Thermal dispersion driven by the spontaneous imbibition process

Applied Mathematical Modelling 34 (2010) 4184–4195

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Applied Mathematical Modelling

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Thermal dispersion driven by the spontaneous imbibition process

O. Bautista a, F. Méndez b,*, E.G. Bautista a

a SEPI – ESIME Azcapotzalco IPN, México, D.F. 02550, Mexicob Facultad de Ingeniería, UNAM, México, D.F. 04510, Mexico

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

Article history:Received 10 March 2009Received in revised form 14 April 2010Accepted 23 April 2010Available online 28 April 2010

Keywords:Spontaneous imbibitionPorous mediumConvective heat transferThermal dispersion

0307-904X/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.apm.2010.04.016

* Corresponding author. Tel.: +52 55 56 22 81 03E-mail address: [email protected] (F.

In this work, we have theoretically analyzed the thermal dispersion process under theinfluence of the spontaneous imbibition of a liquid trapped in a capillary element, consid-ering the presence of a uniform temperature gradient. The capillary element is representedby a porous medium which is initially found at temperature T0 and pressure P0. Suddenly,the lower part of the porous medium touches a liquid reservoir at temperature Tl and pres-sure P0. This contact between both phases, in turn causes spontaneously the imbibitionprocess. Using a one-dimensional formulation of the average conservation laws, we derivethe corresponding nondimensional momentum and energy equations. The numerical solu-tions permit us to evaluate the position and velocity of the imbibition front as well as thetemperature profiles and Nusselt numbers. The above results are shown by taking intoaccount the influence of three dimensionless parameters: the ratio of the characteristicthermal time to the characteristic imbibition time, b, the ratio of the hydrostatic head ofthe imbibed liquid to the characteristic pressure difference for the imbibition front, a,and the ratio of the dispersive thermal diffusivity to the effective thermal diffusivity ofthe medium, X. The predictions show that temperature profiles and the heat transfer pro-cess are strongly dependent on thermal dispersion effects, indicating a clear deviation incomparison with the case of X = 0 that represents the absence of the thermal dispersion.

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1. Introduction

It is well known that porous media offer notable advantages for transporting complex fluids in many engineering fieldssuch as cooling of electronic equipment, heat storage, solid matrix heat exchangers, thermal insulation of buildings, chemicalreactors, geothermal operations, oil exploration, among others. The above applications are mainly concerning with the anal-ysis of convective heat and mass transfer in porous media. Due to the complexity of the mathematical models for under-standing the basic fluid-dynamics and thermal characteristics, most of the earlier studies ([1–3]) use simplified modelswhich frequently neglect possible heterogeneities of the porous medium. However, the presence of static or dynamics het-erogeneities are very important and must be taken into account to develop more realistic models. For instance, the thermaldispersion has a relevant importance on the transport characteristics. In this direction, Khaled and Vafai [4] investigated themain characteristics of the enhanced cooling by in a rectangular channel through control thermal dispersion effects insidethe fluid. Also, Pinson et al. [5] used the Boussinesq approximation to account for small scale turbulence, including the ther-mal dispersion for laminar or turbulent flows in a flat plate heat exchanger. They showed that the dispersive effects stronglypredominate over the macroscopic turbulent heat flux. Recently, Pedras and Lemos [6] focusing on important applications(combustion and in steam injection systems for enhanced oil recovery methods) showed the influence of different thermalconductivity ratios between the solid and fluid phases plays a significant role to modulate the thermal dispersion effect.

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Nomenclature

c specific heat (J/kg K)d pore diameter of the porous medium (m)g gravity acceleration (m/s2)H arbitrary length of the porous medium (m)h(t) height or position of the imbibition front (m)k thermal conductivity (W/m K)K permeability of the porous medium (m2)Nu Nusselt numberP pressure (Pa)T temperature (K)T time (s)U velocity (m/s)u nondimensional velocity defined by Eq. (13)y Cartesian coordinate

Greek symbolsa nondimensional parameter defined by Eq. (15)b nondimensional parameter defined by Eq. (12)c nondimensional wicking front defined by Eq. (13)f nondimensional coordinate defined by Eq. (13)g nondimensional coordinate defined by Eq. (13)h nondimensional temperature defined by Eq. (13)j�1 capillary length (m)l viscosity of the liquid (Pa s)q density of the liquid (kg/m3)r surface tension at the liquid interface (N/m)s nondimensional time defined by Eq. (13)U nondimensional pressure defined by Eq. (13)/ porosity of the porous medium

