Using constant head step tests to determine hydraulic apertures in fractured rock

The initial step in the analysis of contaminant transport in fractured rock requires the consideration of groundwater velocity. Practical methods for estimating the average linear groundwater velocity (vˉ) in fractured rock require determination of hydraulic apertures which are commonly calculated by applying the cubic law using transmissivity (T) values and the number of hydraulically active fractures in the test interval. High-resolution, constant-head step injection testing of cored boreholes in a 100 m thick fractured dolostone aquifer was conducted using inflatable packers to isolate specific test intervals from the rest of the borehole. The steps in each test interval were gradually increased from very low to much higher injection rates. At smaller injection rates, the flow rate vs. applied pressure graph projects through the origin and indicates Darcian flow; non Darcian flow is evident at higher injection rates. Non-Darcian flow results in significantly lower calculated T values, which translates to smaller hydraulic aperture values. Further error in the calculated hydraulic aperture stems from uncertainty in the number of hydraulically active fractures in each test interval. This estimate can be inferred from borehole image and core logs, however, all of the fractures identified are not necessarily hydraulically active. This study proposes a method based on Reynolds number calculations aimed at improving confidence in the selection of the number of active fractures in each test interval.     

Effects of biofilm growth on flow and transport through a glass parallel plate fracture

The effects of biofilm growth on flow and solute transport through a sandblasted glass parallel plate fracture was investigated. The fracture was inoculated using soil microorganisms. Glucose, oxygen and other nutrients were supplied to support growth. The biomass initially formed discrete clusters attached to the glass surfaces, but over time formed a continuous biofilm. From dye tracer tests conducted during biofilm growth, it was observed that channels and low-permeability zones dominated transport. The hydraulic conductivity of the fracture showed a sigmoidal decrease with time. The hydraulic conductivity was reduced by a factor of 0.033, from 18 to 0.6 cm/s, corresponding to a 72% decrease in the hydraulic aperture, from 500 to 140 microm. In contrast, the mass balance aperture, determined from fluoride tracer tests, remained relatively constant, indicating that the impact of biomass growth on effective fracture porosity was much less than the effect on hydraulic conductivity. Analyses of pre-biofilm tracer tests revealed that both Taylor dispersion and macrodispersion were influencing transport. During biofilm growth, only macrodispersion was dominant. The macrodispersion coefficient alpha(macro) was found to increase logarithmically with hydraulic conductivity reduction.     

Hydraulic constraints on the performance of a groundwater denitrification wall for nitrate removal from shallow groundwater

Denitrification walls are a practical approach for decreasing non-point source pollution of surface waters. They are constructed by digging a trench perpendicular to groundwater flow and mixing the aquifer material with organic matter, such as sawdust, which acts as a carbon source to stimulate denitrification. For efficient functioning, walls need to be permeable to groundwater flow. We examined the functioning of a denitrification wall constructed in an aquifer consisting of coarse sands. Wells were monitored for changes in nitrate concentration as groundwater passed through the wall and soil samples were taken to measure microbial parameters inside the wall. Nitrate concentrations upstream of the wall ranged from 21 to 39 g N m(-3), in the wall from 0 to 2 g N m(-3) and downstream from 19 to 44 g N m(-3). An initial groundwater flow investigation using a salt tracer dilution technique showed that the flow through the wall was less than 4% of the flow occurring in the aquifer. Natural gradient tracer tests using bromide and Rhodamine-WT confirmed groundwater bypass under the wall. Hydraulic conductivity of 0.48 m day(-1) was measured inside the wall, whereas the surrounding aquifer had a hydraulic conductivity of 65.4 m day(-1). This indicated that during construction of the wall, hydraulic conductivity of the aquifer had been greatly reduced, so that most of the groundwater flowed under rather than through the wall. Denitrification rates measured in the center of the wall ranged from 0.020 to 0.13 g N m(-3) day(-1), which did not account for the rates of nitrate removal (0.16-0.29 g N m(-3) day(-1)) calculated from monitoring of groundwater nitrate concentrations. This suggested that the rate of denitrification was greater at the upstream face of the wall than in its center where it was limited by low nitrate concentrations. While denitrification walls can be an inexpensive tool for removing nitrate from groundwater, they may not be suitable in aquifers with coarse textured subsoils where simple inexpensive construction techniques result in major decreases in hydraulic conductivity.     

