introduction to modeling of transport phenomena in porous media || heat and mass transport

Chapter 7

Heat and Mass Transport

In this chapter, our primary objective is to construct models of heat transport in porous media. However, since advection (with a fluid moving through the void space), is one of the transport modes, any heat transport model must treat, simultaneously, also the transport of fluid(s) mass. As we shall see below, the coupling between the transport of these two extensive quantities is due also to the fact that both the fluid's density, and, perhaps, more so, its viscosity, depend on the temperature.

Problems of heat and mass transport in porous media are encountered in chemical engineering, where reactions absorb or release heat, in petroleum reservoir engineering, in geothermal reservoir engineering, in projects of energy storage in the unsaturated zone in the soil and in aquifers, and in radioactive waste repositories in deep geological formations, where the waste acts as a large heat source.

Unlike the case of mass transport, where the solid itself is assumed impervious to mass flux, here the solid matrix does conduct heat. The average temperature ofthe solid and of the fluid (or fluids) filling the void space, need not be the same; heat may be exchanged between these phases. However, most of the presentation in this chapter will be based on the assumption of thermal equilibrium between the phases, i.e., the assumption that the (averaged) temperatures are the same for all phases.

The development of heat transport models in this chapter is based on the material presented in Subs. 2.2.2( c), 2.2.2( d) and 2.4.6.

In dealing with advective mass flux, we shall employ all the assumptions that lead to Darcy's law in the form of (2.6.54). Unless otherwise specified, we shall also assume that the solid matrix is rigid and stationary.

449 J. Bear et al., Introduction to Modeling of Transport Phenomena in Porous Media© Kluwer Academic Publishers, Dordrecht, The Netherlands 1990

450 HEAT AND MASS TRANSPORT

Radiation, as a mode of energy transport is significant in gas flow through porous media at very high temperatures. Such temperatures occur, for example, in nuclear fuel rods. This mode of energy transport will not be discussed in this chapter.

The entire discussion is at the macroscopic level. Hence, no special symbol will be used to indicate this fact.

7.1 Fluxes

We consider the simultaneous transport of heat and mass, as both phenomena are always coupled to each other in a porous medium domain under nonisothermal conditions. This is a consequence of the constitutive relations for a fluid viscosity, J1 = J1(p, p'Y, T), for its density, p = p(p, p'Y, T), for the surface tension, "Ywn = "Ywn(P'iv, p~, T) between adjacent fluids that occupy the void space, and for the stress-strain relationship of a thermoelastic solid, Us = O's(es ,1J). Here, p'Y is a symbol that denotes the concentrations of components in a nonhomogeneous fluid phase.

In Subs. 2.3.5, in the process of deriving the average of the gradient of a scalar variable, including the case of a temperature gradient, we saw also coupling between the macroscopic values of a state variable in adjacent phases.

7.1.1 Advective flux

The advective mass flux, pqm, of a fluid phase at a volumetric fraction (), is obtained from (2.6.54). We may write the latter equation in the form

p()Vr = pqj = p( q:;j + ()Vsj)

pkj~ ( op {)z ) -- - + pg- + p()Vsj J1 {)x~ {)x~

pkj~ ( {)p {)z ) pkj~ {)z -- ~ + Pog~ + -(Po - p)g~ + p()Vaj, J1 UXl uX~ J1 UXl

(7.1.1)

in which p = p(p,p'Y,T), e.g., in the form of (2.2.78), J1 = J1(p,p'Y,T), and Po is some reference density (at Po,pJ, To). The last form is often used in discussing natural convection (Sec. 7.5).

In this section, we shall usually approximate qm ~ q, i.e., we shall neglect the diffusive flux of the phase mass.

Fluxes 451

From (7.1.1) it follows that we may interpret the advective flux as produced by two driving forces: one resulting from a gradient in piezometric head, <p = z + p/ Po9, of a fictitions fluid of density, Po, and the second resulting from a buoyancy force, directed vertically, acting on fluid particles of density p, imbedded in the fluid of density po. The two forces are of the orders of magnitude

respectively, where (~<p)c and (~p)c are characteristic piezometric head difference and characteristic density difference, respectively. The ratio between the two is of order O( {(~p)c/ Po}/{(~<p)c/ L~cp)}).

When ({(~p)c/Po}/{(~<p)c/L~CP)}) ~ 1, the flow is governed mainly by the head gradients. The flow regime is then referred to as forced convection.

When ({(~p)c/Po}/{(~<p)c/L~CP)}) ~ 1, the flow is determined mainly by the buoyancy force, and the flow regime is referred to as free, or natural convection. Natural convection is further discussed in Sec. 7.5.

Equation (7.1.1) is valid for a single fluid phase within the void space, or for a fluid phase in a multi phase system, assuming that interphase momentum transfer is negligible (see Subs. 2.6.2).

Following the development in Example 4 of Sec. 3.3, it may be of interest to rewrite (7.1.1) in a dimensionless form. Such a form is given by (3.3.29). Let us assume qm ~ q, Vs = 0, and neglect the effect of concentration changes on the fluid's density. If we select the characteristic pressure increment as

( A) _ /1cqc up c = ( ) ,

(kc/L/ )

equation (3.3.29) can be rewritten in the form

~ __ kij ~ ( * z*) kij (p) Raj 1)7,c * 8z* qt - /1* 8x; p + Eu Fr(p)2 + /1* Da Pe1)H 1)!! T 8x;'

(7.1.2)

where

(7.1.3)


is the Rayleigh number for a fluid continuum, with p ~ Po(1 - (3T(T - To», V~c is the characteristic thermal diffusivity of the fluid, defined by

and

V H _ Aj,c j,c - (pC)j,c (7.1.4)

(7.1.5)

is the Peclet number in a porous medium, with the characteristic thermal diffusivity of a porous medium, V~, defined by

(7.1.6)

and (pC)pm = npCv + (1 - n)psCs (7.1.7)

expresses the heat capacity (per unit volume) of the porous medium as a whole.

In writing (7.1.3) through (7.1.6), we have assumed L~P) = Lc. The symbol AH denotes the coefficient of thermal conduction in the porous medium as a whole.

