OF

PECULIARITIES POROUS M.

LIQUID

FLOW

THROUGH

MEDIA A.

UDC 5.32.546

Sattarov

This study d i s c u s s e s the slow-flow m o d e l of [1] obtained by g e n e r a l i z a t i o n of e x p e r i m e n t a l data and a hypothesis of the e x i s t e n c e of m o l e c u l a r i n t e r a c t i o n effects between a solid and a liquid [2-4]. S o m e of these ideas w e r e u s e d e a r l i e r in c o n s t r u c t i o n of special m o d e l s of flow in c a p i l l a r i e s [5-9]. In [10] e x p e r i m e n t a l data a r e p r e s e n t e d which show the p r e s e n c e of a limiting s h e a r s t r e s s ~'0 f o r a n u m b e r of liquids, including water, while the B u e k i n g h a m - R e i n e r equation [11] is used to d e s c r i b e the flow: v=ki[t-4io/3~+l/~(io/i) ~]

(,~,~)

(0.1)

w h e r e v is the m e a n flow v e l o c i t y in a tube of r a d i u s R, k = p g R 2 / 8 p , p and p a r e the liquid density and v i s cosity, g is the a c c e l e r a t i o n of g r a v i t y , and i 0 and i a r e the initial and effective (variable) p r e s s u r e head g r a d i e n t s . The quantities r 0 and i 0 a r e r e l a t e d by the e x p r e s s i o n 90=0.5Rogi0

(0.2)

However, it was shown in [1] that Eq. (0.1) is not unique in d e s c r i b i n g the flow of v i s c o p l a s t i c bodies through a tube, while the o b s e r v e d s h e a r s t r e n g t h effect might be produced by o t h e r f a c t o r s , in p a r t i c u l a r , the physical p r o p e r t i e s and g e o m e t r y of the f i l t e r i n g m e d i u m . In [12, 13], e x p e r i m e n t a l r e s u l t s a r e p r e sented which p l a c e the e x i s t e n c e of a limiting s h e a r s t r e s s "% for w a t e r in doubt. 1. Anomalous P r o p e r t i e s of Liquids in P o r o u s Media. in c o n t r a s t to m a c r o v o l u m e s , where liquids e x p e r i e n c e the effect only of v i s c o u s f r i c t i o n f o r c e s , in m i c r o s p a c e s , g e n e r a l l y speaking, o t h e r f o r c e s b e c o m e significant, including i n t e r a c t i o n of the liquid with the solid wall, and interaction of the b a s i c liquid (solvent) with s u r f a c e a c t i v e p a r t i c l e s and m o l e c u l e s of other s u b s t a n c e s suspended therein, and with s u s pended p a r t i c l e s and other types of m o l e c u l e in the p o r o u s skeleton. In p o r o u s - m e d i u m m o d e l s c o m p o s e d of c a p i l l a r y tubes and slits, the f o r c e s mentioned above offer resistance to motion in a manner analogous to the usual friction forces and create an addition variable shear stress

t0: ~LN(D

(i.i)

w h e r e N(~) is the m o l e c u l a r - s u r f a c e f o r c e acting on a p a r t i c l e with c o o r d i n a t e ~ {the coordinate origin is located at the c e n t e r of the c a p i l l a r y , slit, etc. c r o s s section), and p is the liquid v i s c o s i t y . A p e c u l i a r i t y of the i n t e r a c t i o n between a solid and a p u r e wetting liquid is the fact that it is p e r p e n d i c u l a r to the solid s u r f a c e , on which it a t t a i n s a m a x i m u m , and d e c r e a s e s r a p i d l y upon r e m o v a l t h e r e f r o m into the liquid depth. The m e c h a n i c s of inhomogeneous liquids and liquids with an unstable s t r u c t u r a l s t a t e depend not only on the i n t e r a c t i o n of the solvent with suspended p a r t i c l e s , but a l s o on the nature of the i n t e r a c t i o n of each c o m p o n e n t of the i n t e r s t i t i a l liquid with the solid boundary. In the s p h e r e of action of t h e s e s u r f a c e f o r c e s the n o r m a l m a c r o v o l u m e p r o p e r t i e s of these liquids m a y change, and a n o m a l i e s m a y a p p e a r , a c c o m p a n y i n g l i b e r a t i o n of the m o r e a c t i v e liquid solvent n e a r the solid s u r f a c e . Known, f o r example, a r e the reduction of v o l u m e v i s c o s i t y of c e r t a i n liquids [3] at the solid wall and o t h e r a n o m a l o u s p r o p e r t i e s of w a t e r in soils [14]. Dushanbe. T r a n s l a t e d f r o m I z v e s t i y a A k a d e m i i Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 163-167, S e p t e m b e r - O c t o b e r , 1973. Original a r t i c l e s u b m i t t e d N o v e m b e r 29, 1971.

