Computers and Geotechnics 36 (2009) 435–445

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Computers and Geotechnics

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Natural convection of compressible and incompressible gases in undeformableporous media under cold climate conditions

Marc Lebeau *, Jean-Marie KonradDepartment of Civil Engineering, Laval University, Quebec, QC, Canada

a r t i c l e i n f o a b s t r a c t

Article history:Received 22 December 2007Received in revised form 18 April 2008Accepted 21 April 2008Available online 11 June 2008

Keywords:Porous mediaNatural convectionCompressibleIncompressiblePhase changeNumerical modeling

0266-352X/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.compgeo.2008.04.005

* Corresponding author. Tel.: +1 418 656 2131x317E-mail address: Marc.Lebeau@gci.ulaval.ca (M. Leb

A numerical model for convective heat and mass transport of compressible or incompressible gas flowswith soil-water phase change is presented. In general, the gaseous phase is considered as compressibleand the model accounts for adiabatic processes of compression heating and expansion cooling. The inher-ently compressible gaseous phase may nevertheless be considered as incompressible by adopting theOberbeck–Boussinesq approximations. The numerical method used to solve the equations that describenatural convection is based on a Galerkin finite element formulation with adaptive mesh refinement anddynamic time step control. As most existing numerical studies have focused on the behavior of incom-pressible fluids, model substantiation examines the influence of fluid compressibility on two-widely usedbenchmarks of steady-state convective heat and mass transport. The relative importance of the effect ofpressure-compressibility cooling is shown to increase as the thermal gradient approaches the magnitudeof the adiabatic gradient. From these results, it may be concluded that pore-air compressibility cannot beneglected in medium to large-sized enclosures at small temperature differentials. After demonstrating itsability to solve fairly complex transient problems, the model is used to further our understanding of thethermal behavior of the toe drain at the LA2-BSU dam in the province of Quebec, Canada.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Thermobuoyant motion, or natural convection, of pore-air re-sults from the tendency of most fluids to expand, and experiencea change in density, when heated. The motion occurs as the densitychanges alter the body forces, and the fluid starts to rise. The risingpore-air is then replaced by surrounding cooler pore-air and theprocess continues, forming a convective current. This phenomenonoccurs in a great number of natural and engineered settings. For in-stance, naturally sloping areas that are covered with a layer ofcoarse rock often display lower ground temperatures as heat is ex-tracted by convective currents during the winter months. This ef-fect may even allow permafrost to form in areas where the meanannual air temperature is above zero centigrade [1–5]. Loweringof ground temperatures by buoyancy-driven flow may also occurbeneath the slopes of engineered structures, such as those of rock-fill dams [6].

Over the last decades, these and other observations have led tothe design of specifically engineered embankments and revet-ments, which limit the thawing of permafrost under roadwaysand railways. Numerical and experimental studies of engineeredembankments have been carried out in permafrost-laden regions

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9; fax: +1 418 656 2928.eau).

of the world, such as Canada [7,8], China [9–16], Russia [17], andthe United States of America [18–22].

Konrad et al. [23] recently identified natural heat convection asa plausible cause for the presence of seepage at the downstreamtoe of the south dam of the Laforge-2 project in the province ofQuebec, Canada. The suspected mechanism was specifically identi-fied as a flow restriction in the toe drain due to the presence ofpore-ice, which could only develop in the presence of convectiveheat transport. In response, the purpose of this paper is to: (1)present a numerical model for the study of convective heat andmass transport for compressible or incompressible flows in porousmedia with phase change, and (2) corroborate field observations ofnatural heat convection in rockfill dams through numerical simu-lations. The paper sets the context and then describes the convec-tive heat and mass transport model. The capabilities of thenumerical model are subsequently documented while assessingthe effects of fluid compressibility. The paper goes on to corrobo-rate field observations and establish the importance of natural con-vection on the thermal behavior of a typical rockfill dam.

2. Study context

Commissioned in 1996, the LA2-BSU dam is a 24-m high, 653-mlong rockfill structure built, for the most part, on a till deposit. Asshown in Fig. 1, the typical cross-section consists of a central till

Nomenclature

B width of the enclosure [L]Ct volumetric heat capacity of the soil at constant pressure

[M L�1 T�2 H�1]Ct;a volumetric heat capacity of air at constant pressure

[M L�1 T�2 H�1]Ct;f volumetric heat capacity of the frozen soil at constant

pressure [M L�1 T�2 H�1]Ct;u volumetric heat capacity of the unfrozen soil at constant

pressure [M L�1 T�2 H�1]Ca

t apparent volumetric heat capacity of the soil[M L�1 T�2 H�1]

Da Darcy dimensionless numberg gravitational acceleration constant [L T�2]~g gravity vector [L T�2]gx; gz components of gravity vector [L T�1]Gr Grashof dimensionless numberH height of the enclosure and reference vertical dimension

[L]Ht volumetric enthalpy [M L�1 T�2]K soil intrinsic permeability tensor [L2]K soil intrinsic permeability [L2]kt soil thermal conductivity tensor [M L T�3 H�1]kt soil thermal conductivity [M L T�3 H�1]kt;a thermal conductivity of air [M L T�3 H�1]kt;f frozen soil thermal conductivity [M L T�3 H�1]kt;u unfrozen soil thermal conductivity [M L T�3 H�1]Lf volumetric latent heat of fusion of water [M L�1 T�2]MMa molar mass of dry air [M mol�1]MMi molar mass of fluid constituents [M mol�1]Nu mean Nusselt dimensionless numberPr Prandtl dimensionless number�qa mean magnitude of the volumetric pore-air flux [L T�1]~qa volumetric flux vector of air [L T�1]qa;x; qa;z components of volumetric air flux [L T�1]R universal gas constant [M L2 T�2 mol�1 H�1]Ra Rayleigh–Darcy dimensionless number

Rac critical threshold Rayleigh–Darcy dimensionless num-ber

t time [T]T absolute temperature [H]Ta absolute ambient air temperature [H]To reference absolute temperature [H]TL absolute temperature of fusion (liquidus) of pore-water

