J. Jaime Gómez-Hernández

Upscaling hydraulic conductivities in heterogeneous media: An overview

Abstract Hydraulic conductivity upscaling is a process that transforms a grid of hydraulic conductivities defined at the scale of the measurements, into a coarser grid of block conductivity tensors amenable for input to a numerical flow simulator. The need for upscaling stems from the disparity between the scales at which measurements are taken and the scale at which aquifers are discretized for the numerical solution of flow and transport. The techniques for upscaling range from the simple averaging of the heterogeneous values within the block to sophisticated inversions, after the solution of the flow equation at the measurement scale within an area embedding the block being upscaled. All techniques have their own advantages and limitations. Recently, the definition of the geometry of the grid has been intimately linked to the upscaling problem; promising results have been obtained using elastic gridding. Also, recently, the need to perform Monte-Carlo analysis, involving many realizations of hydraulic conductivity, has steered the development of methods that generate directly the block conductivities in accordance with the rules of upscaling, yet conditional to the measurement data.

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.

To be or not to be multi-Gaussian? A reflection on stochastic hydrogeology

Abstract The multivariate Gaussian random function model is commonly used in stochastic hydrogeology to model spatial variability of log-conductivity. The multi-Gaussian model is attractive because it is fully characterized by an expected value and a covariance function or matrix, hence its mathematical simplicity and easy inference. Field data may support a Gaussian univariate distribution for log hydraulic conductivity, but, in general, there are not enough field data to support a multi-Gaussian distribution. A univariate Gaussian distribution does not imply a multi-Gaussian model. In fact, many multivariate models can share the same Gaussian histogram and covariance function, yet differ by their patterns of spatial continuity at different threshold values. Hence the decision to use a multi-Gaussian model to represent the uncertainty associated with the spatial heterogeneity of log-conductivity is not databased. Of greatest concern is the fact that a multi-Gaussian model implies the minimal spatial correlation of extreme values, a feature critical for mass transport and a feature that may be in contradiction with some geological settings, e.g. channeling. The possibility for high conductivity values to be spatially correlated should not be discarded by adopting a congenial model just because data shortage prevents refuting it. In this study, three alternatives to a multi-Gaussian model, all sharing the same Gaussian histogram and the same covariance function, but with different continuity patterns for extreme values, were considered to model the spatial variability of log-conductivity. The three alternative models, plus the traditional multi-Gaussian model, are used to perform Monte Carlo analyses of groundwater travel times from a hypothetical nuclear repository to the ground surface through a synthetic formation similar to the Finnsjon site in Sweden. The results show that the groundwater travel times predicted by the multi-Gaussian model could be ten times slower than those predicted by the other models. The probabilities of very short travel times could be severely underestimated using the multi-Gaussian model. Consequently, if field measured data are not sufficient to determine the higher-order moments necessary to validate the multi-Gaussian model — which is the usual situation in practice — other alternative models to the multi-Gaussian one ought to be considered.

ISIM3D: and ANSI-C three-dimensional multiple indicator conditional simulation program

Abstract The indicator conditional simulation technique provides stochastic simulations of a variable that (i) honor the initial data and (ii) can feature a richer family of spatial structures not limited by Gaussianity. The data are encoded into a series of indicators which then are used to estimate the conditional probability distribution (cpdf) of the variable under study at any unsampled location. Once the cpdf has been estimated, any particular simulated value is obtained by straightforward Monte-Carlo drawing. Each new simulated value is included in the conditioning data set so that the next simulated values at other locations be conditioned to it. This technique has the advantage over other more traditional techniques such as the turning bands method in that it is not multiGaussian related. The user has full control of the bivariate (2-point) statistics imposed on the simulated field instead of controlling a mere covariance model. The source code is provided in C according to the ANSI standard.

Joint Sequential Simulation of MultiGaussian Fields

The sequential simulation algorithm can be used for the generation of conditional realizations from either a multiGaussian random function or any non-Gaussian random function as long as its conditional distributions can be derived. The multivariate probability density function (pdf) that fully describes a random function can be written as the product of a set of univariate conditional pdfs. Drawing realizations from the multivariate pdf amounts to drawing sequentially from that series of univariate conditional pdfs. Similarly, the joint multivariate pdf of several random functions can be written as the product of a series of univariate conditional pdfs. The key step consists of the derivation of the conditional pdfs. In the case of a multiGaussian fields, these univariate conditional pdfs are known to be Gaussian with mean and variance given by the solution of a set of normal equations also known as simple cokriging equations. Sequential simulation is preferred to other techniques, such as turning bands, because of its ease of use and extreme flexibility.

