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.     

Estimation of anisotropic hydraulic conductivity using geophysical data in a coastal aquifer of Karnataka,India

Estimation of hydraulic parameters is essential to understand the interaction between groundwater flow and seawater intrusion. Though several studies have addressed hydraulic parameter estimation, based on pumping tests as well as geophysical methods, not many studies have addressed the problem with clayey formations being present. In this study, a methodology is proposed to estimate anisotropic hydraulic conductivity and porosity values for the coastal aquifer with unconsolidated formations. For this purpose, the one-dimensional resistivity of the aquifer and the groundwater conductivity data are used to estimate porosity at discrete points. The hydraulic conductivity values are estimated by its mutual dependence with porosity and petrophysical parameters. From these estimated values, the bilinear relationship between hydraulic conductivity and aquifer resistivity is established based on the clay content of the sampled formation. The methodology is applied on a coastal aquifer along with the coastal Karnataka, India, which has significant clayey formations embedded in unconsolidated rock. The estimation of hydraulic conductivity values from the established correlations has a correlation coefficient of 0.83 with pumping test data, indicating good reliability of the methodology. The established correlations also enable the estimation of horizontal hydraulic conductivity on two-dimensional resistivity sections, which was not addressed by earlier studies. The inventive approach of using the established bilinear correlations at one-dimensional to two-dimensional resistivity sections is verified by the comparison method. The horizontal hydraulic conductivity agrees with previous findings from inverse modelling. Additionally, this study provides critical insights into the estimation of vertical hydraulic conductivity and an equation is formulated which relates vertical hydraulic conductivity with horizontal. Based on the approach presented, the anisotropic hydraulic conductivity of any type aquifer with embedded clayey formations can be estimated. The anisotropic hydraulic conductivity has the potential to be used as an important input to the groundwater models.     

Pedotransfer functions estimating soil hydraulic properties using different soil parameters

Estimates of soil hydraulic properties using pedotransfer functions (PTF) are useful in many studies such as hydrochemical modelling and soil mapping. The objective of this study was to calibrate and test parametric PTFs that predict soil water retention and unsaturated hydraulic conductivity parameters. The PTFs are based on neural networks and the Bootstrap method using different sets of predictors and predict the van Genuchten/Mualem parameters. A Danish soil data set (152 horizons) dominated by sandy and sandy loamy soils was used in the development of PTFs to predict the Mualem hydraulic conductivity parameters. A larger data set (1618 horizons) with a broader textural range was used in the development of PTFs to predict the van Genuchten parameters. The PTFs using either three or seven textural classes combined with soil organic mater and bulk density gave the most reliable predictions of the hydraulic properties of the studied soils. We found that introducing measured water content as a predictor generally gave lower errors for water retention predictions and higher errors for conductivity predictions. The best of the developed PTFs for predicting hydraulic conductivity was tested against PTFs from the literature using a subdata set of the data used in the calibration. The test showed that the developed PTFs gave better predictions (lower errors) than the PTFs from the literature. This is not surprising since the developed PTFs are based mainly on hydraulic conductivity data near saturation and sandier soils than the PTFs from the literature. Copyright © 2007 John Wiley & Sons, Ltd.     

Pore size distributions and hydraulic conductivities of rocks derived from Magnetic Resonance Sounding relaxation data using multi-exponential decay time inversion

In hydrogeology there is a variety of empirical formulae available for determination of hydraulic conductivity of porous media, all based on the analysis of grain size distributions of aquifer materials. Sensitivity of NMR measurements to pore sizes makes it a good indicator of hydraulic conductivity. Analogous to laboratory NMR, Magnetic Resonance Sounding (MRS) relaxation data are of a multi-exponential (ME) nature due to the distribution of different pore sizes in an investigated rock layer. ME relaxation behaviour will also arise due to the superposition of NMR signals which originate from different layers. It has been shown, that both kinds of ME behaviour coexist in MRS and can principally be separated by ME inversion of the field data. Only a few publications exist that have proposed approaches to qualitatively and quantitatively estimate petrophysical parameters such as the hydraulic conductivity from MRS measurements, i.e. MRS porosity and decay times. The so far used relations for the estimation of hydraulic conductivity in hydrogeology and NMR experiments are compared and discussed with respect to their applicability in MRS. Taking into account results from a variety of laboratory NMR and MRS experiments mean rock specific calibration factors are introduced for a data-base-calibrated estimation of hydraulic conductivity when no on-site calibration of MRS is available. Field data have been analysed using conventional and ME inversion using such mean calibration values. The results for conventional and ME inversion agree with estimates obtained from well core analysis for shallow depths but are significantly improved using a ME inversion approach for greater depths.     

