{"id":492,"date":"2020-10-26T15:55:59","date_gmt":"2020-10-26T15:55:59","guid":{"rendered":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/?post_type=chapter&p=492"},"modified":"2021-09-17T19:46:23","modified_gmt":"2021-09-17T19:46:23","slug":"governing-equations-for-confined-transient-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-confined-transient-groundwater-flow\/","title":{"raw":"7.2 Governing Equations for Confined Transient Groundwater Flow","rendered":"7.2 Governing Equations for Confined Transient Groundwater Flow"},"content":{"raw":"The law of conservation of mass for flow through a saturated porous medium requires that the flux of fluid mass into the volume equals the flux of fluid mass out of the volume plus the change in mass stored within the volume. Darcy\u2019s Law is represented by the specific discharge, q<\/em>. The mass flux is \u03c1<\/em> q<\/em> (Equation 52).\r\n\r\n\r\n\r\n
<\/td>\r\nMass Flux<\/em> = \u03c1<\/em> q<\/em><\/td>\r\n(52)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nwhere:\r\n\r\n\r\n\r\n\r\n\r\n
Mass Flux<\/em><\/td>\r\n=<\/td>\r\nmass of water passing through a unit area per unit time (M\/(L2<\/sup>T))<\/td>\r\n<\/tr>\r\n
\u03c1<\/em><\/td>\r\n=<\/td>\r\ndensity of water (M\/L3<\/sup>)<\/td>\r\n<\/tr>\r\n
q<\/em><\/td>\r\n=<\/td>\r\nspecific discharge (L\/T)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

One-dimensional Flow<\/h1>\r\nGoverning equations describing groundwater flow are most often presented as representing steady state or transient conditions, and flow in two- or three-dimensional space. However, especially for readers who are not as familiar with differential equations, it is useful to consider one-dimensional flow (along the x<\/em> axis) through a REV first (Figure 52). The mass flux into the REV is the product of the specific discharge and the fluid density (item 1 of Figure 52). If the head is declining, then water is coming out of storage from the porous medium and the mass flux will increase across the REV as shown in item 2 of Figure 52. Multiplying the slope of the mass-flux-vs-distance graph by the distance across the REV (\u0394x<\/em>) determines the increase in mass flux from left to right (item 3 of Figure 52). Adding that increase of flux across the element to the influx produces the outflux (item 4 of Figure 52). It is useful to recall that a flux has units of velocity (L\/T) because it is a flow rate L3<\/sup>\/T divided by the area of flow (L3<\/sup>\/T)\/(L2<\/sup>).\r\n\r\nThe fluxes in and out of the element in Figure 52 are shown in Equations 53 and 54 respectively.\r\n\r\n\r\n\r\n
<\/td>\r\nMass Flux In<\/em> = [latex]\\displaystyle \\rho q_{x}[\/latex]<\/td>\r\n(53)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n
<\/td>\r\nMass Flux Out<\/em> = [latex]\\displaystyle \\rho q_{x}+\\frac{\\Delta (\\rho q_{x})}{\\Delta x}\\Delta x[\/latex]<\/td>\r\n(54)<\/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
Mass Flux<\/em><\/td>\r\n=<\/td>\r\nmass of water passing through a unit area per unit time (M\/ (L2<\/sup>T))<\/td>\r\n<\/tr>\r\n
\u03c1<\/em><\/td>\r\n=<\/td>\r\ndensity of water (M\/L3<\/sup>)<\/td>\r\n<\/tr>\r\n
q<\/em>x<\/em><\/sub><\/td>\r\n=<\/td>\r\nspecific discharge (L\/T)<\/td>\r\n<\/tr>\r\n
\u0394x<\/em><\/td>\r\n=<\/td>\r\nlength of the elementary volume in the x<\/em> direction (L)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[caption id=\"attachment_493\" align=\"alignnone\" width=\"1024\"]\"Figure Figure 52 -<\/strong> Change in mass flux through a representative elementary volume of a porous material with one dimensional flow.[\/caption]\r\n\r\nMass flow rate is obtained by multiplying the fluxes by the cross-sectional area of the elementary volume that they pass through (for one-dimensional flow in the x<\/em> direction, the area is \u0394y<\/em>\u0394z<\/em>), as in Equations 55 and 56.\r\n\r\n\r\n\r\n
