Allan L. Gutjahr

A comparison of seven geostatistically based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow

This paper describes the first major attempt to compare seven different inverse approaches for identifying aquifer transmissivity. The ultimate objective was to determine which of several geostatistical inverse techniques is better suited for making probabilistic forecasts of the potential transport of solutes in an aquifer where spatial variability and uncertainty in hydrogeologic properties are significant. Seven geostatistical methods (fast Fourier transform (FF), fractal simulation (FS), linearized cokriging (LC), linearized semianalytical )LS), maximum likelihood (ML), pilot point (PP), and sequential self-calibration (SS)) were compared on four synthetic data sets. Each data set had specific features meeting (or not) classical assumptions about stationarity, amenability to a geostatistical description, etc. The comparison of the outcome of the methods is based on the prediction of travel times and travel paths taken by conservative solutes migrating in the aquifer for a distance of 5 km. Four of the methods, LS, ML, PP, and SS, were identified as being approximately equivalent for the specific problems considered. The magnitude of the variance of the transmissivity fields, which went as high as 10 times the generally accepted range for linearized approaches, was not a problem for the linearized methods when applied to stationary fields; that is, their inverse solutions and travel time predictions were as accurate as those of the nonlinear methods. Nonstationarity of the “true” transmissivity field, or the presence of “anomalies” such as high-permeability fracture zones was, however, more of a problem for the linearized methods. The importance of the proper selection of the semivariogram of the log10 (T) field (or the ability of the method to optimize this variogram iteratively) was found to have a significant impact on the accuracy and precision of the travel time predictions. Use of additional transient information from pumping tests did not result in major changes in the outcome. While the methods differ in their underlying theory, and the codes developed to implement the theories were limited to varying degrees, the most important factor for achieving a successful solution was the time and experience devoted by the user of the method.

Cross‐correlated random field generation with the direct Fourier Transform Method

This paper presents a computer algorithm that is capable of cogenerating pairs of three-dimensional, cross-correlated random fields. The algorithm produces random fields of real variables by the inverse Fourier transform of a randomized, discrete three-dimensional spectral representations of the variables. The randomization is done in the spectral domain in a way that preserves the direct power and cross-spectral density structure. Two types of cross spectra were examined. One type specifies a linear relationship between the two fields, which produces the same correlation scales for both variables but different variances. The second cross spectrum is obtained from a specified transfer function and the two power spectra, and it produces fields with different correlation scales. For both models the degree of correlation is specified by the coherency. A delay vector can also be specified to produce an out-of-phase correlation between the two fields. The algorithm is very efficient computationally, is relatively easy to use, and does not produce the lineation problems that can be encountered with the turning bands method. Perhaps most important, this random field generator is capable of co-generating cross-correlated random fields.

Joint conditional simulations and the spectral approach for flow modeling

The use of data to condition single random fields has a well-established history. However, the joint use of data from several cross-correlated random fields is not as well developed. For example, the use of both transmissivity and head data in a steady state 2-d stochastic flow problem is essentially an inverse problem that is very important for both flow and transport predictions. This problem is addressed here by using a combination of numerical simulation and analytical methods and its application illustrated. The type of information conveyed by the different data categories is explored. The results presented are especially interesting in that head and transmissivity each give different information: Head values would appear to constrain the geometry of the paths while transmissivity data yields information about travel times. The linearized model is expanded to an iterative procedure and the “true” conditional distribution at several locations is compared with the iterative solution.The problem mentioned above is one with a special transfer function specified by the flow equation. In the second part of the paper a Fast Fourier Transform method for generation and conditioning of two or more random fields is introduced. This procedure is simple to implement, fast and very flexible.

Co-kriging for stochastic flow models

Co-kriging equations for log-transmissivity and heads are derived for a two-dimensional stochastic model. The behavior of the weights as a function of the unknown value of mean hydraulic gradient J are discussed and the procedure is illustrated by studying the ‘screening’ effects of adjacent measurements and added head measurements. In addition, the bias of the estimator for head values is studied when J is also estimated.

General joint conditional simulations using a fast Fourier transform method

A procedure for generating joint statistically homogeneous random fields is examined. The method is based on the spectral representation theorem. It handles large fields easily and is both rapid and flexible. Algorithm development and examples are presented. The procedure is adapted further to include the possibility of generating fields that are jointly conditioned on data from two related fields.

