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Allen M. Shapiro

Publications and source records attributed to Allen M. Shapiro.

At least 37 records · Page 2Linked to original sources
USGS2010

A solute flux approach to transport in heterogeneous formations: 2. Uncertainty analysis

Uncertainty in the mass flux for advection dominated solute movement in heterogeneous porous media is investigated using the Lagrangian framework developed in paper 1 by Dagan et al. (this issue). Expressions for the covariance of the mass flux and cumulative mass flux are derived as functions of the injection volume and sampling area size relative to the scale of heterogeneity. The result is illustrated for solute advection in three types of heterogeneous porous media: stratified formations, two‐ and three‐dimensional porous media; small perturbation approximation is used for the two‐ and three‐dimensional cases. Variances of the mass flux and cumulative mass flux are evaluated as functions of the injection volume (area) scale versus log‐hydraulic conductivity integral scale. The greatest decrease in coefficient of variation (CV) of the mass flux is for the source scale 1–5 times the hydraulic conductivity integral scale; further increase in the source size decreases CV comparatively less. The variance of the cumulative mass flux (or total discharge) indicates that for the source size of 20 hydraulic conductivity integral scales, the transport conditions are almost ergodic. The present results also indicate that the cumulative mass flux is a relatively robust quantity for describing field‐scale solute transport.

Water Resources Research↗
USGS2010

A solute flux approach to transport in heterogeneous formations: 1. The general framework

It is common to represent solute tranport in heterogeneous formations in terms of the resident concentration C (x, t), regarded as a random space function. The present study investigates the alternative representation by q , the solute mass flux at a point of a control plane normal to the mean flow. This representation is appropriate for many field applications in which the variable of interest is the mass of solute discharged through a control surface. A general framework to compute the statistical moments of q and of the associated total solute discharge Q and mass M is established. With x the direction of the mean flow, a solute particle is crossing the control plane at y = η, z = ζ and at the travel (arrival) time τ. The associated expected solute flux value is proportional to the joint probability density function (pdf) g 1 of η, ζ and τ, whereas the variance of q is shown to depend on the joint pdf g 2 of the same variables for two particles. In turn, the statistical moments of η, ζ and τ depend on those of the velocity components through a system of stochastic ordinary differential equations. For a steady velocity field and neglecting the effect of pore‐scale dispersion, a major simplification of the problem results in the independence of the random variables η, ζ and τ. As a consequence, the pdf of η and ζ can be derived independently of τ. A few approximate approaches to derive the statistical moments of η, ζ and τ are outlined. These methods will be explored in paper 2 in order to effectively derive the variances of the total solute discharge and mass, while paper 3 will deal with the nonlinear effect of the velocity variance upon the moments of η, ζ and τ

Water Resources Research↗
USGS2010

Stochastic analysis of solute arrival time in heterogeneous porous media

Longitudinal advective solute movement in heterogeneous porous media is investigated by considering the solute arrival time at a plane perpendicular to the mean fluid velocity. The moments of the solute arrival time are defined in terms of the stochastic properties of a statistically anisotropic hydraulic conductivity field. The flux‐averaged concentration is specified by introducing the moments of the arrival time into a probability density function for the arrival time. The quadratic dependence of the arrival time variance on position in the vicinity of an injection point is indicative of a nondiffusive process. For the assumed spatial correlation of the hydraulic conductivity, the variance of the arrival time asymptotically approaches a linear dependence on the position from the injection point similar to a diffusion process. The impact of assuming solute movement to be a diffusive process from the onset of the solute injection causes erroneous estimates of flux‐averaged concentrations at distances from the injection point that are of the order of the correlation length of the hydraulic conductivity. The arrival time analysis and the particle position analysis given in Dagan (1982, 1984) are complementary interpretations of advective solute movement that yield different definitions of the solute concentration; the position analysis intrinsically defines the resident or volume‐averaged concentration, while the flux‐averaged concentration is defined from the arrival time analysis. The temporal variation of the resident and flux‐averaged concentration are similar at a given position for small values of the variance of the hydraulic conductivity, or at distances from the solute injection point that are large relative to correlation length of the hydraulic conductivity.

