Determining the effective permeability (keff) of geological formations has broad applications to site remediation, aquifer discharge or recharge, hydrocarbon production, and enhanced oil recovery. The objectives of this study are: (1) to explore an approach to estimating keff at the reservoir scale using the critical path analysis (CPA), (2) to evaluate the accuracy of this new approach by comparing the estimated keff to the numerically simulated effective permeability, and (3) to compare the performance of CPA estimates of keff with estimates by three other models i.e., perturbation theory (PT), effective-medium approximation (EMA), and renormalization group theory (RGT). We construct two- and three-dimensional random (uncorrelated) geologic formations based on permeability measurements from the Borden site and assume that the permeability distribution conforms to the log-normal probability density function over a wide range of means and standard deviations. Comparing keff estimated via CPA to keff values derived from numerical flow simulations indicates that CPA provides accurate estimations in both two and three dimensions over a wide range of heterogeneity levels, similar to RGT. Inter-model comparisons show that although PT and EMA provide reasonable keff estimations in rather homogeneous formations, they substantially overestimate the effective permeability in highly heterogeneous formations.

Light- and elevated temperature-induced degradation (LeTID) is assumed to be triggered by the hydrogen content in the crystalline silicon bulk. This article investigates differently thick atomic layer-deposited aluminum oxide (AlOx) layers acting as diffusion barrier for hydrogen originating from a hydrogen-rich silicon nitride (SiNy:H) layer. We demonstrate that the extent of LeTID can be significantly reduced by adjusting the AlOx layer thickness up to 25 nm. To directly measure the diffusing species, a deuterium-rich SiNy:D layer is deposited and the deuterium content is measured in an amorphous Si layer at the back side of the wafer via secondary ion mass spectrometry. Thus, a diffusion length of deuterium in the AlOx layer of $({3.8 \pm 1.6}){\boldsymbol{\ }}\;{\text{nm}}$ is determined at a firing temperature of $({743 \pm 2})^\circ {\rm{C}}$. These results are not only a contribution to determine the LeTID formation dynamics, but also can be used to control LeTID in silicon wafers and solar cells.

A method for quantifying dopant-hydrogen pairs and their formation dynamics in crystalline silicon by means of directly contacted resistance measurements is presented and exemplarily validated. The method can also be applied in-situ in the temperature range where dopant-hydrogen pair formation occurs. Furthermore, the influence of different confounding factors such as a faulty assumption of doping level, unnoticed temperature variations, and unwanted illumination is quantified. It is concluded that the detection limit of dopant-hydrogen pairs of the presented method is most likely limited by unnoticed temperature fluctuations and scales with the actual dopant concentration. For crystalline silicon doped with 10 16 cm -3 as it is usually used for photovoltaic applications, the detection limit is found to be below 10 13 cm -3 .

This study deals with the dynamics of the formation and dissociation of boron–hydrogen (BH) pairs in crystalline silicon during a rapid high‐temperature treatment and subsequent dark annealing between 200 and 300 °C. Highly accurate resistivity measurements are used to detect BH pairs in chemically polished B‐doped float‐zone silicon. It is found that an unexpecteded high amount of hydrogen is present in the as‐purchased wafers. Hydrogen is initially mostly paired to boron but can be dissolved by a short high‐temperature firing step. If a firing step (530 °C) is applied to bare, unpassivated Si wafers, most of the initial BH pairs are dissolved, and hydrogen dimers () form. With increasing peak temperature, an increasing amount of hydrogen leaves the system, while the proportion of BH increases. Additional hydrogen can be introduced by firing a wafer passivated with plasma‐enhanced chemical vapor deposition (PECVD) H. A three‐state model shows a good agreement with the measured data for both bare and coated samples as well as for different annealing temperatures. With increasing dark annealing temperatures, the BH dynamics accelerates, whereas the maximum BH concentration reached decreases. For temperatures above 280 °C, significant changes in the reaction dynamics are observed.

