{"id":485,"date":"2020-10-26T14:58:58","date_gmt":"2020-10-26T14:58:58","guid":{"rendered":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/?post_type=chapter&p=485"},"modified":"2020-12-28T22:56:33","modified_gmt":"2020-12-28T22:56:33","slug":"basis-for-flow-equation-development","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/chapter\/basis-for-flow-equation-development\/","title":{"raw":"7.1 Basis for Flow Equation Development","rendered":"7.1 Basis for Flow Equation Development"},"content":{"raw":"General groundwater flow equations are often referred to as governing equations because they describe the factors that control (i.e., govern) groundwater flow. The equations are based only on Darcy\u2019s Law and the continuity equation for the conservation of mass (of water).\r\n\r\nDevelopment of the governing equations is typically accomplished by conceptualizing groundwater flow through a small volume of fully saturated porous material (REV) that reflects overall hydrogeologic properties of a larger deposit. It can be visualized as a cube of porous material of sufficient size (\u0394x<\/em>, \u0394y<\/em>, \u0394z<\/em>) to encompass macroscopic properties governing groundwater flow (Figure 4<\/a> and Figure 51). The mass of water within the REV (M<\/em>) depends on the density of the water, the porosity of the element (assuming all pores spaces are fully connected), and the volume of the block as shown in Equation 51. Under steady-state flow conditions, the mass of water in the REV is constant, inflow equals outflow, and heads, gradients and flow rates do not vary with time (Figure 51a). When flow is transient, the mass of water in the REV varies with time, inflow does not equal outflow, and heads, gradients, and flow rates vary with time (Figure 51b).\r\n\r\n[caption id=\"attachment_489\" align=\"alignnone\" width=\"1024\"]\"Figure Figure 51 -<\/strong> A small, but representative, volume called a Representative Elementary Volume (REV) of a porous medium has dimensions \u0394x<\/em>, \u0394y<\/em>, \u0394z<\/em>. The confined fully saturated cube of porous material reflects the general properties of a porous earth material. Flow is from left to right. The volumeis fully saturated. a) REV for confined steady-state conditions, the gradient (potentiometric surface illustrated as the light blue plane) does not change with time, mass flow is constant and mass inflow equals mass outflow. b) REV for confined transient conditions, the gradient represented by the potentiometric surface (blue plane) changes with time, \u0394t<\/em>, the flow rate varies with time, mass inflow does not equal mass outflow, and the mass of water (M<\/em>) in the REV changes with time.[\/caption]\r\n\r\n\r\n\r\n
<\/td>\r\nM<\/em> = \u03c1<\/em> n<\/em> \u0394x<\/em> \u0394y<\/em> \u0394z<\/em><\/td>\r\n(51)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nwhere:\r\n\r\n\r\n\r\n\r\n\r\n\r\n
M<\/em><\/td>\r\n=<\/td>\r\nmass of water in the REV (M)<\/td>\r\n<\/tr>\r\n
\u03c1<\/em><\/td>\r\n=<\/td>\r\ndensity of water in the REV (M\/L3<\/sup>)<\/td>\r\n<\/tr>\r\n
n<\/em><\/td>\r\n=<\/td>\r\nfully connected porosity (n<\/em>e<\/em><\/sub>) of the REV (dimensionless)<\/td>\r\n<\/tr>\r\n
\u0394x<\/em>, \u0394y<\/em>, \u0394z<\/em><\/td>\r\n=<\/td>\r\nlength of each side of the REV (L)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>","rendered":"

General groundwater flow equations are often referred to as governing equations because they describe the factors that control (i.e., govern) groundwater flow. The equations are based only on Darcy\u2019s Law and the continuity equation for the conservation of mass (of water).<\/p>\n

Development of the governing equations is typically accomplished by conceptualizing groundwater flow through a small volume of fully saturated porous material (REV) that reflects overall hydrogeologic properties of a larger deposit. It can be visualized as a cube of porous material of sufficient size (\u0394x<\/em>, \u0394y<\/em>, \u0394z<\/em>) to encompass macroscopic properties governing groundwater flow (Figure 4<\/a> and Figure 51). The mass of water within the REV (M<\/em>) depends on the density of the water, the porosity of the element (assuming all pores spaces are fully connected), and the volume of the block as shown in Equation 51. Under steady-state flow conditions, the mass of water in the REV is constant, inflow equals outflow, and heads, gradients and flow rates do not vary with time (Figure 51a). When flow is transient, the mass of water in the REV varies with time, inflow does not equal outflow, and heads, gradients, and flow rates vary with time (Figure 51b).<\/p>\n

\"Figure
Figure 51 –<\/strong> A small, but representative, volume called a Representative Elementary Volume (REV) of a porous medium has dimensions \u0394x<\/em>, \u0394y<\/em>, \u0394z<\/em>. The confined fully saturated cube of porous material reflects the general properties of a porous earth material. Flow is from left to right. The volumeis fully saturated. a) REV for confined steady-state conditions, the gradient (potentiometric surface illustrated as the light blue plane) does not change with time, mass flow is constant and mass inflow equals mass outflow. b) REV for confined transient conditions, the gradient represented by the potentiometric surface (blue plane) changes with time, \u0394t<\/em>, the flow rate varies with time, mass inflow does not equal mass outflow, and the mass of water (M<\/em>) in the REV changes with time.<\/figcaption><\/figure>\n\n\n\n
<\/td>\nM<\/em> = \u03c1<\/em> n<\/em> \u0394x<\/em> \u0394y<\/em> \u0394z<\/em><\/td>\n(51)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n\n\n\n
M<\/em><\/td>\n=<\/td>\nmass of water in the REV (M)<\/td>\n<\/tr>\n
\u03c1<\/em><\/td>\n=<\/td>\ndensity of water in the REV (M\/L3<\/sup>)<\/td>\n<\/tr>\n
n<\/em><\/td>\n=<\/td>\nfully connected porosity (n<\/em>e<\/em><\/sub>) of the REV (dimensionless)<\/td>\n<\/tr>\n
\u0394x<\/em>, \u0394y<\/em>, \u0394z<\/em><\/td>\n=<\/td>\nlength of each side of the REV (L)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"author":1,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-485","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\/485","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":10,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/485\/revisions"}],"predecessor-version":[{"id":1150,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/485\/revisions\/1150"}],"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\/485\/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=485"}],"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=485"},{"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=485"},{"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=485"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}