Probability Distribution Of Travel Times Research Articles

Three-dimensional modelling of flow and reactive transport in heterogeneous porous media : M. A. Cushey, A. Bellin & Y. Rubin, in: Groundwater quality: remediation and protection. Proc. conference, Prague, 1995, ed K. Kovar & J. Krasny, (IAHS; Publication, 225), 1995, pp 257–261

Abstract A method is described for simulating three-dimensional (3D) flow fields for use in transport modelling of contaminant migration in statistically anisotropic heterogeneous aquifers. This method differs from previous approaches in that the flow field is generated directly, reducing the computational requirements considerably by bypassing the interim steps of the more traditional techniques. The approach discussed here provides a more computationally efficient and verifiable means of modelling transport in a 3D formation. The method will enable the generation of concentration and travel time probability distribution functions (PDFs) as well as 3D images of the contaminant plume. INTRODUCTION Models which can simulate the migration of reactive solutes in a three dimensional (3D) flow field within anisotropic heterogeneous aquifers are necessary to more accurately estimate the movement of contaminants in the environment and to better understand the underlying processes which govern contaminant transport. Estimates limited to only the mean and variance of the solute concentrations are often insufficient since contaminant concentration distributions are typically non-Gaussian and not fully characterized by their first two moments (Smith & Schwartz, 1980; Bellin et al., 1994; Rubin et al., 1994). Until recently, the complete modelling of contaminant transport in heterogeneous porous media has been limited to analysis in two dimensions. Though traditional approaches such as finite element can be applied in three dimensions, the computational requirements are excessive. The method presented here is an alternative to these approaches and requires less computational effort without loss of accuracy. METHODOLOGY This method is an Eulerian-Lagrangian approach and is applied in a model utilizing a stochastic framework to predict the spatial moments and probability distribution functions (PDFs) of the solute concentration and travel time. The fluid velocity is

Eulerian‐Lagrangian Analysis of Transport Conditioned on Hydraulic Data: 3. Spatial Moments, Travel Time Distribution, Mass Flow Rate, and Cumulative Release Across a Compliance Surface

In paper 1 of this series we described an analytical‐numerical approach to predict deterministically solute transport under uncertainty. The approach allows conditioning such predictions on hydraulic measurements and assessing the corresponding reduction in uncertainty. In paper 2 we examined the effects of log transmissivity and hydraulic head data on conditional predictions of concentration due to instantaneous point and nonpoint sources. In this paper we show how the same approach can be used directly to estimate mass flow rate across a “compliance surface,” cumulative mass release, and the probability distribution of travel times across this surface and the associated uncertainty. Contrary to some other methods in the literature, our approach requires neither a special theory for travel times nor a prior assumption about their probability distribution. We also show how one can compute explicitly the second spatial moment of the conditional mean plume about its center of mass, the conditional mean second spatial moment of the actual plume about its center of mass, and the conditional covariance of the plume center of mass. We illustrate these quantities and the effect of conditioning on some of them by considering instantaneous point, line, and area sources in a two‐dimensional, statistically homogeneous, isotropic, mildly varying log transmissivity field under uniform prior (unconditional) mean flow.

DEVELOPING THE SPATIAL MISMATCH HYPOTHESIS: PROBLEMS OF ACCESSIBILITY TO EMPLOYMENT FOR LOW-WAGE CENTRAL CITY LABOR

This research focuses on the spatial mismatch hypothesis as an explanation for the concentration of unemployment in the central cities of the industrial Midwest. It is argued that the immobility of low-wage labor relative to the mobility of employment results in either the exploitation or isolation of low-wage labor. In the contemporary period, the suburbanization of low-wage employment economically isolates the immobile low-wage population of the central cities. This alternative view of the spatial mismatch de-emphasizes race and emphasizes the segmentation of women and minorities into low-wage occupations. To test the spatial mismatch hypothesis, a model of the probability distribution of travel times to work is estimated for three cities in the industrial Midwest: Chicago, Detroit, and Cleveland. The results support the hypothesis that the decentralization of employment reduces access to employment for central city residents as evidenced by the lower frequency of short trips to work for low-wage centra...

Screening of Groundwater Contaminants by Travel‐Time Distributions

Analytical procedures are proposed for estimating probability distributions of travel times for chemical waste loads to groundwater. Travel distance of the chemical is treated as a renewal process, and travel time is given by the number of annual chemical displacements or renewals required for passage through the soil unsaturated zone. The methods were evaluated for two organic chemicals for a range of site conditions near Lexington, Kentucky, and Des Moines, Iowa. It was found that groundwater contamination risk is very dependent on site-specific soil and hydrologic conditions. Probability distributions based on both hydrologic and soil spatial variability differed markedly from those that considered only hydrologic variability. The effect of soil spatial variability is to increase travel-time variability and risk of groundwater contamination. Travel-time probability distributions were combined with biochemical degradation rates to estimate the expected fraction of chemical mass reaching the water table. This fraction serves as a screening criterion to identify and rank combinations of chemicals and sites that are likely to produce groundwater pollution.

Generalized Traffic Equilibrium with Probabilistic Travel Times and Perceptions

The paper studies the problem of predicting traffic equilibrium (TE) in a transportation network within the framework of decision making among discrete choices in a probabilistic and uncertain environment. Conventional approaches to predict TE typically assume that travel times are deterministic and perceived accurately by the travelers; and some new TE models have considered probabilistic travel times or inaccurate perceptions but not both. In our generalized model, which we refer to as GTESP, the travel time on each route is random and each traveler perceives, possibly inaccurately, a travel time probability distribution for each route which may vary from traveler to traveler. Each traveler uses a disutility function of travel time to evaluate each route and chooses that route which minimizes his expected disutility. GTESP is difficult to solve for general nonlinear disutility functions. However, special cases—in particular when arc travel times are statistically independent and the disutility functions to evaluate route travel times are linear, exponential, or quadratic—are solvable, at least approximately. GTESP is general in the sense that most existing TE models can be shown to be special or limiting cases of GTESP. Furthermore, this paper demonstrates, with illustrative examples, that GTESP appears to capture travelers' risk-taking behavior more realistically than existing TE models.

Technical Note—Estimating the Probability Distribution of Travel Times for Urban Emergency Service Systems

We describe an extension of Larson's hypercube queuing model to enable it to calculate the probability distributions of travel times. In addition, we discuss the general need for this capability in emergency service models as a prerequisite for developing second-generation models that may relate direct outcome measures such as damage, loss of life, and arrests to the managerial decisions.