Sundberg, J. 1982: Metoder for bestamning av varmebverforande egenska- per i jord och berg. (Methods ..... 7065), scanner (AR elek- tronik), constant current source (Philips PE 1541) and a switch box. ...... normal moran, klass I I . En betydligt ...
THERMAL PROPERTIES OF SOILS AND ROCKS
Jan Sundberg
GEOLOGISKA INSTITUTIONEN Publ. A 57 1988
GEOLOGISKA INSTITUTIONEN CHALMERS TEKNISKA HOGSKOLA GOTEBORGS
och
UNIVERSITET
THERMAL PROPERTIES OF SOILS AND ROCKS
Jan Sundberg
Publ. A 57 Dissertation ISSN 0348-2367
Gotebot^ 1988
Postadress
Gatuadress
Telefon
Telex
C h a l m e r s University o f Technology
41296
Sven Hultins gata 8
0 3 1- 7 2 1 0 0 0
2369 G H A L B 1 B S
a n d University o fGoteborg Department of Geology S-412 9 6 G O T E B O R G
GOTEBORG
{Gothenburg} SWEDEN
THERMAL PROPERTIES OF SOILS AND ROCKS Jan and
Sundberg, Department o f Geology, Chalmers U n i v e r s i t y o f Technology U n i v e r s i t y o f G o t e b o r g , S-412 9 6G o t e b o r g , Sweden.
ABSTRACT Knowledge o f t h e thermal p r o p e r t i e s o f rock and s o i l i s v a l u a b l e i n ' many d i f f e r e n t areas. Equipment f o r t h e a n a l y s i s o f t h e r m a l c o n d u c t i v i t y has been developed. Laboratory o r i n s i t u d e t e r m i n a t i o n s o f thermal p r o p e r t i e s can b eperformed under s t a t i o n a r y and t r a n s i e n t c o n d i t i o n s b ymany d i f f e r e n t m e t h o d s . Two k i n d s o f p r o b e m e t h o d s , t h e s i n g l e - p r o b e and t h e m u l t i - p r o b e method have been i n v e s t i g a t e d . Theory and d i f f e r e n t s o u r c e s o f p o t e n t i a l e r r o r s , f o r i n s t a n c e l e n g t h diameter r a t i o and i n f l u e n c e o f sample boundary, have been t r e a t e d . S u g g e s t i o n s t o a v o i d such e r r o r s have a l s o been made. D i f f e r e n t types o f t h e o r e t i c a l methods f o r e s t i m a t i n g thermal conduct i v i t y have been described and analyzed. The s e l f - c o n s i s t e n t a p p r o x i mation has been adopted and applied t o c a l c u l a t e t h e thermal c o n d u c t i v i t y o f d i f f e r e n t types o f rock and s o i l . The method has been d i r e c t l y applied t o c r y s t a l l i n e rock and t o extremely porous s o i l . I n m i n e r a l s o i l , s a n d s t o n e and l i m e s t o n e d e t e r m i n a t i o n s , i t was necessary t o modify t h e method and introduce a contact r e s i s t a n c e between t h e g r a i n s . Vapor d i f f u s i o n , unfrozen and f r o z e n c o n d i t i o n s , i n c l u d i n g u n f r o z e n water have a l s o been t r e a t e d . The method has been v e r i f i e d b y t h e r m a l c o n d u c t i v i t y measurements o na number o f c r y s t a l l i n e and s e d i mentary rocks and 600 s o i l s . A method Introduced e a r l i e r f o r computing the thermal c o n d u c t i v i t y o f rock/mineral from measurements o n a m i x t u r e o f p u l v e r i z e d r o c k / m i n e r a l and water, has been e v a l u a t e d . The r e s u l t i n d i c a t e s t h a t t h e u s e o f t h e mean v a l u e o f Hashin-Shtrikman's bounds t o c a l c u l a t e t h e s o l i d phase c o n d u c t i v i t y may I n t r o d u c e l a r g e errors. The s t u d y a l s o t r e a t s t h e content, mineral distribut p r o p e r t i e s o f rock and s o i t a r y volume (REV) o f soil
I n f l u e n c e o f changes i n temperature, water i o n , vapor d i f f u s i o n , e t c ,o nt h e thermal l , and discusses t h e r e p r e s e n t a t i v e elemenand rock.
An e x t e n s i v e s t a t i s t i c a l s t u d y o n c r y s t a l l i n e r o c k has been p e r f o r m e d based on more than 4000 measured and c a l c u l a t e d thermal c o n d u c t i v i t y values. S t a t i s t i c a l i n t e r v a l s were created f o rd i f f e r e n t types o f rock. A guide has been developed f o r c a l c u l a t i n g t h e thermal p r o p e r t i es o f s o i l based o n9 0 0 m e a s u r e m e n t s . Keywords: Thermal c o n d u c t i v i t y , thermal p r o p e r t i e s , rock, s o i l , needle-probe, m u l t i - p r o b e , method, s e l f - c o n s i s t e n t , Hashin and Shtrikman, determination, errors, axial-flow. ISSN 0348-2367 Publ. A57 1988
i
PREFACE Following t h e o i l crisis at the beginning o f the seventies, extensive e f f o r t s ensued t o develop ways o f s u b s t i t u t i n g t h e o i l u t i l i z e d f o r h e a t i n g b y o t h e r e n e r g y f o r m s . T h e E a r t h H e a t Pump G r o u p a t t h e C h a l mers U n i v e r s i t y o f Technology t o o k p a r t i n t h i s work. The group was formed i n t h e l a t e s e v e n t i e s and c o n c e n t r a t e d i t s e f f o r t s on d e v e l o p i n g h e a t pump s y s t e m s c o m b i n e d w i t h h e a t s t o r a g e i n g r o u n d . T h i s r e s e a r c h w o r k was made p o s s i b l e t h r o u g h t h e f i n a c i a l s u p p o r t o f t h e Swedish Council f o rB u i l d i n g Research. A s u b s t a n t i a l part o f t h i s work h a s b e e n c a r r i e d o u t u n d e r t h e l e a d e r s h i p o f P r o f . K. G o s t a E r i k s s o n , the Department o f Geology, Chalmers U n i v e r s i t y o f Technology. One o f t h e most i m p o r t a n t f a c t o r s a f f e c t i n g t h e p e r f o r m a n c e o f s u b s u r f a c e h e a t i n g systems a r e t h e thermal p r o p e r t i e s o f s o i l s and r o c k s . S i n c e knowledge o f t h e s e t h e r m a l p r o p e r t i e s was l i m i t e d , work was initiated t o further explore this area. This thesis presents results from t h i s work. The t h e s i s comprises a summary, t h r e e r e p o r t s and one paper. The p r o j e c t was c a r r i e d o u t i n t w o phases. The f i r s t phase was p e r f o r m e d d u r i n g 1 9 7 9 - 1 9 8 5 a n d r e s u l t e d i n r e p o r t n o . 1 , 2 a n d 3. R e p o r t n o . 2 has b e e n w r i t t e n i n c o l l a b o r a t i o n w i t h Jacob J o n s s o n a n d Bo T h u n h o l m . My p a r t o f t h e w o r k i s d e s c r i b e d i n t h e p r e f a c e t o r e p o r t n o . 2. T h i s f i r s t phase, which c o n s t i t u t e s t h e main p a r t o f t h e work, has been f i n a n c i a l l y supported by t h e Swedish Council f o rBuilding Research. The second phase was p e r f o r m e d d u r i n g 1988 and i s r e p o r t e d i n paper no. 1 and t h i s summary. T h i s l a t t e r phase has been funded by t h e Chalmers U n i v e r s i t y o f Technology and t h e Swedish G e o t e c h n i c a l Institute. Since t h r e e o f t h e r e p o r t s a r e w r i t t e n i n Swedish, an e x t e n s i v e summary h a s been made i n E n g l i s h . T h e a i m h a s been t h a t t h e E n g l i s h summary f u l l y cover t h e whole work and make f o r a f r u i t f u l r e a d i n g . I n some cases t h i s summary has been e x t e n d e d r e l a t i v e t o t h e background m a t e r i a l due t o e x p e r i e n c e gained d u r i n g t h e work. T h i s has been p a r t i c u l a r y t r u e when d e a l i n g w i t h e x p e r i m e n t a l methods. T h i s work was p e r f o r m e d a b o u t 8 y e a r s a g o . Some p a r t s o f r e p o r t n o . 1 and 2 w e r e l e s s i m p o r t a n t f o r t h e d i s s e r t a t i o n o r d e s c r i b e d i n a b e t t e r way i n the summary. These p a r t s o f r e p o r t no. 1 and 2 have been excluded f r o m the thesis. I w a n t t o e x p r e s s m y g r a t i t u d e t o P r o f . K. G b s t a E r i k s s o n f o r h i s support and encouragement throughout t h e e n t i r e course o f t h e work. I e s p e c i a l l y w i s h t o t h a n k my a d v i s e r d u r i n g p h a s e 2 , D r . G u n n a r G u s t a f s o n , f o r h i s good guidance, v a l u a b l e s u g g e s t i o n s and c r i t i c a l comments. The
d i s c u s s i o n s w i t h a l l my c o l l e a g u e s a t t h e D e p a r t m e n t
i i1
o f Geology
have been most v a l u a b l e . I n a d d i t i o n t o those already acknowledged i n t h e r e p o r t s , I w o u l d l i k e t o s p e c i a l l y m e n t i o n my c o l l e a g u e s a t t h e E a r t h H e a t Pump G r o u p , I n g v a r R h e n a n d P e t e r W i l e n , who h a v e c o n t i n u a l l y g i v e n me c r i t i c a l s u g g e s t i o n s f o r i m p r o v e m e n t s . I n 1978, i t was p r i m a r i l y D r . S v e n - A k e L a r s o n w h o i n t r o d u c e d me t o t h e w o r l d o f h e a t f l o w a n d t h e r m a l c o n d u c t i v i t y . I w o u l d l i k e t o e x t e n d my t h a n k s f o r his support d u r i n g t h e work. I a l s o thank Dr. L a r s 0. E r i c s s o n who, d r a w i n g o n h i s e x p e r t i s e i n a r e l a t e d f i e l d , h e l p e d me w i t h v a l u a b l e comments. My t h a n k s Institute
a l s o t o Dr. Jan H a r t l e n and t h e Swedish f o rsupport and understanding.
Geotechnical
I a l s o w i s h t o t h a n k a l l o f t h e o t h e r s w h o h a v e h e l p e d me p r o d u c e t h i s m a n u s c r i p t : M r s . Ann-Marie H e l l g r e n , who helped t y p e phase 1 , M r s . Eva Dyrenas, f o r t y p i n g phase 2, M r s . R u t g e r d A b r i n k , who made t h e d r a w i n g s f o r p h a s e 2, M r s . E v a R a l d o w , who c o r r e c t e d my E n g l i s h a n d M r s . Lena K a r l s s o n f o rv a l u a b l e a s s i s t a n c e w i t h l a b o r a t o r y a n a l y s e s .
Linkoping
Jan
i n November 1988
Sundberg
iv
REPORTS AND PAPER COMPRISING THIS THESIS
T h i s t h e s i s c o n t a i n s two p a r t s . The f i r s t p a r t i s a ne x t e n s i v e t e x t w h i c h s u m m a r i z e s , e v a l u a t e s and e x t e n d s some p a r t s o f t h e s e c o n d p a r t . The second p a r t c o n t a i n s t h e f o l l o w i n g r e p o r t s and paper and w i l l b e r e f e r r e d t o i n t h e t e x t b y t h e r e p o r t o r p a p e r number. Some p a r t s o f r e p o r t no. 1 and 2 a r e l e s s i m p o r t a n t f o r t h e d i s s e r t a t i o n o r d e s c r i bed i n a b e t t e r w a y i n t h e summary. These p a r t s o f r e p o r t no. 1 a n d 2 are e x c l u d e d f r o m t h e t h e s i s . The e x c l u d e d p a r t s a p p e a r s f r o m t h e c o n t e n t s o fr e p o r t no. 1 and 2 .
Reports Sundberg, J .1982: Metoder f o r bestamning a vv a r m e b v e r f o r a n d e egenskaper i j o r d och berg. (Methods f o r d e t e r m i n i n g t h e thermal p r o p e r t i e s of r o c k a n ds o i l - i n S w e d i s h ) . C h a l m e r s T e k n i s k a h o g s k o l a , J o r d v a r m e g r u p p e n , R e p o r t No. 5, G b t e b o r g , Sweden. ( R e p o r t no. 1 ) . S u n d b e r g , J . , T h u n h o l m , B., J o h n s o n , J . , 1 9 8 5 : V a r m e b v e r f o r a n d e e g e n skaper i svensk berggrund. (Thermal properties o f Swedish rocks - i n S w e d i s h ) . S w e d i s h C o u n c i l f o r B u i l d i n g R e s e a r c h , R e p o r t R97, S t o c k h o l m , Sweden. (Report no. 2 ) . Sundberg, J., 1986: V a r m e b v e r f o r a n d e egenskaper i svenska j o r d a r t e r . V a r m e k o n d u k t i v i t e t , s p e c i f i k v S r m e k a p a c i t e t och l a t e n t varme. (Thermal p r o p e r t i e s o fSwedish s o i l s . Thermal c o n d u c t i v i t y , thermal c a p a c i t y and l a t e n t h e a t - i n S w e d i s h ) . S w e d i s h C o u n c i l f o r B u i l d i n g R e s e a r c h , R e p o r t R104, S t o c k h o l m , Sweden. ( R e p o r t no. 3 ) .
Paper Sundberg, J., 1988: The s e l f - c o n s i s t e n t a p p r o x i m a t i o n a p p l i e d t o t h e r m a l c o n d u c t i v i t y o fc r y s t a l l i n e r o c k , s e d i m e n t a r y r o c k and s o i l . In m a n u s c r i p t f o r p u b l i s h i n g . (Paper no. 1 ) .
V
CONTENTS ABSTRACT PREFACE R E P O R T S AND CONTENTS
i PAPER COMPRISING T H I S T H E S I S
1.
INTRODUCTION
2.
E X P E R I M E N T A L P R O B E METHODS FOR THERMAL P R O P E R T I E S Measurement technique Theory Equipment Calibration Sources o f error
2.1 2.2 2.3 2.4 2.5 3. 3.1 3.2 3.3 3.4 3.5
i i v v i i i
1 DETERMINING 4 4 5 10 13 13
T H E O R E T I C A L METHODS FOR D E T E R M I N I N G T H E R M A L PROPERTIES Introduction A p p l i c a t i o n t o rock A p p l i c a t i o n t o s o i l and porous rock Accuracy Computing t h e thermal c o n d u c t i v i t y o f rock from measurements on p u l v e r i z e d w a t e r - s a t u r a t e d samples
21 21 23 24 28 28
4. 4.1 4.2 4.3 4.4
T H E R M A L P R O P E R T I E S OF R O C K S AND S O I L S D i f f e r e n t thermal t r a n s p o r t mechanisms Influence of various characteristics Representative elementary volume Thermal p r o p e r t i e s o f rocks and s o i l s
31 31 32 36 37
5.
CONCLUSIONS
42
REFERENCES
44
APPENDIX: REPORT
No. 1
Metoder
f o r bestamning av varmeoverfbrande
egenskaper
i jord och berg
49
REPORT
No. 2
Varmeoverforande egenskaper
i svensk berggrund
REPORT
No. 3
Varmeoverfbrande egenskaper
i svenska
PAPER
No. 1
The s e l f - c o n s i s t e n t approximation applied t o t h e thermal c o n d u c t i v i t y o f c r y s t a l l i n e rock, sedimentary rock and s o i l
vii
jordarter
77 ...
149
279
1
1.
INTRODUCTION
K n o w l e d g e o f t h e t h e r m a l t r a n s p o r t p r o p e r t i e s o f r o c k and s o i l i s l u a b l e i n many d i f f e r e n t a r e a s . Some e x a m p l e s a r e t h e u t i l i z a t i o n storage of ground heat, geothermal heat flow determinations and d e t e r m i n a t i o n s o f h e a t l o s s f r o m b u r i e d c a b l e s and p i p e l i n e s .
vaand
The t h e r m a l p r o p e r t i e s o f a m a t e r i a l depend on a number o f p r o p e r t i e s s o m e o f w h i c h can be t i m e - d e p e n d e n t . The t h e r m a l c o n d u c t i v i t y o f c r y s t a l l i n e r o c k i s m a i n l y i n f l u e n c e d by t h e f o l l o w i n g f a c t o r s : mineral composition temperature isotropy/anisotropy fluid/gas in micro fissures Q u a r t z has a t h e r m a l c o n d u c t i v i t y s e v e r a l t i m e s h i g h e r t h a n t h a t o f o t h e r common r o c k f o r m i n g m i n e r a l s . The q u a r t z c o n t e n t i s t h e r e f o r e i m p o r t a n t f a c t o r . The t h e r m a l c o n d u c t i v i t y o f r o c k d e c r e a s e s as t h e temperature increases.
an
I f the t e x t u r e of the rock i s a n i s o t r o p i c a l , thermal c o n d u c t i v i t y i s a f u n c t i o n of the d i r e c t i o n of the heat flow. I f the micro f i s s u r e s in the rock are f i l l e d w i t h a i r instead of water, the thermal conductivit y d e c r e a s e s r a p i d l y w i t h s m a l l c r a c k p o r o s i t y (< 1 % ) . A t a l a r g e r scale the ordinary cracks also influence heat transport. In a d d i t i o n t o the above mentioned f a c t o r s , the thermal conductivity o f s o i l and s e d i m e n t a r y r o c k i s a f u n c t i o n o f t h e p o r o s i t y and the degree of water s a t u r a t i o n . T h e r m a l c o n d u c t i v i t y d e c r e a s e s as p o r o s i t y i n c r e a s e s . M o r e o v e r t h e r m a l c o n d u t i v i t y s h a r p l y f a l l s when the degree o f s a t u r a t i o n i s below appr o x i m a t e l y 50%. A t u n s a t u r a t e d c o n d i t i o n s and a b o v e r o o m temperature, v a p o r d i f f u s i o n and r a d i a t i o n become m o r e i m p o r t a n t w i t h increasing temperature. B o t h t h e s e h e a t t r a n s p o r t m e c h a n i s m s can be a d d e d t o t h e t h e r m a l c o n d u c t i v i t y and f o r m an e f f e c t i v e t h e r m a l c o n d u c t i v i t y as a function of temperature. M e a s u r e m e n t o f t h e r m a l c o n d u c t i v i t y can be c l a s s i f i e d as i n s i t u m e a s u r e m e n t and l a b o r a t o r y m e a s u r e m e n t s . I n s i t u m e a s u r e m e n t s a r e p e r f o r med a t n a t u r a l a n d u n d i s t u r b e d c o n d i t i o n s . One p r o b l e m a t i n s i t u m e a s u r e m e n t s i s t o k n o w how r e p r e s e n t a t i v e t h e m e a s u r e m e n t i s due t o n a t u r a l c h a n g e s i n e.g. w a t e r c o n t e n t . I f a p r o p e r e v a l u a t i o n can be made on s u c h t i m e d e p e n d e n t v a r i a b l e s , i n s i t u m e a s u r e m e n t s a r e , i n general, preferable. L a b o r a t o r y m e a s u r e m e n t s c o m p r i s e a s m a l l e r s a m p l e v o l u m e . The result of such measurements i s r e l i a b l e , provided the f o l l o w i n g points are
1
2
fulfilled: the sample i s undisturbed the sample volume i s representative o f t h e s o i l / r o c k the volume a f f e c t i n g t h e measurements i s representative of t h e sample c o r r e c t i o n i s made f o r t e m p e r a t u r e d i f f e r e n c e s between l a b o r a t o r y and f i e l d c o r r e c t i o n i s made f o ro t h e r t i m e - d e p e n d e n t v a r i a b l e s ( e . g . water content) Calculations o f t h e thermal conductivity o f earth materials from volume f r a c t i o n s o f m i n e r a l s , pore gas and pore f l u i d o f f e r many advantages. Knowing t h e changes i n , e.g. temperature and water content, i t i s possible t o c a l c u l a t e t h e change i n thermal c o n d u c t i v i t y . E s t i m a t e s c a n be made f r o m t h e r e s u l t o f a g e o t e c h n i c a l i n v e s t i g a t i o n . An a n a l y s i s o f t h e s e n s i t i v i t y o f t h e t h e r m a l c o n d u c t i v i t y c a n be made f r o m p o s s i b l e v a r i a t i o n s 1n t h e v o l u m e f r a c t i o n s . T h e o r e t i c a l c a l c u l a t i o n s o f e l e c t r i c a l t r a n s p o r t p r o p e r t i e s have been performed already during t h e l a s t century. However, t h e o r i e s o f elect r i c a l t r a n s p o r t can be t r a n s f e r r e d i n t o o t h e r areas o f t r a n s p o r t such as h y d r a u l i c c o n d u c t i v i t y a n d t h e r m a l c o n d u c t i v i t y o r v i c e v e r s a . T h i s has been done r a t h e r e x t e n s i v e l y f o rboth rock and s o i l m a t e r i a l However, erfipirical and semi-empirical s o l u t i o n s have dominated t h e f i e l d . An i n t e r e s t i n g t e n d e n c y i s t h a t e x p e r i e n c e g a i n e d i n o n e a r e a of e x p e r t i z e was sometimes d i f f i c u l t t o apply t o t h e work i n other areas. Several
terms that describe thermal
transport
T h e r m a l c o n d u c t i v i t y , A. ( W / ( m . K ) ) : t h e a b i l i t y transport thermal energy. Thermal d i f f u s i v l t y , K (m^/s): temperature differences. Thermal thermal
the ability
a r e defined
below:
of a material t o
o f a material
t o level
capacity, C (J/(m^,K)): t h e capacity o f a material t o store e n e r g y . C=QC, Q : d e n s 1 t y , K g / m ^ , c : t h e r m a l c a p a c i t y , J/(Kg,K).
These thermal
properties
a r e r e l a t e d t o each other
2
as f o l l o w s :
3
A s e l e c t i o n o f d i f f e r e n t methods t o determine thermal c o n d u c t i v i t y is summarized i n t h e t a b l e below: Method
Determining
property
Comment
Multi-probe method
Conductivity Diffusivity
T r a n s i e n t f i e l d and l a b o r a t o r y method. Applicable t o rock and s o i l .
Single-probe method (needle-probe)
Conductivity (Diffusivity)
T r a n s i e n t f i e l d and laboratory method. A p p l i c a b l e t o rock and s o i 1 .
Divided-bar method
Conductivity
Stationary laboratory method. Applicable t o rock.
THS-method (Transient hot strip)
Conductivity Diffusivity
Transient laboratory method. Applicable t o rock, fluid, (soil).
Theoretical calculation
Conductivity Thermal capacity
Calculation from rock m i n e r a l c o n t e n t and soil mineral content, p o r o s i t y and w a t e r c o n t e n t
The a i m o f t h e work has
been:
to develop Instrumentation f o r measuring thermal c o n d u c t i v i t y , primarily of soils to investigate potential probe method
errors using the transient cylindrical
to e v a l u a t e d i f f e r e n t t h e o r e t i c a l methods f o r determining the t h e r m a l p r o p e r t i e s o f s o i l s and r o c k s to evaluate the a p p l i c a b i l i t y o f using thes e l f - c o n s i s t e n t approximation f o r calculating thermal conductivity to recommend t h e r m a l p r o p e r t y v a l u e s f o r rocks the b a s i s o f m e a s u r e m e n t s and c a l c u l a t i o n s t o i n c r e a s e t h e knowledge and u n d e r s t a n d i n g t r a n s p o r t i n s o i l s and r o c k s
3
and s o i l s ,
of thermal
on
4
2.
