Use of Picard and Newton iteration for solving nonlinear ground water flow equations

This study examines the use of Picard and Newton iteration to solve the nonlinear, saturated ground water flow equation. Here, a simple three-node problem is used to demonstrate the convergence difficulties that can arise when solving the nonlinear, saturated ground water flow equation in both homogeneous and heterogeneous systems with and without nonlinear boundary conditions. For these cases, the characteristic types of convergence patterns are examined. Viewing these convergence patterns as orbits of an attractor in a dynamical system provides further insight. It is shown that the nonlinearity that arises from nonlinear head-dependent boundary conditions can cause more convergence difficulties than the nonlinearity that arises from flow in an unconfined aquifer. Furthermore, the effects of damping on both convergence and convergence rate are investigated. It is shown that no single strategy is effective for all problems and how understanding pitfalls and merits of several methods can be helpful in overcoming convergence difficulties. Results show that Picard iterations can be a simple and effective method for the solution of nonlinear, saturated ground water flow problems.     

Uncertainty estimation of pathlines in ground water models

A method is proposed to estimate the uncertainty of the location of pathlines in two-dimensional, steady-state confined or unconfined flow in aquifers due to the uncertainty of the spatially variable unconditional hydraulic conductivity or transmissivity field. The method is based on concepts of the semianalytical first-order theory given in Stauffer et al. (2002, 2004), which allows estimates of the lateral second moment (variance) of the location of a moving particle. However, this method is reformulated in order to account for nonuniform recharge and nonuniform aquifer thickness. One prominent application is the uncertainty estimation of the catchment of a pumping well by considering the boundary pathlines starting at a stagnation point. In this method, the advective transport of particles is considered, based on the velocity field. In the case of a well catchment, backtracking is applied by using the reversed velocity field. Spatial variability of hydraulic conductivity or transmissivity is considered by taking into account an isotropic exponential covariance function of log-transformed values with parameters describing the variance and correlation length. The method allows postprocessing of results from ground water models with respect to uncertainty estimation. The code PPPath, which was developed for this purpose, provides a postprocessing of pathline computations under PMWIN, which is based on MODFLOW. In order to test the methodology, it was applied to results from Monte Carlo simulations for catchments of pumping wells. The results correspond well. Practical applications illustrate the use of the method in aquifers.     

Determining extreme parameter correlation in ground water models

In ground water flow system models with hydraulic-head observations but without significant imposed or observed flows, extreme parameter correlation generally exists. As a result, hydraulic conductivity and recharge parameters cannot be uniquely estimated. In complicated problems, such correlation can go undetected even by experienced modelers. Extreme parameter correlation can be detected using parameter correlation coefficients, but their utility depends on the presence of sufficient, but not excessive, numerical imprecision of the sensitivities, such as round-off error. This work investigates the information that can be obtained from parameter correlation coefficients in the presence of different levels of numerical imprecision, and compares it to the information provided by an alternative method called the singular value decomposition (SVD). Results suggest that (1) calculated correlation coefficients with absolute values that round to 1.00 were good indicators of extreme parameter correlation, but smaller values were not necessarily good indicators of lack of correlation and resulting unique parameter estimates; (2) the SVD may be more difficult to interpret than parameter correlation coefficients, but it required sensitivities that were one to two significant digits less accurate than those that required using parameter correlation coefficients; and (3) both the SVD and parameter correlation coefficients identified extremely correlated parameters better when the parameters were more equally sensitive. When the statistical measures fail, parameter correlation can be identified only by the tedious process of executing regression using different sets of starting values, or, in some circumstances, through graphs of the objective function.     

The practical use of simplicity in developing ground water models

The advantages of starting with simple models and building complexity slowly can be significant in the development of ground water models. In many circumstances, simpler models are characterized by fewer defined parameters and shorter execution times. In this work, the number of parameters is used as the primary measure of simplicity and complexity; the advantages of shorter execution times also are considered. The ideas are presented in the context of constructing ground water models but are applicable to many fields. Simplicity first is put in perspective as part of the entire modeling process using 14 guidelines for effective model calibration. It is noted that neither very simple nor very complex models generally produce the most accurate predictions and that determining the appropriate level of complexity is an ill-defined process. It is suggested that a thorough evaluation of observation errors is essential to model development. Finally, specific ways are discussed to design useful ground water models that have fewer parameters and shorter execution times.     

A methodology for validating numerical ground water models

Ground water validation is one of the most challenging issues facing modelers and hydrogeologists. Increased complexity in ground water models has created a gap between model predictions and the ability to validate or build confidence in predictions. Specific procedures and tests that can be easily adapted and applied to determine the validity of site-specific ground water models do not exist. This is true for both deterministic and stochastic models, with stochastic models posing the more difficult validation problem. The objective of this paper is to propose a general validation approach that addresses important issues recognized in previous validation studies, conferences, and symposia. The proposed method links the processes for building, calibrating, evaluating, and validating models in an iterative loop. The approach focuses on using collected validation data to reduce uncertainty in the model and narrow the range of possible outcomes. This method is designed for stochastic numerical models utilizing Monte Carlo simulation approaches, but it can be easily adapted for deterministic models. The proposed methodology relies on the premise that absolute validity is not theoretically possible, nor is it a regulatory requirement. Rather, the proposed methodology highlights the importance of testing various aspects of the model and using diverse statistical tools for rigorous checking and confidence building in the model and its predictions. It is this confidence that will encourage regulators and the public to accept decisions based on the model predictions. This validation approach will be applied to a model, described in this paper, dealing with an underground nuclear test site in rural Nevada.     

