Renormalization group methods in subsurface hydrology: overview and applications in hydraulic conductivity upscaling

The renormalization group (RG) approach is a powerful theoretical framework, more suitable for upscaling strong heterogeneity than low-order perturbation expansions. Applications of RG methods in subsurface hydrology include the calculation of (1) macroscopic transport parameters such as effective and equivalent hydraulic conductivity and dispersion coefficients, and (2) anomalous exponents characterizing the dispersion of contaminants due to long-range conductivity correlations or broad (heavy-tailed) distributions of the groundwater velocity. First, we review the main ideas of RG methods and their hydrological applications. Then, we focus on the hydraulic conductivity in saturated porous media with isotropic lognormal heterogeneity, and we present an RG calculation based on the replica method. The RG analysis gives rigorous support to the exponential conjecture for the effective hydraulic conductivity [Water Resour. Res. 19 (1) (1983) 161]. Using numerical simulations in two dimensions with a bimodal conductivity distribution, we demonstrate that the exponential expression is not suitable for all types of heterogeneity. We also introduce an RG coarse-grained conductivity and investigate its applications in estimating the conductivity of blocks or flow domains with finite size. Finally, we define the fractional effective dimension, and we show that it justifies fractal exponents in the range 1−2/dα<1 (where d is the actual medium dimension) in the geostatistical power average.     

A method to estimate hydraulic conductivity while drilling

A field test and analysis method has been developed to estimate the vertical distribution of hydraulic conductivity in shallow unconsolidated aquifers. The field method uses fluid injection ports and pressure transducers in a hollow auger that measure the hydraulic head outside the auger at several distances from the injection point. A constant injection rate is maintained for a duration time sufficient for the system to become steady state. Exploiting the analogy between electrical resistivity in geophysics and hydraulic flow two methods are used to estimate conductivity with depth: a half-space model based on spherical flow from a point injection at each measurement site, and a one-dimensional inversion of an entire dataset.

The injection methodology, conducted in three separate drilling operations, was investigated for repeatability, reproducibility, linearity, and for different injection sources. Repeatability tests, conducted at 10 levels, demonstrated standard deviations of generally less than 10%. Reproducibility tests conducted in three, closely spaced drilling operations generally showed a standard deviation of less than 20%, which is probably due to lateral variations in hydraulic conductivity. Linearity tests, made to determine dependency on flow rates, showed no indication of a flow rate bias. In order to obtain estimates of the hydraulic conductivity by an independent means, a series of measurements were made by injecting water through screens installed at two separate depths in a monitoring pipe near the measurement site. These estimates differed from the corresponding estimates obtained by injection in the hollow auger by a factor of less than 3.5, which can be attributed to variations in geology and the inaccurate estimates of the distance between the measurement and the injection sites at depth.     

Assimilation of historical head data to estimate spatial distributions of stream bed and aquifer hydraulic conductivity fields

Management of water resources in alluvial aquifers relies mainly on understanding interactions between hydraulically connected streams and aquifers. Numerical models that simulate this interaction often are used as decision support tools for water resource management. However, the accuracy of numerical predictions relies heavily on unknown system parameters (e.g., streambed conductivity and aquifer hydraulic conductivity), which are spatially heterogeneous and difficult to measure directly. This paper employs an ensemble smoother to invert groundwater level measurements to jointly estimate spatially varying streambed and alluvial aquifer hydraulic conductivity along a 35.6‐km segment of the South Platte River in Northeastern Colorado. The accuracy of the inversion procedure is evaluated using a synthetic experiment and historical groundwater level measurements, with the latter constituting the novelty of this study in the inversion and validation of high‐resolution fields of streambed and aquifer conductivities. Results show that the estimated streambed conductivity field and aquifer conductivity field produce an acceptable agreement between observed and simulated groundwater levels and stream flow rates. The estimated parameter fields are also used to simulate the spatially varying flow exchange between the alluvial aquifer and the stream, which exhibits high spatial variability along the river reach with a maximum average monthly aquifer gain of about 2.3 m³/day and a maximum average monthly aquifer loss of 2.8 m³/day, per unit area of streambed (m²). These results demonstrate that data assimilation inversion provides a reliable and computationally affordable tool to estimate the spatial variability of streambed and aquifer conductivities at high resolution in real‐world systems.     

