Ensemble average and ensemble variance behavior of unsteady, one-dimensional groundwater flow in unconfined, heterogeneous aquifers: an exact second-order model
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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- "stochastic root-water uptake process (Kim et al., 2005.b), stochastic solute transport process in river channels (Liang and Kavvas, 2008), stochastic snow accumulation and melt processes (Ohara et al., 2008), and unconfined groundwater flow process (Cayar and Kavvas, 2009.a and 2009.b). "
ABSTRACT: A stochastic kinematic wave model for open-channel flow under uncertain channel properties is developed. The Fokker-Planck equation (FPE) of the kinematic open-channel flow process under uncertain channel properties is developed by using the method of characteristics. Every stochastic partial differential equation has a one-to-one relationship with a nonlocal Lagrangian-Eulerian FPE (LEFPE). As such, one can develop an LEFPE for the governing equation of any hydrologic or hydraulic process as the physically based stochastic model of the particular process. A linear, deterministic, differential equation in Eulerian-Lagrangian form, LEFPE provides a quantitative description of the evolution of the probability density functions of the state variables of the process at one shot to describe the ensemble behavior of the process instead of the commonly used many Monte Carlo simulations to quantify the same ensemble behavior. Under certain assumptions, the nonlocal LEFPE reduces to the classical local FPE, which is more convenient in practical applications. The numerical solutions of the resulting FPE are validated by Monte Carlo simulations under varying channel conditions. The validation results demonstrated that the developed FPE can express the ensemble behavior of the kinematic wave process under uncertain channel properties adequately.Journal of Hydrologic Engineering 01/2012; 17(1):168-181. DOI:10.1061/(ASCE)HE.1943-5584.0000425 · 1.62 Impact Factor
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- "The basic methodology has developed by Kavvas (2005) for the probabilistic modeling of nonlinear hydrologic conservation equations and by Kim et al. (2005) for numerical modeling of Fokker-Planck equation. The methodology has been shown to be useful in various applications in hydrology (Kim, 2006; Liang and Kavvas, 2008; Ohara et al., 2008; Cayar and Kavvas, 2009; Kim et al., 2011a). As an application, how the soil water probability density function (PDF) will be affected by future climate changes provided by diverse global climate models and future greenhouse gas emission scenarios will be analyzed in an attempt to evaluate future droughts compared to the present drought. "
ABSTRACT: In this study, the effect of climate change on drought occurrence is modeled using a conceptual soil water model and results from the model are analyzed. Future climatic data are prepared by applying climate information drawn by four global climate models in accordance with greenhouse gas emission scenarios A2, A1B and B2 to the spatio-temporal changing factor method which is a relatively simple statistical downscaling method to downscale the climate information to 60 meteorological observation points in Korea. Using a conceptual soil water model that can simulate temporal evolution of the soil water probability density function, changes in the soil water probability density function relative to climate changes are analyzed. Based on these changes in the probability density function, we can see that drought will occur more frequently in Korea in future. Since drought is not an absolute quantitative concept but is a spatiotemporally relative concept and is not an average event in terms of probability, it will be more realistic to define it as an extreme event with a low probability to occur at a site during a period and from this viewpoint, the quantile comparison method (that is, comparing the non-exceedance probability of drought occurrence of now with that of the future based on the reference soil water) applied in this study is expected to be appropriately applicable to the evaluation of the effects of extreme hydrologic events.
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ABSTRACT: Second-order exact ensemble averaged equation for linear stochastic differential equations with multiplicative randomness and random forcing is obtained by using the cumulant expansion ensemble averaging method and by taking the time dependent sure part of the multiplicative operator into account. It is shown that the satisfaction of the commutativity and the reversibility requirements proposed earlier for linear stochastic differential equations without forcing are necessary for the linear stochastic differential equations with forcing when the cumulant expansion ensemble averaging method is used. It is shown that the applicability of the operator equality, which is used for the separation of operators in the literature, is also subjected to the satisfaction of the commutativity and the reversibility requirements. The van Kampen’s lemma, which is proposed for the analysis of nonlinear stochastic differential equations, is modified in order to make the probability density function obtained through the lemma depend on the forcing terms too. The second-order exact ensemble averaged equation for linear stochastic differential equations with multiplicative randomness and random forcing is also obtained by using the modified van Kampen’s lemma in order to validate the correctness of the modified lemma. Second-order exact ensemble averaged equation for one dimensional convection diffusion equation with reaction and source is obtained by using the cumulant expansion ensemble averaging method. It is shown that the van Kampen’s lemma can yield the cumulant expansion ensemble averaging result for linear stochastic differential equations when the lemma is applied to the interaction representation of the governing differential equation. It is found that the ensemble averaged equations given for one the dimensional convection diffusion equation with reaction and source in the literature obtained by applying the lemma to the original differential equation are restricted with small sure part of multiplicative operator. Second-order exact differential equations for the evolution of the probability density function for the one dimensional convection diffusion equation with reaction and source and one dimensional nonlinear overland flow equation with source are obtained by using the modified van Kampen’s lemma. The equation for the evolution of the probability density function for one dimensional nonlinear overland flow equation with source given in the literature is found to be not second-order exact. It is found that the differential equations for the evolution of the probability density functions for various hydrological processes given in the literature are not second-order exact. The significance of the new terms found due to the second-order exact ensemble averaging performed on the one dimensional convection diffusion equation with reaction and source and during the application of the van Kampen’s lemma to the one dimensional nonlinear overland flow equation with source is investigated.Stochastic Environmental Research and Risk Assessment 01/2012; 27(1). DOI:10.1007/s00477-012-0591-z · 2.67 Impact Factor