Markus Tuller

Adsorption and capillary condensation in porous media: Liquid retention and interfacial configurations in angular pores

Conventional models of liquid distribution, flow, and solute transport in partially saturated porous media are limited by the representation of media pore space as a bundle of cylindrical capillaries (BCC). Moreover, the capillary model ignores the dominant contribution of adsorptive surface forces and liquid films at low potentials. We propose two new complementary elements for improving our understanding of liquid configuration in porous media: (1) an approach for considering the individual contributions of adsorptive and capillary forces to the matric potential and (2) a more realistic model for pore space geometry. Modern interface science formalism is applied to determine the thickness of adsorbed liquid films as a function of thermodynamic conditions and specific surface area of the medium. The augmented Young-Laplace (AYL) equation provided the necessary framework for combining adsorptive and capillary processes. A new pore space geometry composed of an angular pore cross section (for capillary processes) connected to slit-shaped spaces with internal surface area (for adsorption processes) offers a more realistic representation of natural porous media with explicit consideration of surface area (absent in the standard BCC model). Liquid-vapor configuration, saturation, and liquid-vapor interfacial area were calculated for different potentials and pore (unit cell) dimensions. Pore dimensions may be easily related to measurable soil properties such as specific surface area and porosity. Rigorous calculations based on the AYL equation were simplified and led to the development of algebraic expressions relating saturation and interfacial area of liquid in the proposed pore space geometry to chemical potential. These simple expressions are amenable to upscaling procedures similar to those presently used with the BCC model.

Liquid Retention and Interfacial Area in Variably Saturated Porous Media: Upscaling from Single Pore to Sample Scale Model

A new model for liquid configuration in angular pore space considering both capillary and adsorptive contributions was proposed as an alternative to the conventional bundle of capillaries representation. In this study we develop a statistical framework for upscaling pore-scale processes to represent a sample-scale response of variably saturated porous medium. The representation of pore size distribution by the gamma distribution enables derivation of closed-form expressions for sample-scale liquid retention and liquid-vapor interfacial area. The statistical framework calculates the expected values of liquid configuration as a function of pore geometry and chemical potential considerations. Media properties are used to estimate upscaling parameters by matching model predictions with measured retention data subject to specific surface area constraint. Additionally, a method for estimating liquid-solid adsorption behavior for the medium is proposed. Model predictions compare favorably with measured retention data, yielding a similar close fit as obtained with the van Genuchten parametric model. Liquid-vapor interfacial area as a function of chemical potential is readily calculated using the estimated retention parameters. Model calculations of liquid-vapor interfacial area for sand show reasonable agreement with measurements obtained with surface-active tracers. The contribution of liquid films dominates the total liquid-vapor interfacial area and often surpasses the capillary contribution (curved menisci) by several orders of magnitude. This illustrates potential limitations in using cylindrical pore network modeling of interfacial area for multiphase flow predictions. The detailed picture of liquid vapor interfaces provides a sound basis for unsaturated hydraulic conductivity calculations in the sample cross section (i.e., neglecting network effects) and offers insights into microbial habitats and related exchange processes in partially saturated porous media.

Hydraulic conductivity of variably saturated porous media: Film and corner flow in angular pore space

Many models for hydraulic conductivity of partially saturated porous media rely on oversimplified representation of the pore space as a bundle of cylindrical capillaries and disregard flow in liquid films. Recent progress in modeling liquid behavior in angular pores of partially saturated porous media offers an alternative framework. We assume that equilibrium liquid-vapor interfaces provide well-defined and stable boundaries for slow laminar film and corner flow regimes in pore space comprised of angular pores connected to slit-shaped spaces. Knowledge of liquid configuration in the assumed geometry facilitates calculation of average liquid velocities in films and corners and enables derivation of pore-scale hydraulic conductivity as a function of matric potential. The pore-scale model is statistically upscaled to represent hydraulic conductivity for a sample of porous medium. Model parameters for the analytical sample-scale expressions are estimated from measured liquid retention data and other measurable medium properties. Model calculations illustrate the important role of film flow, whose contribution dominates capillary flow (in full pores and corners) at relatively high matric potentials (approximately −100 to −300 J kg−1, or −1 to 3 bars). The crossover region between film and capillary flow is marked by a significant change in the slope of the hydraulic conductivity function as often observed in measurements. Model predictions are compared with the widely applied van Genuchten–Mualem model and yield reasonable agreement with measured retention and hydraulic conductivity data over a wide range of soil textural classes.

