Velimir V. Vesselinov

Three‐dimensional numerical inversion of pneumatic cross‐hole tests in unsaturated fractured tuff: 2. Equivalent parameters, high‐resolution stochastic imaging and scale effects

In paper 1 of this two-part series we described a three-dimensional numerical inverse model for the interpretation of cross-hole pneumatic tests in unsaturated fractured tuffs at the Apache Leap Research Site (ALRS) near Superior, Arizona. Our model is designed to analyze these data in two ways: (1) by considering pressure records from individual borehole monitoring intervals one at a time, while treating the rock as being spatially uniform, and (2) by considering pressure records from multiple tests and borehole monitoring intervals simultaneously, while treating the rock as being randomly heterogeneous. The first approach yields a series of equivalent air permeabilities and air- filled porosities for rock volumes having length scales ranging from meters to tens of meters, represented nominally by radius vectors extending from injection to monitoring intervals. The second approach yields a high-resolution geostatistical estimate of how air permeability and air-filled porosity, defined on grid blocks having a length scale of 1 m, vary spatially throughout the tested rock volume. It amounts to three-dimensional pneumatic “tomography” or stochastic imaging of the rock. Paper 1 described the field data, the model, and the effect of boreholes on pressure propagation through the rock. This second paper implements our inverse model on pressure data from five cross-hole tests at ALRS. We compare our cross-hole test interpretations by means of the two approaches with earlier interpretations by means of type curves and with geostatistical interpretations of single-hole test data. The comparisons show internal consistency between all pneumatic test interpretations and reveal a very pronounced scale effect in permeability and porosity at ALRS.

Three‐dimensional numerical inversion of pneumatic cross‐hole tests in unsaturated fractured tuff: 1. Methodology and borehole effects

We describe a three-dimensional numerical inverse model for the interpretation of cross-hole pneumatic tests in unsaturated fractured tuffs at the Apache Leap Research Site (ALRS) near Superior, Arizona. The model combines a finite volume flow simulator, FEHM, an automatic mesh generator, X3D, a parallelized version of an automatic parameter estimator, PEST, and a geostatistical package, GSTAT. The tests are simulated by considering single-phase airflow through a porous continuum, which represents primarily interconnected fractures at the site. The simulator solves the airflow equations in their original nonlinear form and accounts directly for the ability of all packed-off borehole intervals to store and conduct air through the system. Computations are performed in parallel on a supercomputer using 32 processors. We analyze pneumatic cross-hole test data, previously conducted by our group at ALRS, in two ways: (1) by considering pressure records from individual borehole monitoring intervals one at a time, while treating the rock as being spatially uniform, and (2) by considering pressure records from multiple tests and borehole monitoring intervals simultaneously, while treating the rock as being randomly heterogeneous. The first approach yields a series of equivalent air permeabilities and air-filled porosities for the rock volume being tested, having length scales of the order of meters to tens of meters. The second approach yields a high-resolution geostatistical estimate of how air permeability and air-filled porosity, defined on grid blocks having a length scale of 1 m, vary spatially throughout the tested rock volume. It amounts to three-dimensional pneumatic “tomography” or stochastic imaging of the rock, a concept originally proposed by one of us in 1987. The first paper of this two-part series describes the field data, the model, and the effect of boreholes on pressure propagation through the rock. The second paper implements our approach on selected cross-hole test data from ALRS.

Theoretical interpretation of a pronounced permeability scale effect in unsaturated fractured tuff

Received 10 May 2001; revised 8 January 2002; accepted 8 January 2002; published 27 June 2002. [1] Numerous single-hole and cross-hole pneumatic injection tests have been conducted in unsaturated fractured tuff at the Apache Leap Research Site (ALRS) near Superior, Arizona. Single-hole tests have yielded values of air permeability at various locations throughout the tested rock volume on a nominal scale of � 1 m. Cross-hole tests have yielded equivalent air permeabilities (and air-filled porosities) for a rock volume characterized by a length scale of several tens of meters. Cross-hole tests have also provided high-resolution tomographic estimates of how air permeability (and air-filled porosity), defined over grid blocks having a length scale of 1 m, vary throughout a similar rock volume. The results have revealed a highly pronounced scale effect in permeability (and porosity) at the ALRS. We examine the extent to which the permeability scale effect is amenable to interpretation by a recent stochastic scaling theory, which treats the rock as a truncated random fractal. INDEX TERMS: 1869 Hydrology: Stochastic Processes; 1875 Hydrology: Unsaturated Zone; 3260 Mathematical Geophysics: Inverse Theory; 3250 Mathematical Geophysics: Fractals and Multifractals; KEYWORDS: scaling, permeability, fractals, fractured rocks

