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Water percolation within a deep snowpack field investigations at a site on Mt. Seymour, British Columbia Wankiewicz, Anthony Cyril 1976

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WATER PERCOLATION WITHIN A DEEP SNOWPACK FIELD INVESTIGATIONS AT A SITE ON MT. SEYMOUR, BRITISH COLUMBIA by ANTHONY CYRIL WANKIEWICZ M.Sc, U n i v e r s i t y of A l b e r t a , 1971 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE-REQUIREMENTS FOR THE DEGREE OF ' DOCTOR OF' PHILOSOPHY i n the I n t e r d i s c i p l i n a r y Program •in Hydrology We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA MaM&kt, 1976 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements fo r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia , I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e fo r re ference and study. I f u r t h e r agree t h a t pe rmiss ion fo r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by''the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a I ' g a i n s h a l l not be a l lowed without my w r i t t e n p e r m i s s i o n . T The U n i v e r s i t y of B r i t i s h Columbia 20 75 tvesbrook Place Vancouver, Canada V6T 1W5 i i ABSTRACT The c a p i l l a r y p r e s s u r e and f l u x were measured i n an o l d wet snowpack on Mt. Seymour, B r i t i s h C o l u m b i a , u s i n g t e n s i o -meters and t e n s i o n l y s i m e t e r s . The f l u x was r e g u l a t e d from 0.01 t o 10 cm/hr w i t h i n s n o w p l o t s , u s i n g i s o l a t i o n c o v e r s and i r r i g a t i o n a t t h e snow s u r f a c e . The t e n s i o m e t e r s measured s m a l l - s c a l e changes o f c a p i l l a r y p r e s s u r e i n . b o t h space and t i m e . The extremes o f p r e s s u r e measured d u r i n g f i n e weather snowmelt were -U and -14 cm(H 20). The v e r t i c a l g r a d i e n t s o f p r e s s u r e were much s m a l l e r t h a n the g r a v i t a t i o n a l p o t e n t i a l g r a d i e n t w i t h s t e a d y o r d e c r e a s i n g f l o w s (ythe g r a v i t y d r a i n a g e p o s t u l a t e ) but c o u l d be l a r g e r at a w e t t i n g f r o n t . Assuming Darcy's e q u a t i o n d e s c r i b e s u n s a t u r a t e d f l o w , t h e g r a v i t y . d r a i n a g e p o s t u l a t e means t h a t f l u x measurements d u r i n g s t e a d y o r d e c r e a s i n g f l o w c o u l d be i n t e r p r e t e d as h y d r a u l i c c o n d u c t i v -i t i e s . Snow l y s i m e t e r s w i t h an i n t e r f a c e p r e s s u r e ' o f -25 cm (H^O), were used t o measure f l u x e s o v e r the. s m a l l s c a l e nec-e s s a r y f o r c o mparison w i t h p r e s s u r e and l i q u i d w a t e r c o n t e n t changes w i t h i n snow l a y e r s . The h e i g h t o f the- l y s i m e t e r r i m was s e l e c t e d t o t h e o r e t i c a l l y make the f l o w c o l l e c t i o n 100? e f f i c i e n t . The h y d r a u l i c c o n d u c t i v i t y - c a p i l l a r y p r e s s u r e r e l a t i o n s f o r snow d e s c r i b e h y s t e r e s i s l o o p s between boundary d r y i n g and w e t t i n g c u r v e s . The boundary c u r v e s f o l l o w power f u n c t i o n s w i t h an exponent between 11 and 15, the r a t i o o f p r e s s u r e between the two c u r v e s b e i n g 1.4. The d e r i v a t i v e o f h y d r a u l i c c o n d u c t i v i t y w i t h r e s p e c t t o l i q u i d c o n t e n t , when ' i i i expressed as a r a t i o to the hydraulic conductivity i t s e l f (the •frac t i o n a l d e r i v a t i v e ) , was found to be a slowly changing, function of the snow's hydraulic conductivity. The f r a c t i o n a l derivative would equal the r a t i o of flux-wave speed to fl u x i n the kinematic wave analogy. It had values of 60 to' 125 over a 0.05 to 5 cm/hr range of hydraulic conductivity, according to the r e s u l t s from three d i f f e r e n t experimental methods ( l i q u i d water content changes, wetting front speeds, and drainage wave speeds). i v TABLE OP CONTENTS Page CHAPTER 1 : INTRODUCTION 1 1 . 1 Previous Research of P e r c o l a t i o n i n Wet Snowpacks 1 1 . 2 Development of a R i g i d Porous Media Model 3 1 . 3 Research O b j e c t i v e s 5 1 . 4 Mt. Seymour — the Study Area 7 1 . 5 Mt. Seymour — the Old Wet Snowpack 18 1 . 6 Flow Features of the Snowpack 24 CHAPTER 2 : A RIGID POROUS MEDIA MODEL FOR WET SNOW 34 2 . 1 Old Wet Snow as a R i g i d Medium 34 2 . 2 C a p i l l a r y Pressure i n Snow 36 2 . 3 G r a v i t y Drainage P o s t u l a t e 42 2 . 4 Darcy P o s t u l a t e f o r Unsaturated Flow 44' 2 . 5 H y d r a u l i c C o n d u c t i v i t y R e l a t i o n s 47 CHAPTER 3 : SOME DEVELOPMENTS OF THE MODEL 52 3 . 1 P r o f i l e of C a p i l l a r y Pressure Above an I n t e r f a c e 52 3 . 2 P a t t e r n of C a p i l l a r y Pressure i n Layered Snowpacks 57 3 - 3 T r a n s l a t o r y Waves 59 3 . 4 Kinematic Waves 6 l CHAPTER 4 : FIELD INSTRUMENTATION 66 4 . 1 Measurement of C a p i l l a r y Pressure i n Snow 66 4 . 2 Sources of E r r o r With Tensiometer F i e l d 7^ V Measurements Page 74 4 . 3 Snow Lysimeter Theory 78 4.4 Lysimeter Response 82 4 . 5 Tension Snow Lysimeter 88 4 . 6 S t e e l Snow Lysimeter 97 CHAPTER 5 : EXPERIMENTAL PROCEDURES 98 5 . 1 Regulated Snowplots — Experimental I n s t a l l a t i o n s 98 5 . 2 L i q u i d Water Content Measurements 107 5 . 3 Regulated Snowplots — A r t i f i c i a l C o n t r o l of the Flow 111 5.4 S t a i n i n g the Flow Paths 115 5 . 5 Regulated Snowplots — D e s c r i p t i o n o f Layers and Patterns of Flow. 117 5 . 6 N a t u r a l Melt Wave Experiments 124 CHAPTER 6 : PRESENTATION AND INTERPRETATION OF FIELD DATA 127 6 . 1 C a p i l l a r y Pressure i n Regulated Snowplots 127 6 . 2 C a p i l l a r y Pressure During N a t u r a l Melt 138 6 . 3 Test of G r a v i t y Drainage P o s t u l a t e . 144 6.4 R e l a t i o n Between C a p i l l a r y Pressure and H y d r a u l i c C o n d u c t i v i t y of Wet Snow 146 6 . 5 F r a c t i o n a l D e r i v a t i v e — L i q u i d Measurements 157 6 . 6 F r a c t i o n a l D e r i v a t i v e — T r a n s l a t o r y Waves 159 6 . 7 F r a c t i o n a l D e r i v a t i v e — Kinematic Waves 165 CHAPTER 7 : CONCLUSIONS REFERENCES LIST OF TABLES v u Page 4 . 1 - 1 Tenslometer time l a g f o r 70% response to a step change. 75 5 . 5 - 1 Snowpack experiments on Mt. Seymour, June - J u l y , 1 9 7 4 . 118 6 . 3 - 1 Space-average c a p i l l a r y p r e s s u r e g r a d i e n t s . ri the i n the r e g u l a t e d snowplots. 145 6 . 3 - 2 . C a p i l l a r y p r e s s u r e g r a d i e n t i n snow du r i n g drainage. 145 6 . 4 - 1 Parameters f o r the c a p i l l a r y p r e s s u r e -h y d r a u l i c c o n d u c t i v i t y r e l a t i o n . 147 6 . 4 - 2 F l u x d e t e r m i n a t i o n by tensiometer. 155 6 . 5 - 1 Snow l a y e r d e n s i t y measurements. 158 6 . 6 - 1 F r a c t i o n a l d e r i v a t i v e c a l c u l a t e d from wetting f r o n t speed. 164 v i i i LIST OF FIGURES Page 1 . 4 - 1 L o c a t i o n of Mt. Seymour. 8 1 . 4 - 2 Water e q u i v a l e n t , Mt. Seymour snow course. 9 1 . 4 - 3 Snow d e n s i t y , Mt. Seymour snow course. 9 1 . 4 - 4 F i e l d s i t e topography. 10 1 . 4 - 5 - Weather elements at- Brockton Knoitl. 12 1 . 4 - 6 Experimental s i t e , May 10, 1 9 7 4 . 15 1 . 4 - 7 Experimental s i t e , J u l y 3 0 , 1 9 7 4 . 15 1 . 4 - 8 Map of the experimental s i t e . 16 1 . 4 - 9 Snow depth at the experimental s i t e . 17 1 . 5 - 1 Composite of snowpack p r o f i l e s . 20 1 . 6 - 1 The dye s t a i n p a t t e r n at s i t e G, June 2 2 , 1 9 7 3 . 2 5 1 . 6 - 2 The dye s t a i n p a t t e r n at s i t e G, June 2 5 , 1 9 7 3 i n a s e c t i o n p e r p e n d i c u l a r to that i n the p r evious p i c t u r e . 26 1 . 6 - 3 Close-up of the upper p a r t of the center blue p a t t e r n i n f i g u r e 1 . 6 - 1 . 28 1 . 6 - 4 Close-up of the,middle l e f t of a s e c t i o n , cut 10 cm f u r t h e r back i n t o the s e c t i o n shown i n f i g u r e 1 . 6 - 2 . 30 1 . 6 - 5 Close-up of the center blue p a t t e r n i n f i g u r e 1 . 6 - 1 . 31 1 . 6 - 6 . Close-up of the lower l e f t of f i g u r e 1 . 6 - 2 . 33 2.2-1 I d e a l i z e d s e c t i o n of wet snow g r a i n s . 37 2.2-2 L i q u i d r e t e n t i o n c h a r a c t e r i s t i c s f o r snow. 41 i x 2.5-1 Hypothetical hysteres is loops i n the hydraulic Page c o n d u c t i v i t y - c a p i l l a r y pressure r e l a t i o n . 5 0 3.1-1 Height of the t r a n s i t i o n i n water pressure above an i n t e r f a c e wi thin a wet snowpack. 55 4.1-1.. Diagram of a tensiometer. 67 4.1-2 Tensiometer f o r measuring c a p i l l a r y pressure i n wet snowpacks. 69 4.4-1 E f f e c t i v e saturat ion p r o f i l e s above a zero-tension lysimeter 83 4.4- 2 E f f e c t i v e saturat ion p r o f i l e s above a tension lys imeter . 85 4.5- 1 Diagram of a tension l y s i m e t e r . 89 4.5- 2 Tension lysimeter f o r measuring the f l u x i n wet snowpacks over small s c a l e s . 91 4.6- 1 S teel lys imeter . 96 5.1-1 Diagram of the i n s t a l l a t i o n f o r experiment 1. 99 5.1-2 Diagram of the i n s t a l l a t i o n f o r experiment 3• 101 5.1-3 Diagram of the i n s t a l l a t i o n for experiment 4. 102 5.1-4 View of the experiment 1 i n s t a l l a t i o n . 104 5.1-5 View of the experiment 3 i n s t a l l a t i o n . 105 5.1-6 View of the experiment 4 i n s t a l l a t i o n . 106 5.5-1 Plow pattern i n the 8 - 9 l a y e r . 119 5.5-2 Plow pattern i n the regulated snowplot of experiment 3. 121 5.5-3 Plow pattern i n the regulated snowplot of experiment 4. 123 X 5.6-1 Experiment 2 I n s t a l l a t i o n f o r measuring the Page c a p i l l a r y pressure v a r i a t i o n s from d i u r n a l snowmelt. 125 6.1-1 F l u x at 48 cm depth from flow r e g u l a t i o n at the snow s u r f a c e . 128 6.1-2 C a p i l l a r y pressure at 38 cm depth i n the expe-riment 1 r e g u l a t e d snowplot. 129 6.1-3 D e c l i n i n g c a p i l l a r y p r essure f o l l o w i n g p l a c e -ment of the i s o l a t i o n cover at the snow s u r f a c e . 131 6.1-4 C a p i l l a r y p r e s s u r e i n the experiment 3 r e g u l a t e d snowplot. 132 6.1-5 C a p i l l a r y p r essure i n the experiment 4 r e g u l a t e d snowplot. 133 6.1-6 C a p i l l a r y pressure changes i n the W - 40 l a y e r foil-owing a p p l i c a t i o n of dyed water. 136 6.1- 7 C a p i l l a r y p r essure changes i n the 50 - 54 l a y e r f o l l o w i n g a p p l i c a t i o n of dyed water. 137 6.2- 1 Net r a d i a t i o n and l y s i m e t e r flow measured at the weather s t a t i o n on June 19 - 21, 1974. 139 6.2-2 D i u r n a l c a p i l l a r y pressure v a r i a t i o n s at seven l e v e l s w i t h i n the snowpack. 140 6.2-3 Extremes of c a p i l l a r y p r essure i n the snowpack over the d i u r n a l melt c y c l e . 143 6.4-1 H y d r a u l i c c o n d u c t i v i t y - c a p i l l a r y p r essure r e l a t i o n f o r the 8 - 9 snow l a y e r . 148 x i 6 . 4 - 2 H y d r a u l i c c o n d u c t i v i t y - c a p i l l a r y p r e s s u r e Page r e l a t i o n f o r the 50 - 54 snow l a y e r . 151 6 . 4 - 3 H y d r a u l i c c o n d u c t i v i t y measurements f o r the W - 40 (lower t h i r d ) snow l a y e r . 153 6 . 6 - 1 Two wetting f r o n t s produced f o r experiment 1. 160 6 . 6 - 2 The wetting f r o n t produced f o r experiment 3 . l 6 l 6 . 6 - 3 Two we t t i n g f r o n t s produced f o r experiment 4 . 162 6 . 7 - 1 Drainage wave at 56 cm depth. 166 6 . 7 - 2 The f r a c t i o n a l d e r i v a t i v e of the h y d r a u l i c c o n d u c t i v i t y ' with r e s p e c t to l i q u i d water content as a f u n c t i o n of h y d r a u l i c c o n d u c t i v i t y . 167 x i i LIST OF SYMBOLS SYMBOL D L d h bd h. bw h h h, h. M h •R h r h K K c K K" P DEFINITION" UNIT snow l a y e r t h i c k n e s s cm c h a r a c t e r i s t i c l e n g t h cm a c c e l e r a t i o n of g r a v i t y 2 2 ( 9 8 0 cm/sec ) cm/sec d r y i n g b u b b l i n g pressure head we t t i n g b u b b l i n g pressure head c a p i l l a r y p r essure head ambient c a p i l l a r y p r essure head i n t e r f a c e c a p i l l a r y p r essure head manometer r e a d i n g manometer c a p i l l a r y r i s e l y s i m e t e r ri'm height manometer zero o f f s e t <?. h y d r a u l i c c o n d u c t i v i t y ( c o e f f i c i e n t <8f p e r m e a b i l i t y ) cm/hr s a t u r a t e d h y d r a u l i c c o n d u c t i v i t y cm/hr geometric mean of K and K' cm/sec K f o r which w = OJ cm/sec ab s o l u t e a i r pressure dynes/cm c a p i l l a r y p r e s s u r e (matnic pressure) dynes/cm £ cm(H 2 0 ) cm(H 2 0 ) cm(H.20) cm(H 2 0 ) cm(H 2 0 ) cm cm cm cm 2 SECTION 3 . 2 2 . 4 2 . 2 2 . 2 3 . 1 2 . 2 3 - 1 3 . 1 4 . 2 4 . 2 4 . 3 4 . 2 2 . 4 2 . 5 . 2 . 5 2 . 5 2 . 2 2 . 2 x i i i SYMBOL DEFINITION UNIT SECTION 2 P a b s o l u t e water pressure dynes/cm 2 . 2 w Q manometer 'gauge s e n s i t i v i t y ' cmCH^OVcc 4 . 1 R g Reynolds number 2 . 4 2 R n net shortwave r a d i a t i o n gm/cm 5 . 3 2 R n o net shortwave r a d i a t i o n at z g gm/cm 5 . 2 r l 5 R 2 orthogonal r a d i i of curvature cm 2 . 2 S e f f e c t i v e s a t u r a t i o n 2 . 2 * S ambient e f f e c t i v e s a t u r a t i o n 4 . 4 e S^ i n t e r f a c e e f f e c t i v e s a t u r a t i o n 4 . 4 t time hr 3 - 3 t ^ dynamic response time hr 4 . 4 t s t a r t - u p response time hr 4 . 4 t,p cup response time hr „ 4 . 1 U f l u x wave speed cm/hr 3 . 3 V downward volume f l u x cm/hr 2 . 4 V Q i n i t i a l f l u x cm/hr 3 . 3 V-j^ f i n a l f l u x cm/hr 3 . 3 v pore v e l o c i t y cm/hr 2 . 4 W i n t e r f a c e water storage cm 4 . 4 z' height above datum cm 2 . 4 z t r a n s i t i o n height cm 3 . 1 n z snow s u r f a c e height above datum cm 3 - 3 s 3 a very sma l l number cmCR^O) 3 - 1 8 e x t i n c t i o n c o e f f i c i e n t cm ^ 5 - 2 e a pore s i z e d i s t r i b u t i o n index 2 . 5 n a pore s i z e d i s t r i b u t i o n index. 2 . 5 x i v SYMBOL DEFINITION UNIT SECTION 9 l i q u i d water content, by volume 2 . 2 r e s i d u a l l i q u i d water content ( ' i r r e d u c e a b l e ' water content) 2 . 2 X a pore s i z e d i s t r i b u t i o n index 2.2 u dynamic v i s c o s i t y of water ( 0 . 0 1 7 9 cm 2/sec at 0 . 0 ° C ) cm 2/hr 2 . 3 v e x p o n e n t i a l approximation parameter cm ^ 3 - 1 P d d e n s i t y of i c e s k e l e t o n gm/cc 2 . 1 d e n s i t y of i c e ( 0 . 9 1 8 gm/cc at 0 . 0 at 0 . 0 ° C ) gm/cc 2 . 2 p t o t a l d e n s i t y of snow gm/cc 2 . 2 s P w d e n s i t y of water ( 1 . 0 gm/cc at 0 . 0 ° C ) gm/cc 2 . 2 a s u r f a c e t e n s i o n of water ( 7 5 . 6 dynes/cm at 0.0'°C) dynes/cm 2 . 2 a standard d e v i a t i o n cj) <j> p o r o s i t y 2 . 2 (j) e f f e c t i v e p o r o s i t y 2 . 2 co f r a c t i o n a l d e r i v a t i o n of K with 6 2 . 5 . oo - average w over a range of 6 2 . 5 ACKNOWLEDGEMENTS xv I wish to thank my s u p e r v i s o r , Michael Church, of the Department of Geography, f o r h i s advice and encouragement. Mr. Church i n t r o d u c e d the w r i t e r to f i e l d r e s e a r c h . The use of h i s and B r i a n Sagar's Mt. Seymour weather data Is g r e a t f u l l y acknowledged. Dr. Jan de V r i e s of the Department of S o i l S cience, p r o v i d e d expert t h e o r e t i c a l and experimental a s s i s t a n c e making h i s instruments and l a b o r a t o r y f a c i l i t i e s f r e e l y a v a i l -a b l e . Many ideas i n t h i s workhhave r e s u l t e d from l i v e l y d i s c u s s i o n s with c o l l e a g u e s ? s notably with Drs. Len Chou and Narendar Nagpal. Tom N i c h o l a s s i s t e d d u r i n g the 1 9 7 ^ f i e l d season. My w i f e , C l a i r e , typed the t h e s i s and has encouraged me throughout the course of the work. I would l i k e to acknow-ledge the f i n a n c i a l a s s i s t a n c e of the H.R. MacMIllan f a m i l y and of the U n i v e r s i t y of B r i t i s h Columbia. 1 CHAPTER ONE INTRODUCTION Th i s r e s e a r c h i s concerned with the behaviour of l i q u i d phase water i n deep snowpacks. The l i q u i d i n t r o d u c e d at the sur f a c e by e i t h e r r a i n or snowmelt, penetrates the snow and under s u i t a b l e c o n d i t i o n s runs o f f from the base of the pack. Snow i n g e n e r a l d i f f e r s from other l a y e r s comprising the su r f a c e of the e a r t h i n i t s d e n s i t y , t e x t u r e and s t r u c t u r e , mode of formation, albedo, and most n o t a b l y i n i t s low m e l t i n g p o i n t and hence t r a n s i e n t e x i s t e n c e . The seasonal snow cover i s a vast r e s e r v o i r of the winter p r e c i p i t a t i o n , s u p p l y i n g moisture to the atmosphere, to the s o i l and to overland flow. 1 . 1 PREVIOUS RESEARCH OF PERCOLATION IN WET SNOWPACKS L i q u i d flow w i t h i n snow has been measured at snowplots to determine melt i n d i c e s or t u r b u l e n t exchange c o e f f i c i e n t s f o r g e n e r a l i z a t i o n to a range of su r f a c e c o n d i t i o n s (U.S. Army, 1 9 5 6 ; Nyberg, 1 9 6 8 ; Naruse et a l ) 1 9 7 0 ) . Flow was measured from the discharge of flow c o l l e c t o r s , v a r i o u s l y r e f e r r e d to as f u n n e l s , d r i p pans, or l y s i m e t e r s . The r a t e of flow l e a v i n g the s u r f i c i a l l a y e r c o u l d be determined more a c c u r a t e l y t h i s way than from a b l a t i o n measurements ( i . e . u s i n g snow stakes and d e n s i t y data) since the l y s i m e t e r i n t e g r a t e s the flow over a l a r g e area. In order to o b t a i n b e t t e r melt i n f o r m a t i o n over s h o r t e r 2 p e r i o d s , r e s e a r c h on the l i q u i d phase i n snow has been pursued along f o u r l i n e s . The t r a v e l time approach i s concerned w i t h o b s e r v i n g the l a g i n di s c h a r g e from a flow c o l l e c t o r a f t e r an input at the snow s u r f a c e , or by o b s e r v i n g the passage of a t r a c e r put i n l i q u i d added to the snow s u r f a c e (Rockwood et a l , 1954; G e r d e l , 1 9 5 4 ) . The snowpack was u s u a l l y c o n s i d e r e d to be a "black box" which, on warming to i t s m e l t i n g p o i n t and having a ' r e s i d u a l water storage c a p a c i t y ' s a t i s f i e d throughout, allows the use of streamflow r o u t i n g techniques to c a l c u l a t e the t r a n s i e n t water storage and r e s u l t a n t t r a v e l time of flow waves. An opposite approach has been to measure the p a t t e r n of l i q u i d water storage and flow i n s i d e the snowpack. The l i q u i d water content i s measured d i r e c t l y under n a t u r a l c o n d i t i o n s . The l i q u i d water content or 'free water' i s the p r o p o r t i o n of the snow which i s i n ithe l i q u i d phase. Instruments that have been s u c c e s s f u l l y used to date are : the c a l o r i m e t e r ( L e a f , 1 9 6 6 , Wakahama,1968b), the c e n t r i f u g e (Langham 1 9 7 4 a ) , the d i e l e c t r i c meter (Ambach and Howarka, 1966) and the p r o f i l i n g and r a d i o -a c t i v e snow gage (Smith and Halverson, 1 9 6 9 ) . The p a t t e r n of flow d i r e c t i o n s , made v i s i b l e by dyed water, was photographed by Gerde l ( 1 9 5 4 ) , Wakahama (1968b) and Langham ( 1 9 7 4 a ) . Both kinds of experiment p o i n t out that l i q u i d water flow i n snow i s not homogenous' but f o l l o w s h o r i z o n t a l s t r u c t u r e s l a t e r a l l y and o f t e n flows i n separated v e r t i c a l zones. The l a r g e l o c a l v a r i a b i l i t y of l i q u i d water content p l u s experimental d i f f i c u l t i e s i n d i s t i n g u i s h i n g the phases i n snow made r e p r e s e n t a t i v e measurement of the s m a l l changes i n l i q u i d water content very d i f f i c u l t . The changes i n the s o l i d component of snow i n the presence 3 of i t s l i q u i d have been s t u d i e d q u a l i t a t i v e l y i n the f i e l d (Seligman , 1 9 6 3 ; P.A. Shumskii, 1 9 6 4 ) . There have been recent attempts to put t h i s study on a q u a n t i t a t i v e b a s i s through t h i n s e c t i o n photography (Wakahama, 1 9 6 3 5 1 9 6 8 a ) , the development of thermodynamic models of phase i n t e r a c t i o n (Colbeck, 1 9 7 3 b , Langham ,1974b), and by measurement of the i s o t o p e exchange between the phases (Buason, 1 9 7 2 ) . 1.2 DEVELOPMENT OF A RIGID POROUS MEDIA MODEL Snow i s by d e f i n i t i o n a medium with i n t e r - c o n n e c t e d pores, which d i s t i n g u i s h e s i t from other forms of i c e . An analogy to r i g i d porous media can be made so long as the snow s t r u c t u r e remains approximately r i g i d over the d u r a t i o n of the flow process of i n t e r e s t . I t has been g e n e r a l l y observed of snow, through which l i q u i d has been f l o w i n g f o r some time, that i t s morph o l o g i c a l c h a r a c t e r i s t i c s change slowly with time, r a p i d changes only t a k i n g p l a c e i n the m e l t i n g s u r f i c i a l l a y e r . A r i g i d porous media model w i t h slowly v a r y i n g parameters may apply to deep snowpacks over a long p e r i o d of m e l t i n g . I t could even serve as a p o i n t of departure fj©"r a n o n r i g i d model. An e a r l y comparison of snow with other porous media was made by Horton ( 1 9 1 5 ) . Gerdel (1945) s t a t e d that " i n a r i p e pack, i s o t h e r m a l at 0 . 0°C, ' r e t e n t i o n or t r a n s m i s s i o n of water through the snow mantle i s s i m i l a r to the same process w i t h i n a modera-t e l y coarse sand which i s f r e e of h y d r o p h i l i c p a r t i c l e s and s o l u b l e s a l t s " . The v a r i a b l e s of i n t e r e s t i n snow are water f l u x , l i q u i d 4 water content and c a p i l l a r y p r e s s u r e . Wakahama ( 1 9 6 8 c ) measured the l i q u i d water content of snow samples which were wetted at a 3 2 r a t e of about one cm /cm /hr and compared the data with c a l c u l a -t i o n s based on a model of flow through s a t u r a t e d tubes and i n t h i n f i l m s . A s i g n i f i c a n t advance was made by Colbeck ( 1 9 7 2 a ) d u r i n g h i s a p p l i c a t i o n of kinematic wave theory to wet snowpacks. Kinematic wave theory had f i r s t been used to d e s c r i b e f l o o d movements i n long r i v e r s ( L i g h t h i l l and Whitham, 1 9 5 5 ) , ' Colbeck noted that the movement of f l u x waves through a n a t u r a l snowpack (Sharp, 1 9 5 1 ) as w e l l as through homogeneous prepared snow columns (Colbeck and Davidson, 1 9 7 2 b ) bore a c l o s e resemblance to t h a t of kinematic waves i n r i v e r s — a sharp w e t t i n g f r o n t f o l l o w e d by a long drawn-out drainage t a i l . He was able to d e r i v e the same d e s c r i p t i o n from a porous media model f o r uniform snow ( 1 9 7 2 b ) and f o r l a y e r e d snow ( 1 9 7 3 a ) . I t i s open to q u e s t i o n i f such a d e s c r i p t i o n could be d e r i v e d with flow averaged over a l a r g e s c a l e f o r a snowpack i n which three dimensional flow i s t a k i n g p l a c e . Consequently the g e n e r a l a p p l i c a t i o n of the kinematic wave analogy w i l l r e q u i r e a c t u a l f i e l d t e s t i n g i n a range of wet snowpacks. Important as the kinematic wave theory may prove to be f o r the h y d r o l o g i c m o d e l l i n g of flow i n wet snow, the r i g i d porous media model used i n Colbeck's d e r i v a t i o n of i t , i s of more fundamental i n t e r e s t . The r i g i d model can be used and/or t e s t e d i n those p a r t s of the snowpack where the necessary assumptions are c o r r e c t . The model c o n s i s t s of the Darcy equation f o r unsaturated flow, r a t i o n a l f u n c t i o n s between model v a r i a b l e s , and the g r a v i t y drainage p o s t u l a t e . With the help of Darcy's 5 equation, the f l u x i s r e p l a c e d by another v a r i a b l e — the h y d r a u l i c c o n d u c t i v i t y — which, u n l i k e f l u x , can always be r e l a t e d to the l i q u i d water content and c a p i l l a r y pressure v a r i a b l e s at any c a p i l l a r y pressure g r a d i e n t . Colbeck p o s t u l a t e d a power law r e l a t i o n between h y d r a u l i c c o n d u c t i v i t y and l i q u i d water content f o r snow. The c a p i l l a r y pressure could be ign o r e d by the g r a v i t y drainage p o s t u l a t e . The p o s t u l a t e s t a t e s that d u r i n g the drainage-of a porous medium, i . e . steady or decr e a s i n g . f l u x , the magnitude of the c a p i l l a r y pressure g r a d i e n t i s much small e r than that of the g r a v i t a t i o n a l p o t e n t i a l g r a d i e n t . The p o s t u l a t e reduces Darcy's equation to the very simple form of f l u x equals h y d r a u l i c c o n d u c t i v i t y . Colbeck (1974-<)-) measured the dependence of the c a p i l l a r y pressure of kerosene i n a prepared snow sample on i t s l i q u i d water content, to help show tha t the p o s t u l a t e would be v a l i d over the n a t u r a l range of f l u x e s of i n t e r e s t w i t h i n wet snowpacks. He concluded that g r a v i t y drainage occurs d u r i n g steady or d e c r e a s i n g flows except f o r r e l a t i v e l y narrow zones about some i n t e r f a c e s (air/snow surface,"impermeable i c e sheets w i t h i n the snowpack, impermeable ground s u r f a c e ) or f o r r e l a t i v e l y low f l u x r a t e s i n the snow ( < 1 0 ~ 7 cm/hr). '173 RESEARCH OBJECTIVES The aim of t h i s study i s to i n v e s t i g a t e the v a r i a b l e s i n the r i g i d porous media model — l i q u i d water content, c a p i l l a r y p r e s s u r e , and h y d r a u l i c c o n d u c t i v i t y -- f o r a n a t u r a l wet snow-pack. S p e c i f i c a l l y , the r e l a t i o n s between the v a r i a b l e s are sought. A power law r e l a t i o n between c a p i l l a r y pressure and h y d r a u l i c c o n d u c t i v i t y i s t e s t e d . The r e l a t i o n between the f r a c t i o n a l d e r i v a t i v e of h y d r a u l i c c o n d u c t i v i t y with l i q u i d water content, and the h y d r a u l i c c o n d u c t i v i t y i s a l s o determined Knowledge of the d e r i v a t i v e i s s u f f i c i e n t f o r a p p l i c a t i o n to kinematic wave theory where on l y changes i n l i q u i d water content must be considered. The h y d r a u l i c c o n d u c t i v i t y i s determined by a p p l y i n g the g r a v i t y drainage p o s t u l a t e to Darcy's equation, i . e the h y d r a u l i c c o n d u c t i v i t y i s equated to the f l u x d u r i n g steady-s t a t e or d e c r e a s i n g flow events. The p o s t u l a t e i t s e l f - i s t e s t e d i n the n a t u r a l snowpack. The d e t e r m i n a t i o n of the h y d r o l o g i c r e l a t i o n s r e q u i r e s the development of two types of instruments, which have beien used i n s o i l s c i e n c e r e s e a r c h , f o r use i n wet snowpacks. The c a p i l l a r y pressure i s measured wi t h a tensiometer. Since c a p i l l a r y :P>*?essure measurements i n snow have not p r e v i o u s l y appeared i n the l i t e r a t u r e , an o b j e c t i v e of t h i s work i s to determine the space and time v a r i a t i o n s of pressure i n a snowpack and i n v e s t i g a t e i t s value to the understanding of .the flow process. It i s necessary to i n s t a l l the d i f f e r e n t kinds of instruments very c l o s e together i n order to o b t a i n meaningful r e l a t i o n s h i p s between v a r i a b l e s because the flow p a t t e r n i s q u i t e heterogenous over small s c a l e s . A t e n s i o n l y s i m e t e r i s developed,-which measures f l u x w i t h i n the snowpack, while causing minimal d i s t u r -bance to the h y d r o l o g i c regime above the instrument. Research with other porous media have u s u a l l y taken the phenomenological approach by measuring the r e l a t i o n s of h y d r o l o g i c v a r i a b l e s i n a core or at an u n d i s t u r b e d s i t e , as 7 opposed to d e r i v i n g these r e l a t i o n s from the complex i n t e r n a l geometry of the porous medium. The experimental r e l a t i o n s could then be a p p l i e d to r e l a t e d media or s c a l e d to cover a wider range of media. The r a t i o n a l e f o r measuring the corresponding r e l a t i o n s i n snow at a p a r t i c u l a r s i t e i s s i m i l a r . The r e s u l t s from both r e g u l a t e d and n a t u r a l flow snowplots are d i s c u s s e d i n t h i s work. 1 . 4 MT. SEYMOUR — THE STUDY AREA Mt. Seymour, 1 4 5 0 meters h i g h , l i e s at the southwestern end of the B r i t i s h Columbia Coast Range at l a t i t u d e 4 9 ° 2 3 ' N, l o n g i t u d e 1 2 2 ° 5 7 ' W ( f i g u r e 1 . 4 - 1 ) . I t s weather and c l i m a t e are l a r g e l y dominated by P a c i f i c Ocean weather systems. The subalpine Mountain Hemlock Zone i s found above 9 0 0 meters (Brooke, 1 9 6 6 ) and t h i s zone has a w i n t e r snowcover that i s dense and deep, as can be seen from Mt. Seymour snow course data shown i n 'figure.s 1 . 4 - 2 and • 3 • The t r e e cover i s open i n the upper h a l f of the subalpine zone, t r e e s o c c u r i n g i n -isolated groups about the rocky knbikls ( f i g u r e 1 . 4 - 4 ) . F i t z h a r r i s ( 1 9 7 5 ) has s t u d i e d the snow accumulation on t h i s mountain dur i n g years of above and below normal accumulation. His h i g h e s t sampling s i t e was at 1 2 3 0 meters where he found the t o t a l winter p r e c i p i t a t i o n to range from 1 8 5 0 to 3 5 2 0 mm, water e q u i v a l e n t . About 30% of the winter p r e c i p i t a t i o n f e l l as r a i n , r a instorms o c c u r i n g i n almost every month. Hef(Sound that newly f a l l e n snow had an average d e n s i t y of 0 . 2 6 0 gm/cc and c o n s i s t e d most o f t e n of rimed needles or g r a u p e l . The snow cover l a s t e d 8 0 Z5~ 50 75 100 km. i i i i i SCf\LE F i g u r e 1.4-1 L o c a t i o n of Mt.Seymour 9 S3O0 K kl <c zoo ^> •$ V loO ft 2 o •+- /77? M/A/ \ F i g u r e 1.4-2 Water e q u i v a l e n t , Mt. Seymour snow course, 1070 meters ASL e l e v a t i o n .7 .1 u x .3 0 o I HI-73 + int MAX F i g u r e 1.4-3 Snow density,Mt. Seymour snow course Source: B r i t i s h Columbia Snow S u r v e y " B u l l e t i n s 10 0 1 0 0 2 0 0 3 0 0 4 0 0 i 1 i i i SC fl LE : METERS F i g u r e 1 . 4 - 4 F i e l d s i t e topography Contour i n t e r v a l i s 1 0 meters, no r t h i s towards the top G 1 9 7 3 f i e l d experiments H 1 9 7 4 f i e l d experiments M Weather s t a t i o n Base map:" B.C. I n s t i t u t e of Technology about 230 days at h i s h i g h e s t s i t e . The 1 , 0 0 0 meter e l e v a t i o n can be reached q u i c k l y v i a an a l l -weather road and i s 30 km from downtown Vancouver. As part of a c l i m a t o l o g i c a l study, a weather s t a t i o n was operated on Brockton K n o l l at the 1250 meter e l e v a t i o n on Mt. Seymour ( s i t e M i n f i g u r e 1 . 4 - 4 ) between September, 1972 and November 1974 (M. Church and R.B. Sagar, p e r s o n a l communication). The snow p e r c o l a t i o n experiments were c a r r i e d out nearby. A p r e l i m i n a r y study i n June, 19 73 was c a r r i e d out on an open, l e v e l bench of a n o r t h slope ( s i t e G) 1 0 0 meters east of the s t a t i o n and at the same e l e v a t i o n . The primary experiments i n May-July f 1 9 7 4 were done at an open s i t e , s i t e H. I t s snowpack s u r f a c e sloped down at about 10°ttowards 1 1 0 ° W of n o r t h . S i t e H was l o c a t e d 160 meters SW and 30 meters lower than the weather s t a t i o n . Equipment was back-packed to the s i t e s from the road 1 . 