Automated Estimation of Aquifer Parameters from Arbitrary-Rate Pumping Tests in Python and MATLAB.

Inspired by the analysis by Mishra et al. (2012) of variable pumping rate tests using piecewise-linear reconstructions of the pumping history, this article contains a derivation of the convolutional form of pumping tests in which the pumping history may take any possible form. The solution is very similar to the classical Theis (1935) equation but uses the Green's function for a pumped aquifer given by taking the time derivative of the well function W ( u ( t ) ) . This eliminates one integration inside another and renders the convolution including the pumping history about as computationally demanding as calculating the well function alone, so that the convolution can be completed using handy mathematical software. It also allows nonlinear well losses, and because an easily-computed deterministic model exists for all data points and pumping history, an objective function may include all data, so that errors are reduced in calculating any nonlinear-well losses. In addition, data from multiple observation wells may be used simultaneously in the inversion. We provide codes in MATLAB and Python to solve for drawdown resulting from an arbitrary pumping history and compute the optimal aquifer parameters to fit the data. We find that the subtleties in parameter dependencies and constructing an appropriate objective function have a substantial effect on the interpreted parameters. Furthermore, the optimization from step-drawdown tests is typically nonunique and strongly suggests that a Bayesian inversion should be used to fully estimate the joint probability density of the parameter vector.

Reactive particle-tracking solutions to a benchmark problem on heavy metal cycling in lake sediments.

Geochemical systems are known to exhibit highly variable spatiotemporal behavior. This may be observed both in non-smooth concentration curves in space for a single sampling time and also in variability between samples taken from the same location at different times. However, most models that are designed to simulate these systems provide only single-solution smooth curves and fail to capture the noise and variability seen in the data. We apply a recently developed reactive particle-tracking method to a system that displays highly complex geochemical behavior. When the method is made to most closely resemble a corresponding Eulerian method, in its unperturbed form, we see near-exact match between solutions of the two models. More importantly, we consider two approaches for perturbing the model and find that the spatially-perturbed condition is able to capture a greater degree of the variability present in the data. This method of perturbation is a task to which particle methods are uniquely suited and Eulerian models are not well-suited. Additionally, because of the nature of the algorithm, noisy spatial gradients can be highly resolved by a large number of mobile particles, and this incurs negligible computational cost, as compared to expensive chemistry calculations.

Testing the limits of the spatial Markov model for upscaling transport: The role of nonmonotonic effective velocity autocorrelations.

The spatial Markov model is a Lagrangian random walk model, widely and successfully used for upscaling transport in heterogeneous flows across a broad range of problems. It is particularly useful at early or pre-asymptotic times when many other conventional upscaling approaches may not be valid. However, as with all upscaled models, it must have its limits. In particular, the question of what the smallest scale at which it can be legitimately applied, without violating implicit assumptions, remains. Here we address this issue by considering one of the most classical transport upscaling problems: Taylor dispersion in a bounded shear flow. We demonstrate that the smallest scale for the spatial Markov model depends on the transverse width of the domain, the variability of the flow field as quantified by a coefficient of variation, and the competition of longitudinal and transverse diffusion coefficients. We show that this scale is a factor of the Peclet number smaller than the classical Taylor dispersion scale, meaning that for advection-dominated systems where Peclet numbers are large, this model can be applied at much smaller scales than classical Taylor-Aris dispersion theories.

Predicting the enhancement of mixing-driven reactions in nonuniform flows using measures of flow topology.

The ability for reactive constituents to mix is often the key limiting factor for the completion of reactions across a huge range of scales in a variety of media. In flowing systems, deformation and shear enhance mixing by bringing constituents into closer proximity, thus increasing reaction potential. Accurately quantifying this enhanced mixing is key to predicting reactions and typically is done by observing or simulating scalar transport. To eliminate this computationally expensive step, we use a Lagrangian stochastic framework to derive the enhancement to reaction potential by calculating the collocation probability of particle pairs in a heterogeneous flow field accounting for deformations. We relate the enhanced reaction potential to three well known flow topology metrics and demonstrate that it is best correlated to (and asymptotically linear with) one: the largest eigenvalue of the (right) Cauchy-Green tensor.