Finally, with (pC)j (= pCv) denoting the heat capacity (per unit volume) of the fluid, if we select the characteristic fluid velocity such that

VH Vc=_c,

Lc (7.1.8)

the dimensionless expression for the advective flux of the fluid, takes the form

* k'[j { {) (* z*) '* {) z* } qi = - 11* {)x~ P + Eu Fr2 - Ra T {)x~ , r J J

(7.1.9)

where Ra' is now the Rayleigh number of a porous medium defined by

(7.1.10)

In the literature (e.g., Bories, 1987), a Rayleigh number for a porous medium is often defined as

(7.1.11)

Fluxes

with a Prandtl number for a porous medium defined by

Pr = (PC)~cllc Ac

If we now select the characteristic velocity as

v; _ V!! (pC)pm,c c - ncLc (pC) f,c '

453

(7.1.12)

the dimensionless expression for the advective flux, (7.1.9), remains the same, with Ra' replaced by Ra.

In a multiphase system, coupling takes place between the advective fluxes of heat and mass, due to

• The dependence, mentioned above, of fluid density, viscosity and surface tension on temperature. The latter effect was discussed in detail in Subs. 2.6.2 .

• The dependence of the effective permeabilities on temperature through the latter's effect on the porosity, e.g., in a thermoelastic solid matrix, and because of the temperature effect on the configurations of the fluid phases within the void space (through its effect on Iwn, which, in turn, determines the radius of curvature of the microscopic menisci).

The two flux equations in (7.1.14) below, are also coupled to each other by the capillary pressure relationship which in this case, due to the effect of temperature on the surface tension, takes the form

Pn - Pw = Pc(Sw,p~,p7v,T). (7.1.13)

Based on the discussion in Subs. 2.6.2, and assuming Swn ~ Sws, we can write the advective mass fluxes in a two-fluid system, w, n, in the form

(7.1.14)

where we have assumed Tw w == Tnn == T. The coefficients that introduce the effects of'VT and 'V p"Y can be obtained from the presentation in Sub. 2.6.2.


We could use (7.1.13) to write

V'( ) - oPe S OPe 'Y OPe 'Y OPe Pn - Pw - oSw V' w + op7n V' Pw + op~ V' Pn + oT V'T. (7.1.15)

A particular case may serve as an example. In unsaturated flow in the soil, when we consider only the flow of water, w, and assume that the air is at constant pressure, say Pn = 0, we can write the advective flux equation for the water in the form

kwji ( OPe I oSw oz ) -- -- -- - Pw9-J.lw oSw pi",T OXi OXi

+ (kwji OPe I _ '" 'Y .. ) 0 p7n + (kwji OPe I _ ",T .. ) oT !:I 'Y WJ~ !:I !:IT WJ~ !:I . J.lw upw Sw,T UXi J.lw U Sw,pi" UXi

(7.1.16)

Continuing to consider nonisothermal water flow in the unsaturated zone, assuming that the air is at a constant pressure, (7.1.14) is often written for a rigid isotropic porous medium, in the form

where P / Po ~ 1 - f3TT, J.l = J.l(p, T), Kw = kwPo9 / J.l is the effective hydraulic conductivity of the water, related to the reference density, Po, and 'Ij; = 'Ij;((}w,1'wa(T)) = -Pw/Po9 is the fluid's suction, also related to Po.

In (7.1.17), we have introduced the coefficients

Kw((}w) :()~ IT = isothermal water dijJusivity,

Kw((}w) !:I0'lj; I = nonisothermal water dijJusivity. u1'wa Ow

The gravity term in (7.1.17) is often neglected. In developing (7.1.17), the effect of solute concentration has been neglected.

The advective heat flux in a fluid phase is expressed by

q~ = (}pCvTVr == pCvT(q;:j + (}Vsj), (7.1.18)

Fluxes 455

in which the heat capacity, Cv, may be a function of temperature, e.g., in a gas.

For a solid phase, the advective heat flux is given by

(7.1.19)

Henceforth, unless otherwise stated, we shall assume Va == O. In a deformable, e.g., thermoelastic, solid matrix, the solid's velocity is

determined by solving the problem of fluid mass flow through a deformable porous medium, as described in Sees. 4.1.3. and 5.3.3., this time under nonisothermal conditions.

7.1.2 Dispersive flux

The expressions for the dispersive heat and mass fluxes, follow from the presentation in Subs. 2.6.4. A detailed discussion on the dispersive mass flux, is also given in Subs. 6.1.1.

The expression for the dispersive heat flux can be derived from (2.6.152), with eH = pCvT. We obtain

rH = _ Dl.!. 8(pCvT ) J J~ 8X i . (7.1.20)

To obtain an expression for the coefficient of thermal dispersion, DTI, appearing in this equation, we have to insert in (2.6.153), a thermal Peclet number, PeH , defined by

H V~ Pe = A/pCv' (7.1.21)

in which V is the macroscopic velocity of the fluid, ~ is the hydraulic radius, A is the thermal conductivity of the fluid phase, and A/ pCv is the thermal diffusivity. This definition of the Peclet number can be understood by comparing (2.6.118) with (2.6.110).

Thus, the coefficient of thermal dispersion is given by

H H Vk a V£''' H H Daim = aaikim Va f(Pe ,fa/~a), (7.1.22)

with the thermal dispersivity defined by

(7.1.23)


where f~ is the distance of correlation between the velocities, V H , of energy particles. The meaning of VH can be obtained from the expression for the conductive flux of heat at the microscopic level

j~i == e~(V! - V~) = paCVaTa(V! - V~) = -AafjfjT . Xi

We obtain, again at the microscopic level

(7.1.24)

We assume that f~ ~ f~, and, therefore, the thermal dispersivity aH and the component transport dispersivity, a"Y, are approximately the same.

This means that for PeH ~ 1, f(PeH, r) = PeH 1(1 + PeH + f~ Il:1a )

= 0(1), and the coefficients of thermal and mass dispersion are the same. However, when PeH ~ 1, f(PeH,r) ~ PeH/(l +f~/I:1).

