Torben Olesen

Modeling diffusion and reaction in soils : IX. The Buckingham-Burdine-Campbell equation for gas diffusivity in undisturbed soil

Accurate description of gas diffusivity (ratio of gas diffusion coefficients in soil and free air, D s /D 0 ) in undisturbed soils is a prerequisite for predicting in situ transport and fate of volatile organic chemicals and greenhouse gases. Reference point gas diffusivities (R p ) in completely dry soil were estimated for 20 undisturbed soils by assuming a power function relation between gas diffusivity and air-filled porosity (e). Among the classical gas diffusivity models, the Buckingham (1904) expression, equal to the soil total porosity squared, best described R p . Inasmuch as our previous works (Parts III, VII, VIII) implied a soil-type dependency of D s /D 0 (e) in undisturbed soils, the Buckingham R p expression was inserted in two soil- type-dependent D s /D 0 (e) models. One D s /D 0 (e) model is a function of pore-size distribution (the Campbell water retention parameter used in a modified Burdine capillary tube model), and the other is a calibrated, empirical function of soil texture (silt + sand fraction). Both the Buckingham-Burdine-Campbell (BBC) and the Buckingham/soil texture-based D s /D 0 (e) models described well the observed soil type effects on gas diffusivity and gave improved predictions compared with soil type independent models when tested against an independent data set for six undisturbed surface soils (11-46% clay). This study emphasizes that simple but soil-type-dependent power function D s /D 0 (e) models can adequately describe and predict gas diffusivity in undisturbed soil. We recommend the new BBC model as basis for modeling gas transport and reactions in undisturbed soil systems.

Modeling Diffusion and Reaction in Soils: VII. Predicting Gas and Ion Diffusivity in Undisturbed and Sieved Soils

The classical Penman (1940) and Millington-Quirk (1960, 1961) diffusivity models were transformed into general form by introducing a tortuosity parameter, m. Compared with measured diffusivities close to phase saturation (soil-water and soil-air saturation for ion and gas diffusivity, respectively), the Penman (1940) model was superior to the Millington-Quirk models independent of diffusion type. The combined use of the Penman model to predict the diffusivity at phase saturation together with a general Millington-Quirk model to predict relative decrease in diffusivity with decreasing phase content was labeled the Penman-Millington-Quirk (PMQ) model. The best fit of the new PMQ model to measured data was obtained with m = 3 (high tortuosity) and m = 6 (medium tortuosity) for gas diffusivity in undisturbed and sieved soils, respectively, and m = 1 (high tortuosity) for ion diffusivity. Measurements did not suggest a significant difference between ion diffusivity in undisturbed, sieved, or aggregated soils. The differences in m-values between diffusion types are likely caused by different diffusion pathways and geometries for ion and gas diffusivity as well as a large effect of soil heterogeneity and spatial variability on gas diffusivity. The PMQ model predicted gas diffusivity in sieved and undisturbed soil well, but a soil-type dependent model (Part IV ofthis series) was superior for predicting ion diffusivity. The new models seem promising for more accurately predicting gas and ion diffusion and, therefore, for improving simulations of diffusion-constrained chemical and biological reactions in soils.

Gas Permeability in Undisturbed Soils: Measurements and Predictive Models

Accurate prediction of changes in the gas permeability during variable soil-moisture conditions is a prerequisite for improved simulation and design of soil-venting systems for removal of volatile organic chemicals in polluted soils. Air permeability, k, as a function of soil air-filled porosity,

O2 uptake, C metabolism and denitrification associated with manure hot-spots

O2, C and N metabolism in organic hot-spots (sites where the intensity of microbial respiration creates a high O2 demand) was studied with fresh or anaerobically digested liquid cattle manure as substrates. A gel-stabilized mixture of soil and manure, 16 mm thick, was sandwiched between layers of soil with a water content adjusted to field capacity, and incubated at 15°C for up to 3 wk. When fresh manure was used, O2 microprofiles demonstrated an O2 penetration into the hot-spot of < 1 mm after 1–3 d, increasing to ca. 2 mm after 3 wk. During this time, O2 uptake rates decreased from 100–150 to ca. 50 nmol O2 cm−2 h−1. With digested manure, the lower C availability in this substrate resulted in O2 penetration depths of 3–4 mm and O2 uptake rates of <30 nmol O2 cm−2 h−1 throughout the 3 wk. Maximum denitrification rates were also consistently lower with digested manure (4 nmol N cm−2 h−1) than with fresh manure (18 nmol N cm−2 h−1). A numerical model of NO3− transport indicated that denitrification was limited by the availability of NO3− during the first week in the fresh manure treatment, and that the soil was the only significant source of NO3− during at least 3 d; after this time nitrification at the soil-manure interface became increasingly important. After the first week with fresh manure, and throughout the experiment with digested manure, C availability apparently regulated denitrification.