6 km away and 2 0 0 meters below. At the weather s t a t i o n , the instruments were mounted on two separate towers and maintained on a weekly b a s i s . A hygrothermograph ( c a l i b r a t e d p e r i o d i c a l l y ) was housed i n a Stevenson screen, the base of which was a d j u s t e d weekly to be about 1 . 5 meters above the snowpack s u r f a c e . There was a t i p p i n g bucket p r e c i p i t a t i o n gauge — w i t h propane heater and A l t e r w i n d s h i e l d , and a cup anemometer with r e c o r d e r . T h e i r h e i g h t s were ad j u s t e d monthly and were from two to f o u r meters above the snow s u r f a c e . In a d d i t i o n , the s t a t i o n was equipped wi t h a r e c o r d i n g pyranometer f o r measuring incoming short wave r a d i a t i o n . The wi n t e r s n o w f a l l of 1 9 7 3 - 4 produced the h e a v i e s t snow-pack of r e c o r d on Mt. Seymour snow course (see f i g u r e 1 . 4 - 2 ) -30 Figure 1 . 4 - 5 ( a ) Weather elements at Brockton Knoll, Oct. , 1 9 7 3-Mar. , 1 9 7 4 30 lo 1\ M A -10 10 0 I0\ s* : * vs.; H i — / — * 4 w t J J L O i l 1 MfiY JUN? JULY F i g u r e 1 . 4 - 5 ( b ) Weather elements at Brockton K n o l l , A p r . 1 9 7 4 - S e p t . 1 9 7 4 SEPT which provided an extended o p p o r t u n i t y to c a r r y out many experiments the f o l l o w i n g s p r i n g . Data f o r s e l e c t e d weather elements measured at s i t e M are shown i n f i g u r e 1 . 4 - 5 f o r the 1 9 7 3 - 4 h y d r o l o g i c p e r i o d . F o l l o w i n g the c o l d s p e l l of mid-January to e a r l y March, the whole snowpack probably warmed to the m e l t i n g p o i n t very soon a f t e r the heavy r a i n of March 15 due to the p e r c o l a t i n g r a i n and meltwater. Except f o r o c c a s i o n a l n o c t u r n a l snow c r u s t s , and an o c c a s i o n a l new l a y e r of f r e s h c o l d snow, the snowpack r e t a i n e d i t s i s o t h e r m a l c h a r a c t e r over the r e s t of the snow season, and d i d not e n t i r e l y melt away from Brockton K n o l l u n t i l August. T h i s may be considered as a 'warm snowpack', d e f i n e d as one which has an i n t e r i o r temperature, below 30 cm, which stays at or near 0°C through a l l or most of the snow season (Smith, 1 9 7 4 ) . F i g u r e s 1 . 4 - 6 and 1 . 4 - 7 show views of the r e s e a r c h s i t e , s i t e ,H, l o o k i n g SW (downslope), both p i c t u r e s from about the same l o c a t i o n . A map of s i t e H i s shown i n f i g u r e 1 . 4 - 8 . The fence enclosed an area of u n d i s t u r b e d snowpack and a s e r i e s of snowpits which were dug p r o g r e s s i v e l y u p h i l l . The snow height and depth of access p i t s dug i n the snow i s shown i n f i g u r e 1 . 4 - 9 . The p i t s gave access to snow s e c t i o n s w i t h i n which h y d r o l o g i c r e s e a r c h instruments were pl a c e d . The primary snow-p i t was very e x t e n s i v e , and p r o v i d e d a l a r g e w a l l area f o r p r e l i m i n a r y t e s t i n g of apparatus, as w e l l as f a c i l i t a t i n g the d i g g i n g of the subsequent snowpits. The n e c e s s i t y of a l i g n i n g the p i t s so as to have an u p h i l l p i t w a l l p e r p e n d i c u l a r to the payers' slope, i . e . f a c i n g SSW, i n c r e a s e d the problem of p r o t e c t i o n from d i r e c t sunshine. 15 F i g u r e 1 . 4 - 6 Experimental s i t e , May 1 0 , 1974. The depth of snow was 650 cm. Fi g u r e 1 . 4 - 7 Experimental s i t e , J u l y 3 0 , 1974. The depth of snow was 150 cm. The t r e e s on the r i g h t are 10 meters above the ground. Both views look WSW towards Georgia S t r a i t from about the same l o c a t -i o n . ScftLB Zr£KO IS fltfEKflGE HOCK TuRFftCE THE LETTER HAS HEIGHT VARIATIONS- oF ONE Merek.. .s. 1 SITE H I I I I I I I I I I I I 1.1 I [ I I I I I 11 I 11 1 I I I 11 I [ I I I I I [ I I I I I I I I I F 1 1 10 IS MAY io 10 1$ zo JUNE 15 I 5 10 if JULY ZS F i g u r e 1 . 4 - 9 Snow depth-at the experimental s i t e , May-July,- -1974. 18 The decrease i n snow depth d u r i n g the 1974 a b l a t i o n season was measured from a graduated pole ( c e n t e r l e f t of f i g u r e 1 . 4 - 7 ) . The zero of the s c a l e i s based on the average of snow depth measurements made about the s i t e . The decrease i n snow height du r i n g the p e r i o d June 10 - J u l y 3 0 , the p e r i o d i n which the primary experiments were done, was 415 cm. Probably only a few percent o f t h i s f i g u r e i s due to s e t t l i n g about the graduated p o l e . Snow of about constant d e n s i t y equal to 0 . 5 8 gm/cc appeared at the s u r f a c e . The a b l a t i o n was c a l c u l a t e d to be 2 4 l cm of water e q u i v a l e n t , which r e s u l t e d i n a flow of 4 . 8 cm/ day i n t o the snowpack. Rain f e l l on 20 of these days and averaged 0 . 7 4 cm/day over the, whole p e r i o d . 1 . 5 MT. SEYMOUR — THE OLD WET SNOWPACK The only f e a t u r e s normally v i s i b l e i n a s e c t i o n of. the wet snow were i c e sheets and glands (see below). The term'ice sheet' r e f e r s to a l l t h i n sheets w i t h i n the snowpack which appear, i n s e c t i o n , as dark bands as a r e s u l t of t h e i r r e l a t i v e l y h igh i c e content. T h i s e m p i r i c a l d e f i n i t i o n c a r r i e s no i m p l i c a -t i o n as to t h e i r p e r m e a b i l i t y . Most of the i c e sheets c o u l d be f o l l o w e d f o r some d i s t a n c e h o r i z o n t a l l y and were roughly p a r a l l e l to each other. Many of them v a r i e d i n t h i c k n e s s across a s e c t i o n , o f t e n l e n s i n g out. I t was n e v e r t h e l e s s p o s s i b l e to f o l l o w a group of i c e sheets, "an i c e sheet formation", c o n t i -nuously over a long d i s t a n c e . I t was found that i f i c e sheets were numbered i n a s e c t i o n most c o u l d be i d e n t i f i e d i n another s e c t i o n a meter or two away, while the major formations of i c e 19 sheets could be I d e n t i f i e d over 15 meters h o r i z o n t a l l y . These sheets dipped down to the WSW at 5 ° on the average (the range was from l e v e l to 1 2 ° ) . The access p i t s were cut at r i g h t angles to the l a y e r s except below 268 cm (see f i g u r e 1 . 5 - 1 below), the l e v e l of an unconformity. Below t h i s l e v e l , the slope of the sheets changed l a t e r a l l y , so that the p i t could not always be cut p e r p e n d i c u l a r to a l l l a y e r s simultaneously. 'In t h i s case, care was taken to a v o i d having any l a y e r s s l o p i n g away from the snowpit. The great depth of snow and i t s subsequent r a p i d r a t e of a b l a t i o n made i t i m p r a c t i c a l to determine, the ab s o l u t e p o s i t i o n s of snow f e a t u r e s or instruments r e l a t i v e to the rock s u r f a c e below. In a d d i t i o n , each new experiment was c a r r i e d out i n undi s t u r b e d snow. Consequently a l l measurements were made r e l a t i v e to a major i c e sheet or the e q u i v a l e n t p o s i t i o n w i t h i n a major formation of i c e sheets. F i g u r e 1 . 5 - 1 i s a composite of three s e c t i o n s made w i t h i n two meters h o r i z o n t a l s e p a r a t i o n . The s e c t i o n s were matched r e l a t i v e to a major i c e sheet and the composite matched to the snow depth s c a l e of f i g u r e 1 . 4 - 9 from p e r i o d i c measurements of i c e sheet depths below the snow s u r f a c e . Since the rock base had roughness elements of one meter h e i g h t , the base of f i g u r e 1 . 5 - 1 does not occur at the average l e v e l . N e v e r t h e l e s s the ground l e v e l drainage was such that the water t a b l e and c a p i l l a r y f r i n g e t ogether were only 5 cm t h i c k . The i c e sheets were on the average 6 . 4 cm apart and 0 . 3 5 cm t h i c k . (The p r e v i o u s year, June 1 9 7 3 3 at s i t e G., the corresponding f i g u r e s were 6 . 1 cm and 0 . 3 x> H F p e)ctively.) Two t h i r d s of the 1974 i c e sheets were 0 . 3 cm or l e s s i n t h i c k n e s s with one t h i r d Z k -1 A 3 fr • 7 iff f i t r3T 13-S _i2_ SAIOUi SURFAC£\ NfUl SNOU MAY 13-15 MAY f 6S0 HO (2.0 (oo 660 5<K> < ' TffilflOMCTnS] SOO wo Ho ice SHOTS' 0 5 0.6 0.7 ScftLE HEIGHT S~HOU ofA/n ry (fi~/cc) (<w F i g u r e 1 . 5 - l ( a ) Composite of snowpack p r o f i l e s , s i t e H 21 zo ZS ICE GLflNDS (IN SoM£ PHOToi) •-FOREST OF i ICE i c e l e a s e s < ANp 6LAMPS OLD MEAT* -£R££> SORF -ACE i c e LSfiS£S ANJ> tSL.flHPf SHOUT LENTES C J Lf\NpS~ Zl • 30-1 - 3/ 32 Vh 31 37 si; W H <hZ t-30 4-10 310 350 ' 330 <— 310 210' 2 70 EX P. I L e V E L I ISO 2.30 EXP. Z L eve L 1 EXP.Z LEVEL 3 ex P. I LEVEL <t fxp. z LEVEL 5 E x C z LEVEL ( exp. 3 T£NS IO MFT£K°S\ ex P. 2 ' L e ve L 7 ZIO ICE SHEETS °-S 0.6 0.7 SCALE HEICHT SNOU] PENS IT Y (f>~/cc) to„) F i g u r e 1 . 5 - l ( b ) Composite o f snowpack p r o f i l e s , s i t e H (continued) 22 so • $1 5t\ 5* Si 51 i ft' 5 - n to -a— it (3 -a— if at -fr— 41 •3 110 lio I to. 12.0 loo go 60 to — SXP. S LEYE L I £XP. S LEVEL Z £X P. 5 LEYS'L 3 T£NSIoMEraS\ - to fiveRPiCE AOCK SURFACE ~fiT SITE H TOP or •SOMED LftYEf\ .ROCK SUfiEAcA fir PROFILE' LoCft TloN i c e s-Hefrs °'S °-6 ».7 s-CflLf HEIGHT S-NOti PENSITY (f^/ccj fc^j F i g u r e 1 . 5 - l ( c ) Composite of snowpack p r o f i l e s , s i t e H (continued) 23 i n f a c t being about 0 . 1 cm. The t h i c k e r i c e sheets ranged i n t h i c k n e s s up to 2 cm. The t h i c k e s t sheet i n f i g u r e 1 . 5 - 1 ( # 3 4 ) c o n s i s t e d of a h e a v i l y i c e d snow matrix of g r a i n diameter about 0 . 1 cm, with nine l a m i n a t i o n s of a l t e r n a t i n g higher and lower i c e appearance. Tiny i c e glands, a centimeter or l e s s a c r o s s , could be found at some depths. The iqe glands seemed to have a r e l a t i v e l y h igh i c e content, to be roughly c y l i n d r i c a l i n shape and to f o l l o w a s:©mewhat wandering route v e r t i c a l l y through a snow l a y e r . The l a y e r of snow between i c e sheets #33 and #34 was unusual i n having a 'close f o r e s t ' of 1 cm diameter glands, about 1 . 5 cm apart (between adjacent g l a n d w a l l s ) with each gland extending the width of the l a y e r . These glands were s u f f i c i e n t l y s o l i d that i n d i v i d u a l s could be p h y s i c a l l y separated from the surrounding snow, j u s t l i k e i c e sheets could be. A l l sheets and glands which were so i s o l a t e d were q u i t e permeable, a i r c o u l d be blown through them and the l a r g e r pores could be seen(and heard) d r a i n i n g a f t e r they had been immersed i n water. The g r a i n diameter of the snow l a y e r s between the i c e sheets appeared to be roughly uniform at about 0 . 1 cm, on the b a s i s of s e v e r a l spot .measurements. The snow d e n s i t y ( f i g u r e 1 . 5 - 1 ) was determined from three s e c t i o n s , exposed at d i f f e r e n t times. They are not p r e c i s e l y comparable because of p o s s i b l e snow s e t t l i n g or change i n l i q u i d water content between s e c t i o n s . Density was determined by weighing, on a beam balance, 224 cc samples cut from a core produced with a 60 cm l o n g , 5 cm diameter b e v e l l e d s t e e l tube. Mathews and Mackay ( 1 9 6 3 ) measured the l i q u i d water content of Mt. Seymour snow by the c a l o r i m e t e r method i n 1 9 6 1 at Mystery Peak ( f i g u r e 1 . 4 - 4 ) . They found that d u r i n g the p e r i o d March 25 to A p r i l 9 i t rose from 11 to 27$ by weight i n snow of depth 2 0 - 3 0 cm. On June 2 5 , 1 9 7 3 , . the l i q u i d water content of snow i n a l a y e r 60 cm t h i c k h was measured at s i t e G by the author u s i n g a d i f f e r e n t c a l o r i m e t e r (the type of Yos i d a , 1 9 5 9 ) - I t was 20% by weight (±W) i n snow of d e n s i t y 0 . 5 gm/cc and hence 10% l i q u i d water content by volume. In view of the p e r m e a b i l i t y of the i c e sheets and the r e l a t i v e l y s m a l l amount of water they can h o l d by c a p i l l a r y f o r c e , the water content i s e s s e n t i a l l y t h at of the snow between the i c e sheets Since t h i s measurement was done at 2 1 0 0 to 2 2 0 0 hrs and the maximum amount of t r a n s i e n t water i n the Mt. Seymour snow i s r e l a t i v e l y s m a l l (see s e c t i o n 6 . 5 ) , i t seems that t h i s i s e s s e n t i a l l y r e s i d u a l water storage. Most i n v e s t i g a t o r s , have found lower values f o r l i q u i d storage i n d r a i n e d snow (eg. U.S Army Corps o f En g i n e e r s , 1 9 5 6 ) , but s i m i l a r high values are given by De Quervain ( 1 9 7 2 ) . The space and time v a r i a t i o n o f r e s i d u a l l i q u i d water i n snowpacks i s very p o o r l y known. 1 . 6 FLOW FEATURES OF THE SNOWPACK The occurence of r e s i d u a l l i q u i d water s t o r e d throughout the snowpack means th a t when a f l u x wave passes through the pack, i n d i v i d u a l molecules w i l l t r a v e l at a much slower r a t e on the average s i n c e they w i l l mix with those i n the pockets of r e s i d u a l water. I f dyed water i s added c o n t i n u o u s l y at the 25 F i g u r e 1.6-1 The dye s t a i n p a t t e r n at s i t e G, June 22, 1973. The flow has moved towards the observer at the lower p a r t of the ex-posed p r o f i l e because of a slope of about 9°. The maximum slope across the photo, j u s t to the l e f t o f the s y n c l i n e was 7°. On the b a s i s of kinematic wave theory, i t can be shown that the flow wave had p a r t i a l l y d r a i n e d away and what i s seen i s the s t a i n p a t t e r n l e f t behind. The meter r u l e r i s on the right,a60cm i n c h - r u l e r i s near the ce n t e r . S e v e r a l of the i c e sheets are l a b e l l e d beside the photo. F i g u r e 1 . 6 - 2 The dye s t a i n p a t t e r n at s i t e G, June 2 5 , 1 9 7 3 , i n a s e c t i o n p e r p e n d i c u l a r to t h a t In the pre v i o u s f i g u r e , which i n t e r c e p t e d i t at the center blue p a t t e r n t h e r e . Two adjacent c o l o r s had been a p p l i e d t o the snow s u r f a c e . A meter r u l e r i s on the r i g h t and a 6 l cm i n c h - r u l e r i s on the snow s u r f a c e . 27 s u r f a c e , the dye i s c a r r i e d to a l l p a r t s of the snow. The r e s u l t i n g p a t t e r n seen i n a f r e s h cut s e c t i o n shows the d i s t r i -b u t i o n of l i q u i d water i n the snow under the steady s t a t e flow c o n d i t i o n . An instantaneous cut s e c t i o n d u r i n g unsteady flow would show the paths of water i n v a s i o n r a t h e r than l i q u i d water content d i s t r i b u t i o n . In r e a l i t y , s e c t i o n i n g d u r i n g unsteady flow i s not p r a c t i c a l because of the time r e q u i r e d to cut away l a r g e amounts of snow. I f a heavy s h o r t - d u r a t i o n pulse of dyed water i s added to the s u r f a c e , the l i q u i d r a p i d l y enters the snow and soon a f t e r d r a i n s away. The dye, however, remains behind to s t a i n the r e s i d u a l water, only being very slowly leached out by subsequent lower f l o w s . There should be l i t t l e dynamic d i s p e r s i o n a f t e r the wave has passed. M o l e c u l a r d i f f u s i o n from Brownian motion, which r e s u l t e d i n s t a i n i n g the flow paths, i s not s i g n i f i c a n t beyond a few snow g r a i n s . A s e c t i o n cut slater, would show the approximate s p a t i a l p a t t e r n of l i q u i d water d i s t r i b u t i o n i n the snowpack over sm a l l s c a l e s , but the flow paths over l a r g e r s c a l e s . The i n t e r p r e t a t i o n to be given below i s a r e v i s i o n s c a i e s . ^ based on the experience gained from u s i n g tensiometers (see-Chapter Six) . On June 2 2 , 1 9 7 3 p r e l i m i n a r y - t e s t s of the p a t t e r n of water entry i n t o snowpacks were done at s i t e G. About 5 cm of water was a p p l i e d s i m u l t a n e o u s l y to s i x small patches, over a p e r i o d of one or two minutes, to the s u r f a c e of the two meter t h i c k wet snowpack t h e r e . A f t e r 1 1 0 minutes, a s e c t i o n was cut ( f i g u r e 1 . 6 - 1 ) . F i g u r e 1 . 6 - 2 was taken three days l a t e r . I t was a s e c t i o n p e r p e n d i c u l a r to the former, i n t e r c e p t i n g i t at the 28 A -F i g u r e 1.6-3 Close-up of the upper p a r t of the c e n t e r blue p a t t e r n i n f i g -ure 1.6-1. The i c e sheet A does not appear to have a f f e c t e d the p a t t e r n . The i c e sheets B s t o r e water, and f i n g e r s of dye extend below them. The i c e glands are the more h e a v i l y dyed wandering l i n e s w i t h i n the dyed v e r t i c a l s t r i p e s C. p o s i t i o n of the center blue p a t t e r n . These photos show that g r a v i t y i s the dominant f o r c e i n moving the l i q u i d . The s e p a r a t i o n between the dyed zones and the white snow zones i n f i g u r e 1 . 6 - 2 i s an almost v e r t i c a l l i n e i n s p i t e of the l a r g e l a t e r a l d i f f e r e n c e In c a p i l l a r y p r essure when the f l u x wave was moving through the snow. The flow i n t h i s o l d wet pack i s a l s o seen to be s e c o n d a r i l y c o n t r o l l e d by the i c e sheets. F i g u r e 1 . 6 - 3 shows an i c e sheet which has l i t t l e e f f e c t on the flow and s e v e r a l whose e f f e c t i s to s t o r e water and to be the o r i g i n of c o n c e n t r a t i o n s of flow. Many i c e sheets are noted f o r t h e i r e f f e c t i n causing l a t e r a l d i v e r s i o n of the flow. F i g u r e 1 . 6 - 4 shows how water f l o w i n g along an i c e l a y e r c o l l e c t s flow along the way. In a d d i t i o n to d i v e r t i n g the flow, the i c e sheet can produce the flow p a t t e r n t h a t occurs below i t . F i g u r e 1 . 6 - 5 i s an enlargement of a s e c t i o n showing c o n s i d e r a b l e l a t e r a l d i v e r s i o n . Note that i n a l l these cases the dye flows l a t e r a l l y w i t h i n the sheet and not over i t . T h i s can be seen by comparing the u n s t a i n e d and s t a i n e d p o r t i o n s of the same sheet. Neverthe-l e s s , the s p a t i a l s c a l e of l a t e r a l d i v e r s i o n appears to be much small e r than the s c a l e of v e r t i c a l flow ( i . e . the snowpack depth) Because of t h i s f a c t , one dimensional flow models, such as the kinematic wave analogy may apply over h o r i z o n t a l s c a l e s l a r g e r than the snowpack depth. Between the i c e l a y e r s can be seen two kinds of flow p a t t e r n . The f i r s t k i n d i s the fi n g e r ' flow p a t t e r n . When a l a r g e amount of water enters snow, e s p e c i a l l y i f that snow i s dry or c o n t a i n s Ice glands, the water w i l l flow as narrow t r i c k l e s . 30 E -H-F i g u r e 1 .6-4 Close-up of the middle l e f t of a s e c t i o n , cut 10cm f u r t h e r back i n t o the s e c t i o n shown i n f i g u r e 1 . 6 - 2 . Yellow dyed water flowed down to the l e f t along i c e sheet E, accumulating blue dye from above along the way. 31 F i g u r e 1 . 6 - 5 Close-up of middle p a r t of the center blue p a t t e r n i n f i g -ure 1 . 6 - 1 . T h i s f i g u r e overlaps f i g u r e 1 . 6 - 3 above i t . The i c e sheet H d i v e r t e d the flow to the r i g h t . A f i n g e r flow p a t t e r n , with some conspicuous i c e glands, i s found above t h i s i c e sheet, and a sheet flow p a t t e r n i s found below i t . The s c a l e i s i n inches. 32 The t r i c k l e s have a c h a r a c t e r i s t i c wandering appearance, o f t e n branching. Indeed, the i c e glands themselves may have been formed from p a r t i a l f r e e z i n g of t r i c k l e s e n t e r i n g a c o l d pack, s i n c e they look l i k e f r o z e n t r i c k l e s of flow. In time, the water d i s p e r s e s sideways from these t r i c k l e s to wet the snow about them. In f i g u r e 1.6-3 the d i s p e r s e d c o l o r i n g from the o r i g i n a l t r i c k l e s can be seen. The o r i g i n a l t r i c k l e s f o l l o w e d o l d i c e glands which are'the more h e a v i l y dyed wandering l i n e s i n the photos. The p a t t e r n of f i n g e r flow occurs i n the upper two-thirds of the s e c t i o n i n f i g u r e 1.6-1. The second type of flow p a t t e r n commonly found between l a y e r s may be c a l l e d the sheet flow p a t t e r n , which i s e i t h e r a uniform p a t t e r n , or one with v e r t i c a l c o n c e n t r a t i o n s of flow, du:e to the divergence caused by i c e sheets above. The lower one t h i r d of the s e c t i o n i n f i g u r e 1.6-1 i s a sheet flow p a t t e r n . F i g u r e 1.6-5 shows an i c e sheet that has gathered the flow from a f i n g e r flow p a t t e r n , i n t o a sheet £Iow p a t t e r n . In f i g u r e 1.6-6 are the d e t a i l s of a sheet flow p a t t e r n . The two kinds of flow p a t t e r n appear to have c h a r a c t e r i s t i c a l l y d i f f e r e n t l a t e r a l v a r i a t i o n s i n c a p i l l a r y p r e s s u r e . 33 H J F i g u r e 1 . 6 - 6 Close-up of the lower l e f t of f i g u r e 1 . 6 - 2 , showing the det a i l s of a predominantly sheet slow p a t t e r n . An upper l i m i t to the l a t e r a l d i s p e r s i o n i s the width of green v e r t i c a l s t a i n . 3 3 A SUMMARY An understanding of the process of l i q u i d water flow w i t h i n snowpacks would improve our measurement methods f o r f o r e c a s t i n g snow t r a f f i c a b i l i t y and snowmelt r a t e s . Current developments i n remote s e n s i n g methods make t h i s understanding very t i m e l y . Dye t r a c i n g experiments have demonstrated t h a t the flow p a t t e r n w i t h i n snowpacks can be q u i t e complex due to s t r u c t u r e s w i t h i n the snow. However, i n deep snowpacks which have been wet f o r some time ( o l d wet snowpacks), the flow pat-t e r n i s predominately v e r t i c a l i n d i r e c t i o n . The w r i t e r per-formed flow experiments i n 1 9 7 4 at a s i t e on Mt. Seymour near Vancouver at' 1 2 0 0 m MSL i n the Coast Range of B r i t i s h Columbia. The snow there was very deep t h a t year, 6 5 0 cm t h i c k on May 1 0 , and had been wet f o r over a month before the experiments were begun. The r e s u l t s of the experiments have been analyzed i n terms of a r i g i d porous medium model to be d i s c u s s e d i n the next chapter. 34 CHAPTER TWO A RIGID POROUS MEDIA MODEL FOR WET SNOW In developing h i s analogy of kinematic waves to d e s c r i b e n a t u r a l flow waves i n snow, Colbeck ( 1 9 7 2 a , 1 9 7 2 b , 1 9 7 4 ) made use of a r i g i d porous media model. The model c o n s i s t s of the f o l l o w i n g concepts : (a) Wet snow i s a r i g i d porous medium over a short time s c a l e . (b) The Darcy equation a p p l i e s to a l l unsaturated flow s i t u a t i o n s . (c) A g r a v i t y drainage p o s t u l a t e a p p l i e s d u r i n g steady or dec r e a s i n g flows. (d) There are simple r e l a t i o n s between the h y d r a u l i c conduc-t i v i t y and both l i q u i d water content and c a p i l l a r y p r e s s u r e . The o b j e c t i v e s of the w r i t e r ' s work are to t e s t the g r a v i t y drainage p o s t u l a t e and to determine the h y d r a u l i c c o n d u c t i v i t y r e l a t i o n s from corresponding f l u x r e l a t i o n s measured i n a wet snowpack. S p e c i f i c a l l y , a f u n c t i o n based on changes i n l i q u i d water content and a power law f u n c t i o n of c a p i l l a r y pressure are i n v e s t i g a t e d . 2 . 1 OLD WET SNOW AS A RIGID MEDIUM Nat u r a l snowcover s e t t l e s as a r e s u l t of the tendency of snow g r a i n s to reach an e q u i l i b r i u m s i z e and shape by means of molecular t r a n s f e r s through the l i q u i d s t a t e , vapor s t a t e , or along snow g r a i n s u r f a c e s . The changes are p a r t i c u l a r l y r a p i d 35 i n the case of l i q u i d water e n t e r i n g a c o l d snowpack f o r the f i r s t time. However, when l i q u i d water has been f l o w i n g through a snowpack f o r a long time — an o l d wet snowpack — the changes are c o n s i d e r a b l y slower. The snow g r a i n s i z e and shape i s s u f f i c i e n t l y c l o s e to e q u i l i b r i u m that the molecular t r a n s f e r s are e i t h e r very slow or are dominated by e q u a l l y slow t r a n s f e r s w i t h i n the s o l i d s t a t e i t s e l f . I f a wet snowpack i s assumed to s e t t l e by Newtonian creep, i t s r a t e of s e t t l i n g can be c a l c u l a t e d i n terms of a ''compactive v i s c o s i t y ' . T h i s v i s c o s i t y i s the i n v e r s e of the f r a c t i o n a l decrease of l a y e r t h i c k n e s s per u n i t time, per u n i t overburden s t r e s s . In c o n t r a s t to dry snow, l i t t l e i n f o r m a t i o n i s a v a i l -able on t h i s parameter f o r wet snow. Kojima (1966) has shown the v a r i a t i o n with d e n s i t y of the compactive v i s c o s i t y of some i n i t i a l l y low d e n s i t y wet snow samples. The w r i t e r has c a l c u l a t e d the v i s c o s i t y of medium d e n s i t y wet snow, from p u b l i s h e d data on the s e t t l i n g between marked l a y e r s over time-w i t h i n snowpacks In the S i e r r a Nevada and i n the A l p s . The S i e r r a Nevada data i s f o r the deeper snow l a y e r s shown i n f i g u r e ' 8 - 2 i n U.S. Army Corps of E n g i n e e r s " ( 1 9 5 6 ) f o r the p e r i o d beginning one month a f t e r the snow became i s o t h e r m a l ' i n the s p r i n g . The r e s u l t i n g rough estimate f o r the compactive 2 v i s c o s i t y i s 3 2 , 0 0 0 gm-day./cm , f o r snow l a y e r s of 0.5 gm/cc t o t a l d e n s i t y . The Alps data i s i n Hughes and Seligman (1940) 2 and gives a compactive v i s c o s i t y of 2 0 , 0 0 0 gm-day/cm f o r snow of t o t a l d e n s i t y equal to 0.46 gm/cc. In both sources, the snow d e n s i t y was measured near to where the s e t t l i n g was measured. At the w r i t e r s f i e l d s i t e H, the t o t a l snow d e n s i t y was 36 about 0 . 5 9 gm/cc, so that the compactive v i s c o s i t y there may be 2 even g r e a t e r than 3 0 , 0 0 0 gm-day/cm , s i n c e i t probably i n c r e a s e s with d e n s i t y . Each experiment on Mt. Seymour never l a s t e d more 2 than a week i n snow l a y e r s whose over-burden was about 30 gm/cm. The d e n s i t y of the i c e s k e l e t o n , [> > should have changed by » • 0 . 59 X 30 X 7 n n n l n / A p d =2 — 30 ooo = °- 0 0^ 1 S m / c c or < 0 . 7 % d u r i n g each experiment. The assumption that o l d wet snow below the s u r f i c i a l melt l a y e r was a r i g i d porous medium, appears to be a good approximation. 2.2 CAPILLARY PRESSURE IN SNOW An i d e a l i z e d - s e c t i o n of the three dimensional geometry of a i r , s o l i d and l i q u i d i n snow i s shown In f i g u r e 2.2-1. R e s i d u a l l i q u i d water may be found as i s o l a t e d r i n g s surrounding g r a i n c o n t a c t s as shown i n the f i g u r e . When flow i s t a k i n g p l a c e through the snow, the r i n g s are u n i t e d by b r i d g e s , such as those between some of the c a p i l l a r y r i n g s shown. The a i r -water i n t e r f a c e s , whether of r i n g s or b r i d g e s , have curvatures obeying the K e l v i n equation: ( 2 . 2 - 1 ) P - P = o ( - + - ). v ' w a r 1 r 2 The pressure d i f f e r e n c e from atmosphere, P w - P , i s r e l a t e d to the orthogonal r a d i i of curvature of an.air-water i n t e r f a c e , r ^ and r„, and to a, the s u r f a c e t e n s i o n of water (a = 7 5 . 6 37 F i g u r e 2.2-1 I d e a l i z e d s e c t i o n of wet snow g r a i n s 38 dynes/cm at 0.0°C). The c a p i l l a r y p r e s s u r e , P , i s d e f i n e d to be (P - P ). The c a p i l l a r y p r essure head, h , i s d e f i n e d by: W ci C (2.2-2) h c = P c / p w g where p w i s the d e n s i t y of water (1.0 gm/cc at 0.0°C). Por b r e v i t y , h c w i l l be r e f e r r e d to as c a p i l l a r y pressure or j u s t p ressure but the meaning w i l l be c l e a r from the u n i t s which are i n centimeters of water. In unsaturated c o n d i t i o n s , h I s ' c a negative number. In an unsaturated porous medium, the l i q u i d water content depends on the c a p i l l a r y p r e s s u r e . The l i q u i d water content, 0, i s the v o l u m e t r i c f r a c t i o n of the medium occupied by l i q u i d water. The r e l a t i o n of l i q u i d water content to c a p i l l a r y p r e s s u r e , 6(h ) cannot be determined a n a l y t i c a l l y from equation 2.2-1 because of the complex boundaries. In p r a c t i c e , i t has been determined e m p i r i c a l l y f o r each medium of i n t e r e s t . The 8(h ) r e l a t i o n i s not unique but depends on the h i s t o r y of wetting or d r y i n g of the medium. The l i q u i d water content i s g r e a t e r d u r i n g d r y i n g than d u r i n g wetting f o r a giv e n value of c a p i l l a r y p r essure ( h y s t e r e s i s ) . The sets of h y s t e r e s i s loops i n 9(h ) f a l l between two boundary curves which are determined from the extreme cases of we t t i n g an i n i t i a l l y dry medium — the boundary wetting curve (B.W.C.) -- and by d r y i n g an i n i t i - - ,: a l l y s a t u r a t e d medium — the boundary d r y i n g curve (B.D.C.). Brooks and Corey (1966) found f o r a wade range of porous 39 media ( c o n s o l i d a t e d or not, und i s t u r b e d or sieved)' that the r e t e n t i o n curve of o i l (the boundary d r y i n g curve) f o l l o w e d a power furicI7ion U : l - 3 ) S* = (h.,/h ) X f o r h < h. . - v bd c c — bd S* = 1 f o r h > h. . , c bd ' In t h i s equation the e f f e c t i v e s a t u r a t i o n , S , takes the p l a c e of the l i q u i d water content, 9, being expressed i n terms of i t i n (2.2-4") * 9.-9. ( 2 . 2 - 4 ) s = <j> - e . where <j)iis the p o r o s i t y of the medium, and h^d 5 a n ( l ^ are determined by f i t t i n g the data t o equation 2 . 2 - 3 . The method i s give n inddet.all i n Appendix I I of Brooks and Corey ( 1 9 6 4 ). The three parameters are c a l l e d the d r y i n g b u b b l i n g p r e s s u r e , the r e s i d u a l water content, and an index of pore s i z e d i s t r i b u t i o n , r e s p e c t i v e l y . The denominator of equation 2 . 