\/td>\n<\/tr>\n
\u03c1<\/em><\/td>\n=<\/td>\nfluid density (M\/L3<\/sup>)<\/td>\n<\/tr>\n
g<\/em><\/td>\n=<\/td>\ngravitational constant (acceleration of gravity) (L\/T2<\/sup>)<\/td>\n<\/tr>\n
\u03bc<\/em><\/td>\n=<\/td>\ndynamic viscosity (M\/LT)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

rearranging:<\/p>\n\n\n\n
\"\displaystyle k=\frac{K\mu }{g\rho }=\frac{0.052\ \frac{\textup{cm}}{\textup{s}}\ 0.011\ \frac{\textup{gm}\cdot \textup{cm}}{\textup{s}}}{980\ \frac{\textup{cm}}{\textup{s}^{2}}\ 0.998\ \frac{\textup{g}}{\textup{cm}^{2}}}=5.8\times 10^{-7}\ \textup{cm}^{2}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

5c) What is the hydraulic conductivity of the sandstone if the water temperature is 35 \u00b0C (in cm\/s)?<\/em><\/p>\n

If the water temperature in the sandstone is 35 \u00b0C, Equation 31 of this book can be used to calculate, K<\/em>, of the sandstone at that temperature in cm\/s, using the relationships for the physical properties of water and temperature shown as Figure 28<\/a> of this book.<\/p>\n\n\n\n
\"\displaystyle K=\frac{5.8\times 10^{-7}\ {\textup{cm}}^{2}\ 980\ \frac{\textup{cm}}{\textup{s}^{2}}\ 0.994\ \frac{\textup{g}}{\textup{cm}^{3}}}{0.007\ \frac{\textup{g}\cdot \textup{cm}}{\textup{s}}}=\frac{0.08\ \textup{cm}}{\textup{s}}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Return to Exercise 5<\/a><\/p>\n","protected":false},"author":1,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-892","chapter","type-chapter","status-publish","hentry"],"part":873,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/892","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":14,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/892\/revisions"}],"predecessor-version":[{"id":1199,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/892\/revisions\/1199"}],"part":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/parts\/873"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/892\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/wp\/v2\/media?parent=892"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapter-type?post=892"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/wp\/v2\/contributor?post=892"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/wp\/v2\/license?post=892"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}