Mixing-Driven Equilibrium Reactions in Multidimensional Fractional Advection Dispersion Systems.

We study instantaneous, mixing-driven, bimolecular equilibrium reactions in a system where transport is governed by a multidimensional space fractional dispersion equation. The superdiffusive, nonlocal nature of the system causes the location and magnitude of reactions that take place to change significantly from a classical Fickian diffusion model. In particular, regions where reaction rates would be zero for the Fickian case become regions where the maximum reaction rate occurs when anomalous dispersion operates. We also study a global metric of mixing in the system, the scalar dissipation rate and compute its asymptotic scaling rates analytically. The scalar dissipation rate scales asymptotically as t-(d+α)/α , where d is the number of spatial dimensions and α is the fractional derivative exponent.

Communication: A full solution of the annihilation reaction A + B → [empty-set] based on time-subordination.

The connection between the governing equations of chemical reaction and the underlying stochastic processes of particle collision and transformation have been developed previously along two end-member conditions: perfectly mixed and maximally diffusion-limited. The complete governing equation recognizes that in the perfectly mixed case, the particle (i.e., molecular or macro-particle) number state evolution is markovian, but that spatial self-organization of reactants decreases the probability of reactant pairs finding themselves co-located. This decreased probability manifests itself as a subordination of the clock time: as reactant concentrations become spatially variable (unmixed), the time required for reactants to find each other increases and the random operational time that particles spend in the active reaction process is less than the clock time. For example, in the system A + B → [empty-set], a simple approximate calculation for the return time of a brownian motion to a moving boundary allows a calculation of the operational time density, and the total solution is a subordination integral of the perfectly-mixed solution with a modified inverse gaussian subordinator. The system transitions from the well-mixed solution to the asymptotic diffusion-limited solution that decays as t(-d∕4) in d-dimensions.

Fractional calculus in hydrologic modeling: A numerical perspective.

Fractional derivatives can be viewed either as handy extensions of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Lévy motion, and that fractional integration is the corresponding inverse operator. Fractional integration, and its multi-dimensional extensions derived in this way, are intimately tied to fractional Brownian (and Lévy) motions and noises. By following these general principles, we discuss the Eulerian and Lagrangian numerical solutions to fractional partial differential equations, and Eulerian methods for stochastic integrals. These numerical approximations illuminate the essential nature of the fractional calculus.

Hydraulic Conductivity Fields: Gaussian or Not?

Hydraulic conductivity (K) fields are used to parameterize groundwater flow and transport models. Numerical simulations require a detailed representation of the K field, synthesized to interpolate between available data. Several recent studies introduced high resolution K data (HRK) at the Macro Dispersion Experiment (MADE) site, and used ground-penetrating radar (GPR) to delineate the main structural features of the aquifer. This paper describes a statistical analysis of these data, and the implications for K field modeling in alluvial aquifers. Two striking observations have emerged from this analysis. The first is that a simple fractional difference filter can have a profound effect on data histograms, organizing non-Gaussian ln K data into a coherent distribution. The second is that using GPR facies allows us to reproduce the significantly non-Gaussian shape seen in real HRK data profiles, using a simulated Gaussian ln K field in each facies. This illuminates a current controversy in the literature, between those who favor Gaussian ln K models, and those who observe non-Gaussian ln K fields. Both camps are correct, but at different scales.

Anomalous mixing and reaction induced by superdiffusive nonlocal transport.

