Lesson 4 - Defining the Basic Inputs for an Aquifer Pathway
In the previous Lesson we provided a basic introduction to the Aquifer element. In this Lesson we will discuss in more detail how the inputs are specified.
Let’s remind ourselves what the Aquifer dialog looks like:
Let’s start with the most fundamental inputs: the Length and the Area.
Specifying the Length is straightforward. Although in most cases the length is constant for most systems that you will simulate, GoldSim allows it to change during the simulation. This would most commonly be done by shortening the pathway (e.g., in order to simulate the end of the pathway eroding away). When the pathway is shortened, mass that was present in the truncated portion of the pathway is immediately flushed from the pathway (to downstream pathways).
The Area is the cross-sectional area at the outlet end of the pathway (perpendicular to the flow). In the Example we discussed in the previous Lesson specifying the Area was straightforward (since it was a tube, and the area of the tube is well-defined). But in most cases, you will be using the Aquifer element to simulate an environmental component such as a groundwater pathway. In such cases, the appropriate Area to specify may not be obvious, and is therefore worth discussing further.
Let’s first consider how the Area actually impacts the outputs of the pathway (i.e., the mass in the pathway, the mass rate leaving the pathway and the concentration leaving the pathway).
You may recall from the previous Lesson that the expected travel time for a (fully saturated) pathway can be estimated as follows:
Expected Travel Time = n A L/Q
From this, you might initially conclude that choosing a larger Area would result in a longer travel time. That would be correct if the flow rate were independent of Area, as in modeling a fixed flux through pipes of various diameters. But this would be incorrect if we assume that the flow rate (Q) is a linear function of the Area. Since we are modeling a section of an aquifer, our choice of Area does not change the flow velocity. So if you increased the Area, to maintain consistency, you would also need to increase the flow rate. As a result, changing the Area has no impact whatsoever on the travel time. This means that Area also has no impact on the mass rate leaving the pathway (or therefore the total mass in the pathway). What it does impact, however, is the concentration leaving the pathway. In particular, the concentration leaving the pathway is inversely proportional to the Area. This makes sense, of course, since you can think of the concentration as the mass rate divided by the fluid flow rate. As we just stated, changing the Area has no impact on the mass rate, but it directly impacts the flow rate.
To understand the implications of this, it is important to understand what we are really trying to represent with a 1-D groundwater pathway. What we are trying to simulate is essentially a plume. Plumes, of course, are not one-dimensional at all. When mass enters the groundwater, it spreads and disperses not only longitudinally, but also transversely and vertically, and illustrated schematically below:
At any given longitudinal distance from the source, if you were to look at a cross-section of the plume (perpendicular to the flow direction), the concentration would vary across that surface. At that distance you could conceptually define the area based on a concentration contour. But of course the contour you would select is somewhat arbitrary and the area associated with any contour would change with time (unless it reached a steady-state). (As an aside, one outcome of modeling plumes assuming Gaussian disperson is that they have no “edge”. Mathematically, they extend an infinite distance with an infinitesimal concentration at that distance. A plume cannot therefore be shown to have an actual extent, and depiction relies on the use of isopleths, or lines of constant concentration.)
So how do we select an appropriate Area? As pointed out above, the Area impacts the computed concentration leaving the pathway (but does not affect the mass discharge rate). To understand how to select the Area it is important to keep in mind what we are actually trying to calculate. You might say that what we are trying to do is compute a concentration at a specific point in space (at a certain X, Y and Z coordinate). Although as we will discuss in Lesson 13 GoldSim does provide a way to estimate that, it is typically not realistic to try to do so. Real plumes do not look anything like the schematics shown above. Instead, due to heterogeneities in the subsurface environment (as well as the source term), real plumes (even in well-sorted homogeneous materials) tend to be very irregular indeed. As a result, predicting a concentration at a specific coordinate is generally not very meaningful.
Fortunately, for most applications, this is not actually what is required at all. Most often, what we are really interested in is a mass discharge rate into a particular feature or across some plane surface (e.g., a well, a river or some other surface water body), in which case the Area actually has no impact. This mass is then mixed (and diluted) by additional water. Hence, what we are typically interested in is not a concentration at a point in space, but one that is averaged (due to mixing within the downstream feature).
In the case of a well, the well would often be assumed to capture the entire plume as well as water outside the plume (i.e., it would also capture clean water). The concentration of interest is then that in the water being pumped out of the well, which is determined by the mass discharge rate of the pathway into the well (which is independent of Area, since we assume the well captures the entire plume) and the flow rate from the well. Similarly, if the pathway discharges into a stream, the concentration of interest is that in the stream (downgradient of the discharge point) and is determined by the mass discharge rate of the pathway (which is independent of Area) and the flow rate in the stream.
