{"id":510,"date":"2020-10-26T20:08:47","date_gmt":"2020-10-26T20:08:47","guid":{"rendered":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/?post_type=chapter&p=510"},"modified":"2020-12-29T20:38:16","modified_gmt":"2020-12-29T20:38:16","slug":"governing-equations-for-unconfined-groundwater-flow","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/chapter\/governing-equations-for-unconfined-groundwater-flow\/","title":{"raw":"7.3 Governing Equations for Unconfined Groundwater Flow","rendered":"7.3 Governing Equations for Unconfined Groundwater Flow"},"content":{"raw":"Unconfined flow equations are non-linear in that the transmissivity of the aquifer depends on the saturated thickness and the saturated thickness varies in the direction of flow because the water table slopes. By definition, a confined aquifer is fully saturated, so transmissivity values are constant for a confined aquifer of constant thickness. Under water table conditions the water table slopes, flow is parallel to the water table and the saturated thickness decreases in the direction of flow (Figure 55). If the slope of the water table is small, Darcy's Law can be applied to develop governing equations by using the Dupuit simplification or Dupuit assumptions. Dupuit\u2019s simplification uses the approximate gradient (dh\u00a0over\u00a0the\u00a0distance\u00a0x<\/em>, -dh<\/em>\/dx<\/em>) rather than the true gradient (dh\u00a0over\u00a0the\u00a0flow\u00a0path\u00a0length<\/em>, -dh<\/em>\/dL<\/em>) by assuming the flow is horizontal (no vertical components of flow) as shown by the solid arrows in Figure 55 and by Equation 72.\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle Q=-K\\frac{dh}{dx}A[\/latex] \u00a0\u00a0\u00a0 rather than \u00a0\u00a0\u00a0 [latex]\\displaystyle Q=-K\\frac{dh}{dL}A[\/latex]<\/td>\r\n(72)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nExpressing the flow area as the product of the height of the water table and the unit width of the system in the y<\/em> direction, the flow is shown in Equation 73.\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle Q= -K\\frac{dh}{dx}h\\Delta y[\/latex]<\/td>\r\n(73)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[caption id=\"attachment_750\" align=\"alignnone\" width=\"851\"]\"Figure Figure 55 -<\/strong> Dupuit's simplification mathematically approximates unconfined flow as horizontal by using the gradient -dh<\/em>\/dx<\/em> (blue solid arrows) instead of the gradient along the flow path -dh<\/em>\/dL<\/em> (orange dashed lines). The calculated heads and flow rates are sufficiently accurate if the slope of the water table is small.[\/caption]\r\n\r\nIf the bottom of the unconfined aquifer is used as the datum, then the head defines the saturated thickness. To include this dependency in the flow equations, Equation 70 is adjusted so that aquifer thickness, b<\/em>, is replaced with h<\/em>, and the varying value of h<\/em> has to be inside the derivative. In addition, to represent unconfined flow, specific yield, S<\/em>y<\/em><\/sub>, is used as the aquifer storativity. Thus, unconfined, two-dimensional (plan view), transient, anisotropic, heterogeneous<\/em> conditions of groundwater flow are represented by Equation 74.\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(74)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFor unconfined, two-dimensional (plan view), transient, anisotropic, homogeneous<\/em> conditions of groundwater flow, hydraulic conductivities do not need to be within the derivative, resulting in Equation 75.\r\n\r\n\r\n\r\n
[latex]\\displaystyle S_y\\frac{\\partial h}{\\partial t}\\ =K_x\\frac{\\partial}{\\partial x}\\left(h\\frac{\\partial h}{\\partial x}\\right)+K_y\\frac{\\partial}{\\partial y}\\left(h\\frac{\\partial h}{\\partial y}\\right)[\/latex]<\/td>\r\n(75)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nUnconfined, two-dimensional, plan view, transient, isotropic, homogeneous<\/em> flow is represented using only one value of K<\/em> as shown in Equation 76.\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle S_y\\frac{\\partial h}{\\partial t}\\ =\\frac{K}{2}\\left(\\frac{\\partial^{2} h^2}{\\partial x^2}+\\frac{\\partial^{2} h^2}{\\partial y^2}\\right)[\/latex]<\/td>\r\n(76)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>","rendered":"