2 - 4 i s c a l l e d the e f f e c t i v e p o r o s i t y , <j> . h ^ i s approximately the pressure head below which the a i r phase becomes continuous throughout the medium, and above which the medium i s p r a c t i c a l l y s a t u r a t e d . It can be d i r e c t l y measured i n a uniform t e x t u r e d medium by measu-r i n g the height of the c a p i l l a r y f r i n g e f o l l o w i n g immersion of a p o r t i o n of the medium i n water. As 9 approaches 9^, dh c/d8 becomes very l a r g e . As (9 - 8^) ->• 0, 9^ approaches the r e s i d u a l water content (or ' i r r e d u c i b l e water content') where each r i n g of 4 0 water i s i s o l a t e d from each other. X was found to i n c r e a s e with d e c r e a s i n g range of pore s i z e s . The w r i t e r attempted to measure the r e t e n t i o n curve f o r two blocks of snow taken from n a t u r a l snowpacks at the m e l t i n g temperature. Each b l o c k was excavated and then t r a n s p o r t e d i n a j a c k e t which c o u l d be atta c h e d to a base p l a t e i n the l a b o r a t o r y . The base p l a t e supported a l a y e r of f i n e g l a s s beads through which a negative pressure c o u l d be a p p l i e d to the l i q u i d water i n the snow. P o r t i o n s of l i q u i d were drawn o f f between s u c c e s s i v e l y i n c r e a s i n g l e v e l s of s u c t i o n . Because of some indeterminacy i n the l e v e l of s a t u r a t i o n at the s t a r t of the experiments, accumulated water content was c a l c u l a t e d from the r e s i d u a l water content. T h i s was done by choosing 0 . such that 0 - 0 . versus h J • 1 l c was a s t r a i g h t l i n e on l o g - l o g paper as r e q u i r e d by equation 2 . 2 - 3 . S was then c a l c u l a t e d f o r each value of h coby d i v i d i n g (0 - 0 ^ ) by the e f f e c t i v e p o r o s i t y , cf> . Now (J) i s giv e n by : ( 2 . 2 - 5 ) ' * P. ( 2 . 2 - 5 ) = i - - s + (P - - 6 . ) Y p . w p . 1 i l where p , p . , and p are the d e n s i t i e s of the snow, of pure i c e s 1 w and of water, r e s p e c t i v e l y . I f P i s measured when ^ i s c l o s e to S^P^/P^ i . e . when the sample i s q u i t e dry, § i s giv e n a p p r o x i -mately by (1 - P /P . ) . The snow bl o c k and the r e t e n t i o n curve S 1 apparatus were packed i n snow and p r o t e c t e d as much as p o s s i b l e from short wave r a d i a t i o n . The very low flows o b r t a i n e d at the more negative pressures i n d i c a t e s that i n t e r n a l melt was not important d u r i n g t h a t p o r t i o n of the t e s t s . U n f o r t u n a t e l y , the p o s s i b l e s e t t l i n g of the snow bl o c k was not measured. I f i t ( l i n e a r s c a l e s ) ( l o g a r i t h m i c s c a l e s ) F i g u r e 2 . 2 - 2 L i q u i d r e t e n t i o n c h a r a c t e r i s t i c s f o r snow F e b r u a r y , 760 meters e l e v a t i o n , 0° s l o p e , 0°C snow t e m p e r a t u r e . (At end o f the experiment: p = 0.42 gm/cc, 0.15 cm g r a i n s i z e , e = 0.07 * 0.04.) J u l y , 1220 meters e l e v a t i o n , 36°N. s l o p e , 0°C snow t e m p e r a t u r e . (At end o f the expe r i m e n t : P a = 0.56 gm/cc, 6 = 0.19 - 0 . 0 5 . ) o c c u r r e d , i t would probably have taken p l a c e most r a p i d l y near the s a t u r a t e d c o n d i t i o n . The e f f e c t i v e s a t u r a t i o n - c a p i l l a r y p r e ssure r e l a t i o n s , S*(h ), are shown i n f i g u r e 2 . 2 - 2 to be r e p r e s e n t e d by a power f u n c t i o n except near s a t u r a t i o n . In view of the u n c e r t a i n t i e s i n these experiments and the e x i s t e n c e of only two curves, the e x t r a p o l a t i o n of the data to n a t u r a l snow-packs should be done with c a u t i o n . The pore s i z e d i s t r i b u t i o n index, A , of each curve may be compared with X = 3 - 9 given f o r two sandstone cores (Brooks and Corey, 1 9 6 6 ) . Por purposes of c a l c u l a t i o n , the nominal values h ^ = - 4 . 0 cm and X = 4 . 0 w i l l be used i n the f o l l o w i n g s e c t i o n s . . 2 . 3 THE DARCY POSTULATE FOR UNSATURATED FLOW Volume f l u x i s the volume of l i q u i d c r o s s i n g a u n i t area i n u n i t time. The Darcy equation r e p l a c e s the f l u x with another v a r i a b l e - h y d r a u l i c c o n d u c t i v i t y , wich, u n l i k e the f l u x i t s e l f , can be r e l a t e d to the l i q u i d water content and c a p i l l a r y pressure independently of the c a p i l l a r y p r essure g r a d i e n t . The Darcy P o s t u l a t e s t a t e s t h a t the downward f l u x i s p r o p o r t i o n a l to the p o t e n t i a l g r a d i e n t , i . e . f o r v e r t i c a l flow : 9h ( 2 . 3 - D V = K ( g ^ c + 1) . K i s the h y d r a u l i c c o n d u c t i v i t y , z i s the height above a datum, 3h c/9 z i s the c a p i l l a r y p r essure g r a d i e n t . K i s a constant with r e s p e c t to changes, i n 9 h c / 9 z , but i t may be a f u n c t i o n of the l i q u i d water content or of the c a p i l l a r y p r essure i t s e l f . A l a r g e number o f d i v e r s e s o i l water problems have been s u c c e s s f u l l y d e s c r i b e d by u s i n g t h e o r y based on Darcy's p o s t -u l a t e f o r u n s a t u r a t e d f l o w . S w a r t z e n d r u b e r (1966) has e x t e n s -i v e l y d i s c u s s e d the i n s t a n c e s o f d e v i a t i o n from the p r o p o r t -i o n a l e x p r e s s i o n . M o r e - t h a n - p r o p o r t i o n a l b e h a v i o u r appears t o be produced by s u r f a c e e f f e c t s o f c l a y p a r t i c l e s i n t h e media concerned. On t h e o t h e r hand, columns o f c o a r s e sand which were t i l t e d a t d i f f e r e n t a n g l e s showed no change i n K f o r a g i v e n water c o n t e n t ( C h i l d s and C o l l i s - G e o r g e , 1950). L e s s - t h a n - p r o p o r t i o n a l b e h a v i o u r o c c u r s i f t h e i n e r t i a l e f f e c t s o f the f l o w become comparable t o the v i s c o s i t y e f f e c t s The r a t i o o f t h e e f f e c t s i s e x p r e s s e d i n the Reynolds number (2.3-2) R e = vdp w / y where d i s a l e n g t h c h a r a c t e r i s t i c o f t h e f l o w , and y the p dynamic v i s c o s i t y (y = 0.0179 dyne-sec/cm f o r water a t 0.0°C) The pore v e l o c i t y , v, i s g i v e n by (2.3-3) v = V/e . The c r i t i c a l Reynolds number i s about one f o r t h e s a t u r a t e d porous media s t u d i e s i n Pancher, Lewis and Barnes (1933, as g i v e n i n Muskat, 1937) when d . i s d e f i n e d t o be the mean g r a i n d i a m e t e r . The Reynolds number s h o u l d be much l e s s t h a n one under normal melt o r r a i n i n p u t s t o t h e snowpack, s i n c e t h e c h a r a c t e r i s t i c l e n g t h d d u r i n g u n s a t u r a t e d f l o w i s the s m a l l d i m e n s i o n o f r i n g s and b r i d g e s o f the water among t h e snow g r a i n s . But even i f d i s t a k e n t o be as l a r g e as the g r a i n 44 s i z e , say d = 0 . 1 cm and 9 as s m a l l as 0 . 0 1 , a r a i n f a l l r a t e t h a t produces a f l u x i n the snow equal to V = R y9/dp = 1 x 0 . 0 1 7 9 x 0 . 0 1 / 0 . 1 x 1 = 1 . 8 x 10. cm/sec, e w or 6 . 3 cm/hr, i s r e q u i r e d f o r R g to exceed one. A f t e r r e v i e w i n g the use of water flow theory i n s o i l s , K l u t e ( 1 9 7 3 ) concluded t h a t as a f i r s t approximation (and sometimes a very good one) the theory embodied i n [the Darcy p o s t u l - • ate and the c o n t i n u i t y equation] i s s u f f i c i a n t l y v a l i d f o r p r e d i c t i n g unsaturated flow behaviour. Even when i t i s not a good model, i t p r o v i d e s a p o i n t of departure f o r a m o d i f i e d or extended theory to t r e a t more complicated systems." In t h i s work, the Darcy p o s t u l a t e i s assumed to d e s c r i b e unsaturated flow i n snow. The complete Darcy equation i s u s e f u l f o r d i s c u s s i n g flow problems where the c a p i l l a r y p r essure g r a d i e n t s are of com-parable s i z e to the g r a v i t a t i o n a l - p o t e n t i a l g r a d i e n t . Within a w e t t i n g f r o n t t h a t i s e n t e r i n g a snowpack, the c a p i l l a r y p r e ssure g r a d i e n t s may be l a r g e and Colbeck ( 1 9 7 4 ) has a p p l i e d Darcy's equation to d e s c r i b e t h i s problem. 2 . 4 THE GRAVITY DRAINAGE POSTULATE The f l u x can be c o n s i d e r e d as an independent v a r i a b l e i n snow because i t i s u s u a l l y imposed at the snowpack s u r f a c e . When l i q u i d flow at the s u r f a c e ceases, i n t e r n a l drainage continues w i t h i n the snowpack, a process r e f e r r e d to as re-d i s t r i b u t i o n , as l o n g as the r e s i d u a l l i q u i d water content has been s a t i s f i e d . Unsteady flow has been d e s c r i b e d by 45 Colbeck i n columns of a r t i f i c i a l l y packed homogeneous snow (Colbeck and Davidson, 1972b), and i n n a t u r a l f i r n (Colbeck, 1972a). He observed that the l e a d i n g edge, the w e t t i n g f r o n t , of the f l u x wave d i s t o r t e d i n t o a shape t h a t can be approximated as a shock f r o n t and that the shock p a r t f i r s t grew and then decayed with depth. The maximum f l u x of the wave decreased and the minimum i n c r e a s e d with depth u n t i l at a s u f f i c i e n t depth steady flow was approached. The t r a i l i n g edge of the wave, the drainage 'wave', became i n c r e a s i n g l y elongated with depth. A long t r a i l i n g edge f o r the drainage wave i s a s m a l l v e r t i c a l f l u x g r a d i e n t . The l i q u i d water content g r a d i e n t as w e l l as the c a p i l l a r y pressure g r a d i e n t should a l s o be s m a l l s i n c e these v a r i a b l e s are .not s e n s i t i v e to changes i n f l u x . The g r a v i t y drainage p o s t u l a t e s t a t e s t h a t d u r i n g steady or d e c r e a s i n g f l u x , the c a p i l l a r y p r essure g r a d i e n t i s much s m a l l e r than the g r a v i t a t i o n a l p o t e n t i a l g r a d i e n t . From the p o s t u l a t e , the Darcy equation takes the f o l l o w i n g simple form: (2.4-1) V = K . Hence, the v a l i d i t y of the g r a v i t y drainage p o s t u l a t e would allow the d e t e r m i n a t i o n of K from measurements of the f l u x . The d e t e r m i n a t i o n of K i n t h i s work i s based on (2.4-1) from f l u x measurements i n o l d wet snowpacks. An o b j e c t i v e of t h i s work i s to t e s t the p o s t u l a t e . i n the f i e l d . Laboratory measurements have been done by P r i l l et a l (1965) f o r drainage from sand columns. T h e i r work showed that as the columns became unsa t u r a t e d , two zones e s t a b l i s h e d themselves w i t h i n the columns: a lower end zone which had c l o s e to minus-one c a p i l l a r y pressure g r a d i e n t , whose t h i c k n e s s slowly i n c r e a s e d w i t h time; and an upper zone which had only a s m a l l pressure g r a d i e n t . The z o n a t i o n was most c l e a r l y marked i n the coarse g r a i n e d sand columns. There, as long as the c a p i l l a r y pressure at a given height above, the..... water t a b l e i n the column d i d not have a numerical value c l o s e to t h a t of the h e i g h t , g r a v i t y drainage was a good approximation to r e a l i t y . T h e i r experiments a l s o showed t h a t a f t e r an i n i t i a l l y r a p i d establishment of the lower end zone to a height about the order of t h a t of the c a p i l l a r y f r i n g e , i t s continued growth became very slow. They estimated t h a t i t would take two months to a year f o r the whole 1 5 0 c m l e n g t h of the columns to reach e q u i l i b r i u m . I t can be shown from kinematic wave theory that there i s a t h i r d zone d u r i n g drainage, an upper end zone whose t h i c k n e s s i n c r e a s e d from the s u r f a c e downwards with time, where the c a p i l l a r y pressure g r a d i e n t again becomes important (Colbeck, 1 9 7 4 c ) . The t h i c k n e s s of the upper end zone i s shown i n equation 3 - 4 - 1 3 to be a couple of centimeters. Since the f u l l Darcy equation would be r e q u i r e d to d e s c r i b e the two end zones, a l l the w r i t e r ' s experiments were done In the middle ' g r a v i t y drainage' zone. 47 2 . 5 THE HYDRAULIC CONDUCTIVITY RELATIONS The h y d r a u l i c c o n d u c t i v i t y i s r e l a t e d to the l i q u i d water content (or e f f e c t i v e s a t u r a t i o n ) , and hence, to the c a p i l l a r y p ressure i n a porous medium. Bro.o.ks and Corey ( 1 9 6 6 ) s u b s t i t u t e d equation 2 . 2 - 3 i n t o equations which r e l a t e c o n d u c t i v i t y to pore s i z e d i s t r i b u t i o n data (Burdine, 1953), and i n t e g r a t e d the r e s u l t to o b t a i n the f o l l o w i n g power laws f o r K on the boundary d r y i n g curve: ( 2 . 5 - D K = K s S* £ and ( 2 - 5 - 2 ) K = K s ( h b d / h c ) n h c: < h b d K = K h > h, , s c bd which forms a l s e t along with ( 2 . 2 - 3 ) S* = ( h b d / h c ) X h c < h b d S" = 1 h c > h b d i n the v a r i a b l e s K, S and h . Only two of these equations can be independent, ( 2 . 5 - 3 ) n = Xe r e l a t e s the pore s i z e d i s t r i b u t i o n oindices. Brooks and Corey demonstrated that the power laws were c l o s e l y f o l l o w e d (except near the s a t u r a t e d s t a t e ) by a wide range of porous media, when drain e d from the s a t u r a t e d c o n d i t i o n . I t must be noted that 48 t h e i r range i n K d i d not exceed three orders of magnitude whereas i n snow, K at normal flow r a t e s i s four to seven orders of magnitude below the s a t u r a t e d s t a t e . From measurements of drainage waves i n columns of homogeneous snow, Colbeck and Davidson ( 1 9 7 2 b ) measured an average value of e = 3 . 3 . In s e c t i o n 2 . 2 , a value of X = 4 was s e l e c t e d as a nominal value on the b a s i s of l a b o r a t o r y t e s t s . For e = ' 3 . 5 and X - 4, equation 2 . 5 - 3 gives n = 14. That i s , the r e l a t i o n between c a p i l l a r y pressure and h y d r a u l i c c o n d u c t i v i t y i s expected to f o l l o w a power law with a l a r g e exponent. It i s e a s i e r to measure changes i n l i q u i d water content i n snow than to measure i t s a b s o l u t e v a l u e . The d e r i v a t i v e of h y d r a u l i c c o n d u c t i v i t y with l i q u i d water content, i s r e l a t e d to the K(9) f u n c t i o n to w i t h i n a constant and i s adequate f o r many a p p l i c a t i o n s of the t h e o r y , such as f o r the kinematic wave analogy. A convenient q u a n t i t y to use i s the f r a c t i o n a l d e r i v a -t i v e of K with r e s p e c t to 6, which i s given the symbol co. ( 2 . 5 - 4 ) co - K d6 From equation 2 . 5 - 1 , K 1 / £ ( 2 . 5 - 5 ) oo = ^ (/) so that oo v a r i e s as the - 0 . 2 9 power of K f o r e = 3 .5> i . e . a slowly changing f u n c t i o n of K. The small range of oo w i l l be shown e x p e r i m e n t a l l y i n Chapter S i x . The average value of co over a range of l i q u i d water contents, 0 to r e f e r r e d to as co, i s given by: 49 f n r- C s ~ • 1 , 0 ' , I n ( K V K ) ( 2 . 5 - 6 ) b ) = — — f Q = _ A _ _ Z from the d e f i n i t i o n of co. I f two d i f f e r e n t p a i r s of ( 0 , K) values are measured f o r a snow l a y e r , then co can be determined from them. Note that co i s a f u n c t i o n of K and K''' , and that co(K, K) = co(K) f o r any m o n o t o n i c a l l y i n c r e a s i n g f u n c t i o n , K ( 0 ) . For the s p e c i a l case where K ( 0 ) i s an e x p o n e n t i a l f u n c t i o n , co i s the e x p o n e n t i a l c o e f f i c i e n t . For the case where K ( 0 ) i s a power law, as c o n s i d e r e d above, co = co f o r a value of K given by: ( 2 . 5 - 7 ) K f = -ln(K'VK) a f u n c t i o n which i s not very s e n s i t i v e to the exact value of e. Hence values of co(K) can be determined by measuring co, f o r which K i s g i v e n by t h i s equation u s i n g any reasonable estimate of e. In Chapter S i x , f o r the range of K rs c o n s i d e r e d , i s c l o s e l y approximated by K = /KK'r which i s Independent of e. It was noted i n s e c t i o n 2 . 2 that the S (h ) r e l a t i o n shows c h y s t e r e s i s between wetting and d r y i n g curves. In which .(or both) of the h y d r a u l i c c o n d u c t i v i t y r e l a t i o n s does t h i s h y s t e r e s i s appear? For sand, the r e l a t i o n between e f f e c t i v e s a t u r a t i o n and h y d r a u l i c c o n d u c t i v i t y shows e i t h e r l e s s h y s t e r e s i s than the e f f e c t i v e s a t u r a t i o n - c a p i l l a r y pressure r e l a t i o n ( P o u l o v a s s i l l i s , 1 9 6 9 ) , or h a r d l y any at a l l (Talsma, 1 9 7 0 ) . It w i l l be assumed that f o r snow, the r e l a t i o n between K and S i s i n v a r i a n t with -IOOO -100 V -to X 0 -/ : I 1 1 1 1 I I I ~ cRoSS- HftTCHEP BKEfi -DIUdNIU. S~NOU)MELT LOOP 1 1 1 1 1 1 II" 'i r MINI M UM PAY, NT R '_ BOUND? '" J" ' U \l r-MAXIMUM OIURNAL FL0U1 RAINSTORM FLou) HYPKAULIC COfJOUCTI VITy C Afi&ITXFlrl Y SCPiLE) - > F i g u r e 2.5-1 H y p o t h e t i c a l h y s t e r e s i s loops i n the h y d r a u l i c c o n d u c t i v i t y - c a p i l l a r y pressure r e l a t i o n vn o 51 r e s p e c t to wetting and d r y i n g and that h y s t e r e s i s occurs only i n the c a p i l l a r y pressure r e l a t i o n s equations 2 . 5 - 2 and ' 2 . 2 - 3 . The l i q u i d water content would be g r e a t e r d u r i n g d r y i n g than d u r i n g wetting f o r a g i v e n value of h . Corey and Corey ( 1 9 6 7 ) found-that equation 2 . 5 - 2 d e s c r i b e d the boundary wetting curve as w e l l as the boundary d r y i n g curve but with d i f f e r e n t c o e f f i c i e n t s K , h^ and n.. Large h y s t e r e s i s i n the c a p i l l a r y p r essure r e l a t i o n s was a l s o found by P o u l o v a s s i l i s ( 1 9 7 0 ) and by Talsma ( 1 9 7 0 ) . In t h i s r e p o r t a model of h y s t e r e s i s f o r the K(h ) r e l a t i o n w i l l be assumed to c o n s i s t l o f h y s t e r e s i s loops which are co n f i n e d between two boundary curves. The boundary curves are taken to be d e s c r i b e d by the power law equation 2 . 5 - 2 with i d e n t i c a l values of K andun, but with d i f f e r e n t values of h, . s b Drying curves a s y m p t o t i c a l l y approach the B.D.C. arid w e t t i n g curves a s y m p t o t i c a l l y approach the B.W.C. The occurrence of h y s t e r e s i s i n the c a p i l l a r y p r essure r e l a t i o n s , r e s u l t s i n d i f f e r e n c e s over c y c l e s of d r y i n g and we t t i n g , as shown i n f i g u r e 2 . 5 - 1 - During d i u r n a l snowmelt c y c l e s , the K(h ) f u n c t i o n i s probably near the boundary wetting curve i n the l a t e morning and e a r l y a f t e r n o o n . L a t e r , the snowpack d r a i n s and the pressure decreases with the f u n c t i o n a s y m p t o t i c a l l y approaching the boundary d r y i n g curve e a r l y the next morning. I f a r a i n storm occurs while the snow i s near the B.W.C., the c a p i l l a r y pressure a f t e r the storm can be lower (more negative) than b e f o r e , even though the input of water at the snow s u r f a c e may r e t u r n to i t s p r e - r a i n s t o r m r a t e . 51A SUMMARY An o l d wet snowpack can be approximated by a r i g i d porous medium model. The l i q u i d water i n the snow i s at a negative p r e s s u r e r e l a t i v e to atmosphere. The l i q u i d water f l u x i s assumed to be p r o p o r t i o n a l to the h y d r a u l i c g r a d i e n t (Darcy's p o s t u l a t e ) , the constant of p r o p o r t i o n a l i t y c a l l e d the h y d r a u l i c c o n d u c t i v i t y at a given l i q u i d content. During steady or d e c r e a s i n g flows, the h y d r a u l i c g r a d i e n t i s c l o s e to u n i t y throughout much of a deep snowpack ( g r a v i t y drainage p o s t u l a t e ) so that the h y d r a u l i c o n d u c t i v i t y and f l u x values are e q u a l . The r e l a t i o n s of e i t h e r e f f e c t i v e s a t u r a t i o n or of h y d r a u l i c c o n d u c t i v i t y to the c a p i l l a r y pressure d i s p l a y h y s t e r e s i s . The r e l a t i o n s are power f u n c t i o n s on the boundary w e t t i n g or d r y i n g curves. The f r a c t i o n a l d e r i v a t i v e of h y d r a u l i c cond-u c t i v i t y with r e s p e c t to l i q u i d water content i s i n t r o d u c e d to r e p l a c e the h y d r a u l i c c o n d u c t i v i t y - l i q u i d content r e l a t i o n f o r experimental convenience. The model i s f u r t h e r developed i n the next chapter along c e r t a i n l i n e s which are r e q u i r e d f o r a p p l i c a t i o n i n l a t e r chapters. 52 CHAPTER THREE SOME DEVELOPMENTS OF THE MODEL The model d i s c u s s e d i n the previous chapter i s a p p l i e d to s e v e r a l cases of i n t e r e s t . The f i r s t two s e c t i o n s are concerned with the e f f e c t of l y s i m e t e r s and snow l a y e r s on the snowpack. Two methods, which f o l l o w from the g r a v i t y drainage p o s t u l a t e , are d i s c u s s e d i n the remaining s e c t i o n s , f o r determining the f u n c t i o n co(K). These methods g i v e averages of the f r a c t i o n a l ' d e r i v a t i v e , w3 based on the t r a n s l a t o r y wave or kinematic wave a n a l o g i e s , which can be compared with the d i r e c t d e t e r m i n a t i o n of co from measurement of f l u x and l i q u i d water content changes-, i n i n d i v i d u a l l a y e r s . 3 . 1 PROFILE OF CAPILLARY PRESSURE ABOVE AN INTERFACE In t h i s s e c t i o n , the e f f e c t of a lower boundary c o n d i t i o n f o r c a p i l l a r y pressure on the v e r t i c a l p r o f i l e of the c a p i l l a r y pressure w i l l be d i s c u s s e d . The i n t e r f a c e i n the snow may be con s i d e r e d to be e i t h e r between two d i f f e r e n t snow l a y e r s , or between the snow and the contact s u r f a c e of a l y s i m e t e r . The d i s c u s s i o n i s r e s t r i c t e d to the case of a steady 'ambient f l u x ' w i t h i n the snow. In the absence of any flow, the steady s t a t e d i s t r i b u t i o n of c a p i l l a r y pressure i n a porous medium i s a l i n e a r decrease of h with h e i g h t , from the value h = h. at the i n t e r f a c e c • ' c 1 (e.g. h± = 0 53 at the water t a b l e ) . In f a c t , there i s r a r e l y s u f f i c i e n t time f o r the e q u i l i b r i u m p r o f i l e i n h c to become e s t a b l i s h e d . In wet snow, some flow, V, even I f s m a l l , w i l l continue as the r e s u l t of drainage from upper l a y e r s . The c a p i l l a r y pressure w i l l change with height from the value of h^ to a s y m p t o t i c a l l y approach an ambient p r e s s u r e , h . The ambient pressure i s g i v e n by : ( 3 . 1 - D h e = h^(K = V ) . The i n t e r f a c e i s denoted as wetting or d r y i n g , depending on whether lu i s g r e a t e r or l e s s than the ambient p r e s s u r e , r e s p e c -t i v e l y . The t r a n s i t i o n h e i g h t , z , i s d e f i n e d as the height above the i n t e r f a c e above which ( 3 . 1 - 2 ) |h c - h e | < 3 where B i s a very s m a l l v a l u e . In coarse g r a i n e d m a t e r i a l s , the t r a n s i t i o n height i s r e l a t i v e l y small f o r most f l u x e s of i n t e r e s t , as can be seen i n data g i v e n by P o u l o v a s s i l i s ( 1 9 6 9 ) f o r sand. It i s important to determine the t r a n s i t i o n height f o r normal f l u x e s i n snowpacks s i n c e at l a r g e r heights above i n t e r f a c e s , the h y d r a u l i c 'conductivity can be determined by measuring the f l u x , s u b j e c t to the g r a v i t y drainage p o s t u l a t e . The c a p i l l a r y pressure p r o f i l e i n the t r a n s i t i o n zone i s give n by i n t e r g r a t i n g Darcy's equation to get : h h m ( 3 . 1 - 3 ) z z - z. = - / h C ! _ V / K ( h ) i V as a f u n c t i o n of h i s complicated i n g e n e r a l because h y s t e r e s i s K C 54 can produce a d i f f e r e n t ^ f u n c t i o n at each height above the i n t e r f a c e . However, i f - t h e snow i s wetted from a very dry s t a t e , ^7 w i l l be g i v e n by the boundary w e t t i n g curve which corresponds to equation 2 . 5 - 2 . Thus f o r a d r y i n g i n t e r f a c e (lu < h ), or f o r a w e t t i n g i n t e r f a c e with h. < h, 1 D W h ( 3 - 1 - 4 ) z - z = - f C d h c / { l - ( h c / h e ) n } h. l For a wetting i n t e r f a c e (h. > h ) with h. > h, I e I D M h ( 3 - 1 - 5 ) z - z = - / c d h c / { l - ( h c / h e ) n ) bw-The- denominator of the second term i n 3 - 1 - 5 i s c l o s e to one f o r normal flow i n snowpacks. These equations can be i n t e g r a t e d , but f o r high v a l u e s of n, the i n t e r g r a n d c o n s i s t s of so many terms that i t s numerical i n t e g r a t i o n , i s , i n f a c t , e a s i e r to perform. I t i s necessary to o b t a i n an approximate a n a l y t i c a l s o l u t i o n n e v e r t h e l e s s , i n order to determine the t r a n s i t i o n h e i g h t . T h i s i s done by approximating the power law e x p r e s s i o n , equation 2.5'..-2), as an e x p o n e n t i a l f u n c t i o n ( 3 . 1 - 6 ) K - V exp v (h - -h ) where F i g u r e 3.1-1 Height of the t r a n s i t i o n i n water pressure above an i n t e r f a c e w i t h i n a wet snowpack. The t r a n s i t i o n height i s 2ni and the i n t e r f a c e pres-sure i s h|. vl hc-f>e|<0.1 f o r 2 - > Z r , . The c o n s t r u c t i o n near s a t u r a t i o n f o r vht---r i i m p l i c i t l y assumes that y«Ks • INT£RF*c£r VJ1 56 (3.1 - 7 ) v = -n/h e . The approximation i s a good one f o r s e v e r a l orders of magnitude v a r i a t i o n i n K about K = V,. (as long as V << K ). Outside of s t h i s range, the i n t e g r a l i s i n s e n s i t i v e to the form of K(h ) c (When %r <•<• 1 , dz = -dh , even i f h < h, , and when % >> 1 , K c c b K K dz - ^ l h c - 0 . ) I f both the power and e x p o n e n t i a l forms of K(h ) are expanded i n a T a y l o r ' s s e r i e s i t i s e a s i l y seen that the f i r s t few corresponding terms i n the s e r i e s are almost equal as long as n i s l a r g e . For the nominal value of n = 14, v v a r i e s from 3 • 5 cm to 0 . 9 3 cm ^ as h g changes from -4 to - 1 5 cm. I n t e g r a t i o n of equation 3 - 1 - 3 u s i n g the e x p o n e n t i a l form from h. to h = h (z ) g i v e s l n c n & (3.1-8) z = (h - h ) - - ln{exp v(h - h ) -1} n e n v ^ e n + (h. - h ) + - ln{exp v(h - h.) -1} I e v e I J S o l u t i o n s f o r these equations, g i v i n g the t r a n s i t i o n height as a f u n c t i o n of the i n t e r f a c e p r e s s u r e , are shown i n f i g u r e 3.1-1 f o r two values of ambient snowpack p r e s s u r e . At heights g r e a t e r than z , the c a p i l l a r y p r e s s u r e i s almost i n d i s t i n g u i s h -able from the ambient p r e s s u r e . In the f i g u r e , f o r the d r y i n g i n t e r f a c e , the t r a n s i t i o n height does not exceed a c e r t a i n value f o r a given h , no matter how great i s the c o n t r a s t of the i n t e r -face pressure to the ambient p r e s s u r e . This f a c t i s made use of i n d e s i g n i n g t e n s i o n l y s i m e t e r s i n Chapter Four. On the other 57 ' hand, the wet t i n g i n t e r f a c e has a t r a n s i t i o n height which continues t o i n c r e a s e at about 1 cm per cm i n c r e a s e i n pressure c o n t r a s t s , at l e a s t f o r l a r g e c o n t r a s t s . Consequently zero pressure l y s i m e t e r s , which are s a t u r a t e d at the i n t e r f a c e , have l a r g e r t r a n s i t i o n h e i g h t s , i n g e n e r a l . 3 . 2 PATTERN OP CAPILLARY PRESSURE IN LAYERED SNOWPACKS The concept of t r a n s i t i o n height w i l l now be a p p l i e d to l a y e r e d snow. In f i g u r e 3 - 2 - 1 are shown th r e e s e c t i o n s of a h y p o t h e t i c a l snow p r o f i l e . The p r o f i l e c o n s i s t s of l a y e r s , each c h a r a c t e r i z e d by an ambient p r e s s u r e , h given by h (K — V ) , f o r e Q a steady f l u x V through the snowpack. The l a y e r s have t h i c k n e s s D T and a t r a n s i t i o n height z which i s a f u n c t i o n of h_ and hn. at L n c e • . . • l each l a y e r ' s lower i n t e r f a c e . In s e c t i o n s EL and B „ , DT-'<z , i . e . J 1 2 L n the pressure does not have the height to adjust to h g before r e a c h i n g the top of the l a y e r . The pressure at any l e v e l , z, i s determined by the average h g of the l a y e r s f o r some d i s t a n c e about z. The l a y e r values of h g vary randomly or s y s t e m a t i c a l l y with depth i n s e c t i o n s B 1 and r e s p e c t i v e l y . In both cases, d i f f e r e n c e s i n c a p i l l a r y p r e s s u r e measured at d i f f e r e n t h e i g h t s , d i v i d e d by the height d i f f e r e n c e , could be taken as the appro-ximate pressure g r a d i e n t between those l e v e l s . In s e c t i o n A, however, DT£i>. z so that the c a p i l l a r y L n pressure changes i n steps. The i n c l u s i o n of i c e sheets at i n t e r -faces does not a l t e r the argument but may i n c r e a s e or decrease the t r a n s i t i o n height depending on the c h a r a c t e r i s t i c s of the i c e 58 F i g u r e 3-2-1 H y p o t h e t i c a l p r e s s u r e p r o f i l e s i n a snowpack. he(y) i s a snowlayer p r o p e r t y . An impermeable s l o p i n g s u r f a c e i s at the base of the snowpack. 59 sheet. In s e c t i o n A, the d i f f e r e n c e s i n c a p i l l a r y p r e s s u r e s measured at d i f f e r e n t l e v e l s w i l l not give the pressure g r a d i e n t i n g e n e r a l s i n c e t h i s g r a d i e n t i s concentrated at the i n t e r f a c e s and i s zero f o r the r e s t of the s e c t i o n . C a p i l l a r y pressure g r a d i e n t s w i t h i n snow l a y e r s cannot be measured i f the t e n s i o -meters are more than a l a y e r a p a r t , i n s e c t i o n A type snow. What i s measured i n s t e a d , i s the mean g r a d i e n t , averaged between the two l e v e l s at which the tensiometers are p l a c e d . S p a t i a l mean c a p i l l a r y pressure g r a d i e n t s are measured i n the f i e l d experiments. 3.3 TRANSLATORY WAVES The wetting f r o n t of a f l u x wave o f t e n forms i n t o a shock front' where the c a p i l l a r y p r essure g r a d i e n t i s r e l a t i v e l y l a r g e compared to other p a r t s of the wave. As noted i n s e c t i o n 2.4'- the wetting f r o n t of a d i u r n a l melt wave changes i t s shape and s i z e with depth and time. Colbeck (1974.c) obtained an a n a l y t i c s o l u t i o n f o r the we t t i n g f r o n t i n the s p e c i a l case of water entry i n t o dry snow ( i n i t i a l S = 0 ) and us i n g a s p e c i a l form of the K(h c ) f u n c t i o n . He s u b s t i t u t e d the Darcy equation i n t o the equation of c o n t i n u i t y , assumed that the wave f r o n t changes slowly with time while I t t r a n s l a t e s through the snow, i n t e g r a t e d twice, and f i t t e d the matching p o i n t s ahead and behind the f r o n t to the kinematic wave s o l u t i o n . In the w r i t e r ' s r e g u l a t e d snowplots, a much simpler i n p u t wave than the d i u r n a l melt wave was i n v e s t i g a t e d , namely, the up-step f u n c t i o n produced by e i t h e r removing an i s o l a t i o n cover or by i r r i g a t i n g the snow s u r f a c e . A t r a n s l a t o r y wave of f i x e d shape and s i z e w i l l t h e n move i n t o t h e snow whose speed, U, can be d e r i v e d from c o n s e r v a t i o n o f mass: V l " V0 (3.3-1) U = ^ ^ 61 ~ 90 where V Q i s t h e i n i t i a l s t e a d y f l u x i n the snow, V 1 i s t h e f i n a l s t e a d y f l u x , and e Q and e.