Allan D. Woodbury

The geostatistical characteristics of the borden aquifer

A complete reexamination of Sudickys (1986) field experiment for the geostatistical characterization of hydraulic conductivity at the Borden aquifer in Ontario, Canada is performed. The sampled data reveal that a number of outliers (low ln (K) values) are present in the data base. These low values cause difficulties in both variogram estimation and determining population statistics. The analysis shows that assuming either a normal distribution or exponential distribution for log conductivity is appropriate. The classical, Cressie/Hawkins and squared median of the absolute deviations (SMAD) estimators are used to compute experimental variograms. None of these estimators provides completely satisfactory variograms for the Borden data with the exception of the classical estimator with outliers removed from the data set. Theoretical exponential variogram parameters are determined from nonlinear (NL) estimation. Differences are obtained between NL fits and those of Sudicky (1986). For the classical-screened estimated variogram, NL fits produce an ln (K) variance of 0.24, nugget of 0.07, and integral scales of 5.1 m horizontal and 0.21 m vertical along A–A′. For B–B′ these values are 0.37, 0.11, 8.3 and 0.34. The fitted parameter set for B–B′ data (horizontal and vertical) is statistically different than the parameter set determined for A–A′. We also evaluate a probabilistic form of Dagans (1982, 1987) equations relating geostatistical parameters to a tracer clouds spreading moments. The equations are evaluated using the parameter estimates and covariances determined from line A–A′ as input, with a velocity equal to 9.0 cm/day. The results are compared with actual values determined from the field test, but evaluated by both Freyberg (1986) and Rajaram and Gelhar (1988). The geostatistical parameters developed from this study produce an excellent fit to both sets of calculated plume moments when combined with Dagans stochastic theory for predicting the spread of a tracer cloud.

MINIMUM RELATIVE ENTROPY INVERSION : THEORY AND APPLICATION TO RECOVERING THE RELEASE HISTORY OF A GROUNDWATER CONTAMINANT

In this paper we show that given prior information in terms of a lower and upper bound, a prior bias, and constraints in terms of measured data, minimum relative entropy (MRE) yields exact expressions for the posterior probability density function (pdf) and the expected value of the linear inverse problem. In addition, we are able to produce any desired confidence intervals. In numerical simulations, we use the MRE approach to recover the release and evolution histories of plume in a one-dimensional, constant known velocity and dispersivity system. For noise-free data, we find that the reconstructed plume evolution history is indistinguishable from the true history. An exact match to the observed data is evident. Two methods are chosen for dissociating signal from a noisy data set. The first uses a modification of MRE for uncertain data. The second method uses “presmoothing” by fast Fourier transforms and Butterworth filters to attempt to remove noise from the signal before the “noise-free” variant of MRE inversion is used. Both methods appear to work very well in recovering the true signal, and qualitatively appear superior to that of Skaggs and Kabala [1994]. We also solve for a degenerate case with a very high standard deviation in the noise. The recovered model indicates that the MRE inverse method did manage to recover the salient features of the source history. Once the plume source history has been developed, future behavior of a plume can then be cast in a probabilistic framework. For an example simulation, the MRE approach not only was able to resolve the source function from noisy data but also was able to correctly predict future behavior.

Minimum relative entropy: Forward probabilistic modeling

The pioneering work of Jaynes in Bayesian/maximum entropy methods has been successfully explored in many disciplines. The principle of maximum entropy (PME) is a powerful and versatile tool of inferring a probability distribution from constraints that do not completely characterize that distribution. Minimum relative entropy (MRE) is a method which has all the important attributes of the maximum entropy approach with the advantage that prior information may be easily included. In this paper we use MRE to determine the prior probability density function (pdf) of a set of model parameters based on limited information. The resulting pdf is used in Monte Carlo simulations to provide expected values in field variables such as concentration, and confidence limits. We compare the probabilistic results from a traditional advection-dispersion (ADE) model based on volumetric averaging concepts with that of a model based on the assumption that the hydraulic conductivity is a stationary stochastic process. The results suggest that although Naffs (1990) model satisfies the observed data to a better degree than ADE model, the upper and lower confidence bands about the mean value are larger than the ADE results. This result we attribute to the fact that Naffs (1990) model simply contains more parameters, each of which is unknown and has to be estimated. There is no statistical difference between the expected values of second-spatial moments for the two models. The examples presented in this paper illustrate problems associated with assigning Gaussian pdfs as priors in a probabilistic model. First, such an assumption for the input parameters actually injects more information into the problem than may actually exist, whether consciously or unconsciously. This fact is borne out by comparison with minimum relative entropy theory. Second, the output parameters as suggested from the Monte Carlo analysis cannot be assumed to be Gaussian distributed even when the prior pdf is Gaussian in form. In a practical setting, the significance of this result and the approximation of Gaussian form would depend on the toxicity and environmental standards that apply to the problem.

