Conditioning mean steady state flow on hydraulic head and conductivity through geostatistical inversion

Nonlocal moment equations allow one to render deterministically optimum predictions of flow in randomly heterogeneous media and to assess predictive uncertainty conditional on measured values of medium properties. We present a geostatistical inverse algorithm for steady-state flow that makes it possible to further condition such predictions and assessments on measured values of hydraulic head (and/or flux). Our algorithm is based on recursive finite-element approximations of exact first and second conditional moment equations. Hydraulic conductivity is parameterized via universal kriging based on unknown values at pilot points and (optionally) measured values at other discrete locations. Optimum unbiased inverse estimates of natural log hydraulic conductivity, head and flux are obtained by minimizing a residual criterion using the Levenberg-Marquardt algorithm. We illustrate the method for superimposed mean uniform and convergent flows in a bounded two-dimensional domain. Our examples illustrate how conductivity and head data act separately or jointly to reduce parameter estimation errors and model predictive uncertainty.This work is supported in part by NSF/ITR Grant EAR-0110289. The first author was additionally supported by scholarships from CONACYT and Instituto de Investigaciones Electricas of Mexico. Additional support was provided by the European Commission under Contract EVK1-CT-1999-00041 (W-SAHaRA-Stochastic Analysis of Well Head Protection and Risk Assessment).     

Some properties of flows in unsaturated soils with an exponential dependence of the hydraulic conductivity upon the pressure head

A systematic development of the consequences of an exponential dependence of the hydraulic conductivity upon the pressure head is presented. Alternative expressions for the flux are discussed in detail. For steady flows, partial differential equations in terms of the matric flux potential, the pressure head, and the total head are derived. For steady, plane and axially symmetric flows, partial differential equations for the stream function are given. A theoretical basis for the construction of viscous flow analogs for steady, plane and axially symmetric flows is also presented.     

Diagrammatic theory of effective hydraulic conductivity

This work presents a stochastic diagrammatic theory for the calculation of the effective hydraulic conductivity of heterogeneous media. The theory is based on the mean-flux series expansion of a log-normal hydraulic conductivity medium in terms of diagrammatic representations and leads to certain general results for the effective hydraulic conductivity of three-dimensional media. A selective summation technique is used to improve low-order perturbation analysis by evaluating an infinite set of diagrammatic terms with a specific topological structure that dominates the perturbation series. For stochastically isotropic media the selective summation yeilds the anticipated exponential expression for the effective hydraulic conductivity. This expression is extended to stochastically anisotropic media. It is also shown that in the case of non homogeneous media the uniform effective hydraulic conductivity is replaced by a non-local tensor kernel, for which general diagrammatic expressions are obtained. The non-local kernel leads to the standard exponential behavior for the effective hydraulic conductivity at the homogeneous limit.     

Diagrammatic theory of effective hydraulic conductivity

This work presents a stochastic diagrammatic theory for the calculation of the effective hydraulic conductivity of heterogeneous media. The theory is based on the mean-flux series expansion of a log-normal hydraulic conductivity medium in terms of diagrammatic representations and leads to certain general results for the effective hydraulic conductivity of three-dimensional media. A selective summation technique is used to improve low-order perturbation analysis by evaluating an infinite set of diagrammatic terms with a specific topological structure that dominates the perturbation series. For stochastically isotropic media the selective summation yeilds the anticipated exponential expression for the effective hydraulic conductivity. This expression is extended to stochastically anisotropic media. It is also shown that in the case of non homogeneous media the uniform effective hydraulic conductivity is replaced by a non-local tensor kernel, for which general diagrammatic expressions are obtained. The non-local kernel leads to the standard exponential behavior for the effective hydraulic conductivity at the homogeneous limit.     

