Ambler, PA, United States
Ambler, PA, United States

Time filter

Source Type

Patel A.,Camp Dresser and McKee Inc. | Serrano S.E.,HydroScience Inc.
Journal of Hydrology | Year: 2011

The method of decomposition of Adomian is an approximate analytical series to solve linear or nonlinear differential equations. An important limitation is that a decomposition expansion in a given coordinate explicitly uses the boundary conditions in such axis only, but not necessarily those on the others. This paper presents improvements of the method that permit the practical consideration of all of the conditions imposed on multidimensional initial-value and boundary-value problems governed by (nonlinear) groundwater equations, and the analytical modeling of irregularly-shaped heterogeneous aquifers subject to sources and sinks. The method yields simple solutions of dependent variables that are continuous in space and time, which easily permit the derivation of heads, gradients, seepage velocities and fluxes, thus minimizing instability. It could be valuable in preliminary analysis prior to more elaborate numerical analysis. © 2010 Elsevier B.V.


Tiaif S.,University of Bengkulu | Serrano S.E.,HydroScience Inc.
Groundwater | Year: 2015

The unconfined alluvial aquifer at Louisville, Kentucky, is an important source of water for domestic and industrial uses. It has been the object of several modeling studies in the past, particularly via the application of classical analytical solutions, and numerical solutions (finite differences and finite elements). A new modeling procedure of the Louisville aquifer is presented based on a modification of Adomian's Decomposition Method (ADM) to handle irregularly shaped boundaries. The new approach offers the simplicity, stability, and spatial continuity of analytical solutions, in addition to the ability to handle irregular boundaries typical of numerical solutions. It reduces to the application of a simple set of algebraic equations to various segments of the aquifer. The calculated head contours appear in reasonably agreement with those of previous studies, as well as with those from measured head values from the U.S. Geological Survey field measurement program. A statistical comparison of the error standard deviation is within the same range as that reported in previous studies that used complex numerical solutions. The present methodology could be easily implemented in other aquifers when preliminary results are needed, or when scarce hydrogeologic information is available. Advantages include a simple approach for preliminary groundwater modeling; an analytic description of hydraulic heads, gradients, fluxes, and flow rates; state variables are described continuously over the spatial domain; complications from stability and numerical roundoff are minimized; there is no need for a numerical grid or the handling of large sparse matrices; there is no need to use specialized groundwater software, because all calculations may be done with standard mathematics or spreadsheet programs. Nonlinearity, the effect of higher order terms, and transient simulations could be included if desired. © 2014, National Ground Water Association.


Serrano S.E.,HydroScience Inc.
Journal of Hydrologic Engineering | Year: 2016

The propagation of flood waves in rivers is governed by the Saint Venant equations. Under certain simplifying assumptions, these nonlinear equations have been solved numerically via computationally intensive, specialized software. There is a cogent need for simple analytical solutions for preliminary analyses. In this paper, new approximate analytical solutions to the nonlinear kinematic wave equation and the nonlinear dynamic wave equations in rivers are presented. The solutions have been derived by combining Adomian's decomposition method (ADM), the method of characteristics, the concept of double decomposition, and successive approximation. The new solutions compare favorably with independent simulations using the modified finite-element method and field data at the Schuylkill River near Philadelphia. The time to peak calculated by the analytical and numerical methods is in excellent agreement. There appears to be some minor differences in the peak magnitude and recession limb, possibly because of numerical dissipation. Including the momentum equation in the analysis causes a decrease in the magnitude of the flow rate at all times. Except for the flood peak, the nonlinear analytical solution exhibits lower flow rates than the numerical solution. The numerical solution also shows higher dispersion. The new analytical solutions are easy to apply, permit an efficient preliminary forecast under scarce data, and an analytic description of flow rates continuously over the spatial and temporal domains. They may also serve as a potential source of reference data for testing new numerical methods and algorithms proposed for the open channel flow equations. The ADM nonlinear kinematic and nonlinear dynamic wave solutions exhibit the usual features of nonlinear hydrographs, namely, their asymmetry with respect to the center of mass, with sharp rising limbs and flatter recession limbs. Linear approximations of the governing equations usually miss these important features of nonlinear waves. The greatest portion of the magnitude of discharge is given by the initial nonlinear kinematic wave component, which implies that in the lower Schuylkill River the translational components dominate the propagation of flood waves, in agreement with previous research. The nonlinear dynamic wave better predicts the flow rate during peak times and especially during recession and low-flow periods. Thus, while both the nonlinear kinematic and the nonlinear dynamic wave models are based on simple approximate analytical solutions that are easy to implement, the nonlinear kinematic wave equation model requires less data and less computational effort. © 2015 American Society of Civil Engineers.


Serrano S.E.,HydroScience Inc.
Transport in Porous Media | Year: 2012

Many problems in regional groundwater flow require the characterization and forecasting of variables, such as hydraulic heads, hydraulic gradients, and pore velocities. These variables describe hydraulic transients propagating in an aquifer, such as a river flood wave induced through an adjacent aquifer. The characterization of aquifer variables is usually accomplished via the solution of a transient differential equation subject to time-dependent boundary conditions. Modeling nonlinear wave propagation in porous media is traditionally approached via numerical solutions of governing differential equations. Temporal or spatial numerical discretization schemes permit a simplification of the equations. However, they may generate instability, and require a numerical linearization of true nonlinear problems. Traditional analytical solutions are continuous in space and time, and render a more stable solution, but they are usually applicable to linear problems and require regular domain shapes. The method of decomposition of Adomian is an approximate analytical series to solve linear or nonlinear differential equations. It has the advantages of both analytical and numerical procedures. An important limitation is that a decomposition expansion in a given coordinate explicitly uses the boundary conditions in such axis only, but not necessarily those on the others. In this article we present improvements of the method consisting of a combination of a partial decomposition expansion in each coordinate in conjunction with successive approximation that permits the consideration of boundary conditions imposed on all of the axes of a transient multidimensional problem; transient modeling of irregularly-shaped aquifer domains; and nonlinear transient analysis of groundwater flow equations. The method yields simple solutions of dependent variables that are continuous in space and time, which easily permit the derivation of heads, gradients, seepage velocities and fluxes, thus minimizing instability. It could be valuable in preliminary analysis prior to more elaborate numerical analysis. Verification was done by comparing decomposition solutions with exact analytical solutions when available, and with controlled experiments, with reasonable agreement. The effect of linearization of mildly nonlinear saturated groundwater equations is to underestimate the magnitude of the hydraulic heads in some portions of the aquifer. In some problems, such as unsaturated infiltration, linearization yields incorrect results. © 2012 Springer Science+Business Media B.V.


Mathematical models are the means to characterize variables quantitatively in many groundwater problems. Recent advances in applied mathematics have perfected what is now called Adomian's decomposition method (ADM), a simple modelling procedure for practical applications. Decomposition exhibits the benefits of analytical solutions (i.e. stability, analytic derivation of heads, gradients, fluxes and simple programming). It also offers the advantages of traditional numerical methods (i.e. consideration of heterogeneity, irregular domain shapes and multiple dimensions). In addition, decomposition is one of the few systematic procedures for solving nonlinear equations. By far its greatest advantage is its simplicity of application. It may produce simple results for preliminary simulations, or in cases with scarce information. The method is described with simple applications to regional groundwater flow. Many applications in groundwater flow and contaminant transport are available in the literature. © 2013 Copyright 2013 IAHS Press.

Loading HydroScience Inc. collaborators
Loading HydroScience Inc. collaborators