Testing alternative ground water models using cross-validation and other methods

Many methods can be used to test alternative ground water models. Of concern in this work are methods able to (1) rank alternative models (also called model discrimination) and (2) identify observations important to parameter estimates and predictions (equivalent to the purpose served by some types of sensitivity analysis). Some of the measures investigated are computationally efficient; others are computationally demanding. The latter are generally needed to account for model nonlinearity. The efficient model discrimination methods investigated include the information criteria: the corrected Akaike information criterion, Bayesian information criterion, and generalized cross-validation. The efficient sensitivity analysis measures used are dimensionless scaled sensitivity (DSS), composite scaled sensitivity, and parameter correlation coefficient (PCC); the other statistics are DFBETAS, Cook's D, and observation-prediction statistic. Acronyms are explained in the introduction. Cross-validation (CV) is a computationally intensive nonlinear method that is used for both model discrimination and sensitivity analysis. The methods are tested using up to five alternative parsimoniously constructed models of the ground water system of the Maggia Valley in southern Switzerland. The alternative models differ in their representation of hydraulic conductivity. A new method for graphically representing CV and sensitivity analysis results for complex models is presented and used to evaluate the utility of the efficient statistics. The results indicate that for model selection, the information criteria produce similar results at much smaller computational cost than CV. For identifying important observations, the only obviously inferior linear measure is DSS; the poor performance was expected because DSS does not include the effects of parameter correlation and PCC reveals large parameter correlations.     

Estimating ground water recharge using flow models of perched karstic aquifers

The fraction of rain that is annually recharged to ground water is a function of the transient quantities of precipitation (wet vs. dry years) as well as other meteorological and geologic factors, and thus it is very difficult to estimate. In this study, we have used long records (20 to 30 years) of precipitation and spring discharge to reconstruct the transient character of yearly recharge. These data sets were used to calibrate numerical ground water flow models on the less than 3 km(2) scale for four separate perched karstic aquifers in the Judean and Samarian Mountains of Israel. The stratification and karstic character of the local carbonate rock aquifers cause ground water to flow through discrete dissolution channels and to discharge at isolated springs. An innovative, dual-porosity approach was used where a finite-difference solution simulates flow in the rock matrix, while the karstic channels are simulated using computationally simple drains. Perched conditions are also simulated innovatively using MODFLOW by treating the bottom unsaturated layer as if it is saturated, but by assuming zero pressure head throughout the "unsaturated" layer. Best fitting between measured and computed spring hydrograph data has allowed us to develop a set of empirical functions relating measured precipitation to recharge to the aquifer. The generic methodology presented gives insight into the suspected changes in aquifer recharge rates between particularly wet or dry years.     

Validation of numerical ground water models used to guide decision making

Many sites of ground water contamination rely heavily on complex numerical models of flow and transport to develop closure plans. This complexity has created a need for tools and approaches that can build confidence in model predictions and provide evidence that these predictions are sufficient for decision making. Confidence building is a long-term, iterative process and the author believes that this process should be termed model validation. Model validation is a process, not an end result. That is, the process of model validation cannot ensure acceptable prediction or quality of the model. Rather, it provides an important safeguard against faulty models or inadequately developed and tested models. If model results become the basis for decision making, then the validation process provides evidence that the model is valid for making decisions (not necessarily a true representation of reality). Validation, verification, and confirmation are concepts associated with ground water numerical models that not only do not represent established and generally accepted practices, but there is not even widespread agreement on the meaning of the terms as applied to models. This paper presents a review of model validation studies that pertain to ground water flow and transport modeling. Definitions, literature debates, previously proposed validation strategies, and conferences and symposia that focused on subsurface model validation are reviewed and discussed. The review is general and focuses on site-specific, predictive ground water models used for making decisions regarding remediation activities and site closure. The aim is to provide a reasonable starting point for hydrogeologists facing model validation for ground water systems, thus saving a significant amount of time, effort, and cost. This review is also aimed at reviving the issue of model validation in the hydrogeologic community and stimulating the thinking of researchers and practitioners to develop practical and efficient tools for evaluating and refining ground water predictive models.     