Under these conditions, let us examine the relative magnitudes of the advective, dispersive and conductive heat fluxes. When PeH ~ 1, the advective heat flux is much larger than the conductive one. At the same time thermal advection dominates over thermal dispersion as long as

If PeH ~ 1, advection dominates when

(T) Lc (I:1T)c p H --~-- e .

ac Tc

(7.1.25)

(7.1.26)

Usually ac and LF) are represented by the hydraulic radius, 1:1, and the characteristic length of an REV, respectively. The ratio LF) lac is then often taken as 100.

This means that under most practical circumstances, thermal dispersion can be neglected with respect to thermal advection.

7.1.3 Diffusive flux

The diffusive heat flux is usually referred to as heat conduction.

Fluxes 457

Following the discussion presented in Subs. 2.6.3, the (intrinsic phase average) conductive heat flux in a fluid, or solid phase, assuming that all

phases are at thermal equilibrium, i.e., T/ == T/ == T, is expressed by

J~i = -A~ij :~., J

(7.1.27)

where A~ij = AexT~ij is the ijth component of the thermal conductivity in a porous medium. Note that the asterisk (*) here denotes 'in a porous medium' and not 'a dimensionless value of'.

For the case of a single fluid that occupies the entire void space, i.e., a = f, s, if we remove the constraint of thermal equilibrium, the conductive heat flux is given by (2.6.119). The same equation can be employed when a number of fluid phases occupy the void space, but we assume that they have the same temperature (yet one that is different from that of the solid).

For a single fluid phase that occupies the entire void space, the heat flux through the porous medium as a whole, per unit area of the latter, is expressed by

(7.1.28)

where A H =nAj+(1-n)A:

is the thermal conductivity in single phase flow for the porous medium as a whole, when all phases are in thermal equilibrium.

Following (2.2.129) and the discussion in Subs. 2.6.6, the macroscopic conductive flux of heat is coupled to the diffusive mass flux of a I-component by an expression of the type

H HH aT H'YaW~ Jexi = -Lexij ax· - Lexij ax· .

J J

(7.1.29)

Similarly, the flux of a I-component can be expressed by

J 'Y _ f''Y'Y aw~ f''YH aT exi - -J....exij ax. - J....exij ax·'

J J

(7.1.30)

where the coefficient L~~ represents the Dufour effect, and the coefficient

L~~ represents the Soret effect. Unless otherwise stated, the phenomena of coupling between fluxes of

heat and mass of components will be neglected in the remaining sections of this chapter.


When a liquid and a gas occupy the void space under nonisothermal conditions, such that a change of phase from liquid to vapor, by evaporation, or from vapor to its liquid, by condensation, may take place, the diffusive mass flux of the vapor requires special attention. In principle this is a flux of the mass of a component of a phase, and as such it is treated in Subs. 6.1.2. Thus, we could express the diffusive flux of a vapor, as a component in a gaseous phase, by a w

Jw. = -V* .. Pg ... t g' 9'3 ax. '

3

(7.1.31)

where V;ij = V-:T;ij, and we assume that the vapor's concentration in the gas is at saturation, P;' .. t = P;' .. t (p, p"Yi, T), with p"Yi denoting the concentrations of various components (solutes) in the liquid phase.

However, it has been observed in experiments (e.g., Rollins, Spangler and Kirkham, 1954) that in the presence of a temperature gradient, the actual diffusive flux of vapor mass is larger than that predicted by Fick's law of mass diffusion in a gaseous phase. Philip and de Vries (1957) developed a conceptual model of vapor flux that involved also transport of vapor through the liquid phase from low to high temperature, due to the processes of condensation and evaporation.

To understand why the diffusion vapor flux is understimated by Fick's law, (7.1.31), let us consider, for example, the phenomenon of vapor diffusion at the microscopic level in the unsaturated zone in the soil. As vapor moves in the gaseous phase, say, in the x direction, it encounters a local gaswater pocket. The water pocket, say, a pendular ring of size ~x, is bounded at x and at x + ~x, by gas-water boundaries. As we have assumed that (a) the gas is saturated by the vapor, and (b) a local temperature gradient exists between the two gas-water curved surfaces (menisci) that bound the water pocket in the x direction, vapor must condense at the lower temperature boundary, and water evaporates at the higher temperature one. The two rates, of evaporation and condensation, must be identical if the moisture content of the liquid phase is to remain unchanged. The simultaneous condensation-evaporation processes, is then averaged to form an additional macroscopic mass flux of the vapor (through the liquid) in the unsaturated zone. Alternatively, this amounts to an increase in the cross-sectional area available to vapor diffusion, as calculated by Fick's law. For example, Og is increased to Og + f(Og)Ow, with f(Og) = 1 for Og 2: Owo, and f(Og) = Og/Owo for Og ~ Owo.

Philip and de Vries introduce, in addition, another enhancement factor which is due to fact that in the presence of vapor, the thermal conductiv-

Balance Equations 459

ity in the gaseous phase is larger than that corresponding to the average temperature in the soil, which they define as I:(a=g,w,s) Oar;:. This second enhancement factor is, thus, equal to the ratio between the gradient of the temperature in the gas and that of the averaged temperature for the soil as a whole. However, observations indicate that beyond an average temperature of 62°C, the vapor flux is overstimated by (7.1.31), this time because the thermal conductivity of the gas becomes larger than that of the water.

Jury and Letey (1979) supplemented this theory by taking into account the thermal properties of the phases. Cass et al. (1984) conducted experiments at temperatures up to 35°C, and compared results with the various theories. They concluded that:

• The enhancement factor rises exponentially with moisture content up to the moisture content for which the water becomes a continuous phase. Beyond that point, the liquid phase contribution to vapor flux is dropping.

• The enhancement factor decreases with temperature.

Bensabat (1986) developed a model that replaces (7.1.31) for anisotropic porous medium by

J;i = OgJ;i = -{09T;ij + OwfT(T)f8(Ow)6ij} :~., J

(7.1.32)

where fT and f8 are enhancement factors related to temperature and to moisture content, respectively. The latter is such that it vanishes both at low and at high water contents. In (7.1.32), we note the additional flux due to transport through the water. Bensabat (1986) suggests expressions for f8(Ow).

