THE DIFFUSION TIME-LAG IN POROUS MEDIA WITH DEAD-END

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Oct., 1961

DIFFUSION TIME-LAQ IN POROUS MEDIAWITH DEAD-END Pow VOLUME

1709

THE DIFFUSION TIME-LAG IK POROUS MEDIA WITH DEAD-END PORE VOLUME BYRrcmm C. GOODKNIGEXT AND

DGparknmt of M i d Tcchdopp, University of Cahfomia, BerLdcy, Calif&

Bscdirad Ma7e.h 7. lsBl

The tim+lag phenomenon, obaerved during non-steady a t e diffwion or.fl+ flow in po~ws. m e , ki often used to determine the dif€&ty coefficient. The effect of dead-end pore volume on d~Euramtyd e t e m e d m t b way has not been previously deecrhed. It ki shown here that the dif€usivity calculated from the timelag ie a function of the aum of the flow channel volume and the dead-end pore volume. The non-steady atate diffusion-equationb solved for a Bystem with deadend pore volume and with initial and boundary conditiona encountered in the tme-hg measurement. The dead-end pore volume is ahown to influence only the early portion of the time-lag curve.

Introduction When material begins to m u s e across a barrier there is a time delay in achieving steady state transport. As a result of this delay an extrapolation of the steady state portion of the mass t r a n s port versus time curve intercepts the time axis at a finite, positive time. This intercept has been called the time-lag by 33arrer.l The interpretation of the time-lag in tern of the ditTusivity coefficient has been described by Barrer for a constant d8uSiVity and by Frisch:'+ for systems in which diffusivity may be a function of concentration or time. Most time-lag measurements have been applied to molecular diffusion. Fatt6 has shown however that the diffusivity coefficient, k/r#w, which appears in the equation describing non-steady state fluid flow through porous media can be obtained from a time-lag measurement. In this case the delay in establishment, of steady state fluid flow is the time-lag. The simplicity of the time-lag measurement, in both diffusion and fluid flow systems makes it convenient method for obtaining the diffusion coefficient, tortuosity (when diffusion takes place in the fluid contained in a porous medium), and the term k/+pc when fluid flows through a porous medium. Previous treatments of the time-lag method have inot includcd the situation where there is dead-end volume in the medium through which diffusion or fluid flow takes place. Goodknight, Klikoff ,and F5tt (GKF)' solved the general equation for non-steady state diffusion or fluid flow in such systems but did not include the timejag initial and boundary conditions. It is the purmse of t h s paper to present the solutions for the time-lag initial and boundary conditions and to ,mipare these solutions with experimental data OD a laboratory flow system.

compressibility variable concentration concentration in eink (variable) diffusion coefficient effective diffuxion coefficient a constant coefficient pressure gradient multiplied by time permeability of poroua flow channel rmeability of orifice ength of orifice length of porous mediuni dirnenAionless presaure in flow channels, P' = (I3*

p"

PO)/(Pb - PO)

-

-

dimensionlees resure in dead-end pores, PI' (Pzpd/(pb pressure (variable) initial pressure (constant) boundary pressure (constant) pressure in dead-end pore (variable) volumetric flow rate parameter of La Place t r d o r m t,ime time-lag time-lag pore volume of flow chsnnel (not including dead-end volume) total volume of all dead-end pores volume of individual dead-end porerr distance cocirdinate cumulative volume ciirnulativa volume pafit cumulative volume past X = i; when Z L is linear function of time diffiisivity coefficient for liquid, a = k/&pc a constant tortuosity = diffusion path length/L V,/)IL %/A L YIscoslty

-

Theory GKF have shown that equations 1 and 2 k l o w describe non-steady state diffusion in the fluid contained in R porous medium in which there arc dead-end pores.

Bamenclature -1

woes-sectionrrl area of porous flow channel

A,,

croslwrectionrrl area of orifice

R. M . 3arres. .r P h y ~ ('hem. m,35 (1953). (2; H. L. E'nsch. tbrd 61. '13 (1957). (3) H . L. Frrrrch abrd.. 6%. LO1 (1958). 0 ) U 0. Pollack and H. I,. F r k h . r b d . . 63, 1022 (195s). (5) H.L. Friah. abnd., 63, 1249 (1959). , 44') (lllS! 8 . (6) I. Fart, Tram. Am TRlt. Manan@M6t. E n p r ~ . 216, (7) R 6. Goodknight, W A FZLoff, Jr., and P. Pstt, J . Piij's Chm., 64. '762 (1980). (1)

The analogous equations ic'r non-steady state Ifow of a slightly compressible liquid Lhrmgh a pw u s medium containing dead-end pores aw (3)

and

RICHARD C. GOODKNIGHT AND IRVING FATT

1710

(4)

