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In the recent article by Paradis et al. (2018), an error was made when developing a conceptual model and an analytical solution to describe the one-dimensional displacement of the center of mass of a non-reactive tracer during a single-well injection, drift, and extraction test (push-pull test).

The analytical solution describing the one-dimensional displacement of the center of mass of a tracer during an injection, drift, and extraction test (push-pull test) was expanded to account for displacement during the injection phase. The solution was expanded to improve the in situ estimation of effective porosity. The truncated equation assumed displacement during the injection phase was negligible, which may theoretically lead to an underestimation of the true value of effective porosity. To experimentally compare the expanded and truncated equations, single-well push-pull tests were conducted across six test wells located in a shallow, unconfined aquifer comprised of unconsolidated and heterogeneous silty and clayey fill materials. The push-pull tests were conducted by injection of bromide tracer, followed by a non-pumping period, and subsequent extraction of groundwater. The values of effective porosity from the expanded equation (0.6–5.0%) were substantially greater than from the truncated equation (0.1–1.3%). The expanded and truncated equations were compared to data from previous push-pull studies in the literature and demonstrated that displacement during the injection phase may or may not be negligible, depending on the aquifer properties and the push-pull test parameters. The results presented here also demonstrated the spatial variability of effective porosity within a relatively small study site can be substantial, and the error-propagated uncertainty of effective porosity can be mitigated to a reasonable level (< ± 0.5%). The tests presented here are also the first that the authors are aware of that estimate, in situ, the effective porosity of fine-grained fill material.

The Abbreviations: ALL, complete treatment including organic matter oxidation and chemical and physical dispersion; CEC, cation exchange capacity; EDI, effective disruption index; NP, chemical dispersion; PER, pretreatment to oxidize organic matter; PHYS, physical dispersion; SOM, soil organic matter; W, water dispersion.

Interaction of water and organic matter (OM) in pores affects the capacity for long-term C sequestration and water retention in soils. This study investigated the effects of the amount and distribution of OM in microstructures on water retention in soil microaggregates. The focus of the study was on the hysteretic soil water characteristic (i.e., the difference between the main drying and wetting curves), which has been largely ignored in previous studies on water-OM relations. The main drying and wetting branches of the water retention curve were measured using a water activity meter on water-stable microaggregates (53-250 [micro]m), which were collected from surface soils (0-15 cm) subject to different management practices: tallgrass prairie restoration on a Mollisol, and tillage with and without N fertilization on an Alfisol. The results show that the presence of OM in Abbreviations: ANCOVA, analysis of covariance; OC, organic carbon; OM, organic matter; SSA, specific surface area.

Tillage impacts many components and functions of the soil ecosystem. Thus, the prediction of soil structures produced by tillage may be regarded as a major objective in soil science. Brittle fracture is the desired mode of failure in most tillage operations. Mechanistic or phenomenological models based on the probabilistic Weibull 'weakest link' theory are commonly applied to model brittle fracture of air-dry aggregates. The overall objective of this study was to develop a Weibull model to describe the strength of different-sized soil aggregates across a wide range of water contents. Rupture energy data were obtained for aggregates sampled in three field experiments. These included two soil compaction experiments (Bygholm I and II, sandy loam) and a long-term tillage and fertilization experiment (Maury, silt loam). Aggregates were subjected to a crushing test after having been adjusted to matric potentials ranging from -10 kPa to -163 MPa (air dry). Water content strongly affected the characteristic rupture energy (Weibull [alpha] parameter), and this relationship was successfully modeled with a power law function. In contrast, water content had little or no effect on the spread of aggregate strengths (Weibull [beta] parameter). Based on these results, we proposed a three-parameter Weibull brittle fracture model for the tested sandy loam and silt loam soils that takes account of the effect of water content for a single aggregate size fraction. This model, in which only [alpha] depends on water content, explained on average 89% of the total variation in rupture energy. Further research is needed to fully investigate the influence of water content on the rupture energy of different-sized aggregates. Abbreviations: AIC, Akaike's information criterion; cdf, cumulative probability density function; MP-N1, moldboard plowing and 0 kg N [ha.sup.-1] [yr.sup.-1] treatment; MP-N4, moldboard plowing and 336 kg [ha.sup.-1] [yr.sup.-1] treatment; NT-N1, no-rillage and 0 kg N [ha.sup.-1] [yr.sup.-1] treatment; NT-N4, no-tillage and 336 kg N [ha.sup.-1] [yr.sup.-1] treatment; PAC, compacted treatment; REF, reference treatment.

