Unconditional Statistics Research Articles

Assessment of implicit LES modelling for bypass transition of a boundary layer

The implicit LES model recently introduced by Dairay et al. (2017), which is based on numerical dissipation introduced via the discretisation of the viscous term, is assessed for the case of a flat plate boundary layer submitted to external turbulence and undergoing bypass transition. The free stream turbulent fluctuations are generated using a recently proposed linear forcing method adapted to control their intensity and length scale. Two configurations are considered, one with the leading edge of the plate taken into account and one without. LES results for the configuration without leading edge are compared with a DNS, and results for the configuration with leading edge are compared with available experimental and numerical reference data from the literature. Global parameters, unconditional and conditional statistics, as well as Reynolds stress budgets are analysed, and good agreements are obtained with reference data, hence confirming the ability of the model to predict accurately transitional flows. In particular, it is shown that, thanks to its action limited to the smallest resolved scales, the model is only active in the transitional and turbulent regions and therefore does not alter the development of the streaks in the laminar region of the boundary layer. • Implicit LES modelling based on the viscous term discretisation is assessed for bypass transition. • Unconditional and conditional statistics, including Reynolds stress budgets, are analysed with respect to DNS and reference data. • Thanks to the restriction of the numerical viscosity to the smallest resolved scales, the model is shown to be active only in transitional and turbulent regions. • The good agreement with reference data achieved for all the analysed statistics confirms the ability of the model to provide reliable data at a moderate computational cost.

Investigation of deep learning methods for efficient high-fidelity simulations in turbulent combustion

Turbulent combustion modeling often faces a trade-off between the so-called flamelet-like models and PDF-like models. Flamelet-like models, are characterized by a choice of a limited set of prescribed moments, which are transported to represent the manifold of the composition space and its statistics. PDF-like approaches are designed to directly evaluate the closure terms associated with the nonlinear chemical source terms in the energy and species equations. They generate data on the fly, which can be used to accelerate the simulation of PDF-like based models. Establishing key ingredients for implementing acceleration schemes for PDF-like methods by constructing flamelet-like models on the fly can potentially result in computational saving while maintaining the ability to resolve closure terms. These ingredients are investigated in this study. They include a data-based dimensional reduction of the composition space to a low-dimensional manifold using principal component analysis (PCA). The principal components (PCs) serve as moments, which characterize the manifold; and conditional means of the thermo-chemical scalars are evaluated in terms of these PCs. A second ingredient involves adapting a novel deep learning framework, DeepONet, to construct joint PCs’ PDFs as alternative methods to presumed shapes common in flamelet-like approaches. We also investigate whether the rotation of the PCs into independent components (ICs) can improve their statistical independence. The combination of these ingredients is investigated using experimental data based on the Sydney turbulent nonpremixed flames with inhomogeneous inlets. The combination of constructed PDFs and conditional mean models are able to adequately reproduce unconditional statistics of thermo-chemical scalars, and establish acceptable statistical independence between the PCs, which simplify further the modeling of the joint PCs’ PDFs.

Item fit statistics for Rasch analysis: can we trust them?

AimTo compare fit statistics for the Rasch model based on estimates of unconditional or conditional response probabilities.BackgroundUsing person estimates to calculate fit statistics can lead to problems because the person estimates are biased. Conditional response probabilities given the total person score could be used instead.MethodsData sets are simulated which fit the Rasch model. Type I error rates are calculated and the distributions of the fit statistics are compared with the assumed normal or chi-square distribution. Parametric bootstrap is used to further study the distributions of the fit statistics.ResultsType I error rates for unconditional chi-square statistics are larger than expected even for moderate sample sizes. The conditional chi-square statistics maintain the significance level. Unconditional outfit and infit statistics have asymmetric distributions with means slighly below 1. Conditional outfit and infit statistics have reduced Type I error rates.ConclusionsConditional residuals should be used. If only unconditional residuals are available parametric bootstrapping is recommended to calculate valid p-values. Bootstrapping is also necessary for conditional outfit statistics. For conditional infit statistics the adjusted rule-of-thumb critical values look useful.

