On a power series solution to the Boussinesq equation

For certain initial and boundary conditions the Boussinesq equation, a nonlinear partial differential equation describing the flow of water in unconfined aquifers, can be reduced to a boundary value problem for a nonlinear ordinary differential equation. Using Song et al.'s (2007) [7] approach, we show that for zero head initial condition and power-law flux boundary condition at the inlet boundary, the solution in the form of power series can be obtained with Barenblatt's (1990) [2] rescaling procedure applied to the power series solution obtained in Song et al. (2007) [7] for the power-law head boundary condition. Polynomial approximations can then be obtained by taking terms from the power series. Although for a small number of terms the newly obtained approximations may be worse than polynomial approximations obtained by other techniques, any desired accuracy can be achieved by taking more terms from the power series.     

Visualization of aquifer response to periodic forcing

Many studies of periodic forcing and response in aquifers have focused on describing the induced fluctuations in hydraulic head, without much consideration of the time-dependent flows. Visualization techniques presented in this paper can be applied to obtain a more physically intuitive impression of groundwater motion in aquifers that undergo periodic fluctuations of hydraulic head. The concepts of velocity ellipse and displacement ellipse are introduced as methods for visualizing oscillatory flows associated with individual forcing modes. Cyclical trajectories illustrate the potential complexity of flow paths that can arise due to superposition of modal responses. The full periodic motion that results due to superposition of the mean flow and modal flows is visualized using streaklines. An animated time series of streaklines provides an intuitive impression of the flow and affords insight into apparent dispersion phenomenon that can arise due to periodic fluctuations in both the strength and direction of groundwater flow. Electronic animations are available from the authors.     

Analysis of karst aquifer spring flows with a gray system decomposition model

There are approximately 470,000 km(2) of karst aquifers that feed many large springs in North China. Turbulent flow often exists in these karst aquifers, which means that the classical ground water model based on Darcy's law cannot be applied here. Ground water data are rare for these aquifers. As a consequence, it is difficult to quantitatively investigate ground water flow in these karst systems. The purpose of this study is to develop a parsimonious model that predicts karst spring discharge using gray system theory. In this theory, a white color denotes a system that is completely characterized and a black color represents a system that is totally unknown. A gray system thus describes a complex system whose characteristics are only partially known or known with uncertainty. Using this theory, we investigated the karst spring discharge time series over different time scales. First, we identified three specific components of spring discharge: the long-term trend, periodic variation, and random fluctuation. We then used the gray system model to simulate the long-term trend and obtain periodic variation and random fluctuation components. Subsequently, we developed a predictive model for karst spring discharge. Application of the model to Liulin Springs, a representative example of karst springs in northern China, shows that the model performs well. The predicted results suggest that the Liulin Springs discharge will likely decrease over time, with small fluctuations.     

A semi-analytical time integration for numerical solution of Boussinesq equation

A semi-analytical time integration method is proposed for the numerical simulation of transient groundwater flow in unconfined aquifers by the nonlinear Boussinesq equation. The method is based on the analytical solution of the system of ordinary differential equations with constant coefficients. While it is unconditionally stable and more accurate than the finite difference methods, the computational cost is much more expensive than (can be more than 10 times) that of the finite difference methods for a single time step. However, by partitioning the nonlinear parameters into linear and nonlinear parts, the costly computation can be performed only once. With larger and less variable time step sizes, the total computational cost can be significantly reduced. Three examples are included to illustrate the advantages and limitations of the proposed method.     

Efficient conceptual model for simulating the effect of aquifer heterogeneity on natural groundwater discharge to rivers

When linearity can be assumed (linear response of heads to stresses), stream–aquifer flow exchange can be simulated as the drainage of a number of independent linear reservoirs. This conceptual model, which can be mathematically deduced in a univocal way from an eigenvalue solution of the linear groundwater flow problem, facilitates the understanding of the physical phenomenon and the analysis of influencing factors. The number of reservoirs required to simulate stream depletion in some ideal homogeneous cases of stream–aquifer connection was analyzed in detail in a previous investigation using analytical eigenvalue solutions [16]. However, most aquifers are heterogeneous in nature and numerical solutions must be employed to analyze whether they could also be simulated using few reservoirs. This paper presents a stochastic analysis of the influence of heterogeneity on the simulation of natural groundwater discharges in aquifers connected to rivers, as a series of linear reservoirs. A Monte-Carlo approach was employed to perform this study. The results show that, on a monthly time scale, many cases (even heterogeneous aquifers) can be simulated using just a few reservoirs with sufficient accuracy and at minimum computational cost. Therefore, this modeling technique can be useful to efficiently simulate the integrated management of complex water resources systems at the basin scale (with many aquifers, reservoirs, demands, etc.) that need to simultaneously consider surface and groundwater flow and stream–aquifer interaction.     