Brittle fracture of soil aggregates is usually analyzed with the Weibull 'weakest-link' model. Failure is expressed in terms of a probability distribution function (pdf) of aggregate strengths. Traditionally a two-parameter Weibull model is filled to double log-transformed data with the Weibull parameters ([alpha] and [beta]) estimated using linear regression. The main objective of this study was to compare the goodness-of-fit for a three-parameter versus a two-parameter Weibull model. In addition, we compared three common methods of parameter estimation: linear regression, nonlinear regression, and maximum likelihood. The different models and methods of estimation were evaluated using previously published and unpublished aggregate rupture energy data from three contrasting soil types (Bygholm sandy loam, Maury silt loam, and Karnak silty clay). Overall, the goodness-of-fit was not markedly improved by using a three-parameter as compared with a two-parameter Weibull model. The choice of model had a significant effect on the parameter estimates. The three-parameter model produced lower estimates of [beta] than the two-parameter model. The data were always best fitted using nonlinear regression. Nonlinear regression also resulted in a greater power of distinction between management treatments and aggregate sizes for [alpha] on the Maury soil. We recommend fitting aggregate rupture data to a two-parameter Weibull model and estimating the model parameters using nonlinear regression.

Distributions of soil fragments (a mixture of primary aggregates resulting from tillage fragmentation) in seedbeds are known to influence emergence and early shoot and root growth of crops. However, it is not clearly understood which distribution model parameters the roots are responding to when water imbibition and nutrient availability are not limiting factors. The objective of this study was to determine the effect(s) of variation in geometric mean diameter (GMD) and log of the geometric standard deviation (LogGSD) taken from a lognormal model of soil fragment size distribution on maize (Zea mays L.) root elongation over a range of soil bulk density ([ρ.sub.b]) and air-filled porosity (AFP) levels. Root growth, determined 48 h after seedling emergence, was evaluated in a greenhouse experiment with artificially packed soil fragments sieved from a Maury silt loam (fine, mixed, semiactive, mesic Typic Paleudalf) under sod. Two experiments were conducted. The first consisted of a complete factorial combination of four compactive efforts (CEs) (0.0, 26.8, 53.5, and 107.0 kJ [m.sup.-3]) applied over four GMDs (4.3, 5.1, 6.8, and 9.0 mm) at a uniform LogGSD of 0.22. The second experiment was a complete factorial combination of the same four CE treatments applied over three LogGSD (0.22, 0.31, and 0.48) values at a uniform GMD of 5.1 mm. Increasing CE, GMD, or LogGSD caused the [ρ.sub.b] to increase. Maximum root elongation occurred at intermediate (5.1-6.8 mm) GMD values when 26.8 kJ [m.sup.-3] of CE was applied, corresponding to an average [ρ.sub.b] of 1.15 Mg [m.sup.-3]. There was no direct effect of the spread in soil fragment size (LogGSD) on root elongation. Total root length density (RLD) showed a quadratic response to [ρ.sub.b], relative bulk density ([ρ.sub.b]), bulk density divided by maximum Proctor density), or AFP levels reaching maximum elongation at values of 1.12 Mg [m.sup.-3], 0.78 [m.sup.3] [m.sup.-3], or 0.187 [m.sup.3] [m.sup.-3], respectively. These results suggest that maximum radicle elongation depends more on the size of soil fragments rather than on the spread in their size distribution. Bulk density seems to be a more relevant parameter than size or distribution of soil fragments in characterizing compacted seedbeds. Loose or highly compacted seedbeds are inadequate for maximal early growth of maize roots. Abbreviations: [ρ.sub.b], soil bulk density; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], relative bulk density; AFP, air-filled porosity; CE, compactive effort; GMD, geometric mean diameter; GWC, gravimetric soil water content; LogGSD, log of the geometric standard deviation; [P.sub.i], intrafragment porosity; RLD, root length density; TOC, total organic carbon., THE SPATIAL HETEROGENEITY of soil components, that is, soil structure, is a complex condition related to many biogeochemical processes (Dexter, 1988; Kay, 1990; Dexter, 1997; Angers and Caron, 1998; Kay [...]

