dispersion in heterogeneous porous media: the ...

r 18: 71-79 (1988)

Latin American Applied Research

DISPERSION IN HETEROGENEOUS POROUS MEDIA: THE METHOD OF LARGE-SCALE AVERAGING O. A.PLUMB

Department of Mechanical and Materials Engineering Washington State University, Pullman, WA 99164 S. WHITAKER

Department of Chemical Engineering University of California, Davis, CA 95616

Abstract In multiphase transport phenomena, one often encounters systems with severe heterogeneities, i.e. regions in which the structure of the system is significantly different than the average structure. These heterogeneities are clearly evident in petroleum resevoirs, soil, concrete, wood, etc; however, they also exist in packed beds of catalyst pellets. In order to develop reliable theories of transport processes in heterogeneous, multiphase systems, one must be able to predict the effective coefficients that appear in the spatially smoothed transport equations. This can be done by means of the method of large-scale averaging and an associated closure problem. In this paper we illustrate the method of large-scale averaging with a study of dispersion in heterogeneous porous media. The comparison with experimental results is encouraging for both randomly packed beds and stratified porous media.

Introduction We consider the fluid-solid system illustrated in Figure I in which the a-phase represents a rigid, impremeable solid and the j'3-phase represents a fluid. The problem of diffusive and convective transport in the absence of adsorption and chemical reaction can be described by

trated in Figure 2. One assumes that the radius of the averaging volume is large compared to the pore diameter or the particle diameter, i.e. '0 ~ Qa, Q{3. The average concentration can also be defined in terms of a weighting function 2,3,4 so that precise correspondence between the theoretical dependent variable and the comparable quantity measured in the laboratory can be obtained s ,6. In this study we will avoid the use of weighting functions; however, it is a crucial aspect of the method of volume averaging that must be considered more carefully in the future.

Here CA represents the molar concentration of species A'!:(3 represents the fluid velocity, and D represents the molecular diffusivity in the j'3-phase. In the method of local volume averaging) one seeks the governing equation for the average concentration defined by (CA)f3

=~ V{3

f

cAdY

(3)

V{3 Here V{3 represents the volume of the j'3-phase contained within the averaging volume V which is ilIus-

Fig. 1. -

71

Two-phase system.


18: 71-79 (1988)

( 12)

Fig. 2. -

The quasi-steady condition will be valid when the time is constrained by (D{3tIQ~) ~ 1 and the proof of Eq. (12) is given by Carbonell and Whitaker 7. The periodicity condition expressed by Eq. (9) is used in order to solve for the f -field in a unit cell decribed by tl).e lattice vectors) i. The clo~ure problem has been solved 8 for several, simple regular arrays of cylinders and reason· able agreement is obtained with the longitudinal dispersion studies of Gunn and Price 9 for cubic arrays of spheres. The comparison is shown in Figure 3, and there we see that the experimental data for cubic arrays of spheres is greater than the theory for high Peelet numbers, while the experimental data for random packing of spheres is lower than the theory for high Peclet numbers. The Peclet number used in Figure 3 is defmed by

Local averaging volume.

The local volume-averaged from of Eqs. (I) and (2) is given by 7

~

ak A )/3

€/3(v/3l0p

/3

(1-€p)D/3

at + ...Ij • (€/3(v./3)i3 ... (cA) ) =

(13)

Gunn and Pryce

o

in which the total dispersivity tensor takes the form

-f

I

n/3afdAl-(V/3!>~ "W,...,

O*=D/3[I+=:::: V ltv,... /3 A/3a ~

104

Cubic Array (E

=0.48)

0 Random Packing (E - - - Theory

= 0.37 )

(5)

Here v /3 represents the local velocity deviation

...

...

/3

...v/3=v/3-(v/3) ... ...

(6)

and the vector field f is determined by the following boundary value probTem in the {j-phase

B.C. - !!./3a • '11 = ~/3a,

[(! + ~i) = [(!), ([)/3 =

Fig.3.- Comparisun between theory and experiment for random and ordered arrays of spheres.

at the {j-a interface

(8) i

o.

P"

(7)

= 1,2,3

(9) The studies of Oebbas and Rumpf 10 have made it elear that local heterogeneities are to be expected in packed beds of spheres, and the data of Gunn and Price clearly indicate that these effects can influence dispersion. It is important to note that the experimental value of the longitudinal dispersion coefficient for a cubic array of spheres is ten times larger than the experimental value for randomly packed spheres at a Peelet number of about 300. It is also important to note that disorder gives rise to a lower value of the longitudinal dispersion coefficient, a

(10)

The closure problem given by Eqs. (7) through (10) is based on the assumption that the local concentration deviation

(11) is quasi·steady and the idea CA is a linear function of the gradient of the average concentration, i.e. 72

O. A. PLUMB, S. WHirAKER

o Harlell'llft and Rumer 10i

••

a Gunn and Pry" o H.ssermaa and .Oft ROSel",. • Han and C"boMll

.. Plannkucl'l v Ebach and Ihit. • Carberry and Bretton o Edwards and RIchardson • Blac .... 11 .t al

o RIIII

o· ll.

•

102

-II

101

Thtory

Staumd Cylinders

Ro/RI I E/JoO.3J 0

102 100

lal

10'

PI,

o In -lin' Cylinders

Fig. 5.·-

Theoretical and experimental values for the lateral dispersion coefficient in randomly packed beds.