Parameter equivalence for the Gardner and van Genuchten soil hydraulic conductivity functions for steady vertical flow with inclusions

The analytic element method is well suited for the Gardner hydraulic conductivity function, but is limited in describing real soils. Therefore, parameter equivalence between the van Genuchten and Gardner hydraulic conductivity functions is explored for the case of steady vertical flow through a homogeneous medium with a single inclusion, i.e., a binary soil. The inclusion has different hydraulic parameters than the background medium. Equivalence is established using three methods: (1) effective capillary drive; (2) capillary length; (3) and a least-squares optimization method that aims to fit a Gardner function to a corresponding van Genuchten function by minimizing the difference in log conductivity over a specified pressure range. Comparisons between hydraulic models are made based on scatterplots of pressure head and the vertical Darcian flux obtained using a finite-element numerical solution with both constitutive relations. For applicability of an equivalent Gardner function over a broad range of pressure heads, the crossover pressure must be maintained between the two parametric functions. The crossover pressure is defined as the pressure in which the hydraulic conductivity of the inclusion is equal to the background. It can be shown that a hybrid methodology of preserving the crossover pressure exactly and using the effective capillary drive will result in hydraulic parameters that are easily obtained and provide good agreement between the conductivity functions of the GR model to the VG model.     

A concise parameterization of the hydraulic conductivity of unsaturated soils

The parameter n in the well-known expression for hydraulic conductivity K=K₀S_eⁿ (where K₀ is its value at satiation and S_e the effective saturation) is determined as a function of the exponent in the power form of the soil–water retention relationship. The result is validated with an extensive experimental database comprising some 43 soils, collected by Mualem.     

Unsaturated flows in soils with self-similar hydraulic conductivity distribution

In this article, we are concerned with the statistics of steady unsaturated flow in soils with a fractal hydraulic conductivity distribution. It is assumed that the spatial distribution of log hydraulic conductivity can be described as an isotropic stochastic fractal process. The impact of the fractal dimension of this process, the soil pore-size distribution parameter, and the characteristic length scale on the variances of tension head and the effective conductivity is investigated. Results are obtained for one-dimensional and three-dimensional flows. Our results indicate that the tension head variance is scale-dependent for fractal distribution of hydraulic conductivity. Both tension head variance and effective hydraulic conductivity depend strongly on the fractal dimension. The soil pore-size distribution parameter is important in reducing the variability of the unsaturated hydraulic conductivity and of the fluxes.     

Evaluation of vertical variations in hydraulic conductivity in unconsolidated sediments

Detailed information on vertical variations in hydraulic conductivity (K) is essential to describe the dynamics of groundwater movement at contaminated sites or as input data used for modeling. K values in high vertical resolution should be determined because K tends to be more continuous in the horizontal than in the vertical direction. To determine K in shallow unconsolidated sediments and in the vertical direction, the recently developed direct-push injection logger can be used. The information obtained by this method serves as a proxy for K and has to be calibrated to obtain quantitative K values of measured vertical profiles. In this study, we performed direct-push soil sampling, sieve analyses and direct-push slug tests to obtain K values in vertical high resolution. Using the results of direct-push slug tests, quantitative K values obtained by the direct-push injection logger could be determined successfully. The results of sieve analyses provided lower accordance with the logs due to the inherent limitations of the sieving method.     

Matching hydrologic response to measured effective hydraulic conductivity

The objective of this study was to test the practicability of defining hydrologic response units as combinations of soil, land use and topography for modelling infiltration at the hillslope and catchment scales. In an experimental catchment in the East African Highlands (Kwalei, Tanzania), three methods of measuring infiltration were compared for their ability to capture the spatial variability of effective hydraulic conductivity: the constant head (CH) method; the tension infiltration (TI) method; and the mini‐rainfall simulation (RS) method. The three methods yielded different probability distributions of effective hydraulic conductivity and suggested different types of hydrologic response units. Independently from these measurements, the occurrence of infiltration‐excess overland flow was monitored over an area of 6 ha by means of overland flow detectors. The observed pattern of overland flow occurrence did not match any of the patterns suggested by the infiltration measurements. Instead, clusters of spots with overland flow were practically independent from field borders. Geostatistical analysis of the overland flow confirmed the absence of spatial correlation for distances over 40 m. The RS method yielded the pattern closest to the observations, probably because the method simulated better the processes that trigger infiltration‐excess overland flow, i.e. soil sealing and infiltration through macroporosity. The RS hydrologic response unit correlated significantly with observed overland flow frequency. However, the location of clusters and ‘hot spots’ of overland flow remained largely unexplained by land use, soil and topographic variables. It is concluded that using such landscape variables to define hydrologic units may create artificial boundaries that do no correspond to physical realities, especially if the stochastic component within hydrologic units is neglected. Copyright © 2005 John Wiley & Sons, Ltd.     

