Scaling theory of drying in porous media
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PHYSICAL REVIEW E APRIL 1999VOLUME 59, NUMBER 4Scaling theory of drying in porous media
I. N. Tsimpanogiannis and Y. C. Yortsos*Department of Chemical Engineering, University of Southern California, Los Angeles, California 90089-1211
S. Poulou, N. Kanellopoulos, and A. K. StubosNational Center for Scientific Research Demokritos, 15310 Aghia Paraskevi, Greece
~Received 2 November 1998!
Concepts of immiscible displacements in porous media driven by mass transfer are utilized to model dryingof porous media. Visualization experiments of drying in two-dimensional glass micromodels are conducted toidentify pore-scale mechanisms. Then, a pore network approach is used to analyze the advancing drying front.It is shown that in a porous medium, capillarity induces a flow that effectively limits the extent of the front,which would otherwise be of the percolation type, to a finite width. In conjuction with the predictions of amacroscale stable front, obtained from a linear stability analysis, the process is shown to be equivalent toinvasion percolation in a stabilizing gradient. A power-law scaling relation of the front width with a diffusion-based capillary number is also obtained. This capillary number reflects the fact that drying is controlled bydiffusion in contrast to external drainage. The scaling exponent predicted is compatible with the experimentalresults of Shaw@Phys Rev. Lett.59, 1671~1987!#. A framework for a continuum description of the upstreamdrying regimes is also developed.@S1063-651X~99!04604-8#
PACS number~s!: 81.05.Rmn
The drying of porous media is a problem of significascientific and applied interest. Chen and Pei@1# note thatdrying is one of the most energy-consuming processeindustry. Applications include the drying of granular mateals such as soils, rocks, minerals, building materials,ceramic powders; drying processes in the wood~Simpson@2,3#!, paper, and textile industry; coating technology; athe drying of foodstuff~Fortes and Okos@4#! and pharma-ceutical products. In a different context,in situ drying ofporous media is involved in recent methods for the remetion of contaminated soils by soil vapor extraction or sventing ~Ho and Udell @5#!, as well as in the recovery ovolatile hydrocarbons from underground oil reservoirs by ginjection ~Le Romanceret al. @6#, Le Galloet al. @7#!.
The development of a general mathematical framewormodel drying of porous media has been a rather challengresearch topic for several decades~Waananenet al. @8#!. Al-though a plethora of methods have been presented, and windustrial designers are faced with the demand to descomplicated processes regarding real problems, therestill many unresolved questions, even for simpler prolems ~Prat @9#!. Traditionally, the description of drying inporous media is based on phenomenological approachesconsider the medium as a structureless continuum. In thpartial differential equations are postulated that relateevolution in spacetime of volume-averaged quantities, sas moisture content and temperature. Phenomenologicaempirical parameters are then used to relate fluxes to grents, often drawn from an analogy with nonequilibrium thmodynamics~for example, see Luikov@10#!. However, inthese approaches, the pore microstructure and the under
*Author to whom correspondence should be addressed.PRE 591063-651X/99/59~4!/4353~13!/$15.00t
phenomena, which are key to the quantitative understandof the process, are essentially ignored.
On a fundamental level, drying is a phase change o~usually! one-component liquid into a~usually! two-component gas, and involves at various stages the mo~receding! of individual gas-liquid menisci~see schematic oFig. 1!. The menisci reside in the pore space of the poromedium and are subject to the interfacial forces betweenuid, gas, and solid surfaces. Due to the disorder in therespace geometry, however, their distribution is also disdered in general. The physical processes involve evaporaof the volatile liquid in the gas phase, driven by concenttion gradients, countercurrent diffusion in the gas, possliquid flow through connected films, and the accompanyreceding of menisci. All these interactions are influencedthe porous medium microstructure to a significant degree
In general, four different spatial regimes can be distguished during a drying process~Fig. 2!: A far-field regimeconsisting of the initial liquid phase, a regime where tliquid phase is macroscopically connected and where b
FIG. 1. Schematic of liquid-gas interfaces during drying in prous media.4353 1999 The American Physical Society
4354 PRE 59I. N. TSIMPANOGIANNIS et al.gas and liquid phases are macroscopically continuous,third regime in which the liquid phase has been disconnecin individual clusters of variable sizes~blobs! as a result ofdrying, and a fourth regime consisting of the liquid phasethe form of pendular rings or films covering the solid suface, the thickness of which is progressively reduced,wards a totally dry regime. In the last three regimes thgas phase is macroscopically continuous. Shaw@11# has alsopostulated that liquid films may provide hydraulic connectity to the liquid phase in all three regimes. It is apparent tan appropriate account of the various pore-scale events ithese regimes is fundamental to the accurate representof any macroscopic description~Whitaker @12#!.