Unconfined flow equations are non-linear in that the transmissivity of the aquifer depends on the saturated thickness and the saturated thickness varies in the direction of flow because the water table slopes. By definition, a confined aquifer is fully saturated, so transmissivity values are constant for a confined aquifer of constant thickness. Under water table conditions the water table slopes, flow is parallel to the water table and the saturated thickness decreases in the direction of flow (Figure 55). If the slope of the water table is small, Darcy’s Law can be applied to develop governing equations by using the Dupuit simplification or Dupuit assumptions. Dupuit\u2019s simplification uses the approximate gradient (dh\u00a0over\u00a0the\u00a0distance\u00a0x<\/em>, –dh<\/em>\/dx<\/em>) rather than the true gradient (dh\u00a0over\u00a0the\u00a0flow\u00a0path\u00a0length<\/em>, –dh<\/em>\/dL<\/em>) by assuming the flow is horizontal (no vertical components of flow) as shown by the solid arrows in Figure 55 and by Equation 72.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle Q=-K\frac{dh}{dx}A\" \u00a0\u00a0\u00a0 rather than \u00a0\u00a0\u00a0 \"\displaystyle Q=-K\frac{dh}{dL}A\"<\/td>\n(72)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Expressing the flow area as the product of the height of the water table and the unit width of the system in the y<\/em> direction, the flow is shown in Equation 73.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle Q= -K\frac{dh}{dx}h\Delta y\"<\/td>\n(73)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\"Figure
Figure 55 –<\/strong> Dupuit’s simplification mathematically approximates unconfined flow as horizontal by using the gradient –dh<\/em>\/dx<\/em> (blue solid arrows) instead of the gradient along the flow path –dh<\/em>\/dL<\/em> (orange dashed lines). The calculated heads and flow rates are sufficiently accurate if the slope of the water table is small.<\/figcaption><\/figure>\n

If the bottom of the unconfined aquifer is used as the datum, then the head defines the saturated thickness. To include this dependency in the flow equations, Equation 70 is adjusted so that aquifer thickness, b<\/em>, is replaced with h<\/em>, and the varying value of h<\/em> has to be inside the derivative. In addition, to represent unconfined flow, specific yield, S<\/em>y<\/em><\/sub>, is used as the aquifer storativity. Thus, unconfined, two-dimensional (plan view), transient, anisotropic, heterogeneous<\/em> conditions of groundwater flow are represented by Equation 74.<\/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(74)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

For unconfined, two-dimensional (plan view), transient, anisotropic, homogeneous<\/em> conditions of groundwater flow, hydraulic conductivities do not need to be within the derivative, resulting in Equation 75.<\/p>\n\n\n\n
\"\displaystyle S_y\frac{\partial h}{\partial t}\ =K_x\frac{\partial}{\partial x}\left(h\frac{\partial h}{\partial x}\right)+K_y\frac{\partial}{\partial y}\left(h\frac{\partial h}{\partial y}\right)\"<\/td>\n(75)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Unconfined, two-dimensional, plan view, transient, isotropic, homogeneous<\/em> flow is represented using only one value of K<\/em> as shown in Equation 76.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle S_y\frac{\partial h}{\partial t}\ =\frac{K}{2}\left(\frac{\partial^{2} h^2}{\partial x^2}+\frac{\partial^{2} h^2}{\partial y^2}\right)\"<\/td>\n(76)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"author":1,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-510","chapter","type-chapter","status-publish","hentry"],"part":107,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/510","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":11,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/510\/revisions"}],"predecessor-version":[{"id":1198,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/510\/revisions\/1198"}],"part":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/parts\/107"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/510\/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=510"}],"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=510"},{"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=510"},{"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=510"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}