Travel Time Moments Research Articles

On the interplay between hillslope and drainage network flow dynamics in the catchment travel time distribution

AbstractTravel time is a robust measure of water transport dynamics in catchments. At a given control section along a drainage network, travel time distribution results from an interplay between two main processes: (i) the transport through the hillslopes and (ii) transport through the drainage network. The main scope of this work is to quantify this interplay, specifically we aim to identify the relative impact of hillslopes and channels on the travel time at the catchment scale. A theoretical framework is developed following a bottom‐up modelling approach that combines a Boussinesq model for water flow and water travel time in hillslopes, with a geomorphological model for water transport in drainage network. Simple semi‐analytical solutions are derived for the first two moments of the travel time distribution within a flow section. We provide some relevant examples based on synthetic rainfall data, exploring the relative impacts of hillslope and channel properties. As expected, the dynamics of the hillslopes control the travel time distribution at the catchment scale. The drainage network typically introduces a lag in the average travel time exiting the hillslopes and reduces the temporal fluctuations of the mean travel time and its variance. Our theoretical model provides meaningful insights on the investigation of the dominant dynamics taking place in catchments: results suggest that hillslopes and their features are the main driver of travel time in catchments. The temporal fluctuations of the travel time moments show a non‐linear dependence with the recharge time‐series and need to be considered as time‐variant. Variability of water ages collected in a single water sample can be very large, the latter feature may have a significant effect on the water quality and on the tracer data analysis, largely governed by the contact times between water and catchment material.

The Problem of Congestion Unreliability Cost: The Certain Error of a Mean and the Measure of Travel Time Utility Gains from Public Works

This summary-for-students presents the claim of some recent analyses of home-based daily urban trips made in France or in the USA that the first four moments of travel time, not just the first, matter and that travelers react to changed service on road or transit networks by adjusting all four of their constructed trip duration moments, including the three higher moments characterizing time unreliability. Sole reliance on mean time changes to evaluate improved or worsened trip conditions then short-changes the benefit-cost analysis and notably fails to explain fast fill-ups and super-peaks that are part of the demanded trip time profile.

Numerical and analytical modeling of advective travel times in realistic three‐dimensional fracture networks

Travel time distributions obtained from advective transport in multiple realizations of realistic discrete fracture network simulations are analyzed using the truncated one‐sided stable distribution, which has previously been shown to generalize both the advection‐dispersion solution as well as one‐sided stable distributions. Using this model, it is shown that the Fickian assumption inherent in the advection‐dispersion equation generally fails, despite the first two moments of travel time essentially scaling linearly with distance. It is also observed that the equally probable realizations drawn from the ensemble can produce a wide range of behavior under the current configuration, such that Fickian conditions are almost obtained in some cases for increasing scales. On the basis of a small‐scale calibration against particle breakthrough, the model is then shown to successfully predict limiting bounds of transport for a one order of magnitude increase in scale. Correlation in particle velocity is explicitly shown to be significant for scales close to the characteristic Lagrangian segment length. The network configuration is obtained from extensive site characterization data at the Laxemar region in Sweden and represents a block‐scale domain of reasonably sparse background fractures.

Significance of higher moments for complete characterization of the travel time probability density function in heterogeneous porous media using the maximum entropy principle