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.     

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.     

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.     

Inverse upscaling of hydraulic parameters during constant flux infiltration using borehole radar

Modeling unsaturated flow in porous media requires constitutive relations that describe the soil water retention and soil hydraulic conductivity as a function of either potential or water content. Often, the hydraulic parameters that describe these relations are directly measured on small soil cores, and many cores are needed to upscale to the entire heterogeneous flow field. An alternative to the forward upscaling method using small samples are inverse upscaling methods that incorporate soft data from geophysical measurements observed directly on the larger flow field. In this paper, we demonstrate that the hydraulic parameters can be obtained from cross borehole ground penetrating radar by measuring the first arrival travel time of electromagnetic waves (represented by raypaths) from stationary antennae during a constant flux infiltration experiment. The formulation and coupling of the hydrological and geophysical models rely on a constant velocity wetting front that causes critical refraction at the edge of the front as it passes by the antennae. During this critical refraction period, the slope of the first arrival data can be used to calculate (1) the wetting velocity and (2) the hydraulic conductivity of the wet (or saturated) soil. If the soil is undersaturated during infiltration, then an estimate of the saturated water content is needed before calculating the saturated hydraulic conductivity. The hydraulic conductivity value is then used in a nonlinear global optimization scheme to estimate the remaining two parameters of a Broadbridge and White soil.     

Using high hydraulic conductivity nodes to simulate seepage lakes

In a typical ground water flow model, lakes are represented by specified head nodes requiring that lake levels be known a priori. To remove this limitation, previous researchers assigned high hydraulic conductivity (K) values to nodes that represent a lake, under the assumption that the simulated head at the nodes in the high-K zone accurately reflects lake level. The solution should also produce a constant water level across the lake. We developed a model of a simple hypothetical ground water/lake system to test whether solutions using high-K lake nodes are sensitive to the value of K selected to represent the lake. Results show that the larger the contrast between the K of the aquifer and the K of the lake nodes, the smaller the error tolerance required for the solution to converge. For our test problem, a contrast of three orders of magnitude produced a head difference across the lake of 0.005 m under a regional gradient of the order of 10(-3) m/m, while a contrast of four orders of magnitude produced a head difference of 0.001 m. The high-K method was then used to simulate lake levels in Pretty Lake, Wisconsin. Results for both the hypothetical system and the application to Pretty Lake compared favorably with results using a lake package developed for MODFLOW (Merritt and Konikow 2000). While our results demonstrate that the high-K method accurately simulates lake levels, this method has more cumbersome postprocessing and longer run times than the same problem simulated using the lake package.     

Practical issues in imaging hydraulic conductivity through hydraulic tomography

Hydraulic tomography has been developed as an alternative to traditional geostatistical methods to delineate heterogeneity patterns in parameters such as hydraulic conductivity (K) and specific storage (S(s)). During hydraulic tomography surveys, a large number of hydraulic head data are collected from a series of cross-hole tests in the subsurface. These head data are then used to interpret the spatial distribution of K and S(s) using inverse modeling. Here, we use the Sequential Successive Linear Estimator (SSLE) of Yeh and Liu (2000) to interpret synthetic pumping test data created through numerical simulations and real data generated in a laboratory sandbox aquifer to obtain the K tomograms. Here, we define "K tomogram" as an image of K distribution of the subsurface (or the inverse results) obtained via hydraulic tomography. We examine the influence of signal-to-noise ratio and biases on results using inverse modeling of synthetic and real cross-hole pumping test data. To accomplish this, we first show that the pumping rate, which affects the signal-to-noise ratio, and the order of data included into the SSLE algorithm both have large impacts on the quality of the K tomograms. We then examine the role of conditioning on the K tomogram and find that conditioning can improve the quality of the K tomogram, but can also impair it, if the data are of poor quality and conditioning data have a larger support volume than the numerical grid used to conduct the inversion. Overall, these results show that the quality of the K tomogram depends on the design of pumping tests, their conduct, the order in which they are included in the inverse code, and the quality as well as the support volume of additional data that are used in its computation.     

