Abstract
This short paper shows that different choices made for control functions of information in hydrodynamic flow problems can have significant implications for interpretations of the system. Using as a simple illustration the case of a steady-state, one-dimensional flow model with no internal sources or sinks and with the hydraulic conductivity depending on a single parameter and the distance from the origin, it is shown that, even when a continuous, error-free head data set is provided, statements about the uniqueness or not of the inverse solution are conditioned on the choice of the control function. Care has to be exercised in obtaining physically meaningful results and, depending on the model assumptions and the data available, there may not be acceptable models. It is also shown that there may be more than one model behavior that is acceptable. The results have implications for the hydrodynamic upscaling problem for flow in permeable media, for ensemble averaging methods, and for parameter determination for deterministic models of permeable flow.
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REFERENCES
Carrera, J., and Neuman, S. P., 1986a, Estimation of aquifer parameters under transient and steady conditions: 1. Maximum likelihood method of incorporating prior information: Water Resour. Res., v. 22, no. 2, p. 199–210.
Carrera, J., and Neuman, S. P., 1986b, Estimation of aquifer parameters under transitional and steady state conditions: 3. Application to synthetic and field data: Water Resour. Res., v. 22, no. 2, p. 228–242.
Carrera, J., 1987, State of the art of inverse problem applied to the flow and solute transport equations, in Curtodio, E., Gurgui, A., and Lobo Ferreira, J. P., eds., Groundwater flow and quality modeling: NATO ASI Series C: Mathematical and Physical Sciences, Vol. 224: Dordrecht, The Netherlands, p. 549–583.
Chavent, G., 1974, Identification of functional parameters in partial differential equations, in Goodson, R. E., and Polis, M., eds., Identification of parameters in distributed systems: American Society of Mechanical Engineers, New York, p. 31–48.
Chavent, G., 1987, Identifiability of parameters in the output least squares formulation, in Walter, E., ed., Identifiability of parametric models: Pergamon, New York, 1987.
Clifton, P. M., and Neuman, S. P., 1982, Effects of kriging and inverse modeling on conditional simulation of the Avra valley aquifer in southwestern Arizona: Water Resour. Res., v. 18, no. 4, p. 1215–1234.
Desbarats, A. J., 1992, Spatial averaging of hydraulic conductivity in three-dimensional heterogeneous porous media: Math. Geol., v. 24, no. 3, p. 249–267.
Desbarats, A. J., and Dimitrakopoulos, R., 1990, Geostatistical modeling of transmissibility for 2D reservoir studies: SPE Formation Eval., vol. 8, no. 1, p. 437–443.
Gomez-Hernandez, J. J., and Gorelick, S. M., 1989, Effective groundwater model parameter values: Influence of spatial variability of hydraulic conductivity, leakage, and recharge: Water Resour. Res., v. 22, no. 2, p. 199–210.
Kitamura, S., and Nakagiri, S., 1977, Identifiability of spatially-varying and constant parameters in distributed systems of parabolic type: SIAM Jour. Contr. Optimiz., v. 15, no. 5, p. 785–802.
Kitanidis, P. K., and Vomvoris, E. G., 1983, A geostatistical approach to the inverse problem in groundwater modeling (steady state) and one-dimensional simulations: Water Resour. Res., v. 19, no. 3, p. 677–690.
de Marsily, G., Delhomme, J. P., Coudrain-Ribstein, A., and Lavenue, A. M., 2000, Four decades of inverse problems in hydrogeology, in Zhang, D., and Winter, C. L., eds., GSA special paper 348, Theory, modeling, and field investigation in hydrogeology, A special volume in honor of Shlomo P. Neuman's 60th birthday: Geological Society of America, Boulder, CO, p. 1–17.
Neuman, S. P., 1973, Calibration of distributed parameter groundwater flow models viewed as a multiple-objective decision process under uncertainty: Water Resour. Res., v. 9, no. 4, p. 1006–1021.
Neuman, S. P., and Yakowitz, S., 1979, A statistical approach to the inverse problem of aquifer hydrology: 1. Theory: Water Resour. Res., v. 15, no. 4, p. 845–860.
Rubin, Y., and Gomez-Hernandez, J. J., 1990, A stochastic approach to the problem of upscaling of conductivity in disordered media: Theory and unconditional numerical simulations: Water Resour. Res., v. 26, no. 4, p. 691–701.
Samper, F. J., and Neuman, S. P., 1989, Estimation of spatial covariance structures by adjoint state maximum likelihood cross-validation: 1. Theory: Water Resour. Res., v. 25, no. 3, p. 351–362.
Sun, N.-Z., and Yeh, W. W.-G., 1990, Coupled inverse problems in groundwater modeling: 2. Identifiability and experimental design: Water Resour. Res., v. 26, no. 10, p. 2527–2540.
Yeh, W. W.-G., 1986, Review of parameter identification procedures in groundwater hydrology: The inverse problem: Water Resour. Res., v. 22, no. 2, p. 95–108.
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Lerche, I., Paleologos, E.K. Control Function Measures for Hydrodynamic Problems. Mathematical Geology 34, 345–355 (2002). https://doi.org/10.1023/A:1014899024189
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DOI: https://doi.org/10.1023/A:1014899024189