The travel time formulation of advective transport in heterogeneous porous media is of interest both conceptually, e.g., for incorporating retention processes, and in applications where typically the travel time peak, early, and late arrivals of contaminants are of major concern in a regulatory or remediation context. Furthermore, the travel time moments are of interest for quantifying uncertainty in advective transport of tracers released from point sources in heterogeneous aquifers. In view of this interest, the travel time distribution has been studied in the literature; however, the link to the hydraulic conductivity statistics has been typically restricted to the first two moments. Here we investigate the influence of higher travel time moments on the travel time probability density function (pdf) in heterogeneous porous media combining Monte Carlo simulations with the maximum entropy principle. The Monte Carlo experimental pdf is obtained by the adaptive Fup Monte Carlo method (AFMCM) for advective transport characterized by a multi‐Gaussian structure with exponential covariance considering two injection modes (in‐flux and resident) and lnK variance up to 8. A maximum entropy (MaxEnt) algorithm based on Fup basis functions is used for the complete characterization of the travel time pdf. All travel time moments become linear with distance. Initial nonlinearity is found mainly for the resident injection mode, which exhibits a strong nonlinearity within first 5IY for high heterogeneity. For the resident injection mode, the form of variance and all higher moments changes from the familiar concave form predicted by the first‐order theory to a convex form; for the in‐flux mode, linearity is preserved even for high heterogeneity. The number of moments sufficient for a complete characterization of the travel time pdf mainly depends on the heterogeneity level. Mean and variance completely describe travel time pdf for low and mild heterogeneity, skewness is dominant for lnK variance around 4, while kurtosis and fifth moment are required for lnK variance higher than 4. Including skewness seems sufficient for describing the peak and late arrivals. Linearity of travel time moments enables the prediction of asymptotic behavior of the travel time pdf which in the limit converges to a symmetric distribution and Fickian transport. However, higher‐order travel time moments may be important for most practical purposes and in particular for advective transport in highly heterogeneous porous media for a long distance from the source.

Effect of Sorption Heterogeneity on Moments of Solute Residence Time in Convergent Flows

In this paper, we analyze the impact of physical and chemical heterogeneity on solute travel time to a pumping well. We consider a solute undergoing reversible linear instantaneous equilibrium sorption. Both the distribution coefficient, Kd, and the transmissivity field, T, are considered spatially variable, and are modeled as partially correlated spatial random functions. Groundwater flow and solute transport are then solved within the context of a numerical Monte Carlo framework. The results are analyzed on the basis of dimensional analysis techniques. Simple and compact expressions characterizing the dependence of the target travel time moments on relevant dimensionless groups are proposed. The functional form of these expressions is inspired by, and is consistent with, the previous works of Sanchez-Vila and Rubin (Water Resour. Res. 39(4):1086, 2003) and Riva et al. (J. Contam. Hydrol. 82:23–43, 2006) A key result is that the effects of the chemical and physical heterogeneities on the mean travel time can be decoupled consistently with existing analytical results. The relative role of physical and geochemical heterogeneities in travel time variance is more complex, and such a decoupling is not observed. Potential uses of this work include the assessment of aquifer reclamation time by means of a single pumping well.

Moments Analysis for Improving Decision Reliability Based on Travel Time

Decision reliability is becoming an important issue for both researchers and transportation service providers; different models and methods have been proposed to improve decision reliability. A method is proposed for improving decision reliability by analyzing the moments and central moments of historical travel time data. Because of the existence of traffic congestion—especially nonrecurrent traffic congestion—travel time is generally a random variable with an asymmetric historical distribution. Statistical methods are applied to obtain the higher-order moments and central moments of historical travel time, which provide quantitative information about the variability, asymmetry, and leptokurtosis of travel time. This information is useful in making reliable decisions and avoiding scenarios that can potentially lead to extremely high costs. The method is based on advanced traveler information systems, which provide useful historical travel time information for traffic networks. The new method and other me...

Moments Analysis for Improving Decision Reliability Based on Travel Time

Decision reliability is becoming an important issue for both researchers and transportation service providers; different models and methods have been proposed to improve decision reliability. A method is proposed for improving decision reliability by analyzing the moments and central moments of historical travel time data. Because of the existence of traffic congestion—especially nonrecurrent traffic congestion—travel time is generally a random variable with an asymmetric historical distribution. Statistical methods are applied to obtain the higher-order moments and central moments of historical travel time, which provide quantitative information about the variability, asymmetry, and leptokurtosis of travel time. This information is useful in making reliable decisions and avoiding scenarios that can potentially lead to extremely high costs. The method is based on advanced traveler information systems, which provide useful historical travel time information for traffic networks. The new method and other methods are compared with two testing experiments.