Effective hydraulic conductivity of nonstationary aquifers

Assuming that the ln hydraulic conductivity in an aquifer is mathematically approximated by a spatial deterministic surface, or trend, plus a stationary random noise, we treat the problem of finding what the effective hydraulic conductivity of that aquifer is. This problem is tackled by spectral methods applied to a type of diffusion equation of groundwater flow, together with suitable coordinate transformations. Analytical (exact) solutions in terms of elementary functions are presented for one- and three-dimensional finite and infinite domains. Stability criteria are obtained for the solutions, in terms of a critical parameter, that turns out to involve the product of correlation scale and trend gradient. For the case of finite and symmetrical domains, additional provisions to insure the stability of numerical calculations of effective hydraulic conductivity are provided. Effective hydraulic conductivity is an important property, with potential applications in the calibrations of groundwater and transport numerical models.     

Effective hydraulic conductivity of nonstationary aquifers

Assuming that the ln hydraulic conductivity in an aquifer is mathematically approximated by a spatial deterministic surface, or trend, plus a stationary random noise, we treat the problem of finding what the effective hydraulic conductivity of that aquifer is. This problem is tackled by spectral methods applied to a type of diffusion equation of groundwater flow, together with suitable coordinate transformations. Analytical (exact) solutions in terms of elementary functions are presented for one- and three-dimensional finite and infinite domains. Stability criteria are obtained for the solutions, in terms of a critical parameter, that turns out to involve the product of correlation scale and trend gradient. For the case of finite and symmetrical domains, additional provisions to insure the stability of numerical calculations of effective hydraulic conductivity are provided. Effective hydraulic conductivity is an important property, with potential applications in the calibrations of groundwater and transport numerical models.     

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.     

An innovative method to estimate regional-scale hydraulic diffusivity using GRACE data

ABSTRACT

Estimation of hydraulic properties in the field is usually small-scale and not cost-effective. This paper proposes an innovative method for estimating hydraulic diffusivity at regional scale. Monthly groundwater storage change over the period from 2003 to 2013 is first estimated from GRACE-derived terrestrial water storage (TWS). Assuming that the aquifer system is unconfined and the hydraulic properties are uniform in a geographical cell, the water balance principle and Darcy’s law are used to establish a relation between groundwater storage and hydraulic diffusivity. The value of hydraulic diffusivity is then adjusted using the generalized least squares and linear correlation method. This GRACE-derived hydraulic diffusivity estimation method, or GHDE method for short, is first verified with a hypothetical case and then applied in the Beishan area with available field-measured hydraulic conductivity data. The hypothetical case study demonstrates that the method works perfectly if the TWS data are error free. The Beishan case study illustrates that the estimated hydraulic diffusivities using the GHDE method correlate reasonably well with field test results, suggesting that this method is applicable. The accuracy of this method is constrained by the resolution of the GRACE-derived TWS data and is most suitable for very large scale groundwater problems due to the current accuracy of the GRACE data.

EDITOR A. Castellarin ASSOCIATE EDITOR N. Verhoest     

Multiple parameterization for hydraulic conductivity identification

Hydraulic conductivity identification remains a challenging inverse problem in ground water modeling because of the inherent nonuniqueness and lack of flexibility in parameterization methods. This study introduces maximum weighted log-likelihood estimation (MWLLE) along with multiple generalized parameterization (GP) methods to identify hydraulic conductivity and to address nonuniqueness and inflexibility problems in parameterization. A scaling factor for information criteria is suggested to obtain reasonable weights of parameterization methods for the MWLLE and model averaging method. The scaling factor is a statistical parameter relating to a desired significance level in Occam's window and the variance of the chi-squares distribution of the fitting error. Through model averaging with multiple GP methods, the conditional estimate of hydraulic conductivity and its total conditional covariances are calculated. A numerical example illustrates the issue arising from Occam's window in estimating model weights and shows the usefulness of the scaling factor to obtain reasonable model weights. Moreover, the numerical example demonstrates the advantage of using multiple GP methods over the zonation and interpolation methods because GP provides better models in the model averaging method. The methodology is applied to the Alamitos Gap area, California, to identify the hydraulic conductivity field. The results show that the use of the scaling factor is necessary in order to incorporate good parameterization methods and to avoid a dominant parameterization method.     