Subscriptsc capillaryC characteristicconv convectiveD Darcydiff diffusivee effective heat capacity of the liquid–porous matrixf liquidi nodals solidth thermal

Superscriptn time step

O. Bautista et al. / Applied Mathematical Modelling 34 (2010) 4184–4195 4185

For non-isothermal situations, the imbibition phenomenon is a process for which a wetting viscous liquid displaces orpushes aside a less non-wetting viscous one. Sometimes, this action can be only sustained by the action of capillary forces.In this case, we speak of the spontaneous imbibition. In the above discussion, we accept obviously that both fluids are immis-cible. Therefore, in this class of theoretical and experimental studies the front of displacement of the interface is not knownin advance and should be estimated as a part of the problem. From the pioneer work of Washburn [7] for predicting the well-known displacement law, h � t1/2, nowadays the specialized literature has been extended widely in order to take into ac-count the influence of different factors. Recently, Alava et al. [8] presented the state of the art in disordered media, empha-sizing those aspects that have a strong influence on the interface existing between both phases. Therefore, there are manypossible deviations from the simple Lucas–Washburn description and these new fundamental studies serve us to understandbetter a great variety of technological applications. Nevertheless, there is another class of imbibition processes that underdifferent physical conditions can regulate relevant transport phenomena. Recognizing that the literature is scarce in thisdirection, Alava et al. [8] suggested that additional efforts are required to understand simultaneously the imbibition andtransport effects.

4186 O. Bautista et al. / Applied Mathematical Modelling 34 (2010) 4184–4195

Since the practical applications related to the analysis of simultaneous imbibition-transport processes is very vast, herewe have focused on the theoretical study of the spontaneous imbibition and heat transfer in a Darcian porous medium,including thermal dispersion and gravity effects. In spite of the practical importance of this class of processes, few workshave appeared in the past to understand such combined effects. Without the presence of the thermal dispersion effect, Phil-lips [9] noted that uniform geothermal gradients in natural reservoirs can seriously affect the prediction of fluid-dynamicscharacteristics. Also, the work of Babadagli [10,11] simulated – under controlled experimental conditions – a reservoir of oilsubject to a temperature field. He proved that oil production by imbibition can be driven by injecting a hot water vapor.Other interesting study of combined effects was considered by Sánchez et al. [12] to analyze the convective heat transferprocess conducted by the imbibitions in a non-Darcian porous medium.

In an effort to understand the influence of thermal heterogeneities that are present in porous media, in the present workwe study theoretically the non-isothermal imbibition of a liquid-saturated layer into a dry zone composed by a porous med-ium under the influence of a uniform temperature gradient. In particular, we choose the thermal dispersion as a character-istic heterogeneity of the medium. The present analysis is basically organized as follows: we present an equation of motionfor describing the time evolution of the imbibition front. In contrast with other theories, the above equation is coupled to theenergy equation for the wetting phase. This formulation includes as a particular case, the well-known Lucas–Washburn law[7], valid for isothermal imbibition. For simplicity, we assume one-dimensional flow configuration. The resulting governingequations are reduced to the energy equation which has been solved numerically by using the conventional Crank-Nicolsonnumerical scheme. In particular, the numerical solutions show that the transient response of the imbibition front is sensiblycontrolled by the parameter b. The influence of this dimensionless parameter on the corresponding temperature profiles andNusselt numbers dictated by the thermal dispersion, are also shown.

2. Theoretical model

The physical model under study is shown in Fig. 1. A slender piece or sheet that consists of an air-saturated porous med-ium (with porosity /) is found initially at temperature and pressure T0 and P0, respectively. We assume thermal equilibriumbetween the saturated air and the porous matrix. Suddenly, the lower part of the sheet touches a liquid reservoir at temper-ature Tl and the same pressure P0, causing a non-isothermal imbibition process of the liquid into the porous medium. In addi-tion, the imbibition front h = h(t) is characterized by a uniform capillary pressure, Pc. We assume that the temperature of thefront is maintained at temperature T0. This approximation was proved in [12] based on the fact of that the characteristicimbibitions time is larger than the corresponding thermal time. The origin of coordinates is located at the base of the sheet.We adopt one-dimensional formulation; therefore, we can introduce a longitudinal or vertical coordinate y measures up-ward in the direction of the imbibition front.