Systematic Covariations and Emerging Asian Equity Markets’ Diversification Benefits to US Equity Investors

Unconditional and conditional correlations have played a central role in portfolio analysis, optimization, and performance measurement. However, recent studies show these two correlation measures are inappropriate for measuring both financial integration and, therefore, diversification benefits. We use an alternative correlation measure that we refer to by factor model-implied correlation estimated from the systematic (predictable) portion of returns of a multi-factor model with several global risk factors. Estimated implied correlations, covariances, variances, and in-sample (predicted) mean returns are used to calculate optimal US and Asian equities’ asset allocation weights in alternative Global equity portfolios varying by Asian equity market combined with US equities, as well as by whether: (i) implied or unconditional statistics are used; and (ii) portfolios are optimized by Sharpe’s ratio-maximization or variance-minimization. Risk-adjusted returns of alternative actively-managed Global equity portfolios are compared to US equities’ risk-adjusted returns. We find Global equity portfolios with asset allocation weights calculated using factor model-implied statistics uniformly yield higher risk-adjusted returns than US equities and Global equity portfolios with asset allocation weights calculated using unconditional portfolio statistics. In actively-managed Global equity portfolios with asset allocation weights calculated using implied statistics, India and Taiwan consistently rank as top contributors, while South Korea, Singapore, and Hong Kong consistently rank as bottom contributors to enhancing US equities’ risk-adjusted returns. While our analyses are dynamic, they use implied portfolio statistics estimated from historical returns’ distributions. Future studies can extend this research using conditional(out-of-sample) ex-ante estimates of systematic returns, covariances, variances, and correlations in examining emerging markets’ contributions to developed markets’ equities.

Numerical investigation of flow and scalar fields of piloted, partially-premixed dimethyl ether/air jet flames using stochastic multiple mapping conditioning approach

Abstract Turbulent piloted partially-premixed jet flames of dimethyl ether/air have been investigated using the Reynolds-Averaged Navier-Stokes (RANS) based stochastic multiple mapping conditioning (MMC) approach. The MMC model combines the advantages of mapping closure concept, conditional moment closure, and probability density function (PDF) methods. In the present work, the evolution of the mapping function in the reference space is described using a single reference variable based on the mixture fraction. To achieve localness in the composition space, the turbulent mixing is conditioned on the reference space. In addition to localness, MMC preserves the various characteristics of a good mixing model, such as linearity, Gaussianity, independence, and boundedness. A skeletal chemical mechanism for dimethyl ether is used, which consists of 28 species and 24 reversible reactions. Modified Curl's mixing model is adapted to model the micro-mixing term, and standard two-equation k − e turbulence model is used to model the Reynolds stress term with a modified set of constants. The computed conditional and unconditional statistics demonstrate an excellent agreement with the available experimental data even for flame displaying a large degree of local extinction and re-ignition. For these flames, radical species distribution conditioned on mixture fraction confirms the physical separation between the OH and CH2O species, which was earlier reported using simultaneous laser-induced fluorescence measurements of these radicals. Predicted unconditional scalar dissipation rate for DME-D and DME-F flames reaffirm that near the inlet an overlap between OH and CH2O radicals are observed where the scalar dissipation rate is fairly high. Subsequently, a distinct separation between these radicals becomes evident at downstream locations where the scalar dissipation rate decreases. For the flames investigated here, a strong correlation is noticed between the peak heat release rate and the reaction rate indicator, ROH based on the product of concentrations of OH and CH2O radicals.