<\/td>\r\nMass Flow In<\/em> = [latex]\\displaystyle (\\rho q_{x})\\Delta y\\Delta z[\/latex]<\/td>\r\n(55)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n
<\/td>\r\nMass Flow Out<\/em> = [latex]\\displaystyle \\left ( \\rho q_{x}+\\frac{\\Delta (\\rho q_{x})}{\\Delta x} \\Delta x\\right )\\Delta y\\Delta z[\/latex]<\/td>\r\n(56)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nwhere:\r\n\r\n\r\n\r\n\r\n
Mass Flow<\/em><\/td>\r\n=<\/td>\r\nmass of water passing into, or out of, an REV per unit time (M\/T)<\/td>\r\n<\/tr>\r\n
\u0394y<\/em>, \u0394z<\/em><\/td>\r\n=<\/td>\r\nlength of the elementary volume in the y<\/em> and z<\/em> directions (L)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNext, the change in the inflow is examined using and analysis of the inward flow of mass. The inward flow of mass is defined as mass inflow minus outflow, a positive value of inward flow of mass indicates that the inflow exceeds the outflow and water is going into storage, while a negative value indicates outflow exceeds inflow and water is coming out of storage. The inward flow of mass is determined by subtracting the outflow (Equation 56) from the inflow (Equation 55) as shown in Equation 57.\r\n\r\n\r\n\r\n
<\/td>\r\nInward Flow of Mass<\/em> = [latex]\\displaystyle (\\rho q_{x})\\Delta y\\Delta z- \\left ( \\rho q_{x}+\\frac{\\Delta (\\rho q_{x})}{\\Delta x} \\Delta x\\right )\\Delta y\\Delta z[\/latex]<\/td>\r\n(57)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nwhere:\r\n\r\n\r\n\r\n
Inward flow of Mass<\/em><\/td>\r\n=<\/td>\r\nmass of water flowing into the elementary volume per unit time (M\/T)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nExpanding the second term of Equation 57 and subtracting yields Equation 58. The negative sign indicates that water is coming out of groundwater storage, thus inward flow of mass is negative.\r\n\r\n\r\n\r\n
<\/td>\r\nInward Flow of Mass<\/em> = [latex]\\displaystyle -\\frac{\\Delta (\\rho q_{x})}{\\Delta x}\\Delta x\\Delta y\\Delta z[\/latex]<\/td>\r\n(58)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nGiven that the inward flow of mass is the increase in mass in the element per unit time and mass must be conserved, then the inward mass flow per unit time must equal the change in mass storage per unit time. Recall that specific storage is the change in volume of water stored in a unit volume of aquifer for a change in head, so the change in volume with time is shown as Equation 59.\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\frac{\\Delta V}{\\Delta t}=\\rho g(\\alpha +n\\beta )\\Delta x\\Delta y\\Delta z\\frac{\\Delta h}{\\Delta t}[\/latex]<\/td>\r\n(59)<\/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\r\n\r\n
\u0394V<\/em><\/td>\r\n=<\/td>\r\nchange in volume of water in the REV (L3<\/sup>)<\/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
g<\/em><\/td>\r\n=<\/td>\r\ngravitational constant (acceleration of gravity) (L\/T2<\/sup>)<\/td>\r\n<\/tr>\r\n
\u03b1<\/em><\/td>\r\n=<\/td>\r\ncompressibility of the aquifer solid structure (L3<\/sup>\/L3<\/sup>)\/(F\/L2<\/sup>), inverse pressure)<\/td>\r\n<\/tr>\r\n
\u03b2<\/em><\/td>\r\n=<\/td>\r\ncompressibility of water (L3<\/sup>\/L3<\/sup>)\/(F\/L2<\/sup>), inverse pressure<\/td>\r\n<\/tr>\r\n
n<\/em><\/td>\r\n=<\/td>\r\nfully connected total porosity (n<\/em>e<\/em><\/sub>) of the REV (dimensionless)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe change in stored mass for a unit of time is obtained by the product of Equation 59 and the water density (Equation 60).\r\n\r\n\r\n\r\n