Three-dimensional plume source reconstruction using minimum relative entropy inversion

In this paper we extend the minimum relative entropy (MRE) method to recover the source-release history of a three dimensional plume. This extension is carried out in an analytic framework, and in order to qualify as a linear inverse problem the various transport parameters such as dispersivity and the like are considered to be known. In addition, the groundwater flow system is assumed to be steady and uniform. The contributions of this paper include an explanation how MRE can be used as a measure of resolution in linear inversion, a reporting of a three dimensional analytic solution for mass transport in a steady one dimensional velocity field for a variable in-time source loading, an estimation the source-release history for synthetically generated data sets, and an application of the methodology to a case-study problem at the Gloucester Landfill in Ontario, Canada. We found that the relative entropy measure is useful in indicating the reduction in uncertainty between the posterior and prior pdfs as a result of the new information provided by the physical constraints and data. Using the individual model-parameter relative entropies as a measure of resolution, one can make quantitative judgments about which part of the history is likely to be well resolved. Comparing inversion results for synthetic aquifers with one well and two sample points with only one sample point indicates that temporal data at several wells allows for a superior reconstruction of the release history. We investigate the potential benefits of locating sample points on an spatial rather than temporal basis. Results show that early part of the release history is poorly recovered. Comparing these results with one well and two sample points indicates that temporal data at a few wells allows for a better reconstruction of the release history. An incomplete time record is also investigated. Results show that early part of the release history prior to the commencement of measurements is poorly recovered. It is essential that as much as possible of the time histories of plumes be monitored if the entire release history is to be determined. The MRE approach is used to reconstruct the release history of a 1,4-dioxane plume measured at the Gloucester Landfill in Ontario, Canada. The recovered release history is fairly narrow and generally flat in shape, although two peaks are evident. Neither peak is particularly well defined, judging from the resolution curve and analysis of the confidence ranges. One of the peaks coincides with year 1979, close to the year 1978 in which a large spill was noted. However, a considerable variation of possible source releases is possible.

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.

Inversion of the Borden Tracer Experiment data: Investigation of stochastic moment models

Inversion of Dagans two- and three-dimensional stochastic models using Freybergs (1986), Rajaram and Gelhars (1988, 1991), and Barry and Spositos (1990) moment data from the Borden experiment is carried out to examine (1) the validity of the two-dimensional (Dagan, 1982) and three-dimensional (Dagan, 1988) models and (2) the reduction in uncertainty of the spatial moments over nonconditioned estimates. A direct application of Bayesian statistical inference, in conjunction with Monte Carlo integration, is used to produce posterior probability density functions for the parameters. The parameter ranges from all methods show horizontal integral scales λ between 1.85 and 4.04 m, vertical integral scales λz between 0.144 and 0.459 m, and log hydraulic conductivities In (K) between 0.120 and 0.197. These results compare well to the earlier estimates of Woodbury and Sudicky (1991) and Robin et al. (1991). We show that moment prediction uncertainty is substantially reduced when both tracer moment data and prior estimates of the In (K) geostatistical parameters are incorporated into theoretical formulae based on stochastic dispersion theory.

A FORTRAN program to produce minimum relative entropy distributions

Abstract Relative entropy minimization is a general approach of inferring a probability density function (pdf) from constraints which do not uniquely determine that density. In this paper, a general purpose computer program written in FORTRAN is provided that produces a univariate pdf from a series of constraints and a prior probability. Some guidelines for the selection of the prior are presented. The FORTRAN code is based on an algorithm that utilizes a Newton–Raphson approach. In addition, we use Gauss–Legendre quadrature for the determination of the integrals, Gauss elimination for matrix solution and a line search for the most optimal Newton step. We present examples of relative entropy minimization involving functions that are geometric moments of a variable x . With a uniform prior p ( x ), classic solutions of statistics are obtained. We also varied the nature of the prior for illustrative purposes. For the case where the constraints resemble powers of x and logarithmic transformations, minimum relative entropy produces the Gamma distribution.