Travel time and trajectory moments of conservative solutes in two-dimensional convergent flows

We address advective transport of a solute traveling toward a single pumping well in a two-dimensional randomly heterogeneous aquifer. The two random variables of interest are the trajectory followed by an individual particle from the injection point to the well location and the particle travel time under steady-state conditions. Our main objective is to derive the predictors of trajectory and travel time and the associated uncertainty, in terms of their first two statistical moments (mean and variance). We consider a solute that undergoes mass transfer between a mobile and an immobile zone. Based on Lawrence et al. [Lawrence, A.E., Sánchez-Vila, X., Rubin, Y., 2002. Conditional moments of the breakthrough curves of kinetically sorbing solute in heterogeneous porous media using multirate mass transfer models for sorption and desorption. Water Resour. Res. 38 (11), 1248, doi:10.1029/2001WR001006.], travel time moments can be written in terms of those of a conservative solute times a deterministic quantity. Moreover, the moments of solute particles trajectory do not depend on mass transfer processes. The resulting mean and variance of travel time and trajectory for a conservative species can be written as functions of the first, second moments and cross-moments of trajectory and velocity components. The equations are developed from a consistent second order expansion in σ Y (standard deviation of the natural logarithm of hydraulic conductivity). Our solution can be completely integrated with the moment equations of groundwater flow of Guadagnini and Neuman [Guadagnini, A., Neuman, S.P., 1999a. Nonlocal and localized analyses of conditional mean steady state flow in bounded, randomly non uniform domains 1. Theory and computational approach. Water Resour. Res. 35(10), 2999–3018.,Guadagnini, A., Neuman, S.P., 1999b. Nonlocal and localized analyses of conditional mean steady state flow in bounded, randomly non uniform domains 2. Computational examples. Water Resour. Res. 35(10), 3019–3039.], it is free of distributional assumptions regarding the log conductivity field, and formally includes conditioning. We present analytical expressions for the unconditional case by making use of the results of Riva et al. [Riva, M., Guadagnini, A., Neuman, S.P., Franzetti, S., 2001. Radial flow in a bounded randomly heterogeneous aquifer. Transport in Porous Media 45, 139–193.]. The quality of the solution is supported by numerical Monte Carlo simulations. Potential uses of this work include the determination of aquifer reclamation time by means of a single pumping well, and the demarcation of the region potentially affected by the presence of a contaminant in the proximity of a well, whenever the aquifer is very thin and Dupuit–Forchheimer assumption holds.

Travel time and trajectory moments of conservative solutes in three dimensional heterogeneous porous media under mean uniform flow

We present expressions satisfied by the first statistical moments (mean and variance–covariance) of travel time and trajectory of conservative solute particles advected in a three-dimensional heterogeneous aquifer under uniform in the mean flow conditions. Closure of the model is obtained by means of a consistent second-order expansion in σ Y (standard deviation of the log hydraulic conductivity) of (statistical) moments of quantities of interest. As such, the results obtained are nominally limited to mildly non-uniform fields, with σ Y < 1. Resulting mean and variance of particles travel time and trajectory are functions of first and second moments and cross-moments of trajectory and velocity components. Our solution is applicable to infinite domains and is free of distributional assumptions. As an important application of the methodology we obtain closed-form expressions for the unconditional mean and variance of travel time and particle trajectory for isotropic log-conductivity domain characterized by an exponential variogram. This allows us to recover the non linear behavior of mean travel time versus distance, in agreement with numerical results published in the literature, as well as a non-linear effect in the mean trajectory. The analysis of trajectory variance allows recovering some known results regarding transverse macro-dispersion, evidencing some limitations typical of perturbation theory.

MODELING UNCERTAINTY IN MIXED TRAFFIC NETWORK

As the actual condition of road network exhibits different combination of traffic modes in the light of normal day-to-day variation, the network travel time is subject to uncertainty and variability. The model proposed in this paper suggests an approach to investigate the effect of variation of mixed traffic on travel time and its uncertainty by using an analytic method. Based on this method, the first and second order moments of link travel time can be evaluated while we capture the fluctuation of traffic flow by each mode. An illustrative numerical example is reported on a simple network from which some useful observations are presented providing us an insight into how fluctuation of demand, vehicle occupancy, link capacity improvement, and vehicle characteristic affect network performance under a prevailing condition.