Spatially nonlocal transport describes the evolution of solute concentration due to mass transfer over long ranges. Such long-range mass transfer, present in many flow situations, changes the character of mixing and consequent chemical reactions. We study mixing in terms of the scalar dissipation and reaction rates for mixing-limited equilibrium reactions, using the space-fractional advection-dispersion equation (fADE) to model long range mass transfer. The scalar dissipation and global reaction rates decay as power-laws at late time. As opposed to the Fickian (local) transport model, local reaction rates are not zero where the concentration has zero gradient. As α , the fractional derivative exponent, decreases from two in the fADE, the reaction rate grows larger at the position of zero gradient, due to long-range transfer of reactants from distances larger than Fick's law allows. The reaction rates are also greater far from the reactant source for non-Fickian transport; however, the globally integrated reaction rate decreases with smaller α . This behavior may provide a method to investigate spatial nonlocality as a proper model of upscaling: the reaction products would be found in places precluded by Fickian dispersion, and overall reaction rates are suppressed.

Determinants of Thailand household healthcare expenditure: the relevance of permanent resources and other correlates.

Several papers in the leading health economics journals modeled the determinants of healthcare expenditure using household survey or family budgets data of developed countries. Past work largely used self-reported current income as the core determinant, whereas the theoretically correct concept of household resource constraint is permanent or long-run income (á lá Milton Friedman). This paper strives to rectify the theoretical oversight of using current income by augmenting the model with household asset. Using longitudinal data, we constructed 'wealth index' as a distinct covariate to capture the households' tendency to liquidate assets when defraying necessary healthcare liabilities after exhausting cash incomes. (Current income and assets together capture the household expanded resource base). Using 98 632 household observations from Thailand Socio-Economic Surveys (1994-2000 biennial data cycles) we found, using a double-hurdle model with dependent errors, that out-of-pocket healthcare spending behaves as a technical necessity across income quintiles and household sizes. Pre-1997 economic shock income elasticities are smaller than the post-shock estimates across income quintiles for large and small households. Proximity to death, median age, and assets are also among other significant determinants. Our novel findings extend the theoretical consistency of a multi-level decision model in household healthcare expenditure in the developing Asian country context.

Role of volatilization in changing TBA and MTBE concentrations at MTBE-contaminated sites.

Tertiary butyl alcohol (TBA) is commonly found as an impurity in methyl tertiary butyl ether (MTBE) added to gasoline. Frequent observations of high TBA, and especially rising TBA/MTBE concentration ratios, in groundwater at gasoline spill sites are generally attributed to microbial conversion of MTBE to TBA. Typically overlooked is the role of volatilization in the attenuation of these chemicals especially in the vadose zone, which is a source of contamination to groundwater. Here we show that volatilization, particularly through remediation by vapor extraction, can substantially affect the trends in TBA and MTBE concentrations and the respective mass available to impact groundwater aquifers, through the preferential removal of more volatile compounds, including MTBE, and the apparent enrichment of less volatile compounds like TBA. We demonstrate this phenomenon through numerical simulations of remedial-enhanced volatilization. Results show increases in TBA/MTBE concentration ratios consistent with ratios observed in groundwater at gasoline spill sites. Volatilization is an important, and potentially dominant, process that can result in concentration trends similar to those typically attributed to biodegradation.

Predicting the tails of breakthrough curves in regional-scale alluvial systems.

The late tail of the breakthrough curve (BTC) of a conservative tracer in a regional-scale alluvial system is explored using Monte Carlo simulations. The ensemble numerical BTC, for an instantaneous point source injected into the mobile domain, has a heavy late tail transforming from power law to exponential due to a maximum thickness of clayey material. Haggerty et al.'s (2000) multiple-rate mass transfer (MRMT) method is used to predict the numerical late-time BTCs for solutes in the mobile phase. We use a simple analysis of the thicknesses of fine-grained units noted in boring logs to construct the memory function that describes the slow decline of concentrations at very late time. The good fit between the predictions and the numerical results indicates that the late-time BTC can be approximated by a summation of a small number of exponential functions, and its shape depends primarily on the thicknesses and the associated volume fractions of immobile water in "blocks" of fine-grained material. The prediction of the late-time BTC using the MRMT method relies on an estimate of the average advective residence time, t(ad). The predictions are not sensitive to estimation errors in t(ad), which can be approximated by L/v , where v is the arithmetic mean ground water velocity and L is the transport distance. This is the first example of deriving an analytical MRMT model from measured hydrofacies properties to predict the late-time BTC. The parsimonious model directly and quantitatively relates the observable subsurface heterogeneity to nonlocal transport parameters.