Aquifer structure identification using stochastic inversion

This study presents a stochastic inverse method for aquifer structure identification using sparse geophysical and hydraulic response data. The method is based on updating structure parameters from a transition probability model to iteratively modify the aquifer structure and parameter zonation. The method is extended to the adaptive parameterization of facies hydraulic parameters by including these parameters as optimization variables. The stochastic nature of the statistical structure parameters leads to nonconvex objective functions. A multi-method genetically adaptive evolutionary approach (AMALGAM-SO) was selected to perform the inversion given its search capabilities. Results are obtained as a probabilistic assessment of facies distribution based on indicator cokriging simulation of the optimized structural parameters. The method is illustrated by estimating the structure and facies hydraulic parameters of a synthetic example with a transient hydraulic response.

Contaminant remediation decision analysis using information gap theory

Decision making under severe lack of information is a ubiquitous situation in nearly every applied field of engineering, policy, and science. A severe lack of information precludes our ability to determine a frequency of occurrence of events or conditions that impact the decision; therefore, decision uncertainties due to a severe lack of information cannot be characterized probabilistically. To circumvent this problem, information gap (info-gap) theory has been developed to explicitly recognize and quantify the implications of a severe lack of information in decision making. This paper presents a decision analysis based on info-gap theory developed for a contaminant remediation scenario. The analysis provides decision support in determining the fraction of contaminant mass to remove from the environment. An info-gap uncertainty model is developed to characterize uncertainty due to a lack of information concerning the contaminant flux. The info-gap uncertainty model groups nested, convex sets of functions defining contaminant flux over time based on their level of deviation from a nominal contaminant flux. The nominal contaminant flux defines a best estimate of contaminant flux over time based on existing, though incomplete, information. A robustness function is derived to quantify the maximum level of deviation from nominal that still ensures compliance for alternative decisions. An opportuneness function is derived to characterize the possibility of meeting a desired contaminant concentration level. The decision analysis evaluates how the robustness and opportuneness change as a function of time since remediation and as a function of the fraction of contaminant mass removed.

Adaptive hybrid optimization strategy for calibration and parameter estimation of physical process models

A new adaptive hybrid optimization strategy, entitled squads, is proposed for complex inverse analysis of computationally intensive physics-based models. Typically, models are calibrated and model parameters are estimated by minimization of the discrepancy between model simulations characterizing the system and existing observations requiring a substantial number of model evaluations. Squads is designed to be computationally efficient and robust in identification of the global optimum (i.e. maximum or minimum value of an objective function). It integrates global and local optimization using Adaptive Particle Swarm Optimization (APSO) and Levenberg-Marquardt (LM) optimization using adaptive rules based on runtime performance. The global strategy (APSO) optimizes the location of a set of solutions (particles) in the parameter space. The local strategy (LM) is applied only to a subset of the particles at different stages of the optimization based on the adaptive rules. After the LM adjustment of the subset of particle positions, the updated particles are returned to APSO. Therefore, squads is a global strategy that utilizes a local optimization speedup. The advantages of coupling APSO and LM in the manner implemented in squads is demonstrated by comparisons of squads performance against Levenberg-Marquardt (LM), Particle Swarm Optimization (PSO), Adaptive Particle Swarm Optimization (APSO; i.e. TRIBES), and an existing hybrid optimization strategy (hPSO). All the strategies are tested on 2D, 5D and 10D Rosenbrock and Griewank polynomial test functions and a synthetic hydrogeologic application to identify the source of a contaminant plume in an aquifer. Tests are performed using a series of runs with random initial guesses for the estimated parameters. The performance of the strategies are compared based on their robustness, defined as the percentage of runs that identify the global optimum, and their efficiency, quantified by a statistical representation of the number of function evaluations performed prior to identification of the global optimum. Squads is observed to have better performance than the other strategies for the test functions and the hydrogeologic application when both robustness and efficiency are taken into consideration.

An agent-based approach to global uncertainty and sensitivity analysis

A novel sampling approach to global uncertainty and sensitivity analyses of modeling results utilizing concepts from agent-based modeling is presented (Agent-Based Analysis of Global Uncertainty and Sensitivity (ABAGUS)). A plausible model parameter space is discretized and sampled by a particle swarm where the particle locations represent unique model parameter sets. Particle locations are optimized based on a model-performance metric using a standard particle swarm optimization (PSO) algorithm. Locations producing a performance metric below a specified threshold are collected. In subsequent visits to the location, a modified value of the performance metric, proportionally increased above the acceptable threshold (i.e., convexities in the response surface become concavities), is provided to the PSO algorithm. As a result, the methodology promotes a global exploration of a plausible parameter space, and discourages, but does not prevent, reinvestigation of previously explored regions. This effectively alters the strategy of the PSO algorithm from optimization to a sampling approach providing global uncertainty and sensitivity analyses. The viability of the approach is demonstrated on 2D Griewank and Rosenbrock functions. This also demonstrates the set-based approach of ABAGUS as opposed to distribution-based approaches. The practical application of the approach is demonstrated on a 3D synthetic contaminant transport case study. The evaluation of global parametric uncertainty using ABAGUS is demonstrated on model parameters defining the source location and transverse/longitudinal dispersivities. The evaluation of predictive uncertainties using ABAGUS is demonstrated for contaminant concentrations at proposed monitoring wells.

