6.1 Introduction
Let’s review our previous discussion of the groundwater flow equations. We have the main equation that governs three‑dimensional flow of water in a porous medium, assuming that flow is within the limits of Darcy’s law—that is, not so high that flow becomes turbulent, and not so low that it is below the gradient threshold, as expressed in Equation (43).
| [latex]\frac{{\partial q}_{x}}{\partial x} + \frac{{\partial q}_{y}}{\partial y} + \frac{{\partial q}_{z}}{\partial z} = S_{s}\frac{\partial h}{\partial t}[/latex] | (43) |
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where (parameter dimensions are dark green font with mass as M, length as L, time as T):
| [latex]q[/latex] | = | specific discharge (i.e., groundwater flow) in the [latex]x,\ y,\ or\ z[/latex]direction (LT‑1) |
|---|---|---|
| [latex]x,\ y,\ z[/latex] | = | lengths in the principal directions in our coordinate systems |
| [latex]h[/latex] | = | hydraulic head (L) |
| [latex]t[/latex] | = | time (T) |
| [latex]S_{s}[/latex] | = | specific storage (L‑1) |
| [latex]q = - K\frac{dh}{dl}[/latex] | (44) |
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where (parameter dimensions are dark green font with mass as M, length as L, time as T):
| [latex]K[/latex] | = | hydraulic conductivity (LT-1) | ||
|---|---|---|---|---|
| [latex]\frac{dh}{dl}[/latex] | = | hydraulic gradient, or change in head with length ([latex]:[/latex]) | ||
| [latex]\frac{\partial\left( K_{x}\frac{\partial h}{\partial x} \right)}{\partial x} + \frac{\partial\left( K_{y}\frac{\partial h}{\partial y} \right)}{\partial y} + \frac{\partial\left( K_{z}\frac{\partial h}{\partial z} \right)}{\partial z} = S_{s}\frac{\partial h}{\partial t}[/latex] | (45) | |||
This equation is another way of expressing the continuity equation, or [latex]I - O[/latex] (what goes in minus what goes out)[latex]\ =[/latex] [latex]\mathrm{\Delta}Storage[/latex] (change in storage in the system).
In Section 4, we established that the left side of the equation describes the divergence of flux at any given point in the aquifer, which is a fancy way of saying [latex]I - O[/latex] along each of the principal directions in our [latex]x,\ y,\ z[/latex] coordinate system. Then, in Section 5, we put forth the idea that the right side of the equation describes changes in storage in the system, and we discussed the specific storage and other related storage parameters. The storage parameters relate changes in head to changes in the volume of water stored in the aquifer. More specifically, the specific storage describes the volume of water that can be removed from (or added to) a unit volume of aquifer that will result in a unit decline (or rise) in hydraulic head.
The hydraulic head describes the total fluid energy at a specific point in an aquifer. This head changes from point to point, creating a tensor quantity called the hydraulic gradient, which reflects the second law of thermodynamics—systems spontaneously move from higher to lower energy—for example, groundwater flows from high head to low head. Therefore, the hydraulic gradient describes how head changes with respect to location in the aquifer. But heads in natural aquifers do not only change from one point to the next; they also change through time.
This brings us to the final part of the equation for our discussion; specifically, the partial derivative of head versus time, or [latex]\frac{\partial h}{\partial t}[/latex], on the right side of the equation. This term describes changes in head ([latex]h[/latex]) at a point in the aquifer with respect to time ([latex]t[/latex]) and combines with the specific storage to describe volume changes—that is, changes in the volume of water stored in the aquifer—with time. In this section, we discuss time as it relates to groundwater, and then discuss some applications of the flow equations.