Travel Time Probability Density Research Articles

Adjoint subordination to calculate backward travel time probability of pollutants in water with various velocity resolutions

Abstract. Backward probabilities, such as the backward travel time probability density function for pollutants in natural aquifers/rivers, have been used by hydrologists for decades in water quality applications. Calculating these backward probabilities, however, is challenging due to non-Fickian pollutant transport dynamics and velocity resolution variability at study sites. To address these issues, we built an adjoint model by deriving a backward-in-time fractional-derivative transport equation subordinated to regional flow, developed a Lagrangian solver, and applied the model/solver to trace pollutant transport in diverse flow systems. The adjoint model subordinates to a reversed regional flow field, transforms forward-in-time boundaries into either absorbing or reflective boundaries, and reverses the tempered stable density to define backward mechanical dispersion. The corresponding Lagrangian solver efficiently projects backward super-diffusive mechanical dispersion along streamlines. Field applications demonstrate the adjoint subordination model's success with respect to recovering release history, groundwater age, and pollutant source locations for various flow systems. These include systems with upscaled constant velocity, nonuniform divergent flow fields, or fine-resolution velocities in a nonstationary, regional-scale aquifer, where non-Fickian transport significantly affects pollutant dynamics and backward probabilities. Caution is needed when identifying the phase-sensitive (aqueous vs. absorbed) pollutant source in natural media. The study also explores possible extensions of the adjoint subordination model for quantifying backward probabilities of pollutants in more complex media, such as discrete fracture networks.

Link dynamic vehicle count estimation based on travel time distribution using license plate recognition data

This paper proposes a dynamic vehicle count estimation method for signalized links using license plate recognition (LPR) data considering the recognition errors. The framework contains three sub-components. First, travel time probability density function is estimated based on the kernel density model using the LPR data of matched vehicles; Second, the expected travel time is estimated for unmatched vehicles based on the travel time probability distribution; Third, with the complete arrival and departure information, the cumulative arrival-departure curve of the link is reconstructed, and the link dynamic vehicle count (LDVC) can be calculated correspondingly. To explore how different recognition errors affect LDVC estimation performance, the proposed model is validated through an empirical case of a real LPR dataset and a simulation case with different levels of matched rate. The results show that the model has a high level of estimation accuracy in different levels of matched rate of LPR data.

A new transfer function model for the estimation of non-point-source solute travel times

The scope of this work is to present a new fast and reliable transfer function model, which simulates the spatio-temporal distribution of non-point-source solutes along the unsaturated zone, suitable to be used at large scales within a web-based Decision Support System. With the assumptions of a) a gravity induced water flow, b) a non-reactive solute and c) a purely convective flow, the model uses the transfer functions, i.e., the travel time (TT) probability density functions, derived from the unsaturated hydraulic conductivity curve k(θ). The output concentration of a solute is simply the convolution of the transfer functions with the input concentrations to the system. A model sensitivity analysis, based on Monte Carlo simulations, was carried out, showing that saturated water content and the tortuosity parameter τ were the parameters that affected the mean TT more. The model was validated against concentration experiments carried out on four large soil columns. Results were really good for all soils, with the best agreement with R2 = 0.97, RMSE = 0.11 and ME = −0.01. Moreover, the outputs obtained applying the model to 46 soil profiles sampled in the Valle Telesina, in Southern Italy, completely characterised from the hydrological point of view, were compared with those obtained from the Richard-based model Hydrus 1D. The result of the comparisons gave a very high correlation coefficient (above 0.8), a mean absolute error between the two models of around 40 days and a percent bias of −16%. Finally, the application of transfer function model to a large spatial extent is presented, to show its possible use for the groundwater vulnerability assessment.

Assessment of Groundwater Susceptibility to Non-Point Source Contaminants Using Three-Dimensional Transient Indexes.