^  a r e t h e c o r r e s p o n d i n g l i q u i d water c o n t e n t s . I n f a c t the w e t t i n g f r o n t s h o u l d move f a s t e r t h a n p r e d i c t e d u n t i l a d y n a m i c a l l y s t a b l e f r o n t i s d e v e l o p e d from t h e i n i t i a l u p -step wave produced a t t h e s u r f a c e , z . I f t h e a r r i v a l s o f t h e w e t t i n g f r o n t a t depth (z - z) o c c u r s a t t i m e t a f t e r t h e s up-step i n p u t was i n t r o d u c e d a t t h e snow s u r f a c e , t h e wave speed w i l l be g i v e n a p p r o x i m a t e l y by: (3.3-2) U = (z - z ) / t . Note t h a t t h e o n l y assumptions i n u s i n g a t r a n s l a t o r y wave model f o r a w e t t i n g f r o n t a r e t h a t (a) t h e w e t t i n g f r o n t r a p i d l y forms near t h e snow s u r f a c e and t h e n moves t h r o u g h the snowpack a t a r a t e , U, m a i n t a i n i n g a f i x e d shape and s i z e , and, (b) t h e f l o w i s o n e - d i m e n s i o n a l o v e r t h e s c a l e o f i n t e r e s t . Por a f u r t h e r d i s c u s s i o n o f the t r a n s l a t o r y wave model see P h i l i p (1957). I f a p e r i o d o f stea d y i r r i g a t i o n i s t e r m i n a t e d , t h e d r a i n a g e wave w i l l move t h r o u g h the snow a t a f a s t e r speed t h a n the w e t t i n g f r o n t does. When the d r a i n a g e wave c a t c h e s up w i t h t h e w e t t i n g f r o n t t h e assumption o f c o n s t a n t w e t t i n g f r o n t shape would be i n v a l i d . S i n c e V = K ahead and b e h i n d (but not a t ) t h e t r a n s l a t o r y w e t t i n g f r o n t , t h e denominator o f e q u a t i o n 3-3-1 i s g i v e n by e q u a t i o n 2.5-6, so t h a t 61 ( 3 - 3 - 3 ) U - w l n ( K i / K o ) = w i n ( V l / V 0 ) ' which can be used to determine values of co. The d i f f i c u l t y of d i r e c t measurement of the c a p i l l a r y p ressure g r a d i e n t was d i s c u s s e d i n the l a s t s e c t i o n . An i n d i r e c t method of e s t i m a t i n g Sh c/3z at the wetting front, i s by u s i n g the t r a n s l a t o r y wave model. Thus : 3 lie • 9h where U i s the speed of the w e t t i n g f r o n t and dh^/dt can be -determined from time curves of pressure measured d u r i n g the passage of the f r o n t . 3 . 4 KINEMATIC WAVES L i g h t h i l l and Whitham ( 1 9 5 5 ) developed kinematic wave theory f o r the d e s c r i p t i o n of f l o o d movements i n long r i v e r s where the f o r c e of g r a v i t y dominates the pressure f o r c e s produced by the s l o p i n g f l o o d wave. The theory was extended to i n c l u d e a shock f r o n t which was p o s t u l a t e d to occur i n p l a c e s where s i n g u l a r i t i e s occured i n the s o l u t i o n . A kinematic wave analogy can be a p p l i e d to flow i n snow as long as there i s (a) a f i x e d f l u x - l i q u i d water content r e l a t i o n over the s c a l e of i n t e r e s t ; (b) only l i m i t e d kinematic d i s p e r s i o n of the wetting f r-ont. Below the s u r f i c i a l melt l a y e r , c o n s e r v a t i o n of mass of the l i q u i d water i s expressed by the c o n t i n u i t y equation 62 (3 4 - 1 ) i S - = ^ where V i s the downwards f l u x at height z i n the snow at time t I f . ( 3 - 4 - 2 ) V = V ( l ) , i . e . some f u n c t i o n , then where ( 3 . 4 - 4 ) U ( 6 ) =' | | i s the speed of the kinematic wave of f l u x V = V ( 0 ) . The f l u x wave i s d e s c r i b e d by a l l the kinematic waves f o r d i f f e r e n t values of f l u x , V. Consequently knowledge of V ( 9 ) allows the c a l c u l a t i o n of a f l u x wave at any depth f o r a given flow input at the snow s u r f a c e . Colbeck ( 1 9 7 2 a ) f i r s t a p p l i e d kinematic waves to wet snow by showing that the uniqueness of V ( 6 ) , during steady or de c r e a s i n g f l o w s , f o l l o w s ftf?,om the g r a v i t y drainage p o s t u l a t e (see equation 2 . 4 - i ) . Equation 3 . 4 - 4 then becomes ( 3 . 4 - 5 ) U ( e ) = | £ . Colbeck a l s o assumed a power law r e l a t i o n f o r K ( 6 ) , or r a t h e r •for K(S ) : . ( 2 . 5 - D K •= K gS 63 so that ( 3 . 4 - 6 ) U(S ) = (eK./* )S or i n terms of K, ( 3 . 4 - 7 ) U(K) = ( e K ^ / H K 1 - ^ Colbeck a p p l i e d the kinematic wave theory to d e s c r i b e the whole d i u r n a l melt wave, l o c a t i n g the shock f r o n t i n such a way as to conserve the mass of the melt wave. His experimental design of measuring the f l u x waves from d i f f e r e n t lengths of homogeneous snow columns i n Colbeck and Davidson ( 1 9 7 2 b ) provided a check on the p r e d i c t i o n s of the theory. The w r i t e r a r t i f i c i a l l y r e g u l a t e d the input f l u x to the snow s u r f a c e . In t h i s way the a n a l y s i s of the f l u x waves was s i m p l i f i e d s i n c e no model was r e q u i r e d f o r the input f u n c t i o n . When a snowplot i s suddenly covered with an i s o l a t i o n cover, the input of water at the snow s u r f a c e i s suddenly reduced from i t s i n i t i a l steady v a l u e , V Q , to the very low v a l u e , V^. V'1 i s produced by any short wave r a d i a t i o n p e n e t r a t i n g the i s o l a t i o n cover. A l l the kinematic waves of f l u x V <_ , leave the snow s u r f a c e s i m u l t a n e o u s l y , each p e n e t r a t i n g the snow at the speed g i v e n by equation 3 . 4 - 7 f o r V = K. Since the waves leave the s u r f a c e simultaneously, at any depth (z - z) below the s u r f a c e : ( 3 . 3 - 2 ) U(V) - ( z a - z ) / t s 6 4 where t i s the elapsed time s i n c e the i s o l a t i o n cover was a p p l i e d at the s u r f a c e . Hence by equation 3 - 4 - 7 the r a t i o of f l u x between two depths i s gi v e n by a power law : ( 3 . 4 - 8 ) . I1 - I1 = i ^ r ^ f ' ^ V 2 K 2 Z s Z 2 f o r the down-step input f u n c t i o n , p r o v i d e d the flows are w i t h i n the range _^ V <_ V Q . For e = 3 - 5 > the drainage f l u x under these c o n d i t i o n s w i l l i n c r e a s e as the 1 . 4 power of the depth and i s independent of the i n i t i a l or f i n a l s u r f a c e inputs (though l i m i t e d to the range between t h e i r v a l u e s ) . Equation 3 . 4 - 8 w i l l be r e q u i r e d to help i n t e r p r e t measurements made at d i f f e r e n t l e v e l s i n the snow (note that i t s use i s r e s t r i c t e d to cases with down-step i n p u t s ) . An i n t e r n a l check on the se.-if c o n s i s t e n c y of kinematic wave theory i s a r r i v e d at by c a l c u l a t i n g the c a p i l l a r y p r essure g r a d i e n t from the p r e d i c t e d wave form. 3 h „ „ dh 3 z 3z dK From equation 3 - 4 - 8 , (3.4-10) 9 K - " £ K 3 z -e - 1 (z - z) s From equation 2 . 5 - 2 , 65 hence . 3 h . . . . ,. .: h . . < 3 - * - 1 2 > 3 7 ° - F^TFT (z °- z) • s so that f o r e = 3 - 5 and n = 14, the g r a v i t y flow p o s t u l a t e i s not v i o l a t e d as long as ( 3 . 4 - 1 3 ) h c z - z s << 10 that i s a s l l o n g as the f l u x i s not very low and the depth i s not too shallow. The s m a l l e s t value of h c measured i n snow d u r i n g the f i e l d season was -15.4 cm, so that the c o n d i t i o n w i l l not be v i o l a t e d , at l e a s t f o r depths below 1.5 cm. A complete check of the l i m i t s t o a p p l i c a t i o n of kinematic waves to snowpacks must be made i n terms of a d e s c r i p t i o n of the flow i n which c a p i l l a r y pressure g r a d i e n t s are i n c l u d e d . However the parameters r e l a t i n g the s i g n i f i c a n t h y d r o l o g i c v a r i a b l e s must be measured before t h i s can be done. The kinematic wave speed takes on a simple form when co i s •" used. (3.4-14) U(K) = ctfK The use of a down-step input at the snow sur f a c e allows co(K) to be measured form the t r a n s i t times of the kinematic waves to l y s i m e t e r s l o c a t e d at a depth (z - z ) i n the snowpack. The use s of of i s not r e s t r i c t e d to assuming a power law f o r K(6) which may not n e c e s s a r i l y be the proper r e l a t i o n f o r inhomogenous snowpacks Because coT i s expected to be a slowly changing v a r i a b l e i t can • u s e f u l l y d e s c r i b e the r e s u l t s of drainage experiments. 65A SUMMARY In t h i s chapter the r i g i d porous media model i s developed along s e v e r a l l i n e s . The s t e a d y - s t a t e p r o f i l e o f c a p i l l a r y p ressure above an i n t e r f a c e i s a r a p i d change from the i n t e r f a c e p r e ssure over a short height above the i n t e r f a c e ( t r a n s i t i o n h e i g h t ) . t o the ambient value f o r the snow. .This i s a conse-quence of the high c a p i l l a r y p r e s s u r e s at normal flow r a t e s and the coarse g r a i n e d nature of snowpacks. A l i m i t e d number of pressure measurements may gi v e c a p i l l a r y pressure g r a d i e n t s which are averages over these d e t a i l e d l a y e r i n g e f f e c t s . D i f f e r e n t models are used to d e s c r i b e unsteady flow r e s u l t i n g from a r t i f i c i a l s u r f a c e inputs o f water. A t r a n s l a t o r y wave of a we t t i n g shock f r o n t i s used with the up-step i n p u t , while a kinematic wave analogy i s used to d e s c r i b e the d e c r e a s i n g flow f o l l o w i n g a down-step i n p u t . The theory i n the previous two chapters i s used to d e s c r i b e .the experimental r e s u l t s which were obtained with i n s t r u m e n t a t i o n d i s c u s s e d i n the next chapter. CHAPTER FOUR 66 FIELD INSTRUMENTATION Instruments that have been s u c c e s s f u l l y used i n s o i l water s t u d i e s have been adapted f o r use i n snowpacks. These i n s t r u -ments are the tensiometer which measures c a p i l l a r y p r e s s u r e , and the t e n s i o n l y s i m e t e r which measures f l u x across an i n t e r f a c e at which a negative pressure i s e s t a b l i s h e d . Both instruments p r o v i d e d u s e f u l i n f o r m a t i o n f o r wet snowpacks under the weather c o n d i t i o n s of May - J u l y , 1 9 7 4 ( f i g u r e 1 . 4 - 5 ) . 4 . 1 ' MEASUREMENT OF CAPILLARY PRESSURE IN SNOW When g r a v i t y i s the dominating f o r c e , the r e l a t i o n of c a p i l l a r y pressure to f l u x i n snow i s analogous to that of stage to discharge i n r i v e r s . C a p i l l a r y pressure can be mea-sured by means'of tensiometers, which are p l a c e d at ;-the l e v e l of i n t e r e s t i n the snowpack. A tensiometer i s a water f i l l e d porous cup connected to a manometer. The cup remains f i l l e d with water so long as the c a p i l l a r y pressure i s not so negative as to e i t h e r draw a i r . through the pores i n t o the cup, or to rup t u r e the water column i n s i d e the tensiometer. T h i s happens only i f the c a p i l l a r y pressure f a l l s below the b u b b l i n g pressure of the porous m a t e r i a l forming the cup, or f a l l s below one atmosphere, r e s p e c t i v e l y (see R i c h a r d s , 1 9 6 5 ) . The pressure of the water i n the cup r e l a t i v e to atmospheric p r e s s u r e , equals the c a p i l l a r y pressure of the porous medium, when the two are i n e q u i l i b r i u m . 67 SNOUPIT OveRHPNCr DJVSTr)&LL~ LAMP 1.3 on O.P. SOP PORT TUBE POROUS COP F i g u r e 4.1-1 Diagram of a tensiometer i n s e r t e d i n t o an access hole d r i l l e d i n a snowpit w a l l . The sma l l range of c a p i l l a r y p r essure f o r wet snow means that acc-u r a t e measurement requ-i r e s the manometer to be d i r e c t l y mounted to the h o r i z o n t a l support tube. The use of a tensiometer i n a snowpack r e q u i r e d s p e c i a l a t t e n t i o n to the f o l l o w i n g f o u r p o i n t s : F i r s t , the tensiometer must be used only i n a wet snowpack and care must be taken to prevent f r e e z i n g of the exposed manometer. Because of the r e l a t i v e l y coarse t e x t u r e of snow, the range of c a p i l l a r y p ressure i n snow i s smal l — d u r i n g the 1974 f i e l d season the extreme highest and extreme lowest c a p i l l a r y pressure measured by any tensiometers was - 2 . 3 cm under heavy i r r i g a t i o n and - 1 5 . 4 cm a f t e r a prolonged drainage. Consequently, and secondly, the height of the porous cup r e l a t i v e to the manometer s c a l e zero (the zero o f f s e t ) should be a c c u r a t e l y known,aa task made more d i f f i c u l t by m e l t i n g of the snow s u r f a c e and snowpit w a l l s . T h i r d , the l a t e r a l v a r i a t i o n i n flow r e q u i r e s the use of many tensiometers i f the mean c a p i l l a r y pressure of a l a y e r i s r e q u i r e d . F i n a l l y , short wave r a d i a t i o n P e n e t r a t i n g the snow from the snowpit must be minimized so as to produce an i n s i g n i -f i c a n t change i n snow p r o p e r t i e s i n the v i c i n i t y of the porous cup. The- tensiometer designed by the w r i t e r shown i n f i g u r e s 4 . 1 - 1 and 2 , i s i n s t a l l e d i n the w a l l of a snowpit i n t o the l a y e r of I n t e r e s t . In the r e g u l a t e d snowplot i n s t a l l a t i o n s the tensiometer cups were p o s i t i o n e d d i r e c t l y above l y s i m e t e r s (see s e c t i o n 5 - 1 ) . For the melt wave i n s t a l l a t i o n , shown i n f i g u r e 5 . 6 - 1 , the cup ends were p l a c e d 44 cm i n s i d e the snow p i t w a l l . The zero o f f s e t problem i s much reduced by u s i n g a r i g i d tube, i n s e r t e d i n t o a h o r i z o n t a l access h o l e , to support the porous cup at one end and the manometer at the other end. A l i g h t -weight manometer can be used because the smal l range of c a p i l l a r y F i g u r e 4.1-2 Tensiometer f o r measuring c a p i l l a r y pressure i n wet snowpacks. The porous cup at the end of the support tube i s composed of 70 micron s i n t e r e d g l a s s beads. When the cup i s i n s e r t e d i n t o a snowpack, the p r e s s -ure i n the water manometer w i l l equal the pressure of the l i q u i d water i n the snow. The 38 cm l o n g manometer was adequate f o r the range of pressures measured i n the f i e l d . 70 pressure head i n wet snow r e q u i r e s a water manometer of short l e n g t h . The design e l i m i n a t e s the need f o r a s o l i d ground support f o r the manometer and f a c i l i t a t e s the d e t e r m i n a t i o n of the zero o f f s e t . These f e a t u r e s are . e s p e c i a l l y u s e f u l i n deep, r a p i d l y m e l t i n g snowpacks. T h i r t y - o n e tensiometers were made. To make the porous cups, 7 0-micron g l a s s beads were c l o s e packed i n a s t e e l mold by u s i n g a pneumatic v i b r a t o r , then heated to about 650°C i n a furnace u n t i l the beads were f r i t t e d i n t o a porous r i g i d b l o c k . The c y l i n d r i c a l cups were machined from, the bl o c k <3f f r i t t e d g l a s s beads which had a very high h y d r a u l i c c o n d u c t i v i t y . A c o - a x i a l hole i s d r i l l e d i n t o the in n e r end of the cup which i s cemented to the end of the r i g i d support tube. The cups average 2 cm i n l e n g t h and 1 cm i n diameter. They were t e s t e d to be impermeable to a i r f o r c a p i l l a r y p ressures above from - 7 0 to - 1 2 0 cm (E^O). . The h i g h l y conductive cup g r e a t l y f a c i l i t a t e s f i e l d i n s t a l l a t i o n , f o r the tensiometer can be f i l l e d i n seconds,by p l a c i n g the cup i n t o a c o n t a i n e r of water and drawing water i n t o the system by a p p l y i n g s u c t i o n at the a i r vent. The t r a n s p a r e n t connecting tubes c o n s i s t of the 1 . 3 cm O.D. a c r y l i c support tube, a f l e x i b l e 0 . 6 5 cm O.D. tygon tube and short a i r t i g h t rubber connections. The manometer hangs from the support tube by means of a clamp. The manometer support should be f l u s h a g a i n s t the snowpit w a l l and the clamp allows the manometer-cup d i s t a n c e to be a d j u s t e d . The water manometer i s a g l a s s c a p i l l a r y tube, wired to a s c a l e . The- s c a l e , a 2 cm x 2 cm x 38 cm long wooden s t a f f , c a r r i e s a s t r i p of graph paper, s c a l e d from +4 cm to - 3 4 cm, and covered with c l e a r a c r y l i c spray. 71 The l a r g e bore of the c a p i l l a r y tube f a c i l i t a t e s removal of a i r bubbles by means of a t h i n wire. A i r bubbles o c c a s i o n a l l y formed, e x p e c i a l l y a f t e r heavy r a i n , when a i r was trapped a g a i n s t the water column by wind blown drops e n t e r i n g the a i r vent. Any bubbles were removed before r e a d i n g the meniscus l e v e l of the manometer. The 0 . 2 cm bore produces a c a p i l l a r y r i s e of 1 . 5 cm i n the c a p i l l a r y tube so that 1 . 5 cm was s u b t r a c t e d from the manometer r e a d i n g s . The tensiometers were i n s t a l l e d i n a r e c e s s e d part of the snowpit w a l l , which was covered with. 2 . 5 cm t h i c k p o l y s t y r e n e boards to reduce n o c t u r n a l counter r a d i a t i o n from the manometers and exposed connecting tubes. The i n s t a l l a t i o n procedure was as f o l l o w s . A r e c e s s e d p o r t i o n of the snow was c a r e f u l l y smoothed to a v e r t i c a l plane and the l o c a t i o n of the holes to be d r i l l e d noted on the w a l l . A d r i l l guide assembly was used which c o n s i s t s of a handle, and a 15 cm long sleeve (the d r i l l guide) mounted p e r p e n d i c u l a r to a f l a t board, which could be supported a g a i n s t the planed snowpit w a l l . In p r a c t i c e , however, a s p i r i t l e v e l , a l s o mounted on the assembly, proved to be more accurate i n l e v e l l i n g the d r i l l guide. The d r i l l i s a 1 . 3 cm diameter tube with 0 . 1 cm t h i c k w a l l s b e v e l l e d at the c u t t i n g end.' The diameter of the d r i l l was chosen to be the same as that of the tensiometer support tube so that the support tube would s e a l the hole when the tensiometer was i n p l a c e . Transparent a c r y l i c was s e l e c t e d over s t e e l f o r the d r i l l m a t e r i a l s i n c e the process of c u t t i n g and core removal could be more r e a d i l y observed. In a l l cases, the a c r y l i c d r i l l tube removed the snow core c l e a n l y from the tensiometer access hole so that a minimum of d i s t u r b a n c e occured i n the surrounding 72 snow. The d r i l l was e i t h e r pushed or hammered i n t o the snow as r e q u i r e d . The use of the somewhat f l e x i b l e a c r y l i c as the d r i l l m a t e r i a l i n c r e a s e d the zero o f f s e t problem as w i l l be d i s c u s s e d below. The use of a water manometer l i m i t s the response of the instrument to changes i n f i e l d c a p i l l a r y p r essure because of the time r e q u i r e d f o r the manometer water to enter or leave the porous cup. The 'gauge s e n s i t i v i t y ' of the manometer, Q, i s 3 1 . 8 cm pre s s u r e per cc of water. At very h i g h flow r a t e s In snow, the instrument response time Is l i m i t e d i n a d d i t i o n by the conductance but not by the h y d r a u l i c c o n d u c t i v i t y of the snow (see below). The cup response time of the tensiometer, t ^ , was measured by immersing the porous cup i n a c o n t a i n e r of water and timing, the approach of the manometer r e a d i n g , h , to i t s e q u i l i -brium v a l u e , h^, a c c o r d i n g to : h - h - t / t ( 4 . 1 - 1 ) ^ u = e 1 h - h, o 1 where h i s the i n i t i a l manometer val u e . The tensiometers have o cup response times i n water r a n g i n g from 1 . 3 to 12 seconds with a median value of 3 . 0 seconds. The cup response time i n snow i s i n c r e a s e d from the value i n water, however, by the r a t i o of contact area to area of cup w a l l . Contact with the snow was p e r i o d i c a l l y t e s t e d by pushing the end of the support tube and obs e r v i n g any sudden change i n r e a d i n g . The e x i s t e n c e of improper contact on o c c a s i o n r e f l e c t s a f a u l t i n design whereby the cup r a d i u s i s 0 . 2 cm s m a l l e r than the r a d i u s of the tensiometer support tube.' Since the d r i l l i s the same 73 s i z e as the support tube, a gap may have e x i s t e d around the w a l l of the cup unless i n s e r t i o n of the tensiometer had loosened snow from the w a l l s of the hole and f i l l e d the gap. Contact of the end of the cup i s assumed because the tensiometer holes were d r i l l e d 0 . 6 cm s h o r t e r than the len g t h of tensiometer to be pl a c e d i n t o the h o l e . For a 2 cm long cup, 1 cm i n diameter, the response time i s i n c r e a s e d by a f a c t o r of 9 i f only the end i s i n co n t a c t . Thus i f a l l tensiometers had only end contact the median cup response time i n snow would have been 9 x 3 - 0 = 27 seconds. The response time of a tensiometer to step changes i n c a p i l -l a r y p r e ssure depends on whether the exchange of water between the porous medium and the cup i s l i m i t e d by the cuductance of the cup or by that of the medium (Klute and Gardner, 1 9 6 2 ) . The e f f e c t of the cup, gi v e n by equation 4 . 1 - 1 , with a 27 second response time (median cup response time assuming only end contae.t-t with the snow) would r e s u l t i n . t h e tensiometer r e q u i r i n g 32 seconds to r e c o r d 70% of a step change i n c a p i l l a r y p r e s s u r e . When the c o n d u c t i v i t y of the porous medium l i m i t s the t r a n s f e r of water, the response i s gi v e n approximately by : h - h , h - h, o 1 o 1 f o r long c y l i n d r i c a l cups with an i n i t i a l cup pressure h , suddenly i n s e r t e d i n t o a porous medium with c a p i l l a r y pressure h-^, K i s the h y d r a u l i c c o n d u c t i v i t y of the medium (assumed constant) and L i s the l e n g t h of fehej cup. Equation 4 . 1 - 2 i s taken as a crude approximation to the response time f o r the stubby cups 74 d e s c r i b e d In t h i s r e p o r t . Por L = 2 cm and Q = 3 1 . 8 em/ccV 70% recovery w i l l take p l a c e i n the^.same time ( 3 2 seconds) g i v e n by equation 4.1-1 as l o n g as K i s 0..5 cm/hr. For g r a v i t y drainage, (K = V ) , i f water i s f l o w i n g through a snowpack with a f l u x of l e s s than 0 . 5 cm/hr, the response time f o r 70% recovery w i l l be dominated by the f l u x i n the snowpack. For a higher percentage of r e c o v e r y , the e f f e c t of f l u x on the recovery time i s even more important because of the more slowly v a r y i n g time f u n c t i o n i n equation 4 . 1 - 2 compared to equation 4 . 1 - 1 . The times r e q u i r e d f o r 70% response are l i s t e d i n t a b l e 4 . 1 - 1 f o r d i f f e r e n t values of K. The tensiometer response time c o u l d be improved by r e p l a c i n g the water manometer with a n u l l - t y p e device or with a transducer type manometer which would have the a d d i t i o n a l f e a t u r e of a l l o w i n g the pressure changes to be a u t o m a t i c a l l y recorded. The above simple design,hhowever, has the advantage of being rugged, l i g h t w e i g h t , e a s i l y i n s t a l l e d and economical. 4 . 2 SOURCES OF ERROR WITH TENSIOMETER FIELD MEASUREMENTS When the flow r a t e i n the snowpack was steady, the manometer re a d i n g was steady to w i t h i n the e r r o r of the manometer s c a l e , * 0 . 1 cm. The c a p i l l a r y p r e s s u r e , (h > i n cm), i s giv e n by the f o l l o w i n g equation: ( 4 . 2 - 1 ) h c = h M - h R + h z where h., i s the manometer r e a d i n g , h„ i s the c a p i l l a r y r i s e f o r the bore of c a p i l l a r y t u b i n g used i n the manometer ( h R = 1 . 5 cm), 75 Table 4 . 1 - 1 Tensiometer time l a g f o r 70% response to a step change Snow h y d r a u l i c c o n d u c t i v i t y (cm/hr) Cup Response Estimate o f snow response from eqn. 4 . 1 - 2 i n water, by experiment i n snow, end contact only 1 0 3 . 6 sec 3 2 sec- 1 . 6 sec. 5 3 . 6 sec ' 3 2 s e c 3 . 2 sec s-1 3 . 6 sec 3 2 sec 1 6 sec 0 . 5 3 . 6 sec 3 2 sec 3 2 sec. 0 . 1 3 . 6 sec . 3 2 sec 1 6 0 sec. 0 . 0 5 3 . 6 sec , 3 2 sec 5 min. 0 . 0 1 3 . 6 sec • 3 2 s e c 2 7 min. 0 . 0 0 5 3 . 6 sec 3 2 sec 0 . 9 hr 0 . 0 0 1 3 - 6 sec. 3 2 sec. 4 . 4 hr 76 ' and h i s the zero o f f s e t . The zero o f f s e t i s the height of "the z ° manometer s c a l e zero above the c y l i n d r i c a l a x i s of the cup. When the readings of two d i f f e r e n t tensiometers are'to be compared or when the a b s o l u t e c a p i l l a r y p r e s s u r e i s d e s i r e d , . t h e t o t a l zero o f f s e t must be known. The best procedure i s to d i g a second snowpit behind the tensiometer cup and to c a r e f u l l y measure the d i f f e r e n c e i n l e v e l between the cup and manometer zero. T h i s procedure was not f o l l o w e d f o r l a c k of time and a l s o because i t was incompatible with an a u x i l i a r y procedure (dye s t a i n i n g of the. flow p a t h s ) . Instead, an attempt was made to e l i m i n a t e the zero o f f s e t as much as p o s s i b l e by p l a c i n g the tensiometer support tube i n a h o r i z o n t a l l y d r i l l e d hole s i n c e the manometer support i s a l s o the zero of the s c a l e . The e r r o r i n i g n o r i n g the zero o f f s e t i s c a l l e d the zero e r r o r . The zero e r r o r i s a r e s u l t of l a c k of p r e c i s i o n i n t e n s i o -meter i n s t a l l a t i o n and of any systematic e r r o r from bending of the d r i l l by s l o p i n g Ice sheets. The p r e c i s i o n can be c a l c u l a t e d i n the f o l l o w i n g manner. The h e i g h t s of i n d i v i d u a l s w i t h i n each group of tensiometers p l a c e d i n the same l a y e r , and spread l a t e -r a l l y over 20 cm, were measured r e l a t i v e to a short h o r i z o n t a l l i n e scored on the snowpit w a l l . When the experiment was compl-eted, the snow was excavated to expose the ends of the tensiometer holes and the procedure r e p e a t e d , . r e l a t i v e to new short h o r i z o n t a l l i n e s . The v a r i a n c e of change i n height between ends of each tensiometer tube about the mean height change f o r each tensiometer group was c a l c u l a t e d . The mean height change i t s e l f was not known because the r e l a t i o n between the h o r i z o n t a l l i n e s was unknown. A l l the v a r i a n c e s were added to g i v e a measure of the 77 p r e c i s i o n of tensiometer i n s t a l l a t i o n . The r e a l p r e c i s i o n would be somewhat l e s s than the c a l c u l a t e d value because the l a t t e r would a l s o c o n t a i n the measurement e r r o r s of the method j u s t d e s c r i b e d . The c a l c u l a t e d p r e c i s i o n i n i n s t a l l a t i o n of t e n s i o -meters i n a group of 20 cm l a t e r a l spread, i n the same l a y e r (which contained s e v e r a l t h i n i c e sheets) was a = + 0 . 8 cm. Consequently the standard e r r o r i n the d i f f e r e n c e of c a p i l l a r y pressure head between two tensiometers i n the same group i s / 2 x 0 . 8 = 1 . 1 cm d u r i n g steady s t a t e flow. When tensiometers i n d i f f e r e n t groups are compared, or i f an absolute value of c a p i l l a r y pressure i s r e q u i r e d , the systematic e r r o r caused by s l o p i n g i c e sheets may i n c r e a s e the zero e r r o r . One t e s t showed that an i c e l a y e r 2 cm t h i c k could bend the d r i l l so as to produce zero e r r o r s of up to 5 cm. Hence, the i n s t a l l a t i o n of tensiometers near t h i c k i c e sheets was avoided. The p r o g r e s s i v e m e l t i n g back of the snowpit w a l l r e s u l t s i n a s t e a d i l y i n c r e a s i n g torque on the support tube by the mano-meter. Hence there i s a time dependent component i n the zero o f f s e t caused by the i n c r e a s i n g drop i n l e v e l of the manometer zero — the 'droop'. The droop i s bending of the exposed por-t i o n o f - t h e support tube while the r e s t of the tube i s f i r m l y h e l d i n the snow. The r e l a t i o n s h i p between droop of the acry-l i c tubes and the s e p a r a t i o n of manometer support from the p i t was obtained by f i t t i n g the cubic f u n c t i o n r e q u i r e d by the theory of bending of e l a s t i c c y l i n d e r s to the set of droop-s e p a r a t i o n values p e r i o d i c a l l y measured i n the f i e l d . I n d i v i -d u a l droop measurements were not very p r e c i s e (as r e p e t i t i o n of measurements showed)so that the droop i s c o r r e c t e d f o r by d a i l y 78 (or more frequent) manometer to p i t w a l l s e p a r a t i o n s and. u s i n g the d r o o p - s e p a r a t i o n r e l a t i o n s h i p . I f the s c a t t e r . i n t h i s r e l a t i o n s h i p i s due to r e a l s c a t t e r i n the droops, then the droop c o r r e c t i o n i t s e l f has an e r r o r whose standard d e v i a t i o n i s * 0.2 cm. T h i s value i s comparable to the s c a l e e r r o r of the manometer and should not be important i n the measurements. A l l readings are c o r r e c t e d f o r droop, the c o r r e c t i o n rang-i n g between zero and 1 . 6 cm. The droop e r r o r could be reduced by e i t h e r p e r i o d i c a l l y moving the manometer along the support tube (as long as the 'latte r i s a c c u r a t e l y h o r i z o n t a l ) , or by c o n s t r u c -t i n g the support tube of a more r i g i d m a t e r i a l . The best procedure i s to d i g the snowpit i n such a form and o r i e n t a t i o n as to keep the melt of the snowpit w a l l s m a l l . 4 . 