An investigation of numerical grid effects in parameter estimation

Modern ground water characterization and remediation projects routinely require calibration and inverse analysis of large three-dimensional numerical models of complex hydrogeological systems. Hydrogeologic complexity can be prompted by various aquifer characteristics including complicated spatial hydrostratigraphy and aquifer recharge from infiltration through an unsaturated zone. To keep the numerical models computationally efficient, compromises are frequently made in the model development, particularly, about resolution of the computational grid and numerical representation of the governing flow equation. The compromise is required so that the model can be used in calibration, parameter estimation, performance assessment, and analysis of sensitivity and uncertainty in model predictions. However, grid properties and resolution as well as applied computational schemes can have large effects on forward-model predictions and on inverse parameter estimates. We investigate these effects for a series of one- and two-dimensional synthetic cases representing saturated and variably saturated flow problems. We show that "conformable" grids, despite neglecting terms in the numerical formulation, can lead to accurate solutions of problems with complex hydrostratigraphy. Our analysis also demonstrates that, despite slower computer run times and higher memory requirements for a given problem size, the control volume finite-element method showed an advantage over finite-difference techniques in accuracy of parameter estimation for a given grid resolution for most of the test problems.     

Coupling basin- and site-scale inverse models of the Española aquifer

Large-scale models are frequently used to estimate fluxes to small-scale models. The uncertainty associated with these flux estimates, however, is rarely addressed. We present a case study from the Espa?ola Basin, northern New Mexico, where we use a basin-scale model coupled with a high-resolution, nested site-scale model. Both models are three-dimensional and are analyzed by codes FEHM and PEST. Using constrained nonlinear optimization, we examine the effect of parameter uncertainty in the basin-scale model on the nonlinear confidence limits of predicted fluxes to the site-scale model. We find that some of the fluxes are very well constrained, while for others there is fairly large uncertainty. Site-scale transport simulation results, however, are relatively insensitive to the estimated uncertainty in the fluxes. We also compare parameter estimates obtained by the basin- and site-scale inverse models. Differences in the model grid resolution (scale of parameter estimation) result in differing delineation of hydrostratigraphic units, so the two models produce different estimates for some units. The effect is similar to the observed scale effect in medium properties owing to differences in tested volume. More important, estimation uncertainty of model parameters is quite different at the two scales. Overall, the basin inverse model resulted in significantly lower estimates of uncertainty, because of the larger calibration dataset available. This suggests that the basin-scale model contributes not only important boundary condition information but also improved parameter identification for some units. Our results demonstrate that caution is warranted when applying parameter estimates inferred from a large-scale model to small-scale simulations, and vice versa.     

Stochastic inverse method for estimation of geostatistical representation of hydrogeologic stratigraphy using borehole logs and pressure observations

An approach is presented for identifying statistical characteristics of stratigraphies from borehole and hydraulic data. The approach employs a Markov-chain based geostatistical framework in a stochastic inversion. Borehole data provide information on the stratigraphy while pressure and flux data provide information on the hydraulic performance of the medium. The use of Markov-chain geostatistics as opposed to covariance-based geostatistics can provide a more easily interpreted model geologically and geometrically. The approach hinges on the use of mean facies lengths (negative inverse auto-transition rates) and mean transition lengths (inverse cross-transition rates) as adjustable parameters in the stochastic inversion. Along with an unconstrained Markov-chain model, simplifying constraints to the Markov-chain model, including (1) proportionally-random and (2) symmetric spatial correlations, are evaluated in the stochastic inversion. Sensitivity analyses indicate that the simplifying constraints can facilitate the inversion at the cost of spatial correlation model generality. Inverse analyses demonstrate the feasibility of this approach, indicating that despite some low parameter sensitivities, all adjustable parameters do converge for a sufficient number of ensemble realizations towards their “true” values. This paper extends the approach presented in Harp et al. (doi:, 2008) to (1) statistically characterize the hydraulic response of a geostatistical model, thereby incorporating an uncertainty analysis directly in the inverse method, (2) demonstrate that a gradient-based optimization strategy is sufficient, thereby providing relative computational efficiency compared to global optimization strategies, (3) demonstrate that the approach can be extended to a 3-D analysis, and (4) introduce the use of mean facies lengths and mean transition lengths as adjustable parameters in a geostatistical inversion, thereby allowing the approach to be extended to greater than two category Markov-chain models.     

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.     

Numerical inverse interpretation of single-hole pneumatic tests in unsaturated fractured tuff

A numerical inverse method was used to interpret simultaneously multirate injection and recovery data from single-hole pneumatic tests in unsaturated fractured tuff at the Apache Leap Research Site near Superior, Arizona. Our model represents faithfully the three-dimensional geometry of boreholes at the site, and accounts directly for their storage and conductance properties by treating them as high-permeability and high-porosity cylinders of finite length and radius. It solves the airflow equations in their original nonlinear form and yields information about air permeability, air-filled porosity and dimensionless borehole storage coefficient. Some of this is difficult to accomplish with analytical type-curves. Air permeability values obtained by our inverse method agree well with those obtained by steady-state and type-curve analyses.     

From picket fences to iron gates: suburbanization and gated communities in Phoenix,Las Vegas and Seattle

Suburbanization has been a prominent urban process in the United States since the World War II. It has transformed American cities in profound ways in every single aspect of urban development; from population and wealth distributions, through political organization and affiliations, to the built environment. This paper investigates the link between gated communities and the process of suburbanization in the context of socio-economic inequality. It has been shown time and again in the scholarly literature on suburbanization, that suburban neighborhoods in American cities have been traditionally more affluent and less diverse than central cities. The research on gated communities in the US also shows that they are, on average, more affluent compared to other communities in terms of family income and housing values. Are gated communities then simply a new form of suburban communities? Is the gated community in fact a suburban community with the added element of security features? The paper investigates these questions based on segregation and spatial analyses. The research contributes to the long line of studies on suburbanization, gating and the larger issues of urban inequality.     