Coupled inverse modelling of groundwater flow and mass transport and the worth of concentration data

This paper presents the extension of the self-calibrating method to the coupled inverse modelling of groundwater flow and mass transport. The method generates equally likely solutions to the inverse problem that display the variability as observed in the field and are not affected by a linearisation of the state equations. Conditioning to the state variables is measured by an objective function including, among others, the mismatch between the simulated and measured concentrations. Conditioning is achieved by minimising the objective function by gradient-based methods. The gradient contains the partial derivatives of the objective function with respect to: log conductivities, log storativities, prescribed heads at boundaries, retardation coefficients and mass sources. The derivatives of the objective function with respect to log conductivity are the most cumbersome and need the most CPU-time to be evaluated. For this reason, to compute this derivative only advective transport is considered. The gradient is calculated by the adjoint-state method. The method is demonstrated in a controlled, synthetic study, in which the worth of concentration data is analysed. It is shown that concentration data are essential to improve transport predictions and also help to improve aquifer characterisation and flow predictions, especially in the upstream part of the aquifer, even in the case that a considerable amount of other experimental data like conductivities and heads are available. Besides, conditioning to concentration data reduces the ensemble variances of estimated transmissivity, hydraulic head and concentration.

Stochastic simulation of transmissivity fields conditional to both transmissivity and piezometric data 2. Demonstration on a synthetic aquifer

In the first paper of this series a methodology for the generation of transmissivity fields conditional to both transmissivity and piezometric head data was presented. This methodology, termed the self-calibrated approach, consists of two steps: first, the generation of a seed transmissivity field conditioned only to transmissivity data, and second, the perturbation of the seed field up until the piezometric head data are reproduced. The methodology is now demonstrated on a set of controlled numerical experiments carried out on synthetic aquifers. The objective of these experiments is not just to show that the methodology works, but also to explore its robustness under different situations. A total of 12 experiments have analyzed the performance of the method as a function of: (i) the log10 T transmissivity variance (from 0.2 to 2.0); (ii) the number of log10 T conditioning data (from 10 to 30); (iii) the number of piezometric head data (from 30 to 90); (iv) the number of master points (from 25 to 1000); (v) the magnitude of allowed departure of the final T field from the seed field (up to four times the kriging standard deviation). In all cases, the method was able to generate transmissivity fields conditional to both transmissivity and head measurements, at the same time preserving the spatial variability of the transmissivity field. It was found that the performance of the method increases with both the number of log10 T data and the number of master points, whereas it decreases as either the log10 T variance or the number of piezometric head data increases.

Stochastic Imaging of the Wilmington Clastic Sequence

The geometric architecture of the sand-shale sequence in two layers of the Wilmington field is characterized by indicator variograms produced with noncartesian coordinates ensuring stratigraphic conformity. The sequential indicator simulation (SIS) algorithm produces several alternative equiprobably three-dimensional models of the sand-shale sequence, all of them honoring the data at their locations and reproducing the indicator variogram model. These alternative models provide a direct visualization of spatial uncertainty that can be used to assess the need for additional data.

Theory and Practice of Sequential Simulation

Sequential simulation is a powerful stochastic simulation technique the theory of which relies on the ability to determine, for a given multivariate model, the conditional probability of a single random variable given any number of conditioning values. Sequential simulation can be used with those multivariate models for which these conditional probabilities can be determined. In practice, it is not enough to know how to determine the conditional probabilities, the procedure must be feasible from an operational point of view. Because it is not so in most cases, some approximations to the conditional probability distribution function are used in the implementation of the technique. These approximations are shown not to have a large impact in the performance of the technique, at least for the case in which the underlying multivariate model is multiGaussian.

A Blocking Markov Chain Monte Carlo Method for Inverse Stochastic Hydrogeological Modeling

An adequate representation of the detailed spatial variation of subsurface parameters for underground flow and mass transport simulation entails heterogeneous models. Uncertainty characterization generally calls for a Monte Carlo analysis of many equally likely realizations that honor both direct information (e.g., conductivity data) and information about the state of the system (e.g., piezometric head or concentration data). Thus, the problems faced is how to generate multiple realizations conditioned to parameter data, and inverse-conditioned to dependent state data. We propose using Markov chain Monte Carlo approach (MCMC) with block updating and combined with upscaling to achieve this purpose. Our proposal presents an alternative block updating scheme that permits the application of MCMC to inverse stochastic simulation of heterogeneous fields and incorporates upscaling in a multi-grid approach to speed up the generation of the realizations. The main advantage of MCMC, compared to other methods capable of generating inverse-conditioned realizations (such as the self-calibrating or the pilot point methods), is that it does not require the solution of a complex optimization inverse problem, although it requires the solution of the direct problem many times.