We develop a phenomenological model for the effective conductivity of porous media that consist of two distinct materials characterized by different values of hydraulic conductivity. Our focus is two-fold: first, to relate the terms and parameters of macroscale conductivity to the spatial variability of local flow and conductivity observed in computational experiments, and second to develop a reduced order model for effective conductivity based on those observations. Darcys law is assumed to hold at local (mesoscopic) and large (macroscopic) scales. At the mesoscale a composite medium is a configuration of irregular subdomains consisting of two different materials. The resulting heterogeneities force fluid to flow along irregular paths, and this produces spatial variability in the magnitude and the orientation of the Darcy flux. This variability, along with the macroscopic effective conductivity, depends on the proportion of the total volume allocated to each material, the ratio of the two conductivity values, and the spatial connectivity of material subvolumes. Computational experiments indicate that the effects of heterogeneity are most pronounced when the two conductivity values are very different, and when the volume fraction of the more conducive material is near a percolation threshold. The percolation threshold is the critical volume fraction of the more conducive material at which the medium is no longer traversed by connected paths through that material. The percolation threshold determines the existence of three regimes in effective conductivity, two where the effective conductivity obeys power laws and is dominated by one of the materials, and a third intermediate regime that interpolates between the power laws.

Drylands cover over 40% of the global land area and are home to more than 2 billion humans. Here, we use the terrestrial water storage (TWS) anomaly data derived from GRACE satellites to assess water storage changes globally and find that drylands lost similar to 15.9 +/- 9.1 mm of water between April 2002 and January 2017. The TWS trends are more significant and apparent in dry regions than in humid regions. The decrease in TWS occurred mainly in hyperarid and arid regions. Exact causes to the observed declines in TWS remain elusive due to anthropogenic water withdrawals, atmospheric demand (potential evapotranspiration, PET) in contrast to supply (precipitation, P) caused by the warming, and terrestrial ecohydrological responses. Therefore, we use a process-based model forced by climate data to interpret the causes over three selected dryland regions showing the strongest drying trends. We find that the modeled TWS without considering anthropogenic water withdrawals explains most of the declining GRACE TWS over the southwestern North America (NA) and Middle East but underestimates the drying trend over North China. This suggests that TWS declines in the southwestern NA and the Middle East were primarily driven by the contrast between atmospheric demand and supply, whereas anthropogenic water withdrawals may have played a relatively more dominant role in TWS declines over North China. Additional model experiments indicate that terrestrial ecohydrological processes that help extract deep substrate water are critical for providing water supply additional to precipitation to sustain ET in the drying drylands at decadal scales.

Soil surface roughness (SSR) is widely recognized as an important factor influencing water erosion processes. On semiarid hillslopes with stony soils, rock fragments accumulate as the result of preferential erosion of fine materials, often creating a rough, rocky surface. A series of rainfall events were simulated on a stony plot (2x6.1 m) at three slope gradients (5 %, 12 %, and 20 %) and rock cover was measured. Surface elevations were sampled by terrestrial LiDAR at high resolutions. Roughness indices, including random roughness (RR), fractal dimension, crossover length, and generalized fractal dimension, were calculated from LiDAR points directly. Results showed: 1) SSR displayed an increasing trend as the rainfall simulation proceeded for all three slope treatments; 2) the steeper slope developed greater surface roughness; and 3) both the increase of surficial exposed rocks and the formation of erosional features, e.g., rills and depressions, contributed to the spatiotemporal variations of SSR. Results also showed that the fractal dimension was not a good indicator of soil surface roughness, but rather was an index of the form of the surface. Crossover length was a measure of roughness at a scale of a few millimeters, while random roughness was a measure of elevation variations on the scale of the length of the transect measured, and thus encompassed larger morphological features including rills. We also established a new method for multifractal analysis that characterized the heterogeneity of soil surface roughness. These results improve our understanding of the evolution of semiarid stony hillslopes and the dynamic feedback mechanism between erosion, surface morphology and hydraulics.

Light and elevated temperature induced degradation (LeTID) is observed for multicrystalline (mc) Si passivated emitter and rear cell (PERC) solar cells, strongly limiting solar cell parameters under operation conditions. In this contribution, we investigate the effect of surface passivation layer being present during the firing step based on lifetime samples. The LeTID effect is only observed if the surface passivation layer is present during the firing step. Samples without firing step show no LeTID. A re-passivation of the surface significantly changes the LeTID effect, showing that the whole sample treatment, temperature load and hydrogen content of a sample has to be taken into account investigating and evaluating LeTID.