We combined water activity meters and fractal modeling to facilitate rapid, physically based characterization of the soil water retention curve. Desorption data (6 points per sample) were collected in the tension (h) range 2.0 x [10.sup.2] to 1.5 x [10.sup.5] kPa using the gravimetric method to measure water content (w) and water activity meters to measure h. Thirty-two disturbed samples from a long-term nitrogen fertilization and tillage comparison study on a silt loam soil were analyzed. A new form of an established fractal equation was derived: w = [ah.sup.D-3] - 3 [[rho].sub.w]/[[rho].sub.s], where a is a compound parameter including the bulk density ([[rho].sub.b]) and air entry tension ([h.sub.a]), D is the mass fractal dimension, [[rho].sub.w] is the density of water, and [[rho].sub.s] is the particle density. This model was fitted to the measured water retention curves by nonlinear regression analysis. The a and D parameters were estimated, while [[rho].sub.w] and [[rho].sub.s] were fixed at 1.00 and 2.65 Mg [m.sup.-3], respectively. Convergence was always achieved and the equation fitted the data extremely well; residual sums of squares ranged from 1.2 x [10.sup.-6] to 6.7 x [10.sup.-5], with a median value of 2.2 x [10.sup.-5]. Estimates of a (0.62-0.74) and D (2.948-2.963) were physically reasonable, and sensitive to soil management practices. The a parameter increased, r = 0.80 (P < 0.01), (signifying decreasing [[rho].sub.b] and/or [h.sub.a]) whereas D decreased, r = -0.75 (P < 0.01), (signifying more rapid capillary drainage) with increasing soil carbon content. Additional research is needed to test this approach on other soil types, and to assess the influence of soil disturbance and variations in [[rho].sub.s] on the model's performance.

Tillage research has traditionally focused on mean effects; few studies have compared treatments in terms of their spatial variability. We applied several methods of time-space series analysis to investigate the spatial variability of gravimetric water content (w), volumetric water content ([theta]), bulk density ([rho]b) and total C (TC) in the upper 10 cm of long-term conventional-till (CT) and no-till (NT) soil management practices at Lexington, KY. The soil was a Maury silt loam (fine, mixed, semiactive, mesic Typic Paleudalf). Replicate transects were established in untracked interrows parallel to the direction of tillage in the CT practice. Each transect was 48.5 m long with 107 equally spaced sampling points. Soft spatial variability was higher under NT than under CT. Spectral analyses of variance identified significant differences between tillage treatments at frequencies