Estimation of local scale dispersion from local breakthrough curves during a tracer test in a heterogeneous aquifer: the Lagrangian approach

The local scale dispersion tensor, Dd, is a controlling parameter for the dilution of concentrations in a solute plume that is displaced by groundwater flow in a heterogeneous aquifer. In this paper, we estimate the local scale dispersion from time series or breakthrough curves, BTCs, of Br concentrations that were measured at several points in a fluvial aquifer during a natural gradient tracer test at Krauthausen. Locally measured BTCs were characterized by equivalent convection dispersion parameters: equivalent velocity, v(eq)(x) and expected equivalent dispersivity, [lambda(eq)(x)]. A Lagrangian framework was used to approximately predict these equivalent parameters in terms of the spatial covariance of log(e) transformed conductivity and the local scale dispersion coefficient. The approximate Lagrangian theory illustrates that [lambda(eq)(x)] increases with increasing travel distance and is much larger than the local scale dispersivity, lambda(d). A sensitivity analysis indicates that [lambda(eq)(x)] is predominantly determined by the transverse component of the local scale dispersion and by the correlation scale of the hydraulic conductivity in the transverse to flow direction whereas it is relatively insensitive to the longitudinal component of the local scale dispersion. By comparing predicted [lambda(eq)(x)] for a range of Dd values with [lambda(eq)(x)] obtained from locally measured BTCs, the transverse component of Dd, DdT, was estimated. The estimated transverse local scale dispersivity, lambda(dT) = DdT/U1 (U1 = mean advection velocity) is in the order of 10(1)-10(2) mm, which is relatively large but realistic for the fluvial gravel sediments at Krauthausen.     

Applications and implications of direct groundwater velocity measurement at the centimetre scale

Three projects involving point velocity probes (PVPs) illustrate the advantages of direct groundwater velocity measurements. In the first, a glacial till and outwash aquifer was characterized using conventional methods and multilevel PVPs for designing a bioremediation program. The PVPs revealed a highly conductive zone that dominated the transport of injected substances. These findings were later confirmed with a natural gradient tracer test. In the second, PVPs were used to map a groundwater velocity field around a dipole recirculation well. The PVPs showed higher than expected velocities near the well, assuming homogeneity in the aquifer, leading to improved representations of the aquifer heterogeneity in a 3D flow model, and an improved match between the modelled and experimental tracer breakthrough curves. In the third study, PVPs detected subtle changes in aquifer permeability downgradient of a biostimulation experiment. The changes were apparently reversible once the oxygen source was depleted, but in locations where the oxygen source lingered, velocities remained low. PVPs can be a useful addition to the hydrogeologist's toolbox, because they can be constructed inexpensively, they provide data in support of models, and they can provide information on flow in unprecedented detail.     

Interpretation of injection-withdrawal tracer experiments conducted between two wells in a large single fracture

Tracer experiments conducted using a flow field established by injecting water into one borehole and withdrawing water from another are often used to establish connections and investigate dispersion in fractured rock. As a result of uncertainty in the uniqueness of existing models used for interpretation, this method has not been widely used to investigate more general transport processes including matrix diffusion or advective solute exchange between mobile and immobile zones of fluid. To explore the utility of the injection-withdrawal method as a general investigative tool and with the intent to resolve the transport processes in a discrete fracture, two tracer experiments were conducted using the injection-withdrawal configuration. The experiments were conducted in a fracture which has a large aperture (>500 microm) and horizontally pervades a dolostone formation. One experiment was conducted in the direction of the hydraulic gradient and the other in the direction opposite to the natural gradient. Two tracers having significantly different values of the free-water diffusion coefficient were used. To interpret the experiments, a hybrid numerical-analytical model was developed which accounts for the arcuate shape of the flow field, advection-dispersion in the fracture, diffusion into the matrix adjacent to the fracture, and the presence of natural flow in the fracture. The model was verified by comparison to a fully analytical solution and to a well-known finite-element model. Interpretation of the tracer experiments showed that when only one tracer, advection-dispersion, and matrix diffusion are considered, non-unique results were obtained. However, by using multiple tracers and by accounting for the presence of natural flow in the fracture, unique interpretations were obtained in which a single value of matrix porosity was estimated from the results of both experiments. The estimate of porosity agrees well with independent measurements of porosity obtained from core samples. This suggests that: (i) the injection-withdrawal method is a viable tool for the investigation of general transport processes provided all relevant experimental conditions are considered and multiple conservative tracers are used; and (ii) for the conditions of the experiments conducted in this study, the dominant mechanism for exchange of solute between the fracture and surrounding medium is matrix diffusion.     

