Parameter estimation through groundwater tracer tests

Ordinary differential equation (ODE) models are quite popular for modelling complex dynamic processes in many scientific fields, and the parameters in these models are usually unknown, and we need to estimate them using statistical methods. When some observations are contaminated, regular estimation methods, such as nonlinear least-square estimation, will bring large bias. In this paper, robust estimations of both constant and time-varying parameters in ODE models using M-estimators are proposed, and their asymptotic properties are obtained under some mild conditions. We focus on Huber M-estimator, and also provide a method to adjust the Huber parameter automatically to the observations. The proposed method is compared to existing methods in numerical simulations and CD8+ T cell data analysis. It is demonstrated that our method gain substantial efficiency as well as robust properties.

FEMA 356 backbone relations tend to provide conservative estimates of the available strengths and deformation capacities of reinforced concrete components, leading to costly seismic rehabilitation solutions for California hospitals. The conservatism is driven by the lack of test data for older, typically poorly detailed, structural components. Although FEMA 356 does provide for building specific component testing, this approach is not common due to cost concerns and/or schedule constraints. However, a test program on lightly-reinforced, poorly-detailed wall segments was undertaken at UCLA in 2004–2005 to support seismic rehabilitation projects on several southern California hospitals. Presented test results are compared with FEMA 356 backbone relations to highlight the advantages associated with building-specific test programs. In general, the test specimens revealed more strength and deformation capacity than assumed by FEMA 356, and more gradual strength deterioration. The test results, when coupled with the FEMA 356 Nonlinear Static Procedure (NSP) and Nonlinear Dynamic Procedure (NDP), enabled the development of more rational and substantially more economical rehabilitation solutions. The test results were used to help develop new backbone relations that were incorporated into ASCE-41-06 Supplement #1.

Faults in some deep mudstones have poor hydraulic connectivity owing to high normal stress on the fault planes. Designing a method for modelling solute transport pathways in such faults/fractures using available data is a critical issue vis-à-vis the safety assessment of radioactive waste disposal. In this study, faults in deep siliceous mudstones with low swelling capacity are investigated using cross-hole hydraulic and tracer tests between two boreholes. The water pressures observed during the hydraulic test are reproduced via hydraulic simulations, assuming a one-dimensional (1D) flow channel with a length 8–80 times the linear distance (4.5 m) between the two borehole test sections. This flow channel length indicates a tortuosity remarkably higher than that previously reported in discrete fracture network simulations and laboratory experiments using a single fracture (i.e. 1–5). Advection–dispersion simulations using a pipe model with a tortuosity of 8–80 reproduces a time series of tracer concentrations observed during the tracer test. The radius of the pipe simulated in the tracer test was 7–39 times the flow channel radius estimated via the hydraulic test, which is in agreement with the ratio of 5–20 previously reported based on in situ tracer and hydraulic tests of faulted/fractured rocks. The simulated longitudinal dispersivity is 1/9–1/20 of the estimated flow path lengths, in agreement with the laboratory and field tracer study values. These results indicate that transport pathways in faults with low hydraulic connectivity can be modelled using a highly tortuous 1D pipe flow path.


 
 
 Tracer tests are fundamental in characterizing fluid flow in underground formations. However, the setting of appropriate boundary conditions in even simple analytical models has historically been controversial, mainly due to the lack of sufficient physical evidence. Determining the relevance of boundary conditions on tracer test analysis becomes therefore a topic of renewed interest. The subject has been previously disscused by studying the tracer breakthrough curve sensitivity to diverse model parameter values. In our present work we examine the issue from a new practical reservoir characterization perspective. Two well-known equivalent elementary tracer transport models having different boundary conditions are matched to the same tracer test data. The resulting model parameter difference quantifies the effect of boundary conditions on reservoir property determination. In our case a tracer pulse injected in a homogeneous one-dimensional porous medium and moving at constant speed dominated by advection and dispersion is considered. The tracer transport models to be used yield conditions (i) on the tracer concentration, and (ii) on the tracer flux. Three data sets from tracer tests performed in different oil reservoirs are used to fit the models and determine the parameter values. We found that boundary conditionsbecome more important as Peclet number gets smaller. The cases analyzed have Peclet numbers 25.0, 5.4 and 3.7. The largest parameter difference obtained in each case is 5%, 18% and 37% respectively. These differences are large for laboratory experiments, but less relevant for tracer tests in oil fields, where data variability is frequently high. Nevertheless, attention should be paid when small Peclet numbers are present.
 