E X P E R I M E N T A L P R O B E M E T H O D S FOR
T h e m o s t common m e t h o d f o r d e t e r m i n i n g s o i l i s t h e p r o b e m e t h o d . Some r e a s o n s
DETERMINING THERMAL PROPERTIES the thermal conductivity of for i t s popularity are:
the t h e o r y i s s i m p l e i t can be e v a l u a t e d g r a p h i c a l l y short time o f measurement easy insertion i n soft m a t e r i a l a p p l i c a b l e t o both f i e l d and l a b o r a t o r y b o t h c o n d u c t i v i t y and d i f f u s i v i t y ( t r a n s i e n t m e t h o d ) can be termined. The probe method i s a l s o o f t e n used f o r f i e l d measurements o f w h i l e t h e m o s t common l a b o r a t o r y m e t h o d f o r m e a s u r i n g t h e r m a l t i e s o f rock m a t e r i a l s i s t h e s t a t i o n a r y divided bar method .
de-
rock, proper-
Report no. 1 t r e a t s d i f f e r e n t methods o f e s t i m a t i n g thermal c o n d u c t i v i t y . The main p a r t o f t h e work i s concentrated on d i f f e r e n t probe methods. 2.1
Measurement
technique
A heat generating probe i s i n s e r t e d into t h e ground. A temperature measuring gauge i s i n s t a l l e d i n t h e probe a t h a l f l e n g t h . A t t i m e t = 0 , a constant e l e c t r i c a l power i s turned on. The increase i n temperature w i t h t i m e i s r e g i s t e r e d . A f t e r a s u f f i c i e n t t i m e , t h e power i s turned o f f and t h e t h e r m a l p r o p e r t i e s a r e e v a l u a t e d f r o m measurement data and a mathematical expression. The s i n g l e - p r o b e method i s f i r s t d e s c r i b e d i n t h e l i t e r a t u r e by t h e two Swedes S t S l h a n e and Pyk ( 1 9 3 1 ) . Today, n e a r l y 60 y e a r s l a t e r , e s s e n t i a l l y t h e same method i s used. The measurement t e c h n i q u e has o f course been f u r t h e r d e v e l o p e d , e s p e c i a l l y d u r i n g t h e l a s t t e n y e a r s . The method has been used and described i n t h e l i t e r a t u r e s e v e r a l t i m e s , s t a r t i n g i n t h e f i f t i e s and l a t e r o n . I n Sweden, S a a r e and Wenner (1957) made a v a l u a b l e c o n t r i b u t i o n t o f i e l d m e a s u r e m e n t s o f the t h e r m a l c o n d u c t i v i t y o f d i f f e r e n t s o i l s . The m u l t i - p r o b e method i s a v a r i a n t on t h e s i n g l e - p r o b e method d e s c r i bed a b o v e . The m e t h o d was d e v e l o p e d a t t h e d e p a r t m e n t o f G e o l o g y , C h a l m e r s U n i v e r s i t y o f T e c h n o l o g y , f r o m an i d e a o f Dr D a v i d M a l m q v i s t . The m e t h o d was f i r s t d e s c r i b e d by L a n d s t r o m e t a l (1979) and was s u b s e q u e n t l y e x a m i n e d and e l a b o r a t e d by t h e a u t h o r ( S u n d b e r g , 1979). The m e a s u r e m e n t t e c h n i q u e i s e s s e n t i a l l y t h e same as t h a t o f t h e s i n g l e - p r o b e method. However, t h e temperature m e a s u r i n g gauge i s p l a c e d a t a c e r t a i n d i s t a n c e away f r o m t h e p r o b e s u r f a c e , see f i g u r e 1. Of course i t i s a l s o p o s s i b l e t o use a c o m b i n a t i o n o f t h e s i n g l e p r o b e and t h e m u l t i - p r o b e method .
4
5
POWER-SUPPLY
REFERENCE TEMPERATURE MEASURING PROBE M E A S U R E M E N T OF A POTENTIAL DAILY T E M P E R A T U R E W A V E )
0,06-0,20m
Figure
1.
( D E P E N D S ON T E S T I N G
MATERIAL!
The m u l t i - p r o b e method.
Another v a r i a n t o f t h e s i n g l e - p r o b e method i s t h e s o - c a l l e d h a l f space probe method. The method i s simple t o use on hard rock. Since i n such m a t e r i a l i t i s d i f f i c u l t t o d r i l l holes f o r t h e probe, t h e design cons i s t e d o f encapsulating a needle probe i n a m a t e r i a l w i t h a low t h e r m a l c o n d u c t i v i t y . The m a t e r i a l i s g r i n d e d away u n t i l t h e probe i s f l u s h w i t h a f l a t surface. The sample o f rock material i s also prepared w i t h a f l a t s u r f a c e and placed d i r e c t l y a g a i n s t t h e h a l f space probe t o measure c o n d u c t i v i t y .A d e t a i l e d d e s c r i p t i o n o f t h e h a l f space probe i s performed by Sass e t a l (1984b). S i m i l a r measurement e q u i p m e n t i s c o m m e r c i a l l y a v a i l a b l e u n d e r t h e name Q u i c k T h e r m a l Cond u c t i v i t y M e t e r (QTM). T h e QTM-method 1s e v a l u a t e d by S a s s e t a l (1984a).
2.2
Theory
The i n f i n i t e l i n e source t h e o r y f o r t h e s i n g l e - p r o b e and t h e m u l t i probe method described i n r e p o r t no. 1 i s v a l i d , provided t h e following conditions a r e met: c o n s t a n t h e a t i n g power homogeneous t e m p e r a t u r e d i s t r i b u t i o n measurement i n f i n i t e line source.
5
at the start of the
6
If
heat
and
starts
the
to
distance
T
=
j "
^
at
t=0,
the
X
thermal
power,
conductivity,
W/(m,K)
K
=
thermal
diffusivity,
m^/s
r
=
radial
- n o . l , eq
(1)
i s derived
eq.
u
a
=
lin-log
u
Euler's
i s small
T
In
B-[-ln
=
(1)
(long
B-(-1n
-
the
time
u
diagram,
in
figure
slope
of
the
asymptote
eq.
(3)
the
1n
figure
2.
V
-
from
the
following
s:" n=1
constant=
shown
From
t
m
n =
time
W/m
distance,
Rewriting
Y
the
t
thermal
T
at
^
=
report
T
- f - ^ x
A. =
equation.
If
temperature
(1)
u
4
In
produced
B-E^(u)
E, ( u ) =
q
be
r i s :
-
or
heat is
conduction obtained.
n ]
(2)
n-n! 0.5772156649....
small
r)
eq.
(2)
can
be
simplified:
v).
eq.
The
general
expression
(3)
(3)
results
in
a
straight
thermal
conductivity
and
thermal
the
diffusivity
can
be
can
be
diffusivity
determined
by
line.
This
evaluated from
the
using
T=0
is
from
the
intercept.
and
2:
= t -2.^46 0
However, the figure
2,
due
the
to
a
determination small
error
logarithmic
in
of the
K
I Srather slope
scale.
6
will
uncertain. have
a
As
strong
I s seen effect
in on
t^
7
T
— —^—-.—if-1n(4K/r^) Figure
2.
l n ( t ) * Y
Temperature r i s e v s 1 n ( t ) f o rt h e case where no thermal contact resistance exists.
When u s i n g t h e m u l t i - p r o b e m e t h o d i n i n s i t u m e a s u r e m e n t s , t h e d i s t a n ce r i s n o r m a l l y b e t w e e n 0 . 0 5 a n d 0 . 2 m d e p e n d i n g o n t h e t y p e o f t e s t m a t e r i a l . However, t h e s i m p l i f i e d eq. ( 3 ) can n o t be used w i t h i n a reasonable measuring time. The procedure used t o c a l c u l a t e t h e thermal c o n d u c t i v i t y and t h e t h e r m a l d i f f u s i v i t y i s based on eq. ( 2 ) . By m i n i m i z i n g t h e f u n c t i o n f ( T , \, x, t ) , w e d e t e r m i n e t h e v a l u e s f o r X. a n d
f(T.
X., X . t ) = [ T ( X . , X , t ) - T ^ j j g ] ^
^obs
~
temperature
A d e t a i l e d d e s c r i p t i o n o f t h e procedure i s given i n report no. 1 . I f r can be d e t e r m i n e d e x a c t l y , a good v a l u e f o r t h e t h e r m a l d i f f u s i v i t y can be d e r i v e d . B l a c k w e l l ( 1 9 5 4 ) h a s d e v e l o p e d a n e q u a t i o n t h a t i n v o l v e s b o t h t h e mat e r i a l i n t h e probe and a p o s s i b l e contact r e s i s t a n c e a t t h e probe surface. Blackwell also presented a long-time solution (see report no. 1 ) . I fu i s s m a l l enough t h e l o n g - t i m e s o l u t i o n can be s i m p l i f i e d : T = B - ( - l n u - Y + 2-A./(rH)) H = t h e conductance a t t h e probe surface,
The l o n g t i m e s o l u (3) p r o v i d e d t h a t sistance i s low). c i t y o f t h e probe.
(5) W/(m^,K)
t i o n s i m p l i f i e d t o eq. ( 5 ) i s transformed into eq. t h e contact conductance i s high (thermal contact r e Equation ( 5 ) i s n o t Influenced by t h e thermal capaThis e f f e c t i s i n t h e higher terms o f eq. ( 5 ) , see
7
8
eq. 5 . 2i n r e p o r t no. 1 . I n c l u d i n g t h e t h e r m a l c a p a c i t y o f t h e p r o b e i n t h e e x p r e s s i o n makes i t p o s s i b l e t o use a s h o r t e r measurement period, e s p e c i a l l y f o r f i e l d probes w i t h a large diameter.
T
A T DEPENDS ON THE CONTACT RESISTANCE ( 1 / H )
-ln(4x/r^) F i g u r e 3.
+ y
Temperature r i s e v s l n ( t )w i t h a thermal resistance at the probe surface.
contact
As c a n b e s e e n i n f i g u r e 3 , a c o n t a c t r e s i s t a n c e a t t h e p r o b e s u r f a c e o n l y r e s u l t s i n p a r a l l e l movement o f t h e slope when u s i n g t h e s i n g l e probe method. However, the time u n t i l a l i n e a r r e l a t i o n s h i p emerges i s i n c r e a s e d . The t h e r m a l c o n t a c t r e s i s t a n c e i s t h e o r e t i c a l l y presumed to be a thin skin with vanishing thermal capacity. S e v e r a l a u t h o r s have t r i e d t o c o m p u t e b o t h c o n d u c t i v i t y and d i f f u s i v i t y t a k i n g i n t o account the contact r e s i s t a n c e . These e f f o r t s were based o n e i t h e r a p p r o x i m a t i v e s o l u t i o n s , ( B l a c k w e l l , 1954) o r curve f i t t i n g p r o c e d u r e s (Beck e t a l , 1956). I n both cases, t h e determinat i o n o f c o n d u c t i v i t y was r e l i a b l e b u t t h e d i f f u s i v i t y r e s u l t w a s s t r o n g l y d e p e n d e n t o n t h e c o n t a c t r e s i s t a n c e . L i n and L o v e ( 1 9 8 5 ) , have analyzed i n s i t u thermal p r o p e r t y d e t e r m i n a t i o n s i n cased boreholes. As a n e x a m p l e o f p r a c t i c a l d e t e r m i n a t i o n s . B e c k ( 1 9 7 1 ) has u s e d B l a c k w e l l ' s long time s o l u t i o n t o estimate the thermal c o n d u c t i v i t y of rock from i n s i t u measurements i n cased boreholes. F r o m a n e x p r e s s i o n g i v e n i n C a r s l a w and J a e g e r ( 1 9 5 9 ) , L i n d q v i s t ( 1 9 8 3 ) d e r i v e d a n i n t e g r a l s o l u t i o n and has u s e d t h i s t o d e t e r m i n e the thermal c o n d u c t i v i t y i n lake bottom sediments using a large diameter probe. This s o l u t i o n excludes the contact resistance at the probe s u r f a c e , which i s probably a v e r y good assumption c o n s i d e r i n g the method of a p p l i c a t i o n .
8
9
K r i s t i a n s e n ( 1 9 8 2 ) s o l v e d t h e i n i t i a l e x p r e s s i o n g i v e n by Blacl25
(u25 (dashed line).
T o p r e v e n t a n e f f e c t o f t h e s a m p l e b o u n d a r y , a 2 cm d i a m e t e r c i e n t f o r a c l a y s a m p l e ( K = 3 . 2 - 1 0 ' ^ m^/s, t = 1 5 0 s, r.=5-10"'* a n d a 4 cm d i a m e t e r i s f o r a r o c k m a t e r i a l (K=10'*m^/s).
Not
only
is m)
suffi-
conduction
O t h e r t y p e s o f t h e r m a l t r a n s p o r t can o c c u r , p a r t i c u l a r l y i n s o i l . E x a m p l e s a r e v a p o r d i f f u s i o n , r a d i a t i o n and c o n v e c t i o n . T h e s e t y p e s o f t h e r m a l t r a n s p o r t a r e d e s c r i b e d i n r e p o r t n o . 3. To a v o i d unnecessary i n f l u e n c e on t h e t h e r m a l c o n d u c t i v i t y . I t i s i m p o r t a n t t o p e r f o r m t h e measurement under c o n d i t i o n s s i m i l a r t o real c o n d i t i o n s . For example, t o m i n i m i z e v a p o r d i f f u s i o n and r a d i a t i o n i n a l a b o r a t o r y m e a s u r e m e n t , t h e t e m p e r a t u r e can be l o w e r e d .
A potential
error
i n water-saturated coarse
material
L a b o r a t o r y measurements on sand r e v e a l e d a tendency t o w a r d s a l o w e r thermal c o n d u c t i v i t y at water s a t u r a t i o n , compared t o t h e c o n d u c t i v i t y o b t a i n e d f o r a s l i g h t r e d u c t i o n i n water c o n t e n t . T h i s i s shown f o r s o m e s a m p l e s i n r e p o r t n o . 3, f i g u r e 8 . 9 . T h e m e a s u r e d c o n d u c t i v i t y a t w a t e r s a t u r a t i o n can be 1 0 - 1 5 % l o w e r . T h i s f i n d i n g i s i n c o n t r a s t t o w h a t was e x p e c t e d , s i n c e w a t e r i s a much b e t t e r c o n d u c t o r t h a n a i r .
19
20
One e x p l a n a t i o n c a n be d e c r e a s e d c o n t a c t w i d t h b e t w e e n t h e m i n e r a l g r a i n s . Because o f a low v e r t i c a l pressure i n t h e upper p a r t o f t h e sample, t h e grains a r e " f l o a t i n g " I n t h e water. Since t h e thermal cond u c t i v i t y p r o b e m a i n l y u s e d i s o n l y 4 cm l o n g , t h e m e a s u r e m e n t i s m a d e o n l y 2 cm b e l o w t h e s a m p l e s u r f a c e . Another p o s s i b i l i t y i s a sort o f thermal contact resistance between p r o b e and sand. T h e p r o b e d i a m e t e r i s a b o u t t h e same as t h e g r a i n s i n a medium grained sand. T h e r e f o r e , an o v e r r e p r e s e n t a t i o n o f water close to t h e probe i s possible. This results i n a longer measuring time before t h e probe can sence t h e real c o n d u c t i v i t y i n t h e sand. I f t h i s I s n o t t a k e n I n t o a c c o u n t , t h e t h e r m a l c o n d u c t i v i t y may be u n d e r estimated. T h e s e p o t e n t i a l e r r o r s may be r e d u c e d by u s i n g a s l i g h t l y l a r g e r p r o b e and e x e r t i n g a s l i g h t p r e s s u r e on t h e sample s u r f a c e . The f i r s t e x p l a n a t i o n seems t o be t h e most p r o b a b l e t o t h e above m e n t i o n e d o b s e r v a tion. T h e r e i s a p o s s i b i l i t y t h a t t h e measurements a l s o could be i n f l u e n c e d by a n o n - r a d i a l heat f l o w , d i s c u s s e d above. However, t h e s e c o n d i t i o n s can n o t account f o r such a b i g d i f f e r e n c e I n thermal c o n d u c t i v i t y .
20
21
3.
THEORETICAL METHODS FOR DETERMINING THERMAL PROPERTIES
3.1
Introduction
The t h e r m a l c a p a c i t y o f r o c k a n d s o i l c a n b e e a s i l y computed f r o m a volume i n t e g r a t i o n . The thermal c a p a c i t i e s o f d i f f e r e n t m i n e r a l s a r e tabulated i n report no. 2 . The t h e r m a l c o n d u c t i v i t y o f c o m p o s i t e m a t e r i a l s , such a s s o i l a n d r o c k , i s much m o r e c o m p l i c a t e d t o c a l c u l a t e . P a p e r n o . 1 i n c l u d e s a n overview o f d i f f e r e n t approaches t o t h e s u b j e c t . The bounds suggested b y H a s h i n a n d S h t r i k m a n ( 1 9 6 2 ) a r e c o n s i d e r e d t o be t h e b e s t b o u n d s f o r t h e t h e r m a l c o n d u c t i v i t y o f a n i s o t r o p i c c o m p o s i t e m a t e r i a l . H o r a i a n d Simmons ( 1 9 6 9 ) s u g g e s t e d t h e mean o f H a s h i n a n d S h t r i k m a n ' s , u p p e r (x.^^) a n d l o w e r (A.^) b o u n d a s a n e f fective thermal conductivity:
x.g
= {X^ + \^)/2
"
(l-^max^ax) +
^1
( 1 4 )
\ \
^min '^"^min%in'
^max =
^ )
'^min =
\
^max "
^max'_^
''min ^ '•^ * - m i n ' '^max = ^ ^ " ^ - W " ^
^ i n
=^^i
* W ' ^
O^-^min'"'' * ^min'"^
^'N'^^in' v^= v o l u m e
fraction
D i f f e r e n t t y p e s o f d i l u t e s u s p e n s i o n t h e o r i e s have been s u g g e s t e d , ( M a x w e l l , 1 8 9 1 and R a y l e i g h , 1 8 9 2 ) . A disadvantage o f d i l u t e suspension t h e o r i e s i s that they a r e o n l y v a l i d i f t h e volume f r a c t i o n o f one o f t h e c o m p o n e n t s i s much l o w e r t h a n 1 i n a t w o - p h a s e s y s t e m .
21
22
The g e o m e t r i c mean h a s o f t e n been u s e d a s a good a p p r o x i m a t i o n o f t h e e f f e c t i v e thermal c o n d u c t i v i t y o f rocks and s o i l s . However, compared t o t h e other e q u a t i o n s mentioned, t h emethod lacks a r e l i a b l e physical background.
G
= n
K. i
(15)
1
1=1
The s e l f - c o n s i s t e n t a p p r o x i m a t i o n ( h e r e a f t e r named SCA) o f a 2-phase m a t e r i a l was suggested by Bruggeman f X S S f ) . T h i s has l a t e r been redeveloped f o r n-phase m a t e r i a l . The method assumes each g r a i n t o be surrounded by a u n i f o r m medium w i t h t h ee f f e c t i v e thermal conductivity ( f i g u r e 8 ) . I n a n-phase m a t e r i a l , t h ee f f e c t i v e thermal conductivity can be e s t i m a t e d f r o m t h e f o l l o w i n g e x p r e s s i o n b y a number o f I t e r a tions: 1
"
"i
'i=i"«-i'-VS m = The dimensionality
Figure
8.
o f t h e problem
A r e a l composite medium and t h e s e l f - c o n s i s t e n t approximat i o n w i t h an e f f e c t i v e medium s u r r o u n d i n g t h e g r a i n .
D a g a n ( 1 9 7 9 ) c o m p a r e d t h e r a t i o b e t w e e n t h e SCA a n d t h e g e o m e t r i c mean equation applied t o hydraulic conductivity. A very i n t e r e s t i n g observ a t i o n made t h e r e b y w a s t h a t t h e g e o m e t r i c m e a n c o i n c i d e d w i t h SCA f o r 2 dimensions when t h ec o n d u c t i v i t y i s l o gnormally d i s t r i b u t e d , s e e f i g u r e 9. T h e g e o m e t r i c mean i s t h u s a s s o c i a t e d w i t h t h e r m a l transport in 2 dimensions.
22
23
s t d - dev. F i g u r e 9.
The r a t i o between d i f f e r e n t equations and t h e geometric mean v e r s u s t h e s t a n d a r d d e v i a t i o n f o r a l o g - n o r m a l d i s t r i b u t e d c o n d u c t i v i t y . A r i t h m e t i c a l ( 1 ) and harmonic ( 6 ) mean. Hashin and S h t r i k m a n ' s upper ( 2 ) and lower ( 5 ) b o u n d s . 2 - d i m SCA a n d 3-d1m SCA CM.
H 3.2
3
Application t o rock
I n r e p o r t no. 2 t h e mean o f Hashin and S h t r i k m a n ' s bounds i s a p p l i e d to t h e thermal c o n d u c t i v i t y o f rock. Measured c o n d u c t i v i t y values were compared w i t h c a l c u l a t e d values, estimated from t h e m i n e r a l content. The t h e r m a l c o n d u c t i v i t y v a l u e s w e r e d e r i v e d f r o m o u r own m e a s u r e ments, r e s e a r c h work on geothermal energy a t CJialmers U n i v e r s i t y o f Technology and f r o m measurements performed by Horai and B a l d r i d g e (1972b). The r e s u l t s showed g e n e r a l l y good agreement between measured and c a l c u l a t e d c o n d u c t i v i t i e s . However, t h e f i n d i n g s o f Horai and B a l d r i d g e s h o w e d a d i s c r e p a n c y o f a b o u t 103! b e t w e e n m e a s u r e d a n d c a l c u l a t e d v a l u e s . Horai and B a l d r i g e concluded t h a t eq. ( 1 4 ) o v e r e s t i m a t e d t h e t h e r m a l c o n d u c t i v i t y b y 5%. The bounds o f Hashin and S h t r i k m a n seem t o be t h e best e s t a b l i s h e d b o u n d s f o r a m a c r o s c o p i c a l l y h o m o g e n e o u s a n d i s o t r o p i c m a t e r i a l . However, t h e s u g g e s t i o n by Horai and Simmons (1969) t o e s t i m a t e t h e e f f e c t i v e c o n d u c t i v i t y f r o m t h e mean v a l u e o f t h e bounds, i s n o t necessarily true. The s e l f - c o n s i s t e n t a p p r o x i m a t i o n has a r e l i a b l e p h y s i c a l background. I n o r d e r t o e x a m i n e t h e a c c u r a c y o f t h e m e t h o d , we c a n c a l c u l a t e t h e thermal c o n d u c t i v i t y o f c r y s t a l l i n e rock. A comparison w i t h measured v a l u e s i s made i n p a p e r n o . 1 . The no.
c o m p a r i s o n i s p a r t l y based on t h e same m a t e r i a l as u s e d i n r e p o r t 2. T h e m a t e r i a l was s u p p l e m e n t e d w i t h t h e w o r k o f E r i c s s o n (1985)
?3
24
and D r u r y a n d J e s s u p ( 1 9 8 3 ) . B o t h SCA a n d t h e m e a n o f H a s h i n a n d Shtrikman's bounds a r ei ngood agreement w i t h measured v a l u e s . T h e o n l y exception i sthe measurements by Horai and Baldridge mentioned e a r l i e r . A p o s s i b l e e x p l a n a t i o n i sdiscussed i ns e c t i o n 3.5.
