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A new method to compare radiation view factors and a study of bare soil evaporation using microlysimeters Streicher, John James 1986

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A NEW METHOD TO COMPUTE RADIATION VIEW FACTORS AND A STUDY OF BARE SOIL EVAPORATION USING MICROLYSTJMETERS By JOHN JAMES STRETCHER B.S., The University of C i n c i n n a t i , 1980 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF SOIL SCIENCE We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA NOVEMBER 1986 © John James Streicher, 1986 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v ailable for reference and study. I further agree that permission for extensive copying of this thesis for s c h o l a r l y purposes may be granted by the head of my department or by h i or her representatives. It i s understood that copying or publication o t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of S o i l Science The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, B.C. Canada V6T 1W5 Date November 28, 1986 DE-6 (2/79) - i i -ABSTRACT In Chapter 1, a numerical computer technique is developed to determine radiation view factors between planar surfaces whose geometry is sufficiently regular so as to be defined by algebraic equations. This technique does not require spherical, cylindrical or rectangular symmetry, although such symmetries may be exploited when they exist. Once the essential geometric problem is formulated, enough generality can be built into the solutions so that certain "new" configurations, derived from translations or rotations of one surface relative to the other, can be solved as a matter of course. In Chapter 2, a model of bare s o i l evaporation is tested against measured flux from lysimeters obtained in the Peace River region of British Columbia and Alberta. Hydraulic diffusivity characteristics, measured from separate, adjacent f i e l d samples, were used in the model. Certain procedural d i f f i c u l t i e s in the measurement of hydraulic diffusivity are examined in detail, and recommendations for improvement are made. The degree to which evaporation simulation agrees with measured flux is discussed. - i i i -TABLE OF CONTENTS Page A b s t r a c t i i T a b l e o f C o n t e n t s i i i L i s t o f T a b l e s v L i s t o f F i g u r e s v i L i s t o f Symbols x i A c k n o w l e d g e m e n t s x i v D e d i c a t i o n x v I n t r o d u c t i o n 1 C h a p t e r I A New Method to Compute R a d i a t i o n V i e w F a c t o r s 2 A . I n t r o d u c t i o n 3 B . M e t h o d o l o g y 5 C . D i s c u s s i o n 21 D . Case S t u d i e s 23 E . Summary and C o n c l u s i o n 50 C h a p t e r I I A S t u d y o f B a r e S o i l E v a p o r a t i o n U s i n g M i c r o l y s i m e t e r s 51 A . I n t r o d u c t i o n 52 B . T h e o r e t i c a l C o n s i d e r a t i o n s 55 C . M e t h o d o l o g y 58 1. S o i l s and S a m p l i n g 58 2 . L a b o r a t o r y Methods 59 3 . F i e l d E x p e r i m e n t s 60 4 . M o d e l l i n g E v a p o r a t i o n 63 - i v -Page D. Results 68 1. General Comments 68 2. Predicting Stage I Duration 68 3. Modelling Stage II Flux 74 4. Modelling Moisture P r o f i l e s 76 E. Discussion 90 1. D i f f u s i v i t y Measurements 90 2. Improvements to Stage II Flux Modelling 95 F. Summary and Conclusion 100 References 101 Appendices 103 A. Longwave Radiation and the Concept of View Factor 103 B. L i s t i n g of Pla n e - P a r a l l e l Disks View Factor Program... 108 C. Hydraulic D i f f u s i v i t y Data 113 D. Hydraulic D i f f u s i v i t y Functions 131 E. Microlysimeter S o i l Parameters 133 F. P a r t i c l e Size Analysis 136 G. Retention C h a r a c t e r i s t i c and Structure 138 - v -LIST OF TABLES Table Page 2.1 Microlysimeter mean water content at the time of transition: Measured vs Modelled 97 - v i -LIST OF FIGURES Figure Page 1.1 S p h e r i c a l coordinate system 6 1.2 S2 projected onto hemispherical s h e l l of S\ 7 1.3 Point Pi on Si r a d i a t e s toward point P2 on S2 9 1.4 YTIP i s w i t h i n the true y-boundary, but outside the modified y-boundary. The ray shown crossed the plane of S2 at point Q, outside the true y-boundary of S2« When YTIP i s tested f o r boundary l i m i t s , the c o r r e c t negative r e s u l t i s obtained only when the modified y-boundary i s used 13 1.5 Radiating d i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , i s d i s p l a c e d a distance d from w a l l base (d expressed i n u n i t s of w a l l h e i g h t ) , r a d i a t e s to w a l l of height H 24 1.6 View f a c t o r f o r F i g . 1.5 25 1.7 Radiating d i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , i s d i s p l a c e d along s t r i p c e n t r e - l i n e a distance d from s t r i p base (d expressed i n u n i t s of s t r i p w i d t h ) , r a d i a t e s to s t r i p of width w 26 1.8 View f a c t o r for F i g . 