 

This paper presents the numerical analysis results of a field tracer test. The objective of this test was to measure the dispersivity and the effective porosity of a permeable sedimentary layer. The tracer test was performed using two vertical bore holes 90 meters in depth and located 7 meters apart. The sedimentary layer, composed of sandstone, conglomerate, and weathered granite, was 10 meters thick and started 80 meters below the surface. One bore hole was used to inject groundwater with tracer and the other to recover to recirculate groundwater. We used a tracer solution containing Br. Before the tracer was injected, groundwater was circulated through the sedimentary layer between the two bore holes to establish a steady state of groundwater flow. The circulation flow rate and Br concentration initially injected were determined by using Gelhar's analytical solution so that the detectable tracer concentration (several ppm) could be recovered within one month. When the groundwater level at the recovery hole became stable, the tracer was injected into the injection hole and the concentration of Br was analyzed at the recovery hole. The circulation flow rate was 7800cm³/ min. The initially injected Br- tracer concentration was 8300 ppm. Thirty eight hours later, the concentration of Br at the recovery hole reached a maximum level of 10 ppm.The convection-dispersion equation was used for the tracer test's numerical analysis. This differential equation was transformed by F. E. M. Half of the horizontal section of the analytical region was used for numerical analysis and meshes like a semi-circle were used for the circulation-line above the surface between the two bore holes. A dispersion coefficient depends upon flow velocity, dispersivity, and diffusion. Initially, the flow velocity distribution and the permeability of layer were analyzed using F. E. M. as a seepage problem. The analyzed permeability of the layer was 2.2×10-5cm/sec.According to the F. E. M. analysis result, using the breakthrough curve of Br at the recovery hole and the matching technique, the effective porosity was determined to be 7% and the dispersivity 3.5 meters. The effective porosity was determined to be 4.5% and the dispersivity 3.5 meters by Gelhar's analytical solution. The 7% or 4.5% effective porosity value was smaller than the 30% porosity value with the boring core samples. This suggests that the measurement of effective porosity using a field tracer test is important in evaluating groundwater flow velocity and mass transport. The relationship between the dispersivity 3.5 meters and the distance 7 meters agreed well with those of Leonhart (1985).

Tracer tests have been widely used in the petroleum industry to determine qualitative information about underground reservoirs, such as the existence of flow barriers, directional flow paths, etc. This paper describes how well-to-well tracer tests can be used to obtain quantitative information about the geostatistical parameters (variance and correlation scale) of permeability. Both theoretical and experimental results are presented. In the theoretical work, a catalog of tracer production curves for different sets of geostatistical parameters for five-spot flooding patterns is produced. This catalog is computed with the help of a new numerical technique that is stable and devoid of any numerical dispersion. The effects of variance of the natural logarithm of permeability (ranging from .14 to 2.25), dimensionless correlation scale (ranging from .2 to .5), and of numerous realizations are quantified. The general effect of areal heterogeneity is that multiple peaks appear in the output tracer concentration curve, even when the flow is confined to a single layer. An increase in variance generally increases the number of prominent peaks in the output concentration profile, decreases the time between the emergence of peaks, and increases the time span of the occurrence of the tracer. An increase in correlation scale generally decreases the number of peaks. By using a sufficient number of permeability realizations, an attempt is made to assess variance and correlation length of the permeability field from the tracer and output concentration curves. In the experimental work, a slab of rock (Arizona flagstone) representing a quarter of a five-spot pattern was selected. A simple minipermeameter was used to measure areal permeability on grids of points on both the top and bottom surfaces of the rock. A tracer test was performed and the tracer output concentrations were measured at six intervals along the depth of the rock at the production well. Six different types of concentration curves were obtained, showing that downhole sampling of the tracer would be important in achieving adequate information about the reservoir rock. Permeabilities measured with the minipermeameter were used as input into the simulator and the numerical production curves were compared with the experimental profiles. The occurrences of similar patterns of multiple peaks confirms the numerical technique and strongly suggests that well-to-well tracer tests in oil fields and aquifers could be used to estimate the geostatistical parameters of the permeability field.