Groundwater susceptibility to non-point source contamination is typically quantified by stable indexes, while groundwater quality evolution (or deterioration globally) can be a long-term process that may last for decades and exhibit strong temporal variations. This study proposes a three-dimensional (3-d), transient index map built upon physical models to characterize the complete temporal evolution of deep aquifer susceptibility. For illustration purposes, the previous travel time probability density (BTTPD) approach is extended to assess the 3-d deep groundwater susceptibility to non-point source contamination within a sequence stratigraphic framework observed in the Kings River fluvial fan (KRFF) aquifer. The BTTPD, which represents complete age distributions underlying a single groundwater sample in a regional-scale aquifer, is used as a quantitative, transient measure of aquifer susceptibility. The resultant 3-d imaging of susceptibility using the simulated BTTPDs in KRFF reveals the strong influence of regional-scale heterogeneity on susceptibility. The regional-scale incised-valley fill deposits increase the susceptibility of aquifers by enhancing rapid downward solute movement and displaying relatively narrow and young age distributions. In contrast, the regional-scale sequence-boundary paleosols within the open-fan deposits “protect” deep aquifers by slowing downward solute movement and displaying a relatively broad and old age distribution. Further comparison of the simulated susceptibility index maps to known contaminant distributions shows that these maps are generally consistent with the high concentration and quick evolution of 1,2-dibromo-3-chloropropane (DBCP) in groundwater around the incised-valley fill since the 1970s’. This application demonstrates that the BTTPDs can be used as quantitative and transient measures of deep aquifer susceptibility to non-point source contamination.

Data-driven distributionally robust optimization approach for reliable travel-time-information-gain-oriented traffic sensor location model

Travel time is one of the most intuitive pieces of traffic information to help decision makers to control real-time traffic conditions and to guide travelers to choose a reasonable route. An optimal sensor location scheme can obtain reliable route travel time information. Most current travel-time-oriented sensor location models are deterministic and assume a given and correct travel time probability density function. Nevertheless, due to widespread observational and systematic errors, prior travel time information is not accurate or reliable. In our study, a novel data-driven link-based network sensor location method is proposed to maximize travel time information gain. The effect of route differentiation is considered, and the sensors are located at links rather than at nodes. In addition, to account for the uncertainty in the prior travel time distribution, the distributionally robust travel time information gain sensor location (DRTTIGSL) model is presented. The prior distribution information is taken into account based on a statistical measure called ϕ-divergence. The ϕ-divergence is used to construct the uncertainty set. The reformulation of DRTTIGSL is dependent on the choice of ϕ-divergence and is tractable. Extensive numerical experiments are conducted to verify the effectiveness of the DRTTIGSL model. Compared with the optimal solutions for the deterministic model, the optimal solutions for the DRTTIGSL model can reduce the worst-case situation with a small price of the average objective value, especially when the total budget is not large.

Synthesizing Route Travel Time Distributions from Segment Travel Time Distributions

This paper examines a way to synthesize route travel time probability density functions (PDFs) on the basis of segment-level PDFs. Real-world data from I-5 in Sacramento, California, are employed. The first finding is that careful filtering is required to extract useful travel times from the raw data because trip times, not travel times, are observed (i.e., the movement of vehicles between locations). The second finding is that significant correlations exist between individual vehicle travel times for adjacent segments. Two analyses are done in this regard: one predicts downstream travel times on the basis of upstream travel times, and the second checks for correlations in travel times between upstream and downstream segments. The results of these analyses suggest that strong positive correlations exist. The third finding is that comonotonicity, or perfect positive dependence, can be assumed when route travel time PDFs are generated from segment PDFs. Kolmogorov–Smirnov tests show that travel times synthesized from the segment-specific data are statistically different only under highly congested conditions, and even then, the percentage differences in the distributions of the synthesized and actual travel times are small. The fourth finding, somewhat tangential, is that there is little variation in individual driver travel times under given operating conditions. This is an important finding, because such an assumption serves as the basis for all traffic simulation models.

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.