Dispersivity ([alpha]) is a required input parameter in solute-transport models based on the advection-dispersion equation (ADE). Normally is obtained from miscible-displacement experiments. This dependency on inverse procedures imposes a severe limitation on our predictive capability. If solute breakthrough curves and soil hydraulic properties were measured simultaneously, pedotransfer functions could be developed to predict [alpha] from independent measurements. In this study, short (6 cm long) undisturbed columns were employed to investigate the relationship between [alpha] and the water-retention curve as parameterized by the air-entry value ([[psi]sub.a]) and Campbell exponent (b). We worked with 69 columns from six soil types ranging in texture from loamy sand to silty clay, conventional-till and no-fill management practices, steady-state saturated flow conditions, and a step decrease in [CaCl.sub.2] concentration from 0.009 to 0.001 M. Breakthrough curves were measured by monitoring changes in effluent electrical conductivity using a computerized data acquisition system. Estimates of [alpha] (calculated using the method of moments) ranged from 1 to 192 mm for the six soil types. Stepwise multiple-regression analysis explained ~50% of the total variation in [alpha], and indicated that dispersion increased as [[psi]sub.a] and b increased. Since both [[psi]sub.a] and b increase with increasing clay content, [alpha] also increases moving from coarse- to fine-textured soils. Our regression equation can be used as a pedotransfer function to predict [alpha] from existing databases of soil hydraulic properties. Further research is needed to independently validate its predictive capability, and to develop strategies for upscaling the model predictions.

The compactability of a soil can be determined from parameters derived from laboratory compaction curves generated using the Proctor test. However, this is a destructive, time-consuming, and labor-expensive procedure. Our objective was to evaluate a new more rapid test to determine the maximum dry bulk density ([[rho].sup.Max.sub.b]) and soil water content at ([[rho].sup.Max.sub.b]) (SW[C.sub.Max]) from laboratory compaction curves using a mold with smaller dimensions than established for the standard procedure. Laboratory compaction curves were developed for nine soils with clay contents ranging from 232 to 385 g [kg.sup.-1] using the standard Proctor test procedure and the proposed procedure, which uses a 54-mm diam. mold and a 24.4 N rammer dropped from a height of 305 mm to produce a compactive effort (CE) of 109 kJ [m.sup.-3] [drop.sup.-1]. At a CE of 545 kJ [m.sup.-3], which is similar to the 540 kJ [m.sup.-3] produced by the standard test, the ([[rho].sup.Max.sub.b]) and SW[C.sub.Max] parameters derived from the proposed procedure were positively and significantly correlated with those derived from the standard Proctor test. With both procedures, ([[rho].sup.Max.sub.b]) decreased and SW[C.sub.Max], increased as the soil clay content increased. The small mold method did not affect the linear relationship between soil clay content and ([[rho].sup.Max.sub.b]). However, different relationships between SW[C.sub.Max] and clay content were observed, depending on the compaction procedure. Use of the small mold procedure requires less dry sieved soil, saves time, and labor in evaluating soil compactability. Based only on the reduction in rammer drops, use of the small mold procedure involves 15 times less labor requirements than the standard Proctor test.

Brittle fracture of heterogeneous materials such as dry soil aggregates is not well understood. The flaws in such materials occur as voids across a wide range of length scales. As a result, failure is an inherently variable phenomenon that is best analyzed using weakest-link (Weibull) theory combined with fractal geometry. According to this probabilistic approach, samples to be fractured are assumed to be exactly the same size. However, sample size variation can result in biased estimates of the model parameters, log([Alpha]), [Beta], and D. Our objective was to derive and test an equation for eliminating this bias by adjusting survival probabilities to a common value of sample length. Rupture energies were determined on 1920 air-dry soil peds collected from three long-term tillage treatments on a Maury silt loam (fine, mixed, mesic Typic Paleudalf). Following adjustment for sample size variation, characteristic strength, as parameterized by log([Alpha]), typically decreased with increasing ped size. Unbiased estimates of [Beta] indicated a wider spread of strengths than reported previously. Unbiased estimates of D, the fractal scaling parameter, were lower than those obtained using other methods, possibly because of renormalization with respect to the distribution of fragile elements instead of voids. Tillage treatment significantly influenced all three parameters, with the strength of no-till peds decreasing more rapidly with increasing size than conventional-till peds. Analysis of the relative errors associated with parameter estimates computed using the adjusted survival probabilities indicated that at least 20 samples per sieved fraction should be used as a standard in future brittle fracture studies on heterogeneous materials.