3 SNOW LYSIMETER THEORY A l y s i m e t e r i s a s t r u c t u r e c o n t a i n i n g a mass of s o i l , designed to permit the measurement of water d r a i n i n g through the s o i l . There are three types ( H a r r o l d and D r e i b e l b i s , 1 9 5 1 ) • an enclosed tank i n t o which e i t h e r a uniform ' s o i l ' i s p l a c e d , or i n t o which a l a r g e u n d i s t u r b e d b l o c k of s o i l i s p l a c e d , and a 'shallow pan or f u n n e l i n s e r t e d at a d e s i r e d depth under undisturbed s o i l h o r i z o n s ' , the Ebermeyer type. The Ebermeyer type, or unenclosed l y s i m e t e r , has been used most o f t e n f o r snowmelt s t u d i e s s i n c e the snow su r f a c e i s l e f t u n d i s t u r b e d (e.g. U.S. Army Corps of Engineers, 1 9 5 6 ) . The snowmelt r a t e i s q u i t e s e n s i t i v e to such d i s t u r b a n c e . The unenclosed l y s i m e t e r has the disadvantage, from 79 the melt measurement p o i n t of view, of a l l o w i n g u n r e s t r i c t e d l a t e r a l flow above the l y s i m e t e r . However t h i s becomes an advan-tage - i f - t h e n a t u r a l flow process w i t h i n the snowpack i s of i n t e r e s t . Snow l y s i m e t e r s c o n s i s t of an i n t e r f a c e , a rim. of h e i g h t , h , above the l y s i m e t e r I n t e r f a c e , a drainage system and measurement device f o r a n a l y z i n g the p e r c o l a t e , and o c c a s i o n a l y some p r o v i -s i o n f o r measuring the v e r t i c a l f o r c e of the o v e r l y i n g snowpack on the l y s i m e t e r . Snow l y s i m e t e r s can a l s o be c l a s s i f i e d a c c o r d i n g to the area of the l y s i m e t e r i n t e r f a c e , the method of c o n s t r u c t i o n of the i n t e r f a c e , and the pressure of the water at the i n t e r f a c e . The simplest d e s i g n , a f u n n e l or pan c o l l e c t o r , i s an impervious s u r f a c e over which the p e r c o l a t e flows toward a d r a i n at the lowest p o i n t i n the s u r f a c e . The lowest l a y e r of snow,, i n e f f e c t becomes part of the drainage system through which the p e r c o l a t e moves, which i s somewhat u n d e s i r a b l e from the p o i n t of view of a n a l y s i s of instrument response. A second method of c o n s t r u c t i o n i s to support the snow on a p e r f o r a t e d base through which the p e r c o l a t e can d r a i n over the whole i n t e r f a c e , a zero t e n s i o n l y s i m e t e r . A snow-air i n t e r f a c e i s u s u a l l y used, though a snow-water i n t e r f a c e would be more l i k e l y to e x h i b i t uniform drainage over the whole i n t e r f a c e area. When i n f o r m a t i o n on flow waves on a s m a l l s c a l e and with a minimum of d i s t o r t i o n i s r e q u i r e d , a snow-water i n t e r f a c e at a n e g a t i v e pressure must be used. A t e n s i o n l y s i m e t e r i s a l y s i -meter with i t s i n t e r f a c e under a negative p r e s s u r e to more c l o s e l y match the c a p i l l a r y p r essure of the snow above i t . The f e a t u r e s of t e n s i o n l y s i m e t e r s are best d e s c r i b e d by comparison with the 80 zero t e n s i o n l y s i m e t e r . To f a c i l i t a t e the comparison, the drainage i s assumed to occur u n i f o r m l y over the whole l y s i m e t e r i n t e r f a c e so that the problem can be considered i n only one dimension. F i r s t the e f f e c t of rim height on l y s i m e t e r e f f i c i e n c y i s c o n s i d e r e d and then the response of the two kinds of l y s i m e t e r w i l l be compared. The r o l e of the r i m does not seem to be s u f f i c i e n t l y under-stood i n snow l y s i m e t e r design, the r i m height o f t e n being too sm a l l t o ensure the best e f f i c i e n c y of the instrument. The r i m should be at l e a s t as high as the t r a n s i t i o n h e i g h t , z , given approximately by equation 3.1-8, i n o o r d e r that the mismatch, ( h e - h ), between the p r e v a i l i n g c a p i l l a r y pressure head of the o v e r l y i n g snow, h , and the c a p i l l a r y p r e s s u r e at r i m height of the snow i n the l y s i m e t e r , h , be i n s i g n i f i c a n t . Otherwise l a t e r a l flow away from the l y s i m e t e r w i l l be induced by the presence of the instrument i t s e l f and the e f f i c i e n c y of the l y s i m e t e r w i l l be l e s s than 100%. The extent towwhich the l y s i -meter e f f i c i e n c y i s a f f e c t e d by nonzero (h - h ) depends on the perimeter to area r a t i o of the l y s i m e t e r , becoming l e s s f o r l y s i -meters with l a r g e r area f o r a giv e n type of snow. On the other hand, too high a r i m r e s u l t s i n unnecessary d i s t u r b a n c e of the o v e r l y i n g snow and i t s p a t t e r n of flow.. Take the t r a n s i t i o n height to be such that the mismatch i n c a p i l l a r y pressure between the snow o u t s i d e the l y s i m e t e r and the snow i n s i d e the l y s i m e t e r at t h i s height i s |h - h | = 0 . 1 cm, a f i g u r e which should produce very l i t t l e l a t e r a l flow i n compa-r i s o n to the ambient v e r t i c a l flow, V, f o r l y s i m e t e r s s e v e r a l centimeters or more a c r o s s . I f the r i m height i s g r e a t e r than 81 the t r a n s i t i o n h e i g h t , the water'approaching the l y s i m e t e r w i l l not experience any s i g n i f i c a n t d i s t u r b a n c e i n the c a p i l l a r y p r e ssure g r a d i e n t u n t i l the water penetrates below the l e v e l of the top of the rim. ^Consequently, the e f f i c i e n c y of the l y s i m e t e r should be p r a c t i c a l l y 100%. The minimum height of r i m r e q u i r e d by equation 3.1-8 was shown i n f i g u r e 3-1-1 f o r two d i f f e r e n t ambient c a p i l l a r y p r essures i n snow. The height of r i m required, i n c r e a s e s with d e c r e a s i n g wetness of the snow. The zero t e n s i o n l y s i m e t e r r e q u i r e s a s i g n i f i c a n t l y h i g h e r rim.above the i n t e r f a c e than does a l y s i m e t e r operated at a t e n s i o n below the lowest t e n s i o n o c c u r i n g i n the snowpack. The s h o r t e r r i m means that the t e n s i o n l y s i m e t e r produces l e s s d i s t u r b a n c e to the snowpack and to i t s flow p a t t e r n . Of p r a c t i c a l importance i s that the low r i m permits the l o c a t i o n of tensiometers only a short d i s t a n c e above the l y s i m e t e r i n t e r f a c e to Improve the c o r r e l a t i o n between the two measurements (see f i g u r e 5.1-1). The f e a s i b i l i t y of improving the e f f i c i e n c y of snow l y s i m e t e r s by I n s t a l l i n g a r i m of s u f f i c i e n t height to reduce the d i s t u r b i n g e f f e c t of the presence of the l y s i m e t e r i n the snow, i s due to the r e l a t i v e l y short r i m r e q u i r e d , which i n t u r n r e f l e c t s the coarse g r a i n e d nature of the snowpack. S o i l s are g e n e r a l l y much f i n e r g r a i n e d and would r e q u i r e a r i m of such a height that the r e s u l t would be, i n f a c t , an enclosed tank l y s i m e t e r . Unenclosed s o i l l y s i m e t e r s , f o r which rims are i m p r a c t i c a l , are c h a r a c t e r i z e d by c o n s i d e r a b l e l a t e r a l flow away from the l y s i m e t e r . 4 . 4 LYSIMETER•RESPONSE 82 The response of sno.w l y s i m e t e r s w i l l be d i s c u s s e d i n terms of dimensional c o n s i d e r a t i o n s s i n c e i t s g e n e r a l a n a l y t i c d e s c r i p -t i o n i s q u i t e complex. A 'response time' i s d e f i n e d i n terms of the change i n water storage above a l y s i m e t e r i n t e r f a c e between two steady f l u x e s , an approach which was used by Colbeck (1974) to d i s c u s s the response of zero t e n s i o n l y s i m e t e r s . The e q u i l i -brium moisture p r o f i l e f o r a steady f l u x , V, can be c a l c u l a t e d , f o r the case of wetting the snow from a very dry s t a t e , from equations 3.1-4 and 3.1-5. S u b s t i t u t i n g i n t o these equations the corresponding wetting curve r e l a t i o n s - to equation;' -2.2-3 and 2/5-3,we get, f o r a d r y i n g i n t e r f a c e ( S i < S g ); or a w e t t i n g s * * i n t e r f a c e (S < S. ) with S < 1 : e l . s 1 •+• X (4.4-1) z - z. = Y / b,dS S • / (1 - (S /S ) £} 1 S. 6 - l and f o r a we t t i n g i n t e r f a c e with S^ = 1 1 + h, (4.4-2) z - z. = r D W / b dS«S • / {1 - ( S V S ) £} l A ^ K? e + (h. - h K ) / ( l - S °) v l bw e S. i s the e f f e c t i v e s a t u r a t i o n at the i n t e r f a c e and S i s the I e ambient e f f e c t i v e s a t u r a t i o n corresponding to V. The moisture p r o f i l e above zero t e n s i o n l y s i m e t e r s ( i . e . h. = 0), are shown f o r three values of S i n f i g u r e 4.4-1 f o r 1 S snow of e = 3.5, X - 4 and h b w = -3 cm. In order to convert F i g u r e 4 . 4 - 1 E f f e c t i v e s a t u r a t i o n p r o f i l e s ..above a zero-tensibn'"'lysimeter i n snow o f e=3-5, ^ = 4 , and U B W=-3 cm. Curve S* W( 0*= 0 . 3 5 ) 1 0. 0021 1. 4cm.-2 0. 049 1.2cm 3 0. 1 3 1. 1cm ./ .Z .3 -f £Ff=£ CTIV£ S~flTURATtON co U) 84 values of • S i n t o values of K u s i n g equation 2.5-1, an estimate of the s a t u r a t e d c o n d u c t i v i t y , K , i s needed. A value of K g equal to 0 . 8 9 cm/sec has been measured with 0 . 1 cm diameter g l a s s beads (DeVries, p e r s o n a l communication). For K = 1 cm/ s sec, the corresponding values of K i n f i g u r e 4 . 4 - 1 are 1 . 5 x - 6 10 , 0 . 0 9 4 and . 2 . 8 cm/hr f o r curves 1 , 2 , and 3 r e s p e c t i v e l y . The e x t r a l i q u i d water s t o r e d by the l y s i m e t e r i n t e r f a c e above that i n undisturbed snow W, i s almost constant, being dominated by the s a t u r a t e d s e c t i o n i n the c a p i l l a r y f r i n g e (z <_ | h^ | = 3 cm). W i s d e f i n e d by Moisture p r o f i l e s f o r the same three values of S but over a t e n s i o n l y s i m e t e r with a constant i n t e r f a c e pressure of - 2 5 cm are shown i n f i g u r e 4 . 4 - 2 f o r the same kind of snow. Note the expanded s c a l e s i n the diagram. Here W i s s e v e r a l orders of magnitude sma l l e r (and i s negative i n s i g n ) . It i s t h e r e f o r e to be expected that the output of the t e n s i o n l y s i m e t e r w i l l more f a i t h f u l l y r e f l e c t the t r u e f l u x regime. Two s i t u a t i o n s are of i n t e r e s t i n comparing the two kinds of l y s i m e t e r s . The s t a r t - u p response time, t , d e f i n e d as ( 4 . 4 - 3 ) * oo , * X W = <j> / (S - S )dz 1 e s ( 4 . 4 - 4 ) t = W/V i s an index of the response of the l y s i m e t e r to being i n s e r t e d i n t o the snow i n which the f l u x i s Y. The dynamic response time, t.;, d e f i n e d as z.%\ l . t l fe F i g u r e 4 . 4 - 2 E f f e c t i v e s a t u r a t i o n , p r o f i l e s above a t e n s i o n l y s i m e t e r o f - 2 5 cm. i n t e r f a c e pressure i n snow of £ = 3 . 5 , ^X=4, and h F C K =-3 cm. Curve S W(^*=0 .35) 1 0. 0021 - 3 "xlO ' T r c m 2 0. 049 - 3 - 2 x l 0 - 3 cm 3 0. 1 3 - 6 . 4 x l 0 - 3 cm .Ol .oz .03 .of- os ET££JfC TI V£ SVJ TURfl TI ON .06 .07 .0% .Of .10 .11 .IZ .13 co ui 86 W., - W 0 ( 4 . 4 - 5 ) t d 0 i s an index of the time r e q u i r e d f o r a snow l y s i m e t e r to respond to a sudden change of f l u x from to when (W^ - WQ) i s the corresponding change i n e x t r a i n t e r f a c e water. ' Since W i s almost constant f o r zero t e n s i o n l y s i m e t e r s , t h e i r s t a r t - u p time i s roughly i n v e r s e l y p r o p o r t i o n a l to the flow r a t e . Assuming <J> to equal 0 . 3 5 , t would be 20 years ( I ) , 2 . 5 hours, and - 4 . 7 minutes s f o r the flow r a t e s of curves 1 , 2 , and 3 i n f i g u r e 4 . 4 - 1 . Remember that the three curves are s o l u t i o n s f o r a snowpack that had been wetted from an i n i t i a l l y very dry s t a t e s i n c e the boundary we t t i n g curve has been used. Output of a l y s i m e t e r can only be con s i d e r e d r e p r e s e n t a t i v e of the f l u x e n t e r i n g the l y s i -meter f o r times l a r g e compared to the s t a r t - u p time. A l t e r n a -t i v e l y , a zero t e n s i o n l y s i m e t e r can only be considered to have e q u i l i b r a t e d with the snowpack f o l l o w i n g i t s exposure to an 3 2 accumulated flow of s e v e r a l cm /cm from snowmelt or r a i n f a l l . The t r a n s i t time of water between the l y s i m e t e r i n t e r f a c e and the measuring d e v i c e i s not i n c l u d e d In the above d i s c u s s i o n . The s t a r t - u p response time of the t e n s i o n l y s i m e t e r Is probably s e v e r a l orders of magnitude s h o r t e r , which would make t h i s k i n d of snow l y s i m e t e r u s e f u l f o r measurement of low flow regimes. The s t a r t - u p response time of the t e n s i o n l y s i m e t e r cannot be c a l c u l a t e d from f i g u r e 4 . 4 - 2 . The i n t r o d u c t i o n of a t e n s i o n l y s i m e t e r with hu < h , i n t o a snowpack which i s on the boundary wetting curve, i n t r o d u c e s complex h y s t e r e s i s e f f e c t s 87 j u s t above the l y s i m e t e r . However, i f the h y s t e r e s i s i s ignored, f i g u r e 4.4-2 would give values of t f o r curves 1, 2, and 3 which s would be equal to 39 hours'i;,23 seconds, and 1.6 seconds r e s p e c -t i v e l y . The dynamic response time, equation 4.4 - 5 , i s always negative as shown by an examination of f i g u r e s 4.4-1 and 4.4-2. | t ^ | i s an index of the time by which a wetting f r o n t a r r i v e s e a r l y . D i s t o r t i o n of the wetting f r o n t by the l y s i m e t e r can only be Ignored f o r time s c a l e s much l a r g e r than | "fc [ Three examples of wetting f r o n t s are c o n s i d e r e d based on the moisture p r o f i l e s g i v e n above : (1) Increase i n S* from 0.0021 to 0.049, (2) Increase i n S g from 0.0021 to 0 . 1 3 , and * (3) Increase i n S g from 0.049 to 0.13. (In a l l three cases, the snow i s assumed to have been wetted to i t s i n i t i a l value from a very dry s t a t e . ) Dynamic response times f o r a zero t e n s i o n l y s i m e t e r are -18 min., -1.20 min.,. and -0 . 5 8 min„, f o r the three w e t t i n g f r o n t s r e s p e c t i v e l y . S u b s t i -t u t i n g equation 3-3-1 i n t o 4.4-5 and u s i n g U = (z - z ) / t , gives s f o r the r a t i o of response time to wetting f r o n t t r a n s i t time from the snow s u r f a c e , -z ,' to the l y s i m e t e r at h e i g h t , z, : s C - " - 6 ) I V t ! " ( 9 , - e n ) ( z - z ) • v 1 0 s ' For a l y s i m e t e r depth of 50 cm below the snow s u r f a c e , the average value of the r a t i o ^ / t f o r the three wetting f r o n t s i s only 1 .5% a c c o r d i n g to equation 4.4-6. Hence zero t e n s i o n l y s i m e t e r s have adequate response f o r measuring the t r a v e l time of wetting f r o n t s , provided the l y s i m e t e r i s not p l a c e d too c l o s e to the s u r f a c e . On the other hand, i f the exact shape of the wetting f r o n t i s r e q u i r e d , a t e n s i o n l y s i m e t e r must be used. Dynamic response times for- the t e n s i o n l y s i m e t e r are 21 seconds, 1 . 6 seconds, and O . 8 5 seconds f o r the three wetting f r o n t s r e s p e c t i v e l y . The r e s o l u t i o n of the l y s i m e t e r may be l i m i t e d by the f l u x measuring device or by the frequency of readings taken. 4 . 5 TENSION SNOW LYSIMETER The w r i t e r b u i l t two kinds of snow l y s i m e t e r . One kind i s the impervious sheet type with a l a r g e area f o r measuring the i n t e g r a t e d f l u x over a l a r g e p l o t , which w i l l be d i s c u s s e d i n the next s e c t i o n . The second k i n d i s the t e n s i o n l y s i m e t e r , of 2 i n t e r f a c e area 110 cm , f o r flow measurement w i t h i n s m a l l p l o t s . A t e n s i o n l y s i m e t e r can only be used i n a wet snowpack and any exposed s e c t i o n s must be p r o t e c t e d f&o»m f r e e z i n g , as i s the case with the tensiometer. S i m i l a r l y , short wave r a d i a t i o n penetra-t i n g the snow should not s i g n i f i c a n t l y change the snow p r o p e r t i e s near the l y s i m e t e r i n t e r f a c e . Tension l y s i m e t e r s have been used i n s o i l s to reduce the i n t e r f a c e end e f f e c t and to 'overcome' the ' r e l u c t a n c e ' of l i q u i d to leave the s o i l (e.g. Cole, 1 9 5 8 ) . I d e a l l y , the negative pressure a p p l i e d should be c o n t i n u o u s l y a d j u s t e d to match the measured pr e s s u r e head i n the s o i l . In p r a c t i c e , the l y s i m e t e r pressure i s h e l d at a constant v a l u e , 89 LIQUID FLOU) TENSION UNIT fl-pjOSTA&LE BOLTS LVSIMETZR. SUPPoR. T F i g u r e 4.5-1 Diagram of a t e n s i o n l y s i m e t e r p l a c e d w i t h i n a snowpack with only the vent and d r a i n expos-ed. The porous ' p l a t e ' i s a 2 cm t h i c k l a y e r of sand sup-ported by a nylon sheet. The l y s i m e t e r r e c e i v e s flow across a 22x5.1 cm area. 90 more negative than the c a p i l l a r y p r essure head expected d u r i n g the w e t t i n g / d r y i n g c y c l e s of the s o i l . U n l i k e the zero t e n s i o n l y s i m e t e r , the t e n s i o n l y s i m e t e r used i n s o i l , w i l l draw water towards i t and overestimate the flow ( e f f i c i e n c y > 1). In order to o b t a i n the proper e f f i c i e n c y , the w r i t e r added a short r i m of c a l c u l a t e d h e i g h t f o r snow. Seven t e n s i o n snow l y s i m e t e r s were b u i l t f o l l o w i n g the design i n f i g u r e 4.5-1. A p i c t u r e of the instrument i s shown In f i g u r e 4.5-2. The l y s i m e t e r c o n s i s t s of a t e n s i o n u n i t , a r i m , an outflow u n i t and a r e c e i v i n g c o n t a i n e r . A wood support keeps the l y s i m e t e r i n t e r f a c e i n contact with the snow and keeps the t e n s i o n u n i t at the c o r r e c t height above the outflow u n i t . The t e n s i o n u n i t i s an a c r y l i c chamber f i l l e d with water which i s maintained at a negative pressure ( i n cm of water) equal to the height of the chamber above the n o z z l e of the outflow u n i t . " The negative pressure produced by the outflow u n i t can be v a r i e d by moving the outflow u n i t along the support s c a l e towhich. i t i s attached by an a d j u s t a b l e clamp. The top of the chamber i s p e r f o r a t e d and supports a l a y e r of f i n e sand which acts as the t e n s i o n ' p l a t e ' across which the water pressure i n the chamber i s communicated to that of the l i q u i d i n the snow. The water chamber remains f i l l e d with water as long as the c a p i l l a r y p r e ssure set by the outflow u n i t i s not so n e g a t i v e as to draw a i r through the sand pores. The sand i s supported by a r n y l o n sheet which i n t u r n i i s supported by a j f i n e screen above the water chamber p e r f o r a t i o n s . A e o l i a n sand, s i e v e d and washed to o b t a i n the 0.104 to F i g u r e 4.5-2 T e n s i o n l y s i m e t e r f o r measuring the f l u x i n wet snowpacks over s m a l l s c a l e s . The r i m would be i n s e r t e d i n t o the r o o f o f a s h o r t t u n n e l i n the snowpit w a l l and kept t h e r e by the wood support and t h e a d j u s t a b l e b o l t s . The porous ' p l a t e ' , a l a y e r o f 0.104-0.246mm. s e i v e d sand, m a i n t a i n s c o n t a c t between t h e l i q u i d water i n t h e snow above and t h a t i n th e water chamber below. The water at the snow/sand I n t e r f a c e was m a i n t a i n e d at -25cm p r e s s u r e by a d j u s t i n g the o u t f l o w u n i t n o z z l e t o be 25cm below I t . 92 0. 246 mm f r a c t i o n i s spread over the water chamber and shaken with a pneumatic v i b r a t o r to o b t a i n a 2 cm t h i c k close-packed l a y e r . The r e s u l t i n g l a y e r has a b u b b l i n g pressure head of -50 cm. The sand serves not only to tr a n s m i t water pressure but a l s o as the contact m a t e r i a l by p e n e t r a t i n g i n t o any spaces i n the snow. The a c r y l i c rim, 6 mm t h i c k , the lower part of which encloses the sand, r i s e s 4 cm above the top of the.sand l a y e r , 1. e. above the l y s i m e t e r i n t e r f a c e . 4 cm i s higher than the g r e a t e s t t r a n s i t i o n height a n t i c i p a t e d , i . e . 3 cm f o r very dry snow of v== 1 cm ^ ( f i g u r e 3.1-1). The l y s i m e t e r area f o r 2 r e c e i v i n g flow i s 112 cm , d e f i n e d by the r e c t a n g u l a r r i m 22 cm x 5.1 cm. The l y s i m e t e r s have a r e l a t i v e l y high perimeter to 2 area r a t i o (0.48 cm/cm ). Two t r a n s p a r e n t p l a s t i c tubes are connected to the base of the water chamber of which one, the i n l e t tube (not shown i n f i g u r e 4.5-1) i s clamped shut i n the f i e l d i n s t a l l a t i o n . The o u t l e t tube i s both a d r a i n f o r the t e n s i o n u n i t , and the hang-ing water column which produces the negative t e n s i o n i n the water chamber. A l l connections are by means of short s e c t i o n s of s o f t rubber t u b i n g which can be clamped shut e a s i l y . The water i n the t e n s i o n u n i t i s p l a c e d under a t e n s i o n In the l a b o r a t o r y because of the care r e q u i r e d to remove any s i g n i f i - . cant a i r bubbles from the water chamber and connecting hoses, and to e s t a b l i s h the c o n s i s t e n c y of the sand l a y e r . I f l a r g e a i r bubb.les migrate to the hanging water column, the c o n t i n u i t y of the water column may be broken and the water chamber could d r a i n . The l y s i m e t e r would then become a zero pressure l y s i - -meter but 93 with a r i m height i n s u f f i c i e n t t o assure 1 0 0 % e f f i c i e n c y . A i r bubbles are removed by sip h o n i n g water through the water chamber from the i n l e t to the o u t l e t tubes, while keeping the water i n the chamber under t e n s i o n . Bubbles are d i s l o d g e d from the chamber p e r f o r a t i o n s by.sharp blows with a t o o l . The t e n s i o n u n i t s can be s t o r e d under t e n s i o n i n d e f i n i t e l y , as long as the end of the o u t l e t tube i s immersed i n a c o n t a i n e r of water whose l e v e l r e l a t i v e t o the water chamber remains w e l l above the b u b b l i n g pressure of the sand, but below the water chamber i t s e l f , so as to r e t a i n the c o n s i s t e n c y of the sand l a y e r . As long as the pore water i n the f i n e sand i s under t e n s i o n , the sand l a y e r i s very r e s i l i e n t , and does, not f r a c -t u r e , even under heavy blows (see below). E v a p o r a t i o n of water from the l y s i m e t e r i n t e r f a c e d u r i n g storage w i l l be r e p l a c e d by water drawn from the c o n t a i n e r of water. The u n i t s were s t o r e d i n the l a b o r a t o r y f o r s e v e r a l days a f t e r being put under t e n s i o n t o check f o r any slow leaks of a i r i n t o the water chamber. These u n i t s were s t o r e d In the same way d u r i n g t r a n s -port to the f i e l d by v e h i c l e , or i n the f i e l d , pending t h e i r i n s t a l l a t i o n . During the l a s t hour of the t r i p to the f i e l d , they were backpacked to the experimental s i t e , the t e n s i o n u n i t s clamped shut and enclosed i n p l a s t i c bags to minimize e v a p o r a t i o n . The sand l a y e r remained s o l i d l y i n plac e even i f the u n i t was i n v e r t e d . The t e n s i o n l y s i m e t e r was i n s t a l l e d i n the snowpack i n the f o l l o w i n g manner. F i r s t , the l a y e r of i n t e r e s t was l o c a t e d on the snowpit w a l l , then the p i t w a l l was r e c e s s e d 30 cm, a dimension l i m i t e d by p o s s i b l e c o l l a p s e of the overhanging snow. Then small t u n n e l s , one f o r each l y s i m e t e r , were dug i n t o the rec e s s e d w a l l , below the snow-layer of i n t e r e s t , 28 cm wide by 35 cm high and 60 cm i n t o the snow. A deeper tunnel would have been more d e s i r a b l e to allow f o r the melt back-.of the snow p i t w a l l d u r i n g warm weather., but 60 cm was the l i m i t of convenient reach. The tu n n e l r o o f was smoothed and l e v e l l e d with a s t r a i g h t edge. The r i m of the t e n s i o n u n i t was f o r c e d up i n t o the tu n n e l r o o f u n t i l the sand l a y e r made contact with the snow. The r i m was not b e v e l l e d ( i n f a c t , the b e v e l had been f i l e d o f f , a f t e r t e s t t r i a l s showed that i t produced shear s t r e s s e s which would crack the r i m d u r i n g i n s t a l l a t i o n ) . The o l d wet snowpack was 'very tough' and the r i m co u l d be f o r c e d i n t o the snow only by r e p e a t e d l y s t r i k i n g a wood bl o c k p l a c e d under the t e n s i o n u n i t , with a hammer. The sand l a y e r r e t a i n e d i t s c o n s i s t e n c y even und§r t h i s treatment. A f t e r the t e n s i o n u n i t was in p p l a e e i n the r o o f of the t u n n e l , i t was h e l d there by p l a c i n g a wooden support under i t and a d j u s t i n g the two b o l t s shown i n f i g u r e 4 . 5 - 1 . The support r e s t e d on a p l a t f o r m board which had been placed on the t u n n e l f l o o r . Any s e t t l i n g of the surrounding snow would only i n c r e a s e the snow-lysimeter c o n t a c t . The i y s i m e t e r support had an ext e n s i o n arm which i n c r e a s e d i t s s t a b i l i t y and provided a means of mounting a s c a l e to which the outflow u n i t could be clamped. The height of the s c a l e was a d j u s t e d so that the s c a l e zero was l e v e l with the l y s i m e t e r i n t e r f a c e . The tu n n e l was then back-f i l l e d with snow to e l i m i n a t e the snow-air i n t e r f a c e at the t u n n e l r o o f . The center of the l y s i m e t e r was i n s t a l l e d about 46 cm i n s i d e the r e c e s s e d p i t w a l l with i t s long s i d e p a r a l l e l to the w a l l . ( C a r e f u l l y s i m e t e r i n s t a l l a t i o n takes about the same amount of time r e g a r d l e s s of whether the l y s i m e t e r i s of the t e n s i o n v a r i e t y or not.) The i n t e r f a c e pressure was permanently set at - 2 5 cm, a value w e l l above the sand l a y e r b u b b l i n g pressure ( - 5 0 cm), yet below the minimum snow c a p i l l a r y pressure measured d u r i n g the f i e l d season ( -15.4 cm). The outflow u n i t can be b u r i e d i n the snowpit w a l l to p r o t e c t the water i n i t from f r e e z i n g , with only the outflow u n i t d r a i n and a i r vent exposed. The r e c e i v i n g c o n t a i n e r was a graduated c y l i n d e r , which was r e p l a c e d by a one l i t r e c o l l e c t i o n b o t t l e when the s i t e was not manned. The r e c e s s was covered with a p o l y s t y r e n e board. When the l y s i m e t e r s were dug out of the snowpack ( r a t h e r than 'melted out') at the end of an experiment, they a l l appeared t'o be i n good contact with the snow. None of the l y s i m e t e r s ' t e n s i o n was broken while w i t h i n the snow d u r i n g the 204 l y s i m e t e r - h o u r s of o p e r a t i o n . A l a y e r of snow of f i n e r pore s i z e was sometimes found r i g h t a g a i n s t the l y s i m e t e r sand l a y e r which may have been produced by pressure d u r i n g i n s t a l l a t i o n . According to equation 25 i n Colbeck ( 1 9 7 3 b ) , the negative t e n s i o n i n the l y s i m e t e r would be expected to i n h i b i t f r e e z i n g of the snow at the l y s i m e t e r i n t e r f a c e ( c o n v e r s e l y , zero t e n s i o n l y s i m e t e r s would encourage some i c e formation at the i n t e r f a c e ) . 96 F i g u r e 4.6-1 Diagram of a s t e e l l y s i m e t e r . There i s no rim along the upslope edge. A d e b r i s screen i s p l a c e d along the downslope edge. 4 . 6 STEEL SNOW LYSIMETER 97 p The w r i t e r a l s o b u i l t a l a r g e area ( 1 0 , 0 0 0 cm ) metal l y s i -meter of the impermeable sheet type. The l y s i m e t e r i s c o n s t r u c t e d from sheet metal about one meter square with a 1 2 . 5 cm high r i m on three s i d e s . The l y s i m e t e r i s i n s t a l l e d so that the sheet slopes down towards the snowpit w a l l as i n f i g u r e 4 . 6 - 1 . The t h r e e - w a l l e d l y s i m e t e r i s much e a s i e r to i n s t a l l i n snow ' i n s i t u ' and a l s o much s a f e r , s i n c e only a narrow s l o t must be cut i n the snow f o r the l y s i m e t e r to be i n s e r t e d . The w r i t e r has not analyzed the e f f i c i e n c y of t h i s d e s i g n . The snowpit w a l l i s re c e s s e d 30 cm back and then a s l o t , one meter across and one meter back i s cut i n t o the p i t w a l l with the help of c o r i n g tubes and a snow saw. The l y s i m e t e r i s then s l i d i n t o the s l o t and the r e c e s s i s b a c k f i l l e d with snow, except f o r the area around the o u t l e t . The l y s i m e t e r o u t l e t i s a d r a i n f i x t u r e covered on top with a d e b r i s screen and emptying below i n t o the r e c e i v i n g c o n t a i n e r . The flow was f i r s t measured i n a 30 l i t r e p a i l i n s i d e which a s c a l e was attached. Much data was l o s t because the p a i l was too s m a l l , so that a t i p p i n g bucket system was used i n a l a t e r experiment. Four metal l y s i m e t e r s were made and back-packed to the experimental s i t e . The l a r g e r area of the metal l y s i m e t e r s should make the f l u x e s they measure more r e p r e s e n t a t i v e of the melt r a t e o c c u r i n g at a s i t e than that of the small t e n s i o n l y s i m e t e r s , although the absence of an u p h i l l r i m has an unknown e f f e c t . The metal l y s i m e t e r took longer to i n s t a l l than the t e n s i o n l y s i m e t e r d i d . The near end of a metal l y s i m e t e r , being melted out can be seen i n f i g u r e 5 - 1 - 4 . •97 A SUMMARY A rugged and economical tensiometer has been designed to measure the c a p i l l a r y pressure i n snowpacks. The s m a l l s c a l e of the tensiometer porous cup allows the s m a l l s c a l e flow f e a t u r e s to be observed. The tensiometer has a good reponse time, with some l a g expected only at the steeper w e t t i n g f r o n t s . The manometer i s mounted d i r e c t l y to the porous cup support tube to minimize the zero e r r o r i n the cup's h e i g h t . A d i s -c u s s i o n i s given of the rim h e i g h t s r e q u i r e d f o r a snow l y s -imeter i n order that i t w i l l measure the f l u x with 100% e f -f i c i e n c y . A t e n s i o n l y s i m e t e r has been designed to measure the f l u x i n snowpacks over s m a l l s c a l e s . The short rim height t h a t the t e n s i o n l y s i m e t e r r e q u i r e s allows a c l o s e p o s i t i o n i n g of the snow instruments to each other so t h a t inhomogeneities i n the snow can be avoided. The t e n s i o n l y s i m e t e r has a much f a s t e r reponse time than the z e r o - t e n s i o n l y s i m e t e r . The i n s t a l l a t i o n of sets of these instruments i n t o experimental snowplots i s d i s c u s s e d i n the f o l l o w i n g chapter. CHAPTER FIVE 98 EXPERIMENTAL- PROCEDURES The i n i t i a l p i l o t experiments at the beginning of the 1974 f i e l d season suggested that to o b t a i n good r e l a t i o n s between the h y d r o l o g i c v a r i a b l e s , the d i f f e r e n t - v a r i a b l e s should a l l be measured c l o s e t ogether i n space, because of .the l a t e r a l v a r i a -t i o n s i n flow produced by i c e sheets. S i m i l a r l y , these measure-ments should be made over a narrow range of h e i g h t , because of the many l a y e r s i n the snow pack. A l s o , i t appeared that to o b t a i n a range of flows l a r g e enough to r e v e a l the f u n c t i o n a l r e l a t i o n s h i p above the 'noise' produced by the l a t e r a l v a r i a t i o n s i n flow, i t would be necessary to a r t i f i c i a l l y r e g u l a t e the f l u x . Consequently experiments were performed i n a s e r i e s of r e g u l a t e d snowplots at s i t e H — areas of snow where the heat and mass exchange at the snow s u r f a c e c o u l d be r e g u l a t e d . In a d d i t i o n , two experiments were done to measure c a p i l l a r y p r e ssure changes with d i u r n a l melt c y c l e s , i n order tosstudy them under n a t u r a l flow r a t e s . 5.1 REGULATED SNOWPLOTS — EXPERIMENTAL INSTALLATION The r e g u l a t e d snowplots were instrumented fro-m snowpits. The p i t s were dug p r o g r e s s i v e l y upslope i n t o the f e n c e d - o f f area, shown i n f i g u r e 1 . 4 - 8 of u n d i s t u r b e d snow f o r each experiment. Each snowplot was u p h i l l from a snowpit and extended 200 cm HQ UO It* Uo StO vi; ™ 500 MO \ 1 1 1 1 1 i > 1 i i 1 \ i F i g u r e 5 . 