PI,

we see poor agreement between theory and experiment with the theory predicting values that are much lower than those measured in the laboratory. At this point it seems clear that local heterogeneities 14 are of importance to a wide range of transport phenomena in porous media. Volkov et al. 15 have cited the problem of hot spots in catalytic reactors having a scale associated with several particle diameters which Schwartz 16, among many other, has discussed the influence of heterogeneities having a length scale on the order of millions of particle diameters on the process of dispersion in groundwater systems. Clearly the N-scale problem described by Cushman 17 must be confronted; however, there is much to be learned from a detailed analysis of the two-scale problem. This is presented elsewhere 18,19 and a summary of that work is given in the next section.

Fig.4.- Comparison between theory and experiment for randomly packed particles.

result that is consistent with the analysis of Koch and Brady 11 • The experimental data of several investigators is shown in Figure 4 for randomly packed particles, and the experimental values of the longitudinal dispersion coefficient are in the same neighborhood as the theoretical values predicted by the solution of Eqs. (7) through (10). The behavior of the experimental data at high Peelet numbers is important to understand. On the basis of both theoretical 12 and experimental l3 studies, it has been demonstrated that the quasi-steady condition for CA is actually satisfied when (D(3t/Q~) ~ 0.10. When this condition is achieved, D:x is constant and has attained a maximum value. However, for short times, i.e., t < O. 10 Q~/D(3, the dispersion coefficient is less than the maximum value and decreases with decreasing time. In experimental studies using a fixed measuring station, the time decreases with increasing Peclet number since changes in the Peclet number are generally brought about by changes in the velocity. This means that D!x is influenced by two effects whenever the time is less than 0.10 QyD(3. The first effect is dominated by the source, !:.(3, in Eq. (7) and this causes the dispersion coefficient to increase with increasing Peclet number. The second effect is the failure of the CA -field, and therefore the [-field, to have reached the quasi-steady state, and this causes the dispersion coefficient to decrease with increasing Peclet number. The net result is the diminished Peelet number dependence shown in Figure 4 for Peclet numbers larger than 10 4 • While disorder tends to reduce the value of the longitudinal dispersion coefficient, just the opposite is true for lateral dispersion coefficients. In Figure 5 we have shown a comparison between lateral dispersion coefficients for randomly packed beds and the theoretical calculations of Eidsath et al. 8 and there

Large-scale averaging In this development we are concerned with heterogeneous systems such as the one illustrated in Figure 6. That layered system is composed of locally homogeneous porous media which we have identified as w-regions and 71-regions. The Darcy's law permeability tensors associated with these regions are denoted by K(3w and !5(311 and elsewhere 29 we have shown how the large-scale Darcy's law permeability tensor, K, can be determined theoretically. In this work we wish to develop the large-scale form of the dispersion equation and a closure scheme that allows for the direct theoretical prediction of the large-scale dispersion tensor. The definition of the large-scale averaged concentration is given by

{(CA>(3 =_1_ V~

J

(14)

v~

and the large-scale form of Eq. (4) can be expressed as

73


18: 71-79 (1988)

Fig.6.-

Layered porous medium.

(15) The analysis of the closure problem 21 indicates that the large-scale concentration deviation can be expressed as

The analysis of this equation involves the decompositions*

D* = {D*}+ 0, :c::;

~

~

(16)

and after a bit of albebraic effort

{€f3}

:t

{(CA)f3}

18

When this result is used in equation (17) and various small terms are discarded, we find

one can arrive at

+ {€f3}{(.!::f3)f3} " Y{(CA)f3}+

a .

+ at {E13C A}+ + '1"

[{Ef3 CA}{t

= {Ef3f} + {~f3)f3}{Ef3S} + + {Ef3}{£f3 S}+ {Ef3£f3 S}

EJ3Dyy DJ3

(20)

EJ3Djy DJ3

and the large-scale dispersion tensor is given by

= {Ef3}{!?*} + {E~f?} - {~f3)f3}{Ei'}-

{Ef3}!?**

- {Ef3}{£f3f} - {Ef3£f3f}·

-

E

J3

PeO. 6

'

0.7 Ef3,

Pep

Pep

~

1

< I.

(30a)

(30b)

(21)

The closure problem for a spatially periodic system is given byl8. .

Problem 1: {Ef3}£f3

0) STRATIFIED SYSTEM

+ Ef3
+ Ef3(!:f3 l

{Ef3£f3} +

· Y.E = Y. (EJ3!?* • YD + + y. (Ef3!?* + {Ef3}f?)

f(!: + gil = .E(!:), {f}=

i

(22)

= 1,2.3

o.

(23) (24)

Problem II:

EJ3 + EJ3(~:/ • ~ S = ~ . (EJ3!?* • ~ S) i

= 1,2,3

{s}= O.

b) TWO- DIMENSIONAL SYSTEM

(25) Fig. 7. -

(27)

As in the case of the closure problem given by Eqs. (7) through 10, the closure proble~ given by Eqs. (22) through (27), requires that the ca-field be quasisteady. This leads to a constraint on the time given by

D*! ~ 1.

(28) QH Here QH represents the length scale of the heterogeneities and 0* should be thought of as the smallest value of A . D* . A where A is an arbitrary unit ~ ..... vector. In order to solve Eqs. (22)-(27) one must have a source of values for D*, and in our study of the stratified system in Figure 7a, we used the experimental data for packed beds which can be thought of as relatively homogeneous. The data for longitudinal and lateral dispersion coefficients can be represented reasonably well by

-

--

Pe p

Pep

~