[latex]\\displaystyle \\frac{\\Delta M}{\\Delta t}=\\rho \\left ( \\rho g(\\alpha +n\\beta )\\Delta x\\Delta y\\Delta z\\frac{\\Delta h}{\\Delta t}\\right )[\/latex]<\/td>\r\n(60)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nwhere:\r\n\r\n\r\n\r\n
\u0394M<\/em><\/td>\r\n=<\/td>\r\nchange in mass of water in the REV (M)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nEquating the change in mass storage (Equation 60) with the inward flow of mass (Equation 58) produces Equation 61.\r\n\r\n\r\n\r\n
[latex]\\displaystyle \\rho \\left ( \\rho g(\\alpha +n\\beta )\\Delta x\\Delta y\\Delta z\\frac{\\Delta h}{\\Delta t}\\right )=-\\frac{\\Delta (\\rho q_{x})}{\\Delta x}\\Delta x\\Delta y\\Delta z[\/latex]<\/td>\r\n(61)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAlthough it is recognized that the water density changes slightly in response to head changes (i.e., pressure changes), the amount of compression or expansion is small enough to assume constant density for the vast majority of applications so, \u03c1<\/em>, can be taken out of the delta-term due to its minimal change with time (Equation 62).\r\n\r\n\r\n\r\n
[latex]\\displaystyle \\rho \\left ( \\rho g(\\alpha +n\\beta )\\Delta x\\Delta y\\Delta z\\frac{\\Delta h}{\\Delta t}\\right )=-\\rho \\frac{\\Delta q_{x}}{\\Delta x}\\Delta x\\Delta y\\Delta z[\/latex]<\/td>\r\n(62)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nDividing both sides by \u03c1<\/em> and by \u0394x<\/em> \u0394y<\/em> \u0394z<\/em> , then substituting the specific storage, S<\/em>s<\/em><\/sub>, for \u03c1g<\/em>(\u03b1<\/em> + n\u03b2<\/em>) provides Equation 63.\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle S_{s}\\frac{\\Delta h}{\\Delta t}=-\\frac{\\Delta q_{x}}{\\Delta x}[\/latex]<\/td>\r\n(63)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe specific discharge can be represented by Darcy\u2019s Law, where q<\/em>x<\/em><\/sub> = \u2013 K<\/em>x<\/em><\/sub>\u0394h<\/em>\/\u0394x<\/em>, assuming the principal direction of the component of hydraulic conductivity is aligned with the x-axis, as represented in Equation 64.\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle S_{s}\\frac{\\Delta h}{\\Delta t}=-\\frac{\\Delta }{\\Delta x}\\left ( -K_{x}\\frac{\\Delta h}{\\Delta x} \\right )[\/latex]<\/td>\r\n(64)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe \u0394\u2019s in the equation describe discrete changes across a small elementary volume. This allows translation of this relationship into a differential form when the discrete change is infinitesimal by replacing \u0394 with d<\/em>, providing the derivative of a smooth function (Equation 65).\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle S_{s}\\frac{dh}{dt}=-\\frac{d}{dx}\\left ( -K_{x}\\frac{dh}{dx} \\right )[\/latex]<\/td>\r\n(65)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nEquation 65 is the governing equation for confined<\/em>, one-dimensional<\/em>, transient<\/em>, heterogeneous<\/em> conditions of groundwater flow. Equation 65 can also be written as Equation 66.\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle S_{s}\\frac{dh}{dt}=\\frac{d}{dx}K_{x}\\frac{dh}{dx}[\/latex]<\/td>\r\n(66)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Three-dimensional Flow<\/h1>\r\nIn most cases, the flux will not be one-dimensional along the x<\/em>-axis. It will occur in an arbitrary direction with a component in each of the x<\/em>, y<\/em>, and z<\/em> directions (see for example Section 5.4), thus the change in mass flux across the element needs to be accounted for in each of the three-dimensions as shown in Figure 53. The partial differential symbol (\u2202\/\u2202x<\/em>) is used in order to represent that only a part of the change in flux across the element occurs in the x<\/em> direction, as there are also changes in the y<\/em>- and z<\/em>-directions (\u2202\/\u2202y<\/em>, \u2202\/\u2202z<\/em>); and when the system is transient, with time, \u2202\/\u2202t<\/em>. The groundwater flow equation for three-dimensional flow is the same as the equation for one-dimensional flow with additional flux terms for the y<\/em>- and z<\/em>-directions.