[H]TS absolute temperature of solidification (solidus) of pore-

water [H]Ts-1; Ts-2 absolute temperatures at the upper boundaries of the

railway embankment [H]Ts-3; Ts-4 absolute temperatures at the upper boundaries of the

rockfill dam [H]u pore pressure [M L�1 T�2]ua pore-air pressure [M L�1 T�2]ua;o reference pore-air pressure [M L�1 T�2]vi specific volume of fluid constituents [M�1 L3]w curve shape parameterxi mole fraction of fluid constituentsx; z spatial coordinates [L]a vertical tilt angle of gravity vectorb0 reference volumetric thermal expansion coefficient of

air [T�1]DT applied temperature differential [H]ha volumetric air contenthw volumetric fraction of liquid waterhw;f volumetric fraction of liquid water in the frozen soilhw;u volumetric fraction of liquid water in the unfrozen soilla;o reference dynamic viscosity of air [M L�1 T�1]q density of fluid phase [M L�3]qa density of air [M L�3]qa;o reference density of air [M L�3]qd dry density [M L�3]w stream function [L2 T�1]

436 M. Lebeau, J.-M. Konrad / Computers and Geotechnics 36 (2009) 435–445

core with granular filters, transitions and rockfill shoulders on eachside. The toe drain is located close to the downstream toe and isdesigned to collect seepage from the dam core and the upper partof the till foundation. The collected water follows the slope of thedrain towards the right abutment of the dam.

Since impoundment in 1996, heat outflows have been observedon the downstream face during winter months (Fig. 1). Althoughwidespread during the first winters, the heat outflows have gradu-ally decreased in numbers with time. The decrease in the numberof heat outflows may be ascribed to the gradual heat loss, whichwas accumulated in the earthwork during summer construction.Only a few heat outflows are now observed each year. It must benoted that these heat outflows are a telltale sign of the buoy-ancy-driven stack, or chimney effect, where airflow moves up-wards and out of the rockfill creating a hole in the snow cover sothat colder air can penetrate [5].

At the end of 2001, after five years of operation, a number ofseepage areas appeared at the downstream toe over a length ofapproximately 100 m. These losses resulted in a decrease of themeasured seepage flows, which had initially been equal to the val-ues predicted during the design phase. Based on these and othermeasurements, it has become apparent that the toe drain isblocked and that water is locally overflowing. Since the earthworkis recent and the drains and filters were designed according tomodern criteria, the blockage cannot be imputed to the migration

of fine particles from the core into the drain. Given the region’ssevere climatic conditions and the presence of other cases of per-mafrost in dams impounding the Caniapiscau reservoir, the mostprobable explanation for the observed phenomena is a progressiveblockage of the toe drain by pore-ice [23].

3. Theoretical formulations

The continuum assumption considers the fluid phase as a con-tinuous medium with density q taken to be well-defined at theREV (representative elementary volume) scale. Fluid density isgenerally not uniform as it varies with mixture composition, tem-perature and pore pressure. In a formal manner, fluid density is tobe regarded as a thermodynamic variable for which an equation ofstate may be defined, i.e., q ¼ f ðMMi; T;u; vi; xiÞ, where MMi, vi andxi are the molar mass, specific volume and mole fraction of the fluidconstituents. In the case of a homogeneous or single-componentfluid, density may be expressed solely in terms of temperatureand pressure. As is well-known, most fluids tend to expand whenheated and contract when compressed. In the context of an essen-tially gas filled porous matrix, these dimensional changes may oc-cur in response to barometric pressure fluctuations and/or ambienttemperature fluctuations. Let us recall that barometric pressurefluctuations may occur diurnally due to global atmospheric tides,

Rockfill shoulder

Toe drain Foundation

FilterTransition

Rockfill shoulder

FilterTransition

Core

Convective heat pattern

Heat outflow

Fig. 1. Typical cross-section of the LA2-BSU dam with a schematic representation of the wintertime convective current.

M. Lebeau, J.-M. Konrad / Computers and Geotechnics 36 (2009) 435–445 437

which are on the order of a few tenths of a kilopascal, whereas re-gional scale weather patterns may produce longer term barometricpressure variations that are on the order of a few kilopascals.

In mathematical terms, the dimensional changes alter fluiddensity, which interacts with the gravity vector to produce fluidmotion. Description of the transport of heat and mass resultsfrom the combination of the principle of conservation (energyand mass), the equation of state of the gas, and phenomenolog-ical equations describing heat conduction and gas flow throughporous media (Fourier’s law and Darcy’s law). The resulting par-tial differential equations differ according to the form of theseconstituents and the use of simplifying assumptions. The litera-ture therefore presents several approaches to solving transientnatural convection in porous media with a single mobile fluid[24–28].

3.1. Governing equations

3.1.1. Compressible gaseous phaseAlthough heat transport in multiphase systems may be de-

scribed by separate equations for each phase, it is herein consid-ered that the various phases are in local thermal equilibrium andthat the temperatures of the various phases are the same withinthe representative elementary volume. In this manner, the convec-tive heat transport equation results from the insertion of Fourier’slaw for heat conduction as well as the Gibbs equations and theMaxwell relation for the system enthalpy into the equation of en-ergy conservation for a cubic volume of porous media with soil-water phase change, from liquid to solid. Mostly concerned withthe transport of the gaseous phase, soil-water is herein treated asimmobile. In the presence of a compressible gaseous phase, adia-batic processes of compression heating and expansion cooling oc-cur due to changes in the gas pressure. Although the workaccomplished by these pressure changes is taken into account bythe last terms of the heat transport equation stated below, thepore-air velocities are considered small enough to neglect viscousdissipation heating that results from the work done against viscousstresses. The equation of mass transport is the result of combiningDarcy’s law with the equation of mass conservation for a cubic vol-ume of porous media. Although several authors have proposedextensions of Darcy’s law [29], it is herein considered that fluidmomentum can be described by a single viscous loss term whileomitting inertial loss. The pore-air is assumed to conform to thepostulates of the kinetic-molecular theory or collision theory andtherefore, obey the equation of state for an ideal gas. Other thandensity, pore-air properties are assumed constant and the equa-tions describing natural convection can be written as follows[27,28,30,31]:

Heat: ~r � ½kt � ~rT� ¼ Cat

oTot þ Ct;a~qa � ~rT � ðoua

ot þ~qa � ~ruaÞMass: ~r � ½qa

Kla;o� ~rðua þ qa~gzÞ� ¼ oðqahaÞ

ot

State: qa ¼ uaMMaRT

ð1Þ

where Ct is the volumetric heat capacity of the soil at constant pres-sure, Ct,a is the volumetric heat capacity of air at constant pressure,Ca

t ¼ Ct þ Lfohw=oT is the apparent volumetric heat capacity of thesoil, ~g is the gravity vector, K is the soil intrinsic permeability ten-sor, kt is the soil thermal conductivity tensor, Lf is the volumetric la-tent heat of fusion of water, MMa is the molar mass of dry air,~qa ¼ �K=la;o � ~rðua þ qa~gzÞ is the volumetric flux vector of air, R isthe universal gas constant, t is the time, T is the absolute tempera-ture, ua is the pore-air pressure, x, z are the spatial coordinates, ha isthe volumetric air content, hw is the volumetric fraction of liquidwater, la,o is the reference dynamic viscosity of air and qa is thedensity of air.