Flow and travel time statistics in highly heterogeneous porous media

In this paper we present flow and travel time ensemble statistics based on a new simulation methodology, the adaptive Fup Monte Carlo method (AFMCM). As a benchmark case, we considered two‐dimensional steady flow in a rectangular domain characterized by multi‐Gaussian heterogeneity structure with an isotropic exponential correlation and lnK variance σY2 up to 8. Advective transport is investigated using the travel time framework where Lagrangian variables (e.g., velocity, transverse displacement, or travel time) depend on space rather than on time. We find that Eulerian and Lagrangian velocity distributions diverge for increasing lnK variance due to enhanced channeling. Transverse displacement is a nonnormal for all σY2 and control planes close to the injection area, but after xIY = 20 was found to be nearly normal even for high σY2. Travel time distribution deviates from the Fickian model for large lnK variance and exhibits increasing skewness and a power law tail for large lnK variance, the slope of which decreases for increasing distance from the source; no anomalous features are found. Second moment of advective transport is analyzed with respect to the covariance of two Lagrangian velocity variables: slowness and slope which are directly related to the travel time and transverse displacement variance, which are subsequently related to the longitudinal and transverse dispersion. We provide simple estimators for the Eulerian velocity variance, travel time variance, slowness, and longitudinal dispersivity as a practical contribution of this analysis. Both two‐parameter models considered (the advection‐dispersion equation and the lognormal model) provide relatively poor representations of the initial part of the travel time probability density function in highly heterogeneous porous media. We identify the need for further theoretical and experimental scrutiny of early arrival times, and the need for computing higher‐order moments for a more accurate characterization of the travel time probability density function. A brief discussion is presented on the challenges and extensions for which AFMCM is suggested as a suitable approach.

Stochastic analysis of transport in a combined heterogeneous vadose zone–groundwater flow system

First‐order analysis was used to investigate the effect of a few characteristics of the vadose zone and the solute source at the soil surface on the expected breakthrough curves (BTCs) of tracer solutes in a combined unsaturated‐saturated spatially heterogeneous formation with a three‐dimensional, statistically anisotropic structure. The analysis is based on a stochastic continuum representation of the stationary and nonstationary Eulerian velocity vectors in the unsaturated and saturated domains, respectively, and a general Lagrangian description of the solute mass flux in steady state flow. Results of the present analysis suggest that the length scales and the flow characteristics of the spatially heterogeneous, unsaturated zone that affect the travel time probability density function between the soil surface and the water table also affect the expected solute BTC at a control plane located downstream in the groundwater. The flow in the unsaturated zone might enhance the early arrival time and the spreading of the expected solute BTC in the groundwater, especially when the vertical extent of the unsaturated zone is relatively large; when the unsaturated zone consists of relatively dry, coarse‐textured, stratified soil material; and when the horizontal extent of the solute source in the direction of the mean groundwater flow is relatively large. The effect of the characteristics of the vadose zone and the solute source at the soil surface on the expected solute BTC in the groundwater decreases as the distance to the control plane increases and as the process of the solute breakthrough is more advanced. The relatively good agreement between the results of the present first‐order, stochastic analysis with the results of detailed, numerical analyses of flow and transport in a three‐dimensional heterogeneous, combined vadose zone–groundwater flow domain suggests that the first‐order stochastic analyses may be sufficiently reliable to show appropriate trends which might occur in realistic combined vadose zone–groundwater flow systems.

Forward and backward temporal probability distributions of sorbing solutes in groundwater

The travel time of a contaminant particle in an aquifer can be used to identify the release time of contamination from a known source, to delineate captures zones around wells, and for other applications. The travel time is uncertain because of spatial variability of hydrogeologic properties and fluctuations of groundwater velocity and therefore is often represented as a random variable. A forward travel time probability density function (pdf) describes the random time that a particle will reach a downgradient location, while the backward travel time pdf describes the time in the past when the particle was at an upgradient location, such as the source location. For sorbing solutes, the particle may have been in either the aqueous phase or the sorbed phase at the location of interest. Since a particle can only travel through the aquifer in the aqueous phase, the random travel time accounts only for particles in the aqueous phase. We develop new probabilistic expressions to characterize the random time in the past that a contaminant particle was sorbed at a location of interest. For a solute that exhibits linear equilibrium sorption, we develop forward and backward pdfs for the time of sorption, defined as the random time at which the particle is at a location of interest in the sorbed phase. For the linear nonequilibrium sorption case, we develop expressions for the forward and backward sorbed phase probability, defined as the discrete probability that a particle will be sorbed at a location of interest at a deterministic time. Although forward pdfs and probabilities produce the same information as backward pdfs and probabilities, backward probabilistic expressions can be calculated more efficiently when the number of observations is small relative to the number of possible source locations or release times.