7.2 Balance Equations

Energy balance equations can be written in terms of temperature, T, internal energy, I, or enthalpy, h = I +p/ p. In what follows, we shall demonstrate all three options.

7.2.1 Single fluid phase

For a single fluid phase that occupies the entire void space, the mass balance equation is (4.1.1), noting its underlying assumptions. Under nonisothermal


conditions, the density is temperature dependent. The porosity may also be temperature-dependent.

The macroscopic heat balance equations for the case of a single fluid that occupies the entire void space, were presented in Subs. 2.4.6. The heat balance equation for the porous medium as a whole, assuming T/ == Ts s == T, and a thermoelastic solid matrix, is (2.4.41). We rewrite this equation in the form

a at (pC)pmT

(7.2.1)

where the coefficient A*H, is defined by

nJ7 + (1- n)J~ + nJjH = _A*H·V'T.

This expression combines the conductive heat fluxes in the fluid and in the solid, and the dispersive heat flux in the fluid.

The last term on the r.h.s. of (7.2.1) expresses a heat sink due to the thermal expansion of the fluid and the solid. External sources, e.g., due to injection of hot water, or internal ones, due, for example, to exothermic chemical reactions, can be added to the r.h.s. of (7.2.1).

When the assumption of thermal equilibrium between the fluid and the solid cannot be made, e.g., in a fractured rock, we write separate heat balance equations for the fluid and for the solid. Each equation will then contain a term that expresses the rate of heat transferred from one phase to the other (per unit volume of porous medium). Often, a transfer coefficient, a*T, is introduced to express this transfer, say, from the fluid to the solid matrix, in the form a*T(Tj - Ts). Regarding this coefficient as a constant, is often questionable.

When (2.4.41) is combined with the mass balance equations for the fluid and for the solid, the resulting equation takes the form

aT (pC)pmfit

_{fh V'.q + 1]~(1- n)csk}T. j3p at (7.2.2)

The term that expresses the advective heat flux by the moving solid is, usually negligible. The case of heat transport in a deformable porous medium is discussed in Subs. 7.2.3.


Following the methodology developed in Sec. 3.3, we may rewrite (7.2.2), without the term that expresses the advective heat flux by the solid, and without the term that expresses fluid and solid compressibility, in a dimensionless form. Let us select the characteristic velocity, Ve , as defined by (7.1.8), but with the porous medium diffusivity, 1JH, defined with A*H, Le., with dispersion, instead of A H, a characteristic time increment, (Llt )e, such that Le = Vc(Llt)e, and (pC)pm,e = ne(pC)j,e, the heat balance equation takes the dimensionless form

(7.2.3)

where (A *H)* is the dimensionless value of the coefficient A *H . For an isotropic porous medium, AiJI = A *HOjj .

If we wish to use the Rayleigh number, Ra, we select Vc according to (7.1.12), and (Llt)e = 1J~ / L~.

In the above discussion, we have used a single characteristic length, L e ,

such that L~T) '= L~V) = L~p) = Le. The three dimensionless coefficients appearing in this balance equation,

can be removed by selecting their corresponding characteristic values equal to their actual ones (under the questionable assumption that the latter are constant in time and uniform throughout the porous medium domain). We then obtain

(7.2.4)

Some authors make use of this assumption to obtain (7.2.4), within the framework of the method of inspectional analysis (Bear, 1972). For a homogeneous, isotropic porous medium, they select the characteristic time and velocity

7.2.2 Multiple fluid phases

We shall present this topic by considering two cases:

(a) Heat and mass transport in the unsaturated zone.

(b) Heat and mass transport in a geothermal reservoir.


(a) Heat and mass transport in the unsaturated zone. In this case, the fluid phases are a liquid, water (w), and a gas (g). We shall assume that the water is a single component phase. The gas is composed of two components: water vapor (w), and 'dry' air (a). The material presented as CASE A in Subs. 5.3.5 is applicable. The model contains mass and energy balance equations for the water, and for the gas.

In the conceptual model, we also include the assumptions that

• The diffusive-dispersive flux of both the water and the gas masses can be neglected.

• There exist no external sources and sinks of water or gas.

• The rate of heat transfer by conduction between the phases is much faster than the rate of transfer by advection of heat through each phase. Consequently, the two phases may be assumed to be in thermal equilibrium, i.e., no heat is exchanged between them.

• Mass balance equation for the water. With /::_g expressing the rate of evaporation (= negative condensation), this balance equation takes the form

(7.2.5)

• Mass balance equation for dry air. This equation is

{)(}gP; a ~= -V·Pgqg· (7.2.6)

• Mass balance equation for the vapor. This equation is

{)(}gP~ ( W () W) fW ~ = -v· Pgqg+ gJhg + w_g'. (7.2.7)

where Jh'/ == J~ + J;W) denotes the sum of diffusive and dispersive fluxes of the water vapor.

The varios fluxes are discussed in Sec. 7.1. We usually assume that the gaseous phase is vapor saturated, i.e., P~

_ P~ .. " with P~ .. t being a function of the pressure and temperature (see (7.1.31)).


• Energy balance equation for the gas. Let us use this opportunity to demonstrate the use of specific internal energy as a state variable. Here, the specific internal energy of the gas, per unit volume of gas, pgIg, is the sum of the corresponding specific energies of the 'dry' air, p;I;, and of the vapor p'; r:. In order to derive the macroscopic energy balance equation for the gas, let us start by considering the microscopic equation, and average it over an REV.

The microscopic energy balance equation for the gas is obtained by summing up those of the 'dry' air and of the water Vapor. we obtain

(7.2.8)

where (pI)g == p'; r: + p;I;, j'; + j; = 0, and we have assumed

By averaging this equation, we obtain the macroscopic equation

(7.2.9)

in which (pI) 9 ,.... pwg Iwg + pag Ia9 g-gg gg'

__ 9 0 9 H 009 000 9

Ip/V·Vg 1 ~ Ipg(V·Vg) I, J* = (pI)g Vg + (1; - I:)J~ ,

fL .. g = - U1 f {(pI)g(Vg - U) + (r;' - I;)j'; + j:}.z,gdS. o }S9W

In view of the assumptions included in the conceptual model, e.g., neglecting the effect of gas compressibility, the energy balance equation for the gas reduces to

in which Lva.p is the latent heat of vaporization.