Applied tracer tests in fractured rock: Can we predict natural gradient solute transport more accurately than fracture and matrix parameters?

Applied tracer tests provide a means to estimate aquifer parameters in fractured rock. The traditional approach to analysing these tests has been using a single fracture model to find the parameter values that generate the best fit to the measured breakthrough curve. In many cases, the ultimate aim is to predict solute transport under the natural gradient. Usually, no confidence limits are placed on parameter values and the impact of parameter errors on predictions of solute transport is not discussed. The assumption inherent in this approach is that the parameters determined under forced conditions will enable prediction of solute transport under the natural gradient. This paper considers the parameter and prediction uncertainty that might arise from analysis of breakthrough curves obtained from forced gradient applied tracer tests. By adding noise to an exact solution for transport in a single fracture in a porous matrix we create multiple realisations of an initial breakthrough curve. A least squares fitting routine is used to obtain a fit to each realisation, yielding a range of parameter values rather than a single set of absolute values. The suite of parameters is then used to make predictions of solute transport under lower hydraulic gradients and the uncertainty of estimated parameters and subsequent predictions of solute transport is compared. The results of this study show that predictions of breakthrough curve characteristics (first inflection point time, peak arrival time and peak concentration) for groundwater flow speeds with orders of magnitude smaller than that at which a test is conducted can sometimes be determined even more accurately than the fracture and matrix parameters.     

A comparison of hydraulic and transport parameters measured in low-permeability fractured media

Data from 90 tracer experiments performed in low-permeability fractured media have been studied to explore correlations among parameters controlling flow and transport. The original data had been interpreted by different authors using different models, which prevents direct comparison of their estimated parameters. In order to produce comparable parameters, the data have been reexamined using simple models (homogeneous domain, steady-state flow regime, single porosity). Specifically, hydraulic conductivity has been derived as the ratio of water flux to head gradient and apparent porosity as the ratio of water velocity to water flux; the former estimated from both first and peak arrival times. Hydraulic conductivity and porosity correlate along a straight line of slope 1:3 in log scale. While the regression is too noisy to be of predictive use, it lends some support to the use of a generalized cubic law. The fact that correlation for first arrival time porosity (0.77) is larger than for peak arrival porosity (0.62) suggests that first arrival is controlled by the same flow paths as hydraulic conductivity. Apparent porosity derived from peak arrival time is found to grow with travel time along a line of 0.55 slope (again log scale). The correlation coefficient ranges between 0.73 and 0.80 (depending on the data set) for hard rocks. The fact that this correlation is maintained when varying the flow rate at a given site leads us to suggest that it is caused by diffusion mechanisms. This conclusion is further supported by the increase of apparent porosity with the matrix porosity of the rock on which the experiments were performed.     

Use of tandem circulation wells to measure hydraulic conductivity without groundwater extraction

Conventional methods to measure the hydraulic conductivity of an aquifer on a relatively large scale (10-100 m) require extraction of significant quantities of groundwater. This can be expensive, and otherwise problematic, when investigating a contaminated aquifer. In this study, innovative approaches that make use of tandem circulation wells to measure hydraulic conductivity are proposed. These approaches measure conductivity on a relatively large scale, but do not require extraction of groundwater. Two basic approaches for using circulation wells to measure hydraulic conductivity are presented; one approach is based upon the dipole-flow test method, while the other approach relies on a tracer test to measure the flow of water between two recirculating wells. The approaches are tested in a relatively homogeneous and isotropic artificial aquifer, where the conductivities measured by both approaches are compared to each other and to the previously measured hydraulic conductivity of the aquifer. It was shown that both approaches have the potential to accurately measure horizontal and vertical hydraulic conductivity for a relatively large subsurface volume without the need to pump groundwater to the surface. Future work is recommended to evaluate the ability of these tandem circulation wells to accurately measure hydraulic conductivity when anisotropy and heterogeneity are greater than in the artificial aquifer used for these studies.     

