El problema inverso de la hidrología subterránea: estado del arte y método de solución

El problema inverso de la hidrología subterránea, que consiste en la obtención de los parámetros hidraúlicos de un acuífero a partir de datos de niveles piezométricos, ha sido objeto de un intenso proceso de investigación en los últimos años. En la primera parte de este artículo, se pasa revista a los trabajos más relevantes de dicho proceso. Ello da pie, en la segunda parte, a proponer un método, basado en la teoría máxima verosimilitud, que permite obtener transmisividades, coeficientes de almacenamiento y goteo en recarga y caudales y niveles en los contornos, con datos en régimen transitorio y/o estacionario en dominios bi- o cuasitri-dimensionales. El metodo se basa en la teoría del estado adjunto, lo cual se traduce en un ahorro considerable de memoria y tiempo de ordenador. La eficacia del algoritmo se muestra con un ejemplo.Peer Reviewe

Inverse modeling of unsaturated flow using clusters of soil texture and pedotransfer functions

Characterization of heterogeneous soil hydraulic parameters of deep vadose zones is often difficult and expensive, making it necessary to rely on other sources of information. Pedotransfer functions (PTFs) based on soil texture data constitute a simple alternative to inverse hydraulic parameter estimation, but their accuracy is often modest. Inverse modeling entails a compromise between detailed description of subsurface heterogeneity and the need to restrict the number of parameters. We propose two methods of parameterizing vadose zone hydraulic properties using a combination of k-means clustering of kriged soil texture data, PTFs, and model inversion. One approach entails homogeneous and the other heterogeneous clusters. Clusters may include subdomains of the computational grid that need not be contiguous in space. The first approach homogenizes within-cluster variability into initial hydraulic parameter estimates that are subsequently optimized by inversion. The second approach maintains heterogeneity through multiplication of each spatially varying initial hydraulic parameter by a scale factor, estimated a posteriori through inversion. This allows preserving heterogeneity without introducing a large number of adjustable parameters. We use each approach to simulate a 95 day infiltration experiment in unsaturated layered sediments at a semiarid site near Phoenix, Arizona, over an area of 50 × 50 m2 down to a depth of 14.5 m. Results show that both clustering approaches improve simulated moisture contents considerably in comparison to those based solely on PTF estimates. Our calibrated models are validated against data from a subsequent 295 day infiltration experiment at the site

Nonlocal and localized analyses of conditional mean transient flow in bounded, randomly heterogeneous porous media

We consider the numerical prediction of transient flow in bounded, randomly heterogeneous porous media driven by random sources, initial heads, and boundary conditions without resorting to Monte Carlo simulation. After applying the Laplace transform to the governing stochastic flow equations, we derive exact nonlocal (integro‐differential) equations for the mean and variance‐covariance of transformed head and flux, conditioned on measured values of log conductivity Y = ln K. Approximating these conditional moment equations recursively to second order in the standard deviation σY of Y, we solve them by finite elements for superimposed mean‐uniform and convergent flows in a two‐dimensional domain. An alternative conditional mean solution is obtained through localization of the exact moment expressions. The nonlocal and localized solutions are obtained using a highly efficient parallel algorithm and inverted numerically back into the time domain. A comparison with Monte Carlo simulations demonstrates that the moment solutions are remarkably accurate for strongly heterogeneous media with σY2 as large as 4. The nonlocal solution is only slightly more accurate than the much simpler localized solution, but the latter does not yield information about predictive uncertainty. The accuracy of each solution improves markedly with conditioning. A preliminary comparison of computational efficiency suggests that both the nonlocal and localized solutions for mean head and its variance require significantly less computer time than is required for Monte Carlo statistics to stabilize when the same direct matrix solver is used for all three (we do not presently know how using iterative solvers would have affected this conclusion). This is true whether the Laplace inversion and Monte Carlo simulations are conducted sequentially or in parallel on multiple processors and regardless of problem size. The underlying exact and recursive moment equations, as well as the proposed computational algorithm, are valid in both two and three dimensions; only the numerical implementation of our algorithm is two‐dimensional

Field determination of the hydraulic properties of leaky multiple aquifer systems