• Energy balance equation for the water. This equation is derived by following steps similar to those undertaken above for developing the energy balance equation for the gaseous phase. In terms of the specific (i.e., per unit mass of water) internal energy, I w , and in view of the simplifying assumptions included in the conceptual model, the energy balance equation for the water takes the form

f)(JwPw1w ( H w fJt = -\7. Pw1wqw + (JwJhw) - Lva.p!w.-g· (7.2.11)

• Energy balance equation for the solid matrix. In the simplified model, the only transfer is by conduction. Hence, the energy balance equation takes the simplified form

f)(JsPs1s = _"O.() JH f)t v Ss· (7.2.12)

(b) Heat and mass transport in a geothermal reservoir. First, let us briefly explain what is a geothermal reservoir. This is a geological formation which contains hot (liquid) water and/or steam, as a result of natural high temperatures that prevail in it. The term hydrothermal system is also often used. Sometime hot water from such a reservoir emerges at ground surface in the form of hot springs. Similarly, in many parts of the world, steam from such a reservoir appears, naturally, at ground surface through fissures, or faults. When artificially brought to the surface through wells, the hot water, or the steam, from a geothermal reservoir, serve as a source of energy, directly, or as an input to electricity generation. The planning and management of hot water or steam production from a geothermal reservoir require models that describe the simultaneous transport of both the mass and the energy of these fluids within the reservoir. Geothermal reservoirs are classified according to the dominant fluid phase which they contain: vapor dominated reservoirs, or hot water dominated ones. The heat content of a fiuid, or enthalpy, h, will serve as a convenient state variable. Figure 7.2.1 shows a pressure-enthalpy diagram for pure water and vapor. We note three domains: one in which only compressed water exists, one in which both steam and water coexist, and one in which we have only superheated steam. In the second domain, temperature is a function of pressure only.

The core of the model contains the mass and heat balance equations of the phases and components that are being transported in the reservoir. Here we need mass balance equations for the liquid water and the steam

Balance Equations

100

>.. ~ ...... '" ~ Cll

o I I I I ""

Cll I~ ~ 10 I ~ ~ I ~

0.. lID

I~ I §

1 1 0

() c o o ..-l

Steam and water

o 100 200 300 400 500 600 700 800 Enthalpy, Cat / gm

465

Figure 7.2.1: Pressure-enthalpy diagram for pure water and vapor (modified from White

et al., 1971).

(=vapor, v) phases, and heat balance equations for the water, the steam and the solid matrix. We shall assume that the three phases are in thermal equilibrium, and that the solid matrix is rigid and stationary, i.e., q: == O. Following Faust and Mercer (1979), Enthalpy, h, will be used as the state variable .

• Mass balance equation for the water. Making use of (5.3.1), this balance equation takes the form

fJ(8wPw) V ( m) fW 8 rw fJt = - . Pwqw - w~v - wPw , (7.2.13)

where f;::~v denotes the rate of vaporization, in mass per unit volume of porous medium, and 8w pw rW (> 0) denotes the rate at which hot water is produced from the reservoir. We have neglected the dispersive-diffusive flux


of the water mass .

• The mass balance equation for the steam (=vapor, v) is

a(8vpv) V' ( m) fV 8 rv at = - . pvqv - v-+w - vPv , (7.2.14)

in which f~-+w is the rate of condensation, and 8vPvrV(> 0) is the rate of steam production.

The specific discharges of the water, nSw V w == qw, and of the steam, nSv Vv == qv, are obtained from (7.1.14), in which w denotes the water, we replace subscript n by v, to denote the steam, and K~ = 0. Often, (5.2.1) and (5.2.2) are used instead of (7.1.14).

The macroscopic energy balance equations for the water and for the steam, expressed as enthalpy balances, are obtained by writing (2.2.48) for each phase, and averaging the resulting equations. We shall neglect the term that expresses viscous energy dissipation .

• Energy balance equation for the steam is

:/vPvhv = -V'·8v(Pvhv V;:: - D~.V'T - A~·V'T) a8vpv "("7 fE 8 r h' +~ + qv· v Pv - V-+W,s - vpv v v' (7.2.15)

where h~ = hv for the rate of production, r v > 0, and h~ = hVini for the rate of injection, r v < 0, A~ denotes the thermal conductivity of the steam (= vapor) in the void space,

represents the dispersive enthalpy flux in the steam, and we have introduced the assumptions that

---o-V

IPvhv I ~ Ipvvh;' I and IV~·V'p/1 ~ IV~v·V'Pvvl·

Recalling that because we have assumed thermal equilibrium between the phases, no heat is transferred across their common boundaries by conduction, the total rate of energy, E, transferred across the steam-water and steamsolid interfaces, is expressed by


All terms in (7.2.15) are at the macroscopic level, with Pv == p,;;', Pv == p:;;u, hv == Ii:;, etc.

• Energy balance equation for the water. Following similar steps and approximations, this equation takes the form

where h'w = hw for r w > 0, and h'w = hw.inj for r w < 0, A~ denotes the thermal conductivity of the water in the void space

represents the dispersive enthalpy flux in the water, and we have introduced the assumptions that

• Energy balance equation for the solid matrix. This equation takes the form

(7.2.17)

A: denotes the thermal conductivity of the solid matrix, and we have assumed 8(1- n)ps/8t ~ 0.

• The energy balance equation for the porous medium as a whole. This equation is obtained by adding up the three phase energy balance equations

8 8t {8vPv hv + 8wPwhw + (1- n)pshs}

= -V·( OvPvhv V: + OwPwhw V:- - A;;·h. VT)

+ %/8vPv + 8wPw) - 8vpvrvh~ - 8wPwrwh'w, (7.2.18)

where


is a coefficient that combines the thermal conductivities of the three phases in a porous medium and the coefficient of enthalpy dispersion in the two fluid phases.