Travel time distribution in a hillslope: Insight from numerical simulations

Solute transport in a three‐dimensional, heterogeneous hillslope is analyzed through a series of detailed, three‐dimensional numerical simulations. The investigation focuses on the transport of a pulse of a passive solute, taking into account realistic features of the relevant flow domain (i.e., the spatial heterogeneity of the soil hydraulic properties and records of the time‐dependent meteorological data and evapotranspiration). The scope of the present work is to analyze a few issues regarding the travel time probability density function (pdf) f(τ) of solute, with particular reference to the following: (1) the suitability of the time invariance assumption and the circumstance under which it may represent a valid approximation, (2) the shape of the resulting travel time pdf f(τ), and (3) the difference between f(τ) and the instantaneous unit hydrograph. It is found that in many cases of practical interest the transport of contaminants in catchments can be analyzed by assuming the time‐invariant approximation for f(τ), provided that the calendar time is replaced by a flow‐corrected time. Here f(τ) is calculated through the analysis of the breakthrough curve represented in terms of flux‐averaged concentration versus cumulate discharge. The rescaling of time with respect to the cumulated outflow takes care in an approximate way of the transient processes occurring in the porous medium. The derived f(τ) is weakly dependent on various attributes, like the level of heterogeneity, presence of evaporation or transpiration, and injection period. The main exception regards the cases in which plant transpiration is intense in the vicinity of the channel after relatively long periods of low rain. The impact of the parameters' heterogeneity on f(τ) is generally quite limited, the dispersion being ruled by the distribution of length paths within the hillslope. The derived f(τ) seems compatible with the Gamma distribution, being characterized by both fast and slow responses, with a pronounced power law early peak and an exponential‐like tail. Comparison of f(τ) with the often employed Gamma instantaneous unit hydrograph emphasizes the differences between water and solute dynamics after rainfall events.

Identifying sources of a conservative groundwater contaminant using backward probabilities conditioned on measured concentrations

If contamination is observed in an aquifer, a backward probability model can be used to obtain information about the former position, source location, or release time of the observed contamination. A backward location probability density function (PDF) describes the possible former positions of an observed contaminant particle at a specified time in the past and can be used to identify the location of the contaminant source. A backward travel time probability density function describes the possible travel times of an observed contaminant particle from a known upgradient location to the observation location and can be used to determine the time of release of contamination from the source. Neupauer and Wilson (1999, 2001, 2002) obtained backward PDFs that depend on the observation locations and sampling times, in addition to the transport properties of the aquifer and solute. We present an approach for conditioning these backward PDFs on measured concentrations. We focus on identifying the location or release time of an instantaneous point source of contamination. We derive the conditioning equations and demonstrate some important features of the technique through the use of a hypothetical example.

Solute Transport Measurement Under Transient Field Conditions Using Time Domain Reflectometry

Measurement of transport properties of field soil remains a challenge. Time domain reflectometry (TDR) has been used for rapid and nondestructive measurement of the movement of conservative tracers in controlled laboratory and field experiments. Measuring transport under transient conditions in the laboratory using TDR has also been reported, but obtaining appropriate calibration relationships remains a challenge. We present a method for rapid and nondestructive measurement of field transport of an electrolytic tracer (applied to the soil surface) under transient rainfall and evapotranspiration conditions with net drainage using TDR without any a priori calibrations. The method uses TDR probes in plots with and without a tracer applied. The simultaneous TDR measurements of apparent impedance and dielectric constant in the paired plots were used to calculate the relative solute mass remaining to a given depth (i.e., TDR probe depth) as a function of time during a 270‐d field experiment under natural rainfall and evapotranspiration conditions of net drainage. The measurements give relative solute mass flux that is equivalent to the solute travel time probability density function.

Backward probability model using multiple observations of contamination to identify groundwater contamination sources at the Massachusetts Military Reservation

Backward location and travel time probability density functions characterize the possible former locations (or the source location) of contamination that is observed in an aquifer. For an observed contaminant particle the backward location probability density function (PDF) describes its position at a fixed time prior to sampling, and the backward travel time probability density function describes the amount of time required for the particle to travel to the sampling location from a fixed upgradient position. The backward probability model has been developed for a single observation of contamination (e.g., Neupauer and Wilson, 1999). In practical situations, contamination is sampled at multiple locations and times, and these additional data provide information that can be used to better characterize the former position of contamination. Through Bayes' theorem we combine the individual PDFs for each observation to obtain a PDF for multiple observations that describes the possible source locations or release times of all observed contaminant particles, assuming they originated from the same instantaneous point source. We show that the multiple‐observation probability density function is the normalized product of the single‐observation PDFs. The additional information available from multiple observations reduces the variances of the source location and travel time probability density functions and improves the characterization of the contamination source. We apply the backward probability model to a trichloroethylene (TCE) plume at the Massachusetts Military Reservation (MMR). We use four TCE samples distributed throughout the plume to obtain single‐observation and multiple‐observation location and travel time PDFs in three dimensions. These PDFs provide information about the possible sources of contamination. Under assumptions that the existing MMR model is properly calibrated and the conceptual model is correct the results confirm the two suspected sources of contamination and reveal that one or more additional sources is likely.

