{"id":901,"date":"2020-12-09T05:03:55","date_gmt":"2020-12-09T05:03:55","guid":{"rendered":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/?post_type=chapter&p=901"},"modified":"2020-12-10T21:10:01","modified_gmt":"2020-12-10T21:10:01","slug":"solution-to-exercise-8","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/chapter\/solution-to-exercise-8\/","title":{"raw":"Solution to Exercise 8","rendered":"Solution to Exercise 8"},"content":{"raw":"8) Show that the units of Ss<\/sub> are 1\/L.\r\n\r\nUsing Equation 45 in this book\r\n\r\n\r\n\r\n
S<\/em>s<\/em><\/sub> = \u03c1g<\/em> (\u03b1<\/em> + n<\/em>e<\/em><\/sub>\u03b2<\/em>)<\/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
S<\/em>s<\/em><\/sub><\/td>\r\n=<\/td>\r\nspecific storage (1\/L)<\/td>\r\n<\/tr>\r\n
\u03b1<\/em><\/td>\r\n=<\/td>\r\ncompressibility of the aquifer solid structure (T2<\/sup>L\/M)<\/td>\r\n<\/tr>\r\n
n<\/em>e<\/em><\/sub><\/td>\r\n=<\/td>\r\neffective porosity (dimensionless)<\/td>\r\n<\/tr>\r\n
\u03b2<\/em><\/td>\r\n=<\/td>\r\ncompressibility of water (T2<\/sup>L\/M)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nchecking units:\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]\\displaystyle S_{s}: \\frac{\\textup{M}}{\\textup{L}^3}\\frac{\\textup{L}}{\\textup{T}^2}\\left(\\frac{\\textup{T}^2\\textup{L}}{\\textup{M}}+\\frac{\\textup{L}^3}{\\textup{L}^3}\\frac{\\textup{T}^2\\textup{L}}{\\textup{M}}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]\\displaystyle S_{s}: \\frac{\\textup{M}}{\\textup{L}^{2}}\\frac{1}{\\textup{T}^2}\\left(\\frac{\\textup{T}^2\\textup{L}}{\\textup{M}}+\\frac{\\textup{T}^2\\textup{L}}{\\textup{M}}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]\\displaystyle S_{s}: \\frac{\\textup{M}}{\\textup{L}^2\\textup{\\textup{T}}^2}\\left(\\frac{\\textup{T}^2\\textup{L}}{\\textup{M}}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]\\displaystyle S_{s}: \\frac{1}{\\textup{L}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Return to Exercise 8<\/a><\/p>","rendered":"

8) Show that the units of Ss<\/sub> are 1\/L.<\/p>\n

Using Equation 45 in this book<\/p>\n\n\n\n
S<\/em>s<\/em><\/sub> = \u03c1g<\/em> (\u03b1<\/em> + n<\/em>e<\/em><\/sub>\u03b2<\/em>)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n\n\n\n
S<\/em>s<\/em><\/sub><\/td>\n=<\/td>\nspecific storage (1\/L)<\/td>\n<\/tr>\n
\u03b1<\/em><\/td>\n=<\/td>\ncompressibility of the aquifer solid structure (T2<\/sup>L\/M)<\/td>\n<\/tr>\n
n<\/em>e<\/em><\/sub><\/td>\n=<\/td>\neffective porosity (dimensionless)<\/td>\n<\/tr>\n
\u03b2<\/em><\/td>\n=<\/td>\ncompressibility of water (T2<\/sup>L\/M)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

checking units:<\/p>\n\n\n\n\n\n\n
\"\displaystyle S_{s}: \frac{\textup{M}}{\textup{L}^3}\frac{\textup{L}}{\textup{T}^2}\left(\frac{\textup{T}^2\textup{L}}{\textup{M}}+\frac{\textup{L}^3}{\textup{L}^3}\frac{\textup{T}^2\textup{L}}{\textup{M}}\right)\"<\/td>\n<\/tr>\n
\"\displaystyle S_{s}: \frac{\textup{M}}{\textup{L}^{2}}\frac{1}{\textup{T}^2}\left(\frac{\textup{T}^2\textup{L}}{\textup{M}}+\frac{\textup{T}^2\textup{L}}{\textup{M}}\right)\"<\/td>\n<\/tr>\n
\"\displaystyle S_{s}: \frac{\textup{M}}{\textup{L}^2\textup{\textup{T}}^2}\left(\frac{\textup{T}^2\textup{L}}{\textup{M}}\right)\"<\/td>\n<\/tr>\n
\"\displaystyle S_{s}: \frac{1}{\textup{L}}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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