1 - 1 Diagram of the I n s t a l l a t i o n f o r experiment 1 . The s e c t i o n i s through instruments i n the r e g u l a t e d snow-p l o t , p a r a l l e l to the snowpit w a l l - Tensiometer Tension l y s i m e t e r Ice sheet -SNOU) PLOT sutrflce JUNE io — JUNE 11 _ — a. f> c d ate J • • .? • • • • • • • •• •? I I i i i i i i ) i 1 1 1 t i l l • f-1 1 4F3 1 i i i -J-* 7 IflYEt ' t-J •i ZOO -t-3oo HORIZONTAL POSITION (cm) +500 VO 100 u p h i l l and 400 cm a c r o s s . The snowplot s u r f a c e was lowered, i f necessary, to be about 46 cm above the tensiometers at the beginning of the experiment. uAt such a depth, the instruments were below a l l but a n e g l i g i b l e p o r t i o n of the p e n e t r a t i n g short wave r a d i a t i o n , yet c l o s e enough to the s u r f a c e to experience a wide range of f l u x e s . The snowplots were instrumented by p l a c i n g l y s i m e t e r s and tensiometers c l o s e t ogether i n the same l a y e r . The i n s t a l l a t i o n f o r experiment 1 i s shown i n f i g u r e 5 - 1 - 1 . T h i s type of i n s t a l l a t i o n r e p r e s e n t s a systematic sampling of the flow regime at the p l o t s c a l e s i n c e , once the approximate l o c a t i o n of the snowplot was s e l e c t e d , the f i r s t instrument was l o c a t e d by chance and the other instruments at r e g u l a r i n t e r v a l s r e l a t i v e to the f i r s t (Peterson and C a l v i n , 1 9 6 5 ) . The l o c a t i o n s of the i c e sheets and l y s i m e t e r s were determined when the instruments were i n s t a l l e d . The l o c a t i o n s of the tensiometer cups were measured r e l a t i v e to the l y s i m e t e r s when the instruments melted out. The experimental i n s t a l l a t i o n s ( i n e l e v a t i o n ) f o r experiments 3 and 4 are shown i n f i g u r e s 5 . 1 - 2 and 3 r e s p e c t i v e l y . For these two experiments, a l l the measurements were taken from photographs a f t e r the p l o t s were cut open to exhume the instruments. The p o s i t i o n of the l a y e r i n g was deduced from the s t a i n p a t t e r n l e f t by the passage of a wave of dyed water. The i n s t a l l a t i o n s c o n s i s t e d of f o u r or f i v e l y s i m e t e r s at the,same l e v e l ( r e l a t i v e to the l a y e r i n g ) about 31 cm apart rim to rim. The long s i d e of the l y s i m e t e r was p a r a l l e l to the snowpit w a l l , and the tensiometers were p l a c e d i n a row — the 'tensiometer l e v e l ' — about 10 cm above the l y s i m e t e r i n t e r f a c e t-w\ 31,0 340 , Ho ^ zzo 110\ F i g u r e 5.1-2 Diagram of the i n s t a l l a t i o n f o r experiment 3. The s e c t i o n i s through instruments i n the r e g u l a t e d snowplot, p a r a l l e l to the snowpit w a l l . Tensiometer Tension l y s i m e t e r O Density core ~ Ice sheet lio -loo L- y SIME TE/? THlN ICE #WJ sneers) -IOO O HORIZONTAL POSITION (C*r<) +100 o \ i i I r i — r 1 — r T P 220 2001 I HO I to S 1*0 /20 /00 Figure 5-1-3 Diagram of the i n s t a l l a t i o n - . f o r experiment 4. The s e c t i o n i s through instruments i n the r e g u l a t e d snowplot j p a r a l l e l to the snowpit w a l l . Tensiometer Tension l y s i m e t e r Ice sheet o Density core o C e n t r i f u g e core LAVE £ -zoo -ISO -too -so HORIZONTAL POSITION o ro 103 ( i . e . 6 cm above the r i m ) . A few tensiometers were a l s o l o c a t e d i n a second row, about 5 cm above the f i r s t so that the c a p i l l a r y p r e ssure g r a d i e n t c o u l d be determined. The tensiometer cups were estimated to be l o c a t e d r i g h t over the l l y s i m e t e r i n the u p h i l l d i r e c t i o n . The r e l a t i v e l o c a t i o n of the tensiometers i n the. c r o s s - h i l l d i r e c t i o n can be seen i n f i g u r e s 5 . 1 - 1 to 3 - The space between and on e i t h e r s i d e of the l y s i m e t e r s .was r e s e r v e d f o r t a k i n g cores from the snow f o r d e n s i t y measurements. The s i z e of the space l i m i t e d the number of cores that c o u l d be taken. The placement of the instruments was i n a r e c e s s of the p i t w a l l , so that a short snow ' c o r n i c e ' overhung about 20 cm from the w a l l (the top of which was not part of the snowplot). The r e c e s s was boarded up with p o l y s t y r e n e boards when readings were not being f r e q u e n t l y taken. In a d d i t i o n a p o l y s t y r e n e l e a n - t o was b u i l t a g a i n s t the instrumented p i t w a l l . The p i t w a l l was very s u s c e p t i b l e to m e l t i n g back because i t faced west-south-west, an unavoidable s i t u a t i o n a r i s i n g from the d i r e c t i o n of slope of the l a y e r i n g i n the snowpack. The instrumented r e c e s s was f u r t h e r p r o t e c t e d i n experiment 4 , by c o v e r i n g the r e c e s s with a sheet of f l e x i b l e b l a c k p l a s t i c , which e l i m i n a t e d the short wave r a d i a t i o n t h at c o u l d penetrate i n t o the snow. The use of black p l a s t i c i s recommended i f short wave r a d i a t i o n i s of concern. Views of the r e g u l a t e d p l o t s are shown i n f i g u r e s 5 - 1 - 4 to 6 , with the p l o t s i n v a r i o u s stages of experiment. Three kinds of i n f o r m a t i o n could be obtained from t h i s p a t t e r n of i n s t a l l a t i o n — the h y d r o l o g i c r e l a t i o n s i n the instrumented l a y e r by d i r e c t measurement; the average h y d r o l o g i c 104 F i g u r e 5 . 1 - 4 View of the experiment 1 i n s t a l l a t i o n , l o o k i n g ESE on June 1 5 j 1 9 7 4 . The instruments were r a p i d l y m e l t i n g out from the unprotected snowplot, which was c o n s i d e r a b l y higher than the surrounding snow s u r f a c e on t h i s date. The second set of tensiometers from the l e f t of the snowplot, near the meter s t i c k , had been removed from the snowpit w a l l . A s t e e l l y s i m e t e r i s seen to the l e f t o f the snowplot. 105 F i g u r e 5 - 1 - 5 View of the experiment 3 I n s t a l l a t i o n and r e g u l a t e d snowplot l o o k i n g east on J u l y 1 2 , 1 9 7 4 . The snowplot s u r f a c e had o r i g i n a l l y been cut lower than the surrounding snow f o r t h i s experiment. A patch of dyed water i s seen on the snowplot s u r f a c e . The dye s t a i n p a t t e r n w i l l be shown i n f i g u r e 5 - 5 - 2 . The d i r t bands at the i c e sheets d i d not extend more than a few centimeters i n s i d e the snowpit w a l l as can be seen by comparing the two photos. A meter s t i c k Is on the lower r i g h t . 106 F i g u r e 5.1-6 View of the experiment 4 i n s t a l l a t i o n on J u l y 2 5 s 1974. The instruments had j u s t been i n s t a l l e d i n t h i s snowplot. The b l a c k p l a s t i c and the p o l y s t y r e n e covers had been t e m p o r a r i l y removed f o r the photo. The two l i t r e c o n t a i n e r s were r e p l a c e d with graduated c y l i n d e r s when d e t a i l e d f l u x measurements were being made. 107 r e l a t i o n s between the instruments and. the snowplot s u r f a c e , from a n a l y s i s of f l u x waves from a r t i f i c i a l l y produced s u r f a c e i n p u t s ; and a d e t e r m i n a t i o n of the c a p i l l a r y pressure g r a d i e n t between the two l e v e l s of tensiometers. JAs p o i n t e d out i n s e c t i o n 3.2, only the space^ averaged c a p i l l a r y pressure g r a d i e n t can be measured between the two l e v e l s of tensiometers. The g r a d i e n t was determined f o r each i n s t a l l a t i o n as a whole and not f o r each l y s i m e t e r w i t h i n the i n s t a l l a t i o n because of the high zero o f f s e t e r r o r and sampling e r r o r f o r s m a l l numbers of tensiometers. The c a p i l l a r y pressure g r a d i e n t of the i n s t a l l a t i o n i s the mean of the d i f f e r e n c e i n pressure between the two rows of tensiometers f o r each l y s i m e t e r ( e x c l u d i n g any tensiometers which were placed below a n o t h e r ) , d i v i d e d by the mean s e p a r a t i o n of the two rows. 5-2 LIQUID WATER CONTENT MEASUREMENTS The measurement of the phase composition of unsaturated snow i s d i f f i c u l t because of the high accuracy r e q u i r e d f o r a coarse g r a i n e d medium. The amount of l i q u i d i s only a s m a l l f r a c t i o n of the t o t a l mass, and d u r i n g sample c o l l e c t i o n i t i s q u i t e d i f f i c u l t to a v o i d m e l t i n g some of the snow. The intended system f o r measurement of water content changes was to core the l a y e r of i n t e r e s t and measure the water content of the cores with a hot water c a l o r i m e t e r (Yosida, 1959). The f i r s t step i n the c a l o r i m e t r y procedure was to take cores of known volume so that the water content by weight measured by the c a l o r i m e t e r c o u l d be converted to a water content by volume. 108 The c a l o r i m e t r y r e s u l t s are not used because the l i q u i d water content values e x h i b i t e d s e v e r a l v a r i a t i o n s which appeared to r e f l e c t the ambient wind and a i r temperature r a t h e r than the a r t i f i c i a l l y r e g u l a t e d flow c o n d i t i o n s . Only the f i r s t step i n the c a l o r i m e t r y measurement i s used -- t h a t i s , the c o l l e c t i o n of a core of known volume,;.and the measurement of i t s weight. Changes i n l i q u i d water content (by volume), are gi v e n by : ( 5 . 2 - 1 ) A6 = A p s - A p d where p i s the t o t a l snow d e n s i t y while p i s the d e n s i t y of S CL the i c e s k e l e t o n . I f the l a y e r i n q u e s t i o n i s below the l e v e l of s i g n i f i c a n t snowmelt, and i f snow metamorphism i s very slow, then ( 5 - 2 - 2 ) A p d - 0 . The cores were cut from deep i n s i d e the l a y e r i n the spaces among the other Instruments. JA 6 l cm, b e v e l l e d s t e e l c o r i n g tube, 0 . 6 l cm w a l l t h i c k n e s s , and 2 . 5 cm I.D. was pushed, or hammered as r e q u i r e d , h o r i z o n t a l l y i n t o the snowpit w a l l . A blow to the sid e of the c o r i n g tube separated the end of the core from the snow. The withdrawn core was pushed out of the c o r e r with a ram (the deepest 2 . 5 cm having been d i s c a r d e d ) i n t o the c a l o r i m e t e r can u n t i l the core was f e i t to be i n contact with the base of the c a l o r i m e t e r can. The core was cut with a blade l e v e l with the top of the c a m g i v i n g a core-segment of le n g t h 15 cm (gross core-segment'volume 7 2 . 4 5 c c ) . The core segment volume was c o r r e c t e d f o r compression of the core d u r i n g sampling, as estimated from the r a t i o of the length of c o r i n g 109 tube i n s i d e the snow to the l e n g t h . o f the core. Core-segments which broke while being p l a c e d i n the c a l o r i m e t e r were d i s c a r d e d . The core-segment'was weighed on a f i e l d balance. The number of core-segments from a sampled l a y e r was s m a l l (average of four), because of the small space a v a i l a b l e f o r sampling among the instruments and because of the number of broken core-segments. The change i n mean sample d e n s i t y was used to c a l c u l a t e the change i n l i q u i d water content f o r the snow l a y e r . The s c a t t e r of d e n s i t y values f o r a g i v e n flow s i t u a t i o n a r i s e s from random e r r o r s i n the measurement of core-segment l e n g t h , compression, and weight, as w e l l as from l a t e r a l v a r i a t i o n s i n d e n s i t y w i t h i n the snow l a y e r . Density samples were taken only d u r i n g e i t h e r steady or d e c r e a s i n g flows. The standard e r r o r of the mean d e n s i t y , of course, decreases with the number of core-segments, n : ( 5 . 2 - 3 ) a = a //nn where a i s the standard d e v i a t i o n of i n d i v i d u a l d e n s i t y P'f measurements. The d e n s i t y values may a l s o be i n e r r o r systema-t i c a l l y d u r i n g the course of a g i v e n flow s i t u a t i o n but randomly between flow s i t u a t i o n s . These mixed e r r o r s , such as the f i e l d balance zero e r r o r , i n as much as they are amenable to c a l c u -l a t i o n , are s m a l l e r than those g i v e n by equation 5 . 2 - 3 . I t i s assumed ( 1 ) that i f there i s a change i n compression along the core the^n t h i s change i s not water content dependent; ( 2 ) that the e r r o r i n c c u t t i n g the core segment le n g t h i s not water 110 content dependent; (3) that the l i q u i d water content was not i n f l u e n c e d by being sampled near other sampling h o l e s ; (4) that equation 5 . 2 - 2 i s a r e a l i s t i c approximation i n an o l d wet snow-pack over short periods of time. The average s e p a r a t i o n ( c e n t e r -t o - c e n t e r ) of cores from other instruments or from other core holes was 6 cm. Core holes were r e f i l l e d . The e f f e c t of p e n e t r a t i n g short wave r a d i a t i o n on changing by i n t e r n a l melt can only be estimated, s i n c e the e x t i n c t i o n c o e f f i c i e n t of the snow at the- experimental s i t e i s not known. A value f o r the e x t i n c t i o n c o e f f i c i e n t , 3 , of . 0 . 1 7 cm i s given f o r a wet snow sample with 0.1 cm g r a i n s i z e i n M e l l o r ( 1 9 6 5 ) . The core-segment samples were withdrawn from an average of 40 cm i n s i d e the snow from both the snow s u r f a c e and snowpit w a l l . The change i n i c e s k e l e t o n d e n s i t y , Ap^, can be approximated as -Bx -4 Ap, - Be ER t = 1 . 9 x ' 1 0 R t gm/cc where x i s taken to be "a no no & 40 cm, B = 0 . 1 7 cm ^ (the v a r i a t i o n i n x was l e s s thanVthe u n c e r t a i n t y due to B). R n Q i s the net shortwave r a d i a t i o n , roughly estimated from the measured flow r a t e s or r a t e of melt back of the snowpit w a l l , and t i s the time. The average of the Ap^ over each experimental p e r i o d , i s estimated to have been -4 2 x 10 gm/cc ( e x c l u d i n g one set of data, which was d i s r e g a r d e d because i t s I n c l u s i o n would i n c r e a s e Ap^ by an order of magnitude). In s p i t e of the apparent i n s i g n i f i c a n t change i n p^, the e f f e c t of short wave r a d i a t i o n cannot be r u l e d out i n view of the u n c e r t a i n t y i n the e x t i n c t i o n c o e f f i c i e n t . I l l 5 . 3 REGULATED SNOWPLOTS — ARTIFICIAL CONTROL OF THE FLOW To. adequately d e f i n e the h y d r o l o g i c r e l a t i o n s f o r a snowpack, a l a r g e range of flows i s most h e l p f u l . Rainstorms can produce l a r g e changes of flow, e s p e c i a l l y i f the snowpack i s i n i t i a l l y q u i t e dry. A very heavy r a i n s t o r m on May 2 2 - 2 5 , 1974 was measured at the p r e c i p i t a t i o n gauge of s i t e M to average 0 . 5 cm/ hr over an eighty hour p e r i o d . Very heavy r a i n occured d u r i n g short i n t e r v a l s , up to 19 cm/hr over one 15 minute p e r i o d . How-ever the occurrence of rainstorms i s not s u f f i c i e n t l y p r e d i c t a b l e to a llow f o r advance p r e p a r a t i o n and d u r i n g such p e r i o d s bad weather makes c a r e f u l f i e l d work i m p o s s i b l e . From May 2 to June 5 the weather could be c h a r a c t e r i z e d as c o o l and cloudy with frequent r a i n or snow. This p e r i o d was used to g a i n experience by p i l o t s t u d i e s with instruments. In the f o l l o w i n g nine weeks, four weeks were sunny and very warm and p r o v i d e d n a t u r a l d i u r n a l flow v a r i a t i o n s between day and night whose form was c o n v e n i e n t l y p r e d i c t a b l e . The d i u r n a l melt c y c l e s of June 19 and J u l y 3 0 , f o r example, r e s u l t e d i n flows at the metal l y s i m e t e r , l o c a t e d under undisturbed snow at 70 cm depth, which rose from a minimum of 0 . 0 2 cm/hr i n the e a r l y morning to 0 . 3 4 cm/hr i n the a f t e r n o o n . This range of flow was extended at both the high and'low end. During high melt p e r i o d s , the snowplot was i r r i g a t e d f o r short p e r i o d s at a r a t e of 7 cm/ hr. The i r r i g a t i o n was done by s p r a y i n g i c e water, as u n i f o r m l y as p o s s i b l e , over the p l o t from a 4 . 6 l i t e r watering can f o r 112 about h a l f an hour. To o b t a i n a longer drainage p e r i o d than that p r o v i d e d by drainage o v e r n i g h t , an i s o l a t i o n cover was put over the snowplot. The I s o l a t i o n cover c o n s i s t e d of a snow covered f l e x i b l e p l a s t i c sheet. The i s o l a t i o n , covers were a l s o used to provide a sharp wetting f r o n t , by suddenly removing them i n the middle of a warm day. The p l a s t i c sheet covered the r e g u l a t e d p l o t and was tucked i n about 15 cm at the edges. The sheet was l a i d down on a s l i g h t slope towards the snowpit so that meltwater aboveeit would flow away from the p l o t . The sheet was covered with about 30 cm of snow so as to i n s u l a t e the r e g u l a t e d p l o t from heat. The sheet and l a y e r of a p p l i e d snow r e q u i r e d about 20 minutes f o r i n s t a l l a t i o n or f o r removal. A l a y e r of 30 cm of snow reduced the p e n e t r a t i n g short wave r a d i a t i o n to the f o l l o w i n g f r a c t i o n : ( 5 . 3 - D R /R = e ~ 3 x = e " 0 , 1 7 x 3 0 = 0 . 0 0 6 1 w n no where 3 = 0 . 1 7 c m - 1 has been assumed as i n s e c t i o n 5 . 2 ' . With' R = 0 . 3 cm/hr, R would be 0 . 0 0 2 cm/hr under steady s t a t e no 3 n J c o n d i t i o n (the net r a d i a t i o n s are expressed i n terms of e q u i v a l e n t m e l t r a t e s ) . The highest flow r a t e s c o u l d not be produced by i r r i g a t i o n alone,, but only i f done on warm sunny days when the snow was already q u i t e wet. Then, with l i m i t e d i r r i g a t i o n , a high f l u x c o u l d reach the instrument l e v e l without being attenuated on the way. The supply of water was l i m i t e d by what was a v a i l a b l e i n the metal l y s i m e t e r c o l l e c t i o n p a i l s u n t i l the beginning of J u l y , 113 when m e l t i n g snow uncovered, a smal l p o o l nearby that was used as the water supply t h e r e a f t e r . The minimum drainage flow i s that of the kinematic wave whose t r a v e l time from the snow s u r f a c e to the l y s i m e t e r l e v e l , z, i s equal to the time between the onset of negative snow surf a c e thermal balance i n the evening and the a r r i v a l of the next day's melt wave at z. The minimum flow can be decreased by p l a c i n g the l y s i m e t e r c l o s e r to the snowplot s u r f a c e , or by a r t i f i c i a l l y i n c r e a s i n g the drainage p e r i o d at the l y s i m e t e r l e v e l . The i s o l a t i o n covers were used f o r up to 50 hours at a td>me, d u r i n g which the flow measured by some l y s i m e t e r s dropped to about 0 . 0 0 2 cm/hr. A r t i f i c i a l c o n t r o l of the flow i n an unenclosed p l o t w i t h i n the snowpack i s p o s s i b l e as long as the f l o w i i s dominated by g r a v i t y f o r c e s , and takes p l a c e i n a v e r t i c a l d i r e c t i o n . In. uniform snow, the l a t e r a l d i s p e r s i o n by c a p i l l a r y f o r c e s to or from the surrounding snow i s l i m i t e d to the edge of the p l o t , except f o r very long times. The instrument i n s t a l l a t i o n i n the r e g u l a t e d p l o t s were such that a margin of 50 cm e x i s t e d on e i t h e r s i d e and 90 cm u p h i l l t o the edge of the snowplot. Nevertheless the minimum flows under the i s o l a t i o n cover were highe r from l y s i m e t e r s near the upslope margin i n some cases so that some t r a n s f e r of water i n the p l o t may have taken p l a c e . L a t e r a l flow Is estimated d u r i n g i r r i g a t i o n by comparing the known i r r i g a t i o n r a t e with the mean flow measured at the l y s i -meter l e v e l . Two steady s t a t e flow r a t e s c o u l d be obtained d u r i n g the 114 wetting process : the f l u x f o l l o w i n g removal of the i s o l a t i o n cover on a warm, sunny day, and the i r r i g a t i o n f l u x . I t i s assumed that the f l u x at the l y s i m e t e r l e v e l d u r i n g steady flow i s the same as t h e . f l u x at the tensiometer — d e n s i t y core l e v e l , 10 cm above. Drainage was produced by immediately f o l l o w i n g the t e r m i n a t i o n of i r r i g a t i o n with replacement of the i s o l a t i o n cover. The f l u x i n c r e a s e s with depth, d u r i n g drainage produced i n t h i s way, a c c o r d i n g to equation 3 . 4 - 8 . The r a t i o of t e n s i o -meter to l y s i m e t e r depths averaged 0 . 8 2 i n the r e g u l a t e d p l o t s , which f o r e = 3 - 5 g i v e s a f l u x r a t i o of 0 . 7 6 at the two l e v e l s , d u r i n g d e c r e a s i n g flows. No c o r r e c t i o n i s made to the l y s i m e t e r f l u x v a l u e s , to r e t a i n the o b j e c t i v e nature of the f i e l d v a l u e s . The treatment schedule i s g i v e n here f o r r e f e r e n c e . On June 1 0 , i r r i g a t i o n was a p p l i e d f o r 26 minutes to the m e l t i n g snowplot of experiment 1 , s t a r t i n g at 1130 P a c i f i c Standard Time (P.S.T.). When the i r r i g a t i o n was stopped, the i s o l a t e d cover was put over the snowplot and kept i n p l a c e f o r 49 hours. On June 1 2 , by 1 3 1 5 » the cover had been removed. A second, 26 minute i r r i g a t i o n was then commenced at 1720 f o l l o w i n g which the p l o t was covered f o r 17 hours. During June 10 - 1 2 , the weather was sunny and warm, as shown i n f i g u r e 1 . 4 - 5 . On J u l y 6 , a f t e r the experiment 3 p l o t had been'covered f o r 19 hours., I t s i s o l a t i o n cover was removed by 1 1 1 5 . The i s o l a t i o n cover was r e p l a c e d again by 1639> and the drainage observed f o r s e v e r a l hours. No i r r i g a t i o n was done i n experiment 3 because of g e n e r a l l y u n s u i t a b l e weather. lWarm and sunny weather r e t u r n e d i n the l a s t week of J u l y . The experiment 4 p l o t was i r r i g a t e d 115 on J u l y 26 ( 1 3 0 5 ) j but because the i r r i g a t i o n had been done too soon a f t e r the p l o t s u r f a c e had been lowered, the data were r e j e c t e d . The snowplot was subsequently covered f o r 21 hours, the i s o l a t i o n cover being removed by 1110 on J u l y 2 7 . The p l o t was i r r i g a t e d f o r 28 minutes, beginning at 1 5 3 8 , and then the i s o l a t i o n cover was immediately r e p l a c e d and maintained f o r 49 hours. 5 . 4 STAINING THE FLOW PATHS When dyed water i s added to the snow s u r f a c e , the f l u x wave of the e x t r a l i q u i d added w i l l pass through the snow. Some of the dye i n the added water w i l l be t r a n s f e r e d to the r e t e n t i o n water i n the pack, so that a s t a i n w i l l remain behind to mark the path of the f l u x wave f o r some time a f t e r the wave has d r a i n e d away. I f a s e c t i o n of snow i s cut open, the s t a i n p a t t e r n can be photographed. The s t a i n p a t t e r n w i l l d e p i c t the flow paths of the added water, and not the flow paths d u r i n g n a t u r a l f l o w s , but i t i s not unreasonable to expect a c l o s e resemblance between the p a t t e r n s . As noted i n s e c t i o n 1 . 6 , food c o l o r i n g of unknown s t r e n g t h was used to dye the water i n the 1973 f i e l d experiment. Although food c o l o r i n g was again used i n May, 1 9 7 4 , subsequent d y e . t e s t s used rhodamine WT.diluted to such a s t r e n g t h as to have n e g l i g i b l e e f f e c t on the snowpack but s t i l l be v i s i b l e i n the cut s e c t i o n . The rhodamine WTddye t e s t s were done to show not only the flow p a t t e r n but a l s o the r e l a t i o n between the s t a i n p a t t e r n and the c a p i l l a r y p r e ssure 116 v a r i a t i o n produced by the p a s s i n g f l u x wave of dyed water. A f t e r the tensiometers i n d i c a t e d passage of the f l u x wave, the t e n s i o -meters were removed from the snow and a bean was pushed to the end of each tensiometer access h o l e . The s e c t i o n c o n t a i n i n g the beans was then exposed by c u t t i n g away the snow, to show t h e i r r e l a t i o n to the dye s t a i n p a t t e r n . About 30 minutes elapsed between the s t a r t of ex c a v a t i o n and when the s e c t i o n was f i r s t photographed. Approximate c a l c u l a t i o n s based on kinematic wave theory i n d i c a t e that the f l u x wave had d r a i n e d and only the remaining s t a i n was photographed. The rhodamine WT was d i l u t e d to a c o n c e n t r a t i o n of about _5 1 . 5 x 10 by weight and s p r i n k l e d over an area of about 1 x 3 meters (the long s i d e p a r a l l e l to the exposed s e c t i o n ) at about 7 cm/hr f o r about 6 minutes. A dyed patch i s shown i n f i g u r e 5 . 1 - 5 . The e f f e c t of the presence of rhodamine WT s a l t i n the snow can be c a l c u l a t e d i f i t s p r o p e r t i e s are assumed to be s i m i l a r totfehose of rhodamine B f o r which data Is given i n Gurr ( 1 9 7 1 ) . Rhodamine B has a molecular weight of 479 so t h a t a - 5 - 5 1 . 5 x 10 dye c o n c e n t r a t i o n r e s u l t s i n 6 . 2 6 x 10 moles of ions/kg of water f o r the b i n a r y s a l t . The f r e e z i n g p o i n t of the snow would be depressed by 1 . 8 6 x 6 . 2 6 x 1 0 ~ 5 = 1/2 x I O - 4 °C where 1 . 8 6 deg kgm/mole i s the f r e e z i n g p o i n t d e p r e s s i o n constant f o r water. By equation 25 i n Colbeck ( 1 9 7 3 b ) t h i s i s equal t o t t h e f r e e z i n g p o i n t d e p r e s s i o n produced by lowering the c a p i l l a r y pressure head by 1 . 5 cm. 'The dye c o n c e n t r a t i o n i s reduced s t i l l f u r t h e r as the added water mixes with the water a l r e a d y i n the snowpack. The e f f e c t of such a low c o n c e n t r a t i o n 117 of dye on the d e n s i t y and s u r f a c e t e n s i o n should be n e g l i g i b l e . 5.5 REGULATED- SNOWPLOTS — DESCRIPTION OP LAYERS AND PATTERNS OF FLOW The l a y e r s of snow i n 1974 which c o u l d be s e l e c t e d f o r i n s t r u m e n t a t i o n i n a r e g u l a t e d p l o t were l i m i t e d to a c e r t a i n range of depth below the l e v e l of snow s u r f a c e e x i s t i n g at the time of each experiment. N e v e r t h e l e s s , the l a y e r s s e l e c t e d seemed to cover the range of i c e sheet d e n s i t y and flow p a t t e r n that the 1973 f i e l d experiments had ledo- one to expect f o r the snowpacks i n the area. The type of l a y e r i n g was determined from the snow p r o f i l e s shown as a composite i n f i g u r e 1 .5-1 as w e l l as from o b s e r v a t i o n s of snow l a y e r s at each snowplot. The p a t t e r n of flow was determined from dye s t a i n t e s t s . In a d d i t i o n , the approximate b u b b l i n g pressure was determined by measuring the height of the c a p i l l a r y f r i n g e i n cubes of snow (10 cm edge) cut from the l a y e r . The cubes were p o s i t i o n e d , h a l f submerged i n a co n t a i n e r of water, a f t e r having been put i n t o or withdrawn from the water to o b t a i n the wetting and d r y i n g b u b b l i n g p r e s s u r e s , r e s p e c t i v e l y . The c h a r a c t e r i s t i c s of the l a y e r , flow p a t t e r n , and i n f o r m a t i o n on the experimental arrangement aVe giv e n f o r each experiment i n t a b l e 5 - 5 - 1 . The l y s i m e t e r l e v e l f o r experiment 1 was near the base of what seemed to be the most un-ifeSrm s e c t i o n of wet snow i n the e n t i r e s i x meter p r o f i l e of f i g u r e 1 . 5 - 1 . Only a few t h i n i c e sheets were found i n the 8 - 9 l a y e r d u r i n g the i n s t a l l a t i o n 118 Table 5 - 5 - 1 Snowpack experiments on Mt. Seymour, June-July, 1 9 7 4 Experiment One Two Three Four F i v e 1974 Ju 6-17 J u l 8 - 2 9 Jy 1 -12 J y 2 2 - 3 0 J y 2 8 - 3 0 Plow r e g u l a t -ed n a t u r a l r e g u l a t -ed r e g u l a t -ed n a t u r a l § l e v e l s 1 7 1 1 3 Layer 8-9 1 9 - 4 3 W-40 low er t h i r d 5 0 - 5 4 4 3 - 5 4 (?) Snow d e n s i t y (gm/cc) • 54 . 5 4 - . 6 8 . 6 2 . 5 7 . 5 9 - . 6 1 Snow f e a t u r e s uniform (?) many i c e sheets, t h i n i c e glands many t h i n i c e sheets many i c e sheets some i c e sheets Plow p a t t e r n sheet f i n g e r f i n g e r sheet Dye s t a i n photo f i g 5 . 5 - l f i g 5 - 5 - 2 f i g 5 . 5 - 3 Exp. photo f i g 5 . 1 - 4 f i g 5 . 6 - l f i g 5 . I - 5 f i g 5 . 1 - 6 Exp. diagram f i g 5 . 1 - l f i g 5 . 1 - 2 f i g 5 - l - 3 Tensiometer l e v e l (cm) 516 221-414 256 128 140 - 2 0 0 (?) Tensiometer depth (cm) 4 4 - 3 6 3 9 - 2 2 l e v e l 1 5 4 - 5 0 47-42 6 7 - 4 4 l e v e l 1 # tensiometers 20 27 25 20 # t e n s i o n l y s i m e t e r s 5 0 5 4 0 # s t e e l l y s i m e t e r s 0 3 0 0 1 # i r r i g a t i o n s 2 1 1 3 0 # i s o l a t i o n cov - e r placements 2 0 2 2 0 # i s o l a t i o n cov -er removals 1 0 2 2 0 F i g u r e 5 -5-1 Flow p a t t e r n i n the 8-9 snow l a y e r . On May 11, 1974, dyed water had been a p p l i e d to a s u r f a c e , l e v e l with the top of the dyed s e c t i o n shown. The 8-9 l a y e r was the same as t h a t i n v e s t i g a t e d i n experiment 1 one month l a t e r . The pole i s marked at 10 cm. i n t e r v a l s and s e v e r a l sheets are l a b e l l e d . 120 (not shown i n : f i g u r e 5-1-1)- It was intended, to study the flow paths induced by the l a y e r i n g i n the snow, by s t a i n i n g the p l o t , at the end of the experiment. However, r a p i d melt of the i n c r e a s i n g l y p r o t r u d i n g snowplot (see f i g u r e 5-1-4) exposed the i c e sheets to i n t e r n a l melt and so the dye s t a i n i n g was not done. However, the same l a y e r had been dyed s e v e r a l meters away, a month e a r l i e r (May 11). At that time the snow had been removed to a l e v e l which, by c o i n c i d e n c e , equaled the snowplot s u r f a c e l e v e l of June 12.- F o l l o w i n g a p p l i c a t i o n of food c o l o r i n g to the cut s u r f a c e a s e c t i o n had been cut open and the photo i n f i g u r e 5 . 5-1 had been taken. The flow p a t t e r n i n the 8 - 9 l a y e r appeared to be of roughly the sheet flow type on May 11. Furthermore i t would appear from t h i s photo that l a t e r a l flow would mainly take p l a c e near the snowplot s u r f a c e i n i c e sheets # 5 3 6 , and 7 - Assuming no change i n i t s b a s i c p h y s i c a l c h a r a c t e r i n the I n t e r v e n i n g month, the 'average' flow i n the surface-instrument l a y e r ( 3 - 9 l a y e r ) i s probably c l o s e to that measured at the l y s i m e t e r l e v e l d u r i n g steady s t a t e flow. The 8 - 9 l a y e r i s t e n t a t i v e l y c h a r a c t e r i z e d as a uniform l a y e r with a sheet flow p a t t e r n . The l y s i m e t e r l e v e l f o r experiment 3 was i n the lower t h i r d of the W - 40 l a y e r of f i g u r e . 5 • 1 - 2 . The s e c t i o n e x h i b i t e d c ross bedding of t h i n i c e sheets below an o l d weathered s u r f a c e (W). Although i c e glands i n t h i s l a y e r were prominent i n the experiment 2 snowpit, 2 . 5 meters away, they were not obvious i n the snowpit w a l l of experiment 3 - On J u l y 12, rhodamine WT dyed water was added to the snowplot s u r f a c e ( f i g u r e 5 - l - 5 ) > then the snowplot was cut open as shown i n f i g u r e 5 . 