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Effect of Temperature on Cap i l l a ry P ressure

Steven A. Grant

U.S. Army Cold Regions Research Engineering Laboratory, Hanover, New Hampshire

Jorg Bachmann

Institute of Soil Science, University of Hannover, Germany.

Th e effect of te mp er at ur e on capill ary pressu re is one of several fasc inati ngprob lems une ar th ed by J.R. Phi lip durin g his long career. In his classic pap erwrit ten with Daniel de Vries, he assumed reasonably, but incorrectly, th atthe relative change in capillary pressure with temperature was equal to thatof th e surface tensio n of water . In fact th e change for capill ary press ure isroughl y four time s as large. Four mechani sms may be pro pose d to expla inthis discrepancy: expansion of water, expansi on of ent rap ped air, soluteeffects on the surface tension of water, and temperature-sensitive contactangles. None of the se expl anat ions describes all of th e per ti nen t dat a. Adefinitive expla nat ion ap pea rs to be as elusive to da y as it has been at anytime.

1. I N T R O D U C T I O N

J o h n P h i l i p ' s i n t e l l e c t , c r e a t i v i t y, and p r o d u c t i v i t yw e r e so p r o te a n t h a t his b o d y of w o r k can be v i e w e d asa s c i e n t i f i c b u l l d o z e r t h a t c r e a t e d a v a s t t e r r a i n of imp o r t a n t r e s u l t s bu t s i m u l t a n e o u s l y le f t n u m e r o u s l e s s e r,b u t f a s c i n a t i n g , p r o b l e m s for his g r a t e f u l s u c c e s s o r s toe x a m i n e in d e t a i l . An i l l u s t r a t i v e e x a m p l e of t h i s s i m i l eis t he s u b j e c t of t h i s c h a p t e r : t he effect of t e m p e r at u r e on c a p i l la r y p r e s s u r e , w h i c h was a m i n o r p o i n t ina s h o r t (yet, n o n e t h e l e s s , v e r y i n f l u e n t i a l ) p a p e r t h a tP h i l i p p u b l i s h e d w i t h D a n i e l de Vr i e s in 1957.

T h r e e a s p e c t s of Philip and de Vries [1957] are not a b l e . F i r s t , g iv e n t he p r o b l e m a d d r e s s e d , the p a p e r isa s t o n i s h i n g l y s h o r t , a s c a n t t en u n d e r s i z e p a g e s . ( T h i s

c h a p t e r , one of m a n y t h a t h a v e d w e lt on a s i n g l e e q u a -

Envi ronmenta l Mechanics : Water, Mass and Energy Transfer inthe BiosphereG e o p h y s i c a l M o n o g r a p h 129C o p y r i g h t 2 0 0 2 by the Amer ican G eophys ica l Union1 0 . 1 0 2 9 / 1 2 9 G M 1 8

t i o n in the p a p e r , is t h r e e t i m e s as l o n g . ) S e c o n d , the

p a p e r ha s b e e n e x t r e m e l y i n f l u e n ti a l b o t h in g e o p h y s i c sb u t no l ess so in e n g i n e e r i n g . A r e c e n t c o m p u t e r i z e di n q u i r y for c i t a t i o n s of the p a p e r f o u n d 473 c i t a t i o n s - as t a t i s t i c t h a t u n d o u b t e d l y u n d e r e s t i m a t e d t he i n f l u e n c eof the p a p e r . G i v e n t he p a p e r ' s age, m a n y m o d e r n aut h o r s h a v e no d o u b t c it e d i n t e r m e d i a t e w o r k s w i t h o u tc i t i n g (or k n o w i n g ) t he o ri g i n a l , s e m i n a l w o r k . Thet h i r d and m o s t f a s c i n a t i n g a s p e c t is t h a t b o t h a u t h o r sl a r g e l y a b a n d o n e d t he t o p i c w i t h t h i s p a p e r. A f t e r l e a vi n g C S I R O , de Vr ie s b e c a m e a p h y s i c s p r o f e s s o r in hisn a t i v e N e t h e r l a n d s . W h i l e hi s p r o f e s s i o n a l o b l i g a t i o n sc o m p e l l e d him to w o r k on o t h e r p r o b l e m s , de Vr i e s ret a i n e d an affinity and i n t e r e s t in th e s u b j e c t . ( A f t e rh i s r e t i r e m e n t he w r o t e a re v i e w p a p e r r e v i s i t i n g thet o p i c in w h i c h he q u o t e d t o u c h i n g l y t he F r e n c h , On re-vient toujours a ses premieres amours [de Vries, 1987].)A s i d e f r o m a p a p e r w r i t t e n d u r i n g a s e c o n d v i s i t byd e Vri e s n e a r l y t h i r t y y e a r l a t e r , P h i l i p a p p e a r s nott o h a v e c o n s i d e r e d t he p r o b l e m f u r t h e r [de Vries andPhilip, 1987] .