The vertical hydraulic conductivity of an aquitard at two spatial scales

Aquitards protect underlying aquifers from contaminants and limit recharge to those aquifers. Understanding the mechanisms and quantity of ground water flow across aquitards to underlying aquifers is essential for ground water planning and assessment. We present results of laboratory testing for shale hydraulic conductivities, a methodology for determining the vertical hydraulic conductivity (K(v)) of aquitards at regional scales and demonstrate the importance of discrete flow pathways across aquitards. A regional shale aquitard in southeastern Wisconsin, the Maquoketa Formation, was studied to define the role that an aquitard plays in a regional ground water flow system. Calibration of a regional ground water flow model for southeastern Wisconsin using both predevelopment steady-state and transient targets suggested that the regional K(v) of the Maquoketa Formation is 1.8 x 10(-11) m/s. The core-scale measurements of the K(v) of the Maquoketa Formation range from 1.8 x 10(-14) to 4.1 x 10(-12) m/s. Flow through some additional pathways in the shale, potential fractures or open boreholes, can explain the apparent increase of the regional-scale K(v). Based on well logs, erosional windows or high-conductivity zones seem unlikely pathways. Fractures cutting through the entire thickness of the shale spaced 5 km apart with an aperture of 50 microns could provide enough flow across the aquitard to match that provided by an equivalent bulk K(v) of 1.8 x 10(-11) m/s. In a similar fashion, only 50 wells of 0.1 m radius open to aquifers above and below the shale and evenly spaced 10 km apart across southeastern Wisconsin can match the model K(v).     

Sensitivity analysis of hydraulic conductivity and Manning's n parameters lead to new method to scale effective hydraulic conductivity across model resolutions

Hydrological modelling is an important tool for research, policy, and management, but uncertainty remains about parameters transferability from field observations made at small scale to models at the catchment scale and larger. This uncertainty compels the need to develop parameter relationships that are translatable across scale. In this study, we compare the changes to modelled processes as resolution is coarsened from 100‐m to 1‐km in a topographically complex, 255‐km² Colorado River headwater catchment. We conducted a sensitivity analysis for hydraulic conductivity (K) and Manning's n parameters across four orders of magnitude. Results showed that K acts as a moderator between surface and subsurface contributions to streamflow, whereas n moderates the duration of high intensity, infiltration‐excess flow. The parametric sensitivity analysis informed development of a new method to scale effective hydraulic conductivity across modelling resolutions in order to compensate for the loss of topographic gradients as resolution is coarsened. A similar mathematical relationship between n and lateral resolution changes was not found, possibly because n is also sensitive to time discretization. This research provides an approach to translate hydraulic conductivity parameters from a calibrated coarse model to higher resolutions where the number of simulations are limited by computational demand.     

Spatial and depth variability of streambed vertical hydraulic conductivity under the regional flow regimes

This study investigated the influence of the regional flow on the streambed vertical hydraulic conductivity (K_v) within the hyporheic zone in three stream reaches of the Weihe River in July 2016. The streambed K_v with two connected depths was investigated at each test reach. Based on the sediment characteristics, the three test reaches could be divided into three categories: a sandy streambed without continuous silt and clay layer, a sandy streambed with continuous silt and clay layer, and a silt–clay streambed. The results demonstrate that the streambed K_v mainly decreases with the depth at the sandy streambed (without continuous silt and clay layer) and increases with the depth at the other two test reaches. At the sandy streambed (with continuous silt and clay layer) where streambed K_v mainly decreases with the depth, the regional upward flux can suspend fine particles and enhance the pore spacing, resulting in the elevated K_v in the upper sediment layers. At another sandy streambed, the continuous silt and clay layer is the main factor that influences the vertical distribution of fine particles and streambed K_v. An increase in streambed K_v with the depth at the silt/clay streambed is attributed to the regional downward movement of water within the sediments that may lead to more fine particles deposited in the pores in the upper sediment layers. The streambed K_v is very close to the bank in the sandy streambed without continuous silt and clay layer and the channel centre in the other two test reaches. Differences in grain size distribution of the sediments at each test reach exercise a strong controlling influence on the streambed K_v. This study promotes the understanding of dynamics influencing the interactions between groundwater and surface water and provides guidelines to scientific water resources management for rivers.     

