{"id":127,"date":"2023-08-23T20:46:48","date_gmt":"2023-08-23T20:46:48","guid":{"rendered":"https:\/\/books.gw-project.org\/stable-isotope-hydrology\/chapter\/solution-exercise-3-hydrograph-separation\/"},"modified":"2024-01-15T14:52:40","modified_gmt":"2024-01-15T14:52:40","slug":"solution-exercise-3-hydrograph-separation","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/stable-isotope-hydrology\/chapter\/solution-exercise-3-hydrograph-separation\/","title":{"raw":"Solution Exercise\u00a03\u00a0-\u00a0Hydrograph Separation","rendered":"Solution Exercise\u00a03\u00a0–\u00a0Hydrograph Separation"},"content":{"raw":"
\r\n

Using the mass balance equation for a 2-component system:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n
\r\n

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

\r\n

\ud835\udc44\ud835\udc5d <\/sub>\u00d7 \ud835\udeff\ud835\udc5d<\/sub> + \ud835\udc44\ud835\udc54 <\/sub>\u00d7 \ud835\udeff\ud835\udc54<\/sub> = \ud835\udc44\ud835\udc60 <\/sub>\u00d7 \ud835\udeff\ud835\udc60<\/sub><\/span><\/p>\r\n<\/td>\r\n

\r\n

<\/p>\r\n<\/td>\r\n<\/tr>\r\n

<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

where:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\r\n

Q<\/em><\/p>\r\n<\/td>\r\n

\r\n

=<\/p>\r\n<\/td>\r\n

\r\n

flow<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

\u03b4<\/p>\r\n<\/td>\r\n

\r\n

=<\/p>\r\n<\/td>\r\n

\r\n

stable isotope composition<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

p<\/em><\/p>\r\n<\/td>\r\n

\r\n

=<\/p>\r\n<\/td>\r\n

\r\n

precipitation<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

g<\/em><\/p>\r\n<\/td>\r\n

\r\n

=<\/p>\r\n<\/td>\r\n

\r\n

groundwater<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

s<\/em><\/p>\r\n<\/td>\r\n

\r\n

=<\/p>\r\n<\/td>\r\n

\r\n

surface water<\/p>\r\n<\/td>\r\n<\/tr>\r\n

<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

and recognizing:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n
\r\n

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

\r\n

\ud835\udc44\ud835\udc5d <\/sub>+ \ud835\udc44\ud835\udc54<\/sub> = \ud835\udc44\ud835\udc60<\/sub><\/span><\/p>\r\n<\/td>\r\n

\r\n

<\/p>\r\n<\/td>\r\n<\/tr>\r\n

<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

and if fractions are used instead of actual flow:<\/p>\r\n \r\n

\ud835\udc44\ud835\udc60<\/sub> = 1 , then\u00a0 <\/span>\ud835\udc44\ud835\udc5d<\/sub>+ \ud835\udc44\ud835\udc54<\/sub> = 1<\/span><\/div>\r\n \r\n
\u00a0<\/span><\/div>\r\n
\u2234\u00a0 \ud835\udc44\ud835\udc5d<\/sub> = 1 \u2212 \ud835\udc44\ud835\udc54<\/sub><\/span><\/div>\r\n
\u2234\u00a0 (1\u2212\ud835\udc44\ud835\udc54<\/sub>)\ud835\udeff\ud835\udc5d<\/sub> + \ud835\udc44\ud835\udc54<\/sub>\ud835\udeff\ud835\udc54<\/sub> = \ud835\udeff\ud835\udc60<\/sub><\/span><\/div>\r\n
\u2234\u00a0 \ud835\udeff\ud835\udc5d<\/sub>\u2212\ud835\udc44\ud835\udc54<\/sub>\ud835\udeff\ud835\udc5d<\/sub> + \ud835\udc44\ud835\udc54<\/sub>\ud835\udeff\ud835\udc54<\/sub> = \ud835\udeff\ud835\udc60<\/sub><\/span><\/div>\r\n
\u2234\u00a0 \ud835\udc44\ud835\udc54<\/sub>(\ud835\udeff\ud835\udc54<\/sub>\u2212\ud835\udeff\ud835\udc5d<\/sub>) = \ud835\udeff\ud835\udc60<\/sub>\u2212 \ud835\udeff\ud835\udc5d<\/sub><\/span><\/div>\r\n
\u2234\u00a0 \ud835\udc44\ud835\udc54<\/sub> = (\ud835\udeff\ud835\udc60<\/sub>\u2212\ud835\udeff\ud835\udc5d<\/sub>) \/ (\ud835\udeff\ud835\udc54<\/sub>\u2212\ud835\udeff\ud835\udc5d<\/sub>) ,<\/span><\/div>\r\n

which, for \u03b42<\/sup>H gives 0.82 for groundwater, which translates to the surface water flow being 82 percent groundwater and 18 percent precipitation derived. For \u03b418<\/sup>O, the answers are 63 percent groundwater and 37 percent precipitation.<\/p>\r\n