We note that (7.2.18) contains no term that expresses an exchange of energy between the individual phases (recalling that earlier, we have already eliminated the conductive exchange of heat across interphase boundaries).

A complete well posed model, will require, in addition to boundary and initial conditiions, also constitutive relations that describe the thermodynamic properties of the phases involved. Among these, we may mention:

• The dependence of water, steam and solid matrix enthalpies on temperature, as follows from the phase rule.

• The dependence of water and steam pressures on their respective temperatures.

• The dependence of water and steam densities on their respective pressures or temperatures.

• In deformable porous media, the dependence of porosity, and through it of other solid matrix properties, on pressure and temperature.

• The dependence of fluid viscosities on temperature, and possibly on pressure.

• Capillary pressure as a function of saturation, possibly with temperature dependency, through the dependence of surface tension on temperature.

• Effective permeability relationships for the two fluid phases.

7.2.3 Deformable porous medium

We consider here heat and mass transport in a deformable porous medium. As an example, we shall consider a single component fluid phase that fully occupies the void space in a thermoelastic solid matrix.

Most of the material needed for constructing the model has been presented in earlier chapters. Therefore, we shall only summarize here the equations that comprise the model (see also Bear and Corapcioglu, 1981).


• Mass balance equation. Because p = pep, T), say as expressed by (2.2.78), the mass balance equation (4.1.44) is replaced by

(Jcsk ap aT V·q + - +nf.l - - nf.lT- - f.lTq ·VT = 0 r at fJp at fJ at fJ r ,

where, to the assumptions leading to (4.1.44), we have added

L~P) l{3pqr·Vpl < I{3Tqr·VTI or (T) ~ 1.

Lc

(7.2.19)

If we assume vertical consolidation only, e.g., in an aquifer, we have csk = csk(p, T), and (7.2.19) becomes

ap aT V·qr + (ap + n{3p) at + (aT - n{3T)Ft - {3Tqr·VT = 0,

(7.2.20)

where aT = (acsk/aT)lp and ap = (acsk/ap)IT. We note that by the assumption Csk = csk(p, T), we have removed the need for coupling with csk, through the mass balance equation for the solid matrix. Equation (7.2.20) involves only p and T as variables .

• Heat balance equation. For the thermoelastic porous medium as a whole, the heat balance equation (7.2.2), is modified, to the form

aT (pC)pmFt

(7.2.21)

in which we have made use of (4.1.37), assumption [A4.9], and Icskl < 1. We often assume also that IVs·VTI < laT/atl. Note here the use of qr, rather than of q.

For vertical consolidation only, i.e., Csk = csk(p, T), with ap defined by (4.1.76), we introduce in (7.2.21)

aCsk = aCsk I ap aCsk I aT _ a ap a aT at - ap T at + aT p at - p at + T at . (7.2.22)

Then, the mass balance equation (7.2.20), and the heat balance one (7.2.21), are two coupled equations to be solved for p and T. For the general case of a


three-dimensional thermoelastic non-deformable porous medium, we have a third state variable, viz., Csk. Hence, we need an additional equation. This is the equilibrium equation (assuming, of course, that inertial affects may be neglected) .

• The equilibrium equation. For the thermoelastic solid matrix considered here, we may follow the discussion presented in Subs. 4.1.3, starting with (4.1.55). For a linearly thermoelastic solid, the constitutive equation to be used is the macroscopic equivalent of (2.2.96).

For an isotropic, linearly thermoelastic solid matrix, the stress-strain relationship (2.2.98), noting the discussion in Subs. 4.1.3, becomes

((T~)ij = 2P:cij + >':cskOij - (3)': + 2P:)aT(T - To)Oij. (7.2.23)

In terms of the displacement, w, this equation takes the form

(T~ij = p: (~:; + ~::) + (>.: ~:: - 17(T - To») Oij,

(7.2.24)

where 17 = (3)': + 2p~)aT. (7.2.25)

Equations (2.2.91) and (4.1.59), into which we insert (7.2.24) provide four equations in the six scalar variables, Wi, pe, Te and Csk. The mass and the energy balance equations provide two additional equations for p, T and Csk. Altogether, we have six equations for the six state variables. We also need information on k = ken), n = n(csk), P = p(p,T) and Pj = pj(p,T).

7.3 Initial and Boundary Conditions

Initial conditions for heat and mass flow problems include information on the distribution of temperatures within the considered porous medium domain. When thermal equilibrium among the (fluid and solid) phases is not assumed, the initial temperature distribution has to be specified separately for each phase.

When specific internal energy, or enthalpy, are employed as state variables, initial conditions have to be specified for them.

In what follows, the discussion on boundary conditions, based on the material in Sec. 2.7, will refer mainly to temperature as the state variable.

We shall assume that all the (fluid and solid) phases are in thermal equilibrium, and that all boundaries are stationary.

Initial and Boundary Conditions 471

7.3.1 Boundary of prescribed temperature

This kind of boundary condition occurs when phenomena that take place in the environment of a considered domain impose a specified temperature, say, h(x, t), on the domain's boundary, independent of what happens within the domain. The boundary condition is written as

T(x, t) = h(x, t) on B. (7.3.1)

This is a Dirichlet, or first kind boundary condition.

7.3.2 Boundary of prescribed flux

Here, the phenomena that take place in the environment impose a certain energy flux through the boundary. This case is discussed in detail in Subs. 2.7.3( d). Denoting this flux by h(x, t), we use (2.7.29) to write the boundary condition for a single fluid that occupies the entire void space, in the form

{pjqrjIj + nJ7 + nJjH + (1- n)J~}v

== (pjqrjIj - A*H.VT).v = h(x, t) on B. (7.3.2)

When, say, for a liquid, Ij == CVjT, this condition takes the form

(7.3.3)

This is a Cauchy, or third kind boundary condition. In the case of multi phase flow, both the advective flux, and the conductive

- dispersive one in (7.3.3) have to be replaced by terms that express their sums over all fluid phases.