A geomorphological explanation of the unit hydrograph concept

AbstractThe water path from any point of a basin to the outlet through the self‐similar river network was considered. This hydraulic path was split into components within the Strahler ordering scheme. For the entire basin, we assumed the probability density functions of the lengths of these components, reduced by the scaling factor, to be independent and isotropic. As with these assumptions, we propose a statistical physics reasoning (similar to Maxwell's reasoning) that considers a hydraulic length symbolic space, built on the self‐similar lengths of the components. Theoretical expressions of the probability density functions of the hydraulic length and of the lengths of all the components were derived. These expressions are gamma laws expressed in terms of simple geomorphological parameters. We validated our theory with experimental observations from two French basins, which are different in terms of size and relief. From the comparisons, we discuss the relevance of the assumptions and show how a gamma law structure underlies the river network organization, but under the influence of a strong hierarchy constraint. These geomorphological results have been translated into travel time probability density functions, through the hydraulic linear hypothesis. This translation provides deterministic explanations of some famous a priori assumptions of the unit hydrograph and the geomorphological unit hydrograph theories, such as the gamma law general shape and the exponential distribution of residence time in Strahler states. Copyright © 2004 John Wiley & Sons, Ltd.

Development of type transfer functions for regional‐scale nonpoint source groundwater vulnerability assessments

Groundwater vulnerability assessments of nonpoint source agrochemical contamination at regional scales are either qualitative in nature or require prohibitively costly computational efforts. By contrast, the type transfer function (TTF) modeling approach for vadose zone pesticide leaching presented here estimates solute concentrations at a depth of interest, only uses available soil survey, climatic, and irrigation information, and requires minimal computational cost for application. TTFs are soil texture based travel time probability density functions that describe a characteristic leaching behavior for soil profiles with similar soil hydraulic properties. Seven sets of TTFs, representing different levels of upscaling, were developed for six loam soil textural classes with the aid of simulated breakthrough curves from synthetic data sets. For each TTF set, TTFs were determined from a group or subgroup of breakthrough curves for each soil texture by identifying the effective parameters of the function that described the average leaching behavior of the group. The grouping of the breakthrough curves was based on the TTF index, a measure of the magnitude of the peak concentration, the peak arrival time, and the concentration spread. Comparison to process‐based simulations show that the TTFs perform well with respect to mass balance, concentration magnitude, and the timing of concentration peaks. Sets of TTFs based on individual soil textures perform better for all the evaluation criteria than sets that span all textures. As prediction accuracy and computational cost increase with the number of TTFs in a set, the selection of a TTF set is determined by a given application.

A note on the effective parameters of the convection‐dispersion equation

We focus on the analysis of the effective parameters of the one‐dimensional convection‐dispersion equation (CDE), estimated from the depth distribution of the solute concentration at a given observation time. This situation should be distinguished from that in which the effective parameters of the one‐dimensional CDE are estimated from the temporal distribution of the solute concentration at a given arbitrary soil depth. In this latter case, both the mean and the variance of the solute travel time probability density function increase linearly with travel distance, and the resultant effective solute velocity, Ve, and dispersion coefficient, De, are constant, independent of travel distance. Results of the present analysis suggest that under both first‐type and third‐type upper boundary conditions, the resultant Veand De are time dependent. Consequently, the effective dispersivity, λe = De/Ve, increases with travel time or distance as the solute body first invades the flow system and tends to approach a constant value (i.e., Fickian behavior) after the solute body has traveled a distance of few tens of pore‐scale dispersivity. These findings, which stem from the Lagrangian nature of the dispersion phenomenon, are consistent with the results of first‐order analyses in saturated and partially saturated heterogeneous formations, based on the theory of dispersion by continuous motions. The findings presented in this technical note have important implications regarding the estimation of parameters of the one‐dimensional CDE from the depth distribution of the solute concentration at a given observation time, and the subsequent prediction of solute movement.