Pore size distributions and hydraulic conductivities of rocks derived from Magnetic Resonance Sounding relaxation data using multi-exponential decay time inversion

In hydrogeology there is a variety of empirical formulae available for determination of hydraulic conductivity of porous media, all based on the analysis of grain size distributions of aquifer materials. Sensitivity of NMR measurements to pore sizes makes it a good indicator of hydraulic conductivity. Analogous to laboratory NMR, Magnetic Resonance Sounding (MRS) relaxation data are of a multi-exponential (ME) nature due to the distribution of different pore sizes in an investigated rock layer. ME relaxation behaviour will also arise due to the superposition of NMR signals which originate from different layers. It has been shown, that both kinds of ME behaviour coexist in MRS and can principally be separated by ME inversion of the field data. Only a few publications exist that have proposed approaches to qualitatively and quantitatively estimate petrophysical parameters such as the hydraulic conductivity from MRS measurements, i.e. MRS porosity and decay times. The so far used relations for the estimation of hydraulic conductivity in hydrogeology and NMR experiments are compared and discussed with respect to their applicability in MRS. Taking into account results from a variety of laboratory NMR and MRS experiments mean rock specific calibration factors are introduced for a data-base-calibrated estimation of hydraulic conductivity when no on-site calibration of MRS is available. Field data have been analysed using conventional and ME inversion using such mean calibration values. The results for conventional and ME inversion agree with estimates obtained from well core analysis for shallow depths but are significantly improved using a ME inversion approach for greater depths.     

On estimating the hydraulic properties of soil,Part 3. Parameters of the Philip infiltration equation

Using a large number of experimental data, the paper estimates the parameters of the Philip infiltration equation. The values of these parameters are obtained using the Philip approach, which employs the hydraulic properties of soil. In the absence of extensive measurements, the tabulated results can be used in modelling infiltration. The paper introduces approximate sorptivity expressions that use a limited number of parameters. Sorptivity may be estimated reasonably well from a knowledge of moisture deficit, a representative capillary pressure, and saturated hydraulic conductivity. The accuracy of prediction is highest for coarser materials. For coarse materials (sands), second-power sorptivity can be approximated as a linear function of initial effective saturation. The proposed expressions for sorptivity can be used as a substitute for the lengthy Philip procedure.     

Porosity and hydraulic conductivity estimation of the basaltic aquifer in Southern Syria by using nuclear and electrical well logging techniques

An alternative approach using nuclear neutron-porosity and electrical resistivity well logging of long (64 inch) and short (16 inch) normal techniques is proposed to estimate the porosity and the hydraulic conductivity (K) of the basaltic aquifers in Southern Syria. This method is applied on the available logs of Kodana well in Southern Syria. It has been found that the obtained K value by applying this technique seems to be reasonable and comparable with the hydraulic conductivity value of 3.09 m/day obtained by the pumping test carried out at Kodana well. The proposed alternative well logging methodology seems as promising and could be practiced in the basaltic environments for the estimation of hydraulic conductivity parameter. However, more detailed researches are still required to make this proposed technique very performed in basaltic environments.     

Determination of Hydraulic Conductivity from Grain‐Size Distribution for Different Depositional Environments

Over 400 unlithified sediment samples were collected from four different depositional environments in global locations and the grain‐size distribution, porosity, and hydraulic conductivity were measured using standard methods. The measured hydraulic conductivity values were then compared to values calculated using 20 different empirical equations (e.g., Hazen, Carman‐Kozeny) commonly used to estimate hydraulic conductivity from grain‐size distribution. It was found that most of the hydraulic conductivity values estimated from the empirical equations correlated very poorly to the measured hydraulic conductivity values with errors ranging to over 500%. To improve the empirical estimation methodology, the samples were grouped by depositional environment and subdivided into subgroups based on lithology and mud percentage. The empirical methods were then analyzed to assess which methods best estimated the measured values. Modifications of the empirical equations, including changes to special coefficients and addition of offsets, were made to produce modified equations that considerably improve the hydraulic conductivity estimates from grain size data for beach, dune, offshore marine, and river sediments. Estimated hydraulic conductivity errors were reduced to 6 to 7.1 m/day for the beach subgroups, 3.4 to 7.1 m/day for dune subgroups, and 2.2 to 11 m/day for offshore sediments subgroups. Improvements were made for river environments, but still produced high errors between 13 and 23 m/day.     

