Symmetry in Nonlinear Hydrologic Dynamics under Uncertainty: Modeling Approach
Publication: Journal of Hydrologic Engineering
Volume 14, Issue 10
Abstract
Symmetry methods can be used to transform almost any kind of linear or nonlinear partial differential equation (PDE) that represents a hydrologic process in any dimension to an equivalent ordinary differential equation (ODE). Meanwhile, Kavvas recently shown in 2003 that the conservation equations of hydrologic processes under uncertainty, expressed as linear or nonlinear stochastic ODEs or PDEs, have a one-to-one correspondence to a mixed Eulerian-Lagrangian nonlocal form of the Fokker-Planck equation (FPE) when the underlying process has finite correlation lengths. Under such correspondence, it is possible to obtain a solution for the ensemble behavior of a particular hydrologic process in terms of the solution of its corresponding FPE for the probability distribution function (PDF) of its state variables under appropriate initial and boundary conditions. A major issue with the resulting FPE in the case of conservation equations in PDE form is that spatial gradients of the process state variables appear in the resulting FPE that prevent its solution. Therefore, a formal algorithm is needed to reduce the PDE of a hydrologic process into an ODE in order to eliminate the original spatial gradients of the process state variables in the corresponding FPE of the process. This is accomplished by the symmetry methods. After such transformation, the resulting FPE, which is a linear deterministic PDE, can be solved to obtain the evolutionary PDF, ensemble average, ensemble variance, and any other statistical function of the hydrologic process being investigated.
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References
Ballio, F., and Guadagnini, A. (2004). “Convergence assessment of numerical Monte Carlo simulations in groundwater hydrology.” Water Resour. Res., 40, W04603.
Bluman, G. W., and Anco, S. C. (2002). Symmetry and integration methods for differential equations, Springer, New York.
Cantwell, B. J. (2002). Introduction to symmetry analysis, Cambridge University Press, U.K.
Cayar, M., and Kavvas M. L. (2008). “Ensemble average and ensemble variance behavior of unsteady, one-dimensional groundwater flow in unconfined, heterogeneous aquifers: An exact second order model.” Stochastic Environ. Res. Risk Assess.
Chen, Z. -Q., Govindaraju, R. S., and Kavvas, M. L. (1994a). “Spatial averaging of unsaturated flow equations under infiltration conditions over areally heterogeneous fields. 1: Development of models.” Water Resour. Res., 30, 523–533.
Chen, Z. -Q., Govindaraju, R. S., and Kavvas, M. L. (1994b). “Spatial averaging of unsaturated flow equations under infiltration conditions over areally heterogeneous fields. 2: Numerical simulations.” Water Resour. Res., 30, 535–548.
Clarkson, P. A. (1995). “Nonclassical symmetry reductions for the Boussinesq equation.” Chaos, Solitons Fractals, 5, 2261–2301.
Dogrul, E. C., Kavvas, M. L., and Chen, Z. -Q. (1998). “Prediction of subsurface stormflow in heterogeneous sloping aquifers.” J. Hydrol. Eng., 3(4), 258–267.
Duffy, C. J. (1996). “A two-state integral-balance model for soil moisture and groundwater dynamics in complex terrain.” Water Resour. Res., 32, 2421–2434.
Duffy, C. J., and Cusumano, J. (1998). “A low-dimensional model for concentration-discharge dynamics in groundwater systems.” Water Resour. Res., 34, 2235–2247.
Edwards, M. P., James, M. H., and Selvadurai, A. P. S. (2008). “Lie group symmetry analysis of transport in porous media with variable transmissivity.” J. Math. Anal. Appl. 341(2), 906–921.
Freeze, R. A. (1975). “A stochastic-conceptual analysis of one-dimensional groundwater flow in nonuniform homogeneous media.” Water Resour. Res., 11, 725–741.
Freundlich, H. (1926). Colloid and capillary chemistry, Methuen, London.
Gandarias, M. L., Romer, J. L., and Diaz, J. M. (1999). “Nonclassical symmetry reductions of a porous medium equation with convection.” J. Phys. A, 32, 1461–1473.
Gandarias, M. L., Venero, P., and Ramirez, J. (1998). “Similarity reductions for a nonlinear diffusion equation.” J. Nonlinear Math. Phys., 5(3), 234–244.
Gelhar, L. W., and Axnes, C. L. (1983). “Three-dimensional stochastic analysis of macrodispersion in aquifers.” Water Resour. Res., 19, 161–180.
Goard, J., and Broadbridge, P. (1996). “Nonclassical symmetry analysis of nonlinear reaction–diffusion equations in two spatial dimensions.” Nonlinear Anal. Theory, Methods Appl., 26, 735–754.
