5.1 Introduction
In this chapter, performance of the two transport models MT3DMS and the conduit transport model in coupled form is investigated. For that, a method of verification is selected which does not require immediate examination of the performance of MT3DMS since this model is a standardised and sophisticated transport model (Zheng & Wang 1999). Especially, the overall performance of the two transport models is examined by testing the proper implementation of the transport solver EMCNOT (Liu et al. 2001, see Chapter 3.2.1) for advection in the conduit system within UMT3D/RUMT3D. The focus is thereby to investigate:
• the transport in the conduit system,
• the interaction with the continuum and
• the initial and boundary conditions of the conduit system.
A benchmark problem that uses a similar pipe network setup as described by Birk (2001) was chosen since for this network with imposed initial and boundary conditions, a “quasi” analytical solution could be determined. Apart from the
“quasi” analytical solution, the performance of the EMCNOT solver within UMT3D/RUMT3D is also compared to the standard finite difference solution of the benchmark problem. This problem setup requires only a description of the features of the conduit system. It is worthwhile to point out that by examining performance of the EMCNOT scheme using different criterions, UMT3D/RUMT3D is benchmarked towards accuracy, numerical stability and computational efficiency.
5.2 Description of the Conduit System
The conduit system used for verification purpose in this chapter is basically a 3-way pipe network with 3 conduit tubes and 4 conduit nodes ( Fig. 5). Conduit tubes 1 through 3 have diameters, di of 0.1, 0.05 and 0.025 m, respectively. All three tubes have a length of 500 m each. Conduit nodes 3 and 4 are supplied with a constant recharge rate, R of 3.15 × 10-4 m3 s-1 each. Only, conduit node 1 allows exchange to the porous matrix. There, mass is permitted to exit the
conduit system with an exchange coefficient of 1.0 × 10-4 m2 s-1. Initially, it is assumed that the concentration in the conduit system is 1 mg L-1. No mass is added to the conduit system, i.e., the concentration of the recharge flux is zero.
Using the recharge rates and pipe diameters, the following pipe flow rates can be calculated for pipes 1 through 3, 0.08, 0.16 and 0.642 m s-1, respectively, which yield residence times of 6233, 3117 and 779 seconds, respectively. The through the “quasi” analytical solution computed time-dependent concentration values at each conduit node could be determined by using the residence times of the different pipes and initial and inflow concentration values to the respective nodes.
One difference between the here described pipe network and the one by Birk (2001) is that, Birk (2001) kept the heads fixed at the outflow conduit node 1 and in the cell where this node was located.
Fig. 5: Configuration of the 3 way pipe network system of the test problem.
5.3 Performance of UMT3D
Breakthrough curves at the conduit node 1 (CON 1, Fig. 5) are used to compare simulation results for the above-described conduit system at the outlet of the pipe network. Fig. 6 shows a comparison between the breakthrough curves at CON 1 using the FD method with two different pipe discretisation intervals of 0.1 and 1 m. The simulation results are also compared with the “quasi” analytical solution in Fig. 6. From Fig. 6, it is obvious that the finer the discretisation of the different pipes, the better the simulation results at CON 1. A pipe discretisation spacing of 0.1 m is rather small compared to a pipe length of 500 m and also results in a rather small transport time step size since a time criterion for the FD method is determined by the minimum of pipe discretisation spacing divided by the flow velocity within a respective pipe multiplied by the Courant number (see Chapter 3.2.2). Such small transport time step sizes can slow down transport calculations considerably in both models. When the concentration values at the conduit nodes are modified, concentration values in the connecting porous matrix cells need to be updated. In other words, in both models the same transport time step
pipe 1 pipe 3
pipe 2
R
R
Γ1
d3= 0.025 m
d2= 0.05 m d1= 0.1 m CON 4
CON 1
CON 3 CON 2 pipe 1
pipe 3
pipe 2
R
R
Γ1
d3= 0.025 m
d2= 0.05 m d1= 0.1 m CON 4
CON 1
CON 3 CON 2
size should be used. Especially for reactive transport problems where many components need to be transported, transport calculations with small transport time step sizes can be very time consuming. Therefore, it was advisable to implement a numerical method that solves advection in the conduit system with little time constraints, while at the same time numerical dispersion is reduced as shown in Fig. 6, particularly with the large pipe discretisation of 1 m. The EMCNOT method is intended to overcome these limitations.
Fig. 6: Comparison of the simulation results using the FD method with pipe discretisations of 1 and 0.1 m.
Fig. 7 shows a comparison of the simulation results at CON 1 using the EMCNOT method with a pipe discretisation of 1 m and different transport time step sizes. The simulation results are also compared with the “quasi” analytical solution in Fig. 7 and with simulation results using the FD method with pipe discretisation of 1 and 0.1 m. Fig. 7 demonstrates that with transport time step sizes of 50% and 10% of the minimum residence in the pipe network (0.5 tr and 0.1 tr, respectively) and a pipe discretisation of 1 m, comparable approximations of the transport solutions to the FD method using a tube discretisation of 1.0 and 0.1, respectively can be obtained. This implies that the EMCNOT scheme achieves time step size improvements by factors of 250 and 500 above the FD method, respectively. Further refinement of the pipe discretisation from 1 to 0.1 m for the different time criteria showed only minor improvements.
0 0.2 0.4 0.6 0.8 1
6300 7300 8300 9300 10300
Time (s) Concentration (mg L-1 )
quasi analytical 1.0 m
0.1 m
1.0 m 0.1 m
Fig. 7: Comparison of the simulation results using the EMCNOT and the FD method.
5.4 Summary
In this chapter, the coupling of the transport models for the continuum and conduit system within UMT3D/RUMT3D was verified by comparison with a semi-analytical solution. Since a standard transport model was used for transport in the porous matrix, only the performance of the conduit transport model needed to be investigated. This was accomplished by testing the proper implementation of the transport solvers for the conduit system. Simulation results obtained with the explicit finite difference (FD) method to solve advection in the conduit system showed large numerical dispersion, which could be improved by using smaller pipe discretisations. However, smaller pipe discretisations are also associated with smaller transport time step sizes for both models. The numerical scheme EMCNOT developed by Liu et al. (2001) showed great improvements in the simulation results allowing much larger transport time step sizes and pipe discretisation. Therefore, the EMCNOT is especially recommended for simulation problems with complex pipe geometries and for simulations with multiple components transported and geochemical reactions. However, further testing of the EMCNOT method with more complex pipe geometries is recommended before it is used for real-world transport problems.
0.45 0.6 0.75 0.9 1.05
6300 6800 7300 7800
Time (s) Concentration (mg L-1 )
quasi analytical FD, 1.0 m FD, 0.1 m
EMCNOT, 1.0 m, dt=0.1*tr EMCNOT, 1.0 m, dt=0.5*tr
EMCNOT dt=0.1*tr
FD 0.1 m EMCNOT
dt=0.5*tr FD
1.0 m
Prediction is very hard, especially when it’s about the future.
Yogi Berra, cited in Michio Kaku, Visions: How Science Will Revolutionize the 21st Century (1997)
Chapter
6 Verification of the Reactive Package within RUMT3D and Plausibility