Water Resources Research↗
USGS2010

Assessing the validity of the channel model of fracture aperture under field conditions

I n recent investigations of fluid and solute movement in discrete fractures, spatial heterogeneity of the fracture aperture has been conceptualized as a series of noninterconnecting constant aperture flow paths, or channels. Two methods of estimating the distribution of the aperture sizes are presented using information from a single-hole pumping test and a radially converging tracer test. The first method uses the transmissivity of the fracture and mean arrival time of the tracer, while the second method uses the mean and variance of the solute arrival time. If the fracture can be conceptualized as a series of nonintersecting flow paths of constant aperture, the two methods should yield identical estimates for the parameters of the distribution. The validity of the channel model can be assessed by comparing the parameters of the distribution that are estimated by the two methods. This technique was demonstrated using hydraulic and tracer tests conducted in a discrete, areally extensive, horizontal fracture in the Silurian dolomite in the northeastern Illinois. A lognormal and a truncated gamma distribution were used to describe the distribution of constant aperture flow paths. The lognormal distribution could not reproduce the abrupt rising limb and maximum rate of mass arrival that characterized the tracer tests. The gamma distribution more accurately predicted the shape of the breakthrough curves; however, the two methods of estimating the parameters of the distribution provided significantly different estimates of the variances of the aperture sizes. The mean aperture as estimated by the two methods was similar. The difference in the variance of the aperture as estimated by the two methods indicates that alternative conceptual models of aperture heterogeneity are required to more accurately describe both fluid and solute movement in this field situation.

Water Resources Research↗
USGS2010

Solute advection in stratified formations

Advection‐dominated solute movement in stratified formations is investigated using a Lagrangian interpretation of particle motion. A probability density function (pdf) for particle position quantifies the expected depth‐integrated resident concentration. A pdf for particle arrival time quantifies the expected depth‐integrated rate of mass arrival, from which the flux‐averaged concentration can be defined. The difference between the flux‐averaged and resident concentrations is shown to be significant for the variability in the hydraulic conductivity that is commonly encountered in field applications. The influence of porosity variations on the advection‐dominated solute movement in stratified porous media is shown to be notable only for large variability in the effective porosity.

Water Resources Research↗
USGS2010

Interpretation of oscillatory water levels in observation wells during aquifer tests in fractured rock

Oscillatory water levels in observation wells have commonly been recorded at the beginning of aquifer tests in highly transmissive fractured formations. In this paper, oscillatory water levels are predicted by the equations coupling the fluid movement in the observation well and the fluid movement in the surrounding formation. The equivalent-porous medium and dual-porosity models of fractured rock are two models considered in this analysis; however, other conceptual models of fractured media can also be coupled with the model presented here for fluid movement in the observation well. Type curves for the response of water levels in observation wells due to pumping in another well are generated by numerical inversion of the Laplace transform solution to the governing equations. Overdamped conditions, where inertial effects are insignificant, and underdamped conditions, where oscillations arise, are predicted by the solution to the governing equations. By matching water level measurements with the appropriate type curve, a conceptual model of the formation can be identified, and aquifer properties can be estimated. This analysis is applied in the interpretation of an aquifer test conducted in a fractured dolomite in northeastern Illinois. If the early time oscillations are ignored, the measured water levels can be explained by an equivalent-porous medium model. By analyzing the early time oscillations, however, the formation is shown to respond as a dual-porosity medium with a storativity of the fracture porosity that is an order of magnitude smaller than the storativity estimated from the equivalent-porous medium model.

Water Resources Research↗
USGS2009

Effects of simplifying fracture network representation on inert chemical migration in fracture-controlled aquifers

[1] While it is widely recognized that highly permeable ‘large‐scale' fractures dominate chemical migration in many fractured aquifers, recent studies suggest that the pervasive ‘small‐scale’ fracturing once considered of less significance can be equally important for characterizing the spatial extent and residence time associated with transport processes. A detailed examination of chemical migration through fracture‐controlled aquifers is used to advance this conceptual understanding. The influence of fracture structure is evaluated by quantifying the effects to transport caused by a systematic removal of fractures from three‐dimensional discrete fracture models whose attributes are derived from geologic and hydrologic conditions at multiple field sites. Results indicate that the effects to transport caused by network simplification are sensitive to the fracture network characteristics, degree of network simplification, and plume travel distance, but primarily in an indirect sense since correlation to individual attributes is limited. Transport processes can be ‘enhanced’ or ‘restricted’ from network simplification meaning that the elimination of fractures may increase or decrease mass migration, mean travel time, dispersion, and tailing of the concentration plume. The results demonstrate why, for instance, chemical migration may not follow the classic advection‐dispersion equation where dispersion approximates the effect of the ignored geologic structure as a strictly additive process to the mean flow. The analyses further reveal that the prediction error caused by fracture network simplification is reduced by at least 50% using the median estimate from an ensemble of simplified fracture network models, and that the error from network simplification is at least 70% less than the stochastic variability from multiple realizations.

