I am using equilibrium model and nonequilibrium (two-region) model in CXTFIT to inversely estimate the parameters of non-reactive tracer transport at saturated state in soil columns with quite amount of earthworm burrows.
Surprisingly, the equilibrium model fitted the observed data very well (BTC with quick response and long tail) and get dispersion coefficient, D with know pore velocity and boundary and initial conditions.
More importantly, I want to get parameters ( mobile water fraction and mass transfer coefficient) to related to my soil structure.
My question is: should I use the estimated D from equilibrium model as known input to nonequilibrium (two-region) to estimate mobile water fraction and mass transfer coefficient? If not, I have three parameters and estimated results highly influenced by initial values (non-uniqueness). If yes, I get very low mobile water fraction.
I highly appreciate if you can give me some suggestion and help.
about two-region model
Just want to add a little more.
Theoretically, I think, the estimated D from equilibrium model should not be used as known input to non-equilibrium (two-region) to estimate mobile water fraction and mass transfer coefficient, for mass transfer coefficient takes account of some dispersion and diffusion.
It seems that using minimum and maximum constraints may help. Mobile water fraction ( 0-1), D, and mass transfer coefficient can be quite varied. So, how to select the best initial values, especially with many columns with different characteristics? How I make sure my results are pretty reliable and consistent?
Look forward to sharing your experience.
Theoretically, I think, the estimated D from equilibrium model should not be used as known input to non-equilibrium (two-region) to estimate mobile water fraction and mass transfer coefficient, for mass transfer coefficient takes account of some dispersion and diffusion.
It seems that using minimum and maximum constraints may help. Mobile water fraction ( 0-1), D, and mass transfer coefficient can be quite varied. So, how to select the best initial values, especially with many columns with different characteristics? How I make sure my results are pretty reliable and consistent?
Look forward to sharing your experience.
Dear lifang:
The two-region mobile-imobile model (MIM) becomes closed to the equilibrium CDE as solutes mix well between mobile and immobile phases for greater time. The corresponding D in terms of the MIM consists of two terms: dispersion in the mobile phase, and transverse diffusion between mobile and immobile phases (De Smedt and Wierenga, 1983: Valocchi, 1985).
I also applied this idea to the dispersion for an unsaturated dune sand (Toride et al., 2003). In the case of the dune sand, relatively reasonable MIM parameter values were obtained from the BTCs since the BTCs showed long tailings for greater time (Fig. 5 in Toride et al., 2003).
If you can also get reasonable MIM parameter values, I would recommend to compare D_MIM (Eq. (14) of Toride et al., 2003) with D based on the equilibrium CDE. However, it is generally difficult to estimate the MIM parameters when the CDE fits well to observed BTCs. If you fail to estimate the MIM parameter values, please let me know. We should further look at your data in details.
Nobuo
De Smedt, F., and P. J. Wierenga. 1984. Solute transfer through columns of glass beads. Water Resour. Res. 20:225-232.
Toride, N., M. Inoue and . F. J. Leij, 2003, Hydrodynamic dispersion in an unsaturated dune sand, Soil Sci. Soc. Am. J. ,67C703-712.
Valocchi, A. J. 1985. Validity of the local equilibrium assumption for modeling sorbing solute transport through homogeneous soils. Water Resour. Res. 46:233-247.
The two-region mobile-imobile model (MIM) becomes closed to the equilibrium CDE as solutes mix well between mobile and immobile phases for greater time. The corresponding D in terms of the MIM consists of two terms: dispersion in the mobile phase, and transverse diffusion between mobile and immobile phases (De Smedt and Wierenga, 1983: Valocchi, 1985).
I also applied this idea to the dispersion for an unsaturated dune sand (Toride et al., 2003). In the case of the dune sand, relatively reasonable MIM parameter values were obtained from the BTCs since the BTCs showed long tailings for greater time (Fig. 5 in Toride et al., 2003).
If you can also get reasonable MIM parameter values, I would recommend to compare D_MIM (Eq. (14) of Toride et al., 2003) with D based on the equilibrium CDE. However, it is generally difficult to estimate the MIM parameters when the CDE fits well to observed BTCs. If you fail to estimate the MIM parameter values, please let me know. We should further look at your data in details.
Nobuo
De Smedt, F., and P. J. Wierenga. 1984. Solute transfer through columns of glass beads. Water Resour. Res. 20:225-232.
Toride, N., M. Inoue and . F. J. Leij, 2003, Hydrodynamic dispersion in an unsaturated dune sand, Soil Sci. Soc. Am. J. ,67C703-712.
Valocchi, A. J. 1985. Validity of the local equilibrium assumption for modeling sorbing solute transport through homogeneous soils. Water Resour. Res. 46:233-247.
Dear Dr.Toride,
Thanks a lot for your response.
