{"id":65,"date":"2022-12-30T18:15:16","date_gmt":"2022-12-30T18:15:16","guid":{"rendered":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/chapter\/governing-equations\/"},"modified":"2023-01-09T16:46:11","modified_gmt":"2023-01-09T16:46:11","slug":"governing-equations","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/variable-density-groundwater-flow\/chapter\/governing-equations\/","title":{"raw":"6.1 Governing Equations","rendered":"6.1 Governing Equations"},"content":{"raw":"
\r\n

For variable-density flow, the mass conservation equations for fluid (groundwater), mass and heat are the basis for numerical models. The mass balance equation for a compressible fluid in a saturated porous medium is shown in Equation\u00a032.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\frac{\\partial (n\\rho )}{\\partial t}=-\\nabla \\cdot (\\rho \\vec{q})+\\rho _{ss}q_{ss}[\/latex]<\/td>\r\n(32)<\/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
t<\/em><\/td>\r\n=<\/td>\r\ntime (T), e.g., s<\/td>\r\n<\/tr>\r\n
[latex]\\vec{q}[\/latex]<\/td>\r\n=<\/td>\r\nspecific discharge as given by Equation\u00a02<\/a> (L\/T), e.g., m\u00a0s\u22121<\/sup><\/td>\r\n<\/tr>\r\n
\u03c1<\/em>ss<\/em><\/sub><\/td>\r\n=<\/td>\r\ndensity of water source, or sink (M\/L3<\/sup>), e.g., kg\u00a0m\u22123<\/sup><\/td>\r\n<\/tr>\r\n
q<\/em>ss<\/em><\/sub><\/td>\r\n=<\/td>\r\ndischarge rate per unit of volume of water source or sink (1\/T), e.g., s\u22121<\/sup><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The term on the left of the equal sign represents the change in fluid mass per unit of time. The first term to the right of the equal sign represents the change in fluid mass due to differences between inflow and outflow, while the second term is the change in fluid mass due to a source or sink. In groundwater, this can be recharge, inflow from an adjacent aquifer or an extraction\/injection well, for example.<\/p>\r\n

The mass balance equation for a solute that instantaneously and linearly partitions between a dissolved phase at concentration C<\/em> in the pores and an adsorbed phase on the rock surface is Equation\u00a033.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\left ( 1+\\frac{\\rho _{b}K_{d}}{n} \\right )\\frac{\\partial (nC)}{\\partial t}=-\\nabla\\cdot (\\vec{q}C)+\\nabla \\cdot (n\\overline{D_{h,C}}\\cdot \\nabla C)+q_{s}C_{s}[\/latex]<\/td>\r\n(33)<\/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
\u03c1<\/em>b<\/em><\/sub><\/td>\r\n=<\/td>\r\ndry bulk density of the rock (M\/L3<\/sup>), e.g., kg\u00a0m\u22123<\/sup><\/td>\r\n<\/tr>\r\n
K<\/em>d<\/em><\/sub><\/td>\r\n=<\/td>\r\ndistribution coefficient (L3<\/sup>\/M), e.g., m3<\/sup>\u00a0kg\u22121<\/sup>, which is the ratio of adsorbed mass and solute mass<\/td>\r\n<\/tr>\r\n
[latex]\\overline{D_{h,C}}[\/latex]<\/td>\r\n=<\/td>\r\nsolute dispersion tensor (L2<\/sup>\/T), e.g., m2<\/sup>s\u22121<\/sup>, which includes the effect of both molecular diffusion and mechanical dispersion<\/td>\r\n<\/tr>\r\n
C<\/em>s<\/em><\/sub><\/td>\r\n=<\/td>\r\nconcentration associated with a source (M\/L3<\/sup>), e.g., kg\u00a0m\u22123<\/sup><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The heat balance equivalent for Equation\u00a033 for temperature being at local instantaneous equilibrium between the pore water and rock matrix is Equation\u00a034.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle \\left ( 1+\\frac{1-n}{n}\\frac{\\rho _{r}c_{s}}{\\rho c} \\right )\\frac{\\partial (nT)}{\\partial t}=-\\nabla \\cdot (\\vec{q}T)+\\nabla \\cdot (\\overline{D_{h,T}}\\cdot \\nabla T)+q_{s}T_{s}[\/latex]<\/td>\r\n(34)<\/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
\u03c1<\/em>r<\/em><\/sub><\/td>\r\n=<\/td>\r\ndensity of the rock (M\/L3<\/sup>), e.g., kg\u00a0m\u22123<\/sup><\/td>\r\n<\/tr>\r\n
c<\/em> and c<\/em>s<\/em><\/sub><\/td>\r\n=<\/td>\r\nspecific heat capacities of the groundwater and rock, respectively (ML2<\/sup>\/(T2<\/sup>\u0398)), e.g., J K\u22121<\/sup>\u00a0kg\u22121<\/sup><\/td>\r\n<\/tr>\r\n
[latex]\\overline{D_{h,T}}[\/latex]<\/td>\r\n=<\/td>\r\ntensor which includes the combined effects of thermal conduction and mechanical dispersion (L2<\/sup>\/T), e.g., m2<\/sup>s\u22121<\/sup><\/td>\r\n<\/tr>\r\n
T<\/em>s<\/em><\/sub><\/td>\r\n=<\/td>\r\ntemperature associated with a source (\u0398), e.g., K<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The terms on the left of the equal sign in Equations\u00a033 and 34 represent the change in solute mass or energy, respectively, per unit of time. For both equations, the terms on the right-hand side of the equal sign represent the change due to:<\/p>\r\n\r\n