The competition between characteristic thermal and dynamics penetration lengths generates a non-isothermal capillaryflow that is developed inside the porous medium. After an elapsed time t, the non-isothermal imbibition front reaches anaverage distance, h(t). Here, the meaning of an average distance h(t) is to accept that the microscopic effects, in a firstapproximation, are neglected [8]. Therefore, the thermal and imbibition effects permit us to introduce two times scales:the thermal penetration scale, tth, and the imbibition scale, tw. An order of magnitude analysis serves us to identify both

Fig. 1. Schematic view of the physical model during the non-isothermal imbibition process in a porous medium.


scales. In order to find both scales, we use the energy equation for the porous medium including the thermal dispersion ef-fect and given by [13]:

oTotþ ðqcÞfðqcÞe

UDoToy¼ o

oyðae þ g1UDÞ

oToy

� �: ð1Þ

where (qc)e represents the effective heat capacity of the liquid-porous matrix system and it is defined as(qc)e = /(qc)f + (1 � /)(qc)s. qf(s) and cf(s) are the density and specific heat of the wetting liquid (or porous medium). Here, the sub-scripts f and s represent wetting and porous-matrix conditions. UD is the Darcy velocity of the liquid in the porous mediumand ke is the effective thermal conductivity. g1 is a constant related to the thermal dispersion in the porous medium. In gen-eral, this constant g1 must be determined experimentally. In our case, we are using a simplified model to describe the exis-tence of thermal dispersion effects that are represented here by a thermal dispersive coefficient given by the term g1 UD ofEq. (1). Therefore, the constant g1 given in units of length indicates or measures the intensity of thermal dispersion effectswhich are at least proportional to the velocity component UD and depends on the physical and geometrical characteristics ofthe porous medium and the physical properties of the fluid. Characteristic values for this constant and some different modelsof thermal dispersion are given in [13] and [17]. ae = ke/(qc)e is the effective thermal diffusivity of the medium. In this work,in addition, we assume that the temperature of the imbibition front is found at thermal equilibrium with the dry zone attemperature T0.

From Eq. (1), we can readily identify by using an order of magnitude analysis, two characteristic time scales associatedwith the thermal energy transported by the imbibed fluid: a convective scale and a conductive scale. The physical impor-tance of these time scales together with the imbibition time scale is to decide which is the most relevant to describe ade-quately the evolution of the imbibition front with the simultaneous presence of thermal and dispersion effects. Therefore,for the estimation of the convective time scale, we use directly a comparison or an energy balance between the transportedthermal energy by the motion of the liquid and the accumulation energy term. A similar argument was used in [12]. In thepresent case, the comparison yields a direct relationship between the first and second term of the left-hand side of Eq. (1)given by,

ðqcÞf ðT1 � T0ÞUCD

hth� ðqcÞeðT1 � T0Þ

tthconv

: ð2Þ

where UCD, hth and tth in the above equation represent the characteristic Darcy volume-average velocity associated with thevelocity of the imbibition front, the characteristic thermal penetration length and the characteristic convective time scale,respectively.

From Eq. (2), we can easily derive that the characteristic convective time scale is given by the relationship,

tthconv �ðqcÞehth

ðqcÞf UCD: ð3Þ

In a similar manner, from Eq. (1), we can perform a dominant balance between diffusive and accumulation terms, which dic-tates that,

ðT1 � T0Þtthdiff

� ðT1 � T0Þðg1UCD þ aeÞh2

th

; ð4Þ

using the above relationship, the characteristic diffusive time scale is then given by,

tthdiff� h2

th

aeðXþ 1Þ ; ð5Þ

where the dimensionless parameter X is given as X ¼ g1UCDae

and represents the competition between the dispersion thermaland thermal diffusivity effects. In order to obtain the order of magnitude of the characteristic thermal length, we can com-pare the diffusive and convective terms of the energy equation, through the relationship

ðqcÞf ðT1 � T0ÞUCD

ðqcÞehth� ðT1 � T0Þ

h2th

ðg1UCD þ aeÞ; ð6Þ

obtaining that the characteristic thermal length is given by

hth �ðqcÞeðqcÞf

ðg1UCD þ aeÞUCD

: ð7Þ

On the other hand, the imbibition volume-average velocity UCD can be easily derived from the driven capillary pressuregradient,