Recent progress in the electronic quality of high-performance (HP) multicrystalline silicon material is reported with measurements and modeling performed at various institutions and research groups. It is shown that recent progress has been made in the fabrication at Trina Solar mainly by improving the high excess carrier lifetimes tau due to a considerable reduction of mid-gap states. However, the high lifetimes in the wafers are still reduced by interstitial iron by a factor of about 10 at maximum power point (mpp) conditions compared to mono-crystalline Cz wafers of equivalent resistivity. The low lifetime areas of the wafers seem to be limited by precipitates, most likely Cu. Through simulations, it appears that dislocations reduce cell efficiency by about 0.25% absolute. The best predictors for PERC cell efficiency from ingot metrology are a combination of mean lifetime and dislocation density because dislocations cannot be improved considerably by gettering during cell processing, while lifetime-limiting impurities are gettered well. In future, the material may limit cell efficiency above about 22.5% if the concentrations of Fe and Cu remain above 10(10) and 10(13)cm(-3), respectively, and if dislocations are not reduced further.

We analyze scale-dependent statistics of correlated random hydrogeological variables and their extremes using neutron porosity data from six deep boreholes, in three diverse depositional environments, as example. We show that key statistics of porosity increments behave and scale in manners typical of many earth and environmental (as well as other) variables. These scaling behaviors include a tendency of increments to have symmetric, non-Gaussian frequency distributions characterized by heavy tails that decay with separation distance or lag; power-law scaling of sample structure functions (statistical moments of absolute increments) in midranges of lags; linear relationships between log structure functions of successive orders at all lags, known as extended self-similarity or ESS; and nonlinear scaling of structure function power-law exponents with function order, a phenomenon commonly attributed in the literature to multifractals. Elsewhere we proposed, explored and demonstrated a new method of geostatistical inference that captures all of these phenomena within a unified theoretical framework. The framework views data as samples from random fields constituting scale mixtures of truncated (monofractal) fractional Brownian motion (tfBm) or fractional Gaussian noise (tfGn). Important questions not addressed in previous studies concern the distribution and statistical scaling of extreme incremental values. Of special interest in hydrology (and many other areas) are statistics of absolute increments exceeding given thresholds, known as peaks over threshold or POTs. In this paper we explore the statistical scaling of data and, for the first time, corresponding POTs associated with samples from scale mixtures of tfBm or tfGn. We demonstrate that porosity data we analyze possess properties of such samples and thus follow the theory we proposed. The porosity data are of additional value in revealing a remarkable cross-over from one scaling regime to another at certain lags. The phenomena we uncover are of key importance for the analysis of fluid flow and solute as well as particulate transport in complex hydrogeologic environments.

We analyze the statistical scaling of structural attributes of virtual porous microstructures that are stochastically generated by thresholding Gaussian random fields. Characterization of the extent at which randomly generated pore spaces can be considered as representative of a particular rock sample depends on the metrics employed to compare the virtual sample against its physical counterpart. Typically, comparisons against features and/patterns of geometric observables, e.g., porosity and specific surface area, flow-related macroscopic parameters, e.g., permeability, or autocorrelation functions are used to assess the representativeness of a virtual sample, and thereby the quality of the generation method. Here, we rely on manifestations of statistical scaling of geometric observables which were recently observed in real millimeter scale rock samples [13] as additional relevant metrics by which to characterize a virtual sample. We explore the statistical scaling of two geometric observables, namely porosity (ϕ) and specific surface area (SSA), of porous microstructures generated using the method of Smolarkiewicz and Winter [42] and Hyman and Winter [22]. Our results suggest that the method can produce virtual pore space samples displaying the symptoms of statistical scaling observed in real rock samples. Order q sample structure functions (statistical moments of absolute increments) of ϕ and SSA scale as a power of the separation distance (lag) over a range of lags, and extended self-similarity (linear relationship between log structure functions of successive orders) appears to be an intrinsic property of the generated media. The width of the range of lags where power-law scaling is observed and the Hurst coefficient associated with the variables we consider can be controlled by the generation parameters of the method.