The measurement errors (given by Hu et al., 2020), especially for precipitation, are so large, it is possible precipitation has the same isotope composition as groundwater, in which case hydrograph separation is not possible. Similarly, the river water could have the same isotope composition as the groundwater, again negating the ability to perform a hydrograph separation. More measurements would lower the error and improve the estimate.<\/p>\r\n

Return to Exercise\u00a03<\/span><\/a><\/strong><\/p>\r\n

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

\n

Using the mass balance equation for a 2-component system:<\/p>\n\n\n\n\n
\n

\n<\/td>\n

\n

\ud835\udc44\ud835\udc5d <\/sub>\u00d7 \ud835\udeff\ud835\udc5d<\/sub> + \ud835\udc44\ud835\udc54 <\/sub>\u00d7 \ud835\udeff\ud835\udc54<\/sub> = \ud835\udc44\ud835\udc60 <\/sub>\u00d7 \ud835\udeff\ud835\udc60<\/sub><\/span><\/p>\n<\/td>\n

\n

\n<\/td>\n<\/tr>\n

<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n\n\n\n\n\n
\n

Q<\/em><\/p>\n<\/td>\n

\n

=<\/p>\n<\/td>\n

\n

flow<\/p>\n<\/td>\n<\/tr>\n

\n

\u03b4<\/p>\n<\/td>\n

\n

=<\/p>\n<\/td>\n

\n

stable isotope composition<\/p>\n<\/td>\n<\/tr>\n

\n

p<\/em><\/p>\n<\/td>\n

\n

=<\/p>\n<\/td>\n

\n

precipitation<\/p>\n<\/td>\n<\/tr>\n

\n

g<\/em><\/p>\n<\/td>\n

\n

=<\/p>\n<\/td>\n

\n

groundwater<\/p>\n<\/td>\n<\/tr>\n

\n

s<\/em><\/p>\n<\/td>\n

\n

=<\/p>\n<\/td>\n

\n

surface water<\/p>\n<\/td>\n<\/tr>\n

<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

and recognizing:<\/p>\n\n\n\n\n
\n

\n<\/td>\n

\n

\ud835\udc44\ud835\udc5d <\/sub>+ \ud835\udc44\ud835\udc54<\/sub> = \ud835\udc44\ud835\udc60<\/sub><\/span><\/p>\n<\/td>\n

\n

\n<\/td>\n<\/tr>\n

<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

and if fractions are used instead of actual flow:<\/p>\n

 <\/p>\n

\ud835\udc44\ud835\udc60<\/sub> = 1 , then\u00a0 <\/span>\ud835\udc44\ud835\udc5d<\/sub>+ \ud835\udc44\ud835\udc54<\/sub> = 1<\/span><\/div>\n

 <\/p>\n

\u00a0<\/span><\/div>\n
\u2234\u00a0 \ud835\udc44\ud835\udc5d<\/sub> = 1 \u2212 \ud835\udc44\ud835\udc54<\/sub><\/span><\/div>\n
\u2234\u00a0 (1\u2212\ud835\udc44\ud835\udc54<\/sub>)\ud835\udeff\ud835\udc5d<\/sub> + \ud835\udc44\ud835\udc54<\/sub>\ud835\udeff\ud835\udc54<\/sub> = \ud835\udeff\ud835\udc60<\/sub><\/span><\/div>\n
\u2234\u00a0 \ud835\udeff\ud835\udc5d<\/sub>\u2212\ud835\udc44\ud835\udc54<\/sub>\ud835\udeff\ud835\udc5d<\/sub> + \ud835\udc44\ud835\udc54<\/sub>\ud835\udeff\ud835\udc54<\/sub> = \ud835\udeff\ud835\udc60<\/sub><\/span><\/div>\n
\u2234\u00a0 \ud835\udc44\ud835\udc54<\/sub>(\ud835\udeff\ud835\udc54<\/sub>\u2212\ud835\udeff\ud835\udc5d<\/sub>) = \ud835\udeff\ud835\udc60<\/sub>\u2212 \ud835\udeff\ud835\udc5d<\/sub><\/span><\/div>\n
\u2234\u00a0 \ud835\udc44\ud835\udc54<\/sub> = (\ud835\udeff\ud835\udc60<\/sub>\u2212\ud835\udeff\ud835\udc5d<\/sub>) \/ (\ud835\udeff\ud835\udc54<\/sub>\u2212\ud835\udeff\ud835\udc5d<\/sub>) ,<\/span><\/div>\n

which, for \u03b42<\/sup>H gives 0.82 for groundwater, which translates to the surface water flow being 82 percent groundwater and 18 percent precipitation derived. For \u03b418<\/sup>O, the answers are 63 percent groundwater and 37 percent precipitation.<\/p>\n

The measurement errors (given by Hu et al., 2020), especially for precipitation, are so large, it is possible precipitation has the same isotope composition as groundwater, in which case hydrograph separation is not possible. Similarly, the river water could have the same isotope composition as the groundwater, again negating the ability to perform a hydrograph separation. More measurements would lower the error and improve the estimate.<\/p>\n

Return to Exercise\u00a03<\/span><\/a><\/strong><\/p>\n

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