MODELING DIFFUSION AND REACTION IN SOILS: X. A UNIFYING MODEL FOR SOLUTE AND GAS DIFFUSIVITY IN UNSATURATED SOIL

Diffusion processes in the soil water and air phases often govern transport and fate of nutrients, pesticides, and toxic chemicals in the vadose zone. This final paper in a 10-part series on diffusion-reaction processes in soils concerns the development of a unifying model platform for predicting solute and gas diffusion coefficients as functions of fluid-phase (water or air) content and pore-size distribution in unsaturated soils. We find that the Buckingham (1904) expression predicts solute diffusivities in water-saturated porous media more accurately than other classical expressions and, extended with a pore-size distribution-based term, yields a new and accurate model for solute diffusivity in unsaturated soil. The same was shown for gas diffusivity in undisturbed soil in Part IX of this series. Thus, the predictive diffusivity models can be rewritten in a common form with two model parameters that vary between solute and gas diffusivity and, in the case of gas diffusivity, also between undisturbed and repacked soil. It is suggested that the two parameters in this unified diffusivity model (UDM) represent porous media (solids-induced) tortuosity (T) and water-induced fluid phase disconnectivity (W), respectively, with W increasing with clay content for solute diffusion but being constant (repacked soil) or decreasing (undisturbed soil) for gas diffusion. Tested against data for 77 soils, the UDM model was markedly more accurate than commonly used soil-type independent models, with 35–50% (gas diffusivity) and 75% (solute diffusivity) reduction in root mean square error of prediction. The use of the new UDM to predict effective diffusion of sorbing chemicals in the soil water and air phases is illustrated. The UDM concept enables a new definition of the relative diffusion coefficient in soil, i.e. relative to the diffusion coefficient in a fluid-saturated porous media instead of in free water or air. This provides new possibilities for analyzing tortuosity phenomena in the soil water and air phases and their effects on diffusive and convective transport parameters in unsaturated soil.

PREDICTIVE-DESCRIPTIVE MODELS FOR GAS AND SOLUTE DIFFUSION COEFFICIENTS IN VARIABLY SATURATED POROUS MEDIA COUPLED TO PORE-SIZE DISTRIBUTION: I. GAS DIFFUSIVITY IN REPACKED SOIL

The soil gas and solute diffusion coefficients and their dependency on soil total porosity (&PHgr;), fluid-phase (air or water) contents, and pore-size distribution largely control chemical release, transport, and fate in soil. The diffusion coefficients hereby play a key role in both local and global environmental issues including spreading, biodegradation and volatilization of hazardous chemicals at polluted soil sites, and soil uptake, production, and emission of greenhouse gases. In a series of papers, we present new advances in describing and predicting the gas and solute diffusion coefficients in variably saturated porous media, carefully distinguishing between repacked and undisturbed media. Also, we establish direct links between gas and solute diffusivity and pore-size distribution, with further links to pore continuity and tortuosity. In this first paper, a porosity correction term is added to a recently presented model for predicting gas diffusivity in repacked soil. The obtained POrosity-Enhanced (POE) model assumes that increased &PHgr; creates additional interconnectivity between air-filled pores. The POE model is tested against data for 18 repacked soils ranging from 0 to 54% clay, including new data measured in this study for both noncompacted and compacted, high-porosity soils. The POE model accurately predicts gas diffusivity across a wide &PHgr; range up to 0.75 m3 m−3, whereas the original model is accurate only for &PHgr; up to 0.55 m3 m−3. A unifying, two-parameter function for gaseous phase pore continuity (fg) is suggested. The fg function illustrates developments in gas diffusivity models during the last century, including assumptions behind the increasingly precise prediction models for repacked soil. Last, the POE model is coupled with the widely used van Genuchten (vG) soil-water characteristic model, hereby establishing an accurate and predictive link between soil gas diffusivity and pore-size distribution. The closed-form POE-vG gas diffusivity model is highly useful to evaluate effects of pore-size distribution and soil type on gas diffusivity and gas transport in repacked soil systems.

Constant Slope Impedance Factor Model For Predicting The Solute Diffusion Coefficient In Unsaturated Soil

Solute diffusivity (ratio of diffusion coefficients in soil and free water, DS/D0) is markedly soil-type dependent. Soil texture and pore size distribution govern the threshold soil-water content (&thetas;th) where DS/D0 approaches zero as a result of discontinuous diffusion pathways. In a recent study (Soil Science 161:633-645), we suggested that &thetas;th can be predicted from the soil-water characteristic curve (SWC) based on the Campbell pore size distribution parameter, b. In this study, the &thetas;th-b expression was recalibrated based on diffusivity data for three soils (Hiroshima sand, Foulum loamy sand, and Yolo loam) measured in this study plus 20 soils reported in the literature, obtaining &thetas;th=0.020b. As the SWC is often not measured, a second &thetas;th expression that requires only knowledge of soil texture and bulk density was calibrated from measured data. A third expression, including both soil texture, bulk density, and Campbell b, was also calibrated and gave the most accurate description of &thetas;th. The solute impedance factor (ratio of diffusivity by volumetric soil-water content), f1 = DS/(&thetas; D0), was shown to increase linearly with the water content available for diffusion, &thetas;a=&thetas;−&thetas;th. The slopes of the f1-&thetas;a relations were similar for most soils and did not exhibit soil-type dependency. Based on this, a so-called constant slope impedance factor (CSIF) model to predict DS(&thetas;a)/D0 is presented. The model can be used in combination with any of the three suggested &thetas;th expressions. Combined with the soil-texture/bulk-density dependent &thetas;th expression, the model accurately predicted solute diffusivities for three independent soils for which the SWC were not known.