On the inadequacies of Long's model for steady two-dimensional stratified flows

Nonlinear analysis of two-dimensional steady flows with density stratification in the presence of gravity is considered. Inadequacies of Long's model for steady stratified flow over topography are explored. These include occurrence of closed streamline regions and waves propagating upstream. The usual requirements in Long's model of constant dynamic pressure and constant vertical density gradient in the upstream condition are believed to be the cause of these inadequacies. In this article, we consider a relaxation of these requirements, and also provide a systematic framework to accomplish this. As illustrations of this generalized formulation, exact solutions are given for the following two special flow configurations: the stratified flow over a barrier in an infinite channel; the stratified flow due to a line sink in an infinite channel. These solutions exhibit again closed-streamline regions as well as waves propagating upstream. The persistence of these inadequacies in the generalized Long's model appears to indicate that they are not quite consequences of the assumptions of constant dynamic pressure and constant vertical density gradient in Long's model, contrary to previous belief.

On the other hand, solutions admitted by the generalized Long's model show that departures from Long's model become small as the flow becomes more and more supercritical. They provide a nonlinear mechanism for the generation of columnar disturbances upstream of the obstacle and lead in subcritical flows to qualitatively different streamline topological patterns involving saddle points, which may describe the lee-wave-breaking process in subcritical flows and could serve as seats of turbulence in real flows. The occurrences of upstream disturbances in the presence of lee-wave-breaking activity described by the present solution are in accord with the experiments of Long (Long, R.R., “Some aspects of the flow of stratified fluids, Part 3. Continuous density gradients”, Tellus 7, 341--357 (1955)) and Davis (Davis, R.E., “The two-dimensional flow of a stratified fluid over an obstacle”, J. Fluid Mech. 36, 127–143 ()).     

On the dependence of the saturated hydraulic conductivity upon the effective porosity through a power law model at different scales

We study the scale dependence of the saturated hydraulic conductivity K_s through the effective porosity n_e by means of a newly developed power‐law model (PLM) which allows to use simultaneously measurements at different scales. The model is expressed as product between a single PLM (capturing the impact of the dominating scale) and a characteristic function κ^? accounting for the correction because of the other scale(s). The simple (closed form) expression of the κ^?‐function enables one to easily identify the scales which are relevant for K_s. The proposed model is then applied to a set of real data taken at the experimental site of Montalto Uffugo (Italy), and we show that in this case two (i.e. laboratory and field) scales appear to be the main ones. The implications toward an important application (solute transport) in Hydrology are finally discussed. Copyright © 2016 John Wiley & Sons, Ltd.     

Measurement of 3-D hydraulic conductivity in aquifer cores at in situ effective stresses

An innovative and nondestructive method to measure the hydraulic conductivity of drill core samples in horizontal and vertical directions within a triaxial cell has been developed. This has been applied to characterizing anisotropy and heterogeneity of a confined consolidated limestone aquifer. Most of the cores tested were isotropic, but hydraulic conductivity varied considerably and the core samples with lowest values were also the most anisotropic. Hydraulic conductivity decreased with increasing effective stress due to closure of microfractures caused by sampling for all core samples. This demonstrates the importance of replicating in situ effective stresses when measuring hydraulic conductivity of cores of deep aquifers in the laboratory.     

On effective conductivity of the ionosphere with random irregularities

The influence of stochastic irregularities of the ionosphere on its effective conductivity has been estimated. The study was carried out for large scale inhomogeneities and quasistationary electromagnetic fields. It is found, that Pedersen conductivity sharply increases in a strong geomagnetic field even for small stochastic ionospheric irregularities of the electron density. This peculiarity has to be taken into account during analysis of ionospheric and magnetospheric measurements.     