3.3
A p p l i c a t i o n to s o i l and porous rock
Farouki (1986) provided an i n t e r e s t i n g d e s c r i p t i o n as w e l l as a c r i t i cal review o f a score o f methods f o r c a l c u l a t i n g the thermal conductiv i t y o f m i n e r a l s o i l s . Farouki compared the c a l c u l a t e d values obtained by a p p l y i n g t h e d i f f e r e n t m e t h o d s w i t h t h e m e a s u r e d v a l u e s r e p o r t e d i n the literature. F a r o u k i i n v e s t i g a t e d methods by Johansen (1975), de V r i e s (1952, 1963), K u n i i a n d S m i t h ( 1 9 6 0 ) , Gemant ( 1 9 5 0 ) . K e r s t e n ( 1 9 4 9 ) , Woodside and M e s s m e r ( 1 9 6 1 ) a n d McGaw ( 1 9 6 9 ) . He f o u n d t h a t , i n g e n e r a l , J o h a n sen's method provided the best agreement w i t h measured values, even i f o t h e r methods, f o r instance de V r i e s ' , were p r e f e r a b l e i nc e r t a i n cond i t i o n s . T h e method b y de V r i e s i sbased o n M a x w e l l ' s (1891) t h e o r y extended t oellipsoidal inclusions. In t h e method b y Johansen, t h e g e o m e t r i c mean i sused t o c a l c u l a t et h e thermal conductivity o f water-saturated soil. I n the unsaturated state, the c o n d u c t i v i t y i sc a l c u l a t e d by i n t e r p o l a t i n gbetween a semie m p i r i c a l e q u a t i o n d e s c r i b i n g t h e d r ys t a t e a n d t h e w a t e r - s a t u r a t e d s t a t e . T h e method by Johansen i sdescribed i n r e p o r t n o . 3. T h e method was e a r l y adopted by the Department o f Geology, Chalmers University o f Technology. Johansen's method i sused i n r e p o r t n o . 3 i norder t o compare 9 0 0 cond u c t i v i t y measurements on soil samples w i t h calculated values. A f t e r an e x t e n s i v e c o r r e l a t i o n , i n c l u d i n g c h a n g e s i n t h e c o n s t a n t s o f J o h a n sen's e x p r e s s i o n , good agreement was achieved when d e a l i n g w i t h mineral soils. Johansen's method has a big advantage i n i t s s i m p l i c i t y . C a l c u l a t i o n is e a s i l y performed i nthe w a t e r - s a t u r a t e d s t a t e . However, the method does n o ttake i n t o account the vapor d i f f u s i o n which i ss u b s t a n t i a l i n h i g h p o r o u s m e d i a a l r e a d y a t 20''c. T h e m e t h o d i s n o t a p p l i c a b l e t o peat and does n o ti n c l u d e v a r i a t i o n s i n the c o n d u c t i v i t y o f t h e m i n e r a l g r a i n s i n t h e d r ys t a t e . T h e l a t t e r h a s s m a l l i n f l u e n c e o n normal p o r o s i t y , b u tbecomes more e s s e n t i a l a t v e r y l o w p o r o s i t i e s . In r e p o r t n o . 3 a good f i t i sachieved between measured a n d c a l c u l a t e d values f o r peat by applying Johansen's method. However, l a t e r research has shown t h a t t h e m e a s u r e m e n t s p r o b a b l y w e r e i n f l u e n c e d b y v a p o r d i f f u s i o n , w h i c h i sw h y t h e good f i t might be j u s t a c o i n c i d e n c e .
24
25
In
order
t o develop
plications
and that
a method
that
resolves
t h e disadvantages
the
self-consistent approximation
The
modified
The
self-consistent approximation
proximity
self-consistent
between
proportion medium,
t h e water
mineral
phase
conductive thermal
passage
more
problems. e
We
p,
defined
and
s o i l .
s t a t i s t i c a l a r e i n
water-saturated
t h e highly
will
conductive
The most thereby
i s a function
grains.
A contact
thermally
reduce I t s
o f t h e
contact
resistance
must
SCA-method.
description
like
and e v a l u a t i o n
no. 1 discusses
distribution
p must
measurements
surface,
p=x-(1-e^)
o f porosity.
Paper
o f t h e contact
t o determine
a t t h e contact
from
size
p
grains
o f t h e size
would
grain
that
a
thickness.
f o rt h e original
t h e pore
a function
is
surrounds
that
, s o l e l y f o rt h e contact
between
factor
is
o r less
t h e mineral
t h e o r e t i c a l treatment
ratio
rock
o f t h e material, which
o f varying
a brief
ap-
method,
o f t h e
SCA-method.
certain ratio,
t o porous
(SCA) assumes
phases
by a f a c t o r
introduced
rock
o f Johansen's
f r a c t i o n s . I n a porous,
layer
no. 1 presents
modified
A
be
i s extended
v i a t h e mineral
between
therefore
A
a
conductivity
resistance
Paper
phase
with
feasible f o rporous
approximation
t h e various
t o t h e volume
i s also
Thus,
p
resistance
some
kind
and t h e grain
I s introduced. i s a function
o f both
f o rt h e porosity
interval
20-95%.
o f p. p
X
porosity
t o t h e contact
means,
diameter.
The factor
a t h e o r e t i c a l determination by empirical
Involves void
i.e. determine t h e
surface
and i s proportional
be determined
o f a
The
conclusion
i s correlated
Outside
and
resistance.
this
t o
Interval,
i s uncertain.
If
t h e heat
flow
dimensional,
a t t h e grain
then
the
influence
the
grain.
o f t h e contact
The harmonic
tion
a t t h e grain
v i t y
represents
named and
contact
t h e harmonic
(figure
contact
can be
phase
1 0 ) , divided
correction
resistance.
assumed
can be used
on t h e thermal
o f t h e mineral
a dimensionless
t h e thermal
surface equation
resistance
mean
contact
mean
conductivity
by t h e grain
factor
The a-values
=
adry
=
c M 1 - p ) A y +
( i / ( p A
g
+ c - ( i - p ) A
a
+
(a),
(17)
{ ^ - c ) { ^ - ^ ) / \ ) ) / x ^ e g
(18)
o f
f r a c -
conducti-
subsequently
f o rt h e
(1-c)(1-p)A^))/A.g
25
t o be 1 calculate
and w a t e r / a i r
t h e d r y state a r e :
a,s a t
t o
saturated
26
X xl x^ x^
= thermal = = =
c o n d u c t i v i t y o f t h e m i n e r a l g r a i n s , W/(m,K) -"w a t e r . W/(m.K) -"d r y a i r , W/(m,K) -"cement i n t h e g r a i n contact r o c k , W/(m,K)
a t porous
p = 1 - 0 . 1 2 8 3 3 - n + 0.064'1-n^ + O . o J g V n ^
f O.IS'
n c c c
= =% = =
porosity fraction r h r sandstone 0.30 limestone 1 soil
Figure
10.
Conditions a tt h e grain contact. The breadth o f t h e section through t h e contact i s represented bya.
For thermal c o n d u c t i v i t y c a l c u l a t i o n s on sedimentary rocks, we i n t r o duce a c o r r e c t i o n f a c t o r c. I t takes i n t o account t h e b e t t e r thermal contact duet o cementation between t h e m i n e r a l g r a i n s a s compared t o s o i l . To c a l c u l a t e f o ra s o i l (c=1) t h e last part o feq. ( 1 7 ) and ( 1 8 ) disappears. The c o n d i t i o n s a t t h e g r a i n c o n t a c t a t d e c r e a s e d w a t e r s a t u r a t i o n a r e a p p r o x i m a t e d w i t h t h e l o g a r i t h m o f t h e degree o fw a t e r s a t u r a t i o n , S^. T h i s i s s u p p o s e d t o b e a r a t h e r g o o d a p p r o x i m a t i o n , s i n c e t h e g r a i n contact i s t h e p a r t o ft h e pore space l a s t drained. «tot = ( A - 1 o g ( S ^ ) ^ l ) ( « 3 3 t - « d r y ' * « d r y A = 0.95 "tot "tot
= "sat = "dry
\ 1 ^ = °
26
'^9'
27
Thus, t h emodified s e l f - c o n s i s t e n tapproximation and 3 p h a s e s c a nb e w r i t t e n a s : K
? v./(2).^ + \ . ) ] - ^ e = 3 " ^ [.^^ 1 e 1
f o r 3 dimensions
(20)
h
= r e l a t i v e humidity (=0 a td r y state, = 1 a twater above t h ew i l t i n g point) A.^ = c o n t r i b u t i o n o f v a p o r d i f f u s i o n t o t h e e f f e c t i v e conductivity o f a i r .
content thermal
The v a p o r d i f f u s i o n c o n t r i b u t i o n a t t h e g r a i n c o n t a c t i s c o n s i d e r e d b y i n t e r p o l a t i n g b e t w e e n a i n c l u d i n g x.^^ I n s t e a d o f x ^ I n e q u a t i o n ( 1 8 ) , ( a ^ g ^ y ) . a n d a i n c l u d i n g x.^, a s u s u a l , f°tot.a'"tot
=
'"tot,v-°tot,a' ' " t o t . a
Ky^Q
= 0 . 2 4 = 10-X.g
'^D
W/(m,K)
X^ = 0 . 2 4 i s o b t a i n e d a t a t e m p e r a t u r e o f a p p r o x i m a t e l y 4o''c. A t t h i s t e m p e r a t u r e , \ h a sv e r y l i t t l e i n f l u e n c e o n x ^ ^ . A t t e m p e r a t u r e s a b o v e 40*'c, a^.^^, i n e q . ( 2 1 ) c a n b e r e p l a c e d b y "tot,V I n a t o t a l l y f r o z e n s t a t e X^ i n e q . ( 1 7 ) i s r e p l a c e d b y A-^^-g. The f u n c t i o n ( A - l o g ( S ^ ) + 1 ) i n eq.(19) i s r e p l a c e d b y due to another form o f drainage. I n f i n e g r a i n e d s o i l i t i s common w i t h u n f r o z e n w a t e r d u e t o a p a r t l y frozen s t a t e . This i s considered by i n t e r p o l a t i n g between t h e thermal conductivity i n t h eunfrozen andfrozen state.
^=^-'Woz- W
(22)
* W
i|i= S h a r e o f u n f r o z e n w a t e r
o ftotal
water
^unfroz"
conductivity i n unfrozen
^froz
conductivity i ntotally
The 1. and res
= Thermal
content state frozen
state
m o d i f i e d s e l f - c o n s i s t e n tapproximation i s evaluated i n paper n o . C a l c u l a t e d values o f s o i l a r e compared w i t h measured a t unfrozen frozen state anda tsaturated, unsaturated andd r y state. The u l t shows good agreement between measured a n d c a l c u l a t e d c o n d u c t i 27
28
v i t y . F a i r l y good agreement i s a l s o achieved saturated with a i r , o i l o r water. Since t h e method i s r a t h e r inconvenient i s u n d e r d e v e l o p m e n t f o r PC-DOS. 3.4
f o r sedimentary
t ouse, a computer
rock,
program
Accuracy
The t h e r m a l c o n d u c t i v i t y o f homogeneous a n d i s o t r o p i c rock i s c o n s i dered t obe determined from t h e mineral content w i t h i n 1 0 %o f accuracy. I f a number o f measurements a n d c a l c u l a t i o n s i s p e r f o r m e d o n d i f f e r e n t samples, t h e margin o fe r r o r should be reduced ( s e e paper n o . 1 ) . I n s a n d s t o n e t h e a c c u r a c y i s g e n e r a l l y + 1 5 % a n d i n l i m e s t o n e ±10% independent o fwhether t h e pore space i s s a t u r a t e d w i t h water, o i l or a i r . In s o i l t h e accuracy depends o n whether t h e m i n e r a l content and s o i l type a r e known. T h e thermal c o n d u c t i v i t y o f a w a t e r - s a t u r a t e d m a t e r i al canbe determined w i t h i n *10% o f accuracy i f t h e mineral content i s known a n dw i t h i n +25% o f accuracy i f t h e m i n e r a l c o n t e n t i s assumed t obe normal according t os o i l type. A t low water content t h e uncertainty i s higher. 3.5
Computing thermal c o n d u c t i v i t y of rock from measurements on p u l v e r i z e d water-saturated samples
Horai a n d Simmons (1969) used a new method t odetermine t h e thermal c o n d u c t i v i t y o fminerals. They measured t h e thermal c o n d u c t i v i t y o fa m i x t u r e o f pulverized mineral andwater using t h e probe method. T h e m e a s u r e d v a l u e w a s s e t equal t o t h e mean o f H a s h i n - S h t r l k m a n ' s bounds ( e q . ( 1 4 ) ) a n d t h e t h e r m a l c o n d u c t i v i t y o f t h e m i n e r a l w a scomputed a t known water c o n d u c t i v i t y a n d p o r o s i t y . T h emethod was a l s o used b y Horai (1971) andwas extended t orock b y Sass e t a l (1971) ( i n s t e a d of u s i n g t h e g e o m e t r i c mean e q u a t i o n ) a n d H o r a i a n d B a l d r i d g e (1972a,b). The thermal c o n d u c t i v i t y values o f di*'ferent minerals (Horai a n d Simmons, 1969. H o r a i , 1971) h a sbeen w i d e l y used. Horai and B a l d r i d g e (1972b) c o m p a r e d i n d i r e c t l y m e a s u r e d v a l u e s o f r o c k w i t h calculated values derived from t h e mineral content. They found a discrepancy o f 1 0 %and suggested t h a t e q . ( 1 4 ) overestimates t h e t h e r m a l c o n d u c t i v i t y b y 5%, s e epaper no.1 . The p o s s i b i l i t y o f m a k i n g a s y s t e m a t i c e r r o r I na p p l y i n g eq. ( 1 4 ) should be considered, since t h e equation i s used three times i nt h e comparison: 1) C a l c u l a t i n g t h e c o n d u c t i v i t y o f t h e s o l i d r o c k phase u s i n g needle probe measurements o n a m i x t u r e o f p u l v e r i z e d rock andwater a n d at a known water c o n d u c t i v i t y .
28
29
2) C a l c u l a t i n g t h e e f f e c t i v e thermal c o n d u c t i v i t y o f t h e rock the c o n d u c t i v i t y o f t h e m i n e r a l s .
from
3) T h e t h e r m a l c o n d u c t i v i t i e s a p p l i e d t o t h e m i n e r a l s i n 2 ) were determined by Horai (1971) i n an e q u i v a l e n t way as i n 1 ) . A s y s t e m a t i c o v e r e s t i m a t i o n o f t h e c o n d u c t i v i t y b y 5% u s i n g e q u a t i o n ( 1 4 ) , as s u g g e s t e d by H o r a i and B a l d r i d g e , w o u l d mean t h e f o l l o w i n g in a comparison e q u i v a l e n t t o t h e one mentioned above: 1) T h e m e a s u r e d r o c k c o n d u c t i v i t y i s u n d e r e s t i m a t e d by 5%. 2) The c a l c u l a t e d rock c o n d u c t i v i t y i s o v e r e s t i m a t e d by 5Z. 3 ) T h e m e a s u r e d m i n e r a l c o n d u c t i v i t y i s u n d e r e s t i m a t e d b y S%. The t o t a l
deviation will
t h e n be - 5 % , d e s p i t e
thecorrection.
Another possible explanation f o rt h e discrepancy i s that t h e e r r o r i n eq. ( 1 4 ) i s n o t c o n s t a n t , b u t depends both on volume f r a c t i o n s and the r a t i o between t h e thermal c o n d u c t i v i t i e s under c o n s i d e r a t i o n . E a r l i e r i n c h a p t e r 3 i thas been shown t h a t a t h e r m a l c o n t a c t r e s i s tance f o rp o r o u s m a t e r i a l s must be i n t r o d u c e d i n t o t h e s e l f - c o n s i s t e n t a p p r o x i m a t i o n . T h e d e v i a t i o n b e t w e e n SCA c o r r e c t e d f o r s o i l a n d the mean o f Hashin-Shtrikman's bounds i s drawn up i n f i g u r e 11 f o r a 2-phase m a t e r i a l as a f u n c t i o n o f t h e volume f r a c t i o n o f t h e water phase and t h e r a t i o between t h e e f f e c t i v e c o n d u c t i v i t y and t h e cond u c t i v i t y o f t h e water phase. I n can be seen i n f i g u r e 11 t h a t t h e e r r o r o b t a i n e d w i t h e q u a t i o n (14) i s h i g h l y dependent on both volume f r a c t i o n and on t h e c o n d u c t i v i t y r a t i o . The 166 d i f f e r e n t d e t e r m i n a t i o n s o f t h e c o n d u c t i v i t y o f d i f f e r e n t m i n e r a l s , p e r f o r m e d b y H o r a i ( 1 9 7 1 ) , w e r e made a t a p o r o s i t y o f approx. 30-35% f o r t h e water-mineral m i x t u r e . The f i g u r e s f o r the p o r o s i t i e s o f i n d i v i d u a l samples a r e no l o n g e r a v a i l a b l e ( K i - i t i Horai 1988, personal communications). Thus t h e c o n d u c t i v i t y o f t h e m i n e r a l s i s underestimated by approx. 0-10%, depending on t h e conduct i v i t y r a t i o between t h e m i n e r a l and water. A p o s s i b l e e x p l a n a t i o n f o r t h e discrepancy i n Horai and Baldridge's (1972b) comparison o f thermal c o n d u c t i v i t y o f rock i s discussed i n paper no. 1 . However, t h e m e t h o d d i s c u s s e d i n w h i c h m e a s u r e m e n t s a r e made o n a s o l i d - w a t e r m i x t u r e , seems t o be a d e q u a t e and r e l i a b l e i f an a c c u r a t e equation i s used f o r t h e c a l c u l a t i o n o f t h e s o l i d phase.
29
30
A ( ,
Figure
11.
HS
mean - SCA HS mean
soil)
D e v i a t i o n b e t w e e n t h e m o d i f i e d SCA, e q . ( 2 0 ) , a n d t h e mean v a l u e o f H a s h l n - S h t r i k m a n ' s bounds, eq. ( 1 4 ) .
30
31
4.
THERMAL PROPERTIES OF ROCKS AND
SOILS
4.1
D i f f e r e n t thermal transport mechanisms
T h e r m a l e n e r g y c a n be t r a n s p o r t e d i n e a r t h m a t e r i a l b y c o n d u c t i o n , r a d i a t i o n , c o n v e c t i o n a n d vapor a n dwater d i f f u s i o n , s e e f i g u r e 12 and r e p o r t n o . 3 . A t ambient t e m p e r a t u r e s , t h e r m a l c o n d u c t i o n i s a b s o l u t e l y dominant ( f o r c e d c o n v e c t i o n d u e t ogroundwater movements disregarded). A t higher temperatures and intermediate degrees o f s a t u r a t i o n , vapor d i f f u s i o n c o n t r i b u t e s more a n dmore t ot h e e f f e c t i v e thermal c o n d u c t i v i t y o f a i r (^gv^^a * ''^ V r i e s , 1 9 5 2 ) . A t a b o u t 60°C, t h e v a p o r d i f f u s i o n i s o f t h e s a m e s i z e a s t h e t h e r m a l c o n d u c t i v i t y o fw a t e r . A t h i g h e r t e m p e r a t u r e s , t h i s means t h a t t h e e f f e c t i v e thermal c o n d u c t i v i t y c a nbe higher a t i n t e r m e d i a t e s a t u r a t i o n compared t o f u l l s a t u r a t i o n ( s e er e p o r t no. 3 and paper no. 1 ) . R a d i a t i o n c a nbe o f importance i ncoarse m a t e r i a l s a n dunder r a t h e r dry c o n d i t i o n s . Natural convection can i n f l u e n c e t h e thermal process at high temperature gradients. Vapor d i f f u s i o n c a na l s o be i n v o l v e d i n d r a s t i c changes i n t h e thermal p r o p e r t i e s . High temperatures, g r a d i e n t s and thus heat f l o w s , a r e common i n b u r i e d d i s t r i c t h e a t i n g p i p e s a n d e l e c t r i c t r a n s m i s s i o n cables. Such c o n d i t i o n s i n sand induce a m o i s t u r e movement i n t h e d i r e c t i o n o f f a l l i n g temperature. This throws t h e system o u to f e q u i l i b r i u m a n d induces a m o i s t u r e movement i n t h e o p p o s i t e d i r e c t i o n , i n the direction o f least water content.
1. C O N D U C T I O N I N SOLID A N D LIQUID 2. C O N D U C T I O N I N A I R 3. R A D I A T I O N P A R T I C L E TO P A R T I C L E A. V A P O U R
DIFFUSION
5. C O N V E C T I O N I N P O R E A I R
Figure
12. Different (1975)
thermal t r a n s p o r t mechanisms, a f t e r
31
Johansen
32
I f t h e heat f l o w and temperature pass a c r i t i c a l p o i n t , unique f o r every s o i l t y p e , t h e m o i s t u r e movement due t o m o i s t u r e g r a d i e n t cannot compensate f o rt h e increased moisture movement under temperat u r e g r a d i e n t . Thus, t h e zone nearest t h e cable/pipe i s d r i e d o u t and the t h e r m a l c o n d u c t i v i t y d r a s t i c a l l y lowered. T h i s can cause a thermal breakdown i n a buried transmission cable. For a d d i t i o n a l det a i l s , s e e r e p o r t n o . 3. It i s possible t o Investigate t h e thermal as b a c k f i l l , b y t h e p r o b e m e t h o d .
4.2
stability of a soil,
used
Influence o f various characteristics
Mineral
content
D i f f e r e n t mineral d i s t r i b u t i o n s can result i n t o t a l l y d i f f e r e n t thermal c o n d u c t i v i t i e s o f both rock and s o i l . However, m i n e r a l s have a much h i g h e r e f f e c t on t h e t h e r m a l c o n d u c t i v i t y o f a rock t h a n on t h e c o n d u c t i v i t y o f s o i l . T h e t h e r m a l c o n d u c t i v i t y o f some common rock f o r m i n g m i n e r a l s i s shown i n t a b l e 1 . A more complete t a b l e can be f o u n d i n r e p o r t n o . 2. As c a n b e seen, q u a r t z h a s 3-4 t i m e s h i g h e r c o n d u c t i v i t y t h a n o t h e r common m i n e r a l s . A s s u m i n g i s o t r o p i c a n d homogeneous c o n d i t i o n s , t h e thermal c o n d u c t i v i t y o f c r y s t a l l i n e rock can be c a l c u l a t e d f r o m t h e m i n e r a l c o n t e n t w i t h r a t h e r h i g h accuracy. T h i s I s -shown i n p a p e r n o . 1 a n d c h a p t e r 3. T h e g e o m e t r i c mean e q u a t i o n underestimates t h e thermal c o n d u c t i v i t y . However, t h e equation can be used as a s i m p l e g u i d e l i n e i f a m u l t i - p h a s e m a t e r i a l can be approximated w i t h a 2-phase m a t e r i a l o f quartz and remaining miner a l s . A c c o r d i n g t o " e p o r t n o . 2, t h e t h e r m a l c o n d u c t i v i t y o f r e m a i n i n g m i n e r a l s i s t h e n i n c r e a s e d compared t o t h e t r u e v a l u e . As a mean v a l u e , 2.4 W/(m,K) i s s u g g e s t e d .
Table
1.
T h e r m a l c o n d u c t i v i t y o f some common r o c k (Moral, 1971)
Mineral Quartz Microcline Plagioclase
(dependent on t h e f r a c t i o n of a n o r t h i t e )
Biotite Muscovite
forming
Conductivity 7.7 2.5 1.9
minerals.
(W/(m,K))
(mean
value)
2.0 2.3
Very l i t t l e has been p u b l i s h e d on q u a n t i t a t i v e m i n e r a l i n v e s t i g a t i o n s in s o i l . I n order t o compile i n f o r m a t i o n on t h i s s u b j e c t , an examinat i o n study has been I n i t i a t e d w i t h i n t h e p r o j e c t (Abrahamson, 1984).
32
33
The t h e o r y i s t h a t t h e m i n e r a l c o n t e n t ( p r i m a r i l y t h e q u a r t z c o n t e n t ) w o u l d d i f f e r d e p e n d i n g o n t h e g r a i n s i z e . I n r e p o r t no. 3, d i a g r a m s have been e s t a b l i s h e d showing the v a r i a t i o n s I n quartz content w i t h g r a i n s i z e f o r d i f f e r e n t s a m p l e s and s o i l t y p e s . No i n f o r m a t i o n a b o u t s a n d was f o u n d and a n i n v e s t i g a t i o n e n s u e d . C o n v e n t i o n a l m i c r o s c o p e t e c h n i q u e was u s e d why n o r e s u l t was p o s s i b l e t o o b t a i n f o r p a r t i c l e s i z e s l e s s t h e n f i n e s a n d . R e s u l t s f r o m c l a y , s a n d and t i l l h a v e been s u m m a r i z e d i n f i g u r e 13. F o r sand, t h e g r a i n s a r e n o t monomineralic a t g r a i n s i z e s l a r g e r t h a n 1 - 2 mm. T h i s p a r t o f t h e d i a g r a m m e s h o w s an a s s u m e d q u a r t z c o n t e n t o f t h e b e d r o c k . CLAY
Figure
13.
SILT
SAND
GRAVEL
V a r i a t i o n r a n g e s o f q u a r t z due t o t h e g r a i n s i z e f r a c t i o n . R e f e r e n c e : See r e p o r t no. 3.