I n Philip and de Vries [1957] t he a u t h o r s d e v e l o p e dw h a t p r o v e d to be a s e m i n a l m o d e l d e s c r i b i n g t he s i m u l -

199

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2 0 0 E F F E C T O F T E M P E R A T U R E O N C A P I L L A RY P R E S S U R E

t a n e o u s t r a n s f e r o f e n e rg y, l i q u i d w a t e r, a n d w a t e r v ap o r i n a n u n s a t u r a t e d , n o n i s o t h e r n i a l p o r o u s m e d i u m .To d e s c r i b e t h e s e p r o c e s s e s , t h e y n e e d e d t o e s t i m a t e t h ec h a n g e i n c a p i l l a r y p r e s s u r e w i t h t e m p e r a t u r e . Philipand de Vries [ 1 95 7 ] t u r n e d n a t u r a l l y t o t h e s o - c a l l e dYo u n g - L a p l a c e e q u a t i o n :

2 7 l g cos 8(1)

w h e r e pc i s c a p i l l a r y p r e s s u r e ( i n p a s c a l s ) ; B , t h e c o nt a c t a n g l e b e t w e e n t h e s o li d a n d t h e l i q u i d - g a s i n t e r f a c e( i n d e g r e e s ) ; 7l g , t h e l i q u i d - g a s i n t e r f a c i a l t e n s i o n ( i nn e w t o n s p e r m e t e r ) ; a n d r , t h e a p p a r e n t p o r e r a d i u s ( i nm e t e r s ) . A t t h a t t i m e it w a s w e ll e s t a b l i s h e d t h a t t h ew a t e r f i l m t h i c k n e s s f o r a g r i c u l t u r a l s o i l s a b o v e t h e p e rm a n e n t w i l t i n g p o i n t w a s e q u i v a l e n t t o m a n y m o l e c u l a rt h i c k n e s s e s [Taylor, 1 9 5 8 ] . T h e c o n v e n t i o n a l w i s d o m oft h e d a y h e l d t h a t t h e c o n t a c t a n g l e o f a l i q u i d - g a s i nt e r f a c e a t a s o l i d w e t t e d w i t h t w o o r m o r e m o l e c u l a r

l a y e r s o f w a t e r w a s z e r o . A s w i l l b e d i s c u s s e d i n S e ct i o n 3 . 4 . 1 , i t h a s o n l y r e c e n t l y b e c o m e a p p a r e n t t h a tt h i s c o n v e n t i o n a l w i s d o m w a s f a l l a c i o u s . A c c o r d i n g l y,i t w a s n o t m e r e l y e x p e d i t i o u s , b u t r e a s o n a b l e a l s o , t oa s s u m e a z e r o c o n t a c t a n g l e a t a p o r o u s s o l i d s o w e t t e da n d , a c c o r d i n g l y, c o s O = 1 . E q u a t i o n ( 1 ) t h e n b e c a m e :

Pc2 7

l g

(2)

Ta k i n g t h e t o t a l d e r i v a t i v e o f e q u a t i o n ( 2) w i t h r e s p e c tt o t e m p e r a t u r e a n d d i v i d i n g t h r o u g h b y pc, t h e y o bt a i n e d :

i dPc _ i d y g

~ d T ~ ^IXT(3)

w h e r e T i s t e m p e r a t u r e ( in k e l v i n s ) .A s t h e o l d c h e s t n u t s a y s , C o m p l e x p r o b l e m s h a v e

s i m p l e , e a s y - t o - u n d e r s t a n d w r o n g a n s w e r s , a n d e q u at i o n ( 3 ) a p p e a r s t o b e w r o n g , o n a v e r a g e , b y a f a c t o rof 4 . For so i l s s tu di ed (l/pc) (dpc/dT)T=298 K r a n g e sf r o m - 0 . 0 0 1 7 2 t o - 0 . 0 2 8 2 9 K_ 1 w i t h a n a v e r a g e v a l u eo f - 0 . 0 0 8 4 4 K - \ w h e r e a s (l /V g) ( d 7 l g / d T ) T = 2 g 8 K forp u r e l i q u i d w a t e r i s - 0 . 0 0 2 1 3 5 K- 1 [Grant, in pre ss ] .T h e d i ff e r e n c e is s m a l l b u t m u c h t o o l a rg e a n d t o o c o ns i s t e n t l y o b s e r v e d t o b e a t t r i b u t a b l e t o e x p e r i m e n t a le r r o r.