7.3.3 Boundary between two porous media

The boundary condition is one of equality of the normal component of the total energy fluxes as the boundary is crossed. For a single fluid that occupies the entire void space, we express this condition in the form

(7.3.4)


where we have made use of the condition [PlQrfh,2'V = 0, together with the one that states no jump in temperature as the boundary is crossed

[Th,2 = 0 on B. (7.3.5)

The last condition follows from assumption [A2.5] of Subs. 2.7.3(a). When a heat transport problem is in a domain with a discontinuity in solid matrix properties, both (7.3.4) and (7.3.5) are required as conditions on the boundary of discontinuity.

In the case of multiple fluid phases, the no-jump condition, (7.3.4), has to be written for the sum of the conductive-dispersive fluxes in all the fluid phases and in the solid matrix.

7.3.4 Boundary with a 'well mixed' domain

Based on the discussion in Subs. 2.7.6, the condition on such a boundary, in single phase flow, is expressed as

(7.3.6)

in which a* is a heat transfer coefficient (for the entire porous medium), and T" is the temperature in the 'well mixed domain'.

Without advection, or when Iq·vl ~ a*, equation (7.3.6) reduces to

a*(T" - TI ) = -A*H.VTI ·V pm pm , (7.3.7)

where we note that T"lfb f= Tlpm on the boundary, i.e., a jump in temperatures takes place on the boundary. As in the case of concentration, this is a consequence of introducing the transition zone and the 'well mixed zone' approximation.

When Iq,vl ~ a*, equation (7.3.6) reduces to

(7.3.8)

7.3.5 Boundary with phase change

This kind of boundary is discussed in Subs. 2.7. 7( c). The applicable condition is (2.7.75), written in terms of specific internal energy, or (2.7.76),

Complete Model 473

writtem in terms of enthalpy. When assumption [A2.16] is applicable, the boundary condition for enthalpy is given by (2.7.77).

One should note that in the case of multiple fluids and multiple components, the boundary condition should take into account the particular behavior of each phase and component.

7.4 Complete Mathematical Model

The content of a complete mathematical model is discussed in Sec. 3.1. In the case of a nonisothermal system, the conceptual model should include statements that refer to the

(a) dependence of state variables and properties of fluid phases and components on temperature,

(b) possibility of thermal expansion of the solid matrix,

(c) possibility of change of phase,

(d) possibility of exothermic and/or endothermic chemical reactions,

(e) advective fluxes caused by temperature effects on surface tension,

(f) thermal dispersion,

(g) conductive fluxes caused by coupled processes, and

(h) possibility of heat released by wetting in a liquid-gas system.

The energy transport problem is, usually, coupled with the one of mass transport of one, or more fluids that occupy the void space. Modeling mass transport of a single fluid phase is discussed in Chap. 4, and of multiple fluids in Chap. 5. In some cases, especially in liquid-gas (hydrothermal) systems with phase change, the presence of solids dissolved in the liquid, may have a significant effect on the entire transport problem.

The complete mathematical model of an energy and mass transport problem, consists of the following items:

(a) Equations of domain boundary surfaces.

(b) List of relevant state variables.


( c) Formulation of energy and mass balance equations for every phase. In the case of multicomponent phases, such equations may have to be written also for every relevant component.

(d) Formulation of the heat and mass flux equations for the relevant phases and components. The total flux may be composed of advection, dispersion and diffusion, or conduction.

( e) For a deformable solid matrix, formulation of the relevant stress-strain relationships.

(f) Equations of state for the fluid phases.

(g) Equations that relate state variables of phases and concentrations of components in the different phases, taking into account the possibility of coexistence of phases and components.

(h) Formulation of various internal sources and sinks of heat (or, enthalpy) and mass of phases.

(i) Formulation of initial and boundary conditions for both energy and mass transport, in terms of the dependent variables.

(j) Numerical values of all relevant energy and mass transport and storage coefficients.

In most cases of practical interest, problems of transport of energy, mass of phases and concentration of components, have to be stated and solved simultaneously.

7.5 Natural Convection

The term natural convection is used to describe fluid motion produced by density variations in a gravity field. Such changes may be caused by changes in solute concentration and/or temperature. The non-uniform density causes motion due to buoyancy effects. The term convective currents better describes the motion produced by such effect.

As an example, consider an initial situation in which a layer of stationary cold (hence, heavier) fluid overlies a layer of stationary warmer (hence, lighter) one. Under certain conditions, to be discussed below, this may be an unstable situation, meaning that even a small disturbance may result in

Natural Convection 475

a completely changed regime, caused by convective currents. Molecular diffusion and thermal conduction tend to reduce these currents by smoothing out density differences.

When buoyancy effects produce motion, the latter encounters resistance due to internal friction within the fluid and to friction at the solid-fluid (microscopic) interfaces. Both are proportional to the fluid's viscosity. The latter resistance is proportional to the inverse of the permeability. Under certain conditions, the resistance to motion is such that the initial motion produced by the disturbance will decay. Under other conditions, it will develop and grow, leading to convective currents.

In the discussion presented in this section, we shall include the following assumptions in the conceptual model:

• The fluid contains no dissolved components.

• The fluid is a 'Boussinesq fluid', Le., one in which the density is assumed to be independent of the temperature, except in the gravity term that appears in the expression for the advective flux.

• in order to write the mathematical model in a dimensionless form, we select fl = flc, and (pCv)c = 1. We then have

(pC)j = 1.

• The solid properties are also independent of temperature.

• The porous medium is homogeneous and isotropic. With k = kc, we then have kij = Oij.

• We neglect effect of thermal dispersion, so that A*H is replaced by AH.

The corresponding heat and fluid mass transport model consists of the following set of dimensionless equations:

• The fluid's mass balance equation is

V*·q* = o. (7.5.1)

• The fluid's motion equation is (7.1.10)

q* = -v*(p* +~) + Ra'T*V*z*. EuFr2

(7.5.2)


• The heat balance equation for the porous medium is

(7.5.3)

Our objective in the following paragraphs is to investigate the conditions under which convective currents will develop in a porous medium domain. In doing so, we shall also demonstrate the methodology of such investigations.