Measures of aggregate size and shape are needed in models to predict soil erosion, seed bed density, tensile strength, and solute diffusion. We hypothesized that size (i.e., equivalent cubic length [L.sub.c] = [[[l.sub.1]([l.sub.2]/[square root of 2])([l.sub.3]/[square root of 3])].sup.1/3], where [l.sub.1], [l.sub.2], and [l.sub.3] are the shortest, intermediate, and longest axial lengths, respectively) and shape (i.e., aspect ratios, [l.sub.1]/[l.sub.3] and [l.sub.2]/[l.sub.3]) vary with soil type and tillage treatment. Caliper measurements of [l.sub.1], [l.sub.2], and [l.sub.3] were made on 1800 aggregates from three sieved fractions (4-8, 8-16, and 16-31.5 mm) of a plowed, plowed plus disked, and no-till Maury silt loam (fine, mixed, mesic Typic Paleudalf) and a no-till Karnak silty clay (fine, montmorillonitic, nonacid, mesic Vertic Haplaquept). Overall means for [l.sub.1]/[l.sub.3] (0.57) and [l.sub.2]/[l.sub.3] (0.78) were close to those for a cube (0.58 and 0.82, respectively). The mean [l.sub.1]/[l.sub.3] for Karnak was 18% lower than the corresponding value for no-till Maury. The [L.sub.c] was lognormally distributed. The mean [log.sub.10]([L.sub.c]) increased from 0.67 in the smallest fraction to 1.19 in the largest fraction. Soil effects on [log.sub.10]([L.sub.c]) were most evident in the 8- to 16-mm fraction (1.01 for Karnak compared with 0.93 for no-till Maury). Tillage effects on [log.sub.10]([L.sub.c]) were most evident in the 4- to 8-mm fraction (0.68 for no-till Maury compared with 0.66 and 0.65 for plowed and plowed plus disked Maury, respectively). The equation [L.sub.c] = [x.sub.l][(m/[m.sub.m]).sup.1/3], where [x.sub.1] is lower sieve aperture, m is aggregate mass, and [m.sub.m] is modal mass, explained 96% of the variation in [L.sub.c]. This equation can be used to predict aggregate size from sieve aperture size and measurements of aggregate mass.

Parameters are needed to quantify tillage-induced changes in soil structure. Our objective was to see if fractal parameters from ped mass- and number-size relations could be applied to quantify such changes. Soil from long-term conventional (moldboard plow, MP, plus two passes with a disk harrow, DH) and no-till (NT) plots on a Maury silt loam (fine, mixed, mesic Typic Paleudalf) was air dried and sieved into six size classes ranging from 0.5-1 to 16-31.5 mm. The mass percentage and weight of 100 peds in each size class were used to calculate the mean mass per ped and number of peds as a function of scale. Fractal equations were then fitted to both data sets ([R.sup.2] always [greater than or equal to] 0.99). The resulting parameter estimates were [D.sub.m] (mass fractal dimension) and [k.sub.m] (mass of an aggregate of unit length) for the mass-size relation, and [D.sub.f] (fragmentation fractal dimension) and [k.sub.f] (number of fragments of unit length) for the number-size distribution. Significant tillage effects were observed for [D.sub.m], [D.sub.f], and [k.sub.f]. The [D.sub.m] increased from 2.88 for NT to [greater than or equal to] 2.94 for MP and DH, suggesting differential compaction of large aggregates by plowing and disking. The [D.sub.f] was always less than the [D.sub.m], as predicted by theory. Soil fragmentation was least for NT ([D.sub.f] = 2.02) and greatest for MP ([D.sub.f] = 2.55); for DH, the [D.sub.f] = 2.15 suggesting that disking broke up large fragments and/or coalesced small fragments following plowing. The [k.sub.f], which was shown to be nonlinearly related to the [D.sub.f], was least for MP and greatest for NT. Similar studies on other soils are needed to test the general applicability of these fractal relations to soil structure.