The q u a r t z c o n t e n t can b e a p p r o x i m a t e l y c a l c u l a t e d f r o m t h e s i z e d i s t r i b u t i o n c u r v e and f i g u r e 1 3 o r r e p o r t no. 3 .
grain
Temperature The t h e r m a l c o n d u c t i v i t y o f c r y s t a l l i n e r o c k d e c r e a s e s w i t h I n c r e a s i n g t e m p e r a t u r e , a p p r o x i m a t e l y lOZ/IOO^C. The t h e r m a l c a p a c i t y I n c r e a s e s a p p r o x i m a t e l y b y 103! a t a t e m p e a t u r e I n c r e a s e o f 100°C. More e x a c t v a l u e s can b e o b t a i n e d f r o m r e p o r t no. 2 , d i f f e r e n t handbooks or a r e v i e w b y Heuze (1983). I n w a t e r - s a t u r a t e d s o i l o r p o r o u s r o c k , a t e m p e r a t u r e d r o p b e l o w 0°C r e s u l t s i n t o t a l l y d i f f e r e n t t h e r m a l p r o p e r t i e s . T h i s I s due t o t h e d i f f e r e n t p r o p e r t i e s o f w a t e r and i c e a s can b e s e e n I n t a b l e 2 .
33
34
Table
2.
\e W/(m,K) 0.57
Physical properties o f water ^
^ i c e
W/(m,K)
MJ/dn^.K)
2.1
4.16
andI c e .
^
^w
MJ/(m^.K)
^\ce
hJ/in^
2.2
333.5
kg/m^ 1000
kq/m^ 917
However, e s p e c i a l l y i n f i n e grained s o i l s , a l lwater does n o t f r e e z e a t O^C. T h e a m o u n t o f u n f r o z e n w a t e r a t a c e r t a i n t e m p e r a t u r e b e l o w 0°C i s r e l a t e d t o t h e w a t e r r e t e n t i o n c a p a c i t y ( p F - c u r v e ) o f t h e s o i l . An a t t e m p t t o r e l a t e t h e u n f r o z e n w a t e r a t d i f f e r e n t t e m p e r a t u r e s i n d i f f e r e n t s o i l t y p e s i s made i n r e p o r t n o . 3. The t h e r m a l c a p a c i t y calculated from
(C) and t h e l a t e n t
'^ = e d - [ V ^ ' V ^ 1 c e ' " - « u ' J
heat
o f f u s i o n (1) can be
'23)
1 = Q^-3.335-10^-(5Wjj/5T)dT
(24)
C c w w^
= t h e r m a l c a p a c i t y , J/(m^,K) = thermal capacity, J/(Kg,K) (g = g r a i n , w = water) = water r a t i o = unfrozen water ratio = d r y d e n s i t y , kg/m^ 1 = l a t e n t h e a t o f f u s i o n , J/m^ (6w^/6T)
= change
Porosity
and
i n unfrozen water
ratio
due t o temperature
pressure
The p o r o s i t y o f a c r y s t a l l i n e rock I s l o w , i n general, l e s s than 1 % . A l a r g e p a r t o f t h e pore space i s i n t h e form o f micro f i s s u r e s . Walsh and Decker (1966) found e x p e r i m e n t a l l y a very small pressure dependence on t h e t h e r m a l c o n d u c t i v i t y p r o v i d e d t h e rock sample was water-saturated. I n d r y rock, however, a r a t h e r strong pressure dep e n d e n c e was o b s e r v e d . T h e w o r k shows how i m p o r t a n t i t i s t o s a t u r a t e samples w i t h water before thermal c o n d u c t i v i t y measurements a r e carried out. Walsh and Decker (1966) suggested t h a t t h e i n f l u e n c e o f micro f i s s u r e s on t h e r m a l c o n d u c t i v i t y o f c r y s t a l l i n e rock could be c a l c u l a t e d using t h e lower l i m i t o f Hashin and Shtrikman's (1962) b o u n d s , s e e c h a p t e r 3. S c h a r l i and Rybach (1984) v e r i f i e d t h i s by measurements and c a l c u l a t i o n on f i v e samples under d r y and w a t e r - s a t u r a t e d c o n d i t i o n s . S c h a r l i and Rybach f o u n d v e r y good r e l a t i v e agreement, w h i l e t h e ab-
34
35
s o l u t e v a l u e s had a l o w e r a c c u r a c y . A t as l o w p o r o s i t i e s as 0 . 8 % , a d e c r e a s e i n t h e r m a l c o n d u c t i v i t y o f 253! was f o u n d i n t h e d r y s t a t e compared w i t h t h e w a t e r - s a t u r a t e d s t a t e . They a l s o suggested t h e f o l lowing expression t o c a l c u l a t e the p o r o s i t y (n) from measurements i n t h e d r y (^-^ip^) a n d w a t e r - s a t u r a t e d s t a t e (J^-gg^.):
n = 8 • l^^^y^ Nat
%
*-dry
(25)
The c a l c u l a t e d p o r o s i t y had an a c c u r a c y o f a b o u t
10%.
The t h e r m a l c o n d u c t i v i t y o f s o i l s d e c r e a s e s w i t h i n c r e a s e d p o r o s i t y . T h e i n f l u e n c e o n a w a t e r - s a t u r a t e d s o i l c a n be a p p r o x i m a t e l y c a l c u l a t e d f r o m t h e g e o m e t r i c mean e q u a t i o n a s :
= '^g'^""'"C
-l
(3.4)
VLED-MIN
VLED-MAX
Figur 3.5
Asklidl iggorande av harmoniskt och aritmetiskt medelvarde vid ett idealfall da mineralen ligger skiktade helt parallellt.
101
18 Ekv.
(3.2) ar det
lens
varmeledningsfbrmciga
lar.
(n=produktsumma).
bestamning Ekv.
geometriska
hansyn
(3.3) ar e t t aritmetiskt
3.5.
om
Denna metod ger
''^^^
( \ l e d - max
Ekv.
(3.4) ar
man en
( \ l e d - min
Ekv.'(3.1)
har
fasmaterial
ovre
och
medelvarde
( 1 9 7 1 ) och
och
resultat
och
Hashin
med
3 mm)* frSn borrning.
* Borrkaxet maste grovt annars
vara
erhcills
en a n r i k n i n g av m i n e r a l i olika Bestamning
fraktioner.
av mineralsamman-
sattning e l l e r ev matning
i
laboratorium. Porositetsbestamning Varmekonduktivitet Specifik
Metod
C:
varmekapacitet
Borrning av
bergvarmebrunn
K y i med k o n s t a n t e f f e k t i hSlet v i a kylslangsystem i nSgon
vecka.* -** Om v a r d e n
130
f o r fryst
brunnsvatten
onskas
kravs langre
provtid.
Berakning
av v a r m e k o n d u k t i v i t e t
och overgSngsmotstSnd
utifrlin
tid-temp-data.
Metoderna SGU
gSr
naturligtvis
a n v i s n i n g a r v i d de noggrannhetsnivci. man
a t t kombinera
och G e o l o g i s k a i n s t i t u t i o n e n ,
dS
erhSller
olika Metod
metoderna. C f S r nog
pli v a l f r i t t
samt overgSngsmotstlind
skan.
Detta
galler
Metoderna anses
mellan
vara
uppmatta.
131
rSd
ar graderade den
efter
brunnsvagg
under f d r u t s a t t n i n g
satt.
a r b e h j a l p l i g a med
en m e d e l v a r m e k o n d u k t i v i t e t
langd rekt
CTH,
basta hela
och
BSde och efter
eftersom borrhalets
koldbararvat-
a t t energifIddena
ar kor-
68 REFERENSER B a l l i n g , N., K r i s t i a n s e n , J . I . , B e i n e r , V . , P a u l s e n , K.D., R a s m u s s e n , R. & S a x o v , S . , 1 9 8 1 : G e o t h e r m a l m e a s u r e m e n t s a n d s u b s u r f a c e t e m p e r a t u r e m o d e l l i n g i n Denmark. G e o s k r i f t e r No. 16, D e p a r t m e n t o f G e o l o g y , A a r h u s U n i v e r s i t y , Denmark. F r i v i k , P - E . & J o h a n s e n , H., 1 9 7 7 : K a l o r i m e t r i s k e m S l n i n g e r a v s p e s i f i k k varme og u f r o s s e t vann f o r m i n e r a l s k e j o r d a r t e r og organiske m a t e r i a l e r . Slutrapport n r 8 - M a t e r i a l e r s varmet e k n i s k e egenskaper. R a p p o r t n r 75 - F r o s t i j o r d . I n s t i t u t t f o r k j ^ l e t e k n i k k . 7 0 3 4 - T r o n d h e i m - NTH. N o r g e . G o r a n s o n , R.W., 1 9 4 2 : H e a t c a p a c i t y ; h e a t o f f u s i o n . ( S e c t i o n 1 6 o f s p e c i a l p a p e r s No. 3 6 , G e o l o g i c a l S o c i e t y o f A m e r i c a . ) "Handbook o f P h y s i c a l c o n s t a n t s " , e d i t e d by F r a n c i s B i r c h , pp. 2 2 3 - 2 4 2 . Guttman, I . , 1970: S t a t i s t i c a l t o l e r a n c e regims: C l a s s i c a l and b a y e s i a n . No 2 6 o f G r i f f i n ' s s t a t i s t i c a l m o n o g r a p h s a n d courses. London. H a s h i n , Z. & S h t r i k m a n , S . , 1 9 6 2 : A v a r i a t i o n a l a p p r o a c h t o t h e theory o f t h e effective magnetic permeability o f multiphase m a t e r i a l s . J . Appl. Rhys. 33, 3125. H o r a i , K., 1 9 7 1 : T h e r m a l c o n d u c t i v i t y o f r o c k - f o r m i n g J. Geophys. Res. 76, 1278.
minerals.
H o r a i , K. & B a l d r i d g e , S . , 1 9 7 2 a : T h e r m a l c o n d u c t i v i t y o f n i n e teen igneous rocks, I A p p l i c a t i o n o f t h e needle probe method to t h e measurement o f t h e thermal c o n d u c t i v i t y o f rock. P h y s . E a r t h P l a n e t , I n t e r i o r s 5, 1 5 1 . H o r a i , K. & B a l d r i d g e , S . , 1 9 7 2 b : T h e r m a l c o n d u c t i v i t y o f n i n e teen igneous rocks, I I Estimation o f t h e thermal conductivi t y o f rock from t h e mineral and chemical compositions. P h y s . E a r t h P l a n e t . I n t e r i o r s 5, 1 5 7 . H o r a i , K. & S i m m o n s , G., ing minerals. Earth
1969: Thermal c o n d u c t i v i t y o f P l a n e t . S c i . l e t t . , 6, 359.
rock-form-
lUGS S u b c o m m i s i o n on t h e S y s t e m a t i c s o f I g n e o u s R o c k s . 1 9 7 3 : C l a s s i f i c a t i o n and n o m e n c l a t u r e o f p l u t o n i c r o c k s . Recommend a t i o n s . N. J . b . M i n e r . M h . , H 4 , 1 4 9 - 1 6 4 . lUGS S u b c o m m i s i o n on t h e S y s t e m a t i c s o f I g n e o u s R o c k s . 1 9 8 0 : C l a s s i f i c a t i o n and nomenclature o f Volcanic rocks, lamprophyres, Carbonatites and M e l i l i t i c rocks. - Geologische Rundschau 69, 194-207. K a p p e l m e y e r , 0 . & H a e n e l , R., 1 9 7 4 : G e o t h e r m i c s w i t h s p e c i a l r e f e r e n c e t o a p p l i c a t i o n . G e o e x p l . Monogr. S e r . l , 4 , 238 pp. K e r s t e n , M.S., 1 9 4 9 : T h e r m a l p r o p e r t i e s o f s o i l s . B u l l e t i n 28. U n i v e r s i t y o f M i n n e s o t a , I n s t i t u t e o f T e c h n o l o g y , e e r i n g e x p e r i m e n t s t a t i o n . V o l . L l l , N o . 2 1 , 2 2 6 p.
132
No. Engin-
69 L a n d o l t - B f i i r n s t e i n , 1 9 6 1 : Z a h l e n w e r t e und F u n k t i o n e n . E i g e n s c h a f ten der M a t e r i e i n i h r e n Aggregatzustanden. 4 . T e i l : Kalor i s c h e Z u s t a n d s g r b s s e n . S p r i n g e r - V e r l a g , B e r l i n 1 9 6 1 . 863 p. L a n d s t r o m , 0 . , L a r s o n , S-A., L i n d , G. & M a l m q v i s t , D., 1979: V a r meflode i b e r g . Chalmers t e k n i s k a hbgskola/Goteborgs u n i v e r s i t e t , G e o l o g i s k a i n s t . Publ B137. L a n d s t r o m , 0 . , L a r s o n , S-A., L i n d , G. & M a l m q v i s t , D., 1980: Geot h e r m a l i n v e s t i g a t i o n s i n t h e Bohus g r a n i t e a r e a i n southw e s t e r n Sweden. T e c t o n o p h y s i c s 6 4 , pp 1 3 1 - 1 6 2 . L o b e r g , B., 1980: G e o l o g i . M a t e r i a l , p r o c e s s e r och S v e r i g e s b e r g g r u n d . 2:a u p p l a g a n , N o r s t e d t s . L u n d e g S r d h , P.H., L u n d q v i s t , J . & L i n d s t r o m , M., 1970: B e r g och j o r d i S v e r i g e . Tredje upplagan. Almqvist & Wiksell Fbrlag AB. S t o c k h o l m P a u l s e n , K.D., S a x o v , . S., B a l l i n g , N. & K r i s t i a n s e n , J . I . , 1 9 8 1 : T h e r m a l c o n d u c t i v i t y measurements on S i l u r i a n l i m e s t o n e s f r o m t h e i s l a n d o f G o t l a n d , Sweden. G e o l . F o r e n . i S t o c k h o l m F b r h . , 103, 349-356. P e t t i j o h n , F . J . , 1975: S e d i m e n t a r y R o c k s . 3:e u p p l a g a n . H a r p e r . SAS, S t a t i s t i c a l A n a l y s i s S y s t e m , 1982: SAS I n s t i t u t e I n c . , SAS U s e r s g u i d e : B a s i c . E d i t i o n . G a r y , N.C.: SAS I n s t i t u t e I n c . 923 pp. S a s s , J.H., L a c h e n b r u c h , A.H. & M u n r o e , R . J . , 1 9 7 1 : T h e r m a l cond u c t i v i t y o f r o c k s f r o m measurements on f r a g m e n t s and i t s a p p l i c a t i o n t o h e a t - f l o w d e t e r m i n a t i o n s . J . Geophys. r e s . v o l . 7 6 , No. 1 4 , pp 3 3 9 1 - 3 4 0 1 . S i b b i t , W.L., D o d s o n , J.G. & T e s t e r , J.W., 1979: T h e r m a l conduct i v i t y o f c r y s t a l l i n e rocks a s s o c i a t e d w i t h energy extract i o n f r o m h o t d r y rock geothermal s y s t e m . J . Geophys. Res., V o l . 8 4 , No B 3 , 1 1 1 7 - 1 1 2 4 . S u n d b e r g , J . , 1980: M e t o d e r f o r b e s t a m n i n g av v a r m e b v e r f b r a n d e e g e n s k a p e r i j o r d och b e r g . R a p p o r t n r 5:1982 f r S n J o r d v a r megruppen, Chalmers t e k n i s k a h b g s k o l a . Gbteborg. W a l s h , J.B., 1966: E f f e c t o f p r e s s u r e and s a t u r a t i n g f l u i d on t h e t h e r m a l c o n d u c t i v i t y o f compact r o c k . J . Geophys. R e s . , V o l . 7 1 , No. 1 2 .
133
1(2) BILAGA 2
ANV«NT U N D E R L A G S M A T E R I A L Referenser t i l l mineralanalyser och varmekonduktivitetsmatningar
1.
B e s k r i v n i n g a r t i l l SGU:s b e r g g r u n d s k a r t o r i s e r i e A f n r : 13-16, 102, 104-105, 107-112, 114-126, 130-132, 135-136, 138, 141 och 144.
2.
B e s k r i v n i n g t i l l SGU:s j o r d a r t s s e r i e A e n r 1 .
3.
Beskrivning till berggrundskarta Ba 2 4 .
4.
B e s k r i v n i n g a r t i l l SGU:s b e r g g r u n d s k a r t o r o v e r K o p p a r b e r g s Ian (Ca 4 0 ) , N o r r b o t t e n s I a n (Ca 4 1 ) samt o p u b l i c e r a t material frSn Vasternorrlands Ian.
5.
Wiking Andersson: s o u t h e r n Sweden. t e t , Lund. 1975.
6.
Pontus Ljunggren: The region o f HSlia i n D a l e c a r l i a , Gbteborg. 1954.
7.
Sven Gavel in: ( t i t e l
8.
0 L a n d s t r b m , S-A L a r s o n , G L i n d & D M a l m q v i s t : V a r m e f l b d e i berg. Chalmers t e k n i s k a hbgskola/Gbteborgs u n i v e r s i t e t , Geol o g i s k a i n s t . Publ B137.1979.
9.
0 L a n d s t r b m , S-A L a r s o n , G L i n d & D M a l m q v i s t : Geothermal investigations i n t h e Bohus g r a n i t e area i s o u t h w e s t e r n S w e d e n . T e c t o n o p h y s i c s 6 4 , pp 1 3 1 - 1 6 2 . 1 9 8 0 .
10.
K Poulsen, S Saxov, N B a l l i n g & J K r i s t i a n s e n : Thermal cond u c t i v i t y measurements on S i l u r i a n l i m e s t o n e s form t h e I s l a n d o f G o t l a n d , Sweden. GFF 1 0 3 , pp 3 4 9 - 3 5 6 . 1 9 8 1 .
11.
A Hasselstrbm: Temperaturmatningar inom svenska g r u v f a l t och i samband darmed bestamning av v a r m e l e d n i n g s f b r m l g a hos m a i mer o c h s i d o b e r g a r t e r f r S n samma g r u v f a l t . STU 7 1 - 5 0 7 / u 4 0 7 . 1972.
12.
N B a l l i n g , J K r i s t i a n s e n , N B r e i n e r , K Poulsen, R Rasmussen & S Saxov: Geothermal measurements and subsurface temperat u r e m o d e l l i n g i n Denmark. Dep. o f Geology, Aarhus Univers i t y . G e o s k r i f t e r 16. 1981.
over
Stockholmstrakten
SGU
Precambrian geology o f t h e Vastana area, Geologiska i n s t i t u t i o n e n , Lunds u n i v e r s i Sweden.
a n n u e j b e s t a m d ) SGU B a 3 2 .
134
1(3) BILAGA 3
V R R M E K O N D U K T I V I T E T OCH
label 1 1
S P E C I F I K VARMEKAPACITET
FOR O L I K A M I N E R A L
V a r m e k o n d u k t i v i t e t ( W / m °C) f o r o l i k a m i n e r a l . ( E n l i g t Horai & Simmons, 1969, och H o r a i , 1971)
Andalusit Albit Amfibol Anortit Biotit
7.5 2.1 3.5 1.7 2.0
Cordierit Diopsid Dolomit Epidot Granat
2.7 4.0 5.5 2.8 3.1
Hematit Hornblainde Kalcit Kalifaltspat Klorit
11.3 2.8 3.6 2.5 5.1
Kvarts Magnetit Mikroklin Muskovit 01ivin*
7.7 5.1 2.5 2.3 4.5
Ortoklas Plagioklas* Pyroxen* Serpentin Sillimanit
2.3 1.9 4.3 3.5 9.1
V a r m e k o n d u k t i v i t e t e n a r beroende av den kemiska ningen hos m i n e r a l e t , se t a b e l l 2.
135
sammansatt-
2(3)
TABELL
2
V a r m e k o n d u k t i v i t e t e n f o r p l a g i o k l a s , o l i v i n och pyroxen beroende sammansattning.
(Horai
and B a l d r i d g e ,
Material
Sammansattning
Plagioklas
An
0 - An
5
2.34
An
(Ab = N a A l S i j O g An =
CaAl2Si20g)
01 i v i n (Fo = Mg2Si04 Fa =
FegSiO^)
Pyroxen Fs =
5 - A n 15
1.92 1.63
An 30 - An 50
1.46
An 50 - An 70
1.46
A n 70 - A n 8 5
1.59
An 85 - An 100
1.72
0 - F a 10
5.10
F a 10. - F a 3 0
4.27
Fa 30 - Fa 50
3.60
Fa 50 - F a 70
3.18
Fa 70 - Fa 90
3.05
Fa 90 - Fa 100
3.14
Fs
(En = MgSiOg FeSiOj)
Varmekonduktivitet
A n 15 - A n 3 0
Fa
0 - F s 10
4.73
F s 10 - F s 3 0
3.93
F s 30 - F s 50
(3.43)
F s 50 - F s 70
(3.18)
Fs 70 - F s 90
(3.14)
Fs 90 - F s 100
(3.22)
136
av
dess
1972.)
(W/m
°C)
3(3)
Tabell 3
Specifik varmekapacitet f o rmineral v i d skilda temp e r a t u r e r , e n l i g t Goranson (1942).
Minenl
C o m p o u n d
-200'
SAb-2An.,
0*
200'
400*
l O *
800*
1.07
1.18
.991
.20
.255
±1:0-900
18
1.09
1.21
1.016
.206
.278
1:0-900
18
.217
.058
1:0-961
71
3; Bfll-1300
71
A f
liquid
A g O
oerargyrite
(.85 a t 60*) .146
.233
.244
78" .256 .279
.318 .251
.354
.408
.462
.271
Uquid AgiAiSi..
proustite
AgiS
acanthi te
.410
AgjSbSi..
pyrargyrite
AliO.
corundum
72
.32 .37
.317
5:0-175
73
.368
5: 175-325
73
(.32 at 60') 0.069
0.72
1.00
L.B. 1.10
1.19
1.26
1.067
0.140
0.289
1.53
1
4:0-1700 2230
.152
.77
1.03
1.11
1.165 1.20
1.136
0.050
0.281
3:0-1300
2
cj'anite
.077
.70
1.00
1.10
1.20
1.27
l.OSi
0.136,
0.313
2: 0-1400
3
aillimanite
.133
.743
1.00
1.08
1.16
1.22
1.054
0.123
0.257
3; 0-1200
4
.97
1.03
1.09
andalusite
mullite
.77
kaolinite
1.02
kaolin
1.17
AI^WT
metakaolin
2(A1F)0-Si0j....
topaz
.71i
11.00 1.10 (.83 a t 50*)
native gold
.127
.133
a-wither! te
1.13
1.35
.140
1.20
1.27
.152
liquid BaCOi.
72
5: 453-533
L.B.
liquid
Au.,
2:0-453
(.34 a t 50*)
argentite
.
10-*e
.70
native silver
•AUSii07-2H!0
Refer ence
.70
oUgoclase
AlsSuOu
E r r o r %; t e m p . raDge *C.
andesine
A«
AUSiiOii
j . / g m . ( T ' K . )
«!.« 4 A b l A n ' .
AhSiO...
'
( . 8 2 a t M ' )
Ubradorita
2Ab-3An.
Constanta i n
Cp (joules p e rg r a m ) f o r t e m > peratures i n * C .
1.03
0.075
0.210
3: O-llOO
5
0.806
0.463
0.0
4:0-300
6
0.641
0.904
0.0
3:0-500
7
1.062
0.151
0.289
2: 0-1300
.119
.0306
.15 .197
.44
.50
.55
.278
^-witherite
2:0-1063
35
5:1063-1300
35 10
5:0-810
10
10: 810-9S0
.64
I
7 8
10,11
BaSO.
barite
BeAlrf).
chrysoberyl
(.84 a t 50')
12
BejAUSUOi.
beryl
(.84 a t 50*)
13
.197
.45
.55
.383
.65
.253
5: 0-1050
diamond
.435
1.06
1.37
1.86
.754
1.067
.4544
4:0-1040
25
^graphite
.635
1.18
1.45. 1.88
.932
.913
.4077
3: 0-1040
26
.06
.2284
2:0-1300
Joly 19
Ca.AM.(SiO.)i..
prehnite
Cat^USiOT
geblenite
.75
CaAl.SuOi..
anorthite glass
C a C O .