\r\n\r\n[caption id=\"attachment_504\" align=\"alignnone\" width=\"1024\"]\"Representative Figure 53 -<\/strong> Representative Elementary Volume showing three-dimensional mass influx and outflux (e.g., \u03c1q<\/em>x<\/em><\/sub>) and an addition or loss of mass flux in the outflow (e.g., + (\u2202\u03c1q<\/em>x<\/em><\/sub>\/\u2202x) \u0394x<\/em>). This schematic represents confined conditions.[\/caption]\r\n\r\nConfined, three-dimensional, transient, anisotropic, heterogeneous<\/em> conditions of groundwater flow are represented by Equation 67.<\/a>\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(67)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhen the values of K<\/em>x<\/em><\/sub>, K<\/em>y<\/em><\/sub> and K<\/em>z<\/em><\/sub> are constants (but not the same value) they can be taken out of the derivative resulting in Equation 68. It describes confined, three-dimensional, transient, anisotropic, homogeneous<\/em> conditions of groundwater flow.\r\n\r\n\r\n\r\n
[latex]\\displaystyle S_s\\frac{\\partial h}{\\partial t}\\ =\\ K_x\\frac{\\partial^2h}{\\partial x^2}+K_y\\frac{\\partial^2h}{\\partial y^2}+K_z\\frac{\\partial^2h}{\\partial z^2}[\/latex]<\/td>\r\n(68)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFor confined, three-dimensional, transient, isotropic and homogeneous<\/em> conditions of groundwater flow, K<\/em>x<\/em><\/sub> = K<\/em>y<\/em><\/sub> = K<\/em>z<\/em><\/sub> = K<\/em>, thus one value of K<\/em> is sufficient to represent the hydraulic conductivity and Equation 69 is formulated.\r\n\r\n\r\n\r\n
[latex]\\displaystyle S_s\\frac{\\partial h}{\\partial t}\\ =\\ K\\left(\\frac{\\partial^2h}{\\partial x^2}+\\frac{\\partial^2h}{\\partial y^2}+\\frac{\\partial^2h}{\\partial z^2}\\right)[\/latex]<\/td>\r\n(69)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFor confined, two-dimensional (plan view), transient, isotropic, homogeneous<\/em> conditions in an aquifer of constant thickness, the z-terms for vertical flow can be omitted from Equation 69. Also, saturated thickness, b<\/em>, is not dependent on head, h<\/em>, (Figure 54) and assuming the aquifer thickness is constant, both sides of Equation 69 can be multiplied by the aquifer thickness leading to Equation 70 that represents horizontal flow in a map view.\r\n\r\n[caption id=\"attachment_505\" align=\"alignnone\" width=\"1024\"]\"Figure Figure 54 -<\/strong> Saturated thickness of confined and unconfined aquifers. a) For confined flow, the saturated thickness remains constant and is independent of hydraulic head, so the flow equations can use b<\/em> for thickness. b) For unconfined flow conditions, saturated thickness depends on head as the water table surface slopes, so thickness is equal to head if the bottom of the aquifer is used as the datum for head measurements.[\/caption]\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle S_sb\\frac{\\partial h}{\\partial t}\\ =\\ Kb\\ \\left(\\frac{\\partial^2h}{\\partial x}+\\frac{\\partial^2h}{\\partial y}\\right)[\/latex]<\/td>\r\n(70)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThen, by using the definition of transmissivity, T<\/em>, as Kb<\/em> and storativity, S<\/em>, as S<\/em>s<\/em><\/sub>b<\/em>, Equation 70 can be written as Equation 71.\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle S\\frac{\\partial h}{\\partial t}\\ =\\ T\\ \\left(\\frac{\\partial^2h}{\\partial x}+\\frac{\\partial^2h}{\\partial y}\\right)[\/latex]<\/td>\r\n(71)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nEquation 70 and Equation 71 represent groundwater flow under confined, two-dimensional (plan view), transient, homogeneous and isotropic<\/em> conditions using S and T<\/em>.","rendered":"