9 19 75 Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the publisher. A copy o f this article is available from the publisher for $15.00.

821

TABLE i Hydrodynamic characteristics of medium and water No. 16 glass capillary, quartz glass capillary, d=102 p d:330 g

Water temp., ~

n

20 30 ~o 50

0.28 t 0.3i8 0.400 1 0.065 0.060

0.37 0.46 0.56

0.485 0.580

0.049 0.029

b

58.6

1.14 3.76 4.48 5.47

67.2 i0i i49

Ko [ 3.35 4A5 5.08 6.02

I 0A7 0.20 0A2 0.i0

21 25 33 4t

E x p e r i m e n t s have c o n f i r m e d the dependence of the r a n g e of action of the s u r f a c e f o r c e s N(~) on the g e o m e t r y of the p o r o u s space, while in s i m u l a t e d m o d e l s , due to the s p a t i a l d i v e r s i t y of the d i r e c t i o n of the n o r m a l to the solid s u r f a c e a weakening of the effect of N(~) on flow d y n a m i c s is seen. Thus p r e l i m i n a r y e x p e r i m e n t showed that motion of p u r e w a t e r in m e d i a p r e p a r e d of s i m p l e g l a s s and quartz at a t e m p e r a t u r e of 18-20~ o c c u r s with an effective head g r a d i e n t i_> 0.001-0.005 in the following models: 1) with a toroid s p a c e between two coaxial tubes if the p o r e d i a m e t e r d_> 30-40 p; 2) in a c a p i l l a r y t u b e if d_~ 2030 p; 3) in a slit if d_> 10-20 p;4) in cubically packed cylindrical b a r s if d ~ 8-15 p; 5) in a m e d i u m of identical s p h e r e s if the m i n i m u m d i a m e t e r d_> 2-7 ]~. Consequently, this m a y explain the c o n t r a d i c t i o n s of the r e s u l t s of a n u m b e r of e x p e r i m e n t s (for e x ample, [10, 13]) on w a t e r f i l t r a t i o n through s p e c i m e n s with v a r i o u s p o r e configurations. 2. Liquid M e c h a n i c s in M i c r o s p a c e s . The h y d r o m e c h a n i c a l g e n e r a l i z a t i o n of [1] leads to the following equation of o n e - d i m e n s i o n a l (~, = 0) and a x i s y m m e t r i c (v =1) liquid flow in ideal m o d e l s of porous m e d i a

i d

(2.1)

H e r e i is the head gradient, f (~) is the s u p p l e m e n t a r y effect of the f o r c e of g r a v i t y a r i s i n g in the s p h e r e of action of the s u r f a c e f o r c e s , ~ is the angle between the g r a v i t y v e c t o r and the flow velocity, ~- = - p d V / d ~ is N e w t o n ' s law, and V is the p a r t i c l e velocity. I n t e g r a t i o n and division by ~v of both s i d e s of Eq. (2.1) g i v e s dY

pg~

t

pg cos (p ~

c

(2,2)

0

In a c a p i l l a r y tube or slit, a c c o r d i n g to the s y m m e t r y condition, C = 0, In the c a s e of ~0=~aR(~/R)~ in c l a s s i f y i n g the liquids a c c o r d i n g to the c h a r a c t e r of t h e i r flow it is s i m p l e to obtain the following e x p r e s sions f o r expenditure Q [1]: f o r p u r e wetting liquids, group A (1 < nr (2.3) f o r group B liquids ( I ~