Water Resources Research↗
USGS2008

Pathogen and chemical transport in the karst limestone of the Biscayne aquifer: 1. Revised conceptualization of groundwater flow

The Biscayne aquifer is a highly transmissive karst limestone that serves as the sole source of drinking water to over two million residents in south Florida. The aquifer is characterized by eogenetic karst, where the most transmissive void space can be an interconnected, touching‐vug, biogenically influenced porosity of biogenic origin. Public supply wells in the aquifer are in close proximity to lakes established by surface mining. The mining of the limestone has occurred to the same depths as the production wells, which has raised concerns about pathogen and chemical transport from these surface water bodies. Hydraulic and forced gradient tracer tests were conducted to augment geologic and geophysical studies and to develop a hydrogeologic conceptual model of groundwater flow and chemical transport in the Biscayne aquifer. Geologic and geophysical data indicate multiple, areally extensive subhorizontal preferential flow zones of vuggy limestone separated by rock with a matrix pore system. The hydraulic response from an aquifer test suggests that the Biscayne aquifer behaves as a dual‐porosity medium; however, the results of the tracer test showed rapid transport similar to other types of karst. The tracer test and concurrent temperature logging revealed that only one of the touching‐vug flow zones dominates transport near the production wells. On the basis of the rising limb of the breakthrough curve, the dispersivity is estimated to be less than 3% of the tracer travel distance, which suggests that the fastest flow paths in the formation are likely to yield limited dilution of chemical constituents.

Water Resources Research↗
USGS2008

Pathogen and chemical transport in the karst limestone of the Biscayne aquifer: 2. Chemical retention from diffusion and slow advection

A tracer experiment, using a nonreactive tracer, was conducted as part of an investigation of the potential for chemical and pathogen migration to public supply wells that draw groundwater from the highly transmissive karst limestone of the Biscayne aquifer in southeastern Florida. The tracer was injected into the formation over approximately 1 h, and its recovery was monitored at a pumping well approximately 100 m from the injection well. The first detection of the tracer occurred after approximately 5 h, and the peak concentration occurred at about 8 h after the injection. The tracer was still detected in the production well more than 6 days after injection, and only 42% of the tracer mass was recovered. It is hypothesized that a combination of chemical diffusion and slow advection resulted in significant retention of the tracer in the formation, despite the high transmissivity of the karst limestone. The tail of the breakthrough curve exhibited a straight‐line behavior with a slope of −2 on a log‐log plot of concentration versus time. The −2 slope is hypothesized to be a function of slow advection, where the velocities of flow paths are hypothesized to range over several orders of magnitude. The flow paths having the slowest velocities result in a response similar to chemical diffusion. Chemical diffusion, due to chemical gradients, is still ongoing during the declining limb of the breakthrough curve, but this process is dwarfed by the magnitude of the mass flux by slow advection.

Water Resources Research↗
USGS2008

Pathogen and chemical transport in the karst limestone of the Biscayne aquifer: 3. Use of microspheres to estimate the transport potential of Cryptosporidium parvum oocysts

The vulnerability of a municipal well in the Northwest well field in southeastern Florida to potential contamination by Cryptosporidium parvum oocysts was assessed in a large‐scale, forced‐gradient (convergent) injection and recovery test. The field study involved a simultaneous pulse introduction of a nonreactive tracer (SF 6 , an inert gas) and oocyst‐sized (1.6, 2.9, and 4.9 μ m diameter) carboxylated polystyrene microspheres into karst limestone of the Biscayne aquifer characterized by a complex triple (matrix, touching‐vug, and conduit) porosity. Fractional recoveries 97 m down gradient were inversely related to diameter and ranged from 2.9% for the 4.9 μ m microspheres to 5.8% for 1.6 μ m microspheres. Their centers of mass arrived at the pumping well approximately threefold earlier than that of the nonreactive tracer SF 6 (gas), underscoring the need for use of colloid tracers and field‐scale tracer tests for these kinds of evaluations. In a modified triaxial cell using near in situ chemical conditions, 2.9 and 4.9 μ m microspheres underestimated by fourfold to sixfold the attachment potential of the less electronegative 2.9–4.1 μ m oocysts in the matrix porosity of limestone core samples. The field and laboratory results collectively suggested that it may take 200–300 m of transport to ensure even a 1‐log unit removal of oocysts, even though the limestone surfaces exhibited a substantive capability for their sorptive removal. The study further demonstrated the utility of microspheres as oocyst surrogates in field‐scale assessments of well vulnerability in limestone, provided that differences in attachment behaviors between oocysts and microspheres are taken into account.

