Groundwater continuity equation

If the groundwater and aquifer are assumed to be incompressible, the continuity equation for water flow is ... 

The three-dimensional flow of groundwater through the subsurface can be described by a combination of Darcy s equation for groundwater flow with a continuity equation (or mass balance equation) and equations of state for the groundwater and the porous medium. A detailed theoretical overview of equations for groundwater flow is given by e.g. Barenblatt et al. (1990), De Marsily (1986), Domenico and Schwartz (1990) and Freeze and Cherry (1979). 

By introducing the equations of state for the groundwater and the porous medium (Equations 1.18, and 1.21 and 1.23, respectively) into the continuity Equation 1.13 gives (e.g. Walton, 1970)... 

For steady-state flow of groundwater, a combination of the continuity Equation 1.11 with Darcy s equation 1.8 yields... 

The influence of the mechanical, thermal and chemical processes on the flow of groundwater can be introduced into the continuity equation for groundwater (Equation 1.12) assuming appropriate simplifying conditions. For example, if the assumptions listed in Section 1.2.2 are met, with the exception of the second assumption which is replaced by... 

Strictly speaking, Equation 1.38 is a continuity equation for groundwater flow through a certain representative elemental volume of porous medium fixed and rigid in space. For small elastic deformations of the porous medium, the equation can be considered to be valid provided that the specific discharge of groundwater is taken as relative to the rock grains (Cooper, 1966 and e.g. Neuzil, 1986 Sharp, 1983). 

For example, the general continuity equation for groundwater (Equation 1.12 5AM/5t = 5np 5t) written in terms of material derivatives in a deforming coordinate system following the motion of the solids, becomes (e.g. Palciauskas and Domenico, 1989 Shi and Wang, 1986)... 

The basis for aU computer models that link groundwater flow to strain through a continuity equation similar to (4) is Gauss theorem. Gauss theorem features a purely mathematical truth. It states (Kreyszig, 1983) that for any vector, r... 

The classic equation for transient groundwater flow (Jacob, 1950) stems from the continuity equation. In turn, the roots of any continuity equation grow from within the divergence theorem of Gauss. 

Let us clarify the above two statements. MOD-FLOW is an example of an internationally respected computer code for three-dimensional groundwater flow. Like other models, it is based on Jacob s development of the continuity equation. As used in MODFLOW, McDonald and Harbaugh s (1988) algebraic formulation of the continuity equation is... 

The goal of subsidence modelers is to reach accurately and with computational speed the left-hand side of Gaussian equation (6), namely to evaluate the displacement field of the skeletal frame ,. Groundwater subsidence modelers have appropriately attempted to reach their goal by starting from continuity equation (11), or its algebraic counterpart (4). They must choose, however, which side of the river to trek on their way towards the left-hand side of equation (6). 

During migration and subsurface storage groundwater comes into contact with crustal rocks that continuously release helium from the decay of uranium and thorium. The basic assumption is made that the water acts as a sink for the helium evolved from the local rocks. The age of groundwater, t, is calculable from the equation given in section 14.2. 

Equation (8.2) can be shown to apply equivalently to either a continuous concentration field or the position probability density of a single particle undergoing Brownian motion [174], This equation is used to model transport processes in a wide range of natural phenomena from population distribution in ecology [146] to pollutant distribution in groundwater [30], One of the earliest (and still important) applications to transport within cells and tissues is to describe the transport of oxygen from microvessels to the sites of oxidative metabolism in cells. 

For unsteady-state groundwater flow, i.e. when the magnitude and direction of the flow change with time, the equation of continuity is... 

The extent of surface weathering of crystalline rocks or of sedimentary rocks such as shales or carbonates, and thus rock permeability (and yield to wells), decreases rapidly with depth. Also, rock weathering is deeper under valley bottoms than on ridges or hill slopes. This reflects the fact that the weathering, which is facilitated by joints, fractures, and faults, tends to create valley bottoms in the first place. Valley bottoms continue to concentrate runoff (R is then a positive term in the infiltration equation) and so remain the locus of deeper development of secondary rock porosity and permeability and thus of enhanced groundwater storing and transmitting capacity. 

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