3.1.2. Isothermally incompressible gaseous phaseAlthough all fluids are compressible to some extent, it is gener-

ally possible to simplify flow analyses by assuming that the fluid isisothermally incompressible and that its density is pressure inde-pendent. Upon neglecting compressibility, the adiabatic heatingand cooling terms of the heat transport equation are suppressedand the pore-air pressure of the equation of state is set equal toa reference pore-air pressure ua,o. As a result, the convective heatand mass transport equations reduce to:

Heat: ~r � ½kt � ~rT� ¼ Cat

oTot þ Ct;a~qa � ~rT

Mass: ~r � ½qaK

la;o� ~rðua þ qa~gzÞ� ¼ oðqahaÞ

ot

State: qa ¼ua;oMMa

RT

ð2Þ

Noting that the density variations that provide the driving forcefor natural convection are usually relatively small, Oberbeck [32]and Boussinesq [33] proposed a powerful simplification to thisdensity-driven flow equation. In the strictest form of the Ober-beck–Boussinesq approximations, it is assumed that viscous dissi-pation is negligible and that fluid properties are constant with theexception of the temperature dependence of the fluid density,which is only taken into account in the gravity term of the momen-tum equation. The Oberbeck–Boussinesq approximations also im-plies that the equation of state is linearized around qa,o (areference density at To and ua,o). For small temperature differences,(T � To), the equation of state may be approximated by the firstterm of its Taylor series expansion [28]. Implementing theseapproximations results in the following equations for convectiveheat and mass transport:

438 M. Lebeau, J.-M. Konrad / Computers and Geotechnics 36 (2009) 435–445

Heat: ~r � ½kt � ~rT� ¼ Cat

oTot þ Ct;a~qa � ~rT

Mass: ~r � Kla;o� ~rðua þ qa~gzÞ

h i¼ 0

State: qa ¼ qa;o½1� boðT � ToÞ�

ð3Þ

where bo is the reference volumetric thermal expansion coefficientof air. Further details on the Oberbeck–Boussinesq approximationscan be found in Spiegel and Veronis [34], Mihaljan [35], Gray andGiorgini [36], and Gartling and Hickox [37].

The description of pore-air motion is herein based on the con-cept of the potential function. In the case of a steady-state flow,one may alternatively assume the existence of streamlines fromwhich a stream function, w, may be derived. The flux componentsmay then be expressed in terms of this stream function. It followsthat the partial differential equation of mass transport of an iso-thermally incompressible gaseous phase may also be expressedas [38]

o2wox2 þ

o2woz2 ¼

oqa;z

ox�

oqa;x

ozð4Þ

where qa,x and qa,z are the components of volumetric air flux in thetwo-dimensional vertical plane of interest. It is to be noted that thestream function is the imaginary part of the complex functionwhereas the potential function is the real part.

3.2. Thermal properties of the porous media

The derivative of the volumetric fraction of liquid water withrespect to temperature is an integral part of the apparent heatcapacity term. According to Bonacina et al. [39], the mathematicalrepresentation of the relationship between the volumetric fractionof liquid water and temperature has little or no influence on thecomputed thermal field. Nonetheless, Bonacina et al. [40] empha-size that the mathematical representation must allow for an exactcomputation of the amount of volumetric enthalpy that is associ-ated with the latent heat effect which occurs over a specific tem-perature range, i.e., DHt ¼

R TLTS

Lfohw=oTdT , where TS and TL are thetemperature of solidification (solidus) and the temperature of fu-sion (liquidus) of pore-water, respectively. In order to meet thisconstraint and have phase change occur over a finite range of tem-peratures, the derivative of the mathematical function between hw

and T must be infinitely small for temperatures other than those inthe specified range. The present study therefore expresses the vol-umetric fraction of liquid water as an exponential function of tem-perature [41]:

hw ¼ ðhw;u � hw;fÞe� T�TL

w

� �2h in o

þ hw;f

hw;u

8<:

T < TL

T P TLð5Þ

where w � (TL � TS)/2 is a curve shape parameter and hw,f, hw,u arethe volumetric fractions of liquid water in the frozen and unfrozensoil, respectively. It is to be noted that the thermal conductivity andvolumetric heat capacity of the soil are also approximated with thisform of equation.

4. Numerical solution

The numerical solutions to the second order partial differen-tial equations that describe natural convection are obtained withFlexPDE [42], a script-driven partial differential equation solverthat uses the finite element method to solve boundary and initialvalue problems. The script describes the equation, domain andauxiliary definitions, which are converted into a Galerkin finiteelement model. So as to reduce spurious oscillations ascribedto convection terms, an artificial diffusion term is added to theGalerkin formulation for incompressible gaseous flows. The

resulting system of equations is solved with a modified New-ton–Raphson iterative procedure. In the case of nonlinear time-dependent problems, a single modified Newton–Raphson step istaken at each time step while an adaptive procedure measuresthe solution curvature in time and adapts the time step to main-tain accuracy. During the solution process, an adaptive meshrefinement procedure measures mesh adequacy and locally re-fines the mesh until a user-defined error tolerance is achieved.This type of dynamic allocation generally eliminates inaccuratesolutions that include such features as oscillation and numericaldispersion [43]. In addition to a preset strategy, FlexPDE alsosupports additional schemes for identifying regions for meshrefinement. This is of particular value in phase change problemsthat are expressed in terms of apparent heat capacity, where apregenerated fixed mesh generally results in an oscillatory pro-gression of the frost front [44]. In the current study, the qua-dratic order triangular elements are split if the temperatures atthe nodes span a range greater than [�w/2,w/2] around(TL + TS)/2. It is to be noted that FlexPDE also contains a stagingfeature that is rather useful when solving specific steady-stateconvective heat and mass transport problems.