Periodic recharge to finite aquifiers from rectangular areas

Analytical solutions for groundwater flow in unconfined rectangular aquifers are presented for the case of recharge from rectangular areas. The linearised differential equation of the flow is solved using Laplace and finite Fourier transforms. The problem concerns the response of finite aquifers to periodic (seasonal) recharge schemes of variable duration. Results can be obtained for various combinations of Dirichlet and Neumann boundary conditions and can be easily used in a preliminary study of water resources management.     

Modifications to the Conduit Flow Process Mode 2 for MODFLOW-2005

As a result of rock dissolution processes, karst aquifers exhibit highly conductive features such as caves and conduits. Within these structures, groundwater flow can become turbulent and therefore be described by nonlinear gradient functions. Some numerical groundwater flow models explicitly account for pipe hydraulics by coupling the continuum model with a pipe network that represents the conduit system. In contrast, the Conduit Flow Process Mode 2 (CFPM2) for MODFLOW-2005 approximates turbulent flow by reducing the hydraulic conductivity within the existing linear head gradient of the MODFLOW continuum model. This approach reduces the practical as well as numerical efforts for simulating turbulence. The original formulation was for large pore aquifers where the onset of turbulence is at low Reynolds numbers (1 to 100) and not for conduits or pipes. In addition, the existing code requires multiple time steps for convergence due to iterative adjustment of the hydraulic conductivity. Modifications to the existing CFPM2 were made by implementing a generalized power function with a user-defined exponent. This allows for matching turbulence in porous media or pipes and eliminates the time steps required for iterative adjustment of hydraulic conductivity. The modified CFPM2 successfully replicated simple benchmark test problems.     

Mutual information analysis to approach nonlinearity in groundwater stochastic fields

In heterogeneous porous media, transmissivity can be regarded as a spatial stochastic variable. Transmissivity fluctuations induce stochasticity in the groundwater velocity field and transport features. In order to model subsurface phenomena, it is important to understand the relationships that exist between the variables that characterize flow and transport. Linear relationships are easier to deal with. Nevertheless, it is well known that flow and transport variables exhibit interdependences that become more and more nonlinear as the heterogeneity increases. The aim of this work is to draw attention to the information contained in nonlinear linkages, and to show that it can be of great relevance with respect to the linear information content. Information theory tools are proposed to detect the presence of nonlinear components. By comparing the cross-covariance function and mutual information, the amount of linear linkage is compared with nonlinear linkage. In order to avoid analytical approximations, data from Monte Carlo simulations of heterogeneous transmissivity fields have been considered in the analysis. The obtained results show that the presence of nonlinear components can be relevant, even when the cross-covariance values are nil.     

GRAPHICAL SOLUTION OF GROUND WATER FLOW PROBLEMS

Abstract

Graphical solutions to linear and radial ground water flow problems are described. These methods are based on finite difference approximations which have been widely used in heat flow problems. The case of the leaky aquiclude and vertical recharge to unconfined aquifers is discussed as well as constant head at a control and constant discharge or recharge. These solutions are useful only when the physical factors are known and an estimate of water level changes under certain conditions is desired. The methods are not intended to replace more rigorous mathematical solutions.     

Models for recession flows in the upper Blue Nile River

Stream‐flow recessions are commonly characterized by the exponential equation or in the alternative power form equation of a single linear reservoir. The most common measure of recession is the recession constant K, which relates to the power function form of the recession equation for a linear reservoir. However, in reality it can be seen that the groundwater dynamics of even the simplest of aquifers may behave in a non‐linear fashion. In this study three different storage–outflow algorithms; single linear, non‐linear and multiple linear reservoir were considered to model the stream‐flow recession of the upper Blue Nile. The recession parameters for the linear and non‐linear models were derived by the use of least‐squares regression procedures. Whereas, for the multiple linear reservoir model, a second‐order autoregressive AR (2) model was applied first in order to determine the parameters by the least‐squares method. The modelling of the upper Blue Nile recession flow performed shortly after the wet season, when interflow and bank storage may be contributing considerably to the river flow, showed that the non‐linear reservoir model simulates well with the observed counterparts. The variation related to preceding flow on a recession parameter of the non‐linear reservoir remains significant, which was obtained by stratification of the recession curves. Although a similar stratification did not show any systematic variation on the recession parameters for the linear and multiple linear reservoir models. Copyright © 2004 John Wiley & Sons, Ltd.     