Computational experiments are performed to evaluate the effects of locally heterogeneous conductivity fields on regional exchanges of water between stream and aquifer systems in the Middle Heihe River basin (MHRB) of northwestern China. The effects are found to be nonlinear in the sense that simulated discharges from aquifers to streams are systematically lower than discharges produced by a base model parameterized with relatively coarse effective conductivity. A similar, but weaker, effect is observed for stream leakage. The study is organized around three hypotheses: (H1) small-scale spatial variations of conductivity significantly affect regional exchanges of water between streams and aquifers in river basins, (H2) aggregating small-scale heterogeneities into regional effective parameters systematically biases estimates of stream–aquifer exchanges, and (H3) the biases result from slow paths in groundwater flow that emerge due to small-scale heterogeneities. The hypotheses are evaluated by comparing stream–aquifer fluxes produced by the base model to fluxes simulated using realizations of the MHRB characterized by local (grid-scale) heterogeneity. Levels of local heterogeneity are manipulated as control variables by adjusting coefficients of variation. All models are implemented using the MODFLOW (Modular Three-dimensional Finite-difference Groundwater Flow Model) simulation environment, and the PEST (parameter estimation) tool is used to calibrate effective conductivities defined over 16 zones within the MHRB. The effective parameters are also used as expected values to develop lognormally distributed conductivity (K) fields on local grid scales. Stream–aquifer exchanges are simulated with K fields at both scales and then compared. Results show that the effects of small-scale heterogeneities significantly influence exchanges with simulations based on local-scale heterogeneities always producing discharges that are less than those produced by the base model. Although aquifer heterogeneities are uncorrelated at local scales, they appear to induce coherent slow paths in groundwater fluxes that in turn reduce aquifer–stream exchanges. Since surface water–groundwater exchanges are critical hydrologic processes in basin-scale water budgets, these results also have implications for water resources management.

Direct numerical simulations of flow through two millimeter-scale rock samples of limestone and sandstone are performed using three diverse fluid dynamic simulators. The resulting steady-state velocity fields are compared in terms of the associated empirical probability density functions (PDFs) and key statistics of the velocity fields. The pore space geometry of each sample is imaged at 5.06−μm voxel size resolution using X-ray microtomography. The samples offer contrasting characteristics in terms of total connected porosity (about 0.31 for the limestone and 0.07 for the sandstone) and are typical of several applications in hydrogeology and petroleum engineering. The three-dimensional fluid velocity fields within the explicit pore spaces are simulated using ANSYS® FLUENT® ANSYS Inc. (2009), EULAG Prusa et al. (Comput. Fluids 37, 1193–1207 2008), and SSTOKES Sarkar et al. (2002). These computational approaches are highly disperse in terms of algorithmic complexity, differ in terms of their governing equations, the adopted numerical methodologies, the enforcement of internal no-slip boundary conditions at the fluid-solid interface, and the computational mesh structure. As metrics of comparison to probe in a statistical sense the internal similarities/differences across sample populations of velocities obtained through the computational systems, we consider (i) integral quantities, such as the Darcy flux and (ii) main statistical moments of local velocity distributions including local correlations between velocity fields. Comparison of simulation results indicates that mutually consistent estimates of the state of flow are obtained in the analyzed samples of natural pore spaces despite the considerable differences associated with the three computational approaches. We note that in the higher porosity limestone sample, the structures of the velocity fields obtained using ANSYS FLUENT and EULAG are more alike than either compared against the results obtained using SSTOKES. In the low-porosity sample, the structures of the velocity fields obtained by EULAG and SSTOKES are more similar than either is to the fields obtained using ANSYS FLUENT. With respect to macroscopic quantities, ANSYS FLUENT and SSTOKES provide similar results in terms of the average vertical velocity for both of the complex microscale geometries considered, while EULAG tends to render the largest velocity values. The influence of the pore space structure on fluid velocity field characteristics is also discussed.

We perform a set of detailed numerical simulations of single-phase, fully saturated flow in stochastically generated, three-dimensional pore structures with diverse porosities (phi) and degrees of connectivity, and analyze the probability density functions (PDFs) of the pore sizes, S, and vertical velocity components, w, which are aligned with the mean flow direction. Both of the PDFs are markedly skewed with pronounced positive tails. This feature of the velocity PDF is dictated by the pore structure and determines the shortest travel times, one of the key transport attributes that underpins the success or the failure of environmental remediation techniques. Using a maximum likelihood approach, we determine that the PDFs of S and w decay according to an exponential and a stretched exponential model, respectively. A strong correlation between the key parameters governing the decay of the upper tails of the two PDFs is found, which provides a quantitative result for this analogy that so far has been stated only qualitatively. The parameter governing the concavity of the tail of the velocity PDF varies linearly with porosity over the entire range of tested values (0.2 <= phi <= 0.6). The parameters controlling the spread of the upper tails of the PDFs of S and w appear to be linked by a power-law relationship.