The law of conservation of mass for flow through a saturated porous medium requires that the flux of fluid mass into the volume equals the flux of fluid mass out of the volume plus the change in mass stored within the volume. Darcy\u2019s Law is represented by the specific discharge, q<\/em>. The mass flux is \u03c1<\/em> q<\/em> (Equation 52).<\/p>\n\n\n\n
<\/td>\nMass Flux<\/em> = \u03c1<\/em> q<\/em><\/td>\n(52)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n\n\n
Mass Flux<\/em><\/td>\n=<\/td>\nmass of water passing through a unit area per unit time (M\/(L2<\/sup>T))<\/td>\n<\/tr>\n
\u03c1<\/em><\/td>\n=<\/td>\ndensity of water (M\/L3<\/sup>)<\/td>\n<\/tr>\n
q<\/em><\/td>\n=<\/td>\nspecific discharge (L\/T)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

One-dimensional Flow<\/h1>\n

Governing equations describing groundwater flow are most often presented as representing steady state or transient conditions, and flow in two- or three-dimensional space. However, especially for readers who are not as familiar with differential equations, it is useful to consider one-dimensional flow (along the x<\/em> axis) through a REV first (Figure 52). The mass flux into the REV is the product of the specific discharge and the fluid density (item 1 of Figure 52). If the head is declining, then water is coming out of storage from the porous medium and the mass flux will increase across the REV as shown in item 2 of Figure 52. Multiplying the slope of the mass-flux-vs-distance graph by the distance across the REV (\u0394x<\/em>) determines the increase in mass flux from left to right (item 3 of Figure 52). Adding that increase of flux across the element to the influx produces the outflux (item 4 of Figure 52). It is useful to recall that a flux has units of velocity (L\/T) because it is a flow rate L3<\/sup>\/T divided by the area of flow (L3<\/sup>\/T)\/(L2<\/sup>).<\/p>\n

The fluxes in and out of the element in Figure 52 are shown in Equations 53 and 54 respectively.<\/p>\n\n\n\n
<\/td>\nMass Flux In<\/em> = \"\displaystyle \rho q_{x}\"<\/td>\n(53)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n
<\/td>\nMass Flux Out<\/em> = \"\displaystyle \rho q_{x}+\frac{\Delta (\rho q_{x})}{\Delta x}\Delta x\"<\/td>\n(54)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n\n\n\n
Mass Flux<\/em><\/td>\n=<\/td>\nmass of water passing through a unit area per unit time (M\/ (L2<\/sup>T))<\/td>\n<\/tr>\n
\u03c1<\/em><\/td>\n=<\/td>\ndensity of water (M\/L3<\/sup>)<\/td>\n<\/tr>\n
q<\/em>x<\/em><\/sub><\/td>\n=<\/td>\nspecific discharge (L\/T)<\/td>\n<\/tr>\n
\u0394x<\/em><\/td>\n=<\/td>\nlength of the elementary volume in the x<\/em> direction (L)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\"Figure
Figure 52 –<\/strong> Change in mass flux through a representative elementary volume of a porous material with one dimensional flow.<\/figcaption><\/figure>\n

Mass flow rate is obtained by multiplying the fluxes by the cross-sectional area of the elementary volume that they pass through (for one-dimensional flow in the x<\/em> direction, the area is \u0394y<\/em>\u0394z<\/em>), as in Equations 55 and 56.<\/p>\n\n\n\n
<\/td>\nMass Flow In<\/em> = \"\displaystyle (\rho q_{x})\Delta y\Delta z\"<\/td>\n(55)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n
<\/td>\nMass Flow Out<\/em> = \"\displaystyle \left ( \rho q_{x}+\frac{\Delta (\rho q_{x})}{\Delta x} \Delta x\right )\Delta y\Delta z\"<\/td>\n(56)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n\n
