Abstract
A new stochastic model for unconfined groundwater flow is proposed. The developed evolution equation for the probabilistic behavior of unconfined groundwater flow results from random variations in hydraulic conductivity, and the probabilistic description for the state variable of the nonlinear stochastic unconfined flow process becomes a mixed Eulerian–Lagrangian–Fokker–Planck equation (FPE). Furthermore, the FPE is a deterministic, linear partial differential equation (PDE) and has the advantage of providing the probabilistic solution in the form of evolutionary probability density functions. Subsequently, the Boussinesq equation for one-dimensional unconfined groundwater flow is converted into a nonlinear ordinary differential equation (ODE) and a two-point boundary value problem through the Boltzmann transformation. The resulting nonlinear ODE is converted to the FPE by means of ensemble average conservation equations. The numerical solutions of the FPE are validated with Monte Carlo simulations under varying stochastic hydraulic conductivity fields. Results from the model application to groundwater flow in heterogeneous unconfined aquifers illustrate that the time–space behavior of the mean and variance of the hydraulic head are in good agreement for both the stochastic model and the Monte Carlo solutions. This indicates that the derived FPE, as a stochastic model of the ensemble behavior of unconfined groundwater flow, can express the spatial variability of the unconfined groundwater flow process in heterogeneous aquifers adequately. Modeling of the hydraulic head variance, as shown here, will provide a measure of confidence around the ensemble mean behavior of the hydraulic head.
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References
Bear J (1972) Dynamics of fluids in porous media. American Elsevier, New York
Ballio F, Guadagnini A (2004) Convergence assessment of numerical Monte Carlo simulations in groundwater hydrology. Water Resour Res. doi:10.1029/2003WR002876
Chang JS, Cooper G (1970) A practical difference scheme for Fokker–Planck equations. J Comput Phys 6:1–16
Cushman JH (1997) The physics of fluids in hierarchical porous media: angstroms to miles. Springer, New York
Dagan G (1989) Flow and transport in porous formations. Springer, New York
Dogrul EC, Kavvas ML, Chen ZQ (1998) Prediction of subsurface stormflow in heterogeneous sloping aquifers. J Hydrol Eng 3(4):258–267
Freeze RA (1975) A stochastic-conceptual analysis of one-dimensional groundwater flow in nonuniform homogeneous media. Water Resour Res 11:725–741
Gardiner CW (2004) Handbook of stochastic methods. Springer, New York
Gelhar LW (1993) Stochastic subsurface hydrology. Prentice-Hall, Upper Saddle River
Hassan AE, Cushman JH, Delleur JW (1998) A Monte Carlo assessment of Eulerian flow and transport perturbation models. Water Resour Res 34:1143–1163
Kavvas ML (2003) Nonlinear hydrologic processes: conservation equations for determining their means and probability distributions. J Hydrol Eng, ASCE 8(2):44–53
Kim S (2003) The upscaling of one-dimensional unsaturated soil water flow model under infiltration and evapotranspiration boundary conditions. PhD dissertation. University of California, Davis
Kim S, Kavvas ML, Chen ZQ (2005) Root-water uptake model at heterogeneous soil fields. J Hydrol Eng 10(2):160–167
Liang L (2003) One-dimensional numerical modeling of the conservation equation for Non- reactive stochastic solute transport by unsteady flow field in stream channels. PhD dissertation. University of California, Davis
Liang L, Kavvas ML (2008) Modeling of solute transport and macrodispersion by unsteady stream flow under uncertain conditions. ASCE J Hydrol Eng (in press)
Ohara N (2003) Numerical and stochastic upscaling of snowmelt process. PhD dissertation. University of California, Davis
Polubarinova-Kochina P (1962) Theory of groundwater movement. Princeton University Press, New Jersey
Serrano S (1992) The form of the dispersion equation under recharge and variable velocity, and its analytical solution. Water Resour Res 28:1801–1808
Sidiropoulos EG, Tolikas PK (1984) Similarity and iterative solutions of Boussineq equation. J Hydrol 74:31–41
Woodbury AD, Sudicky EA (1991) The geostatistical characteristics of the Borden Aquifer. Water Resour Res 27:533–546
Yoon J, Kavvas ML (2003) Probabilistic solution to stochastic overland flow equation. J Hydrol Eng ASCE 8(2):54–63
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Cayar, M., Kavvas, M.L. Ensemble average and ensemble variance behavior of unsteady, one-dimensional groundwater flow in unconfined, heterogeneous aquifers: an exact second-order model. Stoch Environ Res Risk Assess 23, 947–956 (2009). https://doi.org/10.1007/s00477-008-0263-1
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DOI: https://doi.org/10.1007/s00477-008-0263-1