Power of the spacing test for least-angle regression

Recent advances in Post-Selection Inference have shown that conditional testing is relevant and tractable in high-dimensions. In the Gaussian linear model, further works have derived unconditional test statistics such as the Kac–Rice Pivot for general penalized problems. In order to test the global null, a prominent offspring of this breakthrough is the Spacing test that accounts the relative separation between the first two knots of the celebrated least-angle regression (LARS) algorithm. However, no results have been shown regarding the distribution of these test statistics under the alternative. For the first time, this paper addresses this important issue for the Spacing test and shows that it is unconditionally unbiased. Furthermore, we provide the first extension of the Spacing test to the frame of unknown noise variance. More precisely, we investigate the power of the Spacing test for LARS and prove that it is unbiased: its power is always greater or equal to the significance level $\alpha$. In particular, we describe the power of this test under various scenarii: we prove that its rejection region is optimal when the predictors are orthogonal; as the level $\alpha$ goes to zero, we show that the probability of getting a true positive is much greater than $\alpha$; and we give a detailed description of its power in the case of two predictors. Moreover, we numerically investigate a comparison between the Spacing test for LARS, the Pearson’s chi-squared test (goodness of fit) and a numerical testing procedure based on the maximal correlation. When the noise variance is unknown, our analysis unleashes a new test statistic that can be computed in cubic time in the population size and which we refer to as the $t$-Spacing test for LARS. The $t$-Spacing test involves the first two knots of the LARS algorithm and we give its distribution under the null hypothesis. Interestingly, numerical experiments witness that the $t$-Spacing test for LARS enjoys the same aforementioned properties as the Spacing test.

Can foreign aid motivate institutional reform? An evaluation of the HIPC Initiative

PurposeA number of political economy concerns are associated with the provision of foreign aid to developing economies. These concerns suggest that foreign aid is likely to have harmful effects on a recipient’s institutional quality, and that attempts to give aid conditional on policy and institutional reforms are unlikely to succeed. Established in 1996, the Heavily Indebted Poor Country (HIPC) Initiative is a comprehensive, structured attempt to provide multilateral foreign aid conditional on reforms in recipient countries. The purpose of this paper is to evaluate its effectiveness at affecting institutional reform in participating countries.Design/methodology/approachThe authors document how participating countries fared in terms of the quality of their policies and institutions. The authors employ the Fraser Institute’s Economic Freedom of the World index as a measure of economic institutions, and the Freedom House political rights (PR) and civil liberties indices as measures of PR and protections. Based on these measures, the authors report unconditional statistics (e.g. average changes) and also regressions of changes in the measures on HIPC Initiative aid allocations and other controls.FindingsThe authors find that most participating countries experienced either meager increases or outright decreases in institutional quality. The regression results provide no evidence that the Initiative affects meaningful reforms.Originality/valueThe potential for foreign aid to have deleterious effects on the institutional quality of recipient countries has been of increasing concern to students of economic development. Such effects can have important implications for entrepreneurial activity in these countries. The HIPC Initiative is specifically designed to acknowledge and, indeed, overcome these concerns, leading to actual increases in institutional quality of recipient countries. To the authors’ knowledge, this work is the first to assess whether the promise of the HIPC Initiative is being fulfilled.

A multiple mapping conditioning mixing model with a mixture-fraction like reference variable. Part 2: RANS implementation and validation against a turbulent jet flame

Abstract In Part I of this paper, a novel multiple mapping conditioning (MMC) mixing model is proposed and verified using ideal flow test cases. In Part II, the MMC mixing model is validated in the context of inhomogeneous turbulent reactive flows using Sandia flame D–F experimental data. The model is implemented alongside a standard k − ϵ CFD solver in the OpenFOAM code called mmcFoam. The MMC model has two main model constants; Cϕ which has a strong effect on the unconditional statistics and rt which controls the localness in mixture fraction space and hence the level of conditional fluctuations. A detailed series of tests explores the sensitivity of the model predictions towards variations in Cϕ and rt with C ϕ = 2 and r t = 0.935 found to work well in flame F and subsequently in flames D and E as well. With these settings the model accurately predicts the increasing level of localised extinction with increasing jet Reynolds number. Sensitivity to the number of particles per cell and the chemical kinetics scheme is also investigated.

Using Relative Statistics and Approximate Disease Prevalence to Compare Screening Tests.