.50
aragonite
(.84 a t 50*)
.26
calcite C a F .
fluorite
C a M g ( C O ) .
dolomite
.22
.97
1.03
1.09
1.12
1.042
.70
1.05
1.17
1.27
.950
.226
.2313
1:0-1400
18
.68
1.06
1.014
.158
.282
1; 0-700
18
3:0-750
16
.78
1.00
1.13
.823
.497
.1286
.79,
1.00
1.13
.823
.497
.1286
.798
.204
.85
.93 1 . 0 1
(.93 a t 60')
137
1.10
17
5:0-1200
i
IS
4(3)
Compound
(joules p e rg r a m ) f o r t e m peratures i n*C.
Mineral -200'
CaMgSiiOi.
diopeide glaaa
0*
200' 400*
800' 1200'
C,-a
C^nstanU i n + bTer-1 j./gm. (T*K.)
.71
.98 1.06 1.15 1.20 1.053 .999 .98 1.07
•c.
10»6
a
.111 .188
.290 .253
1; 0-1300 1:0-700
.150.
.177,
2:0-1400
(a) p e e u d o w o l lastoaite O ) wollastonite glass
.174
.73
.92
.172
.67
.92 1.00 1.06 1.10 1.007
.074
.269
2; 0-1300
.09
.92 1.03
.834.
.348
.175i
2:0-700
CaSOi CaSO.-2Hrf) CaWO. CdS
anhydrite gypeum Bcheelite greenockite
.52 .58 .322 1.03 ( . 4 0 a t 50°) .445 .50
.60
.64
.569
.675
.048s
5:0-1100
.55
.653
.374
.2605 0
Cu
native copper liquid
.191
.384
.42
.46
.358 .096 .493 0
0 0
.54 .68
.614
.419 .572
0
CaSiOi
cuprite tenonte 2CUO-CO.-HJO . malachite chalcopyrite CuFeSi boumonite CuPbSbSi.. CuiSe
a berzelianite ^ berzelianite
CujS
a chalcocite 0 chalcocite
.255
CuS
covellite dioptase
.228
arsenopyrite siderite hematite
FeiO.
a magnetite 0 magnetite
.42 .41
.41
a troilite 0troilite
.238
pyrite
.075
5:0-950 2:0-537
0 0
5:0-100 5:100-200
.82 .247 .55 0
0 0
3:0-103 10:103-900
.49 .52 (.77 at 34')
.54
.59
.464
.115
0
?; 0-1000
.40 .25
0 0 0 0 0
3:0-755 3:755-903 5: 903-1401 5:1401-1530 5; 1530-1600
.52
( . 4 3 a t S5») .68, .61 .79
.60
0 0 0
.640
.420
.93 1.03
.744 .640
.340 .362
.91
1.095
.69
.392
.635
.66
.71
.594
.69
.83
.606
.500
.33 .46 .63 0.63 .75 .61
.90 1.08
(.94 at 60') .55 .79 ( . 8 0 a l 60°)
FeS
.079
.55
.60
limonite fayaUte hypersthene
0 0
2; 0-I0S4 3:1084-1300
.55
(.73)
.234
.181 .188
138
1.00 1.85 .574 .130 .373
.466
Reference
eat.: 0-1000
.55
.470
.44
2FeK).-3HiO FeiSiOi FfeSiiO.
FeS,
1.07 1.14
.42
a iron 0 iron y iron 4 iron liquid
FeAaS FeCO. FeiOi
.40
.47 .505 .52 .63 ( . 7 4 a t 57°) (.54 a t 50') ( . 3 1 a t 50°)
CuiO CuO
CuSiOiHjO. Fe
.99
.926
E r r o r %; temp, range
.111
3:0-800
.177
3: 0-576 5:576-800
.181
3:0-900
0
0 0
7; 0-138 3; 138-1195
0
7; 0-500
44
5(3)
C^nsUnts in 'j./gm. CTK.) Mineral -200" 0* 200' 400- 800* 1200* a 10* 10-V .406 2.81 .43U .SM .77 pyrrbotite ice .963 2.06 native mercury .138 .138 .138 0 0 a-cinnabar .214 .227 .240 .196 .066 0 (.74,a t60*) leucite (last (.73,a t60') .732 .842 1.00 adularia mierocUne .680 .950 1.04 1.143 .988 .166 .263 orthoclaae .61 .94, 1.05 1.145 1.043 .124 .351 giaaa .70 .97 1.07 1.19 .976 .216t .247 .682 .168 0 .715 .749 .682 .418 sylvite .266 .219 0 .326 o-oiter /5-niter l.lt 1.19 0 0 liquid 1.22 1.22 0 0 (.8Sat 58*) petalite , (joulea per (ram) for temperaturea in "C.
spodumene glass garnet
Error % temp, range 3:0-350 1:0^47 2:0-580
1:0-1100 1:0-1100 2:0-1100 2:0-770 10:0-128 5: 128-338 10:338-410
(.90 at 60') (.91 at 60') (.74 at 58")
.796 1.18 o'boracite ^-boracite 1.41 .805 .84 .87 cbloromagnesite magnesite .161 .864 sellaite .906 1.08 1.206 1.43 brucite (1.30 at 35") periclaae .066 .870 1.09 1.16 1.24 1.30 .752 1.03 1.15 pyroxene amphibole .740 1.03 1.13 1.24 glass .766 1.02 1.14 (1.00 at 9*) kieserite epsomite (1. SI at 32*) olivine (0.79 at 36') talc (0.87 at 59*) rhodochrosit* .203 .70 1.08 1.46 pyrolusite .975 1.00 1.01 (0.74 a 36-) manganite alabandite .322' .569' molybdenite .537 .554 .570 .709 .986 1.085 1.196 albite class .724 1.00 1.U4 1.26
139
.275 1.909 0 .502 1.346 0 .760 .166 0 .857 .542 1.127 .973 1.067 .971
.124 .336 .183 .322
5:0-265 5; 265-100 T; 0-718 .0736
3:0-1000
.217 .233 .281 .226
2:0-1800 1:0-500 1:0-1100 1:0-700
.283i 1.532 .33 X IO-"T« .14.x .924 .227 10-"T« .515 .082 0 1.018 .187 .268 .978 .282 .247
4:0-500 ?;O-500 5:0-456 1:0-1100 1:0-400
6(3)
C o m p o u n d
, (joules per g r a m l f o r t e m permturea i n
Mineral
-200* N»C1..
0'
\200'
.4es .sss
haliU) liquid
400'
800'
11.10 1.29
N»F
viUiaumite
1.034
borax
(.161
N a i A l F ,
cryolite
.DO*
jl.18
NiS
millerite
.lot
I
.S6i
P b C O .
oerussite
.177
PbS
caleoa
.142
I
.221
PbSO.
angleaite
(.364 at
60*)
Pd
palladium
.232
.246
.260
.289
Pt
platinum
.134
.139
.144
.154
S I . .
rhombic
.318
at
I
.207
.300
.773 1.14
1.14
N«B.OilOHK>.
IOH>
1200"
.975 1.095
.S15
Constants i n C , - o + 6r eT-' i./«m. ( T * K . )
0
Error %; temp, range "C.
lff-«c 0
2;0-S00
0
3:800-950
-.184
2; 0-700
.473
1.151
.770
.949
.426
.295
0
3,0-324
.188
.07
0
S:(H!00
.318
.212
.072
0
2; 0-1549
.164
.127
.0249
0
i:(^ieoo
.482
.835
0
3; 0-95.6
.572
.576
0
3:95.6-119
.656
.656
0
T; 119-160
0
?; 160-270
0.0
est.:
35*) 1.39
1.78
.235
sulfur
monocliiiic
.0895
2; 0-1000
sulfur liquid viscous Sb.Si...
stibnite
SiO.....
a-quartz
1.22
.173
.407
.342
I
.375
.698
. 9 6 9 1.129
0.298
1.174 1.327 o-cristobalite
.186
glass
.184
1.074 1.171
1.21 1.34
.70
.95
1.06
.34
.43
.48
SnOt...
cassiterite
SiCOi..
strontianite
T i O i . . .
rutile, brookite
.70
.80
.88
WO.....
tungstite
.33
.355
.382
Z n C O .
smithsonite
Z n O . . . ZnS.... ZrSiOi.
.211
.238
sincite a-w\irtzite ^sphalerite zircon
.607
.763
.383
1.21 .55
1.191
1.6
.168
0-548
1;»-S75 4: 575-1600
0
4:0-250
0
2; 250-1700
.032
.0625
.892
.311,
.021
5: 0-1700
.387
.157
.07
4:0-1100
.619
.395
.022
3:0-450
.289
.14
.536 .44
5: 0-1300
0
.632 .48
.430
.7574 .254
1.01
^ristobalite
0.163
.45 (.61 a t
.58 .53 60')
140
.615 .56
.66 .587
.586 .550
.075
.094
2;0-130O
.041
.084
6:0-900
1(4) BILAGA 4
D V E R E N S S T O M M E L S E M E L L A N U P P M R T T OCH B E R R K N A D (Tillaggsmaterial)
VRRMEKONDUKTIVITET
De a n v a n d a v a r d e n a p a v a i r m e k o n d u k t i v i t e t e n f o r m i n e r a l b y g g e r pS b e s t a m n i n g a r a v H o r a i & S i m m o n s ( 1 9 6 9 ) s a m t H o r a i ( 1 9 7 1 ) . De a n vande s i g d a r av en ny m e t o d i k som i n n e b a r a t tm i n e r a l e t p u l v r i s e r a d e s , v a t t e n m a t t a d e s med d e s t i H e r a t v a t t e n o c h u p p m a t t e s med e n s o n d s m e t o d ( x wff)Darefter korrigerades f o r vatteninnehSllet ( e ) m e d e k v (fj, V a r v i d v a r m e k o n d u k t i v i t e t e n f o r m i n e r a l e t e r h b l l s ( ^ _ ) .Denna e r h b l l s genom m e d e l v a r d e s b i l d n i n g av e t t b v r e o c h e t t ffedre g r a n s v a r d e e n l i g t H a s k i n & S h t r i k m a n ( 1 9 6 2 ) .
dar
W
= i(^b
A.. = X °
"
^)
(i:
1 1 - ^ + e ( T - ^ - + -T; 3A^
m
w
1 )'
w
dar Ekvation
( 1 )m o t s v a r a r
e k v . ( 3 . 1 )f o r2-fas
material.
Sass e t a l ( 1 9 7 1 ) v i s a d e a t t d e t v a r m b j l i g t a t t bestamma vairmek o n d u k t i v i t e t e n a v e n f o r b e r g pa n e d k r o s s a t m a t e r i a l , varefter aven Horai & Baldridge (1972) gjorde motsvarande fbrsbk. D e r a s m e t o d i k v a r d e n s a m m a s o m f o r m i n e r a l b e s t a m n i n g a r o v a n . PS provbiten f r a n motsvarande b e r g a r t e r utfbrdes divided-bar bestamn i n g a r som j a m f b r e l s e . A v v i k e l s e n v a r k o r r e l e r a d t i l l p o r o s i t e t e n . Om h a n s y n t o g s t i l l d e n n a f a n n m a n a t t i n a g o n a v m e t o d e r n a f a n n s e t t s y s t e m a t i s k t f e l pS 5%. Pa d e l v i s m o t s v a r a n d e prover gjordes teoretiska t e t s b e s t a m n i n g a r med t r e s k i l d a m e t o d i k e r : 1) 2) 3)
varmekonduktivi-
utgSende f r S n mineralsammansattning utgaende frcin kemisk sammansattning utgciende f r a n a t o m v i k t och d e n s i t e t .
Man f a n n a t t 1 ) v a r d e n b a s t a . 2 ) k a n med n a g o r l u n d a sakerhet e n d a s t u t f b r a s pS m a g m a t i s k a b e r g a r t e r d a r e n n o r m k a n a n v a n d a s f b r bvergang f r S n kemisk sammansattning till mineralogisk. 3) var den m e s t o s a k r a . Resultaten av jamforelsen mellan matning och berakning Horai & Baldridge, 1972) v i s a s i t a b e l l 3.3.
141
(enligt
2(4) Man fann vid hbgre vardet
jamforelsen beraknat.
en
medelavvikel
se
pS
ca
+10%,
med
det
V i d a n t a g a n d e a v e n s y s t e m a t i s k o v e r s k a t t n i n g med 5% v i d a n v a n d a n d e t av H a s h i n - S h t r i k m a n s samband (ekv. (1) e l l e r ( 3 . 1 ) ) s k u l l e a v v i k e l s e n pS + 1 0 % e n l i g t ovan fbrandras till -5% eftersom ekv. (3.1) anvands 3 gSnger, vid mineralbestamningarna, vid bvergSng av varmekonduktivitet frSn vatten-bergartspulverblandning t i l l bergart samt vid berakning e n l i g t metod (ekv. 3.1). Horai & Baldridge (1972) slutsats b l i r darfor a t t vid praktisk anvandnipg av e k v ( 3 . 1 ) b b r v a r m e k o n d u k t i v i t e t e n r e d u c e r a s med 5%. De u n d e r s o k n i n g a r som ar gjorda vid Geologiska institutionen, T a b e l l 3 . 4 o c h 3 . 5 , v i s a r d o c k pS e n god b v e r e n s s t a m m e l s e mellan u p p m a t t och b e r a k n a d v a r m e k o n d u k t i v i t e t , v a r f b r nagon sadan reducering e j har g j o r t s f o r berakningarna i d e n n a r a p p o r t . Se vidare kapitel 3.3.
142
1(5) BILAGA 5
RESULTAT, BERGARTERS VARMEKONDUKTIVITET, TABELLER
Tabell 1
Medelvarden m m av v a r m e k o n d u k t i v i t e t , W/m C , f o r u r s p r u n g l i g b e r g a r t s k o d . S k i l l n a d e n mellan Nl och N2-N't beror pa a t t under metod 1 a r i n l a g t uppmatt v a r m e k o n d u k t i v i t e t . Metoderna hanfdr s i g i o v r i g t t i l l e k v . (3.1-3.4) i namnd o r d n i n g .
Bergartskod Granit Cranodiorit Tonal i t A p l i t , pegmatit m m Kvartsdiorit Syenit, diorit m m Porfyr Porfyrit Ryolit, dacit Trakyt, basalt m m Kvartsit Ovr. k v a r t i s i t Ovr, omvandlade sed. Omvandlade s e d . , o s p e c . Omvandl, b a s i s k a b e r g a r t e r C n e j s , ospec. Leptit, leptitgnejs m m
Tabell 2
Nl
MV1
stdl
N2
MV2
std2
g4S 255 171 44 122 188 95 59 119 70 36 267 148 197 168 226 742
3.47 3.34 3.16 3.31 2.87 2.67 3.55 2.54 3.37 2.83 6.44 4.65 3.53 3.52 2.58 3.47 3.56
0.380 0.292 0.269 0.477 0.227 0.305 0.463 0.468 0.397 0.347 0.811 0.679 0.478 0.706 0.305 0.466 0.621
714 255 171 8 122 188 34 21 117 70 31 266 141 197 135 202 425
3.JS 3.11 2.94 3.44 2.70 2.56 3.20 2.79 3.18 2.70 6.45 4.41 3.28 3.30 2.56 3.25 3.33
0.35J 0.265 0.239 0.509 0.198 0.273 0.319 0.248 0.357 0.305 0.821 0.686 0.441 0.669 0.251 0.439 0.548
m
5.35 3.82 3.61 4.13 3.13 2.77 3.92 3.31 3.84 3.04 6.88 5.24 4.01 3.99 2.75 3.95 3.98
std3
MV4
std4
0.446 0.358 0.352 0.593 0.263 0.346 0.456 0.410 0.428 0.449 0.592 0.650 0.544 0.841 0.331 0.548 0.674
2.85 2.66 2.54 2.96 2.45 2.42 2.73 2.49 2.74 2.48 5.83 3.63 2.79 2.81 2.44 2.78 2.88
0.248 0.169 0.146 0.380 0.146 0.225 0.216 0.148 0.278 0.213 1.016 0.608 0.311 0.466 0.229 0.309 0.413
Medelvarden av v a r m e k o n d u k t i v i t e t , W/m C, f o r m o d i f i e r a d b e r g a r t s k o d . S k i l l n a d e n mellan N1 och N2-N4 beror pa a t t under metod 1 ar i n l a g t uppmatt v a r m e k o n d u k t i v i t e t . Metoderna hanfor s i g i o v r i g t t i l l e k v . (3.1-3.4) i namnd o r d n i n g .
Bergartskod Granit-ryolit Cranodiorit-ryodacit Tonal i t - d a c i t Kv.syenit-Kv.trakyt Syenit-trakyt Kv,monzonit-Kv.1atit Monzonit-latit Kv.monzondiorit-andesit Kv.diorit-andesit Diorit-andesit Gabbro-basalt P e r i d o t i t , pyroxenit Kvartsit Ovr. k v a r t i s i t Ovr. omvandl. sediment Omvandl, s e d i m e n t , o s p e c . Omvandl. b a s i s k a b e r g a r t e r Gnejs, ospec. Leptit, leptitgnejs m m
Nl
MVl
stdl
N2-4
969 315 332 18 41 63 7 33 50 58 86 8 32 272 122 192 184 227 726
3.49 3.28 3,19 2.94 2.51 2,76 2.68 2.69 2.64 2.34 2.78 4.02 6.61 4.65 3.58 3.54 2.56 3.47 3.58
0.355 0.301 0.395 0.230 0.216 0.196 0.344 0.188 0.183 0.346 0.310 0.162 0.628 0.681 0.488 0.699 0.309 0.465 0.603
738 315 332 18 41 63 7 33 50 18 86 8 27 271 115 192 151 203 409
143
MV2
std2
3.31
0.304 0.273 0.364 0.204 0.199 0.173 0.303 0.160 0.167 0.219 0.289 0.159 0.456 0.689 0.456 0.665 0.267 0.438 0.520
3.07 2.98 2.80 2.44 2.63 2.56 2.56 2.51 2.33 2.66 3.91 6.70 4.41 3.32 3.32 2.53 3.25 3,37
m 3.58 3.75 3.62 3.18 2.58 2.99 2.82 2.91 2.81 2.41 2.85 3.96 7.06 5.24 4.07 4.02 2.72 3.95 4.04
stcl5
MV4
std4
o.m
2.85 2.63 2.57 2.57 2.35 2.42 2.40 2.36 2.34 2.27 2.51 3.83 6.12 3.63 2.81 2.83 2.42 2.78 2.90
0.220 0.173 0.239 0.165 0.167 0.124 0.212 0.116 0.155 0.187 0.256 0.189 0.662 0.609 0.325 0.465 0.243 0.309 0.401
0.376 0.490 0.253 0.272 0.249 0.435 0.227 0.194 0.267 0.323 0.136 0.300 0.654 0.548 0.830 0.342 0.547 0.616
2(5) label 1 3
Toleransinterval1
f o r varmekonduktivitet vid ursprunglig
bergartskod.
A n t a g a n d e om l o g n o r m a l f o r d e l n i n g x.% a v f o r d e l n i n g e n x% av f o r d e l n i n g e n a r med l i g g e r med 9 5 % k o n 95% konfidensfidensgrad inom neg r a d s t o r r e an danstaende internedanst. varde val 1 Bergartskod Granit Granodiorit Tonal i t Aplit, pegmatit m m Kvartsdiorit Syenit, diorit m m Porfyr Porfyrit Ryolit, dacit Trakyt, basalt m m Kvartsit "Svrig kvartsit" Ovriga omvandlade sediment Omvandlade sediment ospec. Omvandlade basiska bergarter Gnejs, ospec. Leptit, leptitgnejs m m
Tabell
4
Toleransinterval1
MV
x=75%
x=S0%
x=75%
5.47 3.34 3.16 3.31 2.87 2.67 3.55 2.54 3.37 2.83 6.61 4.65 3.58 3.54 2.56 3.47 3.58
3.1S 3.10 2.93 2.86 2.67 2.42 3.12 2.10 3.02 2.52 5.94 4.11 3.16 2.98 2.31 3.10 3.13
2.97 2.94 2.78 2.59 2.54 2.25 2.85 1.85 2.80 2.33 5.53 3.75 2.90 2.64 2.14 2.84 2.82
3 . 0 2 -- 3 . 9 3 2 . 9 9 -- 3 . 7 0 2 . 8 3 -- 3 . 5 1 2 . 6 9 -- 4 . 0 0 2.58-3.16 2.30-3.07 2.93-4.21 1 . 9 3 -- 3 . 2 4 2 . 8 6 -- 3 . 9 0 2.40-3.29 5.69-7.62 3 . 8 4 -- 5 . 5 2 2 . 9 8 -- 4 . 2 2 2 . 7 4 -- 4 . 4 2 2 . 1 9 -- 2 . 9 6 2 . 9 2 -- 4 . 0 7 2 . 8 9 -- 4 . 3 1
2 . 8 6 -- 4 . 1 5 2 . 8 5 -- 3 . 8 7 2 . 7 0 -- 3 . 6 7 2.47-4.35 2.48-3.30 2.16-3.26 2 . 7 2 -- 4 . 5 5 1 . 7 3 -- 3 . 6 1 2.68-4.17 2.24-3.53 5.34-8.12 3 . 5 5 -- 5 . 9 7 2 . 7 7 -- 4 . 5 4 2 . 4 7 -- 4 . 9 0 2 . 0 5 -• 3 . 1 5 2 . 7 2 -- 4 . 3 7 2 . 6 6 -- 4 . 7 0
f o r varmekonduktivitet vid modifierad
Granit-Ryolit Cranodiorit-Ryodacit Tonalit-Dacit Kvartssyenit-kv.trakyt Syenit-Trakyt Kv.monzonit-kv.latit Monzonit-latit Kv.monzonit-Andesit Kv.diorit-Andesit Diorit-Andesit Cabbro-Basal t Peridotit, Pyroxenit m
m
metod x% av f o r d , ligger med 95% konfidensgrad inom nedanst. interval 1
x=7S%
x=90%
x=5b%
3.22 3.09 2.94
2.94 2.95 2.73
2.84-4.16 2.85-3.90 2.64-3.65
2.68 2.41 3.18
2.53 2.23 2.67
2 . 4 9 --3.36 2 . 1 3 -- 3 . 3 3 2 . 5 9 -- 4 . 5 1
3.04
2.85
2.79-4.32
4.09 3.17 2.93 2.32 3.13 3.10
3.68 2.82 2.61 2.13 2.88 2.79
3 . 4 9 -- 6 . 0 3 2 . 7 7 -• 4 . 6 6 2.53-5.26 2.02-3.25 2.73-4.66 2.67-4.68
bergartskod.