Modeling and Identification of a Nuclear Reactor

Tracer tests are conducted in geothermal reservoirs as an aid in forecasting thermal breakthrough of reinjection water. To interpret tracer tests, mathematical models have been developed based on the various transport mechanisms in these highly fractured reservoirs. These tracer flow models have been applied to interpret field tests. The resulting matches between the model and field data were excellent and the model parameters were used to estimate reservoir properties. However, model fitting is an indirect process and the model's ability to estimate reservoir properties cannot be judged solely on the quality of the match between field data and model predictions. The model's accuracy in determining reservoir characteristics must be independently verified in a closely controlled environment. In this study, the closely controlled laboratory environment was chosen to test the validity and accuracy of tracer flow models developed specifically for flow in fractured rocks. The laboratory tracer tests were performed by flowing potassium iodide (KI) through artificially fractured core samples. The tracer test results were then analyzed with several models to determine which best fit the measured data. A Matrix Diffusion model was found to provide the best match of the tracer experiments. The core properties, as estimated by the Matrixmore » Diffusion model parameters generated from the indirect matching process, were then determined. These calculated core parameters were compared to the measured core properties and were found to be in agreement. This verifies the use of the Matrix Diffusion flow model in estimating fracture widths from tracer tests.« less

A comparative study of short cut procedures for parameter estimation in differential equations

Hydrological tracer testing is the most reliable diagnostic technique available for the determination of basic hydraulic and geometric parameters necessary for establishing operative solute-transport processes. Tracer-test design can be difficult because of a lack of prior knowledge of the basic hydraulic and geometric parameters desired and the appropriate tracer mass to release. A new efficient hydrologic tracer-test design (EHTD) methodology has been developed to facilitate the design of tracer tests by root determination of the one-dimensional advection–dispersion equation (ADE) using a preset average tracer concentration which provides a theoretical basis for an estimate of necessary tracer mass. The method uses basic measured field parameters (e.g., discharge, distance, cross-sectional area) that are combined in functional relationships that describe solute-transport processes related to flow velocity and time of travel. These initial estimates for time of travel and velocity are then applied to a hypothetical continuous stirred tank reactor (CSTR) as an analog for the hydrological-flow system to develop initial estimates for tracer concentration, tracer mass, and axial dispersion. Application of the predicted tracer mass with the hydraulic and geometric parameters in the ADE allows for an approximation of initial sample-collection time and subsequent sample-collection frequency where a maximum of 65 samples were determined to be necessary for describing the predicted tracer-breakthrough curve (BTC). Inclusion of tracer retardation and decay cause a net increase in tracer-mass estimates so that the preset average tracer concentration will be maintained and there will be a consequent steepening of the BTC, but retardation also causes BTC spreading and a delay in tracer arrival.

The Sun’s polar magnetic field is pivotal in understanding solar dynamo processes and forecasting future solar cycles. However, direct measurements of the polar field have only been available since the 1970s. The chromospheric Ca ii K polar network index (PNI; the fractional area of the chromospheric network regions above a certain latitude) has recently emerged as a reliable proxy for polar magnetic fields. In this study, we derive PNI estimates from newly calibrated, rotation-corrected Ca ii K observations from the Kodaikanal Solar Observatory (1904–2007) and modern data from the Rome Precision Solar Photometric Telescope (2000–2022). We use both of those Ca ii K archives to identify polar network regions with an automatic adaptive threshold segmentation technique and calculate the PNI. The PNI obtained from both the archives shows a significant correlation with the measured polar field from the Wilcox Solar Observatory (Pearson correlation coefficient r > 0.93) and the derived polar field based on an Advective Flux Transport Model (r > 0.91). The PNI series also shows a significant correlation with faculae counts derived from Mount Wilson Observatory observations (r > 0.87) for both Kodaikanal Solar Observatory and Rome Precision Solar Photometric Telescope data. Finally, we use the PNI series from both archives to reconstruct the polar magnetic field over a 119 yr long period, which includes the last 11 solar cycles (Cycles 14–24). We also obtain a relationship between the amplitude of solar cycles (in 13 month smoothed sunspot number) and the strength of the reconstructed polar field at the preceding solar cycle minimum to validate the prediction of the ongoing solar cycle, Cycle 25.