Mass Flow<\/em><\/td>\n=<\/td>\nmass of water passing into, or out of, an REV per unit time (M\/T)<\/td>\n<\/tr>\n
\u0394y<\/em>, \u0394z<\/em><\/td>\n=<\/td>\nlength of the elementary volume in the y<\/em> and z<\/em> directions (L)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Next, the change in the inflow is examined using and analysis of the inward flow of mass. The inward flow of mass is defined as mass inflow minus outflow, a positive value of inward flow of mass indicates that the inflow exceeds the outflow and water is going into storage, while a negative value indicates outflow exceeds inflow and water is coming out of storage. The inward flow of mass is determined by subtracting the outflow (Equation 56) from the inflow (Equation 55) as shown in Equation 57.<\/p>\n\n\n\n
<\/td>\nInward Flow of Mass<\/em> = \"\displaystyle (\rho q_{x})\Delta y\Delta z- \left ( \rho q_{x}+\frac{\Delta (\rho q_{x})}{\Delta x} \Delta x\right )\Delta y\Delta z\"<\/td>\n(57)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n
Inward flow of Mass<\/em><\/td>\n=<\/td>\nmass of water flowing into the elementary volume per unit time (M\/T)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Expanding the second term of Equation 57 and subtracting yields Equation 58. The negative sign indicates that water is coming out of groundwater storage, thus inward flow of mass is negative.<\/p>\n\n\n\n
<\/td>\nInward Flow of Mass<\/em> = \"\displaystyle -\frac{\Delta (\rho q_{x})}{\Delta x}\Delta x\Delta y\Delta z\"<\/td>\n(58)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Given that the inward flow of mass is the increase in mass in the element per unit time and mass must be conserved, then the inward mass flow per unit time must equal the change in mass storage per unit time. Recall that specific storage is the change in volume of water stored in a unit volume of aquifer for a change in head, so the change in volume with time is shown as Equation 59.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle \frac{\Delta V}{\Delta t}=\rho g(\alpha +n\beta )\Delta x\Delta y\Delta z\frac{\Delta h}{\Delta t}\"<\/td>\n(59)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n\n\n\n\n\n
\u0394V<\/em><\/td>\n=<\/td>\nchange in volume of water in the REV (L3<\/sup>)<\/td>\n<\/tr>\n
\u03c1<\/em><\/td>\n=<\/td>\ndensity of water in the REV (M\/L3<\/sup>)<\/td>\n<\/tr>\n
g<\/em><\/td>\n=<\/td>\ngravitational constant (acceleration of gravity) (L\/T2<\/sup>)<\/td>\n<\/tr>\n
\u03b1<\/em><\/td>\n=<\/td>\ncompressibility of the aquifer solid structure (L3<\/sup>\/L3<\/sup>)\/(F\/L2<\/sup>), inverse pressure)<\/td>\n<\/tr>\n
\u03b2<\/em><\/td>\n=<\/td>\ncompressibility of water (L3<\/sup>\/L3<\/sup>)\/(F\/L2<\/sup>), inverse pressure<\/td>\n<\/tr>\n
n<\/em><\/td>\n=<\/td>\nfully connected total porosity (n<\/em>e<\/em><\/sub>) of the REV (dimensionless)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The change in stored mass for a unit of time is obtained by the product of Equation 59 and the water density (Equation 60).<\/p>\n\n\n\n
\"\displaystyle \frac{\Delta M}{\Delta t}=\rho \left ( \rho g(\alpha +n\beta )\Delta x\Delta y\Delta z\frac{\Delta h}{\Delta t}\right )\"<\/td>\n(60)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

where:<\/p>\n\n\n\n
\u0394M<\/em><\/td>\n=<\/td>\nchange in mass of water in the REV (M)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Equating the change in mass storage (Equation 60) with the inward flow of mass (Equation 58) produces Equation 61.<\/p>\n\n\n\n