>-~o) O={ 0 co~o~ cok[ i-- (~+3) io/(n+~+2) --ion-' (i/io) ~],

(2.4) (b
(w=21-~:~vR~+i, k=~'pgRfa/[ (i+v) ~(3+v)~], a= (n+v+2)/(n-i)) w h e r e k is the f i l t r a t i o n coefficient, er is the effective p o r o s i t y a r e a , iR=(i+v)~a/pg , i R and i, a r e the g r a dient v a l u e s above which l i n e a r and turbulent flow occur, r e s p e c t i v e l y , and i 0 is the initial p r e s s u r e g r a d i ent. Although Eqs. (2.3) and (2.4) w e r e obtained f o r the s i m p l e s t c a s e of T0, study of the function Q(i) (the v e l o c i t y function v = Q / w ) l e a d s to the following i n t e r e s t i n g r e s u l t . F o r n--* ~ (filtration in a m e d i u m with negligibly s m a l l s u r f a c e forces), we tend to P o i s e u i l l e flow and D a r c y ' s law, Q=ka~i. F o r n = 0 . 2 5 ( 1 - v ) Eq. (0.1) follows f r o m Eq. (2.4). It is r e m a r k a b l e that we obtain one and the s a m e r e s u l t h e r e f o r different limiting s h e a r s t r e s s e s ; f r o m the c a p i l l a r y m o d e l with To= const and f r o m the slit m o d e l with "~o=~aR(~/R)o.25. T h e c u r v e s of Eq. (2.3) p a s s through the origin and have an a s y m p t o t e . The

822

form of the Q(i) curves depends on v, the geometry of the medium and n, the degree of interaction of the filtering medium with the liquid. In the absence of an interaction effect (n= ~) (for example, in the flow of nonpolar liquids) the medium geometry determines the value k but does not affect the flow dynamics. The curves of Eq. (2.4) may be divided into two groups. The first group is defined by n values corresponding to the region ! _~ n_> - 2- v. For n = 1 and n =- 2- v from Eq. (2.4) we obtain straight lines intersecting the i axis at the points ion (n=l; n =-v -2),while for the remaining n with growth in iLthe Q(i) curves begin to straighten and have the same asymptotes as in the case of group A liquids. The Q(i) curves of the second group are in the interval-v -2 > n > -~ and have no asymptotes, although, like the curves of the first group, they intersect the i axis at points i=i0n~ 0. Thus solution of Eq. (2.1) for the case v0 in a power series leads to more generalized principles of fluid flow. Since for n -~ ~, Q =kwi is valid, consequently Eq. (2.3) describes the flow of a class of generalized Newtonian liquids (A), atn=0.25(l-v) Eq. (2.3) reduces to Eqo (0.i), and the motion described by Eq. (2.4) (i-> n ~- -2-v) correspondsto the flow of a class of viseoplastic (generalized Bingharn) bodies ~). In the interval-2-v > n-~oEq. (2.4) describes the flow of another class of liquids, for example, plastic bodies (C). It is evident that the flow of class B liquids in its fundamental hydrodynamic parameters is close to class C flow, which has an initial head gradient i0. However, with growth in i they take on the characteristics of class A flow. It is obvious that such a unique variable property is possessed only by liquids with an unstable structure, for example, colloidal suspensions of clay and other solids in multicomponent liquids. 3. To verify the conclusions of the theory presented here laboratory ties and flows of liquids in various microspace models were begun.