Abstract. A new field method is proposed for determining the hydraulic properties of aquifers and aquitards in leaky systems. Conventional methods of analyzing leaky aquifers usually rely on drawdown data from the pumped aquifer alone. Such an approach is not sufficient to characterize a leaky system; our new method requires observation wells to be placed not only in the aquifer being pumped but also in the confining layers (aquitards) above and/or below. The ratio of the drawdown in the aquitard to that measured in the aquifer a t the same time and the same radial distance from the pumping well can be used to evaluate the hydraulic properties of the aquitard. The new method is supported by theory and has been applied to the coastal groundwater basin of Oxnard, California. The field results are in good agreement with laboratory measurements. Traditionally, groundwater hydrologists have tended to focus their attention on the more permeable aquifer layers of a groundwater basin in developing water supplies. However, sedimentary groundwater basins usually consist of a series of aquifers separated by confining layers of relatively low permeability, which may act as conduits for the vertical migration of water from one aquifer to another. Since finegrained sediments often tend to be much more compressible than associated coarse-grained aquifer materials, they also can release large quantities of water from storage and thereby increase t,he supply. available t o the aquifer. The combined effects of these phenomena are known as leakage. Usually, when the effects of leakage can be detected by observing drawdown in the aquifer being pumped, the confining beds are called 'aquitards,' and the aquifer is referred t o as being 'leaky.' When such effects cannot be easily detected in the aquifer, the confining beds are called 'aquicludes,' and the aquifer is termed 'slightly leaky' [Neuman and Witherspoon, 19681. Aquitards play an important role in the hydrology of multiple aquifer systems, and we shall mention here only a few examples. Although groundwater recharge is often believed to occur in areas of aquifer outcrops, Gill [I9691 has recently reported that substantial amounts of water produced from the Potomac-RaritanMagothy aquifer system are coming through the aquitards. Earlier, Wdton [I9651 had shown how the Maquoketa formation in Illinois, which is essentially a shale bed, serves as an effective transmitter of water between aquifers. Land subsidence in the San Joaquin Valley and other areas in California has been shown to be associated with water withdrawal from multiple aquifer systems and is generally attributed to the resulting compaction of fine-grained aquitard sediments [Poland and Davis, 19691. Similar situations exist in Venice, Japan, and other parts of the world. For the past 20 years, aquifers at depths below 500 feet have been used for storing natural gas in the United States and Europe. Where the properties of the aquitards were not properly investigated, the gas industry has on occasion witnessed the spectacular and dangerous effects of gas leakage. The storage of other fluids

New scaling model for variables and increments with heavy-tailed distributions

Many hydrological (as well as diverse earth, environmental, ecological, biological, physical, social, financial and other) variables, Y, exhibit frequency distributions that are difficult to reconcile with those of their spatial or temporal increments, ΔY. Whereas distributions of Y (or its logarithm) are at times slightly asymmetric with relatively mild peaks and tails, those of ΔY tend to be symmetric with peaks that grow sharper, and tails that become heavier, as the separation distance (lag) between pairs of Y values decreases. No statistical model known to us captures these behaviors of Y and ΔY in a unified and consistent manner. We propose a new, generalized sub-Gaussian model that does so. We derive analytical expressions for probability distribution functions (pdfs) of Y and ΔY as well as corresponding lead statistical moments. In our model the peak and tails of the ΔY pdf scale with lag in line with observed behavior. The model allows one to estimate, accurately and efficiently, all relevant parameters by analyzing jointly sample moments of Y and ΔY. We illustrate key features of our new model and method of inference on synthetically generated samples and neutron porosity data from a deep borehole

Recent advances in scalable non-Gaussian geostatistics: The generalized sub-Gaussian model

Geostatistical analysis has been introduced over half a century ago to allow quantifying seemingly random spatial variations in earth quantities such as rock mineral content or permeability. The traditional approach has been to view such quantities as multivariate Gaussian random functions characterized by one or a few well-defined spatial correlation scales. There is, however, mounting evidence that many spatially varying quantities exhibit non-Gaussian behavior over a multiplicity of scales. The purpose of this minireview is not to paint a broad picture of the subject and its treatment in the literature. Instead, we focus on very recent advances in the recognition and analysis of this ubiquitous phenomenon, which transcends hydrology and the Earth sciences, brought about largely by our own work. In particular, we use porosity data from a deep borehole to illustrate typical aspects of such scalable non-Gaussian behavior, describe a very recent theoretical model that (for the first time) captures all these behavioral aspects in a comprehensive manner, show how this allows generating random realizations of the quantity conditional on sampled values, point toward ways of incorporating scalable non-Gaussian behavior in hydrologic analysis, highlight the significance of doing so, and list open questions requiring further research

