A review of the literature on hydraulic testing and interpretive methods, particularly in low-permeability media, indicates a need for a comprehensive hydraulic testing interpretive capability. Physical limitations on boreholes, such as caving and erosion during continued drilling, as well as the high costs associated with deep-hole rigs and testing equipment, often necessitate testing under nonideal conditions with respect to antecedent pressures and temperatures. In these situations, which are common in the high-level nuclear waste programs throughout the world, the interpretive requirements include the ability to quantitatively account for thermally induced pressure responses and borehole pressure history (resulting in a time-dependent pressure profile around the borehole) as well as equipment compliance effects in low-permeability intervals. A numerical model was developed to provide the capability to handle these antecedent conditions. Sensitivity studies and practical applications are provided to illustrate the importance of thermal effects and antecedent pressure history. It is demonstrated theoretically and with examples from the Swiss (Nationale Genossenschaft fur die Lagerung radioaktiver Abfalle) regional hydrogeologic characterization program that pressure changes (expressed as hydraulic head) of the order of tens to hundreds of meters can result from 1° to 2°C temperature variations during shut-in (packer isolated) tests in low-permeability formations. Misinterpreted formation pressures and hydraulic conductivity can also result from inaccurate antecedent pressure history. Interpretation of representative formation properties and pressures requires that antecedent pressure information and test period temperature data be included as an integral part of the hydraulic test analyses.

Analysis and interpretation of borehole hydraulic tests in deep boreholes: Principles, model development, and applications

[1] This work focuses on different methods to generate confidence regions for nonlinear parameter identification problems. Three methods for confidence region estimation are considered: a linear approximation method, an F test method, and a log likelihood method. Each of these methods are applied to three case studies. One case study is a problem with synthetic data, and the other two case studies identify hydraulic parameters in groundwater flow problems based on experimental well test results. The confidence regions for each case study are analyzed and compared. Although the F test and log likelihood methods result in similar regions, there are differences between these regions and the regions generated by the linear approximation method for nonlinear problems. The differing results, capabilities, and drawbacks of all three methods are discussed.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2005WR004804

Confidence region estimation techniques for nonlinear regression in groundwater flow: Three case studies

This report presents interpretations of hydraulic tests conducted at 15 well locations in the vicinity of the Waste Isolation Pilot Plant (WIPP) in southeastern New Mexico between 1980 and 1996. The WIPP is a US Department of Energy (DOE) facility to demonstrate safe disposal of transuranic wastes arising form the nation`s defense programs. The WIPP repository lies within bedded halite of the Salado Formation, 2,155 ft below ground surface. The tests reported herein were, with two exceptions, conducted in the Culebra Dolomite member of the Rustler Formation, which overlies the Salado Formation. The remaining tests were conducted in the Magenta Member of the Rustler and in the overlying formation, the Dewey Lake Redbeds. This report completes the documentation of hydraulic-test interpretations used as input to the WIPP Compliance Certification Application (US DOE, 1996).

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Analysis of hydraulic tests of the Culebra and Magenta Dolomites and Dewey Lake Redbeds conducted at the Waste Isolation Pilot Plant Site

Thermodynamic systems analysis—I. A graph-theoretic approach

The calibration of model parameters is an important step in model development. Commonly, system output is measured, and model parameters are iteratively varied until the model out- put is a good match to the measured system output. Optimization algorithms are often used to identify the model parameter values. The presence of noise is difficult to avoid when physical processes are used to calibrate models due to measurement error, model structure error, and errors arising from numerical techniques and approximate solutions. Our study fo- cuses on the effects of noise in parameter identification optimization problems. We generate six test problems, including five perturbations of a smooth problem. A previously studied groundwater parameter identification problem serves as our seventh test problem. We test the Nelder–Mead Algorithm, a combination of the Nelder–Mead Algorithm and Simulated Annealing, and the Shuffled Complex Evolution Method on these test problems. Compar- ison of optimization results for these problems reveals the effects of noise on optimization performance, including an increase in fitness values and a decrease in the number of fit eval- uations. We vary the values of the internal algorithmic parameters to determine the effects of different values and present numerical results that indicate that changing the values of the algorithmic parameters can cause profound differences in optimization results for all three algorithms. A variation of the generally accepted parameter values for the Nelder–Mead Algorithm is recommended, and we determine that the Nelder–Mead/Simulated Annealing Hybrid and Shuffled Complex Evolution Method are too problem dependent for general recommendations for parameter values. Finally, we prove new convergence results for the Nelder–Mead/Simulated Annealing Hybrid in both smooth and noisy cases. Sandia is a multiprogram laboratory operated by Sandia Corporation for the Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000. This research is funded in part by WIPP programs administered by the U.S. Department of Energy.