Schatzkin et al. and other authors demonstrated that the ratios of some conditional statistics such as the true positive fraction are equal to the ratios of unconditional statistics, such as disease detection rates, and therefore we can calculate these ratios between two screening tests on the same population even if negative test patients are not followed with a reference procedure and the true and false negative rates are unknown. We demonstrate that this same property applies to an expected utility metric. We also demonstrate that while simple estimates of relative specificities and relative areas under ROC curves (AUC) do depend on the unknown negative rates, we can write these ratios in terms of disease prevalence, and the dependence of these ratios on a posited prevalence is often weak particularly if that prevalence is small or the performance of the two screening tests is similar. Therefore we can estimate relative specificity or AUC with little loss of accuracy, if we use an approximate value of disease prevalence.

Scalar dissipation rate statistics in turbulent swirling jets

The scalar dissipation rate statistics were measured in an isothermal flow formed by discharging a central jet in an annular stream of swirling air flow. This is a typical geometry used in swirl-stabilised burners, where the central jet is the fuel. The flow Reynolds number was 29 000, based on the area-averaged velocity of 8.46 m/s at the exit and the diameter of 50.8 mm. The scalar dissipation rate and its statistics were computed from two-dimensional imaging of the mixture fraction fields obtained with planar laser induced fluorescence of acetone. Three swirl numbers, S, of 0.3, 0.58, and 1.07 of the annular swirling stream were considered. The influence of the swirl number on scalar mixing, unconditional, and conditional scalar dissipation rate statistics were quantified. A procedure, based on a Wiener filter approach, was used to de-noise the raw mixture fraction images. The filtering errors on the scalar dissipation rate measurements were up to 15%, depending on downstream positions from the burner exit. The maximum of instantaneous scalar dissipation rate was found to be up to 35 s−1, while the mean dissipation rate was 10 times smaller. The probability density functions of the logarithm of the scalar dissipation rate fluctuations were found to be slightly negatively skewed at low swirl numbers and almost symmetrical when the swirl number increased. The assumption of statistical independence between the scalar and its dissipation rate was valid for higher swirl numbers at locations with low scalar fluctuations and less valid for low swirl numbers. The deviations from the assumption of statistical independence were quantified. The conditional mean of the scalar dissipation rate, the standard deviation of the scalar dissipation rate fluctuations, the weighted probability of occurrence of the mean conditional scalar dissipation rate, and the conditional probability are reported.

Probability density functions for the variable solar wind near the solar cycle minimum

AbstractUnconditional and conditional statistics are used for studying the histograms of magnetic field multiscale fluctuations in the solar wind near the solar cycle minimum in 2008. The unconditional statistics involves the magnetic data during the whole year in 2008. The conditional statistics involves the magnetic field time series split into concatenated subsets of data according to a threshold in dynamic pressure. The threshold separates fast‐stream leading edge compressional and trailing edge uncompressional fluctuations. The histograms obtained from these data sets are associated with both multiscale (B) and small‐scale (δB) magnetic fluctuations, the latter corresponding to time‐delayed differences. It is shown here that, by keeping flexibility but avoiding the unnecessary redundancy in modeling, the histograms can be effectively described by a limited set of theoretical probability distribution functions (PDFs), such as the normal, lognormal, kappa, and log‐kappa functions. In a statistical sense the model PDFs correspond to additive and multiplicative processes exhibiting correlations. It is demonstrated here that the skewed small‐scale histograms inherent in turbulent cascades are better described by the skewed log‐kappa than by the symmetric kappa model. Nevertheless, the observed skewness is rather small, resulting in potential difficulties of estimation of the third‐order moments. This paper also investigates the dependence of the statistical convergence of PDF model parameters, goodness of fit, and skewness on the data sample size. It is shown that the minimum lengths of data intervals required for the robust estimation of parameters is scale, process, and model dependent.