We provide a description and computational investigation of an efficient method to stochastically generate realistic pore structures. Smolarkiewicz and Winter introduced this specific method in pores resolving simulation of Darcy flows (Smolarkiewicz and Winter, 2010 [1]) without giving a complete formal description or analysis of the method, or indicating how to control the parameterization of the ensemble. We address both issues in this paper. The method consists of two steps. First, a realization of a correlated Gaussian field, or topography, is produced by convolving a prescribed kernel with an initial field of independent, identically distributed random variables. The intrinsic length scales of the kernel determine the correlation structure of the topography. Next, a sample pore space is generated by applying a level threshold to the Gaussian field realization: points are assigned to the void phase or the solid phase depending on whether the topography over them is above or below the threshold. Hence, the topology and geometry of the pore space depend on the form of the kernel and the level threshold. Manipulating these two user prescribed quantities allows good control of pore space observables, in particular the Minkowski functionals. Extensions of the method to generate media with multiple pore structures and preferential flow directions are also discussed. To demonstrate its usefulness, the method is used to generate a pore space with physical and hydrological properties similar to a sample of Berea sandstone.

The manuscripts reports on a statistical analysis of neutron porosity data using the framework developed by the authors. This framework models the porosity increments as the product of a truncated fractal Brownian motion with lag-dependent variance and a random variable, which here is modeled either by an alpha-stable or lognormal random variable. The variogram of the fractal Brownian motion is modeled as a truncated power variogram. Sections 3-5 are concerned with the estimation of the parameters of the increment models and the determination of sample structure functions. Sections 6 and 7 provide an analysis of the frequency distribution of peak over threshold of the porosity increments and their structure functions. The paper provides an interesting statistical analysis that sheds light on spatial porosity patterns, which may give insight into the spatial distribution of hydraulic conductivity.

Key Points Computational experiments are used to develop generalized pedotransfer functions Kozeny eqn gives good estimates of permeability for porosities in a normal range Power laws give better estimates of permeability across wider ranges of porosity

Finite time Lyapunov exponents are used to determine expanding, contracting, and hyperbolic regions in computational simulations of laminar steady-state fluid flows within realistic three dimensional pore structures embedded within an impermeable matrix. These regions correspond approximately to pores where flow converges (contraction) or diverges (expansion), and to throats between pores where the flow mixes (hyperbolic). The regions are sparse and disjoint from one another, occupying only a small percentage of the pore space. Nonetheless, nearly every percolating fluid particle trajectory passes through several hyperbolic regions indicating that the effects of in-pore mixing are distributed throughout an entire pore structure. Furthermore, the observed range of fluid dynamics evidences two scales of heterogeneity within each of these flow fields. There is a larger scale that affects dispersion of fluid particle trajectories across the connected network of pores and a relatively small scale of nonuniform distributions of velocities within an individual pore.

Heterogeneous flows are observed to result from variations in the geometry and topology of pore structures within stochastically generated three dimensional porous media. A stochastic procedure generates media comprising complex networks of connected pores. Inside each pore space, the Navier-Stokes equations are numerically integrated until steady state velocity and pressure fields are attained. The intricate pore structures exert spatially variable resistance on the fluid, and resulting velocity fields have a wide range of magnitudes and directions. Spatially nonuniform fluid fluxes are observed, resulting in principal pathways of flow through the media. In some realizations, up to 25% of the flux occurs in 5% of the pore space depending on porosity. The degree of heterogeneity in the flow is quantified over a range of porosities by tracking particle trajectories and calculating their attributes including tortuosity, length, and first passage time. A representative elementary volume is first computed so the dependence of particle based attributes on the size of the domain through which they are followed is minimal. High correlations between the dimensionless quantities of porosity and tortuosity are calculated and a logarithmic relationship is proposed. As the porosity of a medium increases the flow field becomes more uniform.