Line sinks are important aspects of flow in porous media because they are used to represent the functions of injection or recovery wells in aquifers or reservoirs. You typically represent a line sink (or source) as an open polygon, a line, which is called an edge in the model. This is the easiest approach, and it is reasonable for representing an actual line source Wells have a finite radius, so they are not actually geometric lines, so they are sometimes represented as cylindrical tubes rather than lines in 3D models.

The objective of this exercise is to learn how to analyze a line sink resembling a well, validate the numerical results using an analytical solution, and then use the numerical results to learn about how line sinks behave.

The set-up for a transient analysis of a line sink in a circular aquifer is described in this video. The Comsol file used in the video is called Jacob example.mph, and it is attached below. The Jacob (1940) solution (Figure 1) can be used to verify numerical analyses of a constant rate line sink in a uniform layer. This solution is not valid at early times, but it is a good approximation of the head in the vicinity of the well at late times. The video shows how to use the Jacob equation to verify the numerical results.

FIgure 1. Jacob solution for the drawdown at a radial distance, r, in the vicinity of a well pumped at constant volumetric rate, Q

Numerical experiments can be a useful way to help understand a system by systematically varying one aspect of the analysis and evaluating how it affects the results. In this example, the simulation outlined in the video was used to evaluate how hydraulic conductivity affects heads in the vicinity of a well. The procedure was to set up a baseline analysis and verify that it is working correctly. Additional analyses were then conducted by varying the hydraulic conductivity. Details are given in the video.

The first plot on the left (Figure 2a) shows that the numerical and Jacob solutions are similar at late times, after t=10⁶ sec. Note that the formatting of the plots caused the label for t=10⁶s to be omitted. It would be the tic between 5 and 20x10⁵. This example was done in v4.2, and later versions of Comsol allow better control of the axes. The results in the first panel also show increasing error at earlier times. This error would continue to increase if the plot was extended to the left because the Jacob solution (green line) is only valid at late times.

The middle panel (b.) uses a smaller hydraulic conducitivity, K. This increases the magnitude of the change in the head; note the change in scale of the y-axis. This means that the slope of the semi-log straight line is steeper than it was when K is larger (a.). The error between the numerical and analytical at early times has also been increased by reducing K. This occurs because the head changes more slowly when K is smaller, so more time is required for the head change to move away from the well. Eventually, the two solutions are similar, but it takes 10x longer for this to occur that it does in panel a.).

Increasing K (in Panel c) reduces the range in the y axis, so the semi-log slope is flatter than it is for solutions using larger values of K. This result is consistent with inspection of the Jacob analysis, which shows that the slope of the head should be inversely proportional to K when the data are plotted as a semi-log function of time. Indeed, this slope is the basis for estimating K from pumping test data.

Figure 2. Drawdown as a function of time at r=50 m from the well. Based on the example problem described in the video. K=1E-7 m/s. Numerical (blue) and analytical solution (green). b.) Drawdown at r=50m when K=1E-8 m/s. c.) Drawdown at r=50m when K=1E-6 m/s.

The large K used in panel c.) causes the numerical results to conform to the analytical results much earlier than they did with smaller values of K. Another even more striking difference is the numerical results flatten out at late time. This occurs because the pressure disturbance caused by the line sink has propagated out to the edge of the aquifer in the numerical model. A constant head boundary condition is used at the outer edge and interaction with this boundary causes the plot of the heads to flatten. The Jacob solution assumes the aquifer is of infinite radial extent, so no outer boundary is considered in the Jacob solution and that is why it differs from the numerical results.

Here is the video

Text files with constants and variables

jacob function.txt
variables for Jacob check revised.txt
constants for Jacob example.txt
variables for Jacob check revised.txt

Comsol files

jacob_example.mph
Jacob check revised reset.mph