I n t e r e s t i n g l y, w h y e q u a t i o n ( 3 ) i s w r o n g h a s n o t b e e nr e s o l v e d a f t e r a l m o s t t h i r t y y e a r s o f d e t a i l e d s t u d y b ys o m e o f t h e m o s t t a l e n t e d m i n d s i n g e o p h y s i c s a n dp e t r o l e u m e n g i n e e r i n g . T h e p r o b l e m w a s a p o p u l a rt o p i c a s s i g n e d t o d o c t o r a l s t u d e n t s i n t h e s e d i s c i pl i n e s[Wilkinson, 1 9 6 0 ; Meeuwig, 1 9 6 4 ; Haridasan, 1 9 7 0 ; Jury,1 9 7 3 ; Okandan, 1 9 7 4 ; Miller, 1 9 8 3 ; Nimmo, 1 9 8 3 ; Hop-mans, 1 9 8 5 ; Salehzadeh, 1 9 9 0 ; She, 1 9 9 7 ] . T h e r e h a v eb e e n n u m e r o u s e x p l a n a t i o n s f or t h i s d i s p a r i t y, a l l o f

w h i c h w e r e b a s e d o n r e a s o n a b l e s u p p o s i t i o n s a b o u t t h en a t u r e o f w e t t i n g l i q u i d s i n p o r o u s m e d i a a n d a l l o fw h i c h f a il e d t o g i v e a c o m p l e t e l y s a t i s f a c t o r y d e s c r i pt i o n o f t h e p h e n o m e n o n . E v e n t h o u g h t h e p h e n o m e n o nh a s n o t b e e n r e s o l v e d , m u c h h a s b e e n l e a r n e d a b o u tp o r o u s m e d i a b e h a v i o r. A g r e a t d e a l o f t h i s k n o w l e d g ew a s d u e t o f a il e d a t t e m p t s t o u n d e r s t a n d t h e i n a d e q u ac i e s o f e q u a t i o n ( 3 ) . I n t h i s P h i l i p w a s w r o n g , b u t i nb e i n g w r o n g , P h i l i p o p e n e d u p r i c h a r e a s o f s t u d y t h a th a s c o m p e l l e d g e o p h y s i c i s t s a n d e n g i n e e r s t o e x p l o r et h e f u n d a m e n t a l s o f t h e i r u n d e r s t a n d i n g of n a t u r a l p h e

n o m e n a .T h e b a l a n c e of t h i s p a p e r w i ll p r e s e n t a s e l e c t i v e s u r

v e y o f t h e e x p e r i m e n t a l d a t a d e s c r i b i n g t h e e ff ec t o ft e m p e r a t u r e o n c a p i l l a r y p r e s s u r e . T h e p a p e r w i l l t h e nr e v i e w t h e e x p l a n a t i o n s p r o p o s e d t o e x p l a i n t h e p h en o m e n o n a n d s u g g e s t i o n s fo r f u t u r e r e s e a r c h .

2 . T H E P H E N O M E N O N

A s fa r a s w e c a n d e t e r m i n e , t h e e f fe c t of t e m p e r a t u r eo n c a p i l l a r y p r e s s u r e w a s f i rs t o b s e r v e d w i t h a r e c o r d i n gt e n s i o m e t e r b y Richards and Neal [ 1 9 3 7 ]. T h e y a c q u i r e dc a p i l l a r y p r e s s u r e c o n t i n u o u s l y i n t h e f i e l d w i t h a s i mp l e c i r c u l a r c h a r t r e c o r d e r, w h i c h s h o w e d t h a t c a p i l l a r yp r e s s u r e s d e c l i n e d i n t h e m o r n i n g , a s t h e s o il w a r m e d .T h e d e c r e a s e in c a p i l l a r y p r e s s u r e w a s m o s t p r o n o u n c e df o r t e n s i o m e t e r c u p s n e a r t h e s u r f a c e . W h i l e s o m e o ft h e p h e n o m e n o n c o u l d h a v e b e e n d u e t o t h e t h e r m a le x p a n s i o n o f li q u i d s in t h e a p p a r a t u s , t h e i r r e s u l t s i n d ic a t e d t h a t c a p i l l a r y p r e s s u r e d e c r e a s e d w i t h i n c r e a s i n gs o i l t e m p e r a t u r e .