A n t a g a n d e om l o g n o r m a l f o r d e l n i n g x% av f o r d e l x% av f o r d e l n i n g e n n i n g e n a r med l i g g e r med 9 5 % k o n 95% konfidensfidensgrad inom neg r a d s t o r r e an danstaende internedanst. varde vall
Bergartskod
Parameterfri x% av f o r d , a r med 9 5 % k o n fidensgrad s t o r r e an n e danst. varde
Parameterfri x% av f o r d , a r med 9 5 % k o n fidensgrad s t o r r e an nedanst. varde
metod x% av f o r d , ligger med 95% konfidensgrad inom nedanst. interval 1
MV
x=75%
x=90%
x=75%
x=90%
x=75%
x=90%
x=90%
3.49 3.28 3.19 2.94 2.51 2.76 2.68 2.69 2.64 2.34 2.78 4.02
3.22 3.04 2.90 2.68 2.30 2.58 2.15 2.SO 2.47 2.02 2.49 3.77
3.02 2.88 2.69 2.52 2.17 2.45 1.89 2.38 2.35 1.83 2.31 3.63
3.07-3.92 2.92-3.65 2.75-3.65 2.58-3.34 2.22-2.82 2.50-3.04 1.99-3.56 2.43-3.98 2.39-2.89 1.89-2.82 2.37-3.21 3.68-4.38
2.91-4.14 2.79-3.83 2.59-3.88 2.44-3.53 2.11-2.97 2.39-3.17 1.76-4.02 2.32-3.11 2.30-3.02 1.74-3.08 2.22-3.42 3.55-4.54
3.24 3.03 2.89
3.03 2.86 2.75
2.90-4.17 2.78-3.82 2.69-4.02
144
3(5) Tabell 5
L a n s v i s r e d o v i s a d v a r m e k o n d u k t i v i t e t f o r u r s p r u n g l i g b e r g a r t s k o d . Endast uppmatta varden samt berakningsmetod e n l i g t e k v . ( 3 . 1 ) medtagen ( s e k a p i t e l 3 . 2 . 2 ) . Lan
Bergartskod
A A A A A A A A A A A A A BD BD BD BD C C C C C C C C C C D D D D D D D D D D D D D E E E E E E E E E E E E E H
GRANIT 2 GRANODIORIT 3 TONALIT 4 KVARTSDIORIT 7 DIORIT MM 8 PORFYR 9 PORFYRIT 10 5VR KVARTSIT 14 OVR OMV SED .15 OMV SED OSPEC 16 OMV BASISKA 17 GNEJS OSPEC 1 8 LEPTITGNEJS 1 9 GRANIT 2 DIORIT MM 8 PORFYR 9 OVR OMV SED 15 GRANIT 2 GRANODIORIT 3 TONALIT k KVARTSDIORIT 7 DIORIT MM 8 DACIT MM 1 1 BASALT MM 12 OVR KVARTSIT 1H OVR OMV SED 15 LEPTITGNEJS 19 GRANIT 2 GRANODIORIT 3 TONALIT 4 KVARTSDIORIT 7 DIORIT MM 8 DACIT MM 1 1 BASALT MM 12 OVR KVARTSIT 14 OVR OMV SED 15 OMV SED OSPEC 16 OMV BASISKA 17 GNEJS OSPEC 1 8 LEPTITGNEJS 19 GRANIT 2 GRANODIORIT 3 TONALIT 4 KVARTSDIORIT 7 DIORIT MM 8 DACIT MM 1 1 BASALT MM 12 OVR KVARTSIT 14 OVR OMV SED 15 OMV SED OSPEC 16 OMV BASISKA 17 GNEJS OSPEC 1 8 LEPTITGNEJS 19 KVARTSIT 13
N
145
80 47 67 18 15 9 19 101 68 73 34 59 166 86 54 61 5 30 13 7 9 9 9 18 23 25 61 125 64 47 9 39 33 20 36 15 80 14 41 57 108 29 6 25 36 14 30 8 9 12 26 10 45 9
MV
3.57 3.29 3.18 2.77 2.85 3.30 2.96 4.49 3.46 3.10 2.58 3.44 3.34 3.20 2.49 3.62 4.15 3.57 3.36 3.21 3.02 2.52 3.43 2.93 4.35 3.44 4.12 3.60 3.44 3.17 2.89 2.69 3.54 2.83 4.64 3.32 3.96 2.66 3.38 3.85 3.49 3.24 3.06 2.95 2.82 3.39 2.80 4.88 3.37 3.66 2.60 3.71 3.59 6.72
Std
0.320 0.281 0.273 0,170 0.402 0.288 0.302 0.617 0.417 0.425 0.408 0.281 0.471 0.425 0.192 0.500 0.468 0.272 0.346 0.238 0.362 0.206 0.193 0.378 0.438 0.458 0.609 0.376 0.311 0.267 0.217 0.317 0.425 0.422 0.567 0.430 0.761 0.299 0.608 0.546 0.280 0.261 0.186 0.152 0.282 0.288 0.255 0.270 0.418 0.611 0.246 0.758 0.511 0.347
4(5) Tabell 5
Fortsattning Lan
N1
Bergart
H K K L L L L L L L L L N N 0 0 0 0 P P P P P P R T T T T T T T T T T T U U W W W W w w w w w w Y Y Y Y Y Y
OVR KVARTSIT 14 GRANODIORIT 3 GNEJS OSPEC 18 GRANIT 2 GRANODIORIT 3 DIORIT MM 8 DACIT MM 11 KVARTSIT 13 OVR KVARTSIT 14 OMV BASISKA 17 GNEJS OSPEC 18 LEPTITGNEJS 19 GRANIT 2 GRANODIORIT 3 GRANIT 2 KVARTSDIORIT 7 OMV BASISKA 17 GNEJS OSPEC 18 GRANIT 2 GRANODIORIT 3 TONALIT 4 KVARTSDIORIT 7 OMV BASISKA 17 GNEJS OSPEC 18 GRANIT 2 GRANIT 2 GRANODIORIT 3 TONALIT 4 APLIT MM 5 PORFYR 9 DACIT MM 11 OVR KVARTSIT 14 OVR OMV SED 15 OMV BASISKA 17 GNEJS OSPEC 18 LEPTITGNEJS 19 GRANIT 2 LEPTITGNEJS 19 GRANIT 2 GRANODIORIT 3 APLIT MM 5 KVARTSDIORIT 7 PORFYR 9 KVARTSIT 13 OVR KVARTSIT 14 OMV BASISKA 17 GNEJS OSPEC 18 LEPTITGNEJS 19 GRANIT 2 GRANODIORIT 3 TONALIT 4 KVARTSDIORIT 7 DIORIT MM 8 OVR KVARTSIT 14
Y Y Y Y
OVR OMV SED 15 OMV SED OSPEC 16 OMV BASISKA 17 LEPTITGNEJS 19 146
MV
11 10 12 40 10 16 33 5 14 15 25 10 6 13 62 11 5 24 54 32 8 11 25 11 8 100 11 11 7 14 23 9 9 15 25 193 17 7 67 5 35 16 7 11 20 26 15 134 59 17 18 11 12 50
5.6.3 3.22 3.13 3.31 3.20 2.89 3.12 7.03 5.47 2.50 3.32 3.28 4.00 3.27 3.44 2.94 2.69 3.84 3.52 3.44 3.18 2.87 2.74 3.52 3.26 3.39 3.46 3.22 3.17 3.42 3.56 4.80 3.97 2.54 3.60 3.66 3.55 3.39 3.46 2.94 3.29 2.75 3.41 6.42 4.57 2.23 3.40 3.34 3.52 3.28 3.07 2.83 2.54 4.66
15 27 12
3.74 3.37 2.68 3.85
Std
0.429 0.218 0.183 0.218 0.340 0.184 0.197 0.346 0.665 0.220 0.289 0.450 0.758 0.144 0.378 0.270 0.077 0.445 0.296 0.292 0.362 0.105 0.238 0.501 0.359 0.465 0.204 0.255 0.338 0.449 0.371 0.530 0.388 0.320 0.511 0.606 0.346 0.355 0.368 0.137 0.454 0.176 0.270 0.910 1.088 0.179 0.340 0.582 0.268 0.222 0.253 0.281 0.265 0.535 0,522 0.371 0.335 0.495
5(5)
Tabell 6
L a n s v i s r e d o v i s a d v a r m e k o n d u k t i v i t e t f o r m o d i f i e r a d b e r g a r t s k o d . Endast uppmatta varden samt berakningsmetod e n l i g t e k v . ( 3 . 1 ) medtagen ( s e k a p i t e l 3 . 2 . 2 ) . Lan
Bergartskod
N
MV
A A A A A A A A A A A BD BD BD BD BD BD BD BD C C C C C C C D D D D D D D D D D D D D E E E E E E E E E E E E E E E
GRANIT 103 GRANODIORIT 104 TONALIT 105 KV MONZONIT 1 1 0 KVARTSDIORIT 114 OVR KVARTSIT 214 DVR OMV SED 215 OMV SED OSP 216 OMV BASISKA 217 GNEJS OSPEC 2 1 8 LEPTITGNEJS 219 GRANIT 103 GRANODIORIT 104 KVARTSSTENIT 108 SYENIT 1 0 9 KV MONZONIT 1 1 0 KVARTSDIORIT 114 GABBRO 116 OVR OMV SED 215 GRANIT 103 GRANODIORIT 104 TONALIT 105 GABBRO 116 QVR KVARTSIT 214 OVR OMV SED 215 LEPTITGNEJS 219 GRANIT 103 GRANODIORIT 104 TONALIT 105 SYENIT 109 KVARTSDIORIT 114 DIORIT 115 GABBRO 116 OVR KVARTSIT 214 OVR OMV SED 215 OMV SED OSP 216 OMV BASISKA 21T GNEJS OSPEC 218 LEPTITGNEJS 219 GRANIT 103 GRANODIORIT 104 TONALIT 105 KV MONZONIT 1 1 0 MONZONIT 1 1 1 KV MONZODIORI 112 KVARTSDIORIT 114 GABBRO 1 1 6 PERIDOTIT MM 117 OVR KVARTSIT 214 OVR OMV SED 215 OMV SED OSP 216 OMV BASISKA 217 GNEJS OSPEC 218 LEPTITGNEJS 219
89 57 93 5 9 101 56 70 32 59 168 108 15 9 25 23 5 11 5 34 14 34 9 23 16 61 125 64 95 9 7 11 23 36 12 79 16 41 57 104 36 36 13 5 15 14 20 7 8 8 12 26 10 45
3.54 3.29 3.09 2.76 2.58 4.49 3.47 3.11 2.50 3.44 3.34 3.55 3.05 2.90 2.41 2.61 2.59 2.51 4.15 3.55 3.45 3.07 2.50 4.35 3.56 4.12 3.59 3.34 3.37 2.68 2.75 2.34 2.82 4.64 3.41 3.99 2.62
147
3.38
3.88 3.50 3.23 3.16 2.90 2.75 2.77 2.66 2.90 3.97 4.88 3.37 3.66 2.60 3.71 3.59
Std
0.323 0.289 0.285 0.081 0.212 0.617 0.445 0.433 0.249 0.281 0.472 0.443 0.245 0.277 0.147
0.201
0.045 0.163 0.468 0.269 0.349
0.313
0.214 0.438 0.485 0.609 0.323 0.322 0.464 0.249 0.218 0.243 0. 185 0.567
0.328
0.732 0.309 0.608 0.536 0.262 0.255 0.331 0.147 0.391 0.205 0.198 0.309 0.091 0.270 0.446 0.611 0.246 0.758 0.511
6(5)
label 1 6
Fortsattning L'an
Bergart
N1
H H K K L L L L L L L L N N 0 0 0 0 0 P P P P P P R T T T T T T T T U U U W W W W W W W W W Y Y Y Y Y Y Y y
KVARTSIT 213 OVR KVARTSIT 214 GRANODIORIT 104 GNEJS OSPEC 2 1 8 GRANIT 103 GRANODIORIT 104 GABBRO 116 KVARTSIT 213 OVR KVARTSIT 214 OMV BASISKA 217 GNEJS OSPEC 218 LEPTITGNEJS 219 GRANIT 103 GRANODIORIT 104 GRANIT 103 GRANODIORIT 104 TONALIT 105 OMV BASISKA 217 GNEJS OSPEC 2 1 8 GRANIT 103 GRANODIORIT 104 TONALIT 105 KV MONZONIT 110 OMV BASISKA 217 GNEJS OSPEC 2 1 8 GRANIT 103 GRANIT 103 GRANODIORIT 104 TONALIT 105 OVR KVARTSIT 214 OVR OMV SED 215 OMV BASISKA 217 GNEJS OSPEC 218 LEPTITGNEJS 219 GRANIT 103 GRANODIORIT 104 LEPTITGNEJS 219 GRANIT 103 GRANODIORIT 1 0 4 KV MONZONIT 110 KV MONZODIORI 112 KVARTSIT 2 1 3 QVR KVARTSIT 214 OMV BASISKA 217 GNEJS OSPEC 218 LEPTITGNEJS 219 GRANIT 103 GRANODIORIT 104 TONALIT 105 KVARTSDIORIT 114 GABBRO 1 1 6 OVR KVARTSIT 214 OVR OMV SED 215 OMV SED OSP 216
Y Y
OMV BASISKA 217 LEPTITGNEJS 219 148
MV
Std
9 11 10 12 71 13 8 5 14 15 26 10 13 6 52 11 9 5 24 64 26 9 5 25 11 7 120 27 20 9 9 14 25 192 16 5 7 102 12 7 6 11 20 26 15 134 56 17 25 5 9 50 15 26
6.72 5.63 3.20 3.13 3.24 3.07 2.97 7.03 5.47 2.50 3.33 3.28 3.44 3.26 3.46 3.31 3.03 2.69 3.84 3.54 3.28 3.14 2.85 2.74 3.52 3.33 3.41 3.44 3.32 4.80 3.97 2.55 3.60 3.66 3.62 3.41 3.39 3.44 2.96 2.83 2.64 6.42 4.57 2.23 3.40 3.34 3.55 3.30 3.04 2.58 2.57 4.66 3.74 3.39
0.347 0.429 0.232 0.183 0.220 0.329 0.121 0.346 0.665 0.220 0.289 0.450 0.314 0.175 0.387 0.297 0.220 0.077 0.445 0.290 0.312 0.355 0.156 0.238 0.501 0.323 0.449 0.225 0.375 0.530 0.388 0.330 0.511 0.608 0.328 0.274 0,355 0.389 0.175 0.063 0.147 0.910 1.088 0.179 0.340 0.582 0.248 0.224 0.247 0.072 0.300 0.535 0.522 0.365
12 50
2.68 3.85
0.335 0.495
R104:1986
VRRMEDVERFDRANDE EGENSKAPER I SVENSKA JORDARTER Varmekonduktivitet, och l a t e n t varme Jan
specifik
varmekapacitet
Sundberg
Denna r a p p o r t h a n f d r s i g t i l l f o r s k n i n g s a n s l a g 810671-8 fran Statens r a d f o r byggnadsforskning t i l l Chalmers tekniska hogskola, Geologiska i n s t i t u t i o n e n , Gdteborg.
149
i FDRORD Denna
rapport
h a n f o r s i g t i l l BFR-projekt 810671-8
varmebverfbrande
egenskaper
del
av p r o j e k t e t
dar
bergdelen
egenskaper
stort
Lena
jordarter.
"Varmebverfbrande
egenskaper
avrapporterad
rapporten
ar
i svensk berggrund"
(BFR-rapport
Ett
i svenska
i
av Sundberg,
och
och Johnson.
antal
1a b o r a t o r i e a n a l y s e r
av j o r d a r t e r
en del
Sven
Jonasson
i L u l e S samt
har bistatt
v i d bestamning
har utfbrts
ral innehall
i olika
Tommy C l a e s s o n ver. Lab.chef
jordarter
har utfbrt
med e t t a n t a l
i vissa
faltmat-
sandprover
av vattenhSl1ande
i e t ts k
egenskaper.
sjalvstandigt
mine-
arbete. sandpro-
G u n n a r T i b b l i n , V I A K A B , h a r b i d r a g i t med
kolvborr-
fran olika
Hellgren har svarat f o r utskriften
Gbteborg, September 1985
Sundberg
Chalmers
fran
m i n e r a l a n a l y s e r pa e t ta n t a l
c y l i n d r a r med l e r a o c h s i l t
Geologiska
av
varmekonduktivitetsmat-
P e t e r A b r a h a m s s o n h a r l a g t n e r m y c k e t mbda da h a n b e h a n d l a t
Jan
berg"
"Varmebverfbrande
Thunholm
n i n g a r . I n g v a r Rhen och P e t e r W i l e n h a r d e l t a g i t
Kallax
i jord
a r en
97:1985).
K a r l s s o n , som ocksci u t f b r t
ningar.
och behandlar Arbetet
institutionen
tekniska
hbgskola
151
delar av landet.
av rapporten.
Ann-Marie
iii INNEHALL Sid. FDRORD
i
INNEHALLSFDRTECKNING
i
SAMMANFATTNING
i
i v
B E T E C K N I N G A R OCH D E F I N I T I O N E R
i x
1.
INLEDNING
1
2.
D V E R S I K T OVER S V E R I G E S JORDARTER
2
2.1 2.2
Jordarters indelning Sveriges jordartsregioner
2 4
3.
JORDARTERS UPPBYGGNAD
9
4.
VRRMETRANSPORTERANDE
4.1
Varmekonduktivitet,
MEKANISMER specifik
12
varmekapacitet och
isbildningsvarme 4.1.1
13
Inledning
13
4.1.2
Jordars
vatteninnehSl1
15
4.1.3
Jordars
mineralinneh^11
21
4.1.4
Frysning
avjord
26
4.2
Straining
30
4.3
Konvektion
4.4
Kopplad varme- o c h f u k t t r a n s p o r t
33 i jord
34
4.4.1
Transport
i vatskefas
34
4.4.2
Transport
i Sngfas
35
4.4.3
Samverkan mellan
4.4.4
Angdiffusionens
4.4.5
Effekter av fuktvandring under temperaturgradient
Sng- o c hv a t s k e f a s inverkan
p3 v a r m e b v e r f b r i n g e n mekanismer
35 36 39
4.5
Sammanfattning - varmetransporterande
5.
METODER
5.1
Matmetoder
44
5.2
Teoretiska metoder
46
F O R VRRMEKONDUKTIVITETSBEST«MNING
43 44
5.2.1
Varmekonduktivitet
46
5.2.2
Specifik varmekapacitet
51
153
IV
6.
M A T N I N G A V V A R M E K O N D U K T I V I T E T , M E T O D I K OCH
ERFARENHETER
53
6.1
Beskrivning av m a t u t r u s t n i n g
53
6.2
Beskrivning av matmetodik
54
6.3
Erfarenheter av matningar
6.3.1
56
Laboratoriematningar
6.3.2
56
Faltmatningar
58
6.4
Insamling
7.
MATRESULTAT
61
7.1
Hela m a t e r i a l e t
61
7.2
Lera
7.2.1
och k l a s s i f i c e r i n g
av d a t a m a t e r i a l e t
59
63 V a r i a t i o n med v a t t e n h a l t o c h d e n s i t e t
63
7.2.2
V a r i a t i o n med p r o v t a g n i n g s d j u p
64
7.2.2.1
Uppmatta
64
7.2.2.2
Beraknade
varden v a r d e n f r S n 10 v a s t k u s t k o m m u n e r
67
7.3
Sand
68
7.4
Silt
71
7.5
Moran
72
7.6
Moranlera
73
7.7
Humusjord
73
8.
ANPASSNING AV T E O R E T I S K BERAKNINGSMODELL MATRESULTATEN
TILL 77
8.1
Mineraljord
78
8.2
Humusjord
89
9.
JORDARS
9.1
Fbrutsattningar
9.2
Hantering
VARMEKONDUKTIVITET - DIAGRAM
91 91
av och indata till diagram
92
9.2.1
Bestamning av j o r d a r t
92
9.2.2
Bestamning av v a t t e n h a l t och d e n s i t e t
93
9.2.3
Arbetsgang
95
REFERENSER
105
BILAGOR
109
154
V
SAMMANFATTNING
Projektet
har
syftat
t i l l a t t bestamma svenska j o r d a r t e r s
overforande
egenskaper.
har
utf'drts
pS
har
v a t t e n h a l t , d e n s i t e t , humushalt,
lande
fbmiciga
sats
Drygt
vanliga
900
svenska
bestamts.
En
varmekonduktivitetsmatningar
jordarter.
for olika
Dessa
diagram
tillsammans
jordarternas vattenhSllande
fbrmaga
och
gbr
a t t granser
varmekapacitet
med
kannedom
f o r variationsomrSdet
och
anpasvar-
vattenmattnads-
over
tivitet,
har
skapas over
t o r r d e n s i t e t och
grad
vattennivS
dessa
vattenhSl-
berakningsmodell
diagram kunnat
v a r i a t i o n med
jordarter.
P a r a l l e l U mad
k o r n s t o r l e k och
teoretisk
t i l l m a t e r i a l e t . Oarmed har
mekonduktivitetens
varme-
av
kurvor
om
grund-
varmekonduk-
l a t e n t varme i en j o r d p r o f i l
kan
ska-
pas.
I
det
fbijande redogbrs
bversiktligt
mekanismer samt f b r r e s u l t a t e t genom
varmeledning,
strllning,
s i o n . Vid lliga temperaturer stciende g r u n d v a t t e n tet.
Vid
inverkan
b l i pataglig
t i o n med komma
konvektion
fbr
ej
kan en
Vid
en
mycket
av
jordartens
jordar.
sankt
varmekonduktivitet.
parameter.
ligt
samre.
sistnamnda
kan
ha
och
Finkorniga
Fbr
radikalt
jordar
innebar sankt
en
viss
plats
egenskaper (lera-silt)
sand
narmast
isbildningsvarmet
vattenhSllande
e g e n s k a p e r medan
sand
kombina-
kraftigt
grundvattenyta.
en
till tar
i
vattenhalt
viktig
en
Inverkan
sankt
vattenhSllande de
Sngdiffusionens
kraftigt
goda
Fbr
stilla-
k o p p l a d e varme- och f u k t t r a n s p o r t e r
bver
ovan grundvattenytan
Sngdiffu-
hbga t e m p e r a t u r e r
varmekonduktivitet, varmekapacitet
stams
att
bbrjar
mattade
ger upphov t i l l en
vattenhalten
genom
det dominerande t r a n s p o r t s a t -
rumstemperatur
jamvikt, varvid
denna
samt
bverfbras
( n a t u r l i g j o r d t e m p e r a t u r ) och
bkande temperatur.
s t o r t varmeflbde
ur
varmekallan
Fbr
ovan
varmeoverforande
p r o j e k t e t . Varme kan
ar varmeledning
temperatur
k r a f t i g t med
av
fbr olika
och
en
samt
mycket
har
betyd-
fbrflyttning
i
behbjd
har
grbvre
vattenhalt. Detta
betydande sasongsvariation
ar
nagon
dm
innebar
varmebverfbrande
egenskaper.
Mineralinnehallet
har
betydelse
framst
fbr
varmekonduktivteten.
K v a r t s h a r b e t y d l i g t h b g r e v a r m e k o n d u k t i v i t e t an b v r i g a
155
vanliga
vi bergartsbildande
m i n e r a l , v a r f o r det
a r av
i n t r e s s e . En
kvartsrik
vitet
an
I p r o j e k t e t har
en
lera.
olika jordarter
Pa
g r u n d av
kornig de
fbrsbk
ha
hbgre
40%
darfdr
till
som
varmekondu.kti-
aven mineralinnehal 1
i
undersbkts.
v i d 0°C.
Detta
energimangder
sattning
ar detta mineral
hbga b i n d n i n g s k r a f t e r f r y s e r e j a l l t
jordart
stora
sand kan
framst
som
kan
a t t approximativt
utfbrts
for
ha
frigbrs
olika
v a t t e n i en f i n -
stor betydelse
da
p3
vatten fryser.
kvantifiera
denna
jordarter direkt
i
grund
Darfbr
av har
fryspunktsnedett
vattenbind-
ningsdiagram.
En
t e o r e t i s k berakningsmodel1
har
genom r e g r e s s i o n s a n a l y s
anpas-
s a t s t i l l r e s u l t a t e n f r S n u t f b r d a m a t n i n g a r och
jordartsanalyser.
Dverensstammelsen Ug
+15%
fidensgrad
Pa
basis
skapats fruset
u n d e r a n t a g a n d e om
av
tillstand ar
ungefarliga erhalls
vad
hbgre som
denna
inom i n t e r v a l l e t
teoretiska berakningsmodel1
torrdensitet,
nedanstSende
kvalitet ar
och
samt f o r l a t e n t
granser
pa
for
sedan
kon-
diagram
i ofruset
varme. IngSngsparametrar i samt
j o r d a r t e r anvandes
jordart. i
och diaOm
diagrammen
tabell.
de
varmebverfbrande
m b j l i g t a t t astadkomma
diagrammen, bbr
har
varmekapacitet
vattenmattnadsgrad nagra
v i d 90%
normalfbrdelning.
over varmekonduktivitet
grammen
Om
vanligen
matningar utfbras.
156
med
de
egenskaperna
bnskas
i rapporten
redovisade
an
Jordart
Varmekonduktivitet
Specifik varmekapacitet
x106 L e r a med lerhalt
0.85-1.1
hog
Latent
C
X"
XI
varrtie
I
o6
x10«
2.0-2.2
3.0-3.5
2.0
2.1-2.5
Torrskorpelera dito
1.1-1
A
1.7-2.3
2.6-3.0
1.7-2.0
1.1-1.6
Siltig lera/ siltskikt
1.1-1.5
2.3-2.8
2.9-3.3
2.0
1.5-2.0
Torrskorpelera dito
1.2-1.6
1.9-2.9
2.5-3.0
1.7-2.0
Silt
2A-i.3
1.1-1
.6
1.2-2.it
2.3-3.2
2.0
0.8-2.0
Sand, grus under grundvattenytan
1.5-2.6 (1.6-2.0)
2.7-3.3 (2.8-3.0)
2.5-3.2 (2.9)
2.0 (2.0)
0.8-1.7 (1.3-1.6)
Sand, grus ovan grundvattenytan
0.6-1.1 (0.7-0.9)
0.7-1.0 (0.8-0.9)
1.2-1.7 {tA)
1.1-1.6 (1.2)
0.1-0.3 (0.2)
Humusjord under grundvattenytan
0.6
1 .7
t.O
2.0
3.1-3.2
Kommentar: fruset vad
+ och
tillstand.
som
ar vanligt
ofruset
tillstSnd.