Lanczos method for the solution of groundwater flow in discretely fractured porous media

Abstract One of the more advanced approaches for simulating groundwater flow in fractured porous media is the discrete-fracture approach. This approach is limited by the large computational overheads associated with traditional modeling methods. In this work, we apply the Lanczos reduction method to the modeling of groundwater flow in fractured porous media using the discrete-fracture approach. The Lanczos reduction method reduces a finite element equation system to a much smaller tridiagonal system of first-order differential equations. The reduced system can be solved by a standard tridiagonal algorithm with little computational effort. Because solving the reduced system is more efficient compared to solving the original system, the simulation of groundwater flow in discretely fractured media using the reduction method is very efficient. The proposed method is especially suitable for the problem of large-scale and long-term simulation. In this paper, we develop an iterative version of Lanczos algorithm, in which the preconditioned conjugate gradient solver based on ORTHOMIN acceleration is employed within the Lanczos reduction process. Additional efficiency for the Lanczos method is achieved by applying an eigenvalue shift technique. The “shift” method can improve the Lanczos system convergence, by requiring fewer modes to achieve the same level of accuracy over the unshifted case. The developed model is verified by comparison with dual-porosity approach. The efficiency and accuracy of the method are demonstrated on a field-scale problem and compared to the performance of classic time marching method using an iterative solver on the original system. In spite of the advances, more theoretical work needs to be carried out to determine the optimal value of the shift before computations are actually carried out.

Lanczos and Arnoldi methods for the solution of convection-diffusion equations

Abstract Two procedures for computing the finite element solution to a convection-diffusion equation, based on the Lanczos and Arnoldi methods, are presented. The Lanczos vectors are generated from the symmetric part of the coefficient matrices, while the Arnoldi vectors are obtained from the unsymmetric matrix. Following a Rayleigh-Ritz procedure, these vectors are then used to reduce the governing system of differential equations to a small system. The Lanczos process produces symmetric tridiagonal and skew-symmetric matrix coefficients for the reduced system. The Arnoldi method results in an upper Hessenberg matrix coefficient for its reduced problem. The finite element solution is then constructed as a linear combination of the Lanczos or Arnoldi vectors. The solution vector for the reduced system holds the components of the approximating solution along the Lanczos or Arnoldi vectors. Each method is applied to a number of different numerical test problems. We conclude that the Lanczos method is preferred for diffusion dominated problems, while the Arnoldi is the method of choice for convection dominated problems.

A Krylov finite element approach for multi-species contaminant transport in discretely fractured porous media

Simulation of non-ideal transport of multi-species solutes in fractured porous media can easily introduce hundreds of thousand to millions of unknowns. In this paper, a Krylov finite element method, the Arnoldi reduction method (ARM), for solving these type problems has been introduced. The Arnoldi reduction technique uses orthogonal matrix transformations to reduce each of the aforementioned coupled systems to much smaller size. In order to speed convergence of the Arnoldi process, an eigenvalue shift in each finite element system is introduced. This approach greatly improves the diagonal dominant properties of the matrices to be solved. This property leads to great enhancement of the iterative solution and the convergence rate for Arnoldi reduction process. In addition, the use of the eigenvalue shift technique greatly relaxes the grid Peclet restrictions. Courant number criteria restrictions are also effectively removed. We utilize an ORTHOMIN procedure to carry out the equation system reductions for discrete fractured media. The proposed numerical method has been verified by comparison against analytical solutions. The developed model is highly efficient in computing time and storage space. Simulations of radioactive decay chain and trichloroethylene transport are made and compared to the Laplace transform Galerkin (LTG) method where appropriate. Examples with about one million unknowns are solved on personal computers and shown that the ARM is even more efficient than the LTG method, by allowing for similar speed increases with multi-components. Therefore, the Arnoldi approach will allow for a variety of complex, high-resolution problems to be solved on small computer platforms.