5-2. The flow 121 F i g u r e 5 - 5 - 2 Flow p a t t e r n i n the r e g u l a t e d snowplot of experiment 3 , 1 5 0 0 , J u l y 1 2 , 1 9 7 4 . Rhodamine WT dyed water had been app-l i e d to the snow s u r f a c e at 1 3 3 6 . The view i s l o o k i n g east across the snowplot. The f i v e l y s i m e t e r s were s t i l l i n the snow but the tensiometers had been removed p r i o r to expos-i n g the s e c t i o n . T h i s i s a f i n g e r flow p a t t e r n , except near the f a r l e f t l y s i m e t e r . A meter s t i c k i s on the r i g h t . 122 p a t t e r n i n the W - 40 l a y e r i s f i n g e r flow f o r the most p a r t with c o n c e n t r a t i o n s of sheet flow at the l e f t edge of the s e c t i o n . L a t e r a l d i v e r s i o n s take p l a c e i n the upper part of the exposed s e c t i o n . The l a y e r i n g i n f i g u r e 5 - 1 - 2 i s based on dye s t a i n photographs. The W - 40 l a y e r i s summarily c h a r a c t e r i z e d .as a l a y e r with many t h i n i c e sheets and a f i n g e r flow p a t t e r n . The b u b b l i n g pressure was measured to be (drying/wetting) : 7 cm/ 4 cm and 6 cm/4 cm inttwo d i f f e r e n t b l o c k s of snow taken from the tensiometer l e v e l of l a y e r W - 40. The l y s i m e t e r l e v e l f o r experiment 4 was In the 50 - 54 l a y e r of f i g u r e 5 - 1 - 3 - T h i s l a y e r had many i c e sheets which could not be completely avoided f o r install£ngn<g the instruments. Lysimeter 6 d i d not g i v e n any flow d u r i n g the s i x days i t was i n s t a l l e d (indeed water evaporated from the outflow u n i t ) even though the l y s i m e t e r remained,under t e n s i o n , and the t e n s i o m e t e r s , 10 cm above i t , d i s p l a y e d the expected changes i n c a p i l l a r y p r e s s u r e . Since f i g u r e 5 . 1 - 3 suggests t h a t the a c c i d e n t a l i n c l u s i o n of i c e sheets• 5 4 a, b, c i n s i d e the l y s i m e t e r r i m has somehow produced a water t i g h t s e a l , the data from.lysimeter 6 are d i s r e g a r d e d . Only a few i c e glands were seen i n the 50 - 54 l a y e r . On J u l y 3 0 , Rhodamine WT dyed water was added to the m e l t i n g snowplot s u r f a c e (the I s o l a t i o n cover had been removed at 1 8 0 0 the evening b e f o r e ) . The snowplot was then cut open and i t s photograph i s shown i n f i g u r e 5 - 5 - 3 - The flow p a t t e r n i n the 50 - 54 payer i s c h a r a c t e r i z e d as a l a y e r with many t h i n i c e sheets and a sheet flow p a t t e r n . The b u b b l i n g pressure was measured to be (drying/wetting) : 2 . 5 cm/1.9 cm, 4 . 0 cm/3 .0 cm, 123 F i g u r e 5 . 5 - 3 Flow p a t t e r n i n the r e g u l a t e d snowplot of experiment 4 , 1 5 0 0 , J u l y 3 0 , 1 9 7 4 . Dyed water had been a p p l i e d to the snow s u r f -ace beginning at 13^7 f o r 8 minutes. The view i s l o o k i n g east across the snowplot. The fo u r l y s i m e t e r s were s t i l l i n the snow. This i s a sheet flow p a t t e r n with some co n c e n t r a t -i o n of the flow. A meter s t i c k i s on the l e f t . 124 and 3 . 5 cm/2.1 cm i n three d i f f e r e n t b l o c k s of snow taken from the tensiometer l e v e l of l a y e r 5 0 - 5 4 . 5.6 NATURAL MELT WAVE EXPERIMENTS In a d d i t i o n to the r e g u l a t e d flow experiments, two e x p e r i -ments (experiments 2 and 5) were done to measure c a p i l l a r y pressure changes with n a t u r a l d i u r n a l melt c y c l e s . Some d e t a i l s are g i v e n In t a b l e 5-5=-l. Experiment 2 was e s t a b l i s h e d and observed from a three meter deep snowpit. Uphillffrom"! the snow-p i t , the n a t u r a l snow s u r f a c e had a slope of.12° (SSW) and 30 cm diameter suncups were, noteed (on June 24). Twenty-seven t e n s i o -meters were p l a c e d at seven l e v e l s under t h i s s u r f a c e over a depth range of two meters i n a s e r i e s of l a y e r s , i n which many i c e glands could be e a s i l y seen. The tensiometer l o c a t i o n s and l a y e r c h a r a c t e r i s t i c s are noted i n f i g u r e 1.5-1. The l e v e l of the second row of tensiometers from the bottom corresponds to the instrument l e v e l of experiment 3 . The snowpit was p r o t e c t e d by a l e a n - t o of p o l y s t y r e n e . A s e r i e s of s e c t i o n s were cut away to r e v e a l the s t a i n p a t t e r n which i n d i c a t e d t h t the flow p a t t e r n below i c e sheet 30 i s f i n g e r fcbow. Ice glands i n l a y e r 33-34 c o u l d be e a s i l y seen. The i n s t a l l a t i o n i s shown i n f i g . 5-6-1. Experiment 5 was the only 197 4 flow experiment not done at s i t e H but at s i t e M i n s t e a d . Nine tensiometers were p l a c e d i n three- rows under a l e v e l u n d i s t u r b e d snow su r f a c e at l e v e l s which correspond, on the b a s i s of the l a y e r i n g sequence and t h e i r height above the snowbase totfche l e v e l s given i n f i g u r e 1.5-1-125 F i g u r e 5 . 6 - 1 Experiment 2 i n s t a l l a t i o n f o r measuring the c a p i l l a r y p r essure v a r i a t i o n s from d i u r n a l snowmelt. The photo was taken on June 20, 1974 and shows the seven l e v e l s of tensiom-e t e r s . The i c e sheets sloped towards the snowpit w a l l at between 5° and 8° . The dark ' d i r t l e n s e s ' extended only a few cm i n s i d e the snowpit w a l l . A meter s t i c k Is i n the middle and a 61 cm i n c h - r u l e r i s at the lower part of the photo. 126 Ice l a y e r s dipped 8 along the snowpit w a l l and 6° towards i t . Large metal l y s i m e t e r s were i n s t a l l e d at s i t e M under u n d i s -turbed snow d u r i n g each of these experiments. In experiment 5 the metal l y s i m e t e r was r i g h t next to the tensiometer a r r a y , l e v e l with the middle row of tensiometers. Experiments 2 and 5 were c a r r i e d out, p a r t l y to take advantage of data provided by concurrent r a d i a t i o n experiments being done on the l e v e l snow su r f a c e at s i t e M to i n v e s t i g a t e elements of the s u r f a c e r a d i a t i o n budget. 126A SUMMARY Pour or f i v e t e n s i o n l y s i m e t e r s were p l a c e d 50 cm below the snowpack s u r f a c e to measure f l u x w i t h i n experimental snowplots at the Mt. Seymour s i t e . Two rows of tensiometers were i n s t a l l e d at 10 and 15 cm above the l y s i m e t e r s to measure the corresponding c a p i l l a r y p r e s s u r e . Snow cores were, per-i o d i c a l l y e x t r a c t e d from the p l o t s to determine the l i q u i d water content changes. The snowplots experienced s e v e r a l c y c l e s o f w e t t i n g and d r y i n g as a r e s u l t o f the r e g u l a t e d input of water at the s u r f a c e from i r r i g a t i o n o r m a n i p u l a t i o n of i s o l a t i o n covers. R e l a t i o n s h i p s amongst the v a r i a b l e s are determined from measurements made d u r i n g steady s t a t e • or slowly d e c r e a s i n g flow. The h y d r a u l i c c o n d u c t i v i t i e s are taken to equal the measured f l u x e s by v i r t u e o f the g r a v i t y drainage p o s t u l a t e . F o l l o w i n g the completion of flow t e s t s at each snowplot, the flow p a t t e r n was determined by dye t r a c i n g and the b u b b l i n g pressure was measured from the c a p i l l a r y f r i n g e height of snow blocks p l a c e d i n water. C a p i l l a r y p r e s s u r e s and t h e i r g r a d i e n t s under n a t u r a l flow c o n d i t i o n s were measured i n und i s t u r b e d p l o t s i n t o which s e v e r a l l e v e l s of tensiometers had been i n s t a l l e d . The r e s u l t s o f these measurements are giv e n i n the next chapter. CHAPTER SIX 127 PRESENTATION AND INTERPRETATION OP FIELD DATA Time and space v a r i a t i o n s of c a p i l l a r y pressure i n a deep snowpack' are d i s c u s s e d infct.he f i r s t f o u r s e c t i o n s . The e x p e r i -mental r e l a t i o n s of f l u x , d u r i n g steady or d e c r e a s i n g flows, to both c a p i l l a r y pressure and to the v a r i a b l e ^ are presented. The g r a v i t y drainage p o s t u l a t e i s shown to be v a l i d f o r the experiments. This means that i f Darcy's equation i s assumed to d e s c r i b e unsaturated flow i n snow, then the measured f l u x r e l a t i o n s can be i n t e r p r e t e d as h y d r a u l i c c o n d u c t i v i t y r e l a t i o n s : K(h ) and *o3'(K). Three d i f f e r e n t methods of determining the f r a c t i o n a l d e r i v a t i v e , ¥ ( K ) , d e f i n e d as - \ are d i s c u s s e d 5 5 K d6 i n the l a s t three s e c t i o n s of the chapter. 6 . 1 CAPILLARY PRESSURE IN REGULATED SNOWPLOTS In f i g u r e 6 . 1 - 1 i s shown pa r t of the f l u x - t i m e curve measured by l y s i m e t e r #4 d u r i n g experiment 1 . Increases i n f l u x due to both removal of the i s o l a t i o n cover and from i r r i g a t i o n can be seen. Also shown i s steady f l u x produced by mid-day melt, and the beginning of a drainage wave f o l l o w i n g t e r m i n a t i o n of i r r i g a t i o n and immediate replacement of the i s o l a t i o n cover. C a p i l l a r y pressure was measured by p l a c i n g tensiometers i n a row, grouped, and about 10 cm., above each l y s i m e t e r . In t h i s way changes i n pressure could be f o l l o w e d along with f l u x produced by the a r t i f i c i a l i n p u t , f l o w s at the s u r f a c e . The 128 .3 ./ .03 ^ .ol X .003 .001 F i g u r e 6.1-1 Flux at 48 cm, depth from flow r e g u l a t i o n at the snow s u r f a c e . experiment 1 t e n s i o n l y s i m e t e r 4 JuNf II 117+ UJ 3 v o > 5 ^  o ^ C; V m fr fr 10 IMITATION ' Uifi TEA 5 5 <><: IX 13 if 15 U 17 IS T/M£~ Chrs, PACIFIC STANDARD TIME) X .03 If .01 -z -9 -6 • 4-0, X - o x —•—• • -• o o- o— b • - 0 ' ° o V Ui x 3! "5: -to -IZ o ^ Q ^ ^ -v 5 O Mi .A" 1179-F i g u r e 6.1-2 C a p i l l a r y p r e s s u r e at 38cm-i n the experiment 1 r e g u l a t snowplot,. as measured by 3 iometers which were 10.2cm, #4 t e n s i o n l y s i m e t e r . depth ed t e n s - _ above -ti 13 14-TIM£ ( hrr. PST) 15 17 17 If vo 130 pressure-time curves, measured j u s t above one of the l y s i m e t e r s i n each r e g u l a t e d flow experiment, are shown i n f i g u r e s 6.1-2, 3 4, and 5. The pressure curves above the other l y s i m e t e r s are s i m i l a r to the ones shown here. By comparing f i g u r e 6.1-2 with the f l u x - t i m e curve of f i g u r e 6.1-1, the connection between c a p i l l a r y pressure and f l u x i s e a s i l y seen. The other f i g u r e s may a l s o be compared with the corresponding l y s i m e t e r curves shown i n s e c t i o n s 6.6 and 6.7. Only i n experiment 4 d i d two of the f o u r l y s i m e t e r s have f l u x curves which d i d not show changes s i m i l a r to those of the pressures measured above them, a f a c t that i s r e a d i l y e x p l a i n e d i n terms of l a t e r a l flow between the two types of instruments, caused by i c e sheets, many of which were noted i n the instrument l a y e r of that experiment. The tensiometers can measure changes on a much sm a l l e r s c a l e than the l y s i m e t e r s . In some cases the f l u x wetting f r o n t can be seen to be composed of s e v e r a l f i n g e r s of flow a r r i v i n g at d i f f e r e n t times. The tensiometer response time, d i s c u s s e d i n s e c t i o n 4.1, i s only s i g n i f i c a n t at the steeper wetting f r o n t s as, f o r example, f o r tensiometer 4b i n the f i r s t steep r i s e i n p r e s s u r e , shown i n f i g u r e 6.1-2. In t h i s example, the asymptoti approach of the pressure to a near steady v a l u e , may be due to tensiometer' response. I f one assumes that the time r a t e of change of pressure i s l a r g e r than the measured valu e , because of nonzero response time, then a lower l i m i t to t h i s q u a n t i t y i s given by the w e t t i n g f r o n t s shown i n the f i g u r e s . In f i g u r e 6.1-2, tensiometers 4b and 4d have w e l l d e f i n e d curves at the f i r s t w e t t i ng f r o n t produced by i s o l a t i o n cover removal at the -z • SKI , . 0 - j va => Q v N> ^ ^ va u, x Hi #• 5 v ° Q 4 J UNE 10 TIME ( h F i g u r e 6.1 - 3 D e c l i n i n g c a p i l l a r y p r essure f o l l -owing placement of the i s o l a t i o n cover at the snow s u r f a c e . The tensiometers were at 46cm depth and were 10 .2cm above #4 l y s i m e t e r d u r i n g experiment 1. If It 17 13 ,4-TlME (hrs. PST) -z • H~CK. o 4-c x U + re ^ ^ H§ ^ ~4 v: v\ va -J CJ o a: •vl tittW-jt-r-(-+• + — + - + - + — •« —+- + --z •50 - X - — x — >d—V—*• 77 6 o b o - o - o - o — d— o — o — » - / o •v V ^ 1 (-o - o -- X — - X — A -~T h--It-'ll JULV Z7 F i g u r e 6.1-5 C a p i l l a r y p r e s s u r e i n the exp-eriment 4 r e g u l a t e d snowplot as measured by 4 tensiometers at 48 cm; depth which were 6.9cm ab-ove #4 l y s i m e t e r . The c o r r e s -ponding f l u x i s shown i n f i g u r e 6.6-3. I2- 13 TIME- Chrf. PST) lit- is 17 It U) 134 snow s u r f a c e . There, the pressure change reaches 50 cm/hr. The wetting f r o n t speed can be estimated from the t r a n s l a t o r y wave speed, measured with l y s i m e t e r 4. The wave speed, U i s 20 cm/hr (see t a b l e 6.6-1). Now (1 v 4 ) : ^ h c I 3 h c 1 / (3'3 4 ) f i = - 7 i t = ~ lo x 5 0 = - 2 - 5 c m / c m . f o r the t r a n s l a t o r y wave. The c a p i l l a r y pressure g r a d i e n t at the wetting f r o n t i s l a r g e r than 2.5 cm pressure/cm, or 2.5 times the g r a v i t a t i o n a l p o t e n t i a l g r a d i e n t . To o b t a i n b e t t e r i n f o r m a t i o n on the wetting f r o n t f o r c e s , a tensiometer with f a s t e r response would be r e q u i r e d . Each group of pressure curves shows c o n s i d e r a b l e displacement among themselves, an e f f e c t that i s p a r t l y due to the zero e r r o r . In s e c t i o n 4.2 the e f f e c t of l a c k of p r e c i s i o n i n tensiometer placement was estimated to produce a zero e r r o r of l e s s than a = + 0.8 cm i n p r e s s u r e . However, a l a r g e part of the d i f f e -rences i n pressure among the curves must be due to r e a l l a t e r a l changes i n p r e s s u r e , s i n c e the d i f f e r e n c e s i n the curves vary with time, an e f f e c t which cannot be produced by the z e r o e e r r o r . The standard d e v i a t i o n of the pressure about the mean pressure of each l y s i m e t e r group, was c a l c u l a t e d from a l l the measurements (not j u s t those shown here) and the a h are l i s t e d f o r the c d i f f e r e n t snow l a y e r s i n t a b l e 6.4-1. The values l i s t e d are f o r spread i n pressure d u r i n g drainage,, but values of a h f o r steady c flow, are s i m i l a r to those l i s t e d . The l a r g e r a h f o r l a y e r c W - 40 (lower t h i r d ) may r e f l e c t the f i n g e r fdowppattern found 135 t h e r e . The pressure-time curves of f l u x waves that were produced by the a p p l i c a t i o n of dyed water to the s u r f a c e , are given i n f i g u r e s 6 . 1 - 6 and 7 f o r experiments 3 and 4 . The corresponding dye s t a i n p a t t e r n s were shown i n f i g u r e s 5 - 5 - 2 and 3 and have been c h a r a c t e r i z e d as f i n g e r flow and sheet flow r e s p e c t i v e l y . The tensiometer access hole l o c a t i o n s (at the cup end) are shown t h e r e . The d i f f e r e n c e s i n flow p a t t e r n are a l s o found i n the pressure curves by comparing both the range of a r r i v a l times f o r the wetting f r o n t , and the range of drainage wave shapes between the two p a t t e r n s . Three c o n c l u s i o n s are drawn from these data. The l a r g e v a r i a t i o n i n the pressure curves f o r the f i n g e r flow p a t t e r n , may be due to the d i f f e r e n c e between the r a p i d p e n e t r a t i o n and drainage i n the snowpack by i n d i v i d u a l t r i c k l e s , and t h e i r slower l a t e r a l d i s p e r s i o n i n t o the i n t e r - t r i c k l e spaces. Only one of the t w e n t y - f i v e tensiometers i n the f i n g e r p a t t e r n f a i l e d to respond to the dye i r r i g a t i o n (though i t responded to i s o -l a t i o n cover removal), so that the f l u x wave must have penetrated to almost a l l p a r t s of the instrumented l a y e r . Consequently, the dye s t a i n c o n t r a s t s seen i n f i g u r e 5 - 5 - 2 , show the l a r g e r l i q u i d water content s t o r e d i n i c e glands and i c e sheets. (A c o n c l u s i o n made use of i n i n t e r p r e t i n g the dye s t a i n p a t t e r n s of s e c t i o n 1 . 6 ) . D e t a i l e d comparison of each tensiometer's pressure curves and the dye p a t t e r n s at that location,; i s i n c o n c l u s i v e , p o s s i b l y because of the d i f f i c u l t y of l o c a l i z i n g the cut s e c t i o n to b e t t e r than w i t h i n a few centimeters of the v e r t i c a l plane T 5 tensiometers above #4 l y s i m e t e r . 4c i s at 44cm, depth, the others are 4 . 2 c m low-er . T 5 tensiometers above #3 l y s i m e t e r . 3b i s at 47cm depth, the others are 6 . 2 c m , low-er . TIME (hn. P r r ) TIME prrj F i g u r e 6 . 1 - 6 C a p i l l a r y pressure changes i n the W',-40 l a y e r f o l l o w i n g the a p p l i c a t i o n o f dyed wat-er to the snow surface between 1336 and 1 3 4 1 on J u l y 1 2 , 1974 (experiment 3 ) . A photo of the r e s u l t i n g dye s t a i n p a t t e r n i s f i g u r e 5 . 5 - 2 . -I v l -a T 5 tensiometers above #4 l y s i m e t e r . 4b i s at 3 9 c m s depth, the others are 4 . 3 c m . low-er . 5 tensiometers above #3 l y s i m e t e r . 3b i s at 3 8 c m v depth, the others are 4 . 5 c m low-er . TIME (hrs. PST) 13 TIME Chrs. PST) F i g u r e 6 . 1 - 7 C a p i l l a r y pressure changes i n the 5 0 - 5 4 l a y e r f o l l o w i n g the a p p l i c a t i o n o f dyed water to the snow surface between 1 3 4 7 and 1355 on J u l y 3 0 , 1 9 7 4 (experiment 4 ) . A photo of the r e s u l t i n g dye s t a i n p a t t e r n i s f i g u r e 5 - 5 - 3 . 138 through the tensiometer cups. 6.2 CAPILLARY PRESSURE DURING NATURAL MELT The r e s u l t s shown so f a r are f o r i n d i v i d u a l l a y e r s i n r e g u l a t e d snowplots. The measurement of the c a p i l l a r y pressure p a t t e r n i n time and depth f o r a snowpack undergoing n a t u r a l melt, i s necessary f o r p u t t i n g the former r e s u l t s i n t h e i r proper p e r s p e c t i v e . T h i s was done i n experiments 2 and 5. F i g u r e 6.2-1 shows the net r a d i a t i o n and f l u x time curves of June 19 - 21, measured at s i t e M. The l a t t e r g i v e s only an index of the flow elsewhere, s i n c e the p o s s i b i l i t y of l a t e r a l flow above the metal l y s i m e t e r cannot be excluded. The p r e s -sures measured by sets of tensiometers p l a c e d at seven l e v e l s , are shown i n f i g u r e 6.2-2. T h e - q u a l i t a t i v e form of the d i u r n a l melt waves can be e a s i l y seen i n the pressure curves, i n c l u d i n g the reduced i n p u t s d u r i n g cloudy p e r i o d s . T h i s i s because the f l u x c l o s e l y approximates the h y d r a u l i c c o n d u c t i v i t y d u r i n g steady flow or drainage. The h i g h value of the exponent n i n the K(h c) r e l a t i o n r e s u l t s i n the l a r g e mid-day drop i n f l u x of June 20 being more subdued i n the pressure curves. Not only do tensiometers show d e t a i l s of flow regime on a s m a l l e r s p a t i a l s c a l e than l y s i m e t e r s , but the measurement i s instantaneous, not depending on a mean i n t e r v a l v a l u e , as does f l u x . 139 I* Ir ^ i 5 / •> K Ii i -' i Id i II K A '•1 v\ IV •—••' i 12. I« JUNE If i IZ JUNE ZO i i2- n JUNE Zl Net r a d i a t i o n towards the l e v e l snow s u r f a c e (courtesy of M. Church and B. Sagar) .6 .5" ^ .H- f - /I % n %; J \ \ / 1 \ JepH J 39 Cm clefHi 30 c~. (, 12. It JUNE II (. iz /? JUNE ZO i 12. If J ONE Zl F l u x i n the snowpack measured by a s t e e l l y s -imeter under a snow s u r f a c e s l o p i n g at 12°W. F i g u r e 6 . 2 - 1 Net r a d i a t i o n and l y s i m e t e r flow measured at the weather s t a t i o n on June 1 9 - 2 1 , 197^ 140 F i g u r e 6.2-2(a) D i u r n a l c a p i l l a r y p r e ssure v a r i a t i o n s at seven l e v e l s w i t h i n the snowpack from melt at the s u r f a c e . The tensiometers shown i n the photo i n f i g u r e 5-6-1 are r e p r e s e n t e d from l e f t to r i g h t , by the symbols: • ° * and (where a p p l i c a b l e ) +,. r e s p e c t i v e l y . The experiment 2 snowplot had a s u r f a c e slope of 12°(SSW). 141 JUNE 11 JUNE ZO JUNE Z l F i g u r e 6.2-2(b) D i u r n a l c a p i l l a r y p r e s s u r e v a r i a t i o n s (continued) The average c a p i l l a r y pressure, f o r each row of tensi o m e t e r s , was calculated, to determine the maximum and minimum mean l e v e l p r essures d u r i n g the d i u r n a l c y c l e . These are p l o t t e d In f i g u r e 6 . 2 - 3 ' f o r June 19 - 20 and f o r J u l y 2 9 . ' The average change i n c a p i l l a r y pressure f o r the whole s e c t i o n over a d i u r n a l melt c y c l e , was 3 . 8 cm ( i . e . - 6 . 7 cm to -1.0.05 cm, June 19 - 2 0 , 1 9 7 5 ) . The average a b s o l u t e pressure d i f f e r e n c e between cons e c u t i v e l e v e l s was 1 . 1 cm d u r i n g peak flow and 1 . 8 cm d u r i n g low flow. TSe whole temporal v a r i a t i o n i s t h e r e f o r e only about three times the average s p a t i a l c o n t r a s t i n c a p i l l a r y p r e s s u r e . Consider the t r a n s i t i o n h e i g h t , z^, g i v e n by equation 3 - 1 - 8 f o r a [h^ - hg.| equal to the 1.1 cm pressure c o n t r a s t between l e v e l s d u r i n g peak flow; and z^, d e f i n e d f o r I n e ~ h n I = u - 5 cm. Using equation 3 . 1 - 7 to c a l c u l a t e v, z n i s 0 . 8 1 cm above wetting i n t e r f a c e s and 0 . 2 1 cm above d r y i n g I n t e r f a c e s . I f the average l a y e r t h i c k n e s s , D T, i s taken to be the average i c e sheet s e p a r a t i o n , 6 cm (see s e c t i o n 1 . 5 ) then ( 6 . 2 - 1 ) z /DT < 0 . 8 1 / 6 = 0.14 . n L ~ • Hence i t would.seem that the snowpack has the type A l a y e r i n g of f i g u r e 3 . 2 - 1 when wetted from a dry s t a t e . C a p i l l a r y pressure g r a d i e n t s cannot be d i r e c t l y measured i f the tensiometers are more than a l a y e r a p a r t . Only the space-average pressure g r a d i e n t can be determined when i n d i v i d u a l l a y e r s cannot be l o c a t e d . R e c o g n i t i o n of the occurrence of r e l a t i v e l y l a r g e s p a t i a l c o n t r a s t s i n c a p i l l a r y p r essure between l a y e r s and of the small t r a n s i t i o n h e i g h t s above the l a y e r i n t e r f a c e s are important 143 F i g u r e 6.2-3 S'OO Extremes o f c a p i l l a r y p r e ssure i n the snowpack over the d i u r n a l melt c y c l e . 4$o *r00 3S0\ V) ^ 300 - J vt u ^ zso\ zoo /sol level 5 level (, level leva le>tl T SN I I I 3 level 3 0(J SUR.F level 7 level l level Z \f)CE JUN i s SNO ul JULY Zt. N 1 £*P. 3 LEVEL J SuA FACE If 7V-J EXP. q-LEVEL -lZ -1° '7 -6 -H-C PlPILL/)KY PlZESfuKE (c^n,H^o) '2. 0 144 f o r the i n t e r p r e t a t i o n of c a p i l l a r y p r essure p a t t e r n s i n snow-packs . 6.3 TEST OF GRAVITY DRAINAGE POSTULATE The space-averaged c a p i l l a r y p r essure g r a d i e n t i n the v e r t i c a l d i r e c t i o n was determined f o r the i n s t a l l a t i o n s i n the r e g u l a t e d p l o t experiments. The g r a d i e n t i s the d i f f e r e n c e i n pressure between the two rows of tensiometers i n an i n s t a l l a -t i o n , d i v i d e d by the mean s e p a r a t i o n of the rows. The g r a d i e n t s are g i v e n i n t a b l e 6 . 3 - l j and are v e r t i c a l space averages over any l a y e r i n g e f f e c t s . The values are from s e l e c t e d i n t e r v a l s d u r i n g which the f l u x was steady or d e c r e a s i n g . (A p e r i o d of steady f l u x i s when a l l tensiometer pressure changes are <_ 0.1 cm.) The standard e r r o r s are determined from the spread i n gra d i e n t values of the l y s i m e t e r groups i n each experiment. Even with the l a r g e r e l a t i v e e r r o r , i t appears t h a t the c a p i l -l a r y pressure g r a d i e n t i s much s m a l l e r than the g r a v i t a t i o n a l p o t e n t i a l g r a d i e n t of u n i t y . These f i g u r e s i n d i c a t e that the g r a v i t y drainage p o s t u l a t e i s c o r r e c t f o r each r e g u l a t e d snow-p l o t experiment as a whole. Thus the h y d r a u l i c c o n d u c t i v i t i e s determined by equating them t o the f l u x e s would on the whole have f r a c t i o n a l e r r o r s equal to the c a p i l l a r y p r essure g r a d i e n t s g i v e n i n the t a b l e . Thehemean c a p i l l a r y pressure g r a d i e n t d u r i n g d e c r e a s i n g flow are g i v e n i n t a b l e 6.3-2 f o r both the r e g u l a t e d snowplot and n a t u r a l melt wave experiments. The standard e r r o r s here r e f e r to the spread i n g r a d i e n t s between the r e g u l a t e d snowplot experiments (not between l y s i m e t e r s ) or between tensiometer l e v e l s i n the d i u r n a l snowmelt p l o t s . The standard e r r o r 1 4 5 Table 6.3-1 Space-average c a p i l l a r y p r essure g r a d i e n t s i n the r e g u l a t e d snowplots Experiment Steady s t a t e ' flow Decreasing flow One Three Pour -0.027*•044 + 0.08 i .1 +0.11 ±.14 -0.009*.11 +0.04 ±.12 +0.10 ±.12 Table 6.3-2 C a p i l l a r y pressure g r a d i e n t i n snow d u r i n g drainage _ Experiment Drainage g r a d i e n t experimental mean l e s s steady flow g r a d i e n t Regulated snowplots D i u r n a l melt snowplots + 0. 04±. 03 -0.01+.02 -0:05 ±.04 -0.002±.02 146 r e f l e c t s both the pressure measurement e r r o r and the v a r i a t i o n i n space average pressure g r a d i e n t produced by l a y e r i n g . To help reduce the e f f e c t of l a y e r i n g , the d i f f e r e n c e between the d e c r e a s i n g flow g r a d i e n t and steady flow g r a d i e n t have a l s o been c a l c u l a t e d . The r e s u l t s , suggest that f o r d e c r e a s i n g flow i n snow, with or without l a y e r i n g e f f e c t s , the c a p i l l a r y pressure g r a d i e n t i s much smal l e r thantthe g r a v i t a t i o n a l p o t e n t i a l , g ra-d i e n t over the range of pre s s u r e s s t u d i e d (h - -5 to -12 cm). 6.4 RELATION BETWEEN HYDRAULIC CONDUCTIVITY AND THE CAPILLARY PRESSURE OF WET SNOW The r e l a t i o n between f l u x and c a p i l l a r y pressure d u r i n g steady or d e c r e a s i n g flow i n a snowpack. was measured from the r e g u l a t e d snowplot experiments. These values of f l u x can be equated to h y d r a u l i c c o n d u c t i v i t i e s ( I . e . V = K) by v i r t u e of the g r a v i t y drainage p o s t u l a t e , i f the Darcy equation d e s c r i b e s unsaturated flow i n snow. The i n t e r p r e t a t i o n of the f l u x r e l a -t i o n i n t h i s s e c t i o n as a h y d r a u l i c c o n d u c t i v i t y r e l a t i o n must be .tentative u n t i l the Darcy equation can be proven i n t h i s medium. The mean p r e s s u r e , h , above each l y s i m e t e r , Is c a l c u l a t e d and i t s standard e r r o r ar- , determined from o\ , and the number h ' h 5 c c of tensiometers per l y s i m e t e r . A l l these q u a n t i t i e s are l i s t e d i n t a b l e 6.4-1. The expected h y s t e r e s i s i n the r e l a t i o n between K and h , r e q u i r e s that measurements be made d u r i n g both d r y i n g and wetting of the snow. The wetting p o i n t s are d i f f i c u l t to o b t a i n s i n c e w e t t i n g i s a s s o c i a t e d with a steep f r o n t , and a 147 Ta b l e 6.4-1 Parameters f o r the C a p i l l a r y P r e s s u r e - H y d r a u l i c C o n d u c t i v i t y R e l a t i o n Experiment One Three Pour L a y e r 8-9 W-40 lower 50-54 # l y s i m e t e r s per l a y e r 5 5 2* # t e n s i o m e t e r s p e r l a y e r 3 4 4 # w e t t i n g p o i n t s 14 0 6 # minimum f l o w p o i n t s 4 1 2 # d r a i n a g e p o i n t s 84 8 26 Prom f crh drainage< cr^ ±0.7cm ±0.4cm ±1.2cm ±0.6cm ±0.9cm ±0-..5cm BWC s l o p e , rj 1-15.0 +10.9 BWC i n t e r c e p t , -5•4cm -5•8cm BDC i n t e r c e p t , -7•6cm —.— -7.3cm htdAow.from i n t e r c e p t s 1.4 1.3 kj/Lw d i r e c t measurement 1.6 1.4 PDC s l o p e + 7.7 + 6.8 PDC i n t e r c e p t a t K'/ on/fir -6.4cm -6.1cm h f c W . d i r e c t measurement -4 cm ,i -2.3cm o t h e r two l y s i m e t e r s were not i n c l u d e d because o f i n t e r f e r e n c e by i c e sh e e t s .01 -I I 10 100 HYPRflVLIC COAJP OCT/V/ry (^tr>/Lr) PORTION OF EXPERIMENTAL CYCLE H y d r a u l i c c o n d u c t i v i t y - c a p i l l a r y pressure r e l a t i o n f o r the 8-9 snow l a y e r . The h y d r a u l i c c o n d u c t i v e V i t y s are equated to f l u x e s , measured during steady or d e c r e a s i n g flows with the t e n s i o n l y s i m e t e r s of experiment 1. The p a i r of l i n e s are boundary dry-i n g and wetting curves f o r the snow l a y e r . 1174- UIETTINC DRYING MINIHUM JUrtE IO 0 • JUNE II m JUNE II 4-149 #4 USIMCTErl #3 L YSiMETElZ #1 £. YS~I M£ T£~/L oo- t»(s.vo u> ?