It is a very nice paper (Toride et al., 2003). You had even more parameters to estimate (I put pore velocity as the known). As you said, it is generally difficult to estimate the MIM parameters when the CDE fits well to the observed BTCs. I saw that traditional CDE modeled your data pretty well too. Did you use minimum and maximum constraints (the range of value that you had expected) to get the reasonable results?
My problem is I have different types of soils. How do you know the estimated results are reasonable, especially to compare the estimated parameters among different soils? If I use minimum and maximum constraints that I choose, I feel the results are subjective and unreliable.
The following table summarizes the results I estimated with MIM model without minimum and maximum constraints for two types of soils (i.e., H-C and H-P). These results were not so desirable.
H-C-2 H-C-3 H-C-4 H-C-5 H-P-2 H-P-3 H-P-4 H-P-5
D.... 6.874 4.4340 1.288 16.110 6.71 6.63 4.81 3.86
beta. 0.080 0.8865 0.130 0.923 0.85 0.55 0.61 0.0001
omega 0.001 0.0012 0.076 0.00 0.00 0.16 0.28 2.88
How to select the reasonable initial value of D, beta and omega? What do you think if I use macroporosity as initial value of Beta (saturated flow)?
Thanks
Lifang
Thanks a lot for your response.
It is a very nice paper (Toride et al., 2003). You had even more parameters to estimate (I put pore velocity as the known). As you said, it is generally difficult to estimate the MIM parameters when the CDE fits well to the observed BTCs. I saw that traditional CDE modeled your data pretty well too. Did you use minimum and maximum constraints (the range of value that you had expected) to get the reasonable results?
My problem is I have different types of soils. How do you know the estimated results are reasonable, especially to compare the estimated parameters among different soils? If I use minimum and maximum constraints that I choose, I feel the results are subjective and unreliable.
The following table summarizes the results I estimated with MIM model without minimum and maximum constraints for two types of soils (i.e., H-C and H-P). These results were not so desirable.
H-C-2 H-C-3 H-C-4 H-C-5 H-P-2 H-P-3 H-P-4 H-P-5
D.... 6.874 4.4340 1.288 16.110 6.71 6.63 4.81 3.86
beta. 0.080 0.8865 0.130 0.923 0.85 0.55 0.61 0.0001
omega 0.001 0.0012 0.076 0.00 0.00 0.16 0.28 2.88
How to select the reasonable initial value of D, beta and omega? What do you think if I use macroporosity as initial value of Beta (saturated flow)?
Thanks
Lifang
Lifang:
Sorry for slow response. The MIM describes the physical nonequlibrium using two parameters: alpha and theta_m/theta. The exchange rate and the mobile fraction cannot be perfectly separated in actual soils. Water should move slowly even in the immobile phase, and alpha is based on the assumption of the immobile phase.
On the other hand, the MIM can describe various types of BTCs with different combinations of omega and beta values. That's a reason why the MIM is widely used for the BTC analysis. However, because of the assumptions of the MIM, it is generally difficult to validate estimated parameter values. As I did in my paper, I usually compare parameter values from different depths and/or water flow rate to verify the estimation. I succeeded to have quite similar values in case of an unsaturated sand regardless of observation depths as shown in Tables 1 & 2 of Toride et al. (2003).
Generally min. and max. constraints do not improve the estimation. It is rather important to give proper initial estimates. It is also important to have accurate data at the tailing part of BTCs as shown in Fig. 5(b) in Toride et al. (2003).
It is difficult to judge the quality of your results only from estimated parameter values. The macroporosity you defined may be different from the mobile fraction. I probably need to have look at your observed BTC as well as experimental conditions in details.
If you need further help, please do not hesitate to ask me.
Regards,
Nobuo
--------------------------------
Nobuo Toride ntoride@bio.mie-u.ac.jp
Sorry for slow response. The MIM describes the physical nonequlibrium using two parameters: alpha and theta_m/theta. The exchange rate and the mobile fraction cannot be perfectly separated in actual soils. Water should move slowly even in the immobile phase, and alpha is based on the assumption of the immobile phase.
On the other hand, the MIM can describe various types of BTCs with different combinations of omega and beta values. That's a reason why the MIM is widely used for the BTC analysis. However, because of the assumptions of the MIM, it is generally difficult to validate estimated parameter values. As I did in my paper, I usually compare parameter values from different depths and/or water flow rate to verify the estimation. I succeeded to have quite similar values in case of an unsaturated sand regardless of observation depths as shown in Tables 1 & 2 of Toride et al. (2003).
Generally min. and max. constraints do not improve the estimation. It is rather important to give proper initial estimates. It is also important to have accurate data at the tailing part of BTCs as shown in Fig. 5(b) in Toride et al. (2003).
It is difficult to judge the quality of your results only from estimated parameter values. The macroporosity you defined may be different from the mobile fraction. I probably need to have look at your observed BTC as well as experimental conditions in details.
If you need further help, please do not hesitate to ask me.
Regards,
Nobuo
--------------------------------
Nobuo Toride ntoride@bio.mie-u.ac.jp