6.2 Steady State versus Transient Groundwater Flow
Steady‑State Flow
Consider the continuity equation; otherwise known as the law of mass conservation, or the continuity principle:
[latex]I - O[/latex] (what goes in minus what goes out)
[latex]= \mathrm{\Delta}Storage[/latex](change in storage in the system) / [latex]\mathrm{\Delta}t[/latex](change in time).
This principle states that mass is conserved in a system, and any change of mass entering a system must be balanced by a corresponding change in either the mass leaving the system, or the mass stored in the system. In other words, if the mass going in equals the mass going out, then the mass in storage will not change, and [latex]\mathrm{\Delta}S/\mathrm{\Delta}t[/latex] will equal zero. In the case where [latex]I[/latex] is equal to [latex]O[/latex], and [latex]\mathrm{\Delta}Storage/\mathrm{\Delta}t[/latex] equals zero, we say that the system is at steady state. If the system is at steady state, heads at any given point in the aquifer do not change through time, the [latex]\partial h[/latex] on the right side of the equation equals zero, as shown by Equation (46) and Equation (47).
| [latex]\frac{{\partial q}_{x}}{\partial x} + \frac{{\partial q}_{y}}{\partial y} + \frac{{\partial q}_{z}}{\partial z} = 0[/latex] | (46) | |
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| [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} = 0[/latex] | (47) |
If we assume the aquifer is homogeneous—that is, the permeability at one point is the same as at any other point—then the hydraulic conductivity ([latex]K[/latex]) will not change with respect to location, and we can remove [latex]K_{x},\ K_{y},[/latex] and [latex]K_{z}[/latex] from inside the differentials as shown in Equation (48).
| [latex]K_{x}\frac{\partial\left( \frac{\partial h}{\partial x} \right)}{\partial x} + K_{y}\frac{\partial\left( \frac{\partial h}{\partial y} \right)}{\partial y} + K_{z}\frac{\partial\left( \frac{\partial h}{\partial z} \right)}{\partial z} = 0[/latex] | (48) |
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If we also assume that the permeability of our aquifer is isotropic, then the hydraulic conductivity ([latex]K[/latex]) at any point in the aquifer will be the same in all directions, and [latex]K_{x}\ = \ K_{y}\ = \ K_{z}\ = \ K[/latex]. In that case, we can factor out [latex]K[/latex] and rewrite it as shown by Equation (49).
| [latex]K\left( \frac{\partial\left( \frac{\partial h}{\partial x} \right)}{\partial x} + \frac{\partial\left( \frac{\partial h}{\partial y} \right)}{\partial y} + \frac{\partial\left( \frac{\partial h}{\partial z} \right)}{\partial z} \right) = 0[/latex] | (49) |
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Also, we can use a simpler notation for the derivatives and rewrite it as shown by Equation (50).
| [latex]K\left( \frac{\partial^{2}h}{{\partial x}^{2}} + \frac{\partial^{2}h}{{\partial y}^{2}} + \frac{\partial^{2}h}{{\partial z}^{2}} \right) = 0[/latex] | (50) |
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Each one of the derivatives in the parentheses is a 2nd order partial differential. This simply means that it isn’t only that head changes with distance, but the change in head with distance changes with distance (or, in other words, the gradient changes with distance). If we then divide both sides by [latex]K[/latex], the equation reduces to Equation (51).
| [latex]\frac{\partial^{2}h}{{\partial x}^{2}} + \frac{\partial^{2}h}{{\partial y}^{2}} + \frac{\partial^{2}h}{{\partial z}^{2}} = 0[/latex] | (51) |
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Equation (51) is called the Laplace equation. This equation describes the steady‑state flow of groundwater (i.e., no changes in storage) in a homogeneous and isotropic aquifer. As we can see, the assumptions of homogeneity and isotropy greatly simplify the equation.
Transient Flow
Steady state means that the amount going in equals the amount going out. In groundwater systems, this means that the amount recharging the system—infiltration of rain, loss from streams, leakage from other formations—is equal to the amount discharging the system—spring flow, discharge to streams, pumping by wells. In field settings, recharge is never exactly equal to discharge—although, in some cases, they may be pretty close over the long term, such that it is often appropriate to assume a system is in a steady state when dealing with regional flow systems. That is discussed more in subsequent sections of this book. Aquifers are in a constant state of dynamic change; recharge varies, discharge varies, the volume in storage varies, and head varies. If it rains more in one year than it usually does, recharge increases and heads increase. If we pump a water from a well, we remove water from storage and heads decline. These scenarios depict transient flow conditions—conditions where heads and the volume of water in storage are changing through time.
The key point to consider here is time. Continuously pumping water from a well will result in ongoing changes in head, and we as hydrogeologists are interested in knowing things like how much we can pump from a well in a day, how fast heads will decline when we pump, and how long it will take for heads to return to their original level if we stop pumping. The right side of the equation essentially represents this aspect of aquifers, so the right side is zero for steady state flow conditions.
When flow varies with time, conditions are transient and the right side of Equation (50) is replaced with Equation (52) which describes the change in storage in the aquifer with time, becoming the transient groundwater flow equation shown in Equation (53).
| [latex]\frac{\mathrm{\Delta}Storage}{t} \approx S_{s}\frac{\partial h}{\partial t}[/latex] | (52) | |
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| [latex]K\left( \frac{\partial^{2}h}{{\partial x}^{2}} + \frac{\partial^{2}h}{{\partial y}^{2}} + \frac{\partial^{2}h}{{\partial z}^{2}} \right) = S_{s}\frac{\partial h}{\partial t}[/latex] | (53) |
It is useful to remember that specific storage is a coefficient, just like [latex]K[/latex] is a coefficient. A coefficient simply relates two quantities to one another:
- [latex]K[/latex] relates discharge to the hydraulic gradient. If we increase the gradient, we increase the discharge, and vice versa; [latex]K[/latex] tells us how much change in discharge we get for a given change in gradient.
- [latex]S_{s}[/latex] relates changes in head to changes in storage in the aquifer. If we remove a volume of water from storage, we lower the head in the aquifer, and vice versa; [latex]S_{s}[/latex] tells us how much head changes for a given change in storage.
The [latex]\frac{\partial h}{\partial t}[/latex] refers to the change in head with time, and the [latex]S_{s}[/latex] is the coefficient that states how much change in storage will occur given that change in head. In summary:
- the three‑dimensional groundwater flow equation is comprised of two parts: 1) the flux (the left side), and 2) the change in storage in the aquifer with time (the right side);
- the left side is composed of 2 parts, 1) the divergence of groundwater flux in each of the three principal directions, and 2) a hydraulic conductivity parameter that describes the ease with which groundwater moves through the medium in response to head gradient; and
- the right side is composed of two parts as well; 1) the change in head with time, and 2) a storage parameter that describes the change in storage resulting from the change in head.