Govindaraju, R. S., Jones, S. E., and Kavvas, M. L. (1990). “Approximate analytical solutions for overland flows.” Water Resour. Res., 26(12), 2903–2912.
Gröbner, W., and Knapp, H. (1967). Contribution to the method of Lie series. B. I. Hochschultaschenbücher 802/802a, Bibliographisches Institut, Mannheim, Germany.
Hassan, A. E., Cushman, J. H., and Delleur, J. W. (1998). “A Monte Carlo assessment of Eulerian flow and transport perturbation models.” Water Resour. Res., 34, 1143–1163.
Horne, F. E., and Kavvas, M. L. (1997). “Physics of the spatially averaged snowmelt process.” J. Hydrol., 191, 179–207.
Kapoor, V., and Gelhar, L. W. (1994). “Transport in three-dimensionally heterogeneous aquifers. 1. Dynamics of concentration fluctuations.” Water Resour. Res., 30, 1775–1788.
Kavvas, M. L. (2001). “General conservation equation for solute transport in heterogeneous porous media.” J. Hydrol. Eng., 6(4), 341–350.
Kavvas, M. L. (2003). “Nonlinear hydrologic processes: Conservation equations for determining their means and probability distributions.” J. Hydrol. Eng., 8(2), 44–53.
Kavvas, M. L., and Karakas, A. (1996). “On the stochastic theory of solute transport by unsteady and steady groundwater flow in heterogeneous aquifers.” J. Hydrol., 179, 321–351.
Kim, S. (2003). “The upscaling of one-dimensional unsaturated soil water flow model under infiltration and evapotranspiration boundary conditions.” Ph.D. dissertation, Univ. of California, Davis, Calif.
Kitanidis, P. K. (1988). “Prediction by the method of moments of transport in a heterogeneous formation.” J. Hydrol., 102, 453–473.
Liang, L. (2003). “One-dimensional numerical modeling of the conservation equation for non-reactive stochastic solute transport by unsteady flow field in stream channels.” Ph.D. dissertation, Univ. of California, Davis, Calif.
Mantoglou, A., and Gelhar, L. W. (1987). “Capillary tension head variance, mean soil moisture content, and effective specific soil moisture capacity of transient unsaturated flow in stratified soils.” Water Resour. Res., 23, 47–56.
Moitsheki, R. J., Broadbridge, P., and Edwards, M. P. (2004). “Systematic construction of hidden nonlocal symmetries for the inhomogeneous nonlinear diffusion equation.” J. Phys. A, 37, 8279–8286.
Myeni, S. M., and Leach, P. G. (2007). “Heuristic analysis of the complete symmetry group and nonlocal symmetries for some nonlinear evolution equations.” Math. Methods Appl. Sci., 30(16), 2065–2077.
Ohara, N. (2003). “Numerical and stochastic upscaling of snowmelt process.” Ph.D. dissertation, Univ. of California, Davis, Calif.
Olver, P. J. (1993). Applications of Lie groups to differential equations, Springer, Berlin.
Rehfeldt, K. R., and Gelhar, L. W. (1992). “Stochastic analysis of dispersion in unsteady flow in heterogeneous aquifers.” Water Resour. Res., 28, 2085–2100.
Rubin, Y., and Dagan, G. (1988). “Stochastic analysis of boundaries effects on head spatial variability in heterogeneous aquifers. 1: Constant head boundary.” Water Resour. Res., 24, 1689–1710.
Serrano, S. (1995). “Analytical solutions of the nonlinear groundwater flow equation in unconfined aquifers and the effect of heterogeneity.” Water Resour. Res., 31, 2733–2742.
Tayfur, G., and Kavvas, M. L. (1994). “Spatially averaged conservation equations for interacting rill-interrill area overland flows.” J. Hydraul. Eng., 120(12), 1426–1448.
Tayfur, G., and Kavvas, M. L. (1998). “Areally-averaged overland flow equations at hillslope scale.” Hydrol. Sci. J., 43(3), 361–378.
Wood, B. D., and Kavvas, M. L. (1999a). “Ensemble-averaged equations for reactive transport in porous media under unsteady flow conditions.” Water Resour. Res., 35, 2053–2068.
Wood, B. D., and Kavvas, M. L. (1999b). “Stochastic solute transport under unsteady flow conditions: Comparison of theory, Monte Carlo simulations, and field data.” Water Resour. Res., 35, 2069–2084.
Zabadal, J. R. S., Vilhena, M. T. M. B., Leite, S. Q. B., and Poffal, C. A. (2005). “Solving unsteady problems in water pollution using Lie symmetries.” Ecol. Modell., 186(3), 271–279.
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© 2009 ASCE.
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Received: Jul 31, 2008
Accepted: Feb 24, 2009
Published online: Mar 2, 2009
Published in print: Oct 2009
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