An Explicit Finite Difference Model for Unconfined Aquifers

Most of the current simulation models for unconfined aquifers are based on the assumption that the free surface variation is small so that it can be combined with permeability to reduce the nonlinear Boussinesq equation to a linear partial differential equation (PDE). One of the most obvious reasons for using the linearization assumption is for the ease of numerical solution. This work presents a simpler alternative which permits an easy direct solution of the Boussinesq equation. A forward in time, central in space (FTCS) explicit finite difference method is used in the simulation model. The model was first validated by comparing its results with known analytical solution. It was then applied to an actual situation in which the short-term responses (from pumping) of an unconfined aquifer were simulated. The study shows that the stability of the model can be easily controlled, and because of the simple algorithm used, the code can be expeditiously developed and economically run on smaller machines. Due to the uncertainties in the calibration results, it is recommended here that more data be collected to improve the calibration before the model is used as a real-time simulation tool.     

Approximating groundwater age distributions using simple streamtube models and multiple tracers

We use a streamtube based decomposition and a recently developed, simple relationship between tracer concentrations and ages to estimate groundwater age distributions. The decomposition assumes that an age distribution can be approximated using a superposition of linearly independent streamtubes. Transport in each streamtube is modeled with inverse Gaussian functions, the parameters of which are inferred from radiometric tracer concentrations. Three simple sampling methods are considered for weakly and moderately heterogeneous aquifers and the method gives reasonable approximations in both systems. The method is sensitive to errors in the measured concentrations but some of these errors are easily identifiable and a range of plausible age distributions can still be found. The method was then tested in a highly heterogeneous system and reasonable estimates of the age distribution were also obtained. The simplicity of this method and its insensitivity to the heterogeneity structure suggest that this approach may be an effective tool for obtaining estimates of age distributions in natural systems.     

Improvements and Corrections to AT123D Code

Although based on exact analytical solutions, semi‐analytical solute transport models can have significant numerical error in applications with high frequency oscillatory source terms and when parameter value combinations cause series solution approximations to converge slowly. Methods for correcting these numerical errors are presented and implemented in the AT123D code, which employs Green's functions to represent point, linear, and rectangular prismatic source zones. In order to increase its computational accuracy, a Romberg numerical integration scheme was added to AT123D with prespecified error criteria, variable time stepping, and partitioning of the integral to handle rapidly changing source terms. More rapidly converging series solution approximations for the Green's functions were also incorporated to improve both accuracy and computational efficiency for finite‐depth aquifers. AT123D also has been modified to eliminate redundant calculations at points where approximate steady‐state conditions have been reached to improve computational efficiency during numerical integration. These modifications help to decrease computer run times that can be excessive for three‐dimensional problems with large numbers of computational points, small time steps, and/or long simulation time periods. Errors in the original AT123D code also were corrected in this modified version, AT123D‐AT, in order to accurately simulate finite‐duration (pulse) source releases.     

Multiple Well Systems with Non‐Darcy Flow

Optimization of groundwater and other subsurface resources requires analysis of multiple‐well systems. The usual modeling approach is to apply a linear flow equation (e.g., Darcy's law in confined aquifers). In such conditions, the composite response of a system of wells can be determined by summating responses of the individual wells (the principle of superposition). However, if the flow velocity increases, the nonlinear losses become important in the near‐well region and the principle of superposition is no longer valid. This article presents an alternative method for applying analytical solutions of non‐Darcy flow for a single‐ to multiple‐well systems. The method focuses on the response of the central injection well located in an array of equally spaced wells, as it is the well that exhibits the highest pressure change within the system. This critical well can be represented as a single well situated in the center of a closed square domain, the width of which is equal to the well spacing. It is hypothesized that a single well situated in a circular region of the equivalent plan area adequately represents such a system. A test case is presented and compared with a finite‐difference solution for the original problem, assuming that the flow is governed by the nonlinear Forchheimer equation.     