}. ts> O0--OOK.vo lr> <J~ «y o ( r - r»tv v. U\ <J~ w i I 1 I i i i | i TI i | | | i i' l T > l l i i t i | , CAPILLARY PRESSURE CAPILLAR Y PR £ SSUtE (<~>) C/lP/LLflty PRESSURE (<*») 150 s c a t t e r i n a r r i v a l times of d i f f e r e n t p a r t s of the wetting f r o n t . T h i s problem is. overcome by r a i s i n g the f l u x i n stages, and making measurements only d u r i n g steady flow. The r e l a t i o n between h y d r a u l i c c o n d u c t i v i t y and pressure i s shown i n f i g u r e 6.4-1 f o r l a y e r 8 - 9 from measurements made with f i v e l y s i m e t e r s . In f i g u r e 6.4-2 i s the r e l a t i o n f o r l a y e r 50 -54 from two l y s i m e t e r s . In a g i v e n l a y e r , the curves measured by d i f f e r e n t l y s i m e t e r s appear to be s h i f t e d i n p r e s s u r e . This s h i f t i s e a s i l y seen r e l a t i v e to the boundary curves, f i x e d f o r each l a y e r , shown i n the f i g u r e s . T h i s e f f e c t can be e x p l a i n e d by sampling e r r o r i n measuring the mean pressure above each l y s i -meter. However, the p o s s i b i l i t y of r e a l l a t e r a l v a r i a t i o n s i n snow p r o p e r t i e s w i t h i n a l a y e r cannot be excluded. I f the curve f o r each l y s i m e t e r i s c a r e f u l l y examined, i t w i l l be seen that repeated c y c l e s of wetting and d r y i n g f o l l o w the same loop. T h i s i s because the wettest and dryest p o i n t s reached are repeated w i t h i n each c y c l e . The h y s t e r e s i s loops f o r a l a y e r would be c o n f i n e d between two boundary curves i n the absence of the pressure sampling e r r o r . The boundary w e t t i n g curve (B.W.C.) i s taken to f o l l o w the power law of equation 2.5-2,. and to be the f u n c t i o n a l r e l a t i o n s h i p * I n d i c a t e d by the wetting p o i n t s ( i - . e. c i r c l e d p o i n t s ) shown f o r a given l a y e r . The value of the exponent n i s g i v e n i n t a b l e 6.4-1, and may be compared with the nominal v a l u e , n = 14, i n t r o d u c e d i n s e c t i o n * The r i g h t b i s e c t o r of the p a i r of r e g r e s s i o n l i n e s p r o v i d e d by l e a s t squares f i t s . ' ' M l 1 1 I I I I III 1 1 I I I I I II 1 I I I I I I II I I I I I I I I I i / 10 100 HYDRAULIC CONPUC I IVITY (C*r,/l,r) F i g u r e 6.4-2 PORTION OF £xfftiiMENTftL CYCLE H y d r a u l i c c o n d u c t i v i t y - c a p i l l a r y pressure r e l a t i o n f o r the 5 0 - 5 4 snow l a y e r . The h y d r a u l i c conduct-i v i t y s are equated to f l u x e s , measured d u r i n g steady or d e c r e a s i n g flows with the t e n s i o n l y s i m e t e r s o f experiment 4 . The p a i r o f l i n e s are boundary dry-i n g and wetting curves f o r the snow l a y e r . I17T OJETTINC PH y/NG MINIMUM JVLY ZL • JULY Z7 e + vJ ULY Z ? ESI JULY zf ® X 152 2.5. Less data i s a v a i l a b l e to estimate the boundary d r y i n g curve (B.D.C.), so that i t i s c a l c u l a t e d by assuming i t has the same exponent n as the B.W.C., and i s l o c a t e d by making a l e a s t squares f i t of i t s p o s i t i o n to the minimum flow p o i n t s . Minimum flow p o i n t s are f o r snow that has been d r a i n i n g under an i s o l a t i o n cover o v e r n i g h t . The p o s s i b i l i t y of a s m a l l amount of water l e a k i n g l a t e r a l l y i n t o the p l o t of experiment 1, cannot be excluded, and because of the time l a g between the tensiometer and l y s i m e t e r l e v e l , the e x t r a flow may not have had time to reach the l a t t e r . In experiments 3 and 4, fehe minimum-flow f l u x r e c o r d was long enough to exclude t h i s p o s s i b i l i t y , but i n experiment 1, the B.D.C. d e f i n e d by these p o i n t s , i s r e a l l y a lower pressure l i m i t to i t s a c t u a l p o s i t i o n on the graph. The r a t i o of p r e s s u r e s on the two boundary curves i s shown i n the t a b l e to be about 1.4. A value 1.4 was d i r e c t l y measured i n the f i e l d f o r the r a t i o of d r y i n g to w e t t i n g b u b b l i n g p r e s s u r e s , f o r l a y e r 50 -54. Hence the r a t i o of pressures of the boundary curves shows that t h i s r a t i o i s about constant r i g h t up to s a t u r a t i o n . There i s no d e t e r m i n a t i o n of the boundary curves f o r l a y e r W - 40 (lower t h i r d ) , because of the poor weather d u r i n g experiment 4. The data a v a i l a b l e i s shown i n f i g u r e 6.4-3. Most of the p o i n t s shown i n the f i g u r e s are taken from the drainage-time curves, where the number of p o i n t s i s l i m i t e d by the number of independent f l u x measurements. As noted i n s e c t i o n 5.3 there i s a l a g i n drainage between the two instrument l e v e l s , and the t r u e f l u x c ould be 24% l e s s than the p l o t t e d p o i n t s . o O x> X -10 -f -g -7 H - 5 -9 -3 r-1. -?> c *> -1 I -7 -i -5 -? I a .01 HYDRAULIC CONDUCTIVITY ( W A r J F i g u r e 6.4-3 H y d r a u l i c c o n d u c t i v i t y measurements f o r the W-40 (lower t h i r d ) snow l a y e r . The h y d r a u l i c conduct-i v i t y s are equated to f l u x e s , measured duri n g s t e -ady or d e c r e a s i n g flows with the t e n s i o n l y s i m e t e r s of experiment 3. 10 100 P0/?T/0Aj Or £Xr°£/i I MENTflL C YCL E The c o r r e c t e d drainage p o i n t s would r e s u l t i n an e q u i v a l e n t pressure drop of only 3% i n the drainage curve. The drainage - p o i n t s should f o l l o w a primary d r y i n g curve, P.D.C, which i s d e f i n e d to be a d r y i n g curve which o r i g i n a t e s on a B.W.C. They should a s y m p t o t i c a l l y approach the B.D.C. In f a c t , the d r y i n g curves appear to be s t r a i g h t l i n e s a long much of t h e i r l e n g t h . The r e g r e s s i o n l i n e f o r the drainage p o i n t s i n l a y e r 8 - 9, and the one f o r l a y e r 50 - 5 4 , were c a l c u l a t e d . The parameters of the P.D.C. are l i s t e d i n the t a b l e . Primary d r y i n g curves that begin from lower peak flows than those produced i n the experiments, may f o l l o w a p a r a l l e l l i n e of s i m i l a r slope f o r the g r e a t e r part of the drainage. The h y s t e r e s i s i n the r e l a t i o n i s seen i n many of the dye water i r r i g a t i o n p r e ssure curves, shown i n s e c t i o n 6 . 1 . Although the f l u x , some time a f t e r the water a p p l i c a t i o n , should be about the same as b e f o r e , the c a p i l l a r y pressure was o f t e n noted to have a much lower value afterwards. Returning again to f i g u r e -6 . 2 - 3 , i t • i s apparent that there are l a r g e v a r i a t i o n s i n pressure between d i f f e r e n t l e v e l s i n the snowpack. That the d i f f e r e n c e s are mainly due to l a y e r d i f f e -rences i n the K(h c) f u n c t i o n , r a t h e r than to f l u x divergence, i s apparent, by comparing the pressures with the K(h c) curves. In f a c t , the pressures appear'to be q u i t e reasonable f o r only a moderate amount of l a t e r a l flow, i f corresponding l a y e r s are compared. L e v e l 6 of June 19 - 20 and L e v e l 3 of J u l y 29 i n f i g u r e 6 . 2 - 3 ) ' are q u i t e near the snow l a y e r s i n v e s t i g a t e d i n the r e g u l a t e d p l o t experiments 3 and 4 r e s p e c t i v e l y . The c a p i l l a r y 155 Table 6 . 4 - 2 Flux determination by tensiometer L i q u i d volume f l u x (cm/hr) From tensiometer measurements From net r a d i a t i o n or kinematic wave theory^ Maximum f l u x June 1 9 , l e v e l 6 - 0 . 6 0. 40 Maximum f l u x J u l y 2 9 , l e v e l 3 1 . 6 0.48 Minimum f l u x June 20,level' 6 depth i s 186 cm ~0,.l 0. 090 Minimum f l u x J u l y 29,level,;.. 3 depth i s 113 cm 0 . 1 1 0. 043 The experimental r e l a t i o n s of f i g u r e s 6.4 - 2 and 6.4-3 are used. The maximum f l u x i s assumed to be equal to measured net r a d i a t i o n equivalent melt. The minimum f l u x i s c a l c u l a t e d from the kine matic wave speed: 2_ _ ^ \y v3 where z i s the tensiometer depth and t i s the drainage i n t e r v a l of 14 hours (equation par-ameters are from Colbeck and Davidson ( 1 9 7 2 b ) ) . 156 pressures at these l e v e l s may be connected to values of K and hence f l u x , u s i n g f i g u r e s 6.4 - 1 and 2 . These 'pressure-deter-mined' f l u x e s are l i s t e d i n t a b l e 6 . 4 - 2 . The maximum f l u x was not measured d i r e c t l y i n the n a t u r a l flow experiments, but a lower l i m i t to the f l u x can be estimated, by t a k i n g i t to be the peak net r a d i a t i o n expressed as equiva-l e n t melt. S i m i l a r l y , the minimum f l u x i s taken to be that of the kinematic wave which takes 14 hours — about the time bet-ween sunset and the a r r i v a l of the next day's wetting f r o n t — to reach the tensiometer l e v e l , z. The value of OJ r e q u i r e d i s give n by the equation d e s c r i b i n g f i g u r e 5 i n Colbeck and Davidson ( 1 9 7 2 b ) . The r e a l minimum f l u x can be l a r g e r than the f i g u r e , given by t h i s c a l c u l a t i o n , i f there i s any overnight m e l t . t a k i n g p l a c e . Comparison of the two types of data i n t a b l e 6 . 4 - 2 shows that much of the pressure v a r i a t i o n with depth can be ex p l a i n e d i n terms of l a y e r d i f f e r e n c e s , with a sma l l e r p a r t of the v a r i a t i o n a pparently due to l a t e r a l flow. In order to transform measured c a p i l l a r y p ressures i n t o f l u x , i t i s thus necessary to know i n d e t a i l the h y d r a u l i c p r o p e r t i e s of each l a y e r . Otherwise, the high value of the exponent n could r e s u l t i n l a r g e e r r o r s i n the estimated f l u x . C a p i l l a r y p r essure measurements should be used only as q u a l i t a -t i v e i n d i c e s of f l u x (and hence K) v a r i a t i o n s , unless the instrumented snow l a y e r i s c a l i b r a t e d by e i t h e r d i r e c t f l u x measurements, or from a b e t t e r understanding of snow p r o p e r t i e s and t h e i r change with time. 157 6 . 5 FRACTIONAL DERIVATIVE — LIQUID MEASUREMENT The r e l a t i o n between the f l u x and changes i n the l i q u i d water content (er the q u a n t i t y ^ d u r i n g steady or d e c r e a s i n g flows was measured ftoom the snowplot experiments. I f the Darcy equation 'describes unsaturated flow i n snow, then the g r a v i t y drainage p o s t u l a t e means that the r e l a t i o n of ^ ^ to V could be i n t e r p r e t e d as the coT'(K) r e l a t i o n , where the f r a c t i o n a l d e r i v a t i v e to Ls T h i s i d e n t i f i c a t i o n i s t e n t a t i v e u n t i l the Darcy K dQ equation i s proven i n snow, as was the i d e n t i f i c a t i o n of the r e l a t i o n between V and h as a K(h ) r e l a t i o n i n the l a s t s e c t i o n , c c R e l a t i v e changes i n l i q u i d water content were measured along with f l u x i n the r e g u l a t e d flow p l o t s . The values of co obtained are f o r the instrumented l a y e r o n l y , and are independent of any l a t e r a l flow that may take p l a c e at h i g h e r l e v e l s (except where the sampling of l a y e r p r o p e r t i e s may be i n f l u e n c e d by the flow p a t t e r n ) . Changes i n l i q u i d water content are taken to equal the changes i n average l a y e r d e n s i t y as measured from sets of snow co r e s . The average co = co over the range 8 to 0' i s / n i - < : \ ' ~ 1 rQ 1 J Q ln(K'/K() ( 2 . 5 - 6 ) <*> = Q i _ Q fa code = Q ; _ fl The value of co, given i n t a b l e 6 . 5 - 1 f o r the three experiments i s c a l c u l a t e d from ( 6 . 5 - D in(vyv) «y - ©s 158 Table 6 . 5 - 1 Snow l a y e r d e n s i t y measurements Experiment One Three Four Layer 8-9 ] W-40 .ower t h i r c 50-54 1 # of f l u x measurements per sample 5 5 2 # of d e n s i t y measurements per sample 3 6 J u l y 2 7 : 3 J u l y ' 3 0 : 2 High Plow Date i Time Plow c h a r a c t e r June 10 15^5-1635 d r y i n g J u l y 6 1500-1600 d r y i n g J u l y 27 I 6 l 0 - l 6 l 5 d r y i n g ' Plux (em/hr) Density (gm/ec) . V, . 0 . 1 6 6 ± 0 .016 0 . 3 2 ±0.13 1.36 : ±0. 44 0. 540 ± 0 .018 0 .613 ± 0 . 0 0 7 3 0 . 5 9 ±-0.0087 Low Plow Date Time Flow c h a r a c t e r Plux J vi . (cm/hr) \ c r ^ D e n s i t y J (gm/cc) 1 CJ>^ June 11 1130-1245 steady J u l y 5 1110-1230 steady J u l y 30 1140-1205 steady 0.0142 . ±0.0028 0.034 ±0.023 0.16 ± 0 . 1 3 0 . 5 2 5 ± 0 . 012-0.597 ±-0.0065 O .56 ±0.0005 . f K=VWi (cm/hr) K f (em/hr) 0.048 0 . 0 5 2 0 . 1 0 0.11 0. 47 0 . 4 9 ~ ... ln(V,/Vz) 164 140 71 Note: The d i f f e r e n c e i n CJ between the l a y e r s i s n o t - s i g n i f i c a n t because of the l a r g e s c a t t e r i n t h e ' f l u x and d e n s i t y measurements. 1 5 9 Each measurement of to p e r t a i n s to the i n d i v i d u a l l a y e r s from which the snow cores were taken. Two d i f f e r e n t means of K, over the range K to K', are g i v e n i n the t a b l e : the geometric mean, — + K; and the corresponding K = K\ f o r which to = co (see equation 2 . 5 - 7 ) . Since the geometric mean i s more o b j e c t i v e ( i . e . not needing an estimate of e ) , and yet. almost the same as the more t t h e o r e t i c a l l y c o r r e c t K , i t - i s p r e f e r a b l e touuse the former. In view of the very l a r g e r e l a t i v e e r r o r s i n both the changes i n l i q u i d water content and i n the f l u x , d i f f e r e n c e s between the values of to are not s i g n i f i c a n t . Since the number of cores that could be withdrawn from the tensiometer l a y e r was l i m i t e d , a n o n d e s t r u c t i v e method of l i q u i d water content d e t e r m i n a t i o n would have been necessary f o r the v a r i a t i o n of - co with K to be determined f o r i n d i v i d u a l l a y e r s i n the snowpack. The geometric mean lo f o r the l i q u i d measurement method was to- = 1 2 0 and the corresponding geometric mean K was K = 0 . 1 3 cm/hr. I f the c o r r e c t i o n given i n s e c t i o n 5 . 3 f o r the d i f f e r e n c e i n f l u x at the two instrument l e v e l s (during drainage) had been used the corresponding values would have been 104 and 0 . 1 1 cm/hr res p e c -t i v e l y . 6.6 FRACTIONAL DERIVATIVE --TRANSLATORY WAVES The f r a c t i o n a l d e r i v a t i v e , to, can be independently measured from wetting f r o n t s i f these f r o n t s are assumed to move as t r a n s l a t o r y waves of constant s i z e and shape through the 160 X .001 It- IS T/ME ( A T . PST) 161 0.1 X X O f f I &X XX XXKXXX 0.01 0.001 juLy 4 X so o y +4++ 4 4 + + + 4 + + + 4-4 4 4 + 4 4 + 4 4 4 4 + 4 + .444 t ++++$ *X X X XxxxJx o • o o o *> Sxxxx X*X X ^ *<iX4s*> O e 0 o o o MX-tof 4 + 4 4 4 xxxxxvvx F i g u r e 6.6-2 Wetting f r o n t produced f o r experiment 3. PEPTH X X X #% L YS 1 METER *L LyS/METER ,s* +++• *<P t-YSlMCT E R. , to • • • *3 LVSIMETEK , 67 0 o o *Z LystMEreA. . io il 12 13 TIME t^rs-.PSTJ l*f IS I 7 162 ./ .ol .001 J ocy z 7| /7 7 ^ so o >c'i x x « J( x x y in < x X x IS <5Q > <3 •5: o H X X X X X X X X * X V X v X X X X X X * X X X X x x x x u X X * x x x x X & X x x x l < X X X X X X X x J O O d F i g u r e 6.6-3 Two w e t t i n g f r o n t s produced f o r exper-iment 4. Depth.55 cm * 3 10 X TIME ( hrs, PST) 17 I? 163 snow. Va r i o u s wetting, f r o n t s measured i n the r e g u l a t e d p l o t experiments are shown i n f i g u r e s 6.6-1, 2 and 3. A one-dimen-s i o n a l model can be used to analyze the data only i f the l y s i -meter i s part of a r e l a t i v e l y uniform l a y e r t h a t extends f o r much of the height between the instruments and the snow surfac e . I f l a t e r a l flow along i c e sheets takes p l a c e , i t must occur r i g h t near, the snow s u r f a c e i n order that the l y s i m e t e r measure corresponding values of f l u x and t r a v e l time. The snow l a y e r above the l y s i m e t e r s i n experiment one, s a t i s f i e s these c o n d i -t i o n s , so that i t s data are used to o b t a i n i n f o r m a t i o n on the (jo(K) r e l a t i o n . In the other r e g u l a t e d snowplot experiments, the occurence of i c e sheets and of l a t e r a l flow above the l y s i -meter l e v e l r e s t r i c t s t h e i r value to the l i q u i d measurement method d i s c u s s e d i n the previous s e d t i o n . F o l l o w i n g removal of the i s o l a t i o n cover, the wetting f r o n t w i l l i n i t i a l l y move f a s t e r than expected u n t i l a dynami-c a l l y s t a b l e f r o n t i s developed, as was noted i n s e c t i o n 3-3-T h i s w i l l r e s u l t In an overestimate of co e s p e c i a l l y i f the l y s i m e t e r i s near the snow s u r f a c e . The speed of the t r a n s -l a t o r y wave was determined from U = (z - z ) / t , where (z - z) s s i s the l y s i m e t e r depth and t i s the time r e q u i r e d f o r the ave-rage of the l y s i m e t e r f l u x e s to reach ( V 1 + V Q.)/2, the mean of the f i n a l and i n i t i a l f l u x e s . The average co over a range of water contents 6Q to 8-^ , was determined from the measurements of f l u x and of wave speed by: U ln(V,/V_) (6.6-1) co — V l " V 0 164 Table 6 . 6 - 1 F r a c t i o n a l d e r i v a t i v e c a l c u l a t e d from w e t t i n g f r o n t speed t U -CJ = V, - Vo K = K f 1974 Method cm cm/hr cm/hr hrs ;.cm/hr — cm/hr cm/hr June 10 i r r i g a t i o n 56 1 . 0 10 0 . 20 2 8 0 72 3 . 4 3 . 2 June 12 i s o l a t i o n cover removal 48 0 . 012 0 . 6 7 2 . 3 5 2 0 . 4 125 0 . 1 1 0 . 0 9 0 The r e s u l t s are shown i n t a b l e 6.6-1 f o r experiment one. Note that the June 12 w value i s c l o s e to the mean to g i v e n f o r the l i q u i d measurement method which has a s i m i l a r estimate of K. 6.7 FRACTIONAL DERIVATIVE — KINEMATIC WAVES When a snowplot i n which the f l u x i s steady i s suddenly covered with an i s o l a t i o n cover, the input at the snow s u r f a c e i s suddenly reduced to a very low v a l u e , V-^,, I f the values of f l u x V^. <_ V < Vg, t r a v e l through the snow as kinematic waves, the f r a c t i o n a l d e r i v a t i v e , co, can be determined by measuring the f l u x and the wave speed. z - z ( 6 . 7 - D =;o) = ^< 5-^ -) (see equation 3-4-14) where (z - z) i s the l y s i m e t e r depth and t i s the time i n t e r v a l s i n c e the i s o l a t i o n cover was p l a c e d over the snowplot. The value of co a p p l i e s to the snowlayer between the s u r f a c e and the l y s i m e t e r l e v e l . The comments i n the l a s t s e c t i o n r e g a r d i n g l a t e r a l flow apply here as w e l l . The kinematic wave method was not used to determine K d u r i n g the other r e g u l a t e d p l o t e x p e r i -ments, because of i c e sheets near the instrument l e v e l . . In f i g u r e 6.7-1 the drainage wave as measured by f i v e l y s i -meters i n a row at the same l e v e l i s shown. The h y d r a u l i c c o n d u c t i v i t y , K, at a given time, i s taken to be the average f l u x , V, of the f i v e l y s i m e t e r s d u r i n g the drainage. The r e s u l t i n g co values f o r experiment 1 are p l o t t e d a g a i n s t K i n F i g u r e 6.7-1 Drainage wave at 56 cm depth from the a r i t h m e t i c average f l u x from f i v e l y s i m e t e r s i n experiment 1, June 10, 1974. 167 IOOO © i-ipuip MEftSUReMfNr, A*E4A(,£ Folf. 3 LA1EHS - KINEMATIC UAVES , Lr\YEH 3-1 •+• THANfLATORy uifl VES , LflVEA. 3-1 KlflJEMAr/C Ulf^V ES , H OHO C £NO\iS SNO UJ F i g u r e 6.7-2 F r a c t i o n a l d e r i v a t i v e of the h y d r a u l i c c o n d u c t i v i t y with r e s p e c t to l i q u i d water content as a f u n c t i o n of the h y d r a u l i c conduct-; i v i t y . Three experimental methods f o r determining these values are compared on the graph. The homogenous snow values are c a l -c u l a t e d from data given i n Colbeck and Davidson (1972b, the l i n e i n t h e i r f i g u r e 5 ) . The f r a c t i o n a l d e r i v a t i v e s and h y d r a u l i c c o n d u c t i v i t y s are equated to - y ^ f and V r e s p e c t i v e l y , when the f l u x r e l a t i o n s are measured d u r i n g steady or d e c r e a s i n g flows. 168 f i g u r e 6 . 7 - 2 as the s o l i d curve. The bump i n the curve of f i g u r e 6 . 7 - 1 means that some l a t e r a l flow probably took p l a c e i n experiment 1 . The p a i r s of (co,K) values obtained by the previous two methods are a l s o p l o t t e d i n f i g u r e 6 . 7 - 2 . The homogeneous snow r e l a t i o n i s c a l c u l a t e d from the drainage data g i v e n i n Colbeck and Davidson ( 1 9 7 2 b , from the l i n e shown i n t h e i r f i g u r e 5 ) • Both the kinematic and t r a n s l a t o r y wave methods use data ' from the same snow l a y e r . I t may be that a l a t e r a l flow e f f e c t has speeded up the h i g h - f l u x waves and hence i n c r e a s e d the co estimate compared to what would be expected from e x t r a p o l a t i o n of e i t h e r the low-flux v a l u e s , or of Colbeck and Davidson's curve. On the other hand, the break i n slope of the s o l i d p o i n t s may i n f a c t r e p r e s e n t a change i n the 0(K) r e l a t i o n from a power law at low f l u x to a l o g a r i t h m i c law at the high f l u x v a l u e s . The p r e c i s e d e f i n i t i o n of the f u n c t i o n and i t s v a r i a t i o n over d i f f e r e n t types of snow r e q u i r e s f u r t h e r measu-rement. F i g u r e 6 . 7 - 2 shows t h a t f o r both homogeneous and l a y e r e d wet snow, co i s about 60 to 250 over the range of K shown. This data may be used to get rough estimates of the speed of f l u x wave movement through o l d wet snowpacks. For d e c r e a s i n g f l o w s , the f r a c t i o n a l d e r i v a t i v e , i s the r a t i o of the speed of the f l u x wave to the f l u x i t s e l f . The f u l l range of v a l i d i t y of kinematic wave theory as a d e s c r i p t i o n of melt waves i n heterogenous snowpacks s t i l l remains to be i n v e s t i g a -t e d . CHAPTER SEVEN CONCLUSIONS Two instruments, which have been used i n s o i l s r e s e a r c h , have been redesigned to i n v e s t i g a t e h y d r o l o g i c r e l a t i o n s i n wet snowpacks. The t e n s i o n l y s i m e t e r and the tensiometer have pro v i d e d new i n f o r m a t i o n on the flow process o c c u r r i n g i n wet snow. The w r i t e r used i s o l a t i o n covers and i r r i g a t i o n , to produce known inputs- to the snow s u r f a c e , so that f l u x waves could be e a s i l y a nalysed. Simultaneous measurements were made of c a p i l l a r y p r e s s u r e , l i q u i d water content changes, and f l u x i n the snowpack. A rugged and economical tensiometer has been designed to measure the c a p i l l a r y pressure i n snowpacks. The changes In p r e s s u r e produced by d i u r n a l melt waves, and even by c l o u d e f f e c t s superimposed on the waves, c o u l d be e a s i l y measured. The sma l l s c a l e of the tensiometer cup allows s m a l l s c a l e flow f e a t u r e s t o be observed, f o r example the wetting f r o n t f l u x curve measured with a l y s i m e t e r , was f r e q u e n t l y r e s o l v e d i n t o f i n g e r s o f ' f l o w whose a r r i v a l was d i s p e r s e d i n time. The response time of the tensiometers, which have water manometers, does not e f f e c t the measurement exceipt at the steeper wetting f r o n t s , where the use of a f a s t e r p r essure transducer i s needed. Large v a r i a t i o n s were found i n pressure over s h o r t d i s t a n c e s . Consequently s e v e r a l tensiometers were r e q u i r e d i n a l a y e r to o b t a i n an adequate sample of the p r e s s u r e . The tensiometer d e s c r i b e d here,. 170 can only be used i n wet snow. From the theory of l y s i m e t r y , i t i s found that even simple zero pressure, instruments can measure f l u x with 100% e f f i c i e n c y , p r o vided that a r i m of proper height i s mounted around the l y s i m e t e r i n t e r f a c e . On the other hand, f o r d i r e c t measurement of the r e l a t i o n s between h y d r o l o g i c v a r i a b l e s , which r e q u i r e very c l o s e j u x t a p o s i t i o n of d i f f e r e n t instruments, the low r i m ( s e v e r a l centimeters h e i g h t ) of the t e n s i o n l y s i m e t e r i s advan-tageous. The t e n s i o n l y s i m e t e r has a s h o r t e r response time, and a s t a r t - u p time that i s s e v e r a l orders of magnitude f a s t e r than the zero t e n s i o n l y s i m e t e r . I t s disadvantage i s that i t can only be i n s t a l l e d i n wet snowpacks, and r e q u i r e s more mainte-nance. Because of the r a p i d change i n ambient pressure between snow l a y e r s , i t i s d i f f i c u l t to measure c a p i l l a r y pressure g r a d i e n t s except as a g r a d i e n t averaged over s e v e r a l snow l a y e r s . The space-average c a p i l l a r y pressure g r a d i e n t was measured to be much sm a l l e r than the g r a v i t a t i o n a l p o t e n t i a l gradient d u r i n g steady or d e c r e a s i n g flow i n the snowpack (the g r a v i t y drainage p o s t u l a t e ) . Hence the r e l a t i o n s of f l u x , measured d u r i n g steady or d e c r e a s i n g f l o w s , to e i t h e r the c a p i l l a r y p r e s s u r e , h , or to the v a r i a b l e ^ might be i n t e r p r e t e d as the h y d r a u l i c conduc-xe V d6' to ^ J t i v i t y r e l a t i o n s K(h c) and oj(K), where oa i s the f r a c t i o n a l d e r i v a t i v e (ir'rrzr i s a slowly changing f u n c t i o n of V and i\ at) V uU i s e a s i e r t o measure than the absolute l i q u i d water content, 0.) The i n t e r p r e t a t i o n f o l l o w s from the assumption that the Darcy equation d e s c r i b e s unsaturated flow i n snow. 171 By p l a c i n g tensiometers j u s t above t e n s i o n l y s i m e t e r s i n the snowpack, the r e l a t i o n between the h y d r a u l i c c o n d u c t i v i t y and the c a p i l l a r y p r essure K(h ), was determined. By r e p e a t i n g the w e t t i n g and d r y i n g c y c l e s of the snow between the same maximum and mimimum values of f l u x , the K(h ) r e l a t i o n was found to repeat the same h y s t e r e s i s loop. The h y s t e r e s i s loop can be d e s c r i b e d as being bounded by two curves, which f o l l o w power f u n c t i o n s of exponent, n i n the range 11 to 1 5 , depending on the snow l a y e r . The value of n, a pore s i z e d i s t r i b u t i o n index, i s r e q u i r e d f o r models of flow i n which c a p i l l a r y pressure e f f e c t s are i n c l u d e d . The boundary d r y i n g curve had pressures that were about h0% h i g h e r than those on the boundary w e t t i n g curve. This r a t i o was the same as that of snow sample b u b b l i n g p r e s -sure s . The high value of n, combined with d i f f e r e n c e s i n the K(h ) r e l a t i o n between snow l a y e r s , shows that the d e t e r m i n a t i o n of f l u x by measurement of c a p i l l a r y p r e s s u r e , may l e a d to l a r g e e r r o r s . Three d i f f e r e n t methods were used to help determine the w(K) r e l a t i o n . The l i q u i d content method f a i l e d to d e f i n e the r e l a t i o n f o r i n d i v i d u a l snow l a y e r s because of l i m i t e d sample s i z e . The t r a n s l a t o r y wave and kinematic wave methods,,which give average values of OJ over s e v e r a l snow l a y e r s , show that co was a s l o w l y changing f u n c t i o n o f K and had values c o n s i s t e n t with the OJ values determined by Colbeck and Davidson ( 1 9 7 2 b ) f o r homogeneous snow. The f r a c t i o n a l d e r i v a t i v e co, i s needed i n the kinematic wave analogy (Colbeck, 1 9 7 2 a ) , where i t i s the r a t i o of the speed of the f l u x wave to the f l u x i t s e l f , d u r i n g d e c r e a s i n g l i q u i d flow w i t h i n a snowpack. 172 Laboratory experiments should be done on columns of snow to measure the r e l a t i o n s between the h y d r a u l i c v a r i a b l e s f o r uniform snow under c a r e f u l l y c o n t r o l l e d c o n d i t i o n s . In p a r t i c u l a r , the Darcy equation f o r unsaturated flow and the l a c k o f h y s t e r e s i s i n the K(Q) f u n c t i o n should be t e s t e d . The f u l l range of v a l i d i t y f o r the kinematic wave theory as a d e s c r i p t i o n of melt waves i n n a t u r a l heterogeneous wet snow packs needs to be i n v e s t i g a t e d . In p a r t i c u l a r , the co f u n c t i o n should be determined, u s i n g perhaps simple z e r o - t e n s i o n l y s i m e t -ers and c o n t r o l l e d s u r f a c e i n p u t s , to see how r e p r e s e n t a t i v e are the present co v a l u e s . The d i r e c t measurement of l i q u i d wat-er content changes u s i n g n o n d e s t r u c t i v e methods could be used to determine co values of i n d i v i d u a l snow l a y e r s . F u r t h e r work could a l s o be done to see i f the tensiometer can be used to determine the f l u x , by measureing the reponse of the manometer water column to an i n i t i a l n o n e q u i l i b r i u m s e t t i n g , as suggested f o r measuring K i n s o i l s by G a r d n e r ( 1 9 6 1 ) , by u s i n g a r e l a t i o n d e r i v e d along l i n e s s i m i l a r to t h a t o f equ-a t i o n 4 . 1 - 2 . The development of such a device f o r use i n snow-packs may provide a simple, r a p i d method of determining the snowmelt r a t e i n the f i e l d . REFERENCES 173 Ambach, W. and F. Howorka, 1 9 6 6 . Avalanche a c t i v i t y and f r e e water content of snow at Obergurgl. I n t e r n a t i o n a l A s s o c i a t i o n of S c i e n t i f i c Hydrology, P u b l i c a t i o n 69 (Davos Symposium) : 65 - 7 2 . Brooke, R.C, 1 9 6 6 . Vegetation-Environment R e l a t i o n s h i p s of Subalpine Mountain Hemlock Zone Ecosystems. (Unpublished Ph.D. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia) 110 pp. Brooks, R.H. and A.T. Corey, 1 9 6 4 . H y d r a u l i c p r o p e r t i e s of porous media* Colorado State U n i v e r s i t y Hydrology Paper 3-(Fort C o l l i n s , Colorado) 2 7 p p . Brooks, Royal H. and Arthur T. Corey, 1 9 6 6 . P r o p e r t i e s of porous media a f f e c t i n g f l u i d flow. 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American Geophysical Union, T r a n s a c t i o n s 26 Part 1 : 8 3 - 9 0 . G e r d e l , R.W. 1 9 5 4 . The t r a n s m i s s i o n of water through snow. American Geophysical Union, T r a n s a c t i o n s 3 5 ( 3 ) : 4 7 5 - 4 8 5 -Gurr, Edward, 1 9 7 1 . S y n t h e t i c Dyes i n B i o l o g y , Medicine, and Chemistry (Academic Press, London) 8 0 7 p p . H a r r o l d , L.L. and F-R. D r e i b e l b i s , 1 9 5 1 . A g r i c u l t u r a l hydrology as e v a l u a t e d by monolith l y s i m e t e r s . T e c h n i c a l B u l l e t i n 1 0 5 0 , United S t a t e s Department of A g r i c u l t u r e , l 4 9 p p . Horton, R.E. 1 9 1 5 . The m e l t i n g of snow. Monthly Weather Review 43 : 599 - 6 0 5 . Hughs, T. and G. Seligman 1 9 4 0 . The temperature, melt water • movement and density, i n c r e a s e i n the neve of an a l p i n e g l a c i e r . Monthly N o t i c e s of the Royal Astronomical S o c i e t y , G eophysical Supplement 4 : 616 - 647. K l u t e , A. and W.R. Gardner 1 9 6 2 . Tensiometer response time. S o i l Science 93 : 204 - 2 0 7 -K l u t e , A. 1 9 7 3 - S o i l water flow theory and i t s a p p l i c a t i o n i n f i e l d s i t u a t i o n s . F i e l d S o i l Water Regime ( S o i l Science S o c i e t y of America, Inc., Madison, Wis.): 9 - 3 5 . 175 Kojima, K e n j i 1 9 6 6 . D e n s l f I c a t i o n of seasonal snow cover. Physics of Snow and Ice 1 - 2 , Oura, H., ed. ( I n s t i t u t e ^ o f Low Temperature S c i e n c e , Hokkaido U n i v e r s i t y ) : 9 2 9 - 9 5 1 . Langham, E . J . 1 9 7 4 a . The occurance and movement of l i q u i d water i n the snowpack. Advanced Concepts and Techniques i n the Study of Snow and Ice Resources. Monterey Conference, ( N a t i o n a l Academy of S c i e n c e s , Washington, B.C.) :6j - 7 5 -Langham, E . J . 1 9 7 4 b . Phase e q u i l i b r i u m of veins i n p o l y c r y s -t a l l i n e i c e . Canadian J o u r n a l of E a r t h Sciences 11 : 1 2 8 0 -1 2 8 7 . Leaf, Charles 1 9 6 6 . Free water content of snowpacks i n s u b a l -pine areas, Western Snow Conference, Proceedings 34 : 17 -24. L i g h t h i l l , M.V. and G.B. Whitham 1 9 5 5 . On kinematic waves — 1 F l o o d movement i n long r i v e r s . Royal S o c i e t y , London, Proceedings Ser. A, 229 ( 1 1 7 7 ' ) . : 2 9 1 •- 3 1 6 . Mathew, W., and J.R. Mackay 1 9 6 3 . Snowcreep s t u d i e s on Mt. Seymour, B.C. — P r e l i m i n a r y f i e l d i n v e s t i g a t i o n s . Dept. of : Mines and T e c h n i c a l . Surveys , .Ottawa, .Geographical-.Bulletin 20 : 58 - 7 5 -M e l l o r , M. 1 9 6 5 - Some o p t i c a l p r o p e r t i e s of snow. I n t e r n a t i o n a l A s s o c i a t i o n of S c i e n t i f i c Hydrology, P u b l i c a t i o n 69 (Davos Symposium) : 1 2 8 - 140. Muskat, M. 1 9 3 7 . The Flow of Homogeneous F l u i d s Through Porous Media (McGraw-Hill, New York) 7 6 3 p p . Naruse, R e n j i , H. Oura, and K Kojima, 1 9 7 0 . 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A, 26 (Japanese, w i t h E n g l i s h summary) : 77 - 86. Yo s i d a , Zyungo 1959• A c a l o r i m e t e r f o r measuring the f r e e water content o f wet snow. Low Temperature S c i e n c e , Ser. A, 18 (Japanese, with E n g l i s h summary) : 1 - 16. 

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