Note that equation 18 is independent of H . Equation 17 then can be written

For convenience the coefficients are defined as k - = 41M

a

(5)

Vol. 65

t

I=--L +

(1

+ k) L a

1

xz

[;-3-B]-

and Equa,tions 3 and 4 are solved here but it is to be understood that the same solutions apply to equations l and 2 since C E P, Cz G P2, (D/S') (DAo/love)3 H

E

Equations 15 and 19 can now be used to obtain an expression for the volume that has flowed past X = L from t = 0 to t. This is

a

The initial and boundary conditions for the time lag measurement are P y x , 0 ) = P2'(X, 0 ) = 0 P'(L, t) = 0 P'(0, t ) = 1, t > 0

( 7) (8 ) ( 9)

As t increases ZL approaches the asymptote

The solution of equations 3 and 4 with the above conditions is carried out in the manner already described by GKF. The result is

ZLS =

- k-4 P

[ ; 4 ( -+ 3K 1 -

+

(21)

This asymptotic straight line in a plot of ZL versus t intercepts the time axis, Z L = 0, at the time-lag

TL.

where

Pn = n-,Lr

n = 1, 2,

-

(11)

If a is replaced by its equivalent, equation 5! we obtain or Equation 10 can be differentiated to obtain the pressure gradient a t any point and time

Let the volume that has flowed past any point X be 2, then z = --IkA

TL = (VI

+ VdPCL 6kA

Equation 20, together with equations 11, 12 and 13, give the complete curve of volume throughput versus time, in the time-lag experiment. Equation 23 or 24 gives the time-lag. If only the time-lag is desired then equatioq 3 can be solved by a simple procedure due to Frisch.s Equation 3 is rearranged to

P

where Integrating both sides Inserting equation 14 into 16 gives

(17)

Then

The first summation on the right-hand side in equation 17 can be simplified by substituting into it SnZfrom equation 12. This summation then becomes

~ _ _ _ _ (181

( 8 ) H . L. Frisch. p r i v a t e c o m m u n i c a t i o n

I ~ I F . E ' ~ : sTIME-LAG I~N I N POROUS MEI)IA WITH DEAD-END PORE VOLUME

Oct., 1961

1711

q* is defined by

for unit pressure difference. Equation 28 now can be rewritten as L

L

lo

+

(P

2

:;

+

P 2 ) dydz = a L 1 ~ aAPt (30)

Rut from equation 29 AP L

IL=

-q*t

q *L,then

=

L

$

-!- --

+

(P

Pt> dydx

(31)

ffL

For the asymptote ILawe get from equation 31 iLa = q*(t - TL)

From equation X Z we find T,, = q*t--

+

ILa -

9*

(32)

(33)

Jf we use equation 31 for I L we ~ obtain

TIME, SECONDS

Fig. 1.-Volume of gas produced at downstream end of linear system as a function of time after applying 0.01 atm. pressure at upstream end. System parameters are shown in insert table. Solid and dashed lines are calculated from theory. Open circles are experimental points. 1

i[' = -

In the limit t

_-_________ ffLq*

(341

we have steady state, then n. Lim P = Lim P2 = 1 - (35) t-+ 1-* m L ==

0:

i

1:quation 34 now beromes

or @ . I

(37)

Hiit from rquation 29 q*

=

l/L then

'This is t,he same result given in equation 22 from the general solution. Equation 20 was evaluated by means of a FOIJTRAN program on an IBM 704 digital computer. The results for various combinations of a, VI, V q and 117 are sh0w.n in Fig. 1 and 2. The time-lags for the t,heoreticalcurves were calculated by fitting a :;t,raight line tlo the calculated volume versus time data,. The line was fitted to points where the incremental volume was constant to within one part, i n 10,000 for a five second time increment. For thc fcwr systems shown in Fig. 1 and I! t,hc time when thjs condition was met was as follows: curve 1, 65 we. ; curve 2 , 230 sec. : curve 3 , 155 sec. ; and curve 110 sec. The time-lags and system pammet,rrs 'P

tnhulated i n Fig. 1.

Experimental 7'he. inbnrstory rnodel hrts been adequately described in jjwvioils r~~i.;ic3t,ioiis.~,7 in brief, the rnodel was a 2" X 2'' x 60'' piece of sandstone coated with epoxy resin m d

'0

?o

40

60

80

100

.__-L---L..

!eo

140

.Mi

TIME, SECONDS.