5. Model substantiation

The geotechnical engineer must show prudence and disciplinewhen using numerical tools to solve boundary and initial valueproblems, and continually question the accuracy of the results.One of the best known methods for assessing the initial capabilitiesof such tools is to compare the numerical solutions to well-definedanalytical, numerical or experimental solutions of simple prob-lems. In general, the comparison process begins with simple stea-dy-state problems and progresses towards more complex time-dependent problems.

As most existing numerical studies have focused on convectiveheat and mass transport of an incompressible fluid, the followingparagraphs examine the role of fluid compressibility while docu-menting the capabilities of the numerical model. The capabilityof the model to solve simple steady-state convective heat and masstransport problems is assessed with two-widely used benchmarks:(1) the finite amplitude analysis of natural convection where theporous media is heated from below; and (2) the shallow layerproblem where the media is heated from the side. The model’sability to solve fairly complex transient convective heat and masstransport problems with soil-water phase change is evaluated bycomparing results with those of a well-documented numericalsolution of a railway embankment.

5.1. Finite amplitude analysis (heated from below)

Consider an infinite layer of air saturated porous media, whichis confined by impervious, rigid and conducting surfaces. As a re-sult of isothermally heating the porous media from below, fluiddensity decreases at the lower boundary and becomes lighter thanthe overlying fluid. Although fluid density increases with elevation,its viscosity and thermal diffusivity have stabilizing effects that canprevent pore-air motion. However, increasing the lower boundarytemperature will eventually result in a convective motion as thetop heavy fluid column becomes unstable. Horton and Rogers[45] were the first to study this porous media analogue to the Ray-leigh–Bénard phenomenon whereas Lapwood [46] theoreticallypredicted the onset of steady-state convective flow. In this andother thermobuoyant flows, the Rayleigh–Darcy number combinesthe time scale of each physical effect (buoyancy, viscous dissipa-tion and thermal dissipation) in order to provide a single instabilitycriterion that can be expressed as follows [31]:

M. Lebeau, J.-M. Konrad / Computers and Geotechnics 36 (2009) 435–445 439

Ra � GrPrDakt;a

kt

¼gð1=ToÞH4ðDT=H � qa;og=Ct;aÞ

ðla;o=qa;oÞ2

la;o=qa;o

kt;a=Ct;a

K

H2

kt;a

ktð6Þ

and, for an isothermally incompressible gaseous phase

Ra � GrPrDakt;a

kt

¼ gboH4DT=H

ðla;o=qa;oÞ2

la;o=qa;o

kt;a=Ct;a

K

H2

kt;a

ktð7Þ

where g is the gravitational acceleration constant, H is the referencevertical dimension, K is the soil intrinsic permeability, kt is the soilthermal conductivity, kt,a is the thermal conductivity of air, To is thereference absolute temperature, DT is the applied temperature dif-ferential, qa,o is the reference density of air, and Ra, Gr, Pr and Da arethe Rayleigh–Darcy, Grashof, Prandtl and Darcy dimensionlessnumbers, respectively. In the case of a compressible gaseous phase,the process of compression cooling affects the apparent verticaltemperature gradient as it adiabatically cools the pore-air when itrises. Including the adiabatic gradient, qag=Ct;a, in the formulationof the Rayleigh–Darcy number is therefore essential to the predic-tion of the onset of convection when heating a porous media frombelow [31].

From various experimental and analytical studies of the Horton–Rogers–Lapwood problem it is noted that the lateral boundaries ofeach convection cell are vertical and adiabatic, and can thus be re-placed by impervious, rigid and insulating surfaces when analysesmust be confined to a finite domain. That is to say that the behaviorof a square enclosure of saturated porous media is numericallyequivalent to that of the Horton–Rogers–Lapwood problem.Numerous numerical analyses of a square enclosure have confirmedthat the onset of steady-state convection occurs at a critical thresh-old Rayleigh–Darcy number, Rac, equal to 4p2 [47–49]. Similar stud-ies have also shown that the layer of porous media undergoes asequence of transitions to chaotic flow with increasing Rayleigh–Darcy number. In fact, the route to turbulence begins with a bifurca-tion from a steady-state to an oscillatory periodic flow at a Ray-leigh–Darcy number of 390, which evolves towards more complextime-dependent flows, and ends in a chaotic state [50–53].

From a mathematical point of view, the single-cell square enclo-sure problem possesses a bifurcation, or loss of uniqueness, at a crit-ical threshold of 4p2. That is to say that the problem always has atrivial solution for which the volumetric flux is equal to zero. Unfor-tunately, iterative finite element methods generally tend towardsthe trivial solution when initial conditions are not chosen appropri-ately. This difficulty has generally been circumvented by turning the

H

a 0u

x

∂=

∂

Z

B

au

z

∂∂

au

z

∂∂

o 2

TT

Δ−

0T

x

∂ =∂

0T

x

∂ =∂

Z

a X

B

303 298 293 288 283 278 273

o 2

TT

Δ+

Fig. 2. Contour plots and vectors (or mesh) for the finite amplitude analysis with anpressure. (c) Stream function.

steady-state problem into a computationally inefficient transientone. Zhao et al. [54], on the other hand, have overcome the difficultyin finding the nontrivial solution of the steady-state problem bygradually increasing the strength of the nonlinear convective term.This is done by solving the problem in stages, where the vertical tiltangle of gravity, a, progresses asymptotically towards zero. In orderto implement this method, referred to as the progressive asymptoticapproach, the gravity vector is decomposed into x and z compo-nents, i.e., gx ¼ g sin a and gz ¼ g cos a, respectively. A tentative non-trivial solution is then found by assuming that the gravity vector tiltsa small angle from vertical. This solution is subsequently used as theinitial condition for the nonlinear system, where the tilt angle is setequal to zero. Hence, the initial tilt angle must be chosen large en-ough to eliminate the strong dependence of the nontrivial solutionon the initial volumetric fluxes and progress asymptotically towardszero to preserve solution accuracy.

