{"id":913,"date":"2020-12-09T05:07:08","date_gmt":"2020-12-09T05:07:08","guid":{"rendered":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/?post_type=chapter&p=913"},"modified":"2020-12-30T04:46:09","modified_gmt":"2020-12-30T04:46:09","slug":"solution-to-exercise-12","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/chapter\/solution-to-exercise-12\/","title":{"raw":"Solution to Exercise 12","rendered":"Solution to Exercise 12"},"content":{"raw":"

12) What conditions does the following governing equation represent? Explain the features of the terms of the equation that indicate each condition.<\/p>\r\n\r\n\r\n\r\n\r\n
[latex]\\displaystyle S_{s}\\frac{\\partial h}{\\partial t}=\\frac{\\partial }{\\partial x}\\left ( K_{x}\\frac{\\partial h}{\\partial x} \\right )+\\frac{\\partial }{\\partial y}\\left ( K_{y}\\frac{\\partial h}{\\partial y} \\right )+\\frac{\\partial }{\\partial z}\\left ( K_{z}\\frac{\\partial h}{\\partial z} \\right )[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis is Equation 67 of this book. It represents transient, confined, groundwater flow where conditions are anisotropic and heterogeneous. The K values are within the differential on the right-hand side of the equation signifying they are variables and can be different values everywhere in the domain (x,y,z), thus the equation is applicable to a heterogeneous and anisotropic system. The left-hand side has a time derivative and thus, this equation represents transient conditions, that is, the heads change with time and water can go into, or come out of, storage. If the left-hand side was set to 0, the equation would represent steady state conditions. If the equation represented unconfined conditions, there would be an h<\/em> inside the derivative, and S<\/em>y<\/em><\/sub> would be used instead of S<\/em>s<\/em><\/sub> as follows (as shown in Equation 74 of this book):\r\n\r\n\r\n\r\n
[latex]\\displaystyle S_y\\frac{\\partial h}{\\partial t}\\ =\\frac{\\partial}{\\partial x}K_x\\left(h\\frac{\\partial h}{\\partial x}\\right)+\\frac{\\partial}{\\partial y}K_y\\left(h\\frac{\\partial h}{\\partial y}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Return to Exercise 12<\/a><\/p>","rendered":"

12) What conditions does the following governing equation represent? Explain the features of the terms of the equation that indicate each condition.<\/p>\n\n\n\n
\"\displaystyle S_{s}\frac{\partial h}{\partial t}=\frac{\partial }{\partial x}\left ( K_{x}\frac{\partial h}{\partial x} \right )+\frac{\partial }{\partial y}\left ( K_{y}\frac{\partial h}{\partial y} \right )+\frac{\partial }{\partial z}\left ( K_{z}\frac{\partial h}{\partial z} \right )\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

This is Equation 67 of this book. It represents transient, confined, groundwater flow where conditions are anisotropic and heterogeneous. The K values are within the differential on the right-hand side of the equation signifying they are variables and can be different values everywhere in the domain (x,y,z), thus the equation is applicable to a heterogeneous and anisotropic system. The left-hand side has a time derivative and thus, this equation represents transient conditions, that is, the heads change with time and water can go into, or come out of, storage. If the left-hand side was set to 0, the equation would represent steady state conditions. If the equation represented unconfined conditions, there would be an h<\/em> inside the derivative, and S<\/em>y<\/em><\/sub> would be used instead of S<\/em>s<\/em><\/sub> as follows (as shown in Equation 74 of this book):<\/p>\n\n\n\n
\"\displaystyle S_y\frac{\partial h}{\partial t}\ =\frac{\partial}{\partial x}K_x\left(h\frac{\partial h}{\partial x}\right)+\frac{\partial}{\partial y}K_y\left(h\frac{\partial h}{\partial y}\right)\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Return to Exercise 12<\/a><\/p>\n","protected":false},"author":1,"menu_order":12,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-913","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\/913","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":7,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/913\/revisions"}],"predecessor-version":[{"id":1210,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/913\/revisions\/1210"}],"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\/913\/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=913"}],"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=913"},{"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=913"},{"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=913"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}