12. Modified Pinder-Sauer surface water/groundwater exchange problem

The modified Pinder-Sauer is a surface-water/groundwater exchange problem that is based on the problem of (Pinder & Sauer, 1971), which has been used as a benchmark test for coupled surface-water/groundwater models (e.g., Hughes et al., 2015; Swain & Wexler, 1996). (Lal, 2001) modified the problem of (Pinder & Sauer, 1971) to make it easier to set up and to include a sinusoidal inflow hydrograph boundary condition at the upstream end of the surface-water system.

The analytical solution of (Lal, 2001) for the discharge at a distance \(x\) from the upstream boundary is

(12.1)\[Q_a = 509.70 + 141.58 \exp \left( \frac{\hat{\lambda}_{1} x}{\Lambda} \right) \sin \left( f_{r} t + \frac{\hat{\lambda}_{2} x}{\Lambda} \right),\]

where \(f_r\) is the characteristic frequency of the system, \(\hat{\lambda}_{1}\) is the amplitude decay constant, \(\Lambda\) is the characteristic length related to the wave number of the water-level disturbance, and \(\hat{\lambda}_{2}\) is a dimensionless wave number. The terms in **equation **12.1 are calculated using model parameters such as friction slope, reach sediment hydraulic conductivity, reach width, etc. and are defined in (Lal, 2001). For the case without aquifer exchange, \(\hat{\lambda}_{1} = -4.779 \times 10^{-2}\), and \(\hat{\lambda}_{2} = -0.3608\). With aquifer exchange, the appropriate values for the variables in **equation **12.1 are \(\hat{\lambda}_{1} = -0.1785\), \(\Lambda = 4894.3\) m, \(f_{r} = 3.49 \times 10^{-4}\) sec\(^{-1}\), \(\hat{\lambda}_{2} = -0.3409\), and all other variables are the same as the case without aquifer exchange.

The model domain represents a flood plane that is 39,624 m long, 427 m across the valley, and underlain by an unconfined aquifer. The flood plane and underlying aquifer are surrounded by impermeable boundaries on all sides. The base of the aquifer is horizontal and specified to be at an elevation of 0.0 m. The flow direction in the model domain is along the long axis of the model from top to bottom.

A total of 65 rows, 15 columns, and 1 layer were used to discretize the model domain. A constant grid spacing of 609.61 m was used for each row. A grid spacing of 28.30 m was used for all columns except the center column (column 8); the grid spacing of column 8 was 30.48 m. The total simulation length was 24 hours and a constant time-step length of 5 minutes was used.

The hydraulic conductivity of the aquifer is 3.048\(\times\)10\(^{-3}\) m/s. The specific yield and specific storage are 0.25, and 1\(\times\)10\(^{-7}\) 1/s, respectively.

Table 12.1 Model parameters for example ex-gwf-sfr-pindersauer.

Parameter

Value

Number of periods

1

Number of time steps

288

Time step length (\(s\))

300.0

Number of layers

1

Number of rows

65

Number of columns

15

Column width (\(m\))

varies

Row width (\(m\))

609.61

Top of the groundwater model (\(m\))

varies

Bottom of the groundwater model (\(m\))

0.0

Starting head (\(m\))

varies

Hydraulic conductivity (\(m/s\))

3.048e-3

Specific storage (\(1/s\))

0.1e-6

Specific yield (unitless)

0.25

Number of reaches

65

Reach bed slope (\(m/m\))

0.001

Mannings roughness coefficient (\(s/m^{1/3}\))

0.03858

Upstream inflow base rate (\(m^3/s\))

509.70

Upstream inflow amplitude (\(m^3/s\))

141.58

Amplitude decay constant

varies

Dimensionless wave number

varies

Characteristic length related to the wave number of the water-level disturbance

4894.3

Characteristic frequency of the system

3.49e-4

A single river channel is located at the center of the aquifer (column 8) parallel to the long axis of the model domain and is simulated using the kinematic-wave approximation option available in the Streamflow Routing (SFR) package. The channel has a bed slope of 0.001, a width of 30.48 m, and a Manning roughness coefficient of 0.03858 s/m\(^{1/3}\). For the case with aquifer exchange, the leakage coefficient is \(1.402 \times 10^{-4}\) sec\(^{-1}\) and seepage is assumed to occur only from the bottom.

Initially, the saturated thickness of the aquifer is 67.05 and 27.43 m at the upstream and downstream ends of the aquifer, respectively. The initial reach stage in each reach was calculated using **equation **12.1, the bed elevation, and Mannings equation using reach parameters.

Table 12.2 Scenario parameters for example ex-gwf-sfr-pindersauer.

Scenario

Scenario Name

Parameter

Value

1

ex-gwf-sfr-pindersauera

lambda val

(-0.04779, -0.3608)

leakance

0.0

2

ex-gwf-sfr-pindersauerb

lambda val

(-0.1785, -0.3409)

leakance

0.000142

The sinusoidal flood hydrograph introduced in the upstream SFR reach is,

(12.2)\[Q = 509.70 + 141.58 \sin \left( \frac{2 \pi t}{T_p} \right),\]

where \(T_p\) is the period of disturbance (sec.) and \(t\) is the simulation time (sec.). A \(T_p\) value of 5 hours (18,000 sec.) was used.

12.1. Example Results

Simulated relative stage and discharge results 15,240 m downstream of the top end of the model domain are shown in **figure **Figure 12.1. Relative stage and discharge results were calculated using the initial stage and discharge of 52.12 m and 509.70 m\(^3\)/s, respectively. For comparison, simulated relative stage and discharge for a simulation without aquifer exchange (leakage coefficient \(= 0.00\) sec\(^{-1}\)) are also shown in **figure **Figure 12.1. Analytical results calculated using **equation **12.1 15,240 m downstream of the top end of the model domain are also shown in **figure **Figure 12.1; SFR results are comparable to the analytical solution.

../_images/ex-gwf-sfr-pindersauer-observations.png

Figure 12.1 (A) Simulated relative stage change 15,240 m downstream of the top end of the model domain with and without aquifer exchange (leakage) for the modified Pinder-Sauer problem. (B) Comparison of relative discharge change 15,240 m downstream of the top end of the model domain and the analytical solution of (Lal, 2001) with and without aquifer exchange (leakage).

12.2. References Cited

Hughes, J. D., Langevin, C. D., & White, J. T. (2015). MODFLOW-based coupled surface water routing and groundwater-flow simulation. Groundwater, 53(3), 452–463.

Lal, A. W. (2001). Modification of canal flow due to stream-aquifer interaction. Journal of Hydraulic Engineering, 127(7), 567–576.

Pinder, G. F., & Sauer, S. P. (1971). Numerical simulation of flood wave modification due to bank storage effects. Water Resources Research, 7(1), 63–70.

Swain, E. D., & Wexler, E. J. (1996). A coupled surface-water and ground-water flow model (MODBRANCH) for simulation of stream-aquifer interaction.

12.3. Jupyter Notebook

The Jupyter notebook used to create the MODFLOW 6 input files for this example and post-process the results is: