Characteristic lengths affecting evaporative drying of porous media

Statistical mechanics of unsaturated porous media

We explore a mean-field theory of fluid imbibition and drainage through permeable porous solids. In the limit of vanishing inertial and viscous forces, the theory predicts the hysteretic "retention curves" relating the capillary pressure applied across a connected domain to its degree of saturation in wetting fluid in terms of known surface energies and void space geometry. To avoid complicated calculations, we adopt the simplest statistical mechanics, in which a pore interacts with its neighbors through narrow openings called "necks," while being either full or empty of wetting fluid. We show how the main retention curves can be calculated from the statistical distribution of two dimensionless parameters λ and α measuring the specific areas of, respectively, neck cross section and wettable pore surface relative to pore volume. The theory attributes hysteresis of these curves to collective first-order phase transitions. We illustrate predictions with a porous domain consisting of a random packing of spheres, show that hysteresis strength grows with λ and weakens as the distribution of α broadens, and reproduce the behavior of Haines jumps observed in recent experiments on an ordered pore network.

Visualization of water drying in porous materials by X-ray phase contrast imaging

We present in this study results from X-ray tomographic microscopy with synchrotron radiation performed both in attenuation and phase contrast modes on a limestone sample during two stages of water drying. No contrast agent was used in order to increase the X-ray attenuation by water. We show that only by using the phase contrast mode it is possible to achieve enough water content change resolution to investigate the drying process at the pore-scale. We performed 3D image analysis of the time-differential phase contrast tomogram. We show by the results of such analysis that it is possible to obtain a reliable characterization of the spatial redistribution of water in the resolved pore system in agreement with what expected from the theory of drying in porous media and from measurements performed with other approaches. We thus show the potential of X-ray phase contrast imaging for pore-scale investigations of reactive water transport processes which cannot be imaged by adding a contrast agent for exploiting the standard attenuation contrast imaging mode.

Exploring the Effects of Atmospheric Forcings on Evaporation: Experimental Integration of the Atmospheric Boundary Layer and Shallow Subsurface

Evaporation is directly influenced by the interactions between the atmosphere, land surface and soil subsurface. This work aims to experimentally study evaporation under various surface boundary conditions to improve our current understanding and characterization of this multiphase phenomenon as well as to validate numerical heat and mass transfer theories that couple Navier-Stokes flow in the atmosphere and Darcian flow in the porous media. Experimental data were collected using a unique soil tank apparatus interfaced with a small climate controlled wind tunnel. The experimental apparatus was instrumented with a suite of state of the art sensor technologies for the continuous and autonomous collection of soil moisture, soil thermal properties, soil and air temperature, relative humidity, and wind speed. This experimental apparatus can be used to generate data under well controlled boundary conditions, allowing for better control and gathering of accurate data at scales of interest not feasible in the field. Induced airflow at several distinct wind speeds over the soil surface resulted in unique behavior of heat and mass transfer during the different evaporative stages.

Simple analytical model of evapotranspiration in the presence of roots

Evaporation of water out of a soil involves complicated and well-debated mechanisms. When plant roots are added into the soil, water transfer between the soil and the outside environment is even more complicated. Indeed, plants provide an additional process of water transfer. Water is pumped by the roots, channeled to the leaf surface, and released into the surrounding air by a process called transpiration. Prediction of the evapotranspiration of water over time in the presence of roots helps keep track of the amount of water that remains in the soil. Using a controlled visual setup of a two-dimensional model soil consisting of monodisperse glass beads, we perform experiments on actual roots grown under different relative humidity conditions. We record the total water mass loss in the medium and the position of the evaporating front that forms within the medium. We then develop a simple analytical model that predicts the position of the evaporating front as a function of time as well as the total amount of water that is lost from the medium due to the combined effects of evaporation and transpiration. The model is based on fundamental principles of evaporation fluxes and includes empirical assumptions on the quantity of open stomata in the leaves, where water transpiration occurs. Comparison between the model and experimental results shows excellent prediction of the position of the evaporating front as well as the total mass loss from evapotranspiration in the presence of roots. The model also provides a way to predict the lifetime of a plant.

Drying by cavitation and poroelastic relaxations in porous media with macroscopic pores connected by nanoscale throats

We investigate the drying dynamics of porous media with two pore diameters separated by several orders of magnitude. Nanometer-sized pores at the edge of our samples prevent air entry, while drying proceeds by heterogeneous nucleation of vapor bubbles--cavitation--in the liquid in micrometer-sized voids within the sample. We show that the dynamics of cavitation and drying are set by the interplay of the deterministic poroelastic mass transport in the porous medium and the stochastic nucleation process. Spatiotemporal patterns emerge in this unusual reaction-diffusion system, with temporal oscillations in the drying rate and variable roughness of the drying front.

Inertial forces affect fluid front displacement dynamics in a pore-throat network model

The seemingly regular and continuous motion of fluid displacement fronts in porous media at the macroscopic scale is propelled by numerous (largely invisible) pore-scale abrupt interfacial jumps and pressure bursts. Fluid fronts in porous media are characterized by sharp phase discontinuities and by rapid pore-scale dynamics that underlie their motion; both attributes challenge standard continuum theories of these flow processes. Moreover, details of pore-scale dynamics affect front morphology and subsequent phase entrapment behind a front and thereby shape key macroscopic transport properties of the unsaturated zone. The study presents a pore-throat network model that focuses on quantifying interfacial dynamics and interactions along fluid displacement fronts. The porous medium is represented by a lattice of connected pore throats capable of detaining menisci and giving rise to fluid-fluid interfacial jumps (the study focuses on flow rate controlled drainage). For each meniscus along the displacement front we formulate a local inertial, capillary, viscous, and hydrostatic force balance that is then solved simultaneously for the entire front. The model enables systematic evaluation of the role of inertia and boundary conditions. Results show that while displacement patterns are affected by inertial forces mainly by invasion of throats with higher capillary resistance, phase entrapment (residual saturation) is largely unaffected by inertia, limiting inertial effects on hydrological properties behind a front. Interfacial jump velocities are often an order of magnitude larger than mean front velocity, are strongly dependent on geometrical throat dimensions, and become less predictable (more scattered) when inertia is considered. Model simulations of the distributions of capillary pressure fluctuations and waiting times between invasion events follow an exponential distribution and are in good agreement with experimental results. The modeling approach provides insights into the rich pore-scale dynamics of displacement fronts; these insights not only improve the basic understanding of these ubiquitous processes, but could shed light on solute dispersion and colloids mobilization at fronts and the mechanical consequences of passing fronts.

Dynamics of unidirectional drying of colloidal dispersions

We investigate the dynamics of unidirectional drying of silica dispersions. For small colloids (radii a < 15 nm), the minute recession of the drying interface inside the growing solid leads to a slowing down of the evaporation rate, as recently proposed by Wallenstein and Russel [J. Phys.: Condens. Matter, 2011, 23, 194104]. We first propose that Kelvin's effect, i.e. the reduction of the partial pressure of water in the presence of highly curved nanomenisci at the drying air-dispersion interface, has to be taken into account, notably for such small colloids. Our model can fit qualitatively the literature measurements, but with a crossover between the linear regime and the slowing down regime that scales as a(2), as compared to the model of Wallenstein and Russel that predicts a linear scaling. We then also present careful measurements of the dynamics of solidification, which clearly demonstrate that both models (taking into account or not Kelvin's effect) do not fit correctly the slowing down. This is consistent with a brief review of similar recent measurements. Nevertheless, the dynamics can be correctly estimated with a significantly lower effective permeability of the solid region. We suggest that this result may come from the polydispersity of the suspensions, and from the inhomogeneity of the flow within the fracturated solid region, as illustrated by infiltration experiments of a coloured dye.

Study of the effect of wind speed on evaporation from soil through integrated modeling of the atmospheric boundary layer and shallow subsurface

In an effort to develop methods based on integrating the subsurface to the atmospheric boundary layer to estimate evaporation, we developed a model based on the coupling of Navier-Stokes free flow and Darcy flow in porous medium. The model was tested using experimental data to study the effect of wind speed on evaporation. The model consists of the coupled equations of mass conservation for two-phase flow in porous medium with single-phase flow in the free-flow domain under nonisothermal, nonequilibrium phase change conditions. In this model, the evaporation rate and soil surface temperature and relative humidity at the interface come directly from the integrated model output. To experimentally validate numerical results, we developed a unique test system consisting of a wind tunnel interfaced with a soil tank instrumented with a network of sensors to measure soil-water variables. Results demonstrated that, by using this coupling approach, it is possible to predict the different stages of the drying process with good accuracy. Increasing the wind speed increases the first stage evaporation rate and decreases the transition time between two evaporative stages (soil water flow to vapor diffusion controlled) at low velocity values; then, at high wind speeds the evaporation rate becomes less dependent on the wind speed. On the contrary, the impact of wind speed on second stage evaporation (diffusion-dominant stage) is not significant. We found that the thermal and solute dispersion in free-flow systems has a significant influence on drying processes from porous media and should be taken into account.

Dynamics of soil water evaporation during soil drying: laboratory experiment and numerical analysis

Laboratory and numerical experiments were conducted to investigate the evolution of soil water evaporation during a continuous drying event. Simulated soil water contents and temperatures by the calibrated model well reproduced measured values at different depths. Results show that the evaporative drying process could be divided into three stages, beginning with a relatively high evaporation rate during stage 1, followed by a lower rate during transient stage and stage 2, and finally maintaining a very low and constant rate during stage 3. The condensation zone was located immediately below the evaporation zone in the profile. Both peaks of evaporation and condensation rate increased rapidly during stage 1 and transition stage, decreased during stage 2, and maintained constant during stage 3. The width of evaporation zone kept a continuous increase during stages 1 and 2 and maintained a nearly constant value of 0.68 cm during stage 3. When the evaporation zone totally moved into the subsurface, a dry surface layer (DSL) formed above the evaporation zone at the end of stage 2. The width of DSL also presented a continuous increase during stage 2 and kept a constant value of 0.71 cm during stage 3.

Pore-scale dynamics of salt precipitation in drying porous media

We study the pore-scale dynamics of salt precipitation in three-dimensional drying porous media, utilizing high resolution x-ray microtomography and scanning electron microscopy. Our results illustrate that the salt precipitation patterns in drying porous media are nonuniform, manifesting the influence of the spatial distribution of pore sizes on the dynamics of salt crystallization and formation of discrete efflorescence. Results reveal that during stage-1 evaporation from saline porous media, the salt precipitation rate initially increases which is followed by a constant precipitation rate. This non-linear behaviour is attributed to the preferential liquid vaporization and salt precipitation in finer pores located at the surface of the porous medium contributing in evaporation according to the pore sizes. We also show that, contrary to common practice, the macroscopic convection-diffusion equation cannot provide accurate predictions for the dynamics of salt precipitation, at least at the early stages, due to the microscale heterogeneity of evaporation sites at the surface that results in salt precipitation exclusively in the finer pores.

Effect of hydrogel particle additives on water-accessible pore structure of sandy soils: a custom pressure plate apparatus and capillary bundle model

To probe the effects of hydrogel particle additives on the water-accessible pore structure of sandy soils, we introduce a custom pressure plate method in which the volume of water expelled from a wet granular packing is measured as a function of applied pressure. Using a capillary bundle model, we show that the differential change in retained water per pressure increment is directly related to the cumulative cross-sectional area distribution f(r) of the water-accessible pores with radii less than r. This is validated by measurements of water expelled from a model sandy soil composed of 2-mm-diameter glass beads. In particular, it is found that the expelled water is dramatically dependent on sample height and that analysis using the capillary bundle model gives the same pore size distribution for all samples. The distribution is found to be approximately log normal, and the total cross-sectional area fraction of the accessible pore space is found to be f(0)=0.34. We then report on how the pore distribution and total water-accessible area fraction are affected by superabsorbent hydrogel particle additives, uniformly mixed into a fixed-height sample at varying concentrations. Under both fixed volume and free swelling conditions, the total area fraction of water-accessible pore space in a packing decreases exponentially as the gel concentration increases. The size distribution of the pores is significantly modified by the swollen hydrogel particles, such that large pores are clogged while small pores are formed.

Drying in porous media with gravity-stabilized fronts: experimental results

In a recent paper [Yiotis et al., Phys. Rev. E 85, 046308 (2012)] we developed a model for the drying of porous media in the presence of gravity. It incorporated effects of corner film flow, internal and external mass transfer, and the effect of gravity. Analytical results were derived when gravity opposes drying and hence leads to a stable percolation drying front. In this paper, we test the theory using laboratory experiments. A series of isothermal drying experiments in glass bead packings saturated with volatile hydrocarbons is conducted. The transparent glass cells containing the packing allow for the visual monitoring of the phase distribution patterns below the surface, including the formation of liquid films, as the gaseous phase invades the pore space, and for the control of the thickness of the diffusive mass boundary layer over the packing. The experimental results agree very well with theory, provided that the latter is generalized to account for the effects of corner roundness in the film region (which was neglected in the theoretical part). We demonstrate the existence of an early constant rate period (CRP), which lasts as long as the films saturate the surface of the packing, and of a subsequent falling rate period (FRP), which begins practically after the detachment of the film tips from the external surface. During the CRP, the process is controlled by diffusion within the stagnant gaseous phase in the upper part of the cells, yielding a Stefan tube problem solution. During the FRP, the process is controlled by diffusion within the packing, with a drying rate inversely proportional to the observed position of the film tips in the cell. Theoretical and experimental results compare favorably for a specific value of the roundness of the films, which is found to be constant and equal to 0.2 for various conditions, and verify the theoretical dependence on the capillary Ca(f), Bond Bo, and Sherwood Sh numbers.

Structure of drying fronts in three-dimensional porous media

Evaporation in a three-dimensional (3D) porous medium, a sand column saturated by water, was studied using synchrotron x-ray tomography. Three-dimensional images of the medium with a resolution of 7 μm were obtained during the evaporation. The entire column was scanned seven times, resulting in nearly 10(4) 2D cross sections and illustrating the spatial distribution of air, liquid, and solid phases at the pore scale. The results were analyzed in order to gain new insights and better understanding of the characteristics of the drying front that was formed when the liquid-filled pores were invaded by air, as well as the structure of the liquid phase as it was dried. The analysis indicates that the liquid phase has a self-similar fractal structure, with its fractal dimension D(f) in all the cross sections being a function of the water content or saturation. In addition, D(f) for the 3D liquid structure, as well as its density correlation function, were computed using the 3D images. A crossover length scale ξ was identified that separates the fractal regime from the compact geometry. For length scales r>ξ, the density correlation function approaches asymptotically the water content of the porous medium. The drying front is shown to be rough and multi-affine, rather than self-affine. Its properties were also computed using the 3D images. The roughness characteristics agree with those for imbibition in porous media, but not with those of fracture surfaces and crack lines.

Analytical solutions of drying in porous media for gravity-stabilized fronts

We develop a mathematical model for the drying of porous media in the presence of gravity. The model incorporates effects of corner flow through macroscopic liquid films that form in the cavities of pore walls, mass transfer by diffusion in the dry regions of the medium, external mass transfer over the surface, and the effect of gravity. We consider two different cases: when gravity opposes liquid flow in the corner films and leads to a stable percolation drying front, and when it acts in the opposite direction. In this part, we develop analytical results when the problem can be cast as an equivalent continuum and described as a one-dimensional (1D) problem. This is always the case when gravity acts against drying by opposing corner flow, or when it enhances drying by increasing corner film flow but it is sufficiently small. We obtain results for all relevant variables, including drying rates, extent of the macroscopic film region, and the demarkation of the two different regimes of constant rate period and falling rate period, respectively. The effects of dimensionless variables, such as the bond number, the capillary number, and the Sherwood number for external mass transfer are investigated. When gravity acts to enhance drying, a 1D solution is still possible if an appropriately defined Rayleigh number is above a critical threshold. We derive a linear stability analysis of a model problem under this condition that verifies front stability. Further analysis of this problem, when the Rayleigh number is below critical, requires a pore-network simulator which will be the focus of future work.

Drying of a model soil

Drying experiments have been carried out with model soils made of different pastes filling granular packings. A detailed information concerning the time evolution of the water saturation distribution inside the sample was obtained from magnetic resonance imaging measurements. This study makes it possible to understand the physical origin of the drying characteristics of these materials. The drying curves exhibit a constant-rate period (CRP) and a falling-rate period (FRP) but the relative durations of these periods depend on the paste structure. With a kaolin suspension the CRP lasts down to very low water densities and is associated with a homogeneous drying of the paste throughout the sample. With a bentonite suspension the CRP is shorter and the drying in the FRP results from a complex process involving fractures progressing downward through the pasty matrix. With a gel the CRP period is even shorter and the drying in the FRP results from the progression of a dry front through the packing as a result of the shrinkage of the gel matrix. This provides an overview of the main possible processes at work when drying a soil as a function of its components along with some practical means for slowing down drying from soils.

Liquid-phase continuity and solute concentration dynamics during evaporation from porous media: pore-scale processes near vaporization surface

Evaporation from porous media involves complex pore scale transport processes affecting liquid phase distribution and fluxes. Often, the initial evaporation rate is nearly constant and supplied by capillary flow from wetted zones below to the surface. Sustaining constant flow against gravity hinges on an upward capillary gradient and on liquid phase continuity with hydraulic conductivity sufficient for supplying evaporative flux. The pore scale liquid phase adjustments during evaporative displacement necessary for maintaining a constant flux have been postulated but rarely measured. In this study we employed detailed imaging using x-ray synchrotron radiation to study liquid phase distribution and dynamics at the most sensitive domain just below the surface of evaporating sand columns. Three-dimensional images at a resolution of 7 microns were obtained from sand column (mean particle size 0.6 mm) initially saturated with calcium iodide solution (4% by mass) to enhance image contrast. Detailed imaging of near-surface liquid phase distribution during evaporation confirmed phase continuity at micrometric scale and provided quantitative estimates of liquid conductance in agreement with values required to supply evaporative flux. Temporal variations in bulk salt concentrations determined from x-ray attenuation were proportional to evaporative water mass loss. Highly resolved salt concentration images revealed existence of evaporating chimneys that supply the bulk of evaporative demand. Delineated mass loss dynamics and salt distribution measured by the x-ray attenuation were in reasonable agreement with a simplified analytical convection-diffusion model for salt dynamics during evaporation from porous media.

Numerical solutions of a generalized theory for macroscopic capillarity

A recent macroscopic theory of biphasic flow in porous media [R. Hilfer, Phys. Rev. E 73, 016307 (2006)] has proposed to treat microscopically percolating fluid regions differently from microscopically nonpercolating regions. Even in one dimension the theory reduces to an analytically intractable set of ten coupled nonlinear partial differential equations. This paper reports numerical solutions for three different initial and boundary value problems that simulate realistic laboratory experiments. All three simulations concern a closed column containing a homogeneous porous medium filled with two immiscible fluids of different densities. In the first simulation the column is raised from a horizontal to a vertical orientation inducing a buoyancy-driven fluid flow that separates the two fluids. In the second simulation the column is first raised from a horizontal to a vertical orientation and subsequently rotated twice by 180 degrees to compare the resulting stationary saturation profiles. In the third simulation the column is first raised from horizontal to vertical orientation and then returned to its original horizontal orientation. In all three simulations imbibition and drainage processes occur simultaneously inside the column. This distinguishes the results reported here from conventional simulations based on existing theories of biphasic flows. Existing theories are unable to predict flow processes where imbibition and drainage occur simultaneously. The approximate numerical results presented here show the same process dependence and hysteresis as one would expect from an experiment.

Evaporation and capillary coupling across vertical textural contrasts in porous media

High and nearly constant evaporation rates from initially saturated porous media are sustained by capillary-driven flow from receding drying front below the evaporating surface. The spatial extent of continuous liquid pathways in homogeneous porous medium is defined by its hydraulically connected pore size distribution. We consider here evaporative losses from porous media consisting of two hydraulically coupled dissimilar domains each with own pore and particle size distributions separated by sharp vertical textural contrast. Evaporation experiments from texturally dissimilar media were monitored using neutron transmission and dye pattern imaging to quantify water distribution and drying front dynamics. Drying front invades exclusively coarse-textured domain while fine-textured domain remains saturated and its surface continuously coupled with the atmosphere. Results show that evaporation from fine-textured surface was supplied by liquid flow from adjacent coarse domain driven by capillary pressure differences between the porous media. A first characteristic length defining limiting drying front depth during which fine sand region remains saturated is deduced from difference in air-entry pressures of the two porous media. A second characteristic length defining the end of high evaporation rate includes the extent of continuous liquid films pinned in the crevices of the pore space and between particle contacts in the fine medium. We established numerically the lateral extent of evaporation-induced hydraulic coupling that is limited by viscous losses and gravity. For certain combinations of soil types the lateral extent of hydraulic coupling may exceed distances of 10 m. Results suggest that evaporative water losses from heterogeneous and coupled system are larger compared with uncoupled or homogenized equivalent systems.