Fig. 2.-Volume of gas produced at downstream end of linear system as a function of time after applying 0.01 atm. pressure at upstream end. Legend same as in Fig. 1 . Xote that curves 2, 3 and 4 coincide at larger times indicating no influence of flow restriction hetween flow chnnnela nine side chambers could be either M, 125 or 17.2 each. The orifice8 could he0.122.0.0615 or 0.0411 Darcy-cm. each. Air pressure st 0.01 atm. was applied to one end of the model by opening a quick-acting toggle valve. A t the samtz time a stopwatch was started. The air emerging from the downstream end moved a slug of water along a calibrated, horizontal glass tube. The volume produced as a function of time was obhined by noting the time the slug passed ureviously calibrated volume markers. For larger volumcs of air than could be handled in the horizontal tube the air x+VItBfed into the bottom of a vertical, water-filled buret set i n a large pan of wat.er. The volume so collected waa recorded AS a function of time after opening the upstream end to air pressure. The experimental data are shonvli as open circles in Fig. 1 . 'The curves in these figures are calculated from equation 20. The esqm-imental time-lags equivalent tlo those for curves 1 , 2 ; 3 and 4 in Fig. I. and 2 are 11.1, 16.9, 16.8 and 16.9 sec., respect,ively. These are in excellent agreement with the theoretically predicted t,ime-lags for the laboratory mode!, In addition, there is good agreement of experiment, arid theory a t the lower, curved portion of the volume v c m u s time curve, as shown in Fig. 1 .

Discussion

quipped with an inlet port, an outlet port, and nine equallyEquation 24 or 38, developed from tile t.heory spaced connectors n. !mg one side. A t these n i n points ~ c(oi~iJ s?i\ms clearly that the time-Iag in a system containt w connected a chambrsr or a chamber and rrifce in 2erii.s. ing dead-end pores j s infliienccti only by the tot.ai or the point could he seaiecl over. The simdstone had a p:r iumhilitv of I .ffi Darcys, and a porosity o f 23.134':~. 'Ygs nore volume (8nw channois plus dead-end j.to:'c=j,

RICHARDC. G O O D K N I AND G ~ IRVINGFAIT

1712

VOl. 65

and not a t all by the resistance to flow between the the time required for the pressure transient to reach flow channels and the dead-end pores. This is the closed end of a linear system after a sudden increase of pressure is applied to the open end.6J1 borne out by the experimental results. Note in Fig. 1 that in a system with deadend This is usually a function of only VI and ie given by pores cmnected to the main flow channels through 0.0471LVM resistance the volume output rises faster than in a (39) kA system of the eame total pore volume but with no resistance between dead-end pores and the main for small values of H. V* does not appear in flow channels. Both systems have the same linear equation 39 kcatuse the first part of the transient to arrive at X = L does not “see” the dead end volume asymptote, however, and the time-lag is identical. These results are not in agreement with some ex- if this volume is connected to the main flow chanperimental data obtained by Fatt,6 and by Barrer nels through a resistance for which H is small. and Gabor.’Jo They found the t i m d a g a t con- Having obtained Vl, Vs and a,various values of H stant total pore volume to be dependent upon H. can be substituted into equation 20 and ZLevaluA possible reason for this discrepancy now can be ated until a match is obtained with experimental advanced. Although the asymptote is independent data. An alternative method for estimating V I , Vs, CY of H, the time required to reach this asymptote is dependent on H. Experiments in which H is finite and H is available through use of the “alternating must be camed out for much longer times to assure flow” technique developed by Stewart, et ad.]* In that the asymptote is reached. Extrapolation of their procedure a sinusoidally alternating gas presdata that had not reached the steady-state asymp sure is applied upstream. The amplitude and phase tote would lead to low time-lags aa observed by shift of the sinusoidally varying output volume, relative to the input pressure, can be interpreted in Fatt. There are two procedures whereby the values of terms of Vl, Vs, (Y and H. VI, Vz,a and H can be estimated from application Acknowledgment.-The authors wish to thank of the theoretically derived equations to experi- the donors of the Petroleum Research Fund, mental data. The sum V1 V 2usually can be ob- administered by the American Chemical Society, tained by a static volumetric measurement. for their support of the research which led to this From V1 Vs and the time-lag, a can be calculated. paper. G. H. Thomas and T. H. Blanco, Jr., The ratio V,/Vl can be calculated only if VI or Vs assisted in the experimental work. can be estimated. This may be done by observing (11) J. A. Putnam. Pb.D. D i s s e r b t i o ~ Univenity of CaIifornia,

+

+

(9) R. M. Barrer and T.Gabor, Proc. Ruv. Soc (Ladrm), U K l , 353 (less:. (10) R. M. Barrer and T. Gabor, ibid., U66,267 (1960)

Berkeley, 1943. (12) C. R. Stewart, A. Lubinaki and E. A. Blenkam. J . Pst. Tech, 1s. 383 (1961).