On the use of peak concentration arrival times for the inference of hydrogeological parameters

Aquifer characterization is subject to large uncertainty due to spatial variability of the hydrologic parameters on the one hand, coupled with data scarcity and measurement error on the other. Traditional characterization methods rely on direct measurements of the hydrologic parameters, which usually are quite few and with large distances in between. In recent years, attempts have been made to evaluate low‐cost, minimally invasive alternative sources of information that can be used to augment the direct measurements. This paper attempts to augment the data available for characterization by looking at tracer experiments. Specifically, we analyze the suitability of peak concentration arrival times to provide information useful for inference of geostatistical models of the conductivity. We found that the peak concentration arrival time is a good proxy for the advection‐dominated travel time and that it is particularly useful for inference through comparison with advection‐dominated, Lagrangian travel time moments when these models are expressed in terms of the geostatistical models of the conductivity. What makes the peak arrival time suitable for this is its limited sensitivity to pore‐scale dispersion at early travel times and its resistance to errors induced by infrequent sampling especially at early and late arrival times. This paper analyzes this concept through analysis of a synthetic case study as well as an analysis of field data collected during a forced‐gradient tracer test.

Uncertainty for solute movement in bounded heterogeneous porous media

The main focus of the present study is to assess the impact of finite flow domain size on solute travel time statistics for a nonreactive conservative solute in a bounded three-dimensional statistically isotropic porous medium. This is accomplished by introducing a modified covariance function of log hydraulic conductivity, accounting for a finite sample size, into the evaluation of solute travel time statistics within the framework of stochastic theory and the spectral perturbation techniques. This study finds that the first two moments of solute travel time for advection-dominated solute movement increase with the size of flow domain.

A fluid queueing model for link travel time moments

AbstractWe analyze the moments of the random time required for a vehicle to traverse a transportation network link of arbitrary length when its speed is governed by a random environment. The problem is motivated by stochastic transportation network applications in which the estimation of travel time moments is of great importance. We analyze this random time in a transient and asymptotic sense by employing results from the field of fluid queues. The results are demonstrated on two example problems. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2004

A three-dimensional numerical method of moments for groundwater flow and solute transport in a nonstationary conductivity field

A three-dimensional numerical method of moments has been developed for solute flux through nonstationary flows in porous media. The solute flux is described as a space–time process where time refers to the solute flux breakthrough and space refers to the transverse displacement distribution at a control plane. Flow nonstationarity may stem from various sources, such as the medium’s conductivity nonstationarity and complex hydraulic boundary conditions. The first two statistics of solute flux are derived using a Lagrangian framework and are expressed in terms of the probability density functions (PDFs). These PDFs are given in terms of one- and two-parcel moments of travel time and transverse locations, and these moments are related to the Eulerian velocity moments. The moment equations obtained analytically for flow and transport are so complex that numerical techniques are used to obtain solutions. In this study, we investigate the influence of various factors, such as the grid resolution relative to correlation length and the number of solute parcels comprising a source, on the accuracy of the calculation results. It has been found that for the computation of means and variances using the developed moment equations, hydraulic head requires at least one numerical grid element per correlation length scale. At least two grid elements are required for velocity, and 1–2 grid elements for the solute flux variance. Five parcels are required per correlation length scale to approximate the initial solute source distribution. The effects of boundary and hydraulic conductivity nonstationarity on flow and transport are also considered. Flow nonstationarity caused by either hydraulic boundary condition or conductivity nonstationarity significantly influences the transport process. The calculation results of numerical method of moments are compared with Monte Carlo simulations. The comparison indicates that the two methods are consistent with each other for head variance, velocity covariance in longitudinal direction, and mean and variance of total solute flux, but numerical method of moment underestimates the velocity variance in transverse direction. The method is applied to an environmental project for predicting the solute flux in the saturated zone below the Yucca Mountain project area, demonstrating the applicability of the method to complex subsurface environments.