Pore-network approaches for describing drying of poromedia were recently proposed by Nowicki, Davis, aScriven@13# and in a series of papers by Prat and co-work~Prat @14,15#, Laurindo and Prat@16,17#!. In a related con-text, Pot et al. @18# used lattice-gas cellular automatasimulate evaporation phenomena in two-dimensional~2D!porous media. Nowicki, Davis, and Scriven@13# developed arather comprehensive pore-network simulation of the pcess, and accounted for both capillary and viscous forHowever, the authors did not delve into the particular pterns that develop or on their effect on the drying rates. P@14,15# and Laurindo and Prat@16,17# studied pattern formation during drying assuming capillary forces only and ignoing viscous forces. The importance of film flows was adiscussed@17#. Based on the assumption of percolation pterns and under isothermal conditions, they proceededcompute evaporation and drying rates by solving a qustatic diffusion equation in the gas phase. However, eadrying experiments in glass-bead packs by Shaw@11# sug-gested that viscous forces are not negligible, and in factneeded to explain the formation of a front width~separatingcontinuous liquid from gas! of a finite size. This, Shaw@11#attempted to scale using scaling expressions obtainedexternal drainage~see below!. Despite the limitation on vis-cous forces, however, Prats studies are important in
FIG. 2. Schematic description of drying regimes in porous mdia, obtained from 2D pore network simulations. Dots and whareas denote liquid-occupied regions, dark shaded areas denotoccupied regions.~Due the 2D topological limitations, a schematof the 2-phase bicontinuum regime is not feasible.!ad
they represent the first attempt to theoretically charactedrying patterns and their rate of change in porous media
As in other processes involving two-phase, immiscibflows in porous media, the following two aspects of dryinpatterns need to be understood:~i! Their geometrical struc-ture, which dictates transport and capacitance and~ii ! theirrates of change. This is the main motivation of this papWe consider the application of a pore-network approachin Nowicki, Davis, and Scriven@13#, Prat@14,15#, and Lau-rindo and Prat@16#, but with specific objectives to understand the structure of drying patterns, particularly in the frotal region. From such an analysis, the derivation of effectmacroscopic models can then be obtained. Drying, involvgas-liquid interfaces, can benefit from advances in the angous problem of external drainage, which is reviewed beloWhat is novel in drying, however, is the additional effectmass transfer in the gas phase, which actually drivesprocess. This needs to be analyzed in some detail.
The paper is organized as follows: First, we give a brreview of recent findings in isothermal drainage processwhich have a direct bearing on drying. Then, experiments2D micromodel geometries are presented to visualize menisms during drying and to help in the development of ttheory. Based on the experimental observations, a theoreapproach is subsequently developed, which combines aments borrowed from isothermal drainage and from thelated bubble growth problem, where mass transfer is aprocess. We use scaling arguments to show that dryinactually a process of invasion percolation in a stabilizigradient~IPSG! ~see below for a definition!, from which wecan infer the scaling of the front width as a function of tdrying parameters. To demonstrate the transition frompercolation-only pattern, which was studied by Prat, tostabilizing gradient and an IPSG, a linear stability analysisthe front in the appropriate geometries must be performThis analysis will precede the main theoretical develoments. We close by providing a framework for a macrscopic description based on transverse averages, and bymenting on the modeling of the other regimes.
