Symmetry in Nonlinear Hydrologic Dynamics under Uncertainty: Ensemble Modeling of 2D Boussinesq Equation for Unsteady Flow in Heterogeneous Aquifers
Publication: Journal of Hydrologic Engineering
Volume 14, Issue 10
Abstract
In this paper, the probabilistic symmetry analysis approach proposed in the previous paper by Cayar and Kavvas in 2009 is adopted and applied to the analysis of nonlinear two-dimensional (2D) groundwater flow subject to random hydraulic conductivity field. The resulting model has the form of a two-dimensional Fokker-Planck equation (FPE). Following the methodology reported by Cayar and Kavvas in 2009, the 2D Boussinessq equation for unconfined groundwater flow in a heterogeneous aquifer is transformed to a system of ordinary differential equations (ODEs) through Lie group symmetry analysis in order to eliminate the spatial derivatives occurring in the governing equation. Next, these derived stochastic ODEs are ensemble averaged with second-order cumulant expansion and converted into a linear, deterministic partial differential equation. This new equation is called the FPE, and describes the evolution of the probability density function (PDF) of the hydraulic head, thereby, describing the ensemble behavior of unconfined aquifer flow in a heterogeneous medium. Once a solution of this FPE is obtained, one can then obtain the ensemble average and ensemble variance behavior of the hydraulic head through an expectation operation. The numerical solutions of the FPE are validated with the Monte Carlo simulations. The validation results indicate that the FPE technique combined with Lie symmetry analysis can provide encouraging results in estimating the time-space behavior of the mean and variance of the hydraulic head in heterogeneous aquifers.
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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.
Bear, J. (1972). Dynamics of fluids in porous media, Elsevier Science, New York.
Bear, J., and Verruijt, A. (1987). Modeling groundwater flow and pollution, Reidel, Dordrecht, The Netherlands.
Bedient, P. B., and Huber, W. C. (1992). Hydrology and floodplain analysis, Addison-Wesley, Reading, Mass.
Bjerg, P. L., Hinsby, K., Christensen, T. H., and Gravesen, P. (1992). “Spatial variability of hydraulic conductivity of an unconfined sandy aquifer determined by a mini slug test.” J. Hydrol., 136, 107–122.
Bluman, G. W., and Anco, S. C. (2002). Symmetry and integration methods for differential equations, Springer, New York.
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.” Special issue in stochastic environmental research and risk assessment “Modeling and prediction of complex environmental systems.”
Cayar, M., and Kavvas, M. L. (2009). “Symmetry in nonlinear hydrologic dynamics under uncertainty: A modeling approach.” J. Hydrol. Eng. (Submitted).
Cantwell, B. J. (2002). Introduction to symmetry analysis, Cambridge University Press, Cambridge, U.K.
Chang, J. S., and Cooper, G. (1970). “A practical difference scheme for Fokker-Planck equations.” J. Comput. Phys., 6, 1–16.
Dagan, G. (1986). “Statistical theory of groundwater flow and transport: Pore to laboratory, laboratory to formation, and formation to regional scale.” Water Resour. Res., 22(09S), 120S–134S.
Dagan, G. (1989). Flow and transport in porous formations, Springer, New York.
Deng, F. W., and Cushman, J. H. (1995). “On higher-order corrections to the flow velocity covariance tensor.” Water Resour. Res., 31(7), 1659–1672.
Gardiner, C. W. (2004). Handbook of stochastic methods, Springer, New York.
Gelhar, L. W. (1977). “Effects of hydraulic conductivity variations on groundwater flows.” Proc., 2nd Int. Symp. On Stochastic Hydraulics, Water Resource Publications, Englewood, Colo., 409–428.
Gelhar, L. W. (1993). Stochastic subsurface hydrology, Prentice-Hall, Upper Saddle River, N.J.
Gelhar, L. W., Welty, C. W., and Rehfeldt, K. R. (1992). “A critical review of data on field scale dispersion in aquifers.” Water Resour. Res., 28(7), 1955–1974.
Grobner, 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.
Hess, K. M., Wolf, S. H., and Celia, M. A. (1992). “Large-scale natural gradient tracer tests in sand and gravel, Cape Cod, Massachusetts. 3: Hydraulic conductivity variability and calculated macrodispersivities.” Water Resour. Res., 28(8), 2011–2027.
Kavvas, M. L. (2003). “Nonlinear hydrologic processes: Conservation equations for determining their means and probability distributions.” J. Hydrologic Eng., 8(2), 44–53.
Kim, S., Kavvas, M. L., and Chen, Z. Q. (2005). “Root-water uptake model at heterogeneous soil fields.” J. Hydrologic Eng., 10(2), 160–167.
Kinzelback, W. (1986). Groundwater modeling, Vol. VII, Elsevier, Amsterdam.
Liang, L., and Kavvas, M. L. (2008). “Modeling of solute transport and macrodispersion by unsteady stream flow under uncertain conditions.” ASCE J. Hydrol. Eng. 11, 510–520.
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.
Polubarinova-Kochina, P. (1962). Theory of groundwater movement, Princeton University Press, Princton, N.J.
Serrano, S. E. (1995). “Analytical solutions of the nonlinear groundwater flow equation in unconfined aquifers and the effect of heterogeneity.” Water Resour. Res., 31, 2733–2742.
Smith, L., and Freeze, R. A. (1979). “Stochastic analysis of steady state groundwater flow in bounded domain: two dimensional simulation.” Water Resour. Res., 15, 1543–1559.
Stauffer, F. (2005). “Uncertainty estimation of pathlines in ground water models.” Ground Water, 43(6), 843–849.
Sudicky, E. A. (1986). “A natural gradient experiment on solute transport in a sand aquifer: Spatial variability of hydraulic conductivity and its role in the dispersion process.” Water Resour. Res., 22, 2069–2082.
Yoon, J. Y., and Kavvas, M. L. (2003). “Probabilistic solution to stochastic overland flow equation.” J. Hydrol. Eng., 8(2), 54–63.
Information & Authors
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Published In
Journal of Hydrologic Engineering
Volume 14 • Issue 10 • October 2009
Pages: 1173 - 1184
Copyright
© 2009 ASCE.
History
Received: Aug 1, 2008
Accepted: Mar 4, 2009
Published online: Sep 15, 2009
Published in print: Oct 2009
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