Colloid and suspended particle migration experiments in a granite fracture

To determine the mobility of colloids (0.001–0.45 μm) and suspended particles (> 0.45 μm) in granite fractures, laboratory particle-migration and conservative tracer studies have been carried out in a natural fracture within a large granite block, with overall dimensions of 83×90×60 cm. Flow fields within this horizontal fracture were controlled through a set of 9 boreholes drilled orthogonally to the fracture. Laboratory experiments were performed using a range of average water velocities which contained values low enough to closely approximate the natural flow velocities of < 2 m yr⁻¹ in plutonic rocks of the Canadian Shield. The particles used had diameters between 0.02 and 22 μm, and included latex spheres, glass spheres and colloidal silica. Migration experiments were carried out with a filtered groundwater, ionic strength of 0.01 mol kg⁻¹, obtained from a granite fracture within the Whiteshell Research Area of Manitoba. Flushing experiments showed that suspended particles as large as 40 μm could be mobilized from the fracture surface. The mobility of suspended particles was significantly less than that of colloids. However, within the size range of colloids used in these studies (0.022–0.090 μm), colloid size did not affect colloid migration. Although, in general, colloids eluted ahead of the conservative tracer, colloid mobility was significantly reduced when the average groundwater velocity dropped below between 32 and 240 m yr⁻¹. Colloid transport was found to be very sensitive to flow path and flow direction.     

Effects of pore volume-transmissivity correlation on transport phenomena

The relevant velocity that describes transport phenomena in a porous medium is the pore velocity. For this reason, one needs not only to describe the variability of transmissivity, which fully determines the Darcy velocity field for given source terms and boundary conditions, but also any variability of the pore volume. We demonstrate that hydraulically equivalent media with exactly the same transmissivity field can produce dramatic differences in the displacement of a solute if they have different pore volume distributions. In particular, we demonstrate that correlation between pore volume and transmissivity leads to a much smoother and more homogeneous solute distribution. This was observed in a laboratory experiment performed in artificial fractures made of two plexiglass plates into which a space-dependent aperture distribution was milled. Using visualization by a light transmission technique, we observe that the solute behaviour is much smoother and more regular after the fractures are filled with glass powder, which plays the role of a homogeneous fault gouge material. This is due to a perfect correlation between pore volume and transmissivity that causes pore velocity to be not directly dependent on the transmissivity, but only indirectly through the hydraulic gradient, which is a much smoother function due to the diffusive behaviour of the flow equation acting as a filter. This smoothing property of the pore volume-transmissivity correlation is also supported by numerical simulations of tracer tests in a dipole flow field. Three different conceptual models are used: an empty fracture, a rough-walled fracture filled with a homogeneous material and a parallel-plate fracture with a heterogeneous fault gouge. All three models are hydraulically equivalent, yet they have a different pore volume distribution. Even if piezometric heads and specific flow rates are exactly the same at any point of the domain, the transport process differs dramatically. These differences make it important to discriminate in situ among different conceptual models in order to simulate correctly the transport phenomena. For this reason, we study the solute breakthrough and recovery curves at the extraction wells. Our numerical case studies show that discrimination on the basis of such data might be impossible except under very favourable conditions, i.e. the integral scale of the transmissivity field has to be known and small compared to the dipole size. If the latter conditions are satisfied, discrimination between the rough-walled fracture filled with a homogeneous material and the other two models becomes possible, whereas the parallel-plate fracture with a heterogeneous fault gouge and the empty fracture still show identifiability problems. The latter may be solved by inspection of aperture and pressure testing.     

A tracer study of headspace ventilation in a collector sewer

A field-scale tracer test was conducted to evaluate in-situ ventilation rates in a major collector sewer. The sewer under study was approximately 11 km long and ranged from 0.61 to 2.1 m in diameter. For the purposes of the tracer testing, the collector was divided into four reaches, each of which was tested individually. The tracer test involved injecting a measured volume of CO gas into a manhole over a short time period. CO concentrations were then measured in the collector headspace at selected manholes along the length of the reach. The technique employed successfully measured average headspace velocities over extended lengths of the collector. In a section that had a relatively stagnant headspace, approximately 1.1 km of sewer could be evaluated, with substantial tracer loss attributed to losses to manholes. In a section of the sewer with elevated headspace velocities, a section approximately 7.0 km long was successfully tested with one injection of tracer gas. The velocities observed in the collector varied substantially with time and location in the collector. The lowest velocities measured were in the upstream sections, with a minimum observed value of 3.8 m/min. The highest velocities were observed in the downstream sections, with a maximum value of 31.5 m/min. The presence of a substantial drop structure appeared to reduce the headspace velocity in the upstream reach. In general, there was an increasing trend in gas-phase flows with distance along the length of the collector. Flows at the discharge end of the collector were almost 2 orders of magnitude greater than those at the beginning.     