In the present study, the side boundaries of a square enclosureare considered perfectly insulated as a constant absolute tempera-ture differential is imposed between the horizontal surfaces, whichare maintained at a constant temperature (Fig. 2). The tempera-tures of the horizontal surfaces are altered by equal amounts rela-tive to the mean reference temperature of the enclosure,To = 288.15 K, while the nominal mean pore-air pressure, ua,o, isset equal to 101.325 kPa. It is to be noted that the initial absolutetemperature is set in accordance with the temperature field of apurely conductive steady-state while the initial pore-air pressurecorresponds to that of a static column of air. The effectiveness ofcombining the progressive asymptotic approach with FlexPDE isshown on Fig. 3, where results are confronted to those of Schubertand Straus [49]. As in other thermobuoyant systems, the principalmeasure of the dynamics of convection is the mean Nusselt dimen-sionless number, Nu, which is the ratio of the heat transports withand without convection:

Nu ¼ �1B

R Bx¼0

oToz dx

� �z¼H

DT=Hð8Þ

where B and H are the width and height of the enclosure, respec-tively. As shown, the solutions of the progressive asymptotic ap-proach are in excellent agreement with the reference numericalsolutions. In quantifiable terms, the difference between the twosolutions translates into a maximum deviation in the order of 1%.

The majority of numerical studies of convective heat transportin porous media have generally neglected the work done by pres-sure changes by adopting the Oberbeck–Boussinesq approxima-tions. In reality, the gaseous phase is compressible, and thevolumetric flux within the enclosure cannot be determined with-out considering the influence of pressure-compressibility cooling

101.325 101.323 101.321 101.319 101.317 101.315 101.313

a 0u

x

∂=

∂ b X

0=

0=

0.0011 0.0007 0.0003 -0.0001 -0.0005 -0.0009 -0.0013

0x

ψ∂ =∂

0x

ψ∂ =∂

Z

c X

B

0z

ψ∂ =∂

0z

ψ∂ =∂

isothermally incompressible gaseous phase, Ra = 60. (a) Temperature. (b) Pore-air

1

10

2

3

5

Mea

n N

usse

lt n

um

ber,

Nu

10 10020 30 50 200 300Rayleigh-Darcy number, Ra

Schubert and Straus [49]

FlexPDE (incompressible-OB)

Fig. 3. Results of the finite amplitude analysis with an isothermally incompressiblegaseous phase.

1

10

2

3

5

Mea

n N

usse

lt n

umbe

r, N

u

10 10020 30 50 200 300Rayleigh-Darcy number, Ra

Typical rockfill enclosure

T HΔ =

0.05 0.03

0.02

1.00

0.015 K/m

Fig. 4. Influence of fluid compressibility on the results of the finite amplitudeanalysis.

440 M. Lebeau, J.-M. Konrad / Computers and Geotechnics 36 (2009) 435–445

and heating processes. Nield [31] has shown that the prime effectof compressibility of an ideal gas is stabilizing whereas other non-Boussinesq effects, such as the variations of fluid properties withtemperature, have a comparatively minor effect. The effectivenessof combining the progressive asymptotic approach with FlexPDEfor a compressible fluid is shown in Fig. 4. Consistency of numeri-cal results is verified by performing a system energy balance,which compares the heat flux at the cold and hot surfaces of theporous enclosure. The energy balance is found to be satisfied with-in 1% for an applied thermal gradient, DT=H, of one while it is with-in 3% for smaller thermal gradients. As foreseen, the effect ofpressure-compressibility is to cool the pore-air adiabatically as itrises, and therefore reduce the mean Nusselt number. The influ-ence of compressibility is also shown to intensify as the tempera-ture difference across the enclosure decreases and/or the height ofthe enclosure increases. In other words, the relative importance ofcompressibility increases as the applied thermal gradient de-creases and approaches the magnitude of the adiabatic gradient,qag=Ct;a, which is equal to 0.0098 K/m. In theory, cooling thepore-air as it rises, and pressure decreases, will counter the fluiddensity decrease that is produced by a thermal gradient in the or-der of one hundredth of a Kelvin per meter.

In the case of a 15 � 15 m enclosure of a typical rockfill material(Table 2), the mean Nusselt number of Fig. 4 is shown to evolve to-wards the solution for incompressible pore-air as the Rayleigh–Darcy number increases, and the applied thermal gradient in-creases. For this particular enclosure, the absolute error of theOberbeck–Boussinesq approximations does not exceed 6%.

H

Z

au

z

∂∂

au

z

∂∂

0T

z

∂ =∂

0T

z

∂ =∂

Z

a X

B

o 2

TT

Δ−

o 2

TT

Δ+ a 0u

x

∂=

∂

B

303 298 293 288 283 278 273

Fig. 5. Contour plots and vectors (or mesh) for the shallow layer problem with an isother(c) Stream function.

5.2. Shallow layer problem (heated from the side)

Let us consider the natural convection that arises in a shallow orsmall H=B aspect ratio enclosure subjected to an end-to-end tem-perature difference. Under these conditions, the fluid rises alongthe isothermally heated wall, travels along the top boundary andthen descends as it releases heat to the isothermally cooled wall.Bejan and Tien [55,56] and Walker and Homsy [57] were the firstto examine pore-air motion in shallow enclosures. Their resultsdemonstrate the dependence of the Nusselt number on the Ray-leigh–Darcy number and the aspect ratio of the enclosure.

In the present study, the top and bottom boundaries of a 0.50aspect ratio enclosure are considered perfectly insulated as a con-stant absolute temperature differential is imposed between thevertical surfaces, which are maintained at a constant temperature(Fig. 5). As in the previous finite amplitude analysis, the initialabsolute temperature is set in accordance with the temperaturefield of a purely conductive steady-state while the initial pore-airpressure corresponds to that of a static column of air. The numer-ical process is fairly straightforward as it involves finding a singleunique solution. The effectiveness of FlexPDE in solving the con-vective flow of isothermally incompressible pore-air is shown onFig. 6, where results are compared to those of Hickox and Gartling[58], Prasad and Kulacki [59], and Peirotti et al. [60]. In this partic-ular configuration, the mean Nusselt dimensionless number is de-fined as follows:

Nu ¼ �1H

R Hz¼0

oTox dz

� �x¼B

DT=Bð9Þ

a 0u

x

∂=

∂b X

0=

0=

0.0012 0.0009 0.0006 0.0003 0.0000 -0.0003 -0.0006

0x

ψ∂ =∂

0x

ψ∂ =∂

Z

c X

0z

ψ∂ =∂

0z

ψ∂ =∂

B

101.325 101.323 101.321 101.319 101.317 101.315 101.313

mally incompressible gaseous phase, Ra = 60. (a) Temperature. (b) Pore-air pressure.