Feasibility of grain-size analysis methods for determination of vertical hydraulic conductivity of streambeds

Accurate estimation of streambed vertical hydraulic conductivity (K_v) is of great importance in the analysis of water quantity exchange and solute transfer between a stream and its sediments. The paper analyzed the inaccuracy of hydraulic conductivity values of sediments derived from grain-size distribution (K_g), which were determined from eight empirical grain-size methods to represent streambed K_v. In this study, the values of K_v for a streambed were derived using falling-head standpipe permeameter tests conducted at eight study sites in the Elkhorn River, Nebraska, and the tested streambed columns were then collected for grain-size analysis by sieving. These empirical methods were used to calculate the K_g values of the streambed from grain-size distribution data of sediments. Unlike many other studies, this study verifies K_g from grain-size distribution with K_v from permeameter tests on the basis of the same samples of streambed sediments. The K_g values derived from the eight empirical methods were larger than the K_v from permeameter tests; there are five methods that give K_g values of about 3–6 times larger than these K_v. The K_g values from the Kozeny formula followed by the Hazen formula give the largest overestimation error if they are used to represent the K_v of the streambed. The USBR and Shepherd formulas generated K_g values close to K_v, but these K_g values are still larger in general than the K_v values. Moreover, the new values of coefficient C for the empirical formulas were revised so that they can be used to calculate the approximate K_v of a streambed. Among the eight methods, the ratios of the original C values to the average new C range from 1.3 to 5.9. It can be hypothesized that smaller C values must be used in the estimation of K_v for general soil samples if these empirical formulas are used to calculate K_v.     

Factors affecting the measurement of the vertical hydraulic conductivity of a streambed sediment using standpipe tests

The vertical hydraulic conductivity (K_v) of a stream or lake sediment is often determined in the field using standpipe tests. Calculation of K_v is based on the assumption that the hydraulic head in the pipe is equal to that of the stream or lake stage. In this work, a modified equation for K_v is developed for the standpipe test which is applicable when this assumption is not valid. The equation involves not only the hydraulic head at different times but also the difference in the hydraulic head (a) between the groundwater level and river stage. The effects of certain factors on K_v, such as the ratio of the hydraulic head at different times (h₁/h₂), the difference a, and the initial water table height (h₀), are also discussed. The results show that when h₁/h₂ is constant, the relative error (E_r) in K_v increases with the ratio a/h₂. Furthermore, if a/h₂ < 0.05, then for any value of h₁/h₂, E_r is less than 5% using the modified equation. Also, if a/h₂ is large, hydraulic head readings with larger h₁/h₂ ratios must be used to avoid large E_r values. The results of a field test also indicate that the error in K_v decreases as the value of h₀ increases. Copyright © 2013 John Wiley & Sons, Ltd.     

Simultaneous scaling of soil water retention and unsaturated hydraulic conductivity functions assuming lognormal pore-size distribution

Using simultaneous scaling, soil spatial variability of hydraulic functions can be described from a single set of scaling factors. The conventional scaling approach is based on empirical curve fitting, without paying much attention to the physical significance of the scaling factors. In this study, the concept of simultaneous scaling of the soil water retention and unsaturated hydraulic conductivity functions is applied to a physically based scaling theory. In this approach, it is assumed that soils are characterized by a lognormal pore-size distribution, which leads directly to lognormally distributed scaling factors. To test this concept, a total of 143 undisturbed soil samples were collected from two soil depths (25 and 50 cm), with each depth divided into two subsets based on the median soil capillary pressure head value, as determined from the lognormal pore-size distribution assumption. Moreover, the theory was compared with the conventional simultaneous scaling method. Both the conventional and physically based simultaneous scaling method performed equally well for all four subsets, as determined from the reduction in weighted root mean squared residual (WRMSR) values after scaling. We showed that the theoretical interpretation of the lognormal scaling factor distribution was applicable to simultaneous scaling of soil hydraulic functions.