Multimodel Bayesian analysis of groundwater data worth

open4We explore the way in which uncertain descriptions of aquifer heterogeneity and groundwater flow impact one’s ability to assess the worth of collecting additional data. We do so on the basis of Maximum Likelihood Bayesian Model Averaging (MLBMA) by accounting jointly for uncertainties in geostatistical and flow model structures and parameter (hydraulic conductivity) as well as system state (hydraulic head) estimates, given uncertain measurements of one or both variables. Previous description of our approach was limited to geostatistical models based solely on hydraulic conductivity data. Here we implement the approach on a synthetic example of steady state flow in a two-dimensional random log hydraulic conductivity field with and without recharge by embedding an inverse stochastic moment solution of groundwater flow in MLBMA. A moment-equations-based geostatistical inversion method is utilized to circumvent the need for computationally expensive numerical Monte Carlo simulations. The approach is compatible with either deterministic or stochastic flow models and consistent with modern statistical methods of parameter estimation, admitting but not requiring prior information about the parameters. It allows but does not require approximating lead predictive statistical moments of system states by linearization while updating model posterior probabilities and parameter estimates on the basis of potential new data both before and after such data are actually collected.openLiang Xue;Dongxiao Zhang;Alberto Guadagnini;Shlomo P. NeumanLiang, Xue; Dongxiao, Zhang; Guadagnini, Alberto; Shlomo P., Neuma

Multimodel Bayesian analysis of dataworth applied to unsaturated fractured tuffs

a b s t r a c t To manage water resource and environmental systems effectively requires suitable data. The worth of collecting such data depends on their potential benefit and cost, including the expected cost (risk) of failing to take an appropriate decision. Evaluating this risk calls for a probabilistic approach to data-worth assessment. Recently we [39] developed a multimodel approach to optimum value-of-information or data-worth analysis based on model averaging within a maximum likelihood Bayesian framework. Adopting a two-dimensional synthetic example, we implemented our approach using Monte Carlo (MC) simulations with and without lead order approximations, finding that the former approach was almost equally accurate but computationally more efficient. Here we apply our methodology to pneumatic permeability data from vertical and inclined boreholes drilled into unsaturated fractured tuff near Superior, Arizona. In an attempt to improve computational efficiency, we introduce three new approximations that require less computational effort and compare results with those obtained by the original Monte Carlo method. The first approximation disregards uncertainty in model parameter estimates, the second does so for estimates of potential new data, and the third disregards both uncertainties. We find that only the first approximation yields reliable quantitative assessments of reductions in predictive uncertainty brought about by the collection of new data. We conclude that, whereas parameter uncertainty may sometimes be disregarded for purposes of analyzing data worth, the same does not generally apply to uncertainty in estimates of potential new data

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

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

Combined Estimation of Hydrogeologic Conceptual Model and Parameter Uncertainty

The objective of the research described in this report is the development and application of a methodology for comprehensively assessing the hydrogeologic uncertainties involved in dose assessment, including uncertainties associated with conceptual models, parameters, and scenarios. This report describes and applies a statistical method to quantitatively estimate the combined uncertainty in model predictions arising from conceptual model and parameter uncertainties. The method relies on model averaging to combine the predictions of a set of alternative models. Implementation is driven by the available data. When there is minimal site-specific data the method can be carried out with prior parameter estimates based on generic data and subjective prior model probabilities. For sites with observations of system behavior (and optionally data characterizing model parameters), the method uses model calibration to update the prior parameter estimates and model probabilities based on the correspondence between model predictions and site observations. The set of model alternatives can contain both simplified and complex models, with the requirement that all models be based on the same set of data. The method was applied to the geostatistical modeling of air permeability at a fractured rock site. Seven alternative variogram models of log air permeability were considered to represent data from single-hole pneumatic injection tests in six boreholes at the site. Unbiased maximum likelihood estimates of variogram and drift parameters were obtained for each model. Standard information criteria provided an ambiguous ranking of the models, which would not justify selecting one of them and discarding all others as is commonly done in practice. Instead, some of the models were eliminated based on their negligibly small updated probabilities and the rest were used to project the measured log permeabilities by kriging onto a rock volume containing the six boreholes. These four projections, and associated kriging variances, were averaged using the posterior model probabilities as weights. Finally, cross-validation was conducted by eliminating from consideration all data from one borehole at a time, repeating the above process, and comparing the predictive capability of the model-averaged result with that of each individual model. Using two quantitative measures of comparison, the model-averaged result was superior to any individual geostatistical model of log permeability considered