1

10

2

3

5

Mea

n N

usse

lt n

umbe

r, N

u

10 10020 30 50 200 300

Rayleigh-Darcy number, Ra

Hickox and Gartling [58]Prasad and Kulacki [59]Peirotti et al. [60]

FlexPDE (incompressible-OB)

Fig. 6. Results of the shallow layer problem with an isothermally incompressiblegaseous phase.

1

10

2

3

5

Mea

n N

usse

lt n

umbe

r, N

u

10 10020 30 50 200 300

Rayleigh-Darcy number, Ra

Typical rockfill enclosure

0.03 0.05 1.00 K/m

T BΔ = 0.0150.02

Fig. 7. Influence of fluid compressibility on the results of the shallow layer problem.

Heat outflows

Subgrade

Rockfill embankment

Convectiveheat pattern

C L

Fig. 8. Railway embankment cross-section with a schematic representation ofwintertime convective currents (photograph reproduced with permission from D.J.Goering).

M. Lebeau, J.-M. Konrad / Computers and Geotechnics 36 (2009) 435–445 441

As expected, the results are in good agreement with referencenumerical data obtained more than two decades ago. In point offact, maximum percent deviation between the numerical solutionsis in the order of 3%. A similar maximum percent deviation is alsoobserved when comparing results with Bejan and Tien’s [55,56]theoretical solution.

Fig. 7 shows the effectiveness of FlexPDE in solving the convec-tive flow of compressible pore-air in the shallow enclosure. Theaccuracy of the results is checked by means of a system energy bal-ance. Although this energy balance generally lies within 1%, it in-creases up to 2% at low applied thermal gradients, i.e.,DT=B = 0.015 K/m. In stark contrast with that observed when heat-ing the porous media from below, the mean Nusselt number is nowshown to increase as the applied thermal gradient decreases. Thatis to say that the effect of pressure-compressibility cooling andheating is destabilizing. In this particular context, the convectivecurrent intensifies as the adiabatic gradient cools the pore-air priorto it releasing heat along the isothermally cooled wall.

In the case of a 15 � 30 m enclosure of a typical rockfill material(Table 2), the mean Nusselt number is shown to evolve towards thesolution for an incompressible gaseous phase as the temperaturedifference across the enclosure increases with increasing Ray-leigh–Darcy numbers. For this enclosure, the absolute error ofthe Oberbeck–Boussinesq approximations recedes below 10% asthe Rayleigh–Darcy number reaches 132, or then again, as the ap-plied thermal gradient equals 0.08 K/m. Hence, pore-air compress-ibility cannot be neglected in medium to large-sized enclosures atsmall temperature differentials.

5.3. Railway embankment problem

Construction of surface infrastructures in permafrost regionsunavoidably alters the thermal regime at the ground surface. Thesechanges may result in warming and eventual thawing of theunderlying permafrost, which if allowed to occur, typically resultin failure of the surface infrastructure due to thaw consolidation.This has led to the design of specifically engineered embankmentsthat limit the thawing process. Goering and Kumar [18,19] andGoering [61,62] were among the first to report experimental andnumerical data on passively cooled roadway embankments. Thesestudies were followed by a well-documented two-dimensionalnumerical analysis of a railway embankment [20]. As schemati-cally shown in Fig. 8, wintertime convection currents arise aswarm pore-air from the railway embankment base rises and is re-placed by colder pore-air from the upper portion of the embank-ment. In sufficiently coarse uniformly graded materials, theconvection currents transport a large amount of heat from the base

of the embankment to the surface, where it is rejected to the coldenvironment. In the summertime, the pore-air is stable as colderpore-air is overlain by warmer pore-air.

The physical domain considered for the validation process isconsistent with that presented by Goering [20] for a model railwayembankment and subgrade. As shown in Fig. 9a, the thermalboundary conditions consist of a combination of prescribed heatfluxes and temperatures. For instance, the geothermal heat flux isimposed at the lower boundary of the computational domain whilevertical boundaries are considered adiabatic. The upper bound-aries, on the other hand, are subjected to annual harmonic temper-ature functions that are determined with surface specific N-factorsin conjunction with air temperature conditions corresponding to asubarctic permafrost zone. The resulting temperature functions aregiven in Table 1, where t is the time in Julian days. Besides the per-vious embankment surface, zero flow or zero normal pressure gra-dient boundary conditions are used throughout. Table 2summarizes the engineering properties of the embankment andsubgrade materials. So as to eliminate the effect of the assumedinitial conditions, the simulation covers a period of twenty-one(21) years. As in the original study, pore-air is assumed to complywith the Oberbeck–Boussinesq approximations.

During the simulation process, the adaptive refinement proce-dure resulted in unstructured meshes ranging from 598 elements(1267 nodes), in summertime, to 22,691 elements (11,230 nodes)during winter. In an inverse manner, time-steps ranged from 6 to0.006 days. Fig. 9 shows the temperature contour plots and streamtraces obtained on specific days during the final years of simula-tion. On the first day of July, stream traces show that the majorityof pore-air loops through the core of the embankment and exits in

a

( )-2

a a o a,o a,o, , , , , ,

sT T

u f g T T u z β ρ=

=

-1

aˆ 0sT T

u

=⋅ ∇ =n Goering [20]

(T = 273.15 K)

a

ˆ 0

ˆ 0

T

u

⋅ ∇ =⋅ ∇ =

n

na

ˆ 0

ˆ 0

T

u

⋅ ∇ =⋅ ∇ =

n

n

17 12 6 0 0

3

6

9

11.5

b

c

d

2aˆ ˆ0.06W m 0T u⋅ ⋅∇ = ⋅∇ =tn k n

250 255 260 265 270 275 280 285 290

Goering [20](T = 265.15 K)

Goering [20](T = 265.15 K)

Goering [20](T = 265.15 K)

17 12 6 0 0

3

6

9

11.5

17 12 6 0 0

3

6

9

11.5

17 12 6 0 0

3

6

9

11.5

Fig. 9. Temperature contour plot and stream traces within the railway embank-ment and subgrade on specific days during the final years of simulation. (a) July 1.(b) November 1. (c) January 1. (d) March 1. Isotherms and stream traces are drawnin black and white, respectively.