UCD �KðP0 � PcÞ

lhi; ð8Þ


where K, l and hi are the permeability of the porous medium, the liquid viscosity and the equilibrium height of the imbibi-tion front, respectively. This last characteristic length hi is the equilibrium height for the imbibition front in a porous med-ium, given as [14]:

hi �j�2

d; ð9Þ

where j�1 ¼ffiffiffiffiffiffiffiffiffiffiffiffir=qg

pis the capillary length; r and q represent the surface tension and the density of the liquid penetrating

into the porous medium; respectively, and d is the pore diameter.Combining Eqs. (7)–(9), the ratio of the characteristic thermal length to the imbibition equilibrium height is given by:

hth

hi� ðqcÞeðqcÞf

laeðXþ 1ÞKðP0 � PcÞ

<< 1: ð10Þ

The above relationship means that the temperature variations are confined to a smaller region than the length defined by theequilibrium height of the imbibition front. This suggests the existence of a thermal boundary layer at the base of the porousmedium.

On the other hand, the imbibition time scale is easily obtained from the relationship

ti �hi

UCP¼ hi/

UCD; ð11Þ

where UCP is an average characteristic velocity, defined by UCP = UCD//. Combining Eqs. (5) and (11), we can define the non-dimensional parameter b as:

b � tth

ti¼ hth

hi

� �2 KðP0 � PCÞl/aeðXþ 1Þ >> 1: ð12Þ

The above inequalities for the ratio hth/hi, given by Eq. (10), and the parameter b, Eq. (12), were just derived for the case ofwater as imbibed fluid and limestone as porous matrix. The fluid and thermal properties of the water were taken from dif-ferent references. For the density, viscosity coefficient and surface tension of water, we take data from [15]. On the otherhand, the specific heats and thermal conductivities of water and limestone were directly taken from [16], together withthe permeability and porosity of the porous medium. All properties were evaluated at T = 300 K and the capillary pressurewas taken as Pc = 19,000 Pa and the atmospheric pressure as Pc = 101,300 Pa. Taking into account the above comments, wehave that hth/hi = 0.446 and b = 5.3. Therefore, the relationships (10) and (12) remain unaltered for the provided data. Rec-ognizing then that tth is larger than ti, the analysis of this transient heat transfer process which is governed by thermal dis-persion effects can be characterized by adopting as characteristic time the thermal penetration scaletth ¼ tthdiff

. The abovepermits us to nondimensionalize adequately the governing equations presented lines below.

For simplicity, we follow the Darcy model for the momentum equation [17], together with the energy equation, Eq. (1),for the wetting region. It is important to emphasize that these governing equations are derived for an incompressible New-tonian fluid flowing into an isotropic and homogeneous porous matrix.

Nevertheless, the existence of the porous medium introduces some additional difficulties to derive these equations. Inparticular, flow instabilities, momentum diffusion caused by the friction, and large-scale variations in porosity and other ef-fects associated with the presence of the porous medium complicate the derivation of these equations. Fortunately, theabove effects may be formulated using the volume-average technique for the continuity, momentum and energy equations[17].

3. Dimensionless governing equations

In order to reduce the physical variables, we prefer to introduce the following nondimensional variables:

s ¼ ttth; g ¼ y

hth; c ¼ hðtÞ

hi; f ¼ y

hðtÞ ; h ¼ T � T0

T1 � T0; U ¼ P � Pc

P0 � Pc; and u ¼ UD

UCD; ð13Þ

In this manner, the mass, momentum and energy nondimensional governing equations can be written as

ouof¼ 0; ð14Þ

oUof¼ �uc� ac; with a ¼ qf ghi

P0 � Pc; ð15Þ

and

ohosþ u

ohog¼ ð1þXuÞ o

2hog2 ; ð16Þ


respectively. It is important to note that the dimensionless parameter a denotes the competition between the hydrostatichead of the imbibed liquid and the characteristic pressure difference for the imbibition front. Since the position of the imbi-bition front c is unknown and considering that u=(1/b)(dc/ds), the corresponding evolution equation for the front is given by:

Fig. 3.

cdcds¼ �b

oUof

��f¼1� abc ð17Þ

and the initial and boundary conditions are:

cðs ¼ 0Þ ¼ 0; Uðf ¼ 0Þ ¼ 1;Uðf ¼ 1Þ ¼ 0; hðg; s ¼ 0Þ ¼ 0;hðg ¼ 0; sÞ ¼ 1 and hðg!1; sÞ ! 0

ð18Þ

On the other hand, a first integration of the mass conservation Eq. (14) dictates that u = u(s) and recognizing also thatc = c(s), the corresponding momentum Eq. (15) can be integrated with the boundary condition U(f = 0) = 1, yielding:

U ¼ �cfu� acfþ 1: ð19Þ

Fig. 2. Position of the imbibition front as a function of the nondimensional time for a = 0.1 and four different values of the parameter b.