Flow in unsaturated fractured porous media: Hydraulic conductivity of rough surfaces

The general trend in models for flow in unsaturated fractured porous media is to regard desaturated fractures as nonparticipating elements that impede flow. Mounting experimental and theoretical evidence shows that fractures retain and conduct liquid in the form of film and partially filled corner flow to a relatively low degree of saturation. A simple geometrical model for rough fracture surfaces is developed offering a tractable geometry for calculations of surface liquid storage due to adsorbed films and capillary menisci. Assuming that under slow laminar flow the equilibrium liquid configurations on the fracture surface are not modified significantly, the average hydraulic conductivities for film and corner flows were derived and used as building blocks for a representative fracture roughness element and an assemblage of statistically distributed surface roughness elements. Calculations for a single representative element yielded excellent agreement with surface storage and unsaturated hydraulic conductivity measurements of Tokunaga and Wan [1997]. A statistical representation of surface roughness using a gamma distribution of pit depths resulted in closed-form expressions for unsaturated hydraulic conductivity averaged across the fracture length (transverse to flow) or weighted by the liquid cross section occupying the fracture surface. An important attribute of the surface roughness model is the direct link between fracture surface and matrix processes unified by the matric potential. The proposed model represents a first step toward development of a comprehensive approach for liquid retention and hydraulic conductivity of unsaturated fractured porous media based on details of liquid configuration for different matric potentials.

Unsaturated hydraulic conductivity of structured porous media: A review of liquid configuration based models

Common approaches for modeling hydraulic functions of unsaturated structured porous media (SPM) rely on macroscopic continuum representation, where parameterization schemes and constitutive relationships originally developed for homogeneous porous media are extended to represent hydraulic behavior of dual (or multi) continuum SPM. Such models often result in inconsistencies due to lack of consideration of structural pore space geometry and the neglect of underlying physical processes governing liquid retention and flow under unsaturated conditions. We review a new framework that considers equilibrium liquid configurations in dual continuum pore space as the basis for calculation of liquid saturation and subsequent introduction of hydrodynamic considerations. The SPM pore space is represented by a bimodal distribution of pore sizes, reflecting two disparate populations of matrix and structural pores. Three steady-state and laminar flow regimes are considered to derive unsaturated hydraulic conductivity functions: (i) flow in completely filled pore spaces, (ii) corner flow in partially filled pores and grooves, and (iii) film flow on solid surfaces. Two key assumptions are used in deriving the average cross-sectional flow velocities in these regimes: (i) that equilibrium liquid–vapor interfaces remain stable under slow laminar flows and (ii) that flow pathways are parallel. Liquid–vapor interfacial configurations for different matric potentials are calculated and statistically upscaled to derive sample-scale saturated and unsaturated hydraulic conductivity from velocity expressions weighted by the appropriate liquid-occupied cross-sectional areas, neglecting three-dimensional (3-D) network effects. Similarly, the hydraulic functions for matrix and structural pores are derived separately and later combined by weighting the individual contributions by the porosities of the associated pore spaces. A parameter estimation scheme was developed to calculate liquid saturation and to predict sample-scale unsaturated hydraulic conductivity. Model evaluation using measured data for homogeneous porous media, fractured welded tuff, and macroporous and aggregated soils shows favorable agreement (within the limitations of model assumptions). Effects of nonequilibrium conditions between matrix and structural pore domains on the hydraulic conductivity and approximate consideration of 3-D network effects are discussed.

Hydraulic functions for swelling soils: pore scale considerations

Abstract Changes in volume and pore space induced by the shrink–swell behavior of clay minerals present a challenge to predictive modeling of hydraulic properties of clayey soils. Despite well-developed theory for crystalline and osmotic swelling of clay minerals at the scale of individual clay lamellae, their translation to prediction of hydraulic properties of swelling soils is limited. In this study we propose a framework that combines physico-chemical processes with pore scale geometrical, hydrostatic, and hydrodynamic considerations toward prediction of constitutive hydraulic relationships for swelling porous media. Variations in pore space are modeled by considering the soil clay fabric as an assembly of colloidal-size tactoids with lamellar structure. The arrangement of clay tactoids and the spacing between individual lamellae are functions of primarily clay hydration state quantifiable via the disjoining pressure that is dominated by a large electrostatic repulsive component. Solution chemistry and clay type are also considered. Silt and sand textural constituents are represented as rigid spheres interspaced by clay fabric in two basic configurations of ‘expansive’ and ‘reductive’ unit cells. Bulk soil properties such as clay content, porosity and surface area serve as constraints for the pore-space geometry. Liquid saturation within the idealized pore space is calculated as a function of chemical potential considering volume changes due to clay shrink–swell behavior. Closed-form expressions for prediction of saturated hydraulic conductivity are derived from calculations of average flow velocities in ducts and between parallel plates, and invoking proportionality between water flux density and unit hydraulic gradient. Preliminary model calculations compare favorably with published data, and show great potential for upscaling considerations.