Evaluation of confidence intervals for a steady-state leaky aquifer model

The fact that dependent variables of groundwater models are generally nonlinear functions of model parameters is shown to be a potentially significant factor in calculating accurate confidence intervals for both model parameters and functions of the parameters, such as the values of dependent variables calculated by the model. The Lagrangian method of Vecchia and Cooley [Vecchia, A.V. & Cooley, R.L., Water Resources Research, 1987, 23(7), 1237–1250] was used to calculate nonlinear Scheffé-type confidence intervals for the parameters and the simulated heads of a steady-state groundwater flow model covering 450 km² of a leaky aquifer. The nonlinear confidence intervals are compared to corresponding linear intervals. As suggested by the significant nonlinearity of the regression model, linear confidence intervals are often not accurate. The commonly made assumption that widths of linear confidence intervals always underestimate the actual (nonlinear) widths was not correct. Results show that nonlinear effects can cause the nonlinear intervals to be asymmetric and either larger or smaller than the linear approximations. Prior information on transmissivities helps reduce the size of the confidence intervals, with the most notable effects occurring for the parameters on which there is prior information and for head values in parameter zones for which there is prior information on the parameters.     

Cauchy prior distribution-based AVO elastic parameter estimation via weakly nonlinear waveform inversion

Cauchy priori distribution-based Bayesian AVO reflectivity inversion may lead to sparse estimates that are sensitive to large reflectivities. For the inversion, the computation of the covariance matrix and regularized terms requires prior estimation of model parameters, which makes the iterative inversion weakly nonlinear. At the same time, the relations among the model parameters are assumed linear. Furthermore, the reflectivities, the results of the inversion, or the elastic parameters with cumulative error recovered by integrating reflectivities are not well suited for detecting hydrocarbons and fuids. In contrast, in Bayesian linear AVO inversion, the elastic parameters can be directly extracted from prestack seismic data without linear assumptions for the model parameters. Considering the advantages of the abovementioned methods, the Bayesian AVO reflectivity inversion process is modified and Cauchy distribution is explored as a prior probability distribution and the time-variant covariance is also considered. Finally, we propose a new method for the weakly nonlinear AVO waveform inversion. Furthermore, the linear assumptions are abandoned and elastic parameters, such as P-wave velocity, S-wave velocity, and density, can be directly recovered from seismic data especially for interfaces with large reflectivities. Numerical analysis demonstrates that all the elastic parameters can be estimated from prestack seismic data even when the signal-to-noise ratio of the seismic data is low.     

Physical splitting of nonlinear effects in high-velocity stable flow through porous media

For a high-velocity stable flow through a periodic corrugated channel representing an element of porous medium, we suggest splitting the overall nonlinear macroscopic effects into two kinds of different physical origin: a pure inertia effect produced by the convective term of Navier–Stokes equations and an inertia–viscous cross effect representing a variation of the viscous dissipation due to a streamline deformation by inertia forces. We will show that the inertia–viscous cross effects may be revealed by simulating a periodic flow, whilst the pure inertia effects are produced by the microscale flow nonperiodicity. We will reveal the individual flow law for each nonlinear component and analyze the relative role of both components numerically by using the finite element method applied to the Navier–Stokes equations. Both the pure inertia and the inertia–viscous cross effects are revealed to be exponential prior to quadratic or cubic ones. The influence of the dead volume is analyzed. The inertia–viscous cross phenomena are shown to be negligible when the flow structure is clearly nonperiodic.     

Periodic forcing in composite aquifers

Observations of periodic components of measured heads have long been used to estimate aquifer diffusivities. The estimations are often made using well-known solutions of linear differential equations for the propagation of sinusoidal boundary fluctuations through homogeneous one-dimensional aquifers. Recent field data has indicated several instances where the homogeneous aquifer solutions give inconsistent estimates of aquifer diffusivity from measurements of tidal lag and attenuation. This paper presents new algebraic solutions for tidal propagation in spatially heterogeneous one-dimensional aquifers. By building on existing solutions for homogeneous aquifers, comprehensive solutions are presented for composite aquifers comprising of arbitrary (finite) numbers of contiguous homogeneous sub-aquifers and subject to sinusoidal linear boundary conditions. Both Cartesian and radial coordinate systems are considered. Properties of the solutions, including rapid phase shifting and attenuation effects, are discussed and their practical relevance noted. Consequent modal dispersive effects on tidal waveforms are also examined via tidal constituent analysis. It is demonstrated that, for multi-constituent tidal forcings, measured peak heights of head oscillations can seem to increase, and phase lags seem to decrease, with distance from the forcing boundary unless constituents are separated and considered in isolation.