Stochastic analysis of transport in unsaturated heterogeneous soils uder transient flow regimes

Approximate analytic solutions are developed to predict the transport of inert solutes in variably saturated, randomly heterogeneous porous media subject to a random boundary flux condition. The mean and variance of solute mass flux past a series of vadose zone control planes are determined using a Lagrangian approach. The method requires the specification of an advective travel time probability density function (pdf) between the soil surface and the control planes and an impulse response function which quantifies the specific flow and transport processes that affect the solute along an individual streamline in the soil. In this study an advective‐dispersive impulse response function for inert solute is assumed, and an advective travel time pdf is derived from the statistics of the boundary flux and soil properties using a one‐dimensional transient vadose zone flow model [Foussereau et al., this issue]. Monte Carlo simulations are conducted using random boundary flux sequences and spatially correlated random hydraulic conductivity fields in a multidimensional variably saturated flow and transport code to test the analytic model assumptions. The analytic model accurately predicts the ensemble mean solute flux breakthrough curves for a variety of soil‐climate combinations. The general behavior of the coefficient of variation of solute flux is also accurately reproduced; however, the minimum coefficient of variation at the mean center of mass was slightly underpredicted in most cases. Results show that for shallow soils in humid climates, boundary flux variability dominates soil variability in determining the uncertainty of solute transport predictions in the vadose zone.

A multiscale approach for estimating solute travel time distributions

This paper uses a multiscale statistical framework to estimate groundwater travel times and to derive conditional travel time probability densities. In the applications of interest here travel time uncertainties depend primarily on uncertainties in hydraulic conductivity. These uncertainties can be reduced if the travel times are conditioned on scattered measurements of hydraulic conductivity and/or hydraulic head. In our approach the spatially discretized log hydraulic conductivity is modeled as a multiscale stochastic process, where each scale describes the process at a different spatial resolution. Related dependent variables such as hydraulic head and travel time are approximated by discrete linear functions of the log conductivity. The linearization makes it possible to incorporate these variables into efficient multiscale estimation and conditional simulation algorithms. We illustrate the application of these algorithms by considering two options for estimating travel time densities: (1) a Monte Carlo technique which only requires linearization of the groundwater flow equation and (2) a Gaussian approximation which also requires linearization of Darcy's law and an implicit particle tracking equation. Both options provide reasonable estimates of the travel time probability density in a synthetic experiment if the underlying log hydraulic conductivity variance is small (0.5). When this variance is increased (to 5.0), the Monte Carlo result is still quite good but the Gaussian approximation is unsatisfactory. The multiscale Monte Carlo option is a very competitive approach for estimating travel time since it provides accurate results over a wide range of conditions and it is more computationally efficient than competing alternatives.

Stochastic Analysis of Immiscible Two-Phase Flow in Heterogeneous Media

Summary We present a stochastic analysis of immiscible two-phase flow (Buckley–Leverett displacement) in heterogeneous reservoirs. Since the detailed spatial variabilities of reservoir properties such as permeability and porosity cannot be described deterministically, both permeability and porosity, or either, are treated as random space functions. In turn, water saturation and oil production rate are also random variables. The main purpose of this study is to estimate the saturation field (by its expected value) and the associated uncertainty (by its standard deviation) for Buckley-Leverett displacement in such random media. The expected value and standard deviation may be used to construct confidence intervals for the saturation field in a reservoir and to evaluate the risk associated with a project caused by the incomplete knowledge of reservoir properties. Through transforming to coordinates attached to streamlines, the evaluation of saturation statistical moments is simplified to that of travel time and transverse displacement probability density functions. The latter do not vary with the two-phase composition but entirely depend on the total velocity under some conditions. Through some one- and two-dimensional examples, we found that the well-known discontinuities existing in saturation profiles and oil production curves for homogeneous media disappear in the case of heterogeneous media. This is due to heterogeneity fingering, or the so-called "heterogeneity induced dispersion." The effect of heterogeneity on oil production is the earlier water breakthrough and later oil arrival. The oil production is expected to take longer in heterogeneous media than in homogeneous media.