Water Resources Research↗
USGS2008

Comment on “Macrodispersion in sand-shale sequences” by A. J. Desbarats

Desbarats [1990] used a particle-tracking scheme to investigate the physics of three-dimensional solute transport in aquifers composed of two porous media of different hydraulic conductivities. The spatially heterogeneous fluid velocity was assumed to be the only mechanism of solute movement; local or pore scale dispersion and molecular diffusion were assumed to be negligible. The particle-tracking scheme used by Desbarats consisted of routing particles from node to node in a finite difference grid. In this scheme, the direction of an individual particle is randomly selected and the prob- ability associated with the particle movement in a given direction is proportional to the fluid flux in that direction. The same method was used by Moreno et al. [1988] to investigate advective transport in a variable-aperture planar fracture.

Water Resources Research↗
USGS2008

Comment on “Flow and tracer transport in a single fracture: A stochastic model and its relation to some field observations” by L. Moreno et al.

Moreno et al. [1988] (hereinafter referred to as MT) used a particle-tracking scheme to investigate the physics of solute movement in a variable-aperture planar fracture. The spatially heterogeneous fluid velocity was assumed to be the only mechanism of solute movement; local or pore scale dispersion and molecular diffusion were assumed to be negligible. The particle-tracking scheme used by MT consisted of routing particles from node to node in a finite difference grid. In this scheme, the direction of an individual particle is randomly selected and the probability associated with the particle movement in a given direction is proportional to the fluid flux in that direction. The same method was used by Desbarats [1990] to investigate advective transport in aquifers composed of two porous media of different hydraulic conductivities.

Water Resources Research↗
USGS2007

Cautions and suggestions for geochemical sampling in fractured rock

Collecting water samples for geochemical analyses in open bedrock boreholes or in discrete intervals of boreholes intersected by multiple fractures is likely to yield ambiguous results for ground water chemistry because of the variability in the transmissivity, storativity, and hydraulic head of fractures intersecting the borehole. Interpreting chemical analyses of water samples collected in bedrock boreholes requires an understanding of the hydraulic conditions in the borehole under the ambient flow regime in the aquifer as well as during sampling. Pumping in open boreholes, regardless of the pumping rate and the location of the pump intake, first draws water from the borehole and then from fractures intersecting the borehole. The time at which the volumetric rate of water entering the borehole from fractures is approximately equal to the pumping rate can be identified by monitoring the logarithm of drawdown in the borehole as a function of the logarithm of time. Mixing of water entering the borehole from fractures with water in the borehole must be considered in estimating the time at which the pump discharge is representative of aquifer water. In boreholes intersected by multiple fractures, after the contribution from the borehole volume has diminished, the contribution of fractures to the pump discharge will be weighted according to their transmissivity, regardless of the location of the pump intake. This results in a flux‐averaged concentration in the pump discharge that is biased by the chemical signature of those fractures with the highest transmissivity. Under conditions where the hydraulic head of fractures varies over the length of the borehole, open boreholes will be subject to ambient flow in the water column in the borehole. In some instances, the magnitude of the ambient flow may be similar to the designated pumping rate for collecting water samples for geochemical analyses. Under such conditions, the contributions to the pump discharge from individual fractures will be a function not only of the transmissivity of the fractures, but also of the distribution of hydraulic head in fractures intersecting the borehole. To reduce or eliminate the deleterious effects of conducting geochemical sampling in open boreholes, a straddle‐packer apparatus that isolates a single fracture or a series of closely spaced fractures is recommended. It is also recommended that open boreholes be permanently outfitted with borehole packers or borehole liners in instances where maintaining the hydraulic and chemical stratification in the aquifer is of importance. In a field example, a comparison of results from sampling in an open borehole and in discrete intervals of the same borehole showed dramatic differences in the concentrations of chemical constituents in the water samples, even though chemical field parameters stabilized prior to both open borehole and discrete interval sampling.

Groundwater Monitoring & Remediation↗
USGS2007

Contamination in fractured-rock aquifers: Research at the former Naval Air Warfare Center, West Trenton, New Jersey

The U.S. Geological Survey and cooperators are studying chlorinated solvents in a fractured sedimentary rock aquifer underlying the former Naval Air Warfare Center (NAWC), West Trenton, New Jersey. Fractured-rock aquifers are common in many parts of the United States and are highly susceptible to contamination, particularly at industrial sites. Compared to 'unconsolidated' aquifers, there can be much more uncertainty about the direction and rate of contaminant migration and about the processes and factors that control chemical and microbial transformations of contaminants. Research at the NAWC is improving understanding of the transport and fate of chlorinated solvents in fractured-rock aquifers and will compare the effectiveness of different strategies for contaminant remediation.

