{"id":105,"date":"2022-12-30T18:15:40","date_gmt":"2022-12-30T18:15:40","guid":{"rendered":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/chapter\/solution-exercise-4\/"},"modified":"2023-01-08T03:46:03","modified_gmt":"2023-01-08T03:46:03","slug":"solution-exercise-4","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/chapter\/solution-exercise-4\/","title":{"raw":"Solution Exercise 4","rendered":"Solution Exercise 4"},"content":{"raw":"
\r\n

The freshwater head can be calculated using [latex]h_{f}=z_{i}+(h_{i}-z_{i})\\frac{\\rho _{i}}{\\rho _{f}}[\/latex], which gives (noting that z<\/em>i<\/em><\/sub>\u00a0=\u00a0100 \u2212 60\u00a0=\u00a040\u00a0m):<\/p>\r\n\r\n\r\n\r\n\r\n\r\n
1.<\/td>\r\n[latex]h_{f,1}=40+(100.4-40)\\frac{1010}{1000}=101\\;\\textrm{m}[\/latex]<\/td>\r\n<\/tr>\r\n
<\/td>\r\n[latex]h_{f,2}=40+(100-40)\\frac{1020}{1000}=101.2\\;\\textrm{m}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The freshwater head is higher at piezometer\u00a02 than at piezometer\u00a01, so the flow is from piezometer\u00a02 to 1. This is opposite to what one would conclude from uncorrected hydraulic head measurements.<\/p>\r\n

2. The specific discharge follows from<\/p>\r\n

[latex]q_{x}=-K_{f}\\frac{\\Delta h_{f}}{\\Delta x}=-10\\times \\frac{101.2-101}{200}=-0.01\\;\\textrm{m}\\;\\textrm{d}^{-1}[\/latex]<\/p>\r\n

which is a negative number because flow is in the direction of declining head.<\/p>\r\n

Return to <\/span>Exercise\u00a0<\/span>4<\/span><\/a><\/p>\r\n

<\/p>\r\n\r\n<\/div>","rendered":"

\n

The freshwater head can be calculated using \"h_{f}=z_{i}+(h_{i}-z_{i})\frac{\rho _{i}}{\rho _{f}}\", which gives (noting that z<\/em>i<\/em><\/sub>\u00a0=\u00a0100 \u2212 60\u00a0=\u00a040\u00a0m):<\/p>\n\n\n\n\n
1.<\/td>\n\"h_{f,1}=40+(100.4-40)\frac{1010}{1000}=101\;\textrm{m}\"<\/td>\n<\/tr>\n
<\/td>\n\"h_{f,2}=40+(100-40)\frac{1020}{1000}=101.2\;\textrm{m}\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The freshwater head is higher at piezometer\u00a02 than at piezometer\u00a01, so the flow is from piezometer\u00a02 to 1. This is opposite to what one would conclude from uncorrected hydraulic head measurements.<\/p>\n

2. The specific discharge follows from<\/p>\n

\"q_{x}=-K_{f}\frac{\Delta h_{f}}{\Delta x}=-10\times \frac{101.2-101}{200}=-0.01\;\textrm{m}\;\textrm{d}^{-1}\"<\/p>\n

which is a negative number because flow is in the direction of declining head.<\/p>\n

Return to <\/span>Exercise\u00a0<\/span>4<\/span><\/a><\/p>\n

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