The planning and execution of chemical tracer tests have been detailed in a recent reference guide developed by the U.S. EPA for water utilities. The objective of this paper is to extend the EPA guidance by presenting a simple, analytical technique to calculate mean residence time (MRT) from chemical tracer data. Case histories of tracer tests in three distribution systems in North Carolina (Raleigh, Greenville, and Cary) are used to illustrate the technique. Examples are given of the time required for sampling and the mileage driven. Tracer responses to positive and negative step input tracer tests are compared. Storage tanks are shown to affect the tracer response at nearby stations and complicate calculation of an MRT value. The MRT values obtained at 22 stations in the Raleigh distribution system compared fairly well to hydraulic model predictions of the average water age.

Characterisation of fractured carbonate aquifers using ambient borehole dilution tests

This work studied tracer and thermal transients during reinjection in geothermal reserviors and developed a new technique which combines the results from interwell tracer tests and thermal injection-backflow tests to estimate the thermal breakthrough times. Tracer tests are essential to determine the degree of connectivity between the injection wells and the producing wells. To analyze the tracer return profiles quantitatively, we employed three mathematical models namely, the convection-dispersion (CD) model, matrix diffusion (MD) model, and the Avodnin (AD) model, which were developed to study tracer and heat transport in a single vertical fracture. We considered three types of tracer tests namely, interwell tracer tests without recirculation, interwell tracer tests with recirculation, and injection-backflow tracer tests. To estimate the model parameters, we used a nonlinear regression program to match tracer return profiles to the solutions.

Mathematical modeling provides a rigorous way to quantify immunological processes and discriminate between alternative mechanisms driving specific biological phenomena. It is typical that mathematical models of immunological phenomena are developed by modelers to explain specific sets of experimental data after the data have been collected by experimental collaborators. Whether the available data are sufficient to accurately estimate model parameters or to discriminate between alternative models is not typically investigated. While previously collected data may be sufficient to guide development of alternative models and help estimating model parameters, such data often do not allow to discriminate between alternative models. As a case study, we develop a series of power analyses to determine optimal sample sizes that allow for accurate estimation of model parameters and for discrimination between alternative models describing clustering of CD8 T cells around Plasmodium liver stages. In our typical experiments, mice are infected intravenously with Plasmodium sporozoites that invade hepatocytes (liver cells), and then activated CD8 T cells are transferred into the infected mice. The number of T cells found in the vicinity of individual infected hepatocytes at different times after T cell transfer is counted using intravital microscopy. We previously developed a series of mathematical models aimed to explain highly variable number of T cells per parasite; one of such models, the density-dependent recruitment (DDR) model, fitted the data from preliminary experiments better than the alternative models, such as the density-independent exit (DIE) model. Here, we show that the ability to discriminate between these alternative models depends on the number of parasites imaged in the analysis; analysis of about [Formula: see text] parasites at 2, 4, and 8h after T cell transfer will allow for over 95% probability to select the correct model. The type of data collected also has an impact; following T cell clustering around individual parasites over time (called as longitudinal (LT) data) allows for a more precise and less biased estimates of the parameters of the DDR model than that generated from a more traditional way of imaging individual parasites in different liver areas/mice (cross-sectional (CS) data). However, LT imaging comes at a cost of a need to keep the mice alive under the microscope for hours which may be ethically unacceptable. We finally show that the number of time points at which the measurements are taken also impacts the precision of estimation of DDR model parameters; in particular, measuring T cell clustering at one time point does not allow accurately estimating all parameters of the DDR model. Using our case study, we propose a general framework on how mathematical modeling can be used to guide experimental designs and power analyses of complex biological processes.