On the Effects of Noise on Parameter Identification Optimization Problems

The Graph-Theoretical Field Model provides a unifying approach for developing numerical models of field and continuum problems. The methodology examines the field problem from the first stages of conceptualization without recourse to the governing differential equations of the field problem; this is accomplished by deriving discrete statements of the physical laws which govern the field behaviour. There are generally three laws, and these are modelled by the “cutset equations”, the “circuit equations”, and the “terminal equations”. In order to establish these three sets of equations it is expedient first to spatially discretize the field in a manner similar to the finite difference method and then to associate a linear graph (denoted as the field graph) with the spatial discretization. The concept of “through” and “across” variables, which underlies the cutset and circuit equations respectively, enables one to define the graph in an unambiguous manner such that each “edge” of the graph identifies a pair of complementary variables. From a knowledge of the constitutive properties and the boundary conditions of the field it is possible to associate terminal equations with sets of edges. Since the resulting sets of equations represent the field equations, these equations provide the basis for a complete (but approximate) solution to the field or continuum problem. In fact, this system approach uses a two part model: one for the components and another for the interconnection pattern of the components which renders the formulation procedures totally independent of the solution procedure. This paper presents the theoretical basis of the model and several graph-theoretic formulations for steady-state problems. Examples from heat conduction and small- deformation elasticity are included.

The graph-theoretic field model—I. Modelling and formulations

This paper is a sequel to a paper entitled “The Graph-Theoretic Field Model—I: Modelling and Formulations” (1). Herein, the Theory of Multi-Terminal Representations is applied to the Graph-Theoretic Field Model to provide mathematical models of finite elements. The element models are obtained solely from the algebraic building blocks of the Graph-Theoretic Field Model, without recourse to any functional mathematics. The theory of Multi-Terminal Representations is developed for both linear and non-linear problems. Examples of the application of the theory to one- and two-dimensional field problems are presented from heat conduction and electrostatics.

The graph-theoretic field model— II. application of multi-terminal representations to field problems

In this paper, we derive discrete versions of Green's identities (which appear in the study of potential field theory) as direct consequences of applying Tellegen's Theorem to the Graph—Theoretic Field Model (GTFM) of a field. The procedure herein is in marked contrast to the existing procedures where Green's Identities are derived from the Divergence Theorem by using some strictly mathematical operations. In particular, Green's third identity, which is the starting point formulation for the Boundary Integral Method, is singled out for special attention in terms of its discrete counterpart in the Graph—Theoretic Field Model. The first discrete identity is used to establish certain properties of solutions for the GTFM and a limiting process is applied to the three discrete identities to derive the traditional vector-calculus forms of Green's identities.

Discrete analogues of Green's identities through the graph—theoretic field model

Quasi-power functionals have been traditionally derived through the Method of Weighted Residuals (MWR) and used as the starting formulation for the finite element method. In this paper, we derive similar but more complete functionals for potential fields through a novel approach. We first form discrete quasi-power functions by applying Tellegen's theorem to the Graph-Theoretic Field Model (GTFM) and then obtain the continuous model quasi-power functionals by applying a limiting process to the discrete counterpart. The procedure herein is in marked contrast to the MWR whereby quasi-power functionals are derived by applying mathematical operations to the partial differential system. The procedure can be used with fields with anisotropy, non-linearities and coupled processes, although the example fields used within are linear and isotropic. This paper is a sequel to similar derivations for Green's identities and variational formulations.

Quasi-power functionals for potential fields from the Graph- Theoretic Field Model

Optimization of grid configurations decreases the discretization error in the numerical solution of field problems. In this paper, the theoretical and algorithmic aspects of optimizing configurations of grids used in equilibrium, potential-type field problems is presented. The criterion for establishing the optimum grid and the steps required to move to the optimum grid are given. Sensitivity network modelling is invoked to systematically form matrices required in the algorithm. One-dimensional examples are presented to illustrate the validity of the approach.

G.J. Savage

Papers

The graph-theoretic field model—I. Modelling and formulations

The graph-theoretic field model— II. application of multi-terminal representations to field problems

Discrete analogues of Green's identities through the graph—theoretic field model

Quasi-power functionals for potential fields from the Graph- Theoretic Field Model

Optimization of configurations of field problem grids using network models