Table 1Annual harmonic temperature functions for the railway embankment problem

N-factor Temperature function (K)

Freeze Thaw

Ta N/A N/A 269:35� 20:00 sin 2 p365 ðt þ 82:25Þ� �

Ts-1 0.50 0.50 271:25� 10:00 sin 2 p365 ðt þ 82:25Þ� �

Ts-2 0.90 1.90 274:25� 26:10 sin 2 p365 ðt þ 82:25Þ� �

Table 2Engineering properties for the railway embankment problem [20]

Property Subgrade Embankment (rockfill)

Ct,f [MJ/(m3 K)] 2.380 1.020Ct,u [MJ/(m3 K)] 3.750 1.020kt,f [W/(m K)] 2.300 0.346kt,u [W/(m K)] 1.500 0.346K [m2] 1 � 10�100 6.3 � 10�7

ha 0.000 0.350hw,u 0.649 0.000qd [kg/m3] 1442 1625

442 M. Lebeau, J.-M. Konrad / Computers and Geotechnics 36 (2009) 435–445

the region of the embankment toe. Large vertical temperature gra-dients are also shown to exist within the embankment while the273.15 K isotherm lies at the embankment/subgrade interface. Asof the first of November, cold air penetrates through the side slopeof the embankment and ascends as it extracts heat from the base ofthe embankment. By the first of January, stream traces show a ser-ies of convection currents that contribute to reducing the temper-ature of the embankment. Remnants of these currents still remainon the first day of March of the final year of simulation. Throughoutthis figure, specific isotherms are generally shown to conform tothose obtained by Goering [20]. The small discrepancies may how-ever be attributed to the differences in the numerical discretiza-tions and, to a limited extend, to the approach used to modelsoil-water phase change. In fact, the model presented in this paperconsiders phase change to occur over a finite range of tempera-tures whereas the reference numerical results relate to an isother-mal phase change process. A detailed comparison of isothermal

240

260

280

300

Ta (

K)

0

4

8

12

16

20

Mea

n m

agni

tude

of

por

e-ai

r fl

ux, q

a (m

/h)

7116 7206 7296 7386 7476Time, t (days)

Goering [20]

Summer SpringWinterAutumn

Fig. 10. Annual variation of ambient air temperature and mean magnitude of pore-air flux.

Table 3Annual harmonic temperature functions for the dam site

N-factor Temperature function (K)

Freeze Thaw

Ta N/A N/A 270:0� 18:1 sin 2�p365 ðt � 66:8Þ� �

Ts-3 0.50 1.00 273:7� 12:4 sin 2�p365 ðt � 67:6Þ� �

Ts-4 0.90 1.90 274:5� 23:0 sin 2�p365 ðt � 67:7Þ� �

Table 4Engineering properties of the dam materials [23]

Property Core and foundation Filter Transition and rockfill

Ct,f [MJ/(m3 K)] 2.000 1.950 1.850Ct,u [MJ/(m3 K)] 2.290 2.130 1.900kt,f [W/(m K)] 1.850 1.480 0.780kt,u [W/(m K)] 1.780 1.690 1.090K [m2] 1 � 10�12 1 � 10�10 5 � 10�7

ha 0.070 0.092 0.138hw,u 0.142 0.088 0.000qd [kg/m3] 2120 2190 2300

M. Lebeau, J.-M. Konrad / Computers and Geotechnics 36 (2009) 435–445 443

and non-isothermal phase change approaches may be found inNixon and McRoberts [63].

As an additional indicator of model agreement, Fig. 10 showsthe mean magnitude of the pore-air flux within the embankmentas a function of time. As shown, the mean pore-air flux varies froma minimum of 3 m/h, at the vernal equinox, to a maximum of19 m/h at the end of autumn. In general, the mean pore-air fluxtends to increase when the ambient air temperature decreases,and it tends to decrease when the air temperature rises. Theembankment material is therefore more effective at removing heatfrom the system during winter than it is at warming the systemduring summer. On the whole, the temporal evolution of the meanmagnitude of the pore-air flux obtained in this study comparesfavorably to that presented in Goering [20].

6. Model application

Given its proven ability to solve various convective heat andmass transport problems, the model is used to further our under-standing of the thermal behavior of the toe drain at the LA2-BSUdam. The area considered for the simulations covers the entiredownstream portion of an approximate representation of thedam (Fig. 11). Seeing as the vertical extent of the physical domainis considerable, pore-air is considered as compressible. As shown,thermal boundary conditions consist of a combination of pre-scribed heat fluxes and temperatures. For instance, annual har-monic temperatures are prescribed at the upper boundaries ofthe computational domain. As these boundaries are in contact withthe ambient environment, harmonic sine functions are determinedwith surface specific N-factors as well as local air temperaturemeasurements (Table 3). Note that the mean annual temperatureof the foundation is equal to 273.7 K or 0.55 �C, indicating that per-mafrost should not be expected in this area. Centerline symmetryand negligible heat fluxes dictate that the normal temperature gra-dient be set equal to zero on the remaining boundaries. Apart fromthe pervious downstream shoulder, boundaries are consideredimpervious and normal pressure gradients are set equal to zero.Table 4 summarizes the engineering properties of the differentmaterials. The simulations cover a period of two (2) years, whichbegins at the end of dam impoundment on August 30, 1996.