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.     

Kinetic aspects of sector zoning in arsenopyrite: A case study

Summary A simple quartz-arsenopyrite vein assemblage formed at temperatures of 280–200°C occurs in the Madan lead-zinc ore district, Bulgaria. Spear-shaped arsenopyrite is bounded by {101}, {010} and {210}; compositionally it is pure FeAs_1-xS_1+X of variable x. The BSE image of an oriented (001) section reveals a pronounced sector zoning defined solely by differences in the S/As ratio. The boundary between the (210) and (010) growth sectors reflects the changes of their growth rates V⁰¹⁰ and V²¹⁰ in perfect correlation with zonal S/As oscillations. The V°10/V²¹⁰ ratios determined in 20 zones range from 0.1 to 2.4, and correspond to S variations from 39 to 34 at.% derived from 7 microprobe point analyses. The S content in the (010) sector is on average 1.5 at.% lower than in the (210) one, the difference decreasing from about 2.5 to less than 0.33 at.% with the decreasing V⁰¹⁰/V²¹⁰. The rate/composition relationship obeys the following rules: the higher the S/As the lower the V⁰¹⁰/V²¹⁰, and the slower the (010) growth. The phenomenon is interpreted in terms of crystal growth theory of a solid solution analyzing the arsenopyrite growth history on hand specimen, microscopic and structural scales. Sector zoning in this particular case is explained by processes taking place very close to the growing crystal surface which are superimposed on those produced by larger-scale variations in the hydrothermal environment.

---

Kinetische Aspekte des sektoriellen Zonenbaues von Arsenkies: Untersuchung eines Wachstumsphdnomens

Zusammenfassung Eine gewöhnliche Quarz-Arsenkies Paragenese im Blei-Zink Erzgebiet von Madan, Bulgarien, bildete sich im Temperatur-Intervall von 280-200°C. Speerförmiger Arsenkies ist begrenzt von den Flächenformen 11011, {010} und {210}, wobei das Mineral der Zusammensetzung FeAs_1-xS_1+X mit variierendem x entspricht. Die Rückstreuelektronenabbildung eines (001)-orientierten Querschnittes zeigt einen deutlichen Sektorzon-enbau, der ausschließlich durch Unterschiede im S/As-Verhaltnis bestimmt wird. Die Grenze zwischen den Wachstumssektoren (210) und (010) spiegelt Veränderungen in ihren Wachstumsgeschwindigkeiten V⁰¹⁰ und V²¹⁰ wieder, die mit den zonenkorrelierten S/As Schwankungen variieren. Die Verhdltnisse V⁰¹⁰/V^21O wurden in 20 Zonen des Kristalls bestimmt; she variieren zwischen 0.1 und 2.4 und entsprechen einer Variation von 39 bis 34 At.% S, die durch 7 Elektronenstrahlmikrosondenanalysen bestimmt wurden. Der S-Gehalt im Sektor (010) ist durchschnittlich mit 1.5 At.% niedriger als im Sektor (210); der entsprechende Unterschied nimmt von ungefdhr 2.5 bis 0.33 At.% mit der Verringerung von V⁰¹⁰/V² ab. Die Beziehung Wachstumsgeschwindigkeit/ Element unterliegt der Regel: je hoher das Verhaltnis S/As; desto niedriger ist V⁰¹⁰/V²¹⁰ und desto langsamer ist das Wachstum von (010). Diese Erscheinung wird im Rahmen der Kristallwachstumstheorie einer festen Lösung gedeutet, wobei die Wachstumsgeschichte des Arsenkieses makroskopisch, mikroskopisch und strukturell analysiert wurde. Auf these Weise wird der Sektorzonenbau als ein Resultat von Prozessen interpretiert, die ganz nah der wachsenden Kristalloberfldche ablaufen und die die Veränderungen des hydrothermalen Milieus überlagern.

     

Maximum likelihood Bayesian averaging of airflow models in unsaturated fractured tuff using Occam and variance windows

We use log permeability and porosity data obtained from single-hole pneumatic packer tests in six boreholes drilled into unsaturated fractured tuff near Superior, Arizona, to postulate, calibrate and compare five alternative variogram models (exponential, exponential with linear drift, power, truncated power based on exponential modes, and truncated power based on Gaussian modes) of these parameters based on four model selection criteria (AIC, AICc, BIC and KIC). Relying primarily on KIC and cross-validation we select the first three of these variogram models and use them to parameterize log air permeability and porosity across the site via kriging in terms of their values at selected pilot points and at some single-hole measurement locations. For each of the three variogram models we estimate log air permeabilities and porosities at the pilot points by calibrating a finite volume pressure simulator against two cross-hole pressure data sets from sixteen boreholes at the site. The traditional Occam’s window approach in conjunction with AIC, AICc, BIC and KIC assigns a posterior probability of nearly 1 to the power model. A recently proposed variance window approach does the same when applied in conjunction with AIC, AICc, BIC but spreads the posterior probability more evenly among the three models when used in conjunction with KIC. We compare the abilities of individual models and MLBMA, based on both Occam and variance windows, to predict space–time pressure variations observed during two cross-hole tests other than those employed for calibration. Individual models with the largest posterior probabilities turned out to be the worst or second worst predictors of pressure in both validation cases. Some individual models predicted pressures more accurately than did MLBMA. MLBMA was far superior to any of the individual models in one validation test and second to last in the other validation test in terms of predictive coverage and log scores.