We show that the scaling so obtained is compatible wShaws @11# experimental results. Thus, although near tleading edge of the front, the displacement pattern will bethe percolation type~assumed by Prat and co-workers tovalid for the entire pattern!, as the width of the front in-creases, viscous forces become increasingly important, ling to a displacement described by IPSG. Our analysis asheds light to a process of liquid flow, termed capillapumping by Le Romanceret al. @6#, in their modeling ofoil recovery from fractured reservoirs by gas injection. Wshow that this effect is actually the consequence of accoing for both capillary and viscous terms in the process. Asprevious pore-scale studies in drying~Nowicki, Davis, andScriven@13#, Prat@15#!, our analysis is restricted to isothemal problems. We also neglect convection in the gas phwhich is expected to be progressively of secondary imptance, and gravity. The effect of the latter can be direcincorporated. However, effects of heat transfer and convtion need a separate analysis, which will be attempted ifuture study.
PRE 59 4355SCALING THEORY OF DRYING IN POROUS MEDIAII. PRELIMINARIES
Drying involves gas-liquid interfaces and mass transand should be subject to an analysis similar to external drage and to bubble growth in porous media. Because ofexpected similarities, we briefly review in this section receadvances in these two areas.
Consider first, drainage, namely, the displacement owetting phase by a nonwetting phase in a porous mediumthe absence of phase change. In drainage, menisci resipores or at the entrance of pore throats with a curvacorresponding to the local capillary pressure, defined asdifference between the nonwetting~gas! and wetting~liquid!phase pressures,
Here,g is the interfacial tension between the fluids,H is themean curvature of the meniscus and a zero contact angleassumed. In the absence of buoyancy and/or viscous fothe capillary pressure, thus the mean curvature, is spatuniform. A meniscus will penetrate a pore throat, adjacenwhich it resides, when the capillary pressure first exceedscapillary entry pressure for that pore throat~roughly equal to2g/r , wherer is the pore throat size!. In drainage at constanrate, the sequence of pore penetration can be modeleinvasion percolation~IP!, where at each time step only onpore throat is invaded, that with the least capillary resista~or, equivalently, the largest radius! among all throats at theperimeter of the interface. During this step, all other menremain stationary or fluctuate slightly. This type of displacment gives rise to a self-similar fractal pattern in the displing phase, which eventually approaches that of the perction cluster. The properties of the latter have been discusin detail in various publications@19,20#. Because of the selfsimilarity involved, however, defining a mean-front positiois not operationally useful.
However, if gravity and/or viscous forces are also imptant, a percolation pattern will not develop over the entregion of displacement. When only gravity forces are imptant, the capillary pressure will vary with the elevationh ofthe interface, sincePc5gxDrh, wheregx is the componentof gravity in the direction of displacement andDr5rw2rnw . Then, the displacement acquires the features odifferent pattern, namely, invasion percolation in a gradi~IPG! @2126#. In this case, the competition between gravand capillary forces is expressed through the Bond num
wherek is the permeability of the porous medium~which isroughly proportional to the square of a mean pore sk;r m
2 , Katz and Thompson,@27#!. Now, one needs to furthedistinguish two cases:
~i! If B.0, for example, in the downwards displacemeof a heavier by a lighter fluid, the two phases are separaby a front of finite widthsG , which scales with the Bondnumber as@21,22#
wheren is the correlation length exponent of percolation. Fa nonzeroB, the front width is finite. Then, the front is noself-similar, but rather self-affine@28# @Fig. 3~a!#. Within thefront, the pattern has the fractal characteristics of the pelation cluster. However, upstream of the front, the patterncompact. Thus, a mean front position can be usefullyfined. In essence, this reflects the transition of the displament from an IP to a pistonlike pattern. The latter is tpattern that develops when only gravity~and not capillarity!acts~and which would be pistonlike in this case!. We shallrefer to this as IPSG. Hulinet al. @23# experimentally dem-onstrated the application of IPG in a drainage problem sbilized by gravity.
~ii ! On the other hand, ifB,0, for example, in the down-wards displacement of a lighter by a heavier fluid, the dplacement is invasion percolation in a destabilizing gradi~IPDG!. This pattern has the different features of capillafingering @Fig. 3~b!#, in which the displacement occurs binvading fingers of a mean width still given by Eq.~3!, thelocal characteristics of which are still controlled by percotion. This regime has been discussed in detail in Fretteet al.@29# and Meakinet al. @30#.
The effect of viscous forces is more complex. At largscales, where viscous effects predominate, two limiting pterns...