Numerical simulation of oxy-fuel jet flames using unstructured LES–CMC

Abstract A finite volume Conditional Moment Closure (CMC) formulation has been developed as an LES sub-grid combustion model. This allows unstructured meshes to be used for both LES and CMC grids making the method more applicable to complex geometry. The method has been applied to an oxy-fuel jet flame. This flame offers new challenges to combustion modelling due to a high CO 2 content in the oxidiser stream and significant H 2 content in the fuel stream. The density ratio of the two streams is of the order 5 and the viscosity of the two streams will also differ. All the flames simulated showed localised extinction in the region around 3–5 jet diameters downstream of the nozzle, which is in very good agreement with the experiment. Trends for conditional and unconditional statistics with changing levels of H 2 in the fuel are correctly captured by the LES–CMC method, although different levels of agreement are observed for different species and temperature and possible reasons for this are discussed. The degree of extinction is also correctly predicted to increase as the H 2 content of the jet is reduced, showing the ability of the CMC method to predict complex turbulence-chemistry interaction phenomenon in the presence of changing fuel composition.

Multifractality and value-at-risk forecasting of exchange rates

Abstract This paper addresses market risk prediction for high frequency foreign exchange rates under nonlinear risk scaling behaviour. We use a modified version of the multifractal model of asset returns (MMAR) where trading time is represented by the series of volume ticks. Our dataset consists of 138,418 5-min round-the-clock observations of EUR/USD spot quotes and trading ticks during the period January 5, 2006 to December 31, 2007. Considering fat-tails, long-range dependence as well as scale inconsistency with the MMAR, we derive out-of-sample value-at-risk (VaR) forecasts and compare our approach to historical simulation as well as a benchmark GARCH(1,1) location-scale VaR model. Our findings underline that the multifractal properties in EUR/USD returns in fact have notable risk management implications. The MMAR approach is a parsimonious model which produces admissible VaR forecasts at the 12-h forecast horizon. For the daily horizon, the MMAR outperforms both alternatives based on conditional as well as unconditional coverage statistics.

Regime transition from premixed to flameless oxidation in turbulent JP-10 flames

An opposed jet configuration featuring fractal generated turbulence was used to characterise premixed fuel lean JP-10 flames with equivalence ratios (ϕ) from 0.20 to 0.80 stabilised against combustion products with a temperature of 1720 K. The cold flow turbulent Reynolds (Ret) number was set to 122 and the estimated Damköhler (Da) numbers varied from below 0.3 (ϕ = 0.20) to 6 (ϕ = 0.80) with laminar flame parameters obtained from computations of the corresponding opposed jet configuration. The gradual transition from a turbulent flame to a supported burning mode related to the Homogeneous Charge Diffusion Ignition (HCDI) flameless oxidation regime was detected with features typical of conventional flames gradually disappearing for ϕ < 0.60. A second mode transition to the corrugated flamelet regime at ϕ = 0.80 is shown to coincide with a gradual change from gradient to counter-gradient turbulent transport. A density segregation based technique was applied to Particle Image Velocimetry (PIV) images to detect the instantaneous location of reaction zones and to provide conditional and unconditional statistics to illustrate the combustion regime transitions. Flame surface area characteristics and turbulent burning velocities were also derived to further characterise the burning modes.

Analysis of Urban Atmosphere Plume Concentration Fluctuations

Concentration variability in the fast-response tracer dataset for continuous, near-surface, point source releases in the urban core from the Joint Urban 2003 field study is analyzed. Concentration variability for conditionally and unconditionally sampled time series is characterized by probability densities, concentration fluctuation intensity, skewness, and kurtosis. Significant day-night differences in plume dispersion are observed. Relative to daytime, nighttime plumes were more likely to have reduced concentration fluctuation intensities, higher normalized surface concentrations, suppressed vertical mixing, and a greater prevalence of Gaussian-like distributions rather than log-normal or mixed mode distributions. This was in spite of the similar stability and turbulence conditions in the urban core for day and night. The potential roles of flow meander and thermal stability in explaining these differences are examined. Probability densities of concentration are found to be a strong function of fluctuation intensity. There are few differences in probability densities between day and night when classified by fluctuation intensity. There are no appreciable differences between conditional and unconditional probability densities and only small differences between conditional and unconditional sampling statistics relative to the larger differences usually observed in more homogeneous settings. Fluctuation intensity, skewness, and kurtosis are higher for the daytime experiments, and closer to the source, but show little difference between conditional and unconditional results over most of their range of values. The log-normal distribution provides a better overall fit to a broader range of the dataset than the exponential or clipped-normal distributions.