Comparison of the dry aggregate strength for soil, using the Weibull measurement model, with the parameters of rupture energy, alphaE and betaE, to the estimated tensile strength, alphaT and betaT, proved to be independent of soil type. In the study aggregate size, sand content and their interaction were responsible for the variation of 93% in log(alphaE) for soil properties and 60% for betaE.

The constant relating the tensile strength (T), specific rupture energy (E) and the strain at rupture epsilon(r) is determined to be 0.0002 cubic meters per gram for the air aggregates used in the study. The T and E could be used for fitting the Weibull's statistical brittle fracture model. The model also reveals that the parameters determined using E are highly correlated and that the relationship between them is linear.

An evaluation of the least limiting water range (LLWR) reveals that LLWR cannot be recommended as an index of structural quantity of soils without additional research. The soil structural form and soil water content yield the impedance, supplies of O2 and water. The least LLWR is the range in the soil water content associated with metric pressure, mechanical resistance and aeration, which are very low. The evaluation does not yield any confirmed results.

A new mass-size-based framework was designed without presuming that for the determination of the fractal dimension D of soil aggregates, the probability of failure, P, and bulk density rho, are scale-invariant. The value of Dr were compared with those obtained through three different techniques. One used manually-estimated aggregate numbers in a number-size relation model. Another assumed scale-invariant rho. The third mass-size-based model had an upper limit to D, and assumed scale-invariant rho.

Investigations of the major soil structural properties of dry aggregate size and strength distributions used multifractals to predict the properties, as a basis for the development of a theoretical framework. A technique to use tensile strength data was evolved to compute the strength of dry aggregates of normalized equivalent cubic length x, which was expressed as a probability of failure (P(x.)) sigma.

The fractal dimension is used to characterize soil aggregate-size distributions. Results show that there is a one-to-one correspondence between the fractal dimensions estimated from mass-size distribution and those computed from actual number-size distribution data. Therefore, the fractal dimension can be estimated from mass-distribution data with the aggregate-size range of 5.0x10(-1) to 3.2x10(1) millimeter.

Concern about soil and groundwater pollution has resulted in numerous studies focused on solute transport. The objectives of our study were to investigate the effect of soil type and land-use management on solute movement. Transport of water and [Cl.sup.-] were measured through intact blocks of Maury (fine, mixed, semiactive, mesic Typic Paleudalf) and Cecil (fine, kaolinitic, thermic Typic Kanhapludult) soils, under steady-state, unsaturated flow conditions. Three replicate blocks for the Maury soil and two replicate blocks for the Cecil soil were studied per land-use treatment. The land-use treatments were conventional-till corn (Zea mays L.) production and long-term grass pasture. Individual blocks were instrumented with time domain reflectometry (TDR) probes at the 5-, 15-, and 25-cm depths. The effluent [Cl.sup.-] and TDR breakthrough curves were fitted using the convection dispersion equation (CDE); the estimated parameters were pore water velocity (v), dispersion coefficient (D), and, for the TDR breakthrough curves, maximum bulk electrical conductivity ([BEC.sub.max]). The CDE fitted the data very well, with model [R.sup.2] values ranging from 0.971 to 0.999. Volumetric water content ([Theta]), total porosity, the soil water retention curve, and saturated hydraulic conductivity were determined on the same blocks. Volumetric water content increased ([R.sup.2] = 0.25) as the slope of the water retention curve decreased. Increasing [Theta] resulted in decreasing v ([R.sup.2] = 0.20) and thus, because of the linear relationship between D and v ([R.sup.2] = 0.26), decreasing D. Structural controls on solute dispersion in this study were mainly indirect, and related to variations in water content produced by differences in pore-size distribution.