-
i
t a b e l l h u v u d e t h a r r o r s i g t i l l 'o f r u s e n
Vardena inom parentes fbrekommande.
157
i t a b e l l e n f o r sand
Vardena avser
helt
fruset
och avser eller
ix B E T E C K N I N G A R OCH
DEFINITIONER
°C
G r a d e r C e l s i u s {°C
= K - 2 7 3 ) (1°C
= 1 K)
c
Specifik
varmekapacitet
J/kg
C
c^^^
Specifik
varmekapacitet
J/m"^
°C
c^
Vattens specifika varmekapcitet
J / k g °C
( 4 . 1 8 . 1 0 - ^ J / k g °C) c^^
Isens specifika varmekapacitet
J / k g °C
( 2 . 2 . 1 0 ' ^ J / k g °C) c^
Mineralpartiklarnas (ca
specifika
varmekapacitet
J / k g °C
7 3 0 J / k g °C)
d
Avstand
m
dp
Partikeldiameter
m 2
D
Diffusionskoefficient
i
Gradient
K
Hydraulisk konduktivitet
1
Vattens isbildningsvarme ( l a t e n t varme)
2
m /s e l m /sK
m/s J/kg
(3.33.10^ J/kg) L
Angbildningsvarme
J/kg
m^^
I s d e l e n s massa
kg
m^
T o r r s u b s t a n s e n s massa
kg
m^
V a t t e n d e l e n s massa
kg
m
T o t a l massa
kg
159
n
Porositet,
n = Vp/V,
n = 1 -
N|^^
Nusselts t a l f o r straining
q
Varmeflbde
P^/p^
W/m
Vatskeflbde
kg/m
2
2
s
2 q^
Angflbde
kg/m
Vattenmattnadsgrad, t T
t i d Temperatur
V
Volymsandel
V|^y
Volymsandel
s
= V,/V„
% s K
°C,
% kvarts
%
Vatskehastighet
m/s
V
Volym
m'^
V, a
Gasvolym
m^ 3
Vp
w
Porvolym
m
P a r t i k e lvolym
m"^
Vattenvolym Vattenkvot, w = Vattenhalt,
w.^
I s k v o t , w.^ Andelen t i l l den
m^ mym^ =
m^/m
= '"is^'^s
ofrusen vattenmassa total a jordmassan
160
i
fbrhallande
w^=m^/m, s
Vattenkvot
av v a t t e n h a l t :
w = Wp^/(l-w^)
V a t t e n h a l t av v a t t e n k v o t : X
Varmekonduktivitet
Vattens (0.57 XJ^
= w/(l+w)
Isens (2.1
varmekonduktivitet
W/m
°C)
varmekonduktivitet W/m
Lufts
°C)
varmekonduktivitet
( 0 . 0 2 3 W/m
°C)
Partikelkonduktivitet
Teoretiskt
beraknad
(kornkonduktivitet)
varmekonduktivitet
efter Haskin & Shtrikman Ag,A^
Dvre resp. beraknad
Ap
( e k v . 5.1)
n e d r e grains f o r t e o r e t i s k t
varmekonduktivitet,
Teoretiskt
beraknad
A^^
varmekonduktivitet.
P a r a l l e l l k o p p l a d , ekv. (5.3) A^g
T e o r e t i s k t beraknad
varmekonduktivitet.
Seriekopplad, ekv. (5.4)
Ag
Teoretiskt beraknad
varmekonduktivitet.
Geometriskt medelvarde, ekv. (5.2) A|^^
Kvarts (7.7
W/m
varmekonduktivitet °C)
A„ m
Varmekonduktivitet
f o r vattenmattat
A^
Varmekonduktivitet
f o r torrt
A^
"Resf'konduktivitet
161
material
material
xii ^ber
Teoretiskt
"^matt
Uppmatt v a r m e k o n d u k t i v i t e t
W/m °C
"^e
Effektiv
W/m
°C
'*'rad
Varmekonduktivitetstillskott
av straining
W/m
°C
•^disp
Varmekonduktivitetstillskott
av dispersion
W/m °C
beraknad varmekonduktivitet
W/m
varmekonduktivitet
°C
2
200
Mjala Grovmjala Finmjala
>600
20- 6
60-20 20- 6 6- 2
2
2-0,6 0,6-0,2
1984.
SGFs laboratoriekommitte
600 - 60
6-
laboratoriekommittes
Bygg,
2 0 0 - 20
2-0,6 0,6-0,2 0,2-0,06
0,2-0,06 0,06-0,02
Silt Grovsilt Mellansilt Finsilt
Ler
SGF:s
0,06-0,02 0,02 - 0,006 0,006 - 0,002 0,02 - 0,006 0,006 - 0,002 < 0,002
< 0,002
Anm J o r d a r t s b e n a m n i n g a r n a o c h a n g i v n a v i k t p r o c e n t i t e x t e n f o l j e r S G F s s y s t e m (13). A l d r e b e n a m n i n g a r st4r dar i n o m parentes.
273
sys-
BILAGA 2 Berakning
av varmekonduktivitet
t e r och d e n s i t e t e r visade
med u t g S n g s p u n k t
i 10 vastkustkommuner.
bvre ochnedre
frSn
vattenkvo-
D a t a f r a n S G I . De r e d o -
granserna f o rvarmekonduktiviteten
baseras
pa 9 5 % k o n f i d e n s g r a d .
label 1 1
Kommun
Medelvarden a v anvanda ingangsvarden samt a v beraknade v a r m e k o n d u k t i v i t e t e r . Antal
W %
Std. Antal %
P , ,Std, k g V 9'
X X m.v. n e d r e W/m°C W/m°C
A bvre W/m C
A L E
6 5 2
7 3 . 8
2 2
6 3 6
1 6 1 9
146
1.03
0 . 8 3
1.26
F K R G E L A N D A
134
4 8 . 8
11.2
151
1 7 4 8
1 1 8
0 . 9 8
1.42
4 5 7
63.7
2 2
4 1 9
1 6 7 8
193
1 . 1 9 1 . 1 0
0 . 8 1
1.43
2 3 5 2
5 2 . 6
21
1929
1 7 0 6
151
1 . 1 4
0 . 8 9
1.43
155
65.5
19.9
123
156
1.08
0 . 8 5
1.33
3 2
5 0 . 2
2 2 . 7
31
1657 1784
2 0 0
1.22
0 . 8 8
1.63
MARK
4 1 7
4 2 . 8
20.8
6 4 2
1 8 5 1
173
1.30
0 . 9 8
1.70
MUNKEDAL
128
5 5 . 8
13.6
1681
2 0 3
1 . 1 1
0 . 8 0
1.49
P A R T I C L E
1674
7 1 . 8
2 6 . 4
1 1 3 1 3 9 2
1579
157
1 . 0 1
0 . 7 9
1.25
5 7 2
6 2 . 8
2 1 . 6
5 9 3
1 6 9 2
162
1 . 1 1
0 . 8 7
1.39
6 5 7 3
6 0 . 9
6 0 2 9
1680
167
1 . 1 0
0 . 8 5
1.39
KUNGiLV L E R U M L I L L A
E D E T
L Y S E K I L
U D D E V A L L A T O T A L T
Tabell 2
Kommun
ALE ALE ALE ALE ALE ALE ALE ALE ALE ALE FARGELANDA
FfiRGELANDA FSRGELANDA FfiRGELANDA FfiRGELANDA FfiRGELANDA FfiRGELANDA FfiRGELANDA KUNGfiLV KUNGfiLV KUNGfiLV KUNGfiLV KUNGfiLV KUNGfiLV KUNGfiLV KUNGfiLV KUNGfiLV KUNGfiLV LERUM LERUM LERUM LERUM LERUM
LERUM LERUM LERUM LERUM LERUM
LILLA EDET LILLA EDET LILLA EDET LILLA EDET LILLA EDET LILLA EDET LILLA EDET LILLA EDET LILLA EDET LYSEKIL LYSEKIL LYSEKIL LYSEKIL LYSEKIL LYSEKIL LYSEKIL
Medelvarden a vvarmekonduktiviteter mun o c h n i v a .
NivS m. V. m
0.75 2.00 3.00 4.00 5.00 8.00
13.00
18.00 23.00 28.00 0.75 2.00 3.00 n.oo 5.00 8.00 13.00 18.00
0.75 2.00 3.00 4.00 5.00 8.00
13.00
18.00 23.00 28.00 0.75 2.00 3.00 4.00 5.00 8.00 13.00 18.00 23.00 28.00 0.75 2.00 3.00 4.00 5.00 8.00 13.00
18.00 23.00 0.75 2.00 3.00 4.00 5.00 8.00 13.00
m.v. W/m°C
nedre W/m°C
1.11 1.02 0.99 1.02 1.00 1.03 1.08 1.06 1.01 1.17 1.50 1.25
bvre W/m°C
0 . 7 8 1.51 0 . 8 1 1.26 0 . 8 1 1.18 0.84 1.22 0.84 1.18 0 . 8 4 1.24 0 . 8 6 1.32 0.87 1.26 0 . 8 9 1.13 1.13 1.20 1.07 2 . 0 4 1.04 1.49 1.04 1.33 0.99 1.32 0 . 9 4 1.34 0 . 9 8 1.44 1.05 1.25
1.18
1.15 1.13 1.20 1.15 1.29
1.18
0 . 9 1 1.50 0 . 8 2 1.41 0 . 8 1 1.34 0 . 8 1 1.40 0 . 8 0 1.46 0 . 8 1 1.38 0 . 8 0 1.45 0 . 8 3 1.84 1.07 1.09 1.04 1.24 0.80 1.61 0 . 8 7 1.53 0 . 8 5 1.50 0 . 8 9 1.45 0 . 9 1 1.36 0 . 9 3 1.35 0.96 1.30 0 . 9 9 1.31 1.00 1.34 1.16 1.38 0.96 1.53 0.92 1.30 0.91 1.16 0 . 8 9 1.30 0 . 8 9 1.22 0 . 8 2 1.35 0 . 8 4 1.28 0.95 1.30
1.09 1.06 1.09 1.11 1.07 1.10 1.27 1.08 1.14 1.16 1.17 1.15 1.15 1.12 1.13 1.12 1.15 1.17 1.27 1.22 1.10 1.03 1.08 1.05 1.07 1.05 1.12 1.00 1.31 1.21 1.22 1.33 1.11 1.20 1.28
0 0 0 0 0 0 276
. . . . . .
90 91 87 79 83 82
1.84 1.56 1.66 2.05 1.44 1.67
f o r v a r j e kom-
Kommun
MARK MARK MARK MARK MARK MARK MARK MARK MARK MUNKEDAL MUNKEDAL MUNKEDAL MUNKEDAL MUNKEDAL MUNKEDAL MUNKEDAL MUNKEDAL PARTILLE PARTILLE PARTILLE PARTILLE PARTILLE PARTILLE PARTILLE PARTILLE PARTILLE PARTILLE UDDEVALLA UDDEVALLA UDDEVALLA UDDEVALLA UDDEVALLA UDDEVALLA UDDEVALLA UDDEVALLA
NivS m . V. m 0.75 2.00 3.00 4.00 5.00 8.00 13.00 18.00 23.00 0.75 2.00 3.00 4.00 5.00 8.00 13.00 18.00 0.75 2.00 3.00 4.00 5.00 8.00 13.00 18.00 23.00 28.00 0.75 2.00 3.00 4.00 5.00 8.00 13.00 18.00
X
m.v. W/m°C
X
nedre W/m°C
1.31 1.30 1.28 1.35 1.32 1.30 1.23 1.37 1.50 1.13 1.12 1.15 1.10 1.05 1.12 1.10 1.23 1.13 1.03 0.99 0.99 1.00 0.99 1.01 1.00 0.97 1.01 1.30 1.21 1.09 1.06 1.08 1.10 1.07 1.09
X
bvre W/m°C
0 . 9 4 1,78 0 . 9 5 1.72 0 . 8 9 1.76 1.03 1.74 0 . 9 8 1.73 0 . 9 9 1.67 0 . 9 5 1.55 1.07 1.72 1.46 1 . 5 4 0 . 7 9 1.56 0 . 7 4 1 . 6 0 0 . 8 3 1.53 0 . 8 2 1.43 0 . 7 5 1.41 0.84 1 . 4 5 0 . 6 3 1.72 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
277
. . . . . . . . . . . . . . . . . .
8 8 1.42 7 9 1.30 7 7 1.24 7 8 1.23 7 7 1.25 7 9 1.20 8 1 1.22 7 9 1.23 8 6 1.08 8 6 1.17 9 6 1.70 9 1 1.56 8 7 1.33 8 8 1.26 8 8 1.30 8 8 1.36 8 3 1.34 8 0 1.42
THE S E L F - C O N S I S T E N T A P P R O X I M A T I O N A P P L I E D TO T H E THERMAL OF C R Y S T A L L I N E ROCK, S E D I M E N T A R Y ROCK AND S O I L .
Jan Sundberg Chalmers U n i v e r s i t y o f T e c h n o l o g y , GOteborg, Sweden Swedish G e o t e c h n i c a l I n s t i t u t e , L i n k S p i n g , Sweden
279
CONDUCTIVITY
1
ABSTRACT D i f f e r e n t types o f t h e o r e t i c a l methods f o r e s t i m a t i n g thermal conduct i v i t y a r e d e s c r i b e d and a n a l y s e d . The s e l f - c o n s i s t e n t a p p r o x i m a t i o n or e f f e c t i v e medium t h e o r y i s a d o p t e d and a p p l i e d t o t h e r m a l c o n d u c t i v i t y on d i f f e r e n t t y p e s o f r o c k and s o i l . The m e t h o d i s d i r e c t l y a p p l i e d t o c r y s t a l l i n e r o c k and t o e x t r e m e l y p o r o u s s o i l . F o r m i n e r a l s o i l , s a n d s t o n e and l i m e s t o n e i t i s n e c e s s a r y t o m o d i f y t h e m e t h o d and introduce a contact r e s i s t a n c e between the g r a i n s . Vapor d i f f u s i o n , u n f r o z e n and f r o z e n c o n d i t i o n s i n c l u d i n g u n f r o z e n w a t e r a r e a l s o t r e a ted. The m e t h o d i s v e r i f i e d by t h e r m a l c o n d u c t i v i t y m e a s u r e m e n t s on a number o f c r y s t a l l i n e and s e d i m e n t a r y r o c k s and 600 s o i l s . A v a l u a t i o n of a method introduced e a r l i e r o f computing the thermal c o n d u c t i v i t y of r o c k / m i n e r a l f r o m m e a s u r e m e n t s on a m i x t u r e o f p u l v e r i z e d r o c k / m i n e r a l and w a t e r , i s t r e a t e d . The r e s u l t s i n d i c a t e t h a t l a r g e e r r o r s may be i n t r o d u c e d .
INTRODUCTION The t h e r m a l c o n d u c t i v i t y o f r o c k and s o i l i s o f i n t e r e s t i n many d i f f e r e n t a r e a s . Some e x a m p l e s a r e g e o t h e r m a l h e a t f l o w d e t e r m i n a t i o n s , t h e r m a l m o d e l l i n g i n t h e o i l i n d u s t r y , u t i l i z a t i o n and s t o r a g e o f ground heat, c a l c u l a t i o n of heat loss from b u i l d i n g s through the g r o u n d and h e a t l o s s f r o m b u r i e d c a b l e s and p i p e l i n e s . I n pace w i t h t h e i n c r e a s e d use o f c o m p u t e r i z e d models i t has become more i n t e r e s t i n g t o determine the e f f e c t i v e thermal p r o p e r t i e s i n a t h e o r e t i c a l way. I n a d d i t i o n , t h e r m a l c o n d u c t i v i t y m e a s u r e m e n t s a r e s o m e t i m e s d i f f i c u l t and e x p e n s i v e t o p e r f o r m . W i t h a good t h e o r y i t I s p o s s i b l e t o draw conclusions concerning the s e n s i t i v i t y o f t h e e f f e c t i v e thermal c o n d u c t i v i t y t o mineralogy, p o r o s i t y , pore f l u i d , degree o f s a t u r a t i o n and t e m p e r a t u r e . I n t h e c o u r s e o f t h e w o r k on t h e t h e r m a l c o n d u c t i v i t y o f b o t h r o c k s o i l , much o f t h e v a l u a b l e e x p e r i e n c e gained f r o m s t u d i e s on t e s t s one m a t e r i a l has l a t e r been a p p l i e d t o o t h e r m a t e r i a l s .
R E P R E S E N T A T I V E ELEMENTARY VOLUME
and on
(REV)
In a t h e o r e t i c a l d e t e r m i n a t i o n o f the e f f e c t i v e thermal c o n d u c t i v i t y and i n c o m p u t e r m o d e l l i n g i t i s o f i n t e r e s t t o know i f t h e v o l u m e used for determination/modelling i s r e p r e s e n t a t i v e of the whole rock/soil mass. T h i s i s u s u a l l y known as e s t i m a t i n g t h e r e p r e s e n t a t i v e e l e m e n t a r y v o l u m e . REV i s d e f i n e d a s t h e e l e m e n t a r y v o l u m e w h o s e p r o p e r t i e s a r e u n c h a n g e d w h e n t h e v o l u m e i s s l i g h t l y i n c r e a s e d . A REV o f r o c k o r s o i l must c o n t a i n enough g r a i n s / p o r e s t o y i e l d a mean v a l u e w i t h some
281
2
s t a t i s t i c a l c e r t a i n t y . F o r I s o t r o p i c normal g r a i n e d r o c k and f i n e o r m e d i u m g r a i n e d s o i l o r s e d l m a n t a r y r o c k , a v o l u m e o f a f e w cm3 w o u l d be e n o u g h t o g u a r a n t e e a m a c r o s c o p i c a l l y h o m o g e n e o u s s a m p l e . H o w e v e r , I n a t i l l w i t h b o u l d e r s o r i f c r a c k s a r e i n c l u d e d f o r a r o c k , t h e REV w o u l d h a v e t o I n c l u d e a m u c h l a r g e r v o l u m e . T h u s , t h e REV c a n be q u i t e d i f f e r e n t d e p e n d i n g o n t h e s c a l e . T h e REV t r e a t e d i n t h i s p a p e r i s j u s t b1g e n o u g h t o l e v e l t h e c h a n g e s I n p o r e s and g r a i n s . The w o r d macro i s a l s o used i n t h i s sense.
T H E O R I E S FOR
DETERMINING E F F E C T I V E THERMAL
CONDUCTIVITY
I n a m u l t i - p h a s e m a t e r i a l , e.g. a r o c k o r s o i l t y p e , i t i s p o s s i b l e t o d e f i n e bounds f o r e f f e c t i v e thermal c o n d u c t i v i t y . These bounds are a f u n c t i o n o f t h e g e o m e t r y o f t h e p h a s e s c o n s i d e r e d , c o n d u c t i v i t y (A..) as w e l l as v o l u m e f r a c t i o n s ( v ^ ) .W i e n e r (1912) e s t a b l i s h e d t h e f o l l o w i n g bounds: n '^h a r The bounds pectively, terms of a rent condu
Vi
n
_ i
^
4'
^ii/i-H= V i
(1)
r e p r e s e n t t h e h a r m o n i c and t h e a r i t h m e t i c a l mean v a l u e , r e o f an n - p h a s e m a t e r i a l . The b o u n d s can be i l l u s t r a t e d i n heat f l o w perpendicular or p a r a l l e l t o laminae w i t h d i f f e ctivities.
I t i s p o s s i b l e t o r e d u c e t h e d i s t a n c e b e t w e e n t h e b o u n d s by i n t r o d u c i n g r e q u i r e m e n t s on t h e i s o t r o p y o f t h e medium. H a s h i n and Shtrikman (1962) derived bounds f o r magnetic p e r m e a b i l i t y I n a m a c r o s c o p i c a l l y h o m o g e n e o u s and I s o t r o p i c m a t e r i a l by m e a n s o f v a r i a t i o n a l theorems. The m e t h o d can be d i r e c t l y a p p l i e d t o t h e r m a l c o n d u c t i v i t y . F o r an n - p h a s e m a t e r i a l we o b t a i n :
u
max
^max
(l"5max'^max'
= Max(\A^
?max "
'3'Niiax'
Amax
Vi-[(Ki-^a^)-1
KH'^'^max)
* ^^^r'^
To d e t e r m i n e t h e l o w e r b o u n d ( \ ^ ) , a l l o f t h e maximum i n d i c e s a r e s u b s t i t u t e d w i t h minima. I n H a s h i n - S h t r i k m a n ' s bounds, t h e number 3 r e p r e s e n t s t h e d i m e n s i o n and s h o u l d be c h a n g e d t o 2 i n a two-dimension a l p r o b l e m . H o r a i and Simmons ( 1 9 6 9 ) s u g g e s t e d t h a t t h e e f f e c t i v e c o n d u c t i v i t y can be c a l c u l a t e d as t h e mean o f t h e u p p e r and l o w e r bounds: \
= ( V
^1^/2
(3)
282
3
Equation ( 3 ) has since been used t o determine t h e thermal c o n d u c t i v i t y o f a l a r g e number o f m i n e r a l s by p e r f o r m i n g needle-probe measurements on w a t e r - m i n e r a l m i x t u r e s ( H o r a i and Simmons, 1 9 6 9 , H o r a i , 1 9 7 1 ) . T h e measured v a l u e h a s t h e r e b y been s e t equal t o e q u a t i o n ( 3 ) and t h em i n e r a l ' s thermal c o n d u c t i v i t y c a l c u l a t e d . The e q u a t i o n has a l s o been applied i n a s i m i l a r way t o measurements o f rocks as w e l l as t o c a l c u l a t i o n s o f t h e r m a l c o n d u c t i v i t i e s o f v a r i o u s rock types based on t h e i r m i n e r a l composition (Horai and Baldridge, 1972a,b). T h i s i s f u r t h e r d i s c u s s e d below. Sundberg e t a l (1985) have a l s o made u s e o f the method t o c a l c u l a t e t h e thermal c o n d u c t i v i t i e s o f about 4000 samples/ o f r o c k based o n m i n e r a l c o m p o s i t i o n and v o l u m e f r a c t i o n s . Some o f t h e s e c a l c u l a t i o n s w e r e c o m p a r e d w i t h m e a s u r e d v a l u e s w i t h good agreement. A t h e o r y developed by Maxwell (1891) f o rd i l u t e suspensions requires t h a t t h e d i s p e r s e d i n c l u s i o n s a r e s o f a r a p a r t t h a t t h e y do n o t i n f l u e n c e e a c h o t h e r (v.«1). M a x w e l l ' s e q u a t i o n w a s o r i g i n a l l y developed f o rs p h e r i c a l i n c l u s i o n s and i s as f o l l o w s :
(4)
Beck (1976) used Maxwell's e q u a t i o n t o c a l c u l a t e t h e t h e r m a l conduct i v i t y o f sedimentary rocks. Maxwell's equation f o rspherical i n c l u sions coincides w i t h Hashln-Shtrikman's lower bound provided \ < \ a n d u p p e r b o u n d p r o v i d e d x.^ < x.^ . Maxwell's e q u a t i o n has l a t e r been e l a b o r a t e d t o be v a l i d f o r e l l i p s o i d a l i n c l u s i o n s a s w e l l , de V r i e s (1952, 1 9 6 3 ) h a s a p p l i e d t h i s equat i o n t o s o i l and f o u n d good agreement w i t h m e a s u r e d v a l u e s , de V r i e s s u g g e s t e d a r a t i o b e t w e e n t h e a x e s o f t h e e l l i p s o i d o f a b o u t 5. S i n c e g r a i n s i n s o i l o r i g i n a t i n g f r o m c r y s t a l l i n e b e d r o c k can u s u a l l y be s a i d t o be s p h e r i c a l ( e x c e p t f o r c l a y p a r t i c l e s ) , i t w o u l d seem t h a t the shape o f t h e e l l i p s o i d also contains a c o r r e c t i o n f o r measured values. R a y l e i g h (1892) have developed a model f o rd i l u t e suspensions. The t h e o r y r e q u i r e s t h a t p h a s e 1 b e s u s p e n d e d i n p h a s e 2 (v^«1) a n d t h a t t h e r e be no c o n t i n u o u s c o n t a c t s u r f a c e t h r o u g h t h e m a t e r i a l . According t o R a y l e i g h , an e x p r e s s i o n f o rs p h e r i c a l i n c l u s i o n s can be w r i t t e n as follows:
Xe
(5)
=
where v
1
«1.