Nondimensional temperature profiles as a function of the nondimensional coordinate g for four different values of the parameter X and with b = 5.


Taking into account the boundary condition U(f = 1) = 0, Eq. (19) can be transformed to the following equation:

Fig. 4.

Fig. 5.

cb

dcds¼ 1� ac; ð20Þ

which satisfies the initial condition c(0) = 0. By solving Eq. (20), we obtain the implicit equation for the time evolution of theimbibition front:

s ¼ �acþ lnð1� acÞa2b

: ð21Þ

As a particular case, neglecting the gravity effect (a ? 0), we obtain in a first approximation the well-known Washburn law,which is easily found by integrating Eq. (20):

c ¼ ð2bsÞ1=2: ð22Þ

Therefore, for finite values of the nondimensional parameter a, the corresponding imbibition fronts represent deviations ofthe classical Washburn front. In this sense, we name the displacement of these fronts as non-Washburnian. Therefore, theenergy equation (16) in terms only of c is reduced to:




ohosþ ð1� acÞ

cohog¼ 1þ 1� ac

c

� �X

� �o2hog2 ; ð23Þ

with the corresponding initial and boundary conditions given by relationships (18). Since the nondimensional parameters a,b and X define the influence of the imbibition process on the corresponding temperature profiles, Eqs. (21) and (23) repre-sent then a typical example of forced convection, where the flow pattern is driven by the combined effects of capillary andgravitational forces. The numerical scheme for the solution of Eq. (21), together with Eq. (23) and the corresponding initialand boundary conditions (18) for c and h, is presented in the following section.

4. Numerical procedure

In this section, we present in brevity the finite-difference method to solve the nondimensional governing equation (23),together with the boundary and initial conditions (18). The finite difference version of Eq. (23) is

hjþ1i ¼ 1þX

1� acc

� �� ðhj

iþ1 � 2hji þ hj

i�1ÞDsDg2 �

1� acc

� �ðhj

iþ1 � hjiÞ

DsDgþ hj

i: ð24Þ

Fig. 6. Time evolution of the temperature for different vertical positions, with b = 25 and X = 1.

Fig. 7. Time evolution of the temperature for different vertical positions, with b = 25 and X = 0.5.


i and j are the nodal and time steps; respectively. In addition, c = c(t) is obtained numerically from Eq. (20) for the imbibitionfront, using a conventional fourth order Runge–Kutta method. With the above procedure, we can obtain the interfacial posi-tion of the imbibition front; i.e. c = c(s; a, b). Introducing the above solution of c into Eq. (24) we obtain the temperatureprofiles for the wetting zone.

5. Results and discussion

The numerical results are plotted in Figs. 2–13. The range of the chosen values for the dimensionless parameters was tak-ing into account real data of imbibed fluids in porous media. Because the range of these parameters can be very large, theassumed values in our numerical results are representative of typical cases. In these figures, al numerical estimations wereperformed with a = 0.1. We have basically emphasized the importance of nondimensional parameters b and X. It should benoted that the solution of Eq. (20) is independent of the temperature field, i.e., there is no influence of the temperature pro-files on the flow pattern. The above situation can be directly appreciated from Eqs. (21) and (23). In these equations is clearthat the imbibition front c is only function of time and dimensionless parameters a and b, while that the temperature profile,h, is dependent explicitly on the dimensionless variables g, c and s together with the dimensionless parameters a and X.Therefore, only the temperature field is coupled with the hydrodynamics of the imbibition front. In Fig. 2, we show the

Fig. 8. Time evolution of the temperature for different vertical positions, with b = 100 and X = 5.

Fig. 9. Time evolution of the temperature for four different values of the parameter X and a fixed position g = 0.5.




non-isothermal imbibition front c as a function of the nondimensional time s and four characteristic values of the nondimen-sional parameter b(= 5, 25, 50 and 100). The influence of the parameter b is notable; in particular, for increasing values of thisparameter, the imbibition process is strongly accelerated.