Microearthquake analysis for hydraulic frac－turing proces

Ｍｉｃｒｏｅａｒｔｈｑｕａｋｅａｎａｌｙｓｉｓｆｏｒｈｙｄｒａｕｌｉｃｆｒａｃ┐ｔｕｒｉｎｇｐｒｏｃｅｓＹＩＮＧ－ＰＩＮＧＬＩ（李应平）ＤｅｐａｒｔｍｅｎｔｏｆＥａｒｔｈ，ＡｔｍｏｓｐｈｅｒｉｃａｎｄＰｌａｎｅｔａｒｙＳｃｉｅｎｃｅｓ，Ｍａｓａｃｈ...     

A method for estimating spatially variable seepage and hydraulic conductivity in channels with very mild slopes

Infiltration along ephemeral channels plays an important role in groundwater recharge in arid regions. A model is presented for estimating spatial variability of seepage due to streambed heterogeneity along channels based on measurements of streamflow‐front velocities in initially dry channels. The diffusion‐wave approximation to the Saint‐Venant equations, coupled with Philip's equation for infiltration, is connected to the groundwater model MODFLOW and is calibrated by adjusting the saturated hydraulic conductivity of the channel bed. The model is applied to portions of two large water delivery canals, which serve as proxies for natural ephemeral streams. Estimated seepage rates compare well with previously published values. Possible sources of error stem from uncertainty in Manning's roughness coefficients, soil hydraulic properties and channel geometry. Model performance would be most improved through more frequent longitudinal estimates of channel geometry and thalweg elevation, and with measurements of stream stage over time to constrain wave timing and shape. This model is a potentially valuable tool for estimating spatial variability in longitudinal seepage along intermittent and ephemeral channels over a wide range of bed slopes and the influence of seepage rates on groundwater levels. Copyright © 2012 John Wiley & Sons, Ltd.     

New graphical methods for estimating aquifer hydraulic parameters using pumping tests with exponentially decreasing rates

Actual pumping tests may involve continuously decreasing rates over a certain period of time, and the hydraulic conductivity (K) and specific storage (S_s) of the tested confined aquifer cannot be interpreted from the classical constant‐rate test model. In this study, we revisit the aquifer drawdown characteristics of a pumping test with an exponentially decreasing rate using the dimensionless analytical solution for such a variable‐rate model. The drawdown may decrease with time for a short period of time at intermediate pumping times for such pumping tests. A larger ratio of initial to final pumping rate and a smaller radial distance of the observation well will enhance the decreasing feature. A larger decay constant results in an earlier decrease, but it weakens the extent of such a decrease. Based on the proposed dimensionless transformation, we have proposed two graphical methods for estimating K and S_s of the tested aquifer. The first is a new type curve method that does not employ the well function as commonly done in standard type curve analysis. Another is a new analytic method that takes advantage of the decreasing features of aquifer drawdown during the intermediate pumping stage. We have demonstrated the applicability and robustness of the two new graphical methods for aquifer characterization through a synthetic pumping test.     

Effect of mineral reactions on the hydraulic properties of unsaturated soils: Model development and application

The selective radius shift model was used to relate changes in mineral volume due to precipitation/dissolution reactions to changes in hydraulic properties affecting flow in porous media. The model accounts for (i) precipitation/dissolution taking place only in the water-filled part of the pore space and further that (ii) the amount of mineral precipitation/dissolution within a pore depends on the local pore volume. The pore bundle concept was used to connect pore-scale changes to macroscopic soil hydraulic properties. Precipitation/dissolution induces changes in the pore radii of water-filled pores and, consequently, in the effective porosity. In a time step of the numerical model, mineral reactions lead to a discontinuous pore-size distribution because only the water-filled pores are affected. The pore-size distribution is converted back to a soil moisture characteristic function to which a new water retention curve is fitted under physically plausible constraints. The model equations were derived for the commonly used van Genuchten/Mualem hydraulic properties. Together with a mixed-form solution of Richards’ equation for aqueous phase flow, the model was implemented into the geochemical modelling framework PHREEQC, thereby making available PHREEQC’s comprehensive geochemical reactions. Example applications include kinetic halite dissolution and calcite precipitation as a consequence of cation exchange. These applications showed marked changes in the soil’s hydraulic properties due to mineral precipitation/dissolution and the dependency of these changes on water contents. The simulations also revealed the strong influence of the degree of saturation on the development of the saturated hydraulic conductivity through its quadratic dependency on the van Genuchten parameter α. Furthermore, it was shown that the unsaturated hydraulic conductivity at fixed reduced water content can even increase during precipitation due to changes in the pore-size distribution.     