283
4
The
g e o m e t r i c
good o f
way
e q u a t i o n
h a s
compared
r e l i a b l e
t h e
m e n t s
o f
o n
w i t h
p h y s i c a l
g e o m e t r i c
made
mean
e q u a t i o n
used
b y
many
t h e r m a l
r o c k s
based good
o n
t h e g e o m e t r i c q u a r t z
f r a c t i o n s .
s t a t e ,
r e s e a r c h e r s
c o n d u c t i v i t y
w i t h mean
d e r i v e d
as
a
p r i m a r i l y
t h e d r y a n d
s o i l s .
Good
The
p h a s e
t h e r m a l
The
t o w a t e r - s a t u r a t e d used
a b o u t
a g r e e m e n t
c o r r e l a t e d
i s s u r r o u n d e d b y
f o r a
2-phase
e x p r e s s i o n
c o n t e x t s . 0 ,
t h e n
s t i t u t e d
F i g u r e
T h e
w i t h
a n
^ g - ' ^ - j -
1 . G r a i n t h e
when
2
f i g u r e
i
« 1 ,
^-A.)A„)
288
4.25
bounds,
(1985).
9
A P P L I C A T I O N TO
POROUS MEDIUM
SCA a s s u m e s a s t a t i s t i c a l p r o x i m i t y b e t w e e n t h e v a r i o u s p h a s e s o f t h e material which are i n proportion t o the volume f r a c t i o n s . In a porous, w a t e r - s a t u r a t e d medium, the water phase instead more or l e s s surrounds the h i g h l y t h e r m a l l y conductive mineral phase w i t h a layer of varying t h i c k n e s s . The m o s t t h e r m a l l y c o n d u c t i v e passage v i a t h e m i n e r a l g r a i n s w i l l t h e r e b y r e d u c e i t s t h e r m a l c o n d u c t i v i t y by a f a c t o r t h a t is a f u n c t i o n of the size of the contact resistance between the m i n e r a l g r a i n s . The t h e o r y f o r t h e s e l f - c o n s i s t e n t m e t h o d i n i t s o r i g i n a l form i s t h e r e f o r e not r e a l l y r e l e v a n t f o r s o i l s . Hence, a c o r r e c t i o n f a c t o r m u s t b e i n t r o d u c e d f o r t h e SCA m e t h o d . Contact
resistance
at the grain contact
surface
A t h e o r e t i c a l treatment of the size of the contact resistance involves c e r t a i n p r o b l e m s . A b e r g ( 1 9 7 8 ) and G u s t a f s o n ( 1 9 8 3 ) h a v e s h o w n t h a t i t is possible to calculate the void ratio in a soil i f the grain d i s t r i b u t i o n f u n c t i o n and a c o n s t a n t t h a t i s m a i n l y a f u n c t i o n o f c o m p a c t i o n a r e k n o w n . I n t h i s c a s e , we w o u l d l i k e t o d e t e r m i n e s o m e k i n d o f a v o i d r a t i o , e^, s o l e l y f o r t h e c o n t a c t s u r f a c e , i . e . d e t e r m i n e t h e r a t i o b e t w e e n t h e p o r e a t t h e c o n t a c t s u r f a c e and t h e g r a i n d i a m e t e r . I f a l a r g e number o f l i n e s (cords) i s drawn through a porous m e d i u m , t h e n t h e l i n e s w o u l d e i t h e r go t h r o u g h a p o r e o r t h r o u g h a g r a i n . The p o r e c o r d s w o u l d t h e r e b y f o r m some k i n d o f d i s t r i b u t i o n function, where the pore cords at the contact surface of the grains s h o u l d l i e w i t h i n a n i n t e r v a l o f 0 a n d x . One o f t h e d i f f i c u l t i e s i s t o d e t e r m i n e t h e x . S e e f i g u r e 3.
p P O R E CORD GRAIN CORDUg)
0
Figure
PORE CORD
3.
dp) (Ig)
LENGTH
Frequency function f o r pore cord
A c c o r d i n g t o f i g u r e 3, lows:
GRAIN CORD
this
can
be
length.
expressed mathematically
as
(11)
289
fol-
10
T h e c o n s t a n t y 1s t h e m i n i m u m g r a i n d i a m e t e r . T h e f u n c t i o n f d ^ ) i s not I n t e g r a t e d t o i n f i n i t y , s i n c e i t i s t h e s m a l l e r o f two adjacent g r a i n s t h a t g o v e r n s t h e s i z e o f t h e c o n t a c t s u r f a c e . I n f i g u r e 4, a s e c t i o n i s t a k e n t h r o u g h t h e c o n t a c t s u r f a c e and t h e m i n e r a l p h a s e . S i n c e l n > > l [ , 1n t h i s s e c t i o n , t h e f r a c t i o n o f t h e p o r e s p a c e can be a p p r o x i m a t e d w i t h e^. The b r e a d t h ( a ) o f t h e s e c t i o n r e p r e s e n t s t h e contact s u r f a c e between t h e m i n e r a l g r a i n s . Hence, t h e m i n e r a l f r a c t i o n can be w r i t t e n as l - e ^ . . I f t h e t o t a l t h e r m a l c o n t a c t r e s i s t a n c e 1s o n l y a f u n c t i o n o f t h e g r a i n s i z e d i s t r i b u t i o n , t h e c o n t a c t r e s i s t a n c e f o r an e v e n l y g r a i n e d m a t e r i a l , w o u l d be c o n s t a n t w i t h a c h a n g e i n p o r o s i t y . H o w e v e r , t h e contact surfaces between the grains are e s s e n t i a l f o r the thermal t r a n s p o r t t h r o u g h t h e m a t e r i a l . The change i n t h e number o f c o n t a c t p o i n t s due t o a c h a n g e i n p o r o s i t y i n an e v e n l y g r a i n e d m a t e r i a l i s probably proportional to the contact resistance. I f this i s true, a f a c t o r p can be d e f i n e d as a f u n c t i o n o f p o r o s i t y and g r a i n s i z e d i s t r i b u t i o n . A lower p o r o s i t y should y i e l d a lower contact resistance t o f i n a l l y a p p r o a c h 0 (p-»1) w h e n p o r o s i t y a p p r o a c h e s 0 . p= x=
x - ( 1 - e j
(12)
f(porosity)
F i g u r e 4. M i n e r a l and
void fractions
o f two
grains in
contact.
To g e t an i d e a , by t h e o r e t i c a l m e a n s , o f t h e p o s s i b l e c h a n g e i n contact r e s i s t a n c e w i t h a change i n p o r o s i t y , important parameters w i l l be t h e n u m b e r o f g r a i n c o n t a c t s u r f a c e s , t h e i r o r i e n t a t i o n and t h e d i r e c t i o n o f t h e h e a t f l o w . F i g u r e 5 shows t h e maximum and m i n i m u m compaction possible between spherical evenly sized grains. There are three d i f f e r e n t l i n e s of symmetry i n the case of l e a s t compaction poss i b l e . I f t h e heat f l o w i s p a r a l l e l t o each o f t h e axes o f symmetry.
290
11
the r e s u l t i n g heat f l o w w i l l depend o n t h e l o c a l i z a t i o n o ft h e g r a i n c o n t a c t s u r f a c e s . Table 6 shows t h e data f o r d i f f e r e n t o r i e n t a t i o n s based o n a geometrical treatment o fa half-sphere. A form o f r e s u l t a n t contact p o i n t s can be c a l c u l a t e d f o r each o r i e n t a t i o n . T h e weighed average value o ft h e d i f f e r e n t o r i e n t a t i o n sw i l l then be (3+3p+4J"3)/10=1.42. Table
6. Geometrical
treatment o f least compaction
Orientation Number o f c o n t a c t p o i n t s Number o f d i r e c t i o n s Resultant i n the direct i o n o ft h e heat f l o w for one contact point
a_ 1 3
1
possible.
b 2 3
c 3 4
1/|2
1 / p
One l i n e o f s y m m e t r y s u f f i c e s f o r t h e c a s e o f maximum c o m p a c t i o n p o s s i b l e . T h e r e s u l t i n g number o f contact p o i n t s i n t h e d i r e c t i o n o f t h e heat f l o w w i l l t h e n be 3'J"2/p=2.45. T h er e s u l t i n g c o n t a c t p o i n t s have b e e n I n c r e a s e d b y a f a c t o r o f 2.45/1.42»1.7, d e p e n d i n g o n t h e c o m p a c t i o n . Factor (1-p) should t h e r e f o r e decrease correspondingly. Based o n s i m p l e g e o m e t r i c a l r e l a t i o n s h i p s i n f i g u r e 5, t h e p o r o s i t y can b e d e t e r m i n e d a t 0 . 2 6 ( = 1-J'2'n/6) f o r maximum c o m p a c t i o n a n d a t 0.476 (=1-n/6) f o r minimum compaction.
F i g u r e 5. Minimum and maximum c o m p a c t i o n p o s s i b l e o f e v e n l y s i z e d spheres, a, b andc represent d i f f e r e n t l i n e s o f symmetry. However, a q u a n t i t a t i v e d e t e r m i n a t i o n seems t ob e p r o b l e m a t i c t o implement t h e o r e t i c a l l y , andshould thus accordingly be performed e m p i r i c a l l y . However, l e tu s f i r s t t r e a t t h e t h e r m a l c o n t a c t r e s i s t a n c e . The t h e r m a l c o n t a c t r e s i s t a n c e b e t w e e n t h e g r a i n s I s d e t e r m i n e d b y the w i d t h o f t h e c o n t a c t s u r f a c e between t h e m i n e r a l g r a i n s , t h e medium a tt h e contact surface ( i c e / w a t e r / a i r ) a s well a s t h e degree o f
291
12
c o m p a c t i o n . I f t h e heat f l o w can be assumed t o be 1 - d i m e n s i o n a 1 a t t h e c o n t a c t s u r f a c e , t h e n t h e h a r m o n i c mean ( e q u a t i o n ( 1 ) ) o f t h e m i n e r a l phase ( i . e . p ) and t h e v o i d f r a c t i o n s ( i . e . 1-p) c a n be used as an a p p r o x i m a t i o n o f t h e e x i s t i n g r e l a t i o n s h i p s . T h i s e q u a t i o n , d i v i d e d by t h e g r a i n c o n d u c t i v i t y r e p r e s e n t s a dimensionless correct i o n f a c t o r , s u b s e q u e n t l y named t h e t h e r m a l c o n t a c t r e s i s t a n c e . Hence, the f o l l o w i n g equations a r e obtained t o describe t h e thermal r e s i s t a n ce i n t h e d r y and i n t h e w a t e r - s a t u r a t e d s t a t e r e s p e c t i v e l y : "dry
= (1/(pAg+(1-p)Aa'>/^g
'13)
ogat = ( l / ( p A g + ( 1 - p ) A w ) ) A g where
Xg = thermal
(14)
conductivity o f themineral
grains,
=
-"-
water,
x^ =
-"-
a i r , W/(m,K)
W/(m,K)
W/(m,K)
An e m p i r i c a l d e t e r m i n a t i o n o f p h a s b e e n made b y m e a n s o f t h e r m a l c o n d u c t i v i t y measurements on d r y ( 7 6 samples) and w a t e r - s a t u r a t e d s o i l s (185 samples). Measurements on d r y s o i l s a r e t a k e n p a r t l y f r o m own i n v e s t i g a t i o n s , and p a r t l y from l i t e r a t u r e Smith (1939,1942) and Johansen (1975). Only i n t h e case o f Johansen's measurements i s t h e m i n e r a l composition known. However, i t has no great e f f e c t a t high p o r o s i t i e s . Measurements a t t h e highest p o r o s i t i e s where g r a i n c o n d u c t i v i t y exh i b i t s t h e g r e a t e s t i n f l u e n c e w e r e based o n own i n v e s t i g a t i o n s o f t i l l . The mineral composition o f a t i l l , o r i g i n a t i n g from c r y s t a l l i n e rocks, should vary w i t h i n r e l a t i v e l y narrow l i m i t s (Sundberg 1986). The c o r r e l a t i o n t o s a t u r a t e d s o i l s has been p e r f o r m e d f o r own m e a s u r e ments on c l a y , c l a y t i l l , t i l l and peat. The c a l c u l a t i o n s f o r t h e g r a i n s ' thermal c o n d u c t i v i t y i s based on t h e v a r i a t i o n o f quartz c o n t e n t w i t h g r a i n s i z e ( S u n d b e r g , 1 9 8 6 ) . Some o f t h e s a m p l e s l a c k e d a grain d i s t r i b u t i o n curve. Values f o rt h e grain thermal c o n d u c t i v i t y o f these samples were thus assumed. Since t h e g r a i n s i n d i f f e r e n t types of sand can have q u i t e d i f f e r e n t thermal c o n d u c t i v i t y , sand samples are e x c l u d e d from t h e c o r r e l a t i o n s , p has been e x p r e s s e d as a t h i r d d e g r e e p o l y n o m i a l i n o r d e r t o be e a s i l y used i n t h e c a l c u l a t i o n s , p has been determined from c o r r e l a t i o n s w i t h measurements on s o i l s i n t h e p o r o s i t y i n t e r v a l 20-95;^. T h e e x p r e s s i o n o f t h e f u n c t i o n f o r p h a s not been v e r i f i e d f o r p o r o s i t i e s o u t s i d e t h i s i n t e r v a l . F i g u r e 6 i l l u s t r a t e s p as a f u n c t i o n o f p o r o s i t y .
p = 1-0.12833n where n = p o r o s i t y
+ 0.06461n^ fraction.
292
+ 0.05491n'
(15)
1,00
0,99
098
0,97
0,95-
0.95
I 0
.
.
.
I
20
.
I
.
I
40
)
)
.
I
50
.
I
.
I
80
.
I
.
100
POROSITY
F i g u r e 6. p as f u n c t i o n o f p o r o s i t y . A t p o r o s i t i e s h i g h e r t h a n a b o u t 50%, t h e n u m b e r o f c o n t a c t s u r f a c e s g r a d u a l l y decreases, which i s why t h e importance o f c o n t a c t r e s i s t a n c e d i m i n i s h e s and t h e t h e o r y f o r t h e o r i g i n a l s e l f - c o n s i s t e n t method becomes i n c r e a s i n g l y more r e l e v a n t . Given p o r o s i t i e s o f 0.26 and 0.48, the r a t i o between t h e s i z e o f t h e g r a i n contacts can be s a i d t o be a p p r o x i m a t e l y 1 . 4 5 . H e n c e , t h i s r a t i o i s l e s s t h a n t h e p r e v i o u s l y d i s c u s s e d v a l u e . The r e a s o n f o r t h i s may be t h a t t h e a s s u m p t i o n o f e v e n l y s i z e d g r a i n s has n o t been s a t i s f i e d . I t i s v e r y d i f f i c u l t t o o b t a i n a p o r o s i t y as l o w as 0.26 i n an even g r a i n e d m a t e r i a l . Compact i o n experiments on f i n e sand have r e s u l t e d i n a d i f f i c u l t y t o reach a p o r o s i t y lower than approx. 0.35. Water a t t h e grain contact
surface
In a pore system t h a t i s drained, t h e most l o o s e l y bound water, i . e . the water i n t h e l a r g e s t pores, w i l l leave f i r s t . At a higher drainage p r e s s u r e , t h e s m a l l e r pores w i l l g r a d u a l l y be d r a i n e d . The c o n t a c t s u r f a c e w i l l be i n i t i a l l y u n a f f e c t e d by t h e d r a i n a g e o f w a t e r , because it i n v o l v e s t h e smallest pores o r p a r t s o f pores. The g r a i n contact s u r f a c e s w i l l t h e r e f o r e be emptied l a s t d u r i n g d r a i n a g e . T h i s can be e x p l a i n e d m a t h e m a t i c a l l y b y a l o g - f u n c t i o n . I n f i g u r e 7, (A-log(S|.) + 1) i s drawn as a f u n c t i o n o f d e g r e e o f s o i l s a t u r a t i o n . T h e same e x p r e s s i o n has a l s o been used by J o h a n s e n ( 1 9 7 5 ) . The f a c t o r A c o r r e c t s a more r a p i d drainage t i o n . The l o g - f u n c t i o between t h e saturated
t h e l o g - f u n c t i o n . A lower value f o rA represents o f t h e contact surface a t low degrees o f saturan above can t h e r e f o r e be used t o i n t e r p o l a t e and t h e u n s a t u r a t e d s t a t e .
293
14 1.0-
0.8-
A= 0.8^ A= 0,6^
0,6-
OAA - log ( S r ) + 1 0,2-
0,0
Figure
7. T h ef u n c t i o n tion.
20
60
40
«tot = ( A - l o g ( S r ) + 1 ) - ( a s a t -
resistance,
«dry' * " d r y
including eq.
(16)
( 9 ) c a nb e w r i t t e n a s f o l l o w s : -1
^e = 3where
100
(A-1og(S^)+1) as function o f water-satura-
In accordance with t h i s , t h e t o t a l thermal (13) a n d ( 1 4 ) , c a nb e w r i t t e n a s f o l l o w s :
and e q u a t i o n
80
(17)
\ «tot' '^g \ '^w
h
= ^a = 1-n =
(l-Sr)-n
Vapor d i f f u s i o n The e f f e c t o f vapor d i f f u s i o n i s highest a t a high p o r o s i t y , intermed i a t e degrees o f s a t u r a t i o n andhigh temperatures. Vapor d i f f u s i o nc a n be v i e w e d a s a n a d d i t i o n t ot h e r m a l c o n d u c t i v i t y i n a i r . A s u m m a t i o n o f these forms an e f f e c t i v e thermal c o n d u c t i v i t y f o r t h e a i r phase, w h i c h c a nb e w r i t t e n a s f o l l o w s ( P h i l i p a n d de V r i e s , 1 9 5 7 ) : ^ v Kg A. h
(18)
= ^a * ^
= t h e r m a l c o n d u c t i v i t y i n d r y a i r , W/(m,K) = t r a n s p o r t o f l a t e n t h e a t t h r o u g h v a p o r d i f f u s i o n , W/(m,K) = r e l a t i v e humidity
294
15
A c t u a l l y , a n a d d i t i o n t o e q u a t i o n (18) s h o u l d b e made f o r t h e c o n t r i b u t i o n o f water d i f f u s i o n d u et o t e m p e r a t u r e g r a d i e n t . However,' t h i s is g e n e r a l l y o f minor importance. Several authors, such as de V r l e s ( 1 9 5 2 ) and S e p a s k h a h and Boersma ( 1 9 7 9 ) , h a v e o b t a i n e d good a g r e e m e n t between measured a n dc a l c u l a t e d v a l u e s o ft h e e f f e c t i v e t h e r m a l conductivity using equation (18). The pore i s s a t u r a t e d w i t h w a t e r vapor t o t h e w i l t i n g p o i n t o f t h e s o i l i n q u e s t i o n (150 mvp n e g a t i v e p r e s s u r e ) . For a c o a r s e l y graded s o i l , t h i s m e a n s a w a t e r c o n t e n t p e r v o l u m e o f a f e w %, w h i l e f o r c l a y t h i s means 1 0 - 4 0 % . A t water c o n t e n t s below t h e s o i l ' s w i l t i n g p o i n t , an a p p r o x i m a t e s t r a i g h t l i n e i n t e r p o l a t i o n can b e p e r f o r m e d . A n e q u a t i o n f o r \ can b e f o u n d i n e.g. d e V r i e s ( 1 9 6 3 ) . T h e e f f e c t i v e a i r c o n d u c t i v i t y , X^^, r e p l a c e s in equation (17). T h er e l a t i o n s h i p sa t the contact s u r f a c e are n a t u r a l l y a l s o a f f e c t e d . The e a r l i e r d e s c r i b e d o - v a l u e I s d e t e r m i n e d b y I n t e r p o l a t i n g between the contact r e s i s t a n c e In the dry andIn the saturated state. The e f f e c t o fvapor d i f f u s i o n means t h a t a n i n t e r p o l a t i o n s h o u l d a l s o be p e r f o r m e d b e t w e e n t h e a l p h a v a l u e s w i t h and w i t h o u t t h e e f f e c t o f vapor d i f f u s i o n . The f o l l o w i n g e x p r e s s i o n i s suggested:
«tot =
'«tot,v-°tot,a' * " t o t , a
«tot.v = ( A - l o g ( S r ) + 1 ) - ( a s a t - «dry,v' * «dry.v
( 2 0 )
«dry,v = n / ( ^ / X g * n - ^ ) / X ^ y ) ) / X g
(21)
X^^Q
= 0.24
= 10-X.g,
W/(m,K)
^ ^ = 0 . 2 4 i s obtained a ta temperature o fapproximately 40''c. A t t h i s t e m p e r a t u r e , x ^ has v e r y s m a l l i n f l u e n c e o n x^^. A t temperature above 40°C, o. . i n eq. ( 1 9 ) can b e r e p a c e d w i t h "tot V e f f e c t o f eq. ( 1 7 ) i n c l u d i n g eq. ( 1 9 ) has o n l y been c o m p a r e d t o m e a s u r e d v a l u e s f o r t e m p e r a t u r e s a r o u n d 20°C. The e f f e c t o f vapor d i f f u s i o n I n t h e p o r o s i t y i n t e r v a l ( 3 5 - 4 5 % ) I sl i m i t e d but c l e a r . The u n c e r t a i n t y i s h i g h e s t a t low degrees o fs a t u r a t i o n . S i n c e the e f f e c t o fvapor d i f f u s i o n r a p i d l y r i s e s w i t h temperature, the t h e r m a l g r a i n c o n t a c t r e s i s t a n c e becomes l e s s i m p o r t a n t a t h i g h temper a t u r e s . A t h i g h p o r o s i t i e s , e.g. i n p e a t , v a p o r d i f f u s i o n i s o f r e l a t i v e l y h i g h i m p o r t a n c e a l r e a d y a t t e m p e r a t u r e s a r o u n d 20°C. Frozen The diff zing zing
state
c o n d i t i o n a tthe contact surface during water content changes I s e r e n t i n the f r o z e n s t a t e as opposed t o the unfrozen s t a t e . Freeo f t h e ground i s a v e r y complex process. The speed o f t h e f r e e process i s v e r y i m p o r t a n t f o r any p o s s i b l e w a t e r r e d i s t r i b u t i o n .
295
16
A
rapid freezing o fthe
situ,
while a slow
front
w i t h consequent
volume
expands
at
degrees
ly
shown
ship
causes
water
b y about
(Kersten
9X.This
1949)
between
a l l o fthe water
the water
that,
causes than
i nt h e
soil
frozen state,
a linear
i si n c o n t r a s t t o t h e
logarithmic correlation
i n the
s u c t i o n b u i l d s u p i nt h e redistribution
starts
t od r a i n out
unfrozen provide
state.
a t higher
a satisfactory
= ^ r
thermal
degrees
medium
when which
that
the
c a nb e w r i t t e n a s :
the
contact
Accordingly,
the
saturated
' 2 2 )
(frozen) andthe
a l l o fthe water
to
from
finer
be placed
capacity sponds When rated be
the
pores i nt h e
alongside
diagram.
Unfrozen
t oa c e r t a i n
soil,
( 2 3 )
^e
' 2 4 )
freezes
i n a soil, finest water
'^froz
" Thermal
same w a y t h a t
t od r a i n out
temperature
required
scale
can the-
i na w a t e r
retention
a ts a t u r a t i o n thus
corre-
zero.
conductivity o fa p a r t i a l l y
frozen state
frozen andunfrozen water
finest
content,
pores
the
ice
should
c o n d u c t i v i t y c a nbe c a l c u l a t e d
water