On estimating the hydraulic properties of soil,Part 1. Comparison between forms to estimate the soil-water characteristic function

The hydraulic properties of soil include the soil-water characteristic function [h(θ), in which θ is water content and h is pressure head (suction)], and the hydraulic conductivity function [K(θ) or K(h)]. These functions are essential to the solution of unsaturated groundwater flow problems. A number of empirical and semiempirical forms have been proposed in the literature to estimate these functions. The present paper employs a nonlinear least-square analysis for comparison between some of the available forms, using a large number of experimental measurements of h(θ) for different classes of soil. Suitability of the forms for predicting the hydraulic conductivity function is examined. In the absence of accurate measurements, the paper facilitates modeling by providing estimates for the parameters of the soil-water characteristic function.     

Comparison of conductivity averaging methods for one-dimensional unsaturated flow in layered soils

One of the important factors influencing the accuracy of the numerical solution of 1D unsaturated flow equation (Richards’ equation) is the averaging method applied to compute hydraulic conductivity between two adjacent nodes of the computational grid. A number of averaging schemes have been proposed in the literature for homogeneous soil, including arithmetic, geometric, upstream and integrated means, as well as more sophisticated approaches, based on the local solution of steady state flow between the neighboring nodes (Darcian means). Another group of methods have been developed for the case when a material interface is present between the nodes. They range from simple arithmetic averaging to more complex schemes using the pressure- and flux-continuity conditions at the interface. In this paper we compare several averaging schemes for a number of steady and unsteady flow problems in layered soils. The first group of methods is applied in the framework of the vertex-centered approach to spatial discretization, where the nodes are placed at the material interfaces, while the second group is used with the cell-centered approach, where the material interfaces are located between computational nodes. The resulting numerical schemes are evaluated in terms of accuracy and computational time. It is shown that the averaging schemes based on Darcian mean principle [19] used in the framework of either vertex-centered or cell-centered approach compare favorably to other methods for a range of test cases.     

Using a non-integer moment of the impulse response to estimate the half-space conductivity

The nth-order moments of the electromagnetic impulse response are useful for interpreting electromagnetic data. We have derived an analytic expression for the half-order moment of a conductive half-space. By inverting this expression, the measured half-order moment can be used to estimate an apparent conductivity of the ground. The first-order moment can also be used to estimate the half-space conductivity. A sensitivity analysis indicates that for an airborne EM configuration, the half-order moment will be most sensitive to material in the top 26–48 m, while the first-order moment will be sensitive to deeper material (down to depths between 66 and 127 m).     

Characterizing hydraulic conductivity with the direct-push permeameter

The direct-push permeameter (DPP) is a promising approach for obtaining high-resolution information about vertical variations in hydraulic conductivity (K) in shallow unconsolidated settings. This small-diameter tool, which consists of a short screened section with a pair of transducers inset in the tool near the screen, is pushed into the subsurface to a depth at which a K estimate is desired. A short hydraulic test is then performed by injecting water through the screen at a constant rate (less than 4 L/min) while pressure changes are monitored at the transducer locations. Hydraulic conductivity is calculated using the injection rate and the pressure changes in simple expressions based on Darcy's Law. In units of moderate or higher hydraulic conductivity (more than 1 m/d), testing at a single level can be completed within 10 to 15 min. Two major advantages of the method are its speed and the insensitivity of the K estimates to the zone of compaction created by tool advancement. The potential of the approach has been assessed at two extensively studied sites in the United States and Germany over a K range commonly faced in practical field investigations (0.02 to 500 m/d). The results of this assessment demonstrate that the DPP can provide high-resolution K estimates that are in good agreement with estimates obtained through other means.     

Comparison of derivative-free optimization methods for groundwater supply and hydraulic capture community problems

Management decisions involving groundwater supply and remediation often rely on optimization techniques to determine an effective strategy. We introduce several derivative-free sampling methods for solving constrained optimization problems that have not yet been considered in this field, and we include a genetic algorithm for completeness. Two well-documented community problems are used for illustration purposes: a groundwater supply problem and a hydraulic capture problem. The community problems were found to be challenging applications due to the objective functions being nonsmooth, nonlinear, and having many local minima. Because the results were found to be sensitive to initial iterates for some methods, guidance is provided in selecting initial iterates for these problems that improve the likelihood of achieving significant reductions in the objective function to be minimized. In addition, we suggest some potentially fruitful areas for future research.