\"\displaystyle \rho \left ( \rho g(\alpha +n\beta )\Delta x\Delta y\Delta z\frac{\Delta h}{\Delta t}\right )=-\frac{\Delta (\rho q_{x})}{\Delta x}\Delta x\Delta y\Delta z\"<\/td>\n(61)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Although it is recognized that the water density changes slightly in response to head changes (i.e., pressure changes), the amount of compression or expansion is small enough to assume constant density for the vast majority of applications so, \u03c1<\/em>, can be taken out of the delta-term due to its minimal change with time (Equation 62).<\/p>\n\n\n\n
\"\displaystyle \rho \left ( \rho g(\alpha +n\beta )\Delta x\Delta y\Delta z\frac{\Delta h}{\Delta t}\right )=-\rho \frac{\Delta q_{x}}{\Delta x}\Delta x\Delta y\Delta z\"<\/td>\n(62)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Dividing both sides by \u03c1<\/em> and by \u0394x<\/em> \u0394y<\/em> \u0394z<\/em> , then substituting the specific storage, S<\/em>s<\/em><\/sub>, for \u03c1g<\/em>(\u03b1<\/em> + n\u03b2<\/em>) provides Equation 63.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle S_{s}\frac{\Delta h}{\Delta t}=-\frac{\Delta q_{x}}{\Delta x}\"<\/td>\n(63)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The specific discharge can be represented by Darcy\u2019s Law, where q<\/em>x<\/em><\/sub> = \u2013 K<\/em>x<\/em><\/sub>\u0394h<\/em>\/\u0394x<\/em>, assuming the principal direction of the component of hydraulic conductivity is aligned with the x-axis, as represented in Equation 64.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle S_{s}\frac{\Delta h}{\Delta t}=-\frac{\Delta }{\Delta x}\left ( -K_{x}\frac{\Delta h}{\Delta x} \right )\"<\/td>\n(64)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The \u0394\u2019s in the equation describe discrete changes across a small elementary volume. This allows translation of this relationship into a differential form when the discrete change is infinitesimal by replacing \u0394 with d<\/em>, providing the derivative of a smooth function (Equation 65).<\/p>\n\n\n\n
<\/td>\n\"\displaystyle S_{s}\frac{dh}{dt}=-\frac{d}{dx}\left ( -K_{x}\frac{dh}{dx} \right )\"<\/td>\n(65)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Equation 65 is the governing equation for confined<\/em>, one-dimensional<\/em>, transient<\/em>, heterogeneous<\/em> conditions of groundwater flow. Equation 65 can also be written as Equation 66.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle S_{s}\frac{dh}{dt}=\frac{d}{dx}K_{x}\frac{dh}{dx}\"<\/td>\n(66)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Three-dimensional Flow<\/h1>\n

In most cases, the flux will not be one-dimensional along the x<\/em>-axis. It will occur in an arbitrary direction with a component in each of the x<\/em>, y<\/em>, and z<\/em> directions (see for example Section 5.4), thus the change in mass flux across the element needs to be accounted for in each of the three-dimensions as shown in Figure 53. The partial differential symbol (\u2202\/\u2202x<\/em>) is used in order to represent that only a part of the change in flux across the element occurs in the x<\/em> direction, as there are also changes in the y<\/em>– and z<\/em>-directions (\u2202\/\u2202y<\/em>, \u2202\/\u2202z<\/em>); and when the system is transient, with time, \u2202\/\u2202t<\/em>. The groundwater flow equation for three-dimensional flow is the same as the equation for one-dimensional flow with additional flux terms for the y<\/em>– and z<\/em>-directions.<\/p>\n

\"Representative
Figure 53 –<\/strong> Representative Elementary Volume showing three-dimensional mass influx and outflux (e.g., \u03c1q<\/em>x<\/em><\/sub>) and an addition or loss of mass flux in the outflow (e.g., + (\u2202\u03c1q<\/em>x<\/em><\/sub>\/\u2202x) \u0394x<\/em>). This schematic represents confined conditions.<\/figcaption><\/figure>\n

Confined, three-dimensional, transient, anisotropic, heterogeneous<\/em> conditions of groundwater flow are represented by Equation 67.<\/a><\/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(67)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

