Numerical Groundwater Flow Modeling, Part 2
Boundaries
This blog is the second in a series on numerical groundwater models. These blogs are meant to be informative on the basic approach to modeling groundwater and what makes a model a good tool. For a tool to be effective, it should be the correct shape and size for the user. This blog describes why model boundaries are important and necessary for a model budget.
Numerical groundwater models are used to compute water levels and flow. Their calculations consider factors that affect groundwater flow. From Part 1 of this series, we know those factors can include precipitation, topography (ground slope), and the properties of the ground through which the water flows (hydraulic conductivity). To solve a groundwater flow problem, the conceptualization of the problem needs to be simple enough so that values can be used to describe all the factors that are important, then the values are used in a system of equations that are finite. Finite means that we are solving a problem in a finite area. We don’t have a method for modeling all the groundwater everywhere all at once in the same model (although one day we might). Therefore, we confine the problem to a specific area and place boundaries around that area. It is important to carefully select boundaries so our solution is not adversely affected by the boundaries chosen.
Figure 1: Groundwater model budget
Bookkeeping is a good method for selecting boundaries because groundwater models keep a tally sheet called a budget. All the water that is inside the model is counted by flow rate (like gallons per day). If there is more water flowing in than flowing out, then that surplus water is “stored” in the model. If more water flows out than comes in, the deficit is a “loss in storage”. All the water needs to be accounted for in the model (Figure 1). If the tally does not balance, we call that a “mass balance error”. The mass balance error in numerical models is never really zero, but it needs to be sufficiently small (less than one one-hundredth of a percent of the flow).
Figure 1: Groundwater model budget.
Boundaries are best in locations where there is little groundwater flow across them (simplified bookkeeping). Let’s revisit the example from Part 1 of the LWS Blog which shows the groundwater beneath the hill that flows to the stream (Figure 2). There is a vertical line (red line) beneath the apex of the hill labeled No-flow boundary. That means that we assume no water flows into the system from the right of that line. This makes sense if you consider that the hill has the highest elevation and highest potential so all the water will flow from the hill toward the stream. Presumably on the other side of the hill there is another stream or even the ocean, but it is outside the boundary and not in our budget.
Figure 2: Vertical no-flow boundaries
There is another vertical No-flow boundary on the left side of Figure 2 beneath the stream and impermeable rock across the bottom so that all the water in the system is: 1) left of the hill, 2) right of the stream, and 3) above the rock. These boundaries make the accounting easy. By setting these horizontal and vertical boundary conditions, the only boundary flows that need to be quantified in our bookkeeping are the infiltration from precipitation (Figure 1 water-in) and groundwater discharge to the stream (water-out). This example has been simplified by the assumption that the No-flow boundaries are fixed in space and that the bedrock does not leak.
Figure 2: Vertical no-flow boundaries.
While this example provides a simplified example of how to set boundaries in a groundwater model, in the real-world boundaries can and do move due to changing groundwater conditions, and bedrock can be leaky. As such, in more complex modeling applications there may need to be a more robust and complex set of assumptions related to setting model boundaries.
In the next LWS blog of this series we will discuss what properties go inside the model boundaries to solve the groundwater flow equation.
If you have any groundwater modeling needs or even general water resources issues, LWS can help; please contact us at 303-350-4090 or by email.
Maura Metheny, Ph.D., P.G., maura@lytlewater.com
Bruce Lytle, P.E., bruce@lytlewater.com
Anna Elgqvist, EI, anna@lytlewater.com
REFERENCES
Figure 2: BCcampus Open Publishing, 2021. Chapter 14 Groundwater. Available online at https://opentextbc.ca/geology/chapter/14-2-groundwater-flow/, (accessed 10/29/21).
