2.7 Flow Nets Provide Insight into Groundwater Flow

Most everyone can envision water discharging at a given rate from an open hose, and knows that if the orifice at the opening of the hose is decreased in size by covering part of it with a finger, a high-speed jet of water will result because the discharge rate remains the same, so the velocity increases. This intuitive concept of mass conservation applies to interpreting flow tubes. Like the hose in the analogy, a flow tube carries a constant volumetric flow rate. If a flow tube becomes narrower, the specific discharge (volumetric flow rate divided by area of flow) must increase. For a homogeneous material, Darcy’s Law says that an increase in hydraulic gradient must accompany an increase in specific discharge. Therefore, in a homogeneous material, where a flow tube becomes narrower, the equipotential lines must be closer together. If the effective porosity is uniform, a higher specific discharge also implies a higher groundwater flow velocity.

The above concepts are illustrated by the two flow systems in Figure 16. In both systems, the confined aquifer is 100 m thick with the top of the aquifer an elevation of 0 meters. The hydraulic conductivity is homogeneous with a value of 0.0002 m/s. Groundwater enters the aquifer across the entire left side at a volumetric flow rate of 0.2 m³/s. Groundwater exits the aquifer on the right side through an outlet, where the hydraulic head is 10 m.

Figure 16 – Groundwater is introduced at the same volumetric rate to the entire left side of both homogeneous systems a) and b), but the outlet on the right is narrower in the system shown in b. In order for the same volumetric rate to flow through the narrow opening in (b), the flow tubes need to be narrower and the gradient needs to be higher than in (a). This results in an increased specific discharge and velocity (specific discharge divided by porosity) near the outlet of (b). The travel times are ~3.1 years in system (a) while travel time is shorter for central flow lines in system (b) (<3 years) and longer for flow lines near the boundaries (>3 years).

In Figure 16a, the outlet spans the entire right side, resulting in straight and uniform flow tubes. In Figure 16b, the outlet is restricted to only a portion of the right side. For groundwater to exit the aquifer, flow tubes narrow near the outlet opening. The narrowing of flow tubes is accompanied by an increase in hydraulic gradient (equipotential lines closer together) in accordance with Darcy’s Law. Hydraulic head contours range from 10 to 21 meters and are labeled in black.

Travel time of a packet of water traversing a flow line is indicated with blue numbers next to the arrow heads of the flow lines. The travel time is 3.1 years from entry to exit all packets traversing system with completely open boundary (Figure 16a). The same volume of water is forced through the system with the narrow outlet, but in this case the central flowlines carry packets of water across the system in a shorter time ~2.7 years, while packets near the boundaries travel more slowly, requiring 3.4 years to reach the exit (Figure 16b).

A graphically constructed flow net cannot be drawn for a heterogeneous system, so a numerical simulation of flow is used to demonstrate the impact of a lens of lower and higher hydraulic conductivity on the system of Figure 16a, as shown in Figure 17. As in Figure 16, the same volumetric flow rate enters the aquifer on the left side and exits the aquifer on the right side in Figure 17. The heterogeneous case has a lower-K zone within the aquifer in Figure 17a and a higher-K zone in Figure 17b.

Figure 17 – Groundwater is introduced at the same volumetric rate to these heterogeneous system as in the homogeneous system shown in Figure 16. a) Within the low-K zone, flow tubes are wider and the gradient is higher (equipotential lines are closer together) than the surrounding higher K material. Packets of water require more time to travel through the low-K zone. b) Within the high-K zone, flow tubes are narrower and the gradient is lower (equipotential lines are further apart) than the surrounding lower K material. Packets of water require less time to travel through the high-K zone.

Flow tubes carry the same volumetric flow at every location along their length, so in the case of the lower-K zone (Figure 17a): (1) flow tubes widen as they enter the lower-K zone and narrow as they exit the zone, and (2) equipotential lines are closer together inside the lower-K zone than outside. Both of these changes are indicated by Darcy’s Law because a higher gradient and larger area serve to carry the same volumetric flow as the flow tube enters the lower-K zone. Overall, the system with the lower-K zone requires a higher overall gradient to drive the same volumetric flow through the aquifer (notice the heads on the left side of Figure 17a are higher than those of Figure 16a). Packets of water require less time to travel across the system in the higher-K zone surrounding the lower-K material. This is indicated by the travel time for a packet of water traversing the central flow line being more than twice as long as the travel time along flow lines that pass around the lower-K material (6.0 years versus 2.7 years). It is useful to remember that, in these examples, the same flow rate is forced to enter the systems on their left side. If instead of constant flow on the left side, the head is held constant on the left side then the system of Figure 17a will have a lower volumetric flow than the system of Figure 16a, but the shape of the flow lines of Figure 17a will be similar with flow diverging at the upgradient side of the low-K zone and converging on the downgradient side.

Flow tubes carry the same volumetric flow at every location along their length, so in the case of the higher-K zone (Figure 17b): (1) flow tubes narrow as they enter the higher-K zone and widen as they exit the zone, and (2) equipotential lines are further apart inside the higher-K zone than outside. Both of these changes are indicated by Darcy’s Law because a lower gradient and smaller area serve to carry the same volumetric flow as the flow tube enters in the higher-K zone. Overall, the system with the higher-K zone requires a lower gradient to drive the same volumetric flow through the aquifer (notice the heads on the left side of Figure 17b are lower than those of Figure 16a). Packets of water require more time to traverse the system in the lower-K zone surrounding the higher-K material, while water packets require less time to traverse the system along the central flow line through the high-K material. This is indicated by the travel time for a packet of water traversing the central flow line being about half that of the travel time along flow lines that pass around the higher-K material (2.3 years versus 4.4 years). It is useful to remember that, in these examples, the same volume of water is forced to enter the systems on their left side. If instead of constant flow on the left side, the head for is held constant on the left side then the system of Figure 17b will have a higher volumetric flow than the system of Figure 16a, but the shape of the flow lines of Figure 17b will be similar with flow converging at the upgradient side of the high-K zone and diverging on the downgradient side.