{"id":883,"date":"2020-12-09T04:59:46","date_gmt":"2020-12-09T04:59:46","guid":{"rendered":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/?post_type=chapter&p=883"},"modified":"2020-12-10T00:10:51","modified_gmt":"2020-12-10T00:10:51","slug":"solution-to-exercise-2","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/chapter\/solution-to-exercise-2\/","title":{"raw":"Solution to Exercise 2","rendered":"Solution to Exercise 2"},"content":{"raw":"2) Show that n<\/em> = e<\/em>\/(1 + e<\/em>), where n<\/em> is porosity and e<\/em> is the void ratio.\r\n\r\nPorosity is defined by Equation 6 of this book. And, when the effective porosity equals the interconnected porosity then V<\/em>I<\/em><\/sub> = V<\/em>v<\/em><\/sub>.\r\n\r\n\r\n\r\n
[latex]\\displaystyle n_e=\\frac{V_I}{V_T}=\\frac{V_V}{V_T}=n[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nVoid Ratio is defined by Equation 7 of this book.\r\n\r\n\r\n\r\n
[latex]\\displaystyle e=\\frac{V_V}{V_S}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe total volume is the sum of the solid and the void volume.\r\n\r\n\r\n\r\n
V<\/em>T<\/em><\/sub> = V<\/em>V<\/em><\/sub> + V<\/em>S<\/em><\/sub><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nUsing the three relationships we can show:\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]\\displaystyle n=\\frac{V_{V}}{V_{V}+V_{S}}[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]\\displaystyle \\frac{1}{n}=\\frac{V_{V}+V_{S}}{V_{V}}=1+\\frac{V_{S}}{V_{V}}=1+\\frac{1}{e}[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]\\displaystyle \\frac{1}{n}=\\frac{e+1}{e}[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]\\displaystyle n=\\frac{e}{1+e}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis matches Equation 8 of this book.\r\n\r\n\r\n\r\n
[latex]\\displaystyle n=\\frac{e}{1+e}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

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

2) Show that n<\/em> = e<\/em>\/(1 + e<\/em>), where n<\/em> is porosity and e<\/em> is the void ratio.<\/p>\n

Porosity is defined by Equation 6 of this book. And, when the effective porosity equals the interconnected porosity then V<\/em>I<\/em><\/sub> = V<\/em>v<\/em><\/sub>.<\/p>\n\n\n\n
\"\displaystyle n_e=\frac{V_I}{V_T}=\frac{V_V}{V_T}=n\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Void Ratio is defined by Equation 7 of this book.<\/p>\n\n\n\n
\"\displaystyle e=\frac{V_V}{V_S}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The total volume is the sum of the solid and the void volume.<\/p>\n\n\n\n
V<\/em>T<\/em><\/sub> = V<\/em>V<\/em><\/sub> + V<\/em>S<\/em><\/sub><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Using the three relationships we can show:<\/p>\n\n\n\n\n\n\n
\"\displaystyle n=\frac{V_{V}}{V_{V}+V_{S}}\"<\/td>\n<\/tr>\n
\"\displaystyle \frac{1}{n}=\frac{V_{V}+V_{S}}{V_{V}}=1+\frac{V_{S}}{V_{V}}=1+\frac{1}{e}\"<\/td>\n<\/tr>\n
\"\displaystyle \frac{1}{n}=\frac{e+1}{e}\"<\/td>\n<\/tr>\n
\"\displaystyle n=\frac{e}{1+e}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

This matches Equation 8 of this book.<\/p>\n\n\n\n
\"\displaystyle n=\frac{e}{1+e}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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