In Figs. 3–5, we show the corresponding temperatures profiles. Accordingly, we present in these figures for different val-ues of a and b, and arbitrary values of the dimensionless time, the nondimensional temperature profile, h, as a function of thenondimensional coordinate g. In these figures, we can appreciate in essence that for increasing values of the thermal disper-sion, here represented by the parameter X, the corresponding values of the temperature are also increased. Therefore, thethermal dispersion induces that the heating process of the porous medium becomes easier.

In Figs. 6–8, we present the time evolution of the dimensionless temperature for different values of the mentioned dimen-sionless parameters, considering fixed values of the nondimensional coordinate g. Obviously, for all cases, the temperature isalways larger for decreasing values of the coordinate g; and for increasing values of the dimensionless time, the temperaturetends to reach a stationary value that depends on the assumed values of the dimensionless parameters.

In addition, in Figs. 9–11, we also present the time evolution of the temperature for fixed values of the coordinate g,showing clearly the influence of the thermal dispersion effect. In these figures, we can again observe that the thermaldispersion effect always increases the values of the temperature. As comparison, in these figures, we have also includedthe case X = 0, which represents the absence of thermal dispersion effects. In Fig. 12, we show also the time evolution of

Fig. 12. Time evolution of the temperature for different positions.

Fig. 13. Nusselt number as a function of the nondimensional time for different values of the parameter X.


the dimensionless temperature h for selected values of the spatial coordinate g and some values of the dimensionless param-eters ab and X. In this case, for fixed values of the dimensionless time, the values of the temperature always reaches highervalues for smaller distances, indicating that the heating process of the porous medium is controlled by the influence of thethermal dispersion effect, here represented by the parameter X.

Finally, in Fig. 13 we have plotted the Nusselt number defined as Nu = �oh/og|g=0, (which represents the nondimensionalheat transfer) as a function of the nondimensional time s and different values of the parameters b. Clearly, for increasingvalues of the dimensionless parameter X, the corresponding Nusselt number is decreasing, indicating as it was previouslycommented, that temperature gradients tend to be small. Therefore, the influence of the thermal dispersion attenuatesthe heat transfer to the porous medium.

In this work, the problem of the non-isothermal imbibition process, including the thermal dispersion and gravity effectsunder a sudden temperature step in a porous medium has been numerically analyzed. The use of an equation of motion forthe imbibition front coupled with the corresponding energy equation for the wetting zone, are shown. The influence of themain parameters involved on the simultaneous imbibition-heat convective process indicates that the thermal dispersion isnot an isolated process. Eqs. (20) and (23) govern this forced convective process. It is really important to note that even thecase X = 0 has not been previously reported in the specialized literature. This last case corresponds to a limit only controlledby diffusive-convective terms of the energy equation. Therefore, the present result has been extended to consider a new


diffusive contribution given by the thermal dispersion term. The practical importance of the present results is directly con-nected with the studies of Babadagli [10,11]. For those cases where the oil production in porous reservoirs is influenced byimbibitions and thermal effects, a special care must be taken into account for including any type of heterogeneity. Therefore,we believe that the present model serve us as a reference to predict and explore new phenomena which remain unexploredin this area, where the transport and the imbibitions processes complicate the well-known regimes purely controlled by dif-fusive and convective terms.

Acknowledgments

This work has been supported by a research Grant No. 58817 by Consejo Nacional De Ciencia y Tecnología at Mexico.

References

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[10] T.J. Babadagli, J. Petrol. Sci. Eng. 14 (1996) 197.[11] T.J. Babadagli, J. Petrol. Sci. Eng. 33 (2002) 223.[12] M. Sánchez, E. Luna, A. Medina, F. Méndez, J. Colloid Interface Sci. 288 (2005) 562.[13] A.D. Nield, A. Bejan, Convection in Porous Media, Springer, New York, 2006.[14] P.-G. De Gennes, W.F. Brochard, D. Queré, Capillarity and Wetting Phenomena Drops, Bubbles, Pearls, Waves, Springer, New York, 2004.[15] F.M. White, Fluid Mechanics, McGraw-Hill, New York, 2003.[16] F.P. Incropera, D.P. DeWitt, Fundamentals of Heat and Mass Transfer, Wiley, New York, 1996.[17] K. Vafai, Handbook of Porous Media, CRC Press, New York, 2005.

Thermal dispersion driven by the spontaneous imbibition process

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