A Tracer Study of Headspace Ventilation in a Collector Sewer

ABSTRACT

A field-scale tracer test was conducted to evaluate in-situ ventilation rates in a major collector sewer. The sewer under study was ~11 km long and ranged from 0.61 to 2.1 m in diameter. For the purposes of the tracer testing, the collector was divided into four reaches, each of which was tested individually. The tracer test involved injecting a measured volume of CO gas into a manhole over a short time period. CO concentrations were then measured in the collector headspace at selected manholes along the length of the reach.

The technique employed successfully measured average headspace velocities over extended lengths of the collector. In a section that had a relatively stagnant headspace, ~1.1 km of sewer could be evaluated, with substantial tracer loss attributed to losses to manholes. In a section of the sewer with elevated headspace velocities, a section ~7.0 km long was successfully tested with one injection of tracer gas. The velocities observed in the collector varied substantially with time and location in the collector. The lowest velocities measured were in the upstream sections, with a minimum observed value of 3.8 m/min. The highest velocities were observed in the downstream sections, with a maximum value of 31.5 m/min. The presence of a substantial drop structure appeared to reduce the headspace velocity in the upstream reach. In general, there was an increasing trend in gas-phase flows with distance along the length of the collector. Flows at the discharge end of the collector were almost 2 orders of magnitude greater than those at the beginning.     

Transverse vertical dispersion in groundwater and the capillary fringe

Transverse dispersion is the most relevant process in mass transfer of contaminants across the capillary fringe (both directions), dilution of contaminants, and mixing of electron acceptors and electron donors in biodegrading groundwater plumes. This paper gives an overview on literature values of transverse vertical dispersivities alpha(tv) measured at different flow velocities and compares them to results from well-controlled laboratory-tank experiments on mass transfer of trichloroethene (TCE) across the capillary fringe. The measured values of transverse vertical dispersion in the capillary fringe region were larger than in fully saturated media, which is credited to enhanced tortuosity of the flow paths due to entrapped air within the capillary fringe. In all cases, the values observed for alpha(tv) were < 1 mm. The new measurements and the literature values indicate that alpha(tv) apparently declines with increasing flow velocity. The latter is attributed to incomplete diffusive mixing at the pore scale (pore throats). A simple conceptual model, based on the mean square displacement and the pore size accounting for only partial diffusive mixing at increasing flow velocities, shows very good agreement with measured and published data.     

Mass removal of chlorinated ethenes from rough-walled fractures using permanganate

In situ chemical oxidation (ISCO) employing permanganate is an emerging technology that has been successful at enhancing mass removal from DNAPL source zones in unconsolidated media at the pilot-scale. The focus of this study was to evaluate the applicability of flushing a permanganate solution across two single vertical fractures in a laboratory environment to remove free phase DNAPL. The fracture experiments were designed to represent a portion of a larger fractured aquifer system impacted by a near-surface DNAPL spill over a shallow fractured rock aquifer. Each fracture was characterized by hydraulic and tracer tests, and the aperture field for one of the fractures was mapped using a co-ordinate measurement machine. Following DNAPL emplacement, a series of water and permanganate flushes were performed. To support observations from the fracture experiments, a set of batch experiments was conducted. The data from both fracture experiments showed that the post-oxidation effluent concentration was not impacted by the oxidant flush; however, changes in the aperture distribution, flow field, and flow rate were observed. These changes resulted in a significant decrease to the mass loading from the fractures, and were attributed to the build-up of oxidation by-products (manganese oxides and carbon dioxide) within the fracture which was corroborated by the batch experiment data and visual examination of the walls of one fracture. These results provide insight into the potential impact that a permanganate solution and oxidation by-products can have on the aperture distribution within a fracture and on DNAPL mass transfer rates. A permanganate flush or injection completed within a fractured rock aquifer may lead to the development of an insoluble product adjacent to the DNAPL which results in the reduction or complete elimination of advective regions near the DNAPL and reduces mass transfer rates. This outcome would have significant implications on the plume generating potential of the remaining DNAPL.     