Fig. 11a shows the temperature contour plot at the beginning ofthe month of November on the second year of simulation withoutconvective heat transport. Note that the temperature distributionis represented by a smooth and continuous variation of color,where red and blue shades roughly correspond to hot and cold

20 0

484.10

475

450

439.50

ˆ ˆ0T⋅ ∇ =n n

-4

aˆ 0sT T

u

=⋅ ∇ =n

A

A’

aq =

250 255 260 265 270

ˆ 0T⋅ ∇ =n

a 57.95 40 20 0

484.10

475

450

439.50

-4sT T=

-3sT T=

Fig. 11. Temperature contour plot and stream traces (drawn in white) for a simplifiedsecond year of simulation. (a) Conductive heat transport. (b) Conductive and convectconvective heat transport with a pervious downstream shoulder.

zones, respectively. The results indicate that the drain remains un-frozen at the onset of the third winter. That is to say that the drain,which freezes completely during wintertime, thaws completelyduring the course of summer. In contrast, the toe drain remainsfrozen throughout the year when convective heat and mass trans-port is considered and the downstream shoulder is assumedimpervious. The stream traces of Fig. 11b clearly show that theclosed-loop convective currents penetrate into the toe drain andmove upwards along the downstream filter, effectively extractingheat from deep within the embankment. The permafrost or year-round frozen soil, shown in light blue, covers the entire horizontalportion of the filter. Changing the pneumatic boundary conditionon the downstream shoulder, from impervious to pervious, ex-tends the frozen region into the dam core (Fig. 11c). The increasein heat extraction may be ascribed to much larger pore-air fluxeswithin the rockfill material. As indicated, the maximum meanmagnitude of the pore-air flux increases from 2.6 to 17.7 m/h asthe pore-air pressure on the upper boundary of the dam is set tovary with ambient air temperature and elevation. Note that thisapproximate assumption of the dependence of pore-air pressureon ambient temperature is only reasonable when the surface spe-cific N-factor for the freezing soil approaches unity.

b 57.95 40

a 0u⋅ ∇ =

-3

aˆ 0sT T

u

=⋅ ∇ =n

Permafrost

2.6 m/h

275 280 285 290

( )-4

a a a a,o,MM , , , ,

sT T

u f g R T u z

=

=

c 57.95 40 20 0

484.10

475

450

439.50

aˆ ˆ0 0T u⋅ ∇ = ⋅ ∇ =n n

-3

aˆ 0sT T

u

=⋅ ∇ =n

Permafrost A

A’

aq = 17.7 m/h

cross-section of the downstream portion of the LA2-BSU dam in November of theive heat transport with an impervious downstream shoulder. (c) Conductive and

Toe drain (rockfill)

Filter

Foundation

455

460

465

470

Ele

vati

on, z

(m

)

260 265 270 275 280

Temperature, T (K)

472.6

455

460

465

470

Ele

vati

on, z

(m

)

260 265 270 275 280

Temperature, T (K)

472.6 Impervious shoulder

Pervious shoulder

a b

Fig. 12. Influence of the type of pneumatic boundary condition on the temperature profile at section AA0 on specific days during the final years of simulation. (a) November 1.(b) May 1.

444 M. Lebeau, J.-M. Konrad / Computers and Geotechnics 36 (2009) 435–445

Fig. 12 shows the impact of the type of pneumatic boundarycondition on the thermal profile at section AA0, which runs throughthe middle of the toe drain. As of the beginning of November, theyear-round frozen soil is shown to extend down to elevation462 m when the embankment shoulder is considered impervious.An even deeper frost penetration, down to elevation 461.1, occurswhen the boundary is considered pervious. Note that accountingfor weather-related and diurnal cycles of barometric pressure, asin Martinez and Nilson [64], was found to have little or no influ-ence on the thermal profile of section AA0.

Although the bottom portion of the toe drain remains frozenthroughout the year, its temperature reaches 272.3 K, or�0.85 �C, at the beginning of the month of November. The toe draintherefore thaws slowly, yet not completely, over the summermonths. This partial blockage of the toe drain may explain theareas of water egress along the downstream toe of the earthworkover the warmer periods of the year. During the rest of the year,the upper portion of the unsaturated toe drain material remainsfrozen. In this frozen and unsaturated state, it is surmised thatthe pore-ice reduces the hydraulic conductivity of the toe drainmaterial, which hinders the longitudinal movement of seepagewater, and produces areas of water egress along the downstreamtoe of the dam.

7. Conclusion

A numerical model was presented for convective heat and masstransport of compressible or incompressible gas flows with phasechange. The solid and gaseous phases were assumed in thermalequilibrium. If desired, the inherently compressible gaseous phasecould be taken as incompressible by adopting the Oberbeck–Bous-sinesq approximations. Otherwise, the gaseous phase was consid-ered compressible and the model accounted for adiabaticprocesses of compression heating and expansion cooling. In thispaper, pore-air properties other than density were taken to beindependent of temperature and pressure. As most existingnumerical studies had previously focused on the behavior of anincompressible fluid, model substantiation examined the influenceof fluid compressibility on two-widely used benchmarks of steady-state convective heat and mass transport. In accordance with pre-vious findings, the prime effect of pressure-compressibility was tocool the pore-air adiabatically as it ascended. Including the adia-batic gradient in the Rayleigh–Darcy number resulted in adequatepredictions of the onset of convection when the porous media washeated from below. In addition, the relative importance of the sta-bilizing effect of pressure-compressibility cooling was shown to in-crease as the thermal gradient approached the magnitude of the

adiabatic gradient, or then again, as the apparent thermal gradientmoved towards zero. In complete contrast with that observedwhen heating the porous media from below, the pressure-com-pressibility cooling and heating processes were shown to be desta-bilizing when the porous media was heated form the side. Fromthese results, it was concluded that pore-air compressibility couldnot be neglected in medium to large-sized enclosures at small tem-perature differentials.

Given its proven ability to solve fairly complex transient con-vective heat and mass transport problems with soil-water phasechange, the model was used to further our understanding of thethermal behavior of the toe drain at the LA2-BSU dam in the prov-ince of Quebec, Canada. This preliminary investigation showed that(1) year-round frozen conditions in the toe drain could only devel-op in the presence of convective heat transport, signs of which hadbeen observed at the dam site; (2) the type of pneumatic boundarycondition applied to the downstream shoulder of the dam wasshown to have considerable influence on the extent of the year-round frozen conditions. This raises an important question regard-ing the role of snow cover in convective heat and mass transportproblems under cold climate conditions; (3) weather-related anddiurnal cycles of barometric pressure were found to have littleinfluence on the thermal behavior of the dam.

Acknowledgments

The authors wish to acknowledge the financial participation ofNSERC (Natural Sciences and Engineering Research Council of Can-ada) Industrial Research Chair in the Operation of InfrastructuresExposed to Freezing. The authors also extend their appreciationto their collaborators at Hydro-Quebec, in particular, Raymond La-det, Pierre Langlois, Serge Larochelle and Marc Smith.

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