Travel time moments for sorbing solutes in heterogeneous domains under nonuniform flow conditions

A methodology for evaluating the unconditional and conditional moments of travel time for a sorbing solute is presented. The approach is applicable for any flow configuration and for a wide range of mass transfer rate‐limited linear processes. The methodology is applicable to the general case of spatially variable hydrological and chemical parameters. The sorption model used to derive the temporal moments is that of a continuous distribution of mass rate coefficients [Haggerty and Gorelick, 1998]. Models such as instantaneous equilibrium, first‐order and two‐site sorption kinetics, among others, can be considered as particular cases of this general model. Using a deterministic approach, the low‐order moments of the breakthrough curves for reactive solutes can be obtained as a function of those for conservative tracers. Using a stochastic approach, the unconditional low‐order statistics of the travel time moments can be obtained. These moments depend on the statistics of two Lagrangian functions, the travel time for a conservative solute, and an integral of the variations of the chemical parameters weighted by the inverse local velocity along the trajectory. Finally, conditional temporal moments are derived. Moments can be conditioned to any type of information, hard or soft, hydraulic or geochemical. Conditioning is found to reduce uncertainty, characterized by a reduction in the variance of the travel time. The general results are particularized for both uniform in the mean and convergent flow conditions and for simple sorption models such as linear instantaneous equilibrium and first‐order kinetics. In all such cases, close‐form results, based on small perturbations expansions, are presented for the travel time moments.

Mean travel time of conservative solutes in randomly heterogeneous unbounded domains under mean uniform flow

We derive a closed‐form expression for mean travel time of a conservative solute migrating under uniform in the mean flow conditions within an infinite stationary field with simple exponential correlation of the natural logarithm of hydraulic conductivity. Our expression is developed from a consistent second‐order expansion in σY (standard deviation of the log hydraulic conductivity) of the equations for moments of travel time and trajectories of conservative solutes in two‐dimensional randomly nonuniform flows of Guadagnini et al. [2001]. As such, it is nominally valid for moderately heterogeneous fields, with σY2 &lt; 1. Its validity for larger heterogeneity degrees is tested against numerical Monte Carlo simulations. Our results clarify the nonlinear effect in the mean travel time with respect to distance that has been observed numerically (and modeled empirically) in the literature.

On the applicability of analytical methods for estimating solute travel time statistics in nonuniform groundwater flow

Groundwater recharge from the unsaturated zone results in a nonuniform mean groundwater flow. We report closed‐form expressions for the mean and variance of advective solute travel time under such nonuniform flow conditions. These expressions are derived directly from velocity statistics, based on an analytical direct evaluation method (DEM). An alternative, indirect evaluation method (IEM) has previously been used for deriving and evaluating the cumulative distribution function of solute travel time in nonuniform mean flow, through its relation to an assumed Gaussian probability density function of solute displacement. For negative groundwater recharge we argue that neither of the two analytical methods DEM and IEM can be considered reliable. For positive groundwater recharge we show that (1) the DEM results reported here remain as reliable for nonuniform as for uniform mean groundwater flow; (2) an assumed lognormal cumulative distribution function based on the DEM‐derived travel time moments is consistent with the IEM‐derived solute travel time cumulative distribution function (cdf), which is in turn consistent with numerical simulation results reported in the literature; and (3) the detailed differences between the DEM‐based lognormal cdf and the IEM‐derived cdf for nonuniform flow are such that the former (DEM) may exhibit even better agreement with reported numerical simulation results than the latter (IEM).