Evaluation of groundwater flow patterns around a dual-screened groundwater circulation well

Dual-screened groundwater circulation wells (GCWs) can be used to remove contaminant mass and to mix reagents in situ. GCWs are so named because they force water in a circular pattern between injection and extraction screens. The radial extent, flux and direction of the effective flow of this circulation cell are difficult to measure or predict. The objective of this study is to develop a robust protocol for assessing GCW performance. To accomplish this, groundwater flow patterns surrounding a GCW are assessed using a suite of tools and data, including: hydraulic head, in situ flow velocity, measured hydraulic conductivity data from core samples, chemical tracer tests, contaminant distribution data, and numerical flow and transport models. The hydraulic head data show patterns that are consistent with pumping on a dual-screened well, however, many of the observed changes are smaller than expected. In situ thermal perturbation flow sensors successfully measured horizontal flow, but vertical flow could not be determined with sufficient accuracy to be useful in mapping flow patterns. Two types of chemical tracer tests were utilized at the site and showed that much of the flow occurs within a few meters of the GCW. Flow patterns were also assessed based on changes in contaminant (trichloroethylene, TCE) concentrations over time. The TCE data clearly showed treated water moving away from the GCW at shallow and intermediate depths, but the circulation of that water back to the well, except very close to the well, was less clear. Detailed vertical and horizontal hydraulic conductivities were measured on 0.3 m-long sections from a continuous core from the GCW installation borehole. The measured vertical and horizontal hydraulic conductivity data were used to construct numerical flow and transport models, the results of which were compared to the head, velocity and concentration data. Taken together, the field data and modeling present a fairly consistent picture of flow and transport around the GCW. However, the time and expense associated with conducting all of those tests would be prohibitive for most sites. As a consequence, a sequential protocol for GCW characterization is presented here in which the number of tools used can be adjusted to meet the needs of individual sites. While not perfect, we believe that this approach represents the most efficient means for evaluating GCW performance.     

A Wind Tunnel Study of Gaseous Pollutants in City Street Canyons

Steady state mean concentrations of tracer gas were measured in a 400:1 scale model of an idealized city with variable geometry placed within a wind tunnel at various orientations to the mean flow for a free stream velocity of 6.8 ft/sec. The tracer gas was released from two parallel line sources to simulate lanes of traffic in an effort to quantify the persistence of pollution as well as the mean values realized at street levels. An aerodynamically rough turbulent boundary layer of neutral thermal stratification was employed to simulate the atmosphere. Values of concentration measured in the model city were converted to prototype concentrations in ppm and compared to National Ambient Air Quality Standards. It was shown that single isolated structures may cause favorable mixing of pollution downwind but very high concentrations exist in the immediate leeward vicinity of the building. Two favorable geometries for city blocks tested were found to reduce pedestrian exposure to pollution both near heavy traffic congestion and downwind. It was concluded that the pollutant dilution was controlled by the mean flow rather than by turbulent diffusion and that the lateral spread of the plume was slight as one proceeded downwind of the line source. The combination of favorable geometry and higher dilution velocities may bring pollution levels down to existing Air Quality Standards. The body of information presented in this paper should interest city planners and air quality monitoring personnel, as well as those researchers attempting to study and model flow in city street canyons. It provides order of magnitude estimates on pedestrian and office worker exposure to pollutants under a wide range of conditions.     

Experimental study of the transport properties of rough self-affine fractures

An experimental study of the transport properties of fluid-saturated joints composed of two complementary rough fracture surfaces, translated with respect to each other and brought in contact, is reported. Quantitative roughness measurements on different fractured granite samples show that the surfaces have a self-affine geometry from which the dependence of the mean aperture on the relative displacement of fracture surfaces kept in contact can be predicted. Variations of the hydraulic and electrical conductances of the joint are measured as functions of its mean aperture. A simple parallel plane model accounts for the global trend of the measurements, but significant deviations are observed when a relative lateral displacement of the surfaces is introduced. A theoretical analysis of their origin shows that they are due both to the randomness of the aperture field and to a nonzero local slope of the surface near the injection hole; the corresponding conductivity fluctuation amplitudes have power law and linear variations with the lateral displacement, and are enhanced by the radial injection geometry.     

First-passage-time transfer functions for groundwater tracer tests conducted in radially convergent flow

Forced-gradient groundwater tracer tests may be conducted using a variety of hydraulic schemes, so it is useful to have simple semi-analytic models available that can examine various injection/withdrawal scenarios. Models for radially convergent tracer tests are formulated here as transfer functions, which allow complex tracer test designs to be simulated by a series of simple mathematical expressions. These mathematical expressions are given in Laplace space, so that transfer functions may be placed in series by simple multiplication. Predicted breakthrough is found by numerically inverting the composite transfer function to the time-domain, using traditional computer programs or commercial mathematical software. Transport is assumed to be dictated by a radially convergent or uniform flow field, and is based upon an exact first-passage-time solution of the backward Fokker–Planck equation. These methods are demonstrated by simulating a weak-dipole tracer test conducted in a fractured granite formation, where mixing in the injection borehole is non-ideal.