New Jersey↗
USGS2007

BAT3 Analyzer: Real-time data display and interpretation software for the multifunction bedrock-aquifer transportable testing tool (BAT3)

The BAT3 Analyzer provides real-time display and interpretation of fluid pressure responses and flow rates measured during geochemical sampling, hydraulic testing, or tracer testing conducted with the Multifunction Bedrock-Aquifer Transportable Testing Tool (BAT3) (Shapiro, 2007). Real-time display of the data collected with the Multifunction BAT3 allows the user to ensure that the downhole apparatus is operating properly, and that test procedures can be modified to correct for unanticipated hydraulic responses during testing. The BAT3 Analyzer can apply calibrations to the pressure transducer and flow meter data to display physically meaningful values. Plots of the time-varying data can be formatted for a specified time interval, and either saved to files, or printed. Libraries of calibrations for the pressure transducers and flow meters can be created, updated and reloaded to facilitate the rapid set up of the software to display data collected during testing with the Multifunction BAT3. The BAT3 Analyzer also has the functionality to estimate calibrations for pressure transducers and flow meters using data collected with the Multifunction BAT3 in conjunction with corroborating check measurements. During testing with the Multifunction BAT3, and also after testing has been completed, hydraulic properties of the test interval can be estimated by comparing fluid pressure responses with model results; a variety of hydrogeologic conceptual models of the formation are available for interpreting fluid-withdrawal, fluid-injection, and slug tests.

Open-File Report↗
USGS2007

Characterizing Hydraulic Properties and Ground-Water Chemistry in Fractured-Rock Aquifers: A User's Manual for the Multifunction Bedrock-Aquifer Transportable Testing Tool (BAT3)

A borehole testing apparatus has been designed to isolate discrete intervals of a bedrock borehole and conduct hydraulic tests or collect water samples for geochemical analyses. This borehole testing apparatus, referred to as the Multifunction Bedrock-Aquifer Transportable Testing Tool (BAT3), includes two borehole packers, which when inflated can form a pressure-tight seal against smooth borehole walls; a pump apparatus to withdraw water from between the two packers; a fluid-injection apparatus to inject water between the two packers; pressure transducers to monitor fluid pressure between the two packers, as well as above and below the packers; flowmeters to monitor rates of fluid withdrawal or fluid injection; and data-acquisition equipment to record and store digital records from the pressure transducers and flowmeters. The generic design of this apparatus was originally discussed in United States Patent Number 6,761,062 (Shapiro, 2004). The prototype of the apparatus discussed in this report is designed for boreholes that are approximately 6 inches in diameter and can be used to depths of approximately 300 feet below land surface. The apparatus is designed to fit in five hard plastic boxes that can be shipped by overnight freight car-riers. The equipment can be assembled rapidly once it is removed from the shipping boxes, and the length of the test interval (the distance between the two packers) can be adjusted to account for different borehole conditions without reconfiguring the downhole components. The downhole components of the Multifunction BAT3 can be lowered in a borehole using steel pipe or a cable; a truck mounted winch or a winch and tripod can be used for this purpose. The equipment used to raise and lower the downhole components of the Multifunction BAT3 must be supplied on site, along with electrical power, a compressor or cylinders of compressed gas to inflate the packers and operate downhole valves, and the proper length of tubing to connect the packers, the submersible pump, and other downhole components to land surface. Borehole geophysical logging must be conducted prior to deploying the Multifunction BAT3 in bedrock boreholes. In particular, it is important to identify the borehole diameter as a function of depth to avoid placing the packers over rough sections of the borehole, where they may be damaged during inflation. In addition, it is advantageous to identify the location of fractures intersecting the borehole wall, for example, using an acoustic televiewer log or a borehole camera. A knowledge of fracture locations is helpful in designing the length of the test interval and the locations where hydraulic tests and geochemical sampling are to be conducted. The Multifunction BAT3 is configured to conduct both fluid-injection and fluid-withdrawal tests. Fluid-injection tests are used to estimate the hydraulic properties of low-permeability fractures intersecting the borehole. The lower limit of the transmissivity that can be estimated using the configuration of the Multifunction BAT3 described in this report is approximately 10-3 square feet per day (ft2/d). Fluid-withdrawal tests are used to collect water samples for geochemical analyses and estimate the hydraulic properties of high-permeability fractures intersecting the borehole. The Multifunction BAT3 is configured with a submersible pump that can support pumping rates ranging from approximately 0.05 to 2.5 gallons per minute, and the upper limit of the of the transmissivity that can be estimated is approximately 104 ft2/d. The Multifunction BAT3 also can be used to measure the ambient hydraulic head of a section of a bedrock borehole, and to conduct single-hole tracer tests by injecting and later withdrawing a tracer solution.

Open-File Report↗