Time-averaging strategies in the finite-volume/particle hybrid algorithm for the joint PDF equation of turbulent reactive flows

The influence of time-averaging on bias is investigated in the finite-volume/particle hybrid algorithm for the joint PDF equation for statistically-stationary turbulent reactive flows. It is found that the time-averaging of the mean fluctuating velocity (TAu) leads to the same variances of the fluctuating velocity before and after the velocity correction, whereas without TAu the estimates are different, and an additional numerical dissipation rate is introduced for the turbulent kinetic energy (TKE). When 100 particles per cell are used without TAu, a large bias error is found to be involved in the unconditional statistics of the statistically-stationary solutions of two tested turbulent flames, the Cabra H2/N2 lifted flame and the Sandia piloted flame E. The use of TAu reduces this bias dramatically for the same number of particles per cell. The conditional statistics in these flames, however, are hardly affected by TAu. To a large extent, the effect of the bias error on the unconditional statistics is similar to the effect of increasing the model constant C ω 1 in the stochastic turbulence frequency model.

A non-hybrid method for the PDF equations of turbulent flows on unstructured grids

In probability density function (PDF) methods of turbulent flows, the joint PDF of several flow variables is computed by numerically integrating a system of stochastic differential equations for Lagrangian particles. A set of parallel algorithms is proposed to provide an efficient solution of the PDF transport equation, modeling the joint PDF of turbulent velocity, frequency and concentration of a passive scalar in geometrically complex configurations. An unstructured Eulerian grid is employed to extract Eulerian statistics, to solve for quantities represented at fixed locations of the domain (e.g. the mean pressure) and to track particles. All three aspects regarding the grid make use of the finite element method (FEM) employing the simplest linear FEM shape functions. To model the small-scale mixing of the transported scalar, the interaction by exchange with the conditional mean model is adopted. An adaptive algorithm that computes the velocity-conditioned scalar mean is proposed that homogenizes the statistical error over the sample space with no assumption on the shape of the underlying velocity PDF. Compared to other hybrid particle-in-cell approaches for the PDF equations, the current methodology is consistent without the need for consistency conditions. The algorithm is tested by computing the dispersion of passive scalars released from concentrated sources in two different turbulent flows: the fully developed turbulent channel flow and a street canyon (or cavity) flow. Algorithmic details on estimating conditional and unconditional statistics, particle tracking and particle-number control are presented in detail. Relevant aspects of performance and parallelism on cache-based shared memory machines are discussed.

Probability density function modeling of scalar mixing from concentrated sources in turbulent channel flow

Dispersion of a passive scalar from concentrated sources in fully developed turbulent channel flow is studied with the probability density function (PDF) method. The joint PDF of velocity, turbulent frequency and scalar concentration is represented by a large number of Lagrangian particles. A stochastic near-wall PDF model combines the generalized Langevin model of Haworth and Pope [Phys. Fluids 29, 387 (1986)] with Durbin's [J. Fluid Mech. 249, 465 (1993)] method of elliptic relaxation to provide a mathematically exact treatment of convective and viscous transport with a nonlocal representation of the near-wall Reynolds stress anisotropy. The presence of walls is incorporated through the imposition of no-slip and impermeability conditions on particles without the use of damping or wall-functions. Information on the turbulent time scale is supplied by the gamma-distribution model of van Slooten et al. [Phys. Fluids 10, 246 (1998)]. Two different micromixing models are compared that incorporate the effect of small scale mixing on the transported scalar: the widely used interaction by exchange with the mean and the interaction by exchange with the conditional mean model. Single-point velocity and concentration statistics are compared to direct numerical simulation and experimental data at Reτ=1080 based on the friction velocity and the channel half width. The joint model accurately reproduces a wide variety of conditional and unconditional statistics in both physical and composition space.

