49 research outputs found
Modeling of Overland Flow by the Diffusion Wave Approach
One of the major issues of present times, i.e. water quality degradation and a need for precise answers to transport of pollutants by overland flow, is addressed with special reference to the evaporator pits located adjacent to streams in the oil-producing regions of Eastern Kentucky. The practical shortcomings of the state-of-the-art kinematic wave are discussed and a new mathematical modeling-approach for overland flows using the more comprehensive diffusion wave is attempted as the first step in solving this problem. A Fourier series representation of the solution to the diffusion wave is adopted and found to perform well. The physically justified boundary conditions for steep slopes is considered and both numerical and analytical schemes are developed. The zero-depth-gradient lower condition is used and found to be adequate. The steady state analysis for mild slopes is found to be informative and both analytical and numerical solutions are found. The effect of imposing transients on the steady state solution are considered. Finally the cases for which these techniques can be used are presented
Fractal scaling analysis of groundwater dynamics in confined aquifers
Groundwater closely interacts with surface water and even climate
systems in most hydroclimatic settings. Fractal scaling analysis of
groundwater dynamics is of significance in modeling hydrological
processes by considering potential temporal long-range dependence
and scaling crossovers in the groundwater level fluctuations. In
this study, it is demonstrated that the groundwater level
fluctuations in confined aquifer wells with long observations
exhibit site-specific fractal scaling behavior. Detrended
fluctuation analysis (DFA) was utilized to quantify the
monofractality, and multifractal detrended fluctuation analysis
(MF-DFA) and multiscale multifractal analysis (MMA) were employed to
examine the multifractal behavior. The DFA results indicated that
fractals exist in groundwater level time series, and it was shown
that the estimated Hurst exponent is closely dependent on the length
and specific time interval of the time series. The MF-DFA and MMA
analyses showed that different levels of multifractality exist,
which may be partially due to a broad probability density
distribution with infinite moments. Furthermore, it is demonstrated
that the underlying distribution of groundwater level fluctuations
exhibits either non-Gaussian characteristics, which may be fitted by
the LĂ©vy stable distribution, or Gaussian characteristics
depending on the site characteristics. However, fractional Brownian
motion (fBm), which has been identified as an appropriate model to
characterize groundwater level fluctuation, is Gaussian with finite
moments. Therefore, fBm may be inadequate for the description of
physical processes with infinite moments, such as the groundwater
level fluctuations in this study. It is concluded that there is
a need for generalized governing equations of groundwater flow
processes that can model both the long-memory behavior and
the Brownian finite-memory behavior
Ensemble modeling of stochastic unsteady open-channel flow in terms of its time–space evolutionary probability distribution – Part 1: theoretical development
The Saint-Venant equations are commonly used as the
governing equations to solve for modeling the spatially varied unsteady flow
in open channels. The presence of uncertainties in the channel or flow
parameters renders these equations stochastic, thus requiring their solution
in a stochastic framework in order to quantify the ensemble behavior and the
variability of the process. While the Monte Carlo approach can be used for
such a solution, its computational expense and its large number of
simulations act to its disadvantage. This study proposes, explains, and
derives a new methodology for solving the stochastic Saint-Venant equations
in only one shot, without the need for a large number of simulations. The
proposed methodology is derived by developing the nonlocal
Lagrangian–Eulerian Fokker–Planck equation of the characteristic form of
the stochastic Saint-Venant equations for an open-channel flow process, with
an uncertain roughness coefficient. A numerical method for its solution is
subsequently devised. The application and validation of this methodology are
provided in a companion paper, in which the statistical results computed by
the proposed methodology are compared against the results obtained by the
Monte Carlo approach
Ensemble modeling of stochastic unsteady open-channel flow in terms of its time–space evolutionary probability distribution – Part 2: numerical application
The characteristic form of the Saint-Venant equations is
solved in a stochastic setting by using a newly proposed Fokker–Planck
Equation (FPE) methodology. This methodology computes the ensemble behavior
and variability of the unsteady flow in open channels by directly solving
for the flow variables' time–space evolutionary probability distribution.
The new methodology is tested on a stochastic unsteady open-channel flow
problem, with an uncertainty arising from the channel's roughness
coefficient. The computed statistical descriptions of the flow variables are
compared to the results obtained through Monte Carlo (MC) simulations in
order to evaluate the performance of the FPE methodology. The comparisons
show that the proposed methodology can adequately predict the results of the
considered stochastic flow problem, including the ensemble averages,
variances, and probability density functions in time and space. Unlike the
large number of simulations performed by the MC approach, only one
simulation is required by the FPE methodology. Moreover, the total
computational time of the FPE methodology is smaller than that of the MC
approach, which could prove to be a particularly crucial advantage in
systems with a large number of uncertain parameters. As such, the results
obtained in this study indicate that the proposed FPE methodology is a
powerful and time-efficient approach for predicting the ensemble average and
variance behavior, in both space and time, for an open-channel flow process
under an uncertain roughness coefficient
Governing equations of transient soil water flow and soil water flux in multi-dimensional fractional anisotropic media and fractional time
In this study dimensionally consistent governing equations of
continuity and motion for transient soil water flow and soil water flux in
fractional time and in fractional multiple space dimensions in anisotropic
media are developed. Due to the anisotropy in the hydraulic conductivities
of natural soils, the soil medium within which the soil water flow occurs is
essentially anisotropic. Accordingly, in this study the fractional
dimensions in two horizontal and one vertical directions are considered to
be different, resulting in multi-fractional multi-dimensional soil space
within which the flow takes place. Toward the development of the fractional
governing equations, first a dimensionally consistent continuity equation
for soil water flow in multi-dimensional fractional soil space and
fractional time is developed. It is shown that the fractional soil water
flow continuity equation approaches the conventional integer form of the
continuity equation as the fractional derivative powers approach integer
values. For the motion equation of soil water flow, or the equation of water
flux within the soil matrix in multi-dimensional fractional soil space and
fractional time, a dimensionally consistent equation is also developed.
Again, it is shown that this fractional water flux equation approaches the
conventional Darcy equation as the fractional derivative powers approach
integer values. From the combination of the fractional continuity and motion
equations, the governing equation of transient soil water flow in
multi-dimensional fractional soil space and fractional time is obtained. It
is shown that this equation approaches the conventional Richards equation as
the fractional derivative powers approach integer values. Then by the
introduction of the Brooks–Corey constitutive relationships for soil water
into the fractional transient soil water flow equation, an explicit form of
the equation is obtained in multi-dimensional fractional soil space and
fractional time. The governing fractional equation is then specialized to
the case of only vertical soil water flow and of only horizontal soil water
flow in fractional time–space. It is shown that the developed governing
equations, in their fractional time but integer space forms, show behavior
consistent with the previous experimental observations concerning the
diffusive behavior of soil water flow