Stochastic joint inversion of hydrogeophysical data for salt tracer test monitoring and hydraulic conductivity imaging

The assessment of hydraulic conductivity of heterogeneous aquifers is a difficult task using traditional hydrogeological methods (e.g., steady state or transient pumping tests) due to their low spatial resolution. Geophysical measurements performed at the ground surface and in boreholes provide additional information for increasing the resolution and accuracy of the inverted hydraulic conductivity field. We used a stochastic joint inversion of Direct Current (DC) resistivity and self-potential (SP) data plus in situ measurement of the salinity in a downstream well during a synthetic salt tracer experiment to reconstruct the hydraulic conductivity field between two wells. The pilot point parameterization was used to avoid over-parameterization of the inverse problem. Bounds on the model parameters were used to promote a consistent Markov chain Monte Carlo sampling of the model parameters. To evaluate the effectiveness of the joint inversion process, we compared eight cases in which the geophysical data are coupled or not to the in situ sampling of the salinity to map the hydraulic conductivity. We first tested the effectiveness of the inversion of each type of data alone (concentration sampling, self-potential, and DC resistivity), and then we combined the data two by two. We finally combined all the data together to show the value of each type of geophysical data in the joint inversion process because of their different sensitivity map. We also investigated a case in which the data were contaminated with noise and the variogram unknown and inverted stochastically. The results of the inversion revealed that incorporating the self-potential data improves the estimate of hydraulic conductivity field especially when the self-potential data were combined to the salt concentration measurement in the second well or to the time-lapse cross-well electrical resistivity data. Various tests were also performed to quantify the uncertainty in the inverted hydraulic conductivity field.     

Empirical relationships for estimating stream depletion by a well pumping near a gaining stream

Siting wells near streams requires an accurate estimate of the quantity of water derived from the river due to pumping. A number of hydrogeological and hydraulic parameters influence this value. This study estimates stream depletion under steady-state conditions for a variety of hydrogeological systems. A finite differences model was used to analyze several hydrogeological situations, and for each of these the stream depletion was estimated using an advective transport method. An empirical equation for stream depletion was obtained for the case of a stream that partially penetrates the aquifer and a pumping well that is screened over a portion of the aquifer. The derived equation, which is valid for both isotropic and anisotropic conditions, expresses stream depletion as a function of the unit inflow to the river, the discharge of the pumping well, the well screen length, the distance between the river and pumping well, the wetted perimeter, and a new parameter called "overlap," which is defined to be the distance between the riverbed and the top of well screen. The overlap parameter makes it possible to consider indirectly the vertical component of flow, which is accentuated when the well is screened below the streambed. The formula proposed here should be useful in deciding where to locate a pumping well and to decide the appropriate length of its screen.     

Inverse modeling of a radial multistep outflow experiment for determining unsaturated hydraulic properties

Modeling flow and solute transport in the unsaturated zone on the basis of the Richards equation requires specifying values for unsaturated hydraulic conductivity and water potential as a function of saturation. The objectives of the paper are to evaluate the design of a transient, radial, multi-step outflow experiment, and to determine unsaturated hydraulic parameters using inverse modeling. We conducted numerical simulations, sensitivity analyses, and synthetic data inversions to assess the suitability of the proposed experiment for concurrently estimating the parameters of interest. We calibrated different conceptual models against transient flow and pressure data from a multi-step, radial desaturation experiment to obtain estimates of absolute permeability, as well as the parameters of the relative permeability and capillary pressure functions. We discuss the differences in the estimated parameter values and illustrate the impact of the underlying model on the estimates. We demonstrate that a small error in absolute permeability, if determined in an independent experiment, leads to biased estimates of unsaturated hydraulic properties. Therefore, we perform a joint inversion of pressure and flow rate data for the simultaneous determination of permeability and retention parameters, and analyze the correlations between these parameters. We conclude that the proposed combination of a radial desaturation experiment and inverse modeling is suitable for simultaneously determining the unsaturated hydraulic properties of a single soil sample, and that the inverse modeling technique provides the opportunity to analyze data from nonstandard experimental designs.     

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.     

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.     

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.