Random walk approximation of fractional-order multiscaling anomalous diffusion.

Random walks are developed to approximate the solutions of multiscaling, fractional-order, anomalous diffusion equations. The essential elements of the diffusion are described by the matrix-order scaling indexes and the mixing measure, which describes the diffusion coefficient in every direction. Two forms of the governing equation (also called the multiscaling fractional diffusion equation), based on fractional flux and fractional divergence, are considered, where the diffusion coefficient and the drift vary in space. The particle-tracking algorithm is also extended to approximate anomalous diffusion with a streamline-dependent mixing measure, using a streamline-projection technique. In this and other general cases, the random walk method is the only known way to solve the nonhomogeneous equations. Five numerical examples demonstrate the flexibility, simplicity, and efficiency of the random walk method.

Embolization of an acute renal arteriovenous fistula following a stab wound: case report and review of the literature.

Surgery has traditionally been the definitive form of invasive management for renal vascular injuries. There is a growing trend in the use of endovascular techniques as an alternative to surgery in the trauma setting. We present the case of a 24-year-old woman with an acute renal arteriovenous fistula caused by a stab wound in the left flank, which was successfully managed with selective arterial embolization. This represents only the second reported case of such an approach in the acute setting.

Stochastic solution of space-time fractional diffusion equations.

Classical and anomalous diffusion equations employ integer derivatives, fractional derivatives, and other pseudodifferential operators in space. In this paper we show that replacing the integer time derivative by a fractional derivative subordinates the original stochastic solution to an inverse stable subordinator process whose probability distributions are Mittag-Leffler type. This leads to explicit solutions for space-time fractional diffusion equations with multiscaling space-fractional derivatives, and additional insight into the meaning of these equations.

Practice trends in the management of prostate disease by family practice physicians and general internists: an internet-based survey.

OBJECTIVES: To assess, using an Internet-based survey, the practice patterns of primary care physicians in the management of prostate disease and to assess the differences between family physicians and general internists. Prostate cancer and benign prostatic hyperplasia are common in the elderly population. The optimal management of these diseases is debated. METHODS: An 18-item survey was designed and administered on an Internet website. Members of the Society of General Internal Medicine and members of the Illinois, North Carolina, and New York chapters of the Academy of Family Physicians were surveyed. RESULTS: A total of 354 responses were obtained from 381 primary care physicians who viewed the survey web page. For patients 50 years old and older, 75% of physicians (87% of family physicians and 69% of general internists; P <0.001) recommended annual prostate cancer screening with digital rectal examination and 49% (67% of family physicians and 40% of general internists; P <0.001) recommended annual prostate cancer screening with prostate-specific antigen measurement. For patients diagnosed with prostate cancer, 76% of the primary care physicians estimated that less than one half of their patients are seen by a radiation oncologist. For the treatment and/or prevention of prostate cancer, 62% of physicians surveyed (76% of family physicians and 55% of general internists; P <0.001) believe there is a potential role for alternative herbal and nutritional therapies. CONCLUSIONS: Compared with general internists, family physicians are more likely to screen for prostate cancer and are more likely to believe there is a role for alternative therapies for prostate cancer.

Governing equations and solutions of anomalous random walk limits.

Continuous time random walks model anomalous diffusion. Coupling allows the magnitude of particle jumps to depend on the waiting time between jumps. Governing equations for the long-time scaling limits of these models are found to have fractional powers of coupled space and time differential operators. Explicit solutions and scaling properties are presented for these equations, which can be used to model flow in porous media and other physical systems.