studies of the anomalous

proper-

The experimental filtration apparatus was a simple Darcy device: two identical piezornetric tubes joined to each other by the specimen to be studied (capillary, slit, etc.). In contrast to [i0] (experiment with pure water, method b), a bundle of m identical capillaries or a narrow slit was used as the specimen, rather than a single capillary. This approach accelerates the rate of liquid rise in the observation tube, increases the flow per unit time into the measurement vessel, and shortens the duration of the experiments, which sharply reduces the effect of extraneous factors dependent on time. In the first stages of the experiment glass (standard and newly drawn) capillary tubes from 2.05" 104 to 4 - 102 mm in length were studied and the absence of an initial head gradient, thus of a limiting shear stress, in pure water was established within the limits of experimental error, which corresponded to a gradient i0= (1-20) - 10-5. For example, in a specimen with 64 capillaries (one with a dimaeter of 100-105 ft, the others less than 100 p) 400 mm in length, complete equalization of levels in the piezometers occurred only after two days, while the last millimeter increase required 15 h. It follows from this that for establishment of level with water flow through a single capillary with diameter 100 #, more than 100 days would be required, with the last millimeter increase requiring about a month. To establish a more reliable dependence of piezometric head h on time t in the observation tube, and of flow Q as a function of i, and to compare the experimental results with theory a special program was "used for solution of the statistical problem of selection of an optimal prediction function from t~he condition of maximum modulus of the correlation coefficient with 20 specially selected functions, We will present some results of this study. 1. Statistical processing of the data reveals that at low gradient values the flow of water in the capillaries considered here is nonlinear and is described with great accuracy by Eq. (2.3). 2. The constant n in Eq. (2.3) is dependent on the microspace dimensions. Thus, at temperatures of 18-20~ for glass capillaries with diameter 0.1 -
(K0(~-l) b n+3

.iR-~, p = ~ ,

4

)

Ko=--k

(3.1)

It is evident, in p a r t i c u l a r , that f o r n< ~ the change in the u s u a l p r o p e r t i e s of p u r e w a t e r (for e x a m ple, m a c r o s c o p i c v i s c o s i t y ) p r o v e d to b e m u c h g r e a t e r in the quartz c a p i l l a r y than in the g l a s s , although

823

the diameter of the latter was at least three times s m a l l e r than the larger. This example indicates the possibilities of the filtration method in studying the interaction of a solid-liquid system and a number of other p r o p e r t i e s of surface effects. The author thanks P. Ya. Polubarinov-Kochin for his interest in the work. LITERATURE 1. 2. 3. 4. 5. 6.

7. 8. 9. 10. 11. 12. 13. 14.

824

CITED

M . A . Sattarov, WSome filtration models in porous media," Dokl. Akad. Nauk SSSR, 203, No. 1 (1972). N.K. Adam, Surface Physics and Chemistry [in Russian], Gostekhizdat, M o s c o w - L e n i n g r a d (1947). B . V . Deryagin, V. V. Karasev, N. N. Z skhavaeva, and V. P. Lazarev, "The mechanism of boundary wetting and pr ope r t i es of a wetted boundary layer," Zh. Tekh. Fiz., 27, No. 5 (1957). A . T . J . Howard and J. D. Isdale, nThe theology of liquids very near to solid boundaries, ~ Brit. J. Appl. Phys., Ser. 2, 2, No. 2 (1969). G.Y. Kovacs, ~Theoretical investigation into mieroseepage," ACTA Techn. Acad. Sci. Hungaricae, 21, No. 1-2 (1958). S . F . Aver'yanov, "Soil permeability as a function of water content," Dokl. Akad. Nauk SSSR, 69, No. 2 (1949). Y. Klausner and S. R. Kraft, "Non-Poseuille flow with axial forces," Israel. J. Technol., 3, No. 2, 152 (1965). A. ]~. Sheidegger, The Physics of Liquid Flow through Porous Media [in Russian], Gostekhizdat,Moscow (1960). Development of Studies in Filtration Theory in the USSR (1917-1967) [in Russian], Nauka, Moscow (1969). N . F . Bondarenko, ~A study of liquid filtration anomalies, ~ in: Collected Studies on Agronomic Physics, No. 14 [in Russian], Kolos, Leningrad (1967). M. Reiner, Deformation and Flow [Russian translation], Gostekhizdat (1963). A . I . Gorshkov, WLimiting shear s t r e s s in liquid water," Zh. Tekh. Fiz., 42, No. 1, 224 (1972). E . C . Childs and E. T r i m a s , " D a r c y ' s law at small potential gradients," J. Soil Sci., 22, No. 3 (1971). A.A. Rode, Basics of Soil Moisture Study. Water P r o p e r t i e s of Soil and Motion of Soil Moisture [in Russian], Vol. 1, Gidrometeoizdat (1965).