When the values of K<\/em>x<\/em><\/sub>, K<\/em>y<\/em><\/sub> and K<\/em>z<\/em><\/sub> are constants (but not the same value) they can be taken out of the derivative resulting in Equation 68. It describes confined, three-dimensional, transient, anisotropic, homogeneous<\/em> conditions of groundwater flow.<\/p>\n\n\n\n
\"\displaystyle S_s\frac{\partial h}{\partial t}\ =\ K_x\frac{\partial^2h}{\partial x^2}+K_y\frac{\partial^2h}{\partial y^2}+K_z\frac{\partial^2h}{\partial z^2}\"<\/td>\n(68)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

For confined, three-dimensional, transient, isotropic and homogeneous<\/em> conditions of groundwater flow, K<\/em>x<\/em><\/sub> = K<\/em>y<\/em><\/sub> = K<\/em>z<\/em><\/sub> = K<\/em>, thus one value of K<\/em> is sufficient to represent the hydraulic conductivity and Equation 69 is formulated.<\/p>\n\n\n\n
\"\displaystyle S_s\frac{\partial h}{\partial t}\ =\ K\left(\frac{\partial^2h}{\partial x^2}+\frac{\partial^2h}{\partial y^2}+\frac{\partial^2h}{\partial z^2}\right)\"<\/td>\n(69)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

For confined, two-dimensional (plan view), transient, isotropic, homogeneous<\/em> conditions in an aquifer of constant thickness, the z-terms for vertical flow can be omitted from Equation 69. Also, saturated thickness, b<\/em>, is not dependent on head, h<\/em>, (Figure 54) and assuming the aquifer thickness is constant, both sides of Equation 69 can be multiplied by the aquifer thickness leading to Equation 70 that represents horizontal flow in a map view.<\/p>\n

\"Figure
Figure 54 –<\/strong> Saturated thickness of confined and unconfined aquifers. a) For confined flow, the saturated thickness remains constant and is independent of hydraulic head, so the flow equations can use b<\/em> for thickness. b) For unconfined flow conditions, saturated thickness depends on head as the water table surface slopes, so thickness is equal to head if the bottom of the aquifer is used as the datum for head measurements.<\/figcaption><\/figure>\n\n\n\n
<\/td>\n\"\displaystyle S_sb\frac{\partial h}{\partial t}\ =\ Kb\ \left(\frac{\partial^2h}{\partial x}+\frac{\partial^2h}{\partial y}\right)\"<\/td>\n(70)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Then, by using the definition of transmissivity, T<\/em>, as Kb<\/em> and storativity, S<\/em>, as S<\/em>s<\/em><\/sub>b<\/em>, Equation 70 can be written as Equation 71.<\/p>\n\n\n\n
<\/td>\n\"\displaystyle S\frac{\partial h}{\partial t}\ =\ T\ \left(\frac{\partial^2h}{\partial x}+\frac{\partial^2h}{\partial y}\right)\"<\/td>\n(71)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Equation 70 and Equation 71 represent groundwater flow under confined, two-dimensional (plan view), transient, homogeneous and isotropic<\/em> conditions using S and T<\/em>.<\/p>\n","protected":false},"author":1,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-492","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\/492","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":35,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/492\/revisions"}],"predecessor-version":[{"id":1212,"href":"https:\/\/books.gw-project.org\/hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow\/wp-json\/pressbooks\/v2\/chapters\/492\/revisions\/1212"}],"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\/492\/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=492"}],"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=492"},{"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=492"},{"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=492"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}