As a result, the concentration output of the Aquifer is typically not very meaningful. And because this is the only output that is affected by the Area, how the Area is specified is not of particular importance! In most cases, you will simply define it so it represents a “reasonable” estimate of the extent of the plume (typically determined by the dimensions of the source and/or perhaps the thickness of the aquifer).
The other four basic inputs for the Aquifer were discussed in the previous Lesson, but several points are worth discussing in a bit more detail here.
- Dispersivity. This is the longitudinal dispersivity of the pathway. It has dimensions of length. You should typically define this as a linear function of the pathway Length (to represent macrodispersion). (Note that in the same way that mass flux is independent of area, it is also independent of vertical and transverse dispersion, since these operate only in directions perpendicular to flow.)
- Number of Cells: As we will discuss in more detail in the next Lesson, the Aquifer element solves the 1D advection-dispersion equation by using a finite difference approximation that involves discretizing the pathway into a series of finite volumes. This represents the number of finite volumes that are used. In order to avoid numerical diffusion, the number of Cells should be no smaller than the Length divided by twice the dispersivity (e.g., if the dispersivity is 10% of the Length, you need at least 5 Cells). If the Number of Cells is too small to represent the dispersivity, GoldSim will write a warning to the Run Log. The maximum number of Cells is 100 (as there would be little benefit in using a number greater than this), and the minimum number of Cells is 4. If you specify a value outside of this range, GoldSim adjusts the number of Cells accordingly and writes a warning message to the Run Log.
- Infill Medium: The Infill Medium is assumed to fill the entire pathway. You can leave this blank, in which case there is no porous medium (e.g., if you were simulating a tube or pipe with no sand, or perhaps a stream). As we noted in the Example discussed in the previous Lesson, the porous medium acts to increase the flow velocity (by reducing the effective flow area) and can act to retard any species that partition onto it.
- Fluid Saturation: This is a dimensionless value that must be greater than 0 and less than or equal to 1. The default is 1 (the porous medium is fully saturated). Decreasing the fluid saturation affects the behavior of the Aquifer in two ways: 1) it increases the flow velocity (by reducing the effective flow area for a given flow rate); and 2) it increases the degree of retardation due to partitioning onto the porous infill (by reducing the effective volume of fluid present in the pathway relative to these solids – all of the Solid is still assumed to impact with the water).
Great care should be taken if you specify that the saturation is less than 1. In particular, if the flows into the pathway are changing with time, the saturation would in nature vary along the flow path (and change dynamically). The Aquifer pathway, however, has no way to represent this (the saturation applies for the entire pathway). As a result, you should generally only use an Aquifer for a partially saturated component if you can assume that the flow system is at a steady-state. Modeling transient unsaturated systems is discussed in Unit 12.
We will discuss the other input fields in this dialog (which represent advanced features) in subsequent Lessons.
Before leaving this Lesson, however, let’s discuss one more fundamental input for an Aquifer; the flow rate. As mentioned in the previous Lesson, all Aquifers must have at least one advective mass flux link to another pathway with a positive Flow Rate defined: it is the only way to define the flow rate through the Aquifer. Otherwise, GoldSim will generate a fatal error message. Several points should be noted regarding flow rates:
- The behavior of the pathway is determined by the total flow rate specified (the sum of all the flows used to create advective mass transfer links from the pathway).
- As mentioned above, this total flow rate should be defined in a way that is consistent with how the Area of the pathway was defined.
- It will not be uncommon for the flow rate entering the pathway to be less than the flow rate leaving the pathway. In fact, for groundwater plumes, this is exactly what must happen. The flow rate in the plume grows along the pathway as more “clean” water is mixed into the plume (i.e., as the plume grows in size). This, of course, results in dilution of the incoming flows. However, if the flow rate entering the pathway is greater than the flow rate leaving the pathway, GoldSim will write a warning message to the Run Log. This is because in most situations this would not be physically realistic (it would imply that “clean” water has somehow exited the pathway, increasing the concentration in the rest of the water).
- If you specify that the flow rate changes suddenly with time, it will change instantaneously throughout the pathway. Although such a response may be appropriate for a confined system, for a phreatic system, flow changes would not physically propagate instantaneously (the response would be much slower and would propagate as a “wave”). As a result, when changing the flow rate (for a phreatic system) you should be aware of how it will be represented in GoldSim (and generally should only change it slowly). Of course, in most realistic situations, unless you are applying an artificial sudden change (e.g., pumping), flow rates will in fact change slowly (not suddenly).