S u b s e q u e n t l y , t h e e f fe c t o f t e m p e r a t u r e o n c a p i l l a r y

p r e s s u r e w a s c ar e f u ll y s t u d i e d i n s e v e r a l l a b o r a t o r y s t u di es . T h e s e s t u d i e s g e n e r a l l y, b u t n o t e x c l u s i v e l y, c o ns i s t e d of d e t e r m i n a t i o n s o f c a p i l l a r y p r e s s u r e s a t u r at i o n r e l a t i o n s d e t e r m i n e d a t m o r e t h a n o n e t e m p e r at u r e . F i g u r e 1 p r e s e n t s a n e x a m p l e of d a t a c o l l ec t e di n t h e s e s t u d i e s , w h i c h i n d i c a t e d t h a t t e m p e r a t u r e w a sh a v i n g a n ef f ec t , t h o u g h b e c a u s e t h e c a p i l l a r y p r e s s u r es a t u r a t i o n r e l a t i o n s w e r e th e m s e l v e s s o c o m p l e x , i t w a sdi ff icu l t to gra sp th e na tu re of th e e ffec t.

I n c o m p a r i s o n , Gardner [ 19 5 5] p l o t t e d c a p i l l a r y p r e ss u r e s o f a s o i l s a m p l e m a i n t a i n e d a t a c o n s t a n t d e g r e eo f s a t u r a t i o n a s i t w a s h e a t e d a n d c o o l e d . T h i s p l o ti s p r e s e n t e d h e r e a s F i g u r e 2 . D u e t o i t s s i m p l i c i t y,F i g u r e 2 d e m o n s t r a t e s c l e ar l y a n d c o n v i n c i n g l y t h e n at u r e o f t h e t e m p e r a t u r e e ff ec t, w h i c h w a s o b s c u r e d b yt h e c o m p l e x i t y o f F i g u r e 1 , t h a t pc d e c r e a s e s l i n e a r l yw i t h t e m p e r a t u r e . T h e d a t a of Gardner [1955] sug ges tt h a t c a p i l l a r y p r e s s u r e a t a p a r t i c u l a r t e m p e r a t u r e isn o t p r e c i s e l y r e p r o d u c i b l e a f t e r c y c l e s o f h e a t i n g a n dc o o l i n g . Faybishenko [ 19 8 3] c o n d u c t e d a s t u d y s i m i l a rt o Gardner [ 1 9 5 5 ], b u t , a s p r e s e n t e d i n F i g u r e 3 , f o u n dt h a t c a p i l l a r y p r e s s u r e s a t h i s o b s e r v a t i o n a l t e m p e r a -

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GRANT AND BACHM ANN 201

~ ~ 1 r -

* 16 18 20 22 24 26 28

Volumetric water content/per cent

F i g u r e 1 . Capi l la ry pressure sa tura t ion re la t ions of aPlainfield sandy loam soil measured at 19.1 (circles), 34.1(squares) , and 49.2 C ( tr iangles) as reported by Nimmoand Miller [1986].

tures were reproducible on cycles of heating and cool

ing and appeared to be linear functions of temperature.

The results of Gardner [1955] an d Faybishenko [1983]

sugge sted tha t the effect of te mpe ra tu re on capillary

pressure could be described by:

Pc = (4)

F i g u r e 3. Capi l la ry pressure measured by Faybishenko[1983] on a loam soi l at three constant water contents, butat a range of t empera tures . Ro ma n numera ls re fer to wate r contents (I : 0 .347 m 3 m ~ 3 , I I : 0 . 3 2 6 m3 m ~ 3 , I I I : 0.3m 3 m - 3 ) . Arabic num bers re fer to s teps in heating-and-cooling cycles.

where a Pc an d b Pc are empiri cal cons tan ts (in pascal s

and pascals per kelvin, respectively).

As with capillary pressure, all known liquid-gas inter-

facial tensi ons are well descri bed as linearl y d ecreas ing

0*