Identification of Pumping Influences in Long‐Term Water Level Fluctuations

Identification of the pumping influences at monitoring wells caused by spatially and temporally variable water supply pumping can be a challenging, yet an important hydrogeological task. The information that can be obtained can be critical for conceptualization of the hydrogeological conditions and indications of the zone of influence of the individual pumping wells. However, the pumping influences are often intermittent and small in magnitude with variable production rates from multiple pumping wells. While these difficulties may support an inclination to abandon the existing dataset and conduct a dedicated cross-hole pumping test, that option can be challenging and expensive to coordinate and execute. This paper presents a method that utilizes a simple analytical modeling approach for analysis of a long-term water level record utilizing an inverse modeling approach. The methodology allows the identification of pumping wells influencing the water level fluctuations. Thus, the analysis provides an efficient and cost-effective alternative to designed and coordinated cross-hole pumping tests. We apply this method on a dataset from the Los Alamos National Laboratory site. Our analysis also provides (1) an evaluation of the information content of the transient water level data; (2) indications of potential structures of the aquifer heterogeneity inhibiting or promoting pressure propagation; and (3) guidance for the development of more complicated models requiring detailed specification of the aquifer heterogeneity.

A computationally efficient parallel Levenberg‐Marquardt algorithm for highly parameterized inverse model analyses

Inverse modeling seeks model parameters given a set of observations. However, for practical problems because the number of measurements is often large and the model parameters are also numerous, conventional methods for inverse modeling can be computationally expensive. We have developed a new, computationally efficient parallel Levenberg-Marquardt method for solving inverse modeling problems with a highly parameterized model space. Levenberg-Marquardt methods require the solution of a linear system of equations which can be prohibitively expensive to compute for moderate to large-scale problems. Our novel method projects the original linear problem down to a Krylov subspace such that the dimensionality of the problem can be significantly reduced. Furthermore, we store the Krylov subspace computed when using the first damping parameter and recycle the subspace for the subsequent damping parameters. The efficiency of our new inverse modeling algorithm is significantly improved using these computational techniques. We apply this new inverse modeling method to invert for random transmissivity fields in 2-D and a random hydraulic conductivity field in 3-D. Our algorithm is fast enough to solve for the distributed model parameters (transmissivity) in the model domain. The algorithm is coded in Julia and implemented in the MADS computational framework (http://mads.lanl.gov). By comparing with Levenberg-Marquardt methods using standard linear inversion techniques such as QR or SVD methods, our Levenberg-Marquardt method yields a speed-up ratio on the order of ∼101 to ∼102 in a multicore computational environment. Therefore, our new inverse modeling method is a powerful tool for characterizing subsurface heterogeneity for moderate to large-scale problems.

Blind source separation for groundwater pressure analysis based on nonnegative matrix factorization

The identification of the physical sources causing spatial and temporal fluctuations of aquifer water levels is a challenging, yet a very important hydrogeological task. The fluctuations can be caused by variations in natural and anthropogenic sources such as pumping, recharge, barometric pressures, etc. The source identification can be crucial for conceptualization of the hydrogeological conditions and characterization of aquifer properties. We propose a new computational framework for model-free inverse analysis of pressure transients based on Nonnegative Matrix Factorization (NMF) method for Blind Source Separation (BSS) coupled with k-means clustering algorithm, which we call NMFk. NMFk is capable of identifying a set of unique sources from a set of experimentally measured mixed signals, without any information about the sources, their transients, and the physical mechanisms and properties controlling the signal propagation through the subsurface flow medium. Our analysis only requires information about pressure transients at a number of observation points, m, where m≥r, and r is the number of unknown unique sources causing the observed fluctuations. We apply this new analysis on a data set from the Los Alamos National Laboratory site. We demonstrate that the sources identified by NMFk have real physical origins: barometric pressure and water-supply pumping effects. We also estimate the barometric pressure efficiency of the monitoring wells. The possible applications of the NMFk algorithm are not limited to hydrogeology problems; NMFk can be applied to any problem where temporal system behavior is observed at multiple locations and an unknown number of physical sources are causing these fluctuations.