Stochastic Formulation for Uncertainty Analysis of Two-Phase Flow in Heterogeneous Reservoirs

Summary In this article we use a direct approach to quantify the uncertainty in flow performance predictions due to uncertainty in the reservoir description. We solve moment equations derived from a stochastic mathematical statement of immiscible nonlinear two-phase flow in heterogeneous reservoirs. Our stochastic approach is different from the Monte Carlo approach. In the Monte Carlo approach, the prediction uncertainty is obtained through a statistical postprocessing of flow simulations, one for each of a large number of equiprobable realizations of the reservoir description. We treat permeability as a random space function. In turn, saturation and flow velocity are random fields. We operate in a Lagrangian framework to deal with the transport problem. That is, we transform to a coordinate system attached to streamlines (time, travel time, and transverse displacements). We retain the normal Eulerian (space and time) framework for the total velocity, which we take to be dominated by the heterogeneity of the reservoir. We derive and solve expressions for the first (mean) and second (variance) moments of the quantities of interest. We demonstrate the applicability of our approach to complex flow geometry. Closed outer boundaries and converging/diverging flows due to the presence of sources/sinks require special mathematical and numerical treatments. General expressions for the moments of total velocity, travel time, transverse displacement, water saturation, production rate, and cumulative recovery are presented and analyzed. A detailed comparison of the moment solution approach with high-resolution Monte Carlo simulations for a variety of two-dimensional problems is presented. We also discuss the advantages and limits of the applicability of the moment equation approach relative to the Monte Carlo approach.

A Full‐Bayesian Approach to parameter inference from tracer travel time moments and investigation of scale effects at the Cape Cod Experimental Site

A method for inverting the travel time moments of solutes in heterogeneous aquifers is presented and is based on peak concentration arrival times as measured at various samplers in an aquifer. The approach combines a Lagrangian [Rubin and Dagan, 1992] solute transport framework with full‐Bayesian hydrogeological parameter inference. In the full‐Bayesian approach the noise values in the observed data are treated as hyperparameters, and their effects are removed by marginalization. The prior probability density functions (pdfs) for the model parameters (horizontal integral scale, velocity, and log K variance) and noise values are represented by prior pdfs developed from minimum relative entropy considerations. Analysis of the Cape Cod (Massachusetts) field experiment is presented. Inverse results for the hydraulic parameters indicate an expected value for the velocity, variance of log hydraulic conductivity, and horizontal integral scale of 0.42 m/d, 0.26, and 3.0 m, respectively. While these results are consistent with various direct‐field determinations, the importance of the findings is in the reduction of confidence range about the various expected values. On selected control planes we compare observed travel time frequency histograms with the theoretical pdf, conditioned on the observed travel time moments. We observe a positive skew in the travel time pdf which tends to decrease as the travel time distance grows. We also test the hypothesis that there is no scale dependence of the integral scale λ with the scale of the experiment at Cape Cod. We adopt two strategies. The first strategy is to use subsets of the full data set and then to see if the resulting parameter fits are different as we use different data from control planes at expanding distances from the source. The second approach is from the viewpoint of entropy concentration. No increase in integral scale with distance is inferred from either approach over the range of the Cape Cod tracer experiment.

Travel time analysis of tracers and reactive solutes in the unsaterated zone

This paper presents a Lagrangian stochastic approach for modeling field scale contaminant transport in the heterogeneous unsaturated geological environments. We derive the probability distribution function of the travel times of tracers and reactive solutes, without limiting it to be either normal or log-normal, and show how it can be used for computing the moments of solute fluxes and breakthrough curves. We also provide closed-form expressions for the travel time moments of tracers and reactive solutes characterized by linear equilibrium and non-equilibrium sorption kinetics. These expressions are useful for prediction and for interpretation of field experiments. In our derivation we account for soil heterogeneity by modeling the soil parameters, such as the saturated hydraulic conductivity and the pore size distribution parameter, as weakly stationary random space functions. Unlike previous studies, we account for the effect of the spatial variability of the water content and refrain from assuming that the saturation is practically constant in the unit gradient flow zone. The derivation is done using a first-order perturbative expansion of the flow equation, and is thus limited to small variability of the input parameters. The effects of spatial variability on the displacement and travel time statistics are discussed and demonstrated.