Hydraulic conductivity of partially saturated fractured porous media : flow in a cross-section

Abstract Standard models for hydraulic functions of partially saturated fractured porous media (FPM) often rely on macroscopic continuum representation and embrace constitutive relationships originally developed for homogeneous porous media to describe hydraulic behavior of dual (or multi) continua FPM. Such approaches lead to inconsistencies due to neglect of underlying physical processes governing liquid retention and flow in the vastly different pore spaces. We propose a framework that considers equilibrium liquid configurations in dual continuum pore space as the basis for calculation of liquid saturation and introduction of hydrodynamic considerations. FPM cross-sectional pore space is represented by a bimodal size distribution reflecting two disparate populations of matrix pores and fracture apertures (with rough-walled surfaces). Three laminar flow regimes are considered, flow in: (1) completely liquid filled pore spaces; (2) partially filled pores or grooves bounded by liquid–vapor interfaces; and (3) surface film flow. Assuming that equilibrium liquid–vapor interfaces remain stable under slow laminar flows, sample-scale unsaturated hydraulic conductivity is derived from average velocity expressions for each flow regime weighted by the appropriate liquid-occupied cross-sectional areas (neglecting 3-D network effects). A parameter estimation scheme was developed and evaluated using two data sets. The results point to the critical need for definitive data sets for improved understanding of flow in partially saturated FPM. Hydraulic conductivity functions for non-equilibrium conditions between matrix and fracture domains are discussed. Approximations for inclusion of network effects are proposed based on direct measurement of saturated hydraulic conductivity supplemented by theoretical considerations applying critical path analysis.

WATER RETENTION AND CHARACTERISTIC CURVE

A soil-water characteristic (SWC) curve describes the amount of water retained in a soil (expressed as mass or volume water content, m or v) under equilibrium at a given matric potential. An SWC is an important hydraulic property, related to size and connectedness of pore spaces, hence strongly affected by soil texture and structure, and by other constituents, including organic matter. Modeling water distribution and flow in partially saturated soils requires knowledge of the SWC, therefore plays a critical role in water management and in prediction of solute and contaminant transport in the environment. Typically a SWC is highly nonlinear and is relatively difficult to obtain accurately. Because the matric potential extends over several orders of magnitude for the range of water contents commonly encountered in practical applications, the matric potential is often plotted on a logarithmic scale. Figure 1 depicts representative SWC curves for soils of different textures, demonstrating the effects of porosity (saturated water content) and

Economical and environmental implications of solid waste compost applications to agricultural fields in Punjab, Pakistan

Application of municipal solid waste compost (MSWC) to agricultural soils is becoming an increasingly important global practice to enhance and sustain soil organic matter (SOM) and fertility levels. Potential risks associated with heavy metals and phosphorus accumulations in surface soils may be minimized with integrated nutrient management strategies that utilize MSWC together with mineral fertilizers. To explore the economic feasibility of MSWC applications, nutrient management plans were developed for rice-wheat and cotton-wheat cropping systems within the Punjab region of Pakistan. Three-year field trials were conducted to measure yields and to determine the economic benefits using three management strategies and two nutrient doses. Management strategies included the application of mineral fertilizers as the sole nutrient source and application of mineral fertilizers in combination with MSWC with and without pesticide/herbicide treatments. Fertilizer doses were either based on standard N, P and K recommendations or on measured site-specific soil plant available phosphorus (PAP) levels. It was found that combining MSWC and mineral fertilizer applications based on site-specific PAP levels with the use of pesticides and herbicides is an economically and environmentally viable management strategy. Results show that incorporation of MSWC improved soil physical properties such as bulk density and penetration resistance. The PAP levels in the surface layer increased by the end of the trials relative to the initial status. No potential risks of heavy metal (Zn, Cd, Cr, Pb and Ni) accumulation were observed. Treatments comprised of MSWC and mineral fertilizer adjusted to site-specific PAP levels and with common pest management showed highest cumulative yields. A basic economic analysis revealed a significantly higher cumulative net profit and value-to-cost ratio (VCR) for all site-specific doses.

Relating soil specific surface area, water film thickness, and water vapor adsorption

Estimation of soil specific surface area (SSA) and dry-end water vapor adsorption are important for porous media characterization and for prediction of water and vapor fluxes in arid environments. The objective of the presented study was to model water adsorption, film thickness, and SSA based on the t-curve theory originally developed for N2 adsorption. Data from 21 source soils with clay contents ranging from 0.6 to 52.2% were used to estimate specific surface area based on water retention, a t-curve type method, the linear prediction method, and a simplified monolayer method. The water retention and the t-curve methods were found to be mathematically analogous and were among the most accurate with regard to correlation coefficient (r = 0.97) and root-mean-square error (RMSE = 11.36 × 103 m2/kg) when compared to measurements obtained with the standard ethylene glycol monoethyl ether (EGME) method. The corrected t-curve method significantly overestimated SSA when compared to EGME data. Comparison of all considered methods with N2-BET (BET) measurements disclosed lower correlation coefficients. For soil studies, the vapor adsorption in conjunction with the t-curve or water retention methods should be preferred for SSA estimation as they show much higher correlation with soil clay content and EGME measurements.