Modeling diffusion and reaction in soils : IV. New models for predicting ion diffusivity

Salt diffusivity as a function of soil-water content and soil-water characteristic curves was measured on three sieved soils of different texture. The data were used together with ion diffusivity data for sieved soils from literature to test different diffusivity models. No significant differences between salt-, counter-, and self-diffusion data were observed, suggesting an insignificant effect of diffusion type in most cases, two new models for predicting ion diffusivity, based on the Campbell soil-water retention model parameter b, were proposed : (i) a model with a b-dependent threshold water content θ th (soil-water content where the ion diffusivity approaches zero) and (ii) a power function model where the power term η, representing the liquid phase tortuosity, is a linear function of b. In both models, the diffusivity at the soil-water content equal to the total porosity was estimated from a simple impedance factor model. Both b-dependent models gave improved predictions of ion diffusivity compared with existing models. The b-dependent power function model with η = 0.3b gave better predictions than both the θ th -dependent model and a recent model for gas diffusivity in undisturbed soil (Part III of this series, Soil Sci. 161 :366-375) where the tortuosity term η was best described as a function of b -1 . For use of the new b-dependent ion diffusivity models in the absence of soil-water characteristic data, it is shown that b can be fairly accurately estimated from soil texture.

PREDICTIVE-DESCRIPTIVE MODELS FOR GAS AND SOLUTE DIFFUSION COEFFICIENTS IN VARIABLY SATURATED POROUS MEDIA COUPLED TO PORE-SIZE DISTRIBUTION: III. INACTIVE PORE SPACE INTERPRETATIONS OF GAS DIFFUSIVITY

Accurate description of the soil-gas diffusion coefficient (DP) as a function of air-filled (&egr;) and total (&PHgr;) porosities is required for studies of gas transport and fate processes. After presenting predictive models for DP in repacked and undisturbed soils (Part I and II), this third paper takes a more descriptive approach allowing for the inclusion of inactive air-filled pore space, &egr;in. Three model-based interpretations of &egr;in are presented: (1) a simple power-law model (labeled Millington-Call) with the exponent (V) taken from Millington (1959; Science 130:100-102), and expanded with a constant &egr;in term (= 0.1 m3 m−3), (2) a model (SOLA) based on analogy with solute diffusion and assuming a linear increase in pore continuity from zero at the threshold air-filled porosity where gas diffusion ceases (&egr;th) to a maximum at &egr; = &PHgr;, (3) a power-law model (VIPS) assuming variable &egr;in that linearly decreases from a maximum at &egr; = &egr;th to zero at &egr; = &PHgr;. Assuming &egr;th = 0.1 m3 m−3, all three models satisfactorily predicted DP in 18 repacked soils. The difference between the three models is mainly pronounced for higher-&PHgr; soils, and each model has its own advantage. The SOLA model together with similar models for solute diffusivity allows a direct comparison of pore continuity in the soil gaseous and liquid phases, suggesting large differences in tortuosity and inactive fluid-phase between the two phases. The low-parameter Millington-Call model could account for variability in measured DP along a field transect (Yolo, California) by varying &egr;in with ±0.03 m3 m−3 and is applicable for stochastic gas transport simulations at field scale. The mathematically flexible VIPS model highly accurately fitted DP(&egr;) data for undisturbed soil, illustrating the large possible variations in &egr;th and V. The VIPS model is coupled with the van Genuchten (vG) soil-water characteristic model, yielding a closed-form expression for DP as a function of soil-water matric potential. The VIPS-vG model is useful to illustrate the combined effects of pore size distribution and inactive pore space on soil-gas diffusivity.

Modified half-cell method for measuring the solute diffusion coefficient in undisturbed, unsaturated soil.

Predictive models for the solute diffusion coefficient, D S , dependency on volumetric soil-water content, 0, are often applied in simulations of solute transport and fate in natural, undisturbed soils. However, all available D S (θ) models have been developed from measurements on sieved, repacked soil. In this study, D S for chloride was measured in both repacked and undisturbed loamy sands at different soil-water contents. The measurements on undisturbed soil were carried out using a modified half-cell method, where the source half-cell is a sieved and repacked soil core and the other half-cell is an undisturbed soil core. Thus, the problems of (i) incomplete contact area at the interface between undisturbed half-cells and (ii) potentially different diffusion properties in undisturbed half-cells can be avoided. The modified half-cell method requires that the diffusion coefficient in sieved, repacked soil is determined separately and that the experimental data is analyzed with a numerical solution to the diffusion equation. No significant difference in chloride D S (θ) between undisturbed and sieved, repacked soil was observed for a Danish (Foulum) loamy sand and a Japanese (Hiroshima) loamy sand. A recently presented soil type dependent D S (θ) model, derived from repacked soil data, shows it to be applicable also for predicting solute diffusion coefficients in natural, undisturbed soils.