Transmissivity upscaling in numerical models of steady aquifer flow: Conditional statistics

Numerical solution of regional scale aquifer flow requires discretizing the transmissivity T. For spatially varying T and by the stochastic approach, T is modeled as a random field. Typically, both the numerical element scale R and the log transmissivity integral scale I are of order of hundreds to thousands meters. Consequently, the upscaled block transmissivity is random and its statistical moments depend on those of the fine scale T, on flow conditions, and on R/I. Modeling Y = ln T as a two‐dimensional normally distributed stationary random field, we have derived [Dagan and Lessoff, Water Resources Research, 43, W05431, doi:10.1029/2006WR005235 2007] the unconditional statistics of the upscaled = ln , accurate to the first‐order in σY2, the log transmissivity variance. Both cases of mean uniform flow [solved previously by Indelman and Dagan, Transport in Porous Media, 12, 161‐183, 1993] and of strongly nonuniform flow in a circular block of radius R centered at a well of radius rw, were considered. Such upscaling applies either to elements in regions of natural gradient flow or those surrounding a projected well, respectively, which are sufficiently far from points of measurement of T. The present article extends the analysis to account for the presence of a measurement removed in the center of the element (e.g., result of a pumping test). The fine scale conditional log transmissivity Yc is modeled as multi‐Gaussian, of nonstationary mean and of two point covariance which depend on the measured value and distance from the measurement location. Upscaling in mean uniform flow is related, for instance, to the case of a pumping well which is not operative, but served to determine T while natural gradient flow conditions prevail in the area. It was found that upscaling has a significant effect upon the mean upscaled log transmissivity that changes from the measured value for R/I ≪ 1 to the effective one 〈Y〉 = ln TG for R/I &gt; 6. Similarly, the conditional log transmissivity variance is greatly reduced compared to the unconditional case. It tends to zero for R/I → 0 or R/I → ∞, and it reaches a maximal value of around 0.15 σY2 for R/I ≃ 1. The second case, of upscaling in well flow, applies to an element with an operative well at its center, which was used to determine the transmissivity Tw. Unlike mean uniform flow, it is found that upscaling has a very small effect for the realistic values of rw/I = 0(10−3) and R/I &lt; 5. In this case the upscaled transmissivity is practically deterministic and equal to the given Tw. A summary of the two articles, (Dagan and Lessoff, 2007) and the present one concludes the paper.

Transmissivity upscaling in numerical aquifer models of steady well flow: Unconditional statistics

Numerical solution of regional‐scale aquifer flow requires discretizing the transmissivity T. Typically, the numerical element scale R and the transmissivity integral scale I are both of the order of hundreds of meters. Then the upscaled block transmissivity is random, and its statistical moments depend on those of T, on flow conditions, and on R/I. Modeling Y = ln T as a two‐dimensional normally distributed stationary random field, we derive unconditional statistics of the upscaled = ln accurate to the first order in σY2 for strongly nonuniform flow in a circular block of radius R centered on a well of radius rw in an unbounded domain. After a proper definition, it was found that the ensemble mean 〈〉 is approximately but not exactly equal to ln TG (the geometric mean) and that the variance ratio 2/σY2, which depends on rw/I and the shape of the correlation ρ, drops slowly from unity for R/I = 0 to approximately 0.6 for R/I = 10. Hence the variance of the upscaled transmissivity in radial flow is larger than that determined previously for uniform flow. Additionally, defining the equivalent Teq as a deterministic value for which the solution of the flow problem renders directly the mean upscaled head drop and specific discharge, we find that in radial flow Teq ≅ TH the harmonic mean and grows slowly with increasing R/I, whereas for mean uniform flow Teq = TG. Application of the procedure is illustrated for an example of aquifer with selected values of parameters.