We shall consider two typical cases of natural convection:

• An infinite horizontal fluid saturated porous medium domain heated from below.

• A fluid saturated porous medium domain in the form of an infinite strip bounded by vertical surfaces at different temperatures.

CASE A. An infinite horizontal layer heated from below. The layer's thickness is chosen equal to the characteristic length, Le. Hence the dimensionless thickness is l.

The boundary conditions are:

z* = 0, z* = 1, q; = 0, z* = 0, T* = 1, (7.5.4) z* = 1, T* = 0.

We begin by noting that the governing equations and these boundary conditions are satisfied by the no-flow solution

q*(O) == 0, T*( 0 ) == 1 - z*, * z* , * ( z*) p +--=Raz 1--EuFr2 2 '

(7.5.5)

which involves no convective currents. N ext we look for the existence of additional solutions to ths same prob

lem, this time with q* f= 0. If we find that a solution with convective currents does exist, we shall investigate the conditions under which the stationary solutions will be unstable.

To find additional solution(s), we perturb the flow regime around the stationary solution

q* q*(O) + £q*',

T* = T*(O) + £T*', p* = p*(O) + £p*',

(7.5.6)


where cq*', cT*' and cp*' are the perturbations, and c is a small parameter. To eliminate the term involving the pressure from the motion equation,

we apply the curl operator to (7.5.2), making use of the identity \7 x (\7 x q) == - \72q, valid when \7.q = O. We obtain

\7*2q* = -Ra' \7* x (\7* x T*\7*z*).

This equation can also be written in the form

\7*2q* = -Ra' £*(T*),

where the operator £* is defined as

{)2 {)2 ( {)2 ()2 ) £* == lx* () () + ly* () () + lz* -() 2 + -() 2 . x* z* y* z* x* y*

(7.5.7)

(7.5.8)

By inserting (7.5.6) into (7.5.3), and (7.5.8), making use of the already known, solution (7.5.5), and neglecting terms of order of magnitude O(c2), we obtain the two linearized equations for the perturbed regime

\7*2 q*' = -Ra'£*(T*'), (7.5.9)

\7*2T*' = -Ra' £*( q*'), (7.5.10)

where we have noted from (7.5.5) that \7*T*(O) == - \7* z* and, therefore

q;~ == q*'. \7* z* == q*(o). \7*T*(O) .

Thus, the solution for the perturbed temperature distribution is coupled only to the vertical component of q*'. We have to solve the pair of equations

n*2q*' = _Ra'n*2 T*' v z* v x • y., (7.5.11)

and (7.5.10). We can decouple these equations, leading to the equations

{ (~ - \7*2) \7*2 - Ra'\7*2 }T*' = 0 {)t* x*y·, (7.5.12)

(7.5.13)

noting that the two equations are, actually, identical. The boundary conditions to be satisfied by the perturbation solution, are

q;~ = T*' = 0, on z* = 0,1. (7.5.14)


The final step is to solve for q;~ and T*', as a superposition of normal modes, and examining the stability with respect to each mode. When this is done, it is shown that the perturbations decay as long as the stability condition

(7.5.15)

is satisfied. The value of 411"2 is referred to as the critical value of Ra.

To summarize CASE A, we have seen that

• In the case of an infinite horizontal layer heated from below, a no-flow solution exists. This solution is stable, as long as the Rayleigh number does not exceed the critical value of 411"2. Then the heat is transferred only by conduction .

• When the Rayleigh number exceeds its critical value, the no-flow solution becomes unstable, disturbances will be amplified and natural convection will develop.

A linear analysis carried out for the case of an infinite layer heated from above, will reveal that the no-flow solution is unconditionally stable. Therefore, convective currents are traditionally associated with heating from below, while the case of heating from above is regarded as a state of no flow, with heat transfer taking place by conduction only.

Obviously, we have demonstrated here only a very simple case. Cheng (1985) and Bories (1987) review more complex cases. Vadasz (1988) considers effects of imperfectly insulated side walls, perfectly conducting side walls and medium heterogeneity on the natural convection in a rectangular porous medium domain heated from below.

CASE B. An infinite vertical strip between two impervious walls at different temperatures. This case (in the vertical xz-plane) is introduced here in order to show that convective currents will always occur whenever a horizontal temperature gradient exists.

Because the domain here is infinite in the z-direction, we have

aq;. = 0 az* '

and aT* -=0. az*

Hence, the mass balance equation reduces to

aq;. = o. ax*

(7.5.16)

(7.5.17)


T = 1

q" = tRa

x=o x == 1

Figure 7.5.1: Temperature and flux distributions in an infinite vertical strip between

two walls at different temperatures.

By combining this equation with the boundary condition

we obtain the solution

° ::; x* ::; 1, -00 < z* < 00, q;. == 0.

From this solution, it follows that

{)p* -- =0, {)x*

* {) ( * z*) R' T* qz' = --{) p + --2 + a . z* EuFr

(7.5.18)

(7.5.19)

(7.5.20)

Then, q;. = 0, {)T* / {)x* == {)2T* / {)x*2 = 0, and the heat balance equation (7.5.3) reduces for the steady state to

{)2T* 8x*2 = 0. (7.5.21)

We note that this equation indicates that in this case, heat transport is governed by conduction only.

The boundary conditions are

x* = 0, T* = 0, x* = 1, T* = 1. (7.5.22)


The solution for the temperature, T*, is

T* = x*. (7.5.23)

In order to determine the specific discharge, we add the constraint

expressing mass conservation at any horizontal cross-section. The solution for the specific discharge is then

q;. = Ra'(x* - ~). (7.5.24)

This magnitude of the convective flux, which always exists, is dictated by the value of the Rayleigh number. The two solutions, for the mass and heat fluxes, are shown in Fig. 7.5.1.

From the results of this analysis, we may conclude that

• the natural convection that develops does not affect the heat transfer which, in this case, is governed by conduction only, and

• the strength of the natural convection is determined by the value of the Rayleigh number.

We note that an analytical solution could be derived in this case, because the two equations could be decoupled.

introduction to modeling of transport phenomena in porous media || heat and mass transport

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