An assessment of aquifer storage recovery using ground water flow models

Owing to increased demands on ground water accompanied by increased drawdowns, technologies that use recharge options, such as aquifer storage recovery (ASR), are being used to optimize available water resources and reduce adverse effects of pumping. In this paper, three representative ground water flow models were created to assess the impact of hydrogeologic and operational parameters/factors on recovery efficiency of ASR systems. Flow/particle tracking and solute transport models were used to track the movement of water during injection, storage, and recovery. Results from particle tracking models consistently produced higher recovery efficiency than the solute transport models for the parameters/properties examined because the particle tracking models neglected mixing of the injected and ambient water. Mixing between injected and ambient water affected recovery efficiency. Results from this study demonstrate the interactions between hydrogeologic and operational parameters on predictions of recovery efficiency. These interactions are best simulated using coupled numerical ground water flow and transport models that include the effects of mixing of injected water and ambient ground water.     

Site-dependent models of earthquake ground motion

Random vibration analyses of structural systems subjected to seismic loading are dependent upon the characterization of earthquake ground motion as a stochastic process. The response of structural systems to earthquakes is dependent strongly on the local geological conditions, which should be incorporated into seismological models of ground motion. In the study presented herein, three previously developed ground-motion models are adapted to incorporate site-dependent characteristics. Records obtained from two recording stations in California are used as a basis for the ground-motion models. Single-degree-of-freedom (SDOF) oscillators are subjected to ensembles of accelerograms generated from these models, and both elastic and inelastic response are considered. Response statistics are compared to those generated by the analysis of structural response to ensembles of recorded motion from the two sites. The important features of the ground motion for effective reproduction of response statistics are identified, and observations are made on the sensitivity of specific response parameters to site-dependent characteristics of the ground motion.     

Arma models for earthquake ground motions

This paper outlines the use of discrete, autoregressive/moving-average (ARMA) models for identification and estimation of parameters in models derived from analysis of uniformly digitized earthquake ground motion acceleration data. Such models are of equal generality as compared to continuous-time models and have a number of significant advantages for purposes of digital analysis and simulation. The structure of ARMA models is briefly described, their relation to continuous models noted, and results of their application to a number of recorded accelerograms summarized.     

基于连续小波变换的自适应面波压制方法

面波干扰的抑制是陆上地震资料处理的主要问题之一.本文根据炮集记录中面波与反射波主要能量在小波域分布区域的不同,以及面波干扰的影响随炮检距变化等特点,提出了一种具有时变、空变特性的自适应面波衰减方法.文中将该方法用于模型及实际炮集资料的处理,并与常用的高通滤波方法进行对比,结果表明,该方法在衰减面波干扰的同时,能更好地保持反射波的振幅及相位信息.     

Dynamics of flood water infiltration and ground water recharge in hyperarid desert

A study on flood water infiltration and ground water recharge of a shallow alluvial aquifer was conducted in the hyperarid section of the Kuiseb River, Namibia. The study site was selected to represent a typical desert ephemeral river. An instrumental setup allowed, for the first time, continuous monitoring of infiltration during a flood event through the channel bed and the entire vadose zone. The monitoring system included flexible time domain reflectometry probes that were designed to measure the temporal variation in vadose zone water content and instruments to concurrently measure the levels of flood and ground water. A sequence of five individual floods was monitored during the rainy season in early summer 2006. These newly generated data served to elucidate the dynamics of flood water infiltration. Each flood initiated an infiltration event which was expressed in wetting of the vadose zone followed by a measurable rise in the water table. The data enabled a direct calculation of the infiltration fluxes by various independent methods. The floods varied in their stages, peaks, and initial water contents. However, all floods produced very similar flux rates, suggesting that the recharge rates are less affected by the flood stages but rather controlled by flow duration and available aquifer storage under it. Large floods flood the stream channel terraces and promote the larger transmission losses. These, however, make only a negligible contribution to the recharge of the ground water. It is the flood duration within the active streambed, which may increase with flood magnitude that is important to the recharge process.     

Implementation of ADI-FDTD subgrids in ground penetrating radar FDTD models

Realistic numerical modeling of ground penetrating radar (GPR) using the finite-difference time-domain (FDTD) method could greatly benefit from the implementation of subgrids – supporting finer spatial resolution – into the conventional FDTD mesh. This is particularly important, when parts of the computational domain need to be modeled in detail or when there are features or regions in the overall computational mesh with values of high dielectric constant supporting propagation of waves at very short wavelengths. A scheme that simplifies the process of implementing these subgrids into the traditional FDTD method is presented. This scheme is based on the combination of the standard FDTD method and the unconditionally stable alternating-direction implicit (ADI) FDTD technique. Because ADI-FDTD is unconditionally stable its time-step can be set to any value that facilitates the accurate calculation of the fields. By doing so, the two grids can efficiently communicate information across their boundary without requiring to use a costly time-interpolation scheme. This paper discusses the performance of ADI-FDTD subgrids when implemented into the traditional FDTD method, using different communication schemes for the information exchange at the boundary of the two grids. The developed algorithm, can handle cases where the subgrid crosses dielectrically inhomogeneous media. In addition, results from the comparison between the proposed scheme and a commonly employed purely FDTD subgridding technique are presented.