What does an instrument measure? Empirical spatial weighting functions calculated from permeability data sets measured on multiple sample supports

With the aid of linear filter theory we analyze 13,824 permeability measurements to empirically address the question, What does an instrument measure? By measure we mean the sample support or sample volume associated with an instrument, as well as how the instrument spatially weights the heterogeneities comprising that sample support. Although the theoretical aspects of linear filter analysis are well documented, physical data for testing the filtering behavior of an instrument, particularly in the context of porous media flow, are rare to nonexistent. Our exploration makes use of permeability data measured with a minipermeameter on a block of Berea sandstone. Data were collected according to a uniform grid that was resampled with tip seals of increasing size (i.e., increasing sample support). Spatial weighting (filter) functions characterizing the minipermeameter measurements were then calculated directly from the permeability data sets. In this paper we limit our presentation to one of the six rock faces, consisting of 2304 measurements, as the general results for each rock face are similar. We found that the empirical weighting functions are consistent with the basic physics of the minipermeameter measurement. They decay as a nonlinear function of radial distance from the center of the tip seal, consistent with the divergent flow geometry imposed by the minipermeameter. The magnitude of the weighting function decreases while its breadth increases with increasing tip seal size, reflecting the increasing sample support. We further demonstrate, both empirically and theoretically, that nonadditive properties like permeability are amenable to linear filter analysis under certain limiting conditions (i.e., small variances). Specifically, the weighting function is independent of the power average employed in its calculation (e.g., arithmetic versus harmonic average). Finally, we examine the implications of these results for other instruments commonly employed in hydraulic testing (e.g., slug and pump tests).

Transport with spatially variable kinetic sorption: recursion formulation

A recursion formulation for the transport of linearly sorbing solutes undergoing nonequilibrium sorption is developed. Constant or spatially varying sorption kinetics can be modeled using the recursion approach. The sorption and desorption rates are modeled as two independent random processes with a prescribed mean and covariance structure with spatial variability in the rate parameters included as well. The recursion solution, in terms of the probability density function for solute travel times, is derived by specifying transition probabilities for moving between the aqueous and sorbed phases. A few simple examples are used to illustrate the approach. The computer implementation leads to a very rapid algorithm that is easily extended to cover cases beyond the basic model presented here. q 1999 Elsevier Science Ltd. All rights reserved.

Spatial variability in subsurface flow and transport: a review

Abstract Stochastic models of spatial variation as they apply to both saturated and unsaturated flow and transport problems are examined in this paper. Both modeling and data interpretive geostatistical approaches are reviewed and an integrated discussion combining the two approaches given. The probabilistic content is of special interest for reliability and risk calculations for waste management and groundwater pollution studies.

Linearized cosimulation of hydraulic conductivity, pressure head, and flux in saturated and unsaturated, heterogeneous porous media

An efficient cosimulator is developed for generating both random hydraulic property fields and related flow regimes under either saturated or unsaturated conditions. This cosimulator combines a spectral random field generator, based on the Fast Fourier Transform technique, and first-order perturbation/spectral solutions for flow in saturated and unsaturated porous media. Owing to the first-order approximation of the simulator, flow regimes in several geological media with different variabilities are simulated to investigate the accuracy of the simulator. For mild and moderately heterogeneous geological media, the simulator is found to be very accurate in terms of pressure head field and flux distributions. In addition, the execution time of the simulator is substantially smaller than that of any classical numerical simulator. However, the accuracy of the simulator deteriorates as the geological medium becomes highly heterogeneous.

Transverse Dispersion of a Kinetically Sorbing Solute

A recursion formulation for the transverse spreading of a solute is developed, and under conditions of steady flow in a stratified aquifer, the transport of a linearly sorbing solute undergoing nonequilibrium sorption is studied. The effect of spatial variability in the velocity field and the sorption kinetics are modeled to see the combined effect of the two processes on the spreading of the solute injected at a point in the aquifer. The main result of this work is a transport model based on a discrete formulation that includes local dispersion and leads to nonasymptotic behavior in the spreading of the plume in a direction normal to the mean flow velocity.