1.7 27 1.9 Disk of radius r r a d i a t e s to c o a x i a l d i s k of Radius R. Separation distance = z 28 1.10 View f a c t o r f o r F i g . 1.9 29 1.11 Ra d i a t i n g d i f f e r e n t i a l area dA, with outward normal along z d i r e c t i o n , i s o f f s e t from d i s k axis by x. The planes of the d i f f e r e n t i a l area and the d i s k are separated by a distance z 30 1.12 View f a c t o r f o r F i g . 1.11 31 1.13 R a d i a t i n g d i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , i s separated from square of side s by distance z, and d i s p l a c e d from square edge by distance x. Both d i f f e r e n t i a l area and square are centered with respect to y - d i r e c t i o n 32 1.14 View f a c t o r for F i g . 1.13 33 - v i i -Figure Page 1.15 Radiating d i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , radiates to 45° - i n c l i n e d square of side s, elevated above plane of dA. both d i f f e r e n t i a l area and square are centered with respect to y - d i r e c t i o n 34 1.16 View factor for F i g . 1.15 35 1.17 D i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , radiates to square of side s 36 1.18 View factor for F i g . 1.17 37 1.19 D i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , radiates to 45° - i n c l i n e d square of side s.. 38 1.20 View factor for F i g . 1.19 39 1.21 D i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , radiates to parabola: z = - y +1 40 1.22 View factor for F i g . 1.21 41 1.23 D i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , radiates to parabola: z = -y +4..... 42 1.24 View factor for F i g . 1.23 43 1.25 D i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , radiates to isosceles triangle of height H and base B... 44 1.26 View factor for F i g . 1.25 45 1.27 D i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , radiates to isosceles t r i a n g l e of height H and base B. dA i s displaced from centre l i n e by B/4... 46 1.28 View factor for F i g . 1.27 47 1.29 D i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , radiates to c i r c l e of radius R i n Y-Z plane. C i r c l e centre i s displaced a distance z from X-Y plane 48 1.30 View factor for F i g . 1.29 49 - v i i i -Figure Page 2.1 Peace River Region 53 2.2 D i f f u s i v i t y function: Fort St. John - Perennial 61 2.3 Modelling stage II cumulative evaporation 65 2.4 Modelling stage II fl u x 66 2.5 Evaporation data: Beaverlodge annual and fallow samples 69 2.6 Evaporation data: Beaverlodge uncultivated and perennial samples 70 2.7 Evaporation data: Dawson Creek uncultivated, perennial, and fallow samples 71 2.8 Evaporation data: Fort St. John uncultivated, perennial, annual, and fallow samples 72 2.9 Evaporation data: UBC Bose series s o i l at 2 bulk densities 73 2.10 Duration of stage I evaporation 75 2.11 Evaporation modelling: Beaverlodge annual 77 2.12 Evaporation modelling: Beaverlodge fallow 78 2.13 Evaporation modelling: Beaverlodge uncultivated 79 2.14 Evaporation modelling: Beaverlodge perennial 80 2.15 Evaporation modelling: Dawson Creek fallow 81 2.16 Evaporation modelling: Dawson Creek perennial 82 2.17 Evaporation modelling: Fort St. John annual fallow 83 2.18 Evaporation modelling: Fort St. John perennial 84 2.19 Time course of volumetri^ water content for Bose s o i l at bulk density 1.2 g/cm 86 2.20 Time course of volumetric water content for Bose s o i l at bulk density 1.0 g/cm3 87 2.21 Predicted time course of volumetric^ water content for Bose s o i l at bulk density 1.2 g/cm 88 - i x -Figure Page 2.22 Predicted time course of volumetric water content for 3 Bose s o i l at bulk density 1.0 g/cm 89 2.23 D i f f u s i v i t y function: Bose s o i l @ 1.2 g/cm3 91 2.24 D i f f u s i v i t y function: Bose s o i l @ 1.2 g/cm3 92 2.25 D i f f u s i v i t y function: Bose s o i l @ 1.0 g/cm3 93 C l D i f f u s i v i t y function: Beaverlodge annual •••• C.2 D i f f u s i v i t y function: Beaverlodge fallow 115 C.3 D i f f u s i v i t y function: Beaverlodge perennial 116 C.4 D i f f u s i v i t y function: Beaverlodge perennial 117 C.5 D i f f u s i v i t y function: Beaverlodge uncultivated 118 C.6 D i f f u s i v i t y function: Dawson Creek fallow 119 C.7 D i f f u s i v i t y function: Dawson Creek fallow 120 C.8 D i f f u s i v i t y function: Dawson Creek perennial 121 C.9 D i f f u s i v i t y function: Dawson Creek perennial 122 C.10 D i f f u s i v i t y function: Fort St. John annual 123 C . l l D i f f u s i v i t y function: Fort St. John fallow 124 C.12 D i f f u s i v i t y function: Fort St. John perennial 125 C.13 D i f f u s i v i t y function: Fort St. John - Perennial 126 C.14 D i f f u s i v i t y function: Beaverlodge B-horizon 127 3 C.15 D i f f u s i v i t y function: Bose s o i l @ 1.2 g/cm 128 3 C.16 D i f f u s i v i t y function: Bose s o i l @ 1.2 g/cm 129 3 C.17 D i f f u s i v i t y function: Bose s o i l @ 1.0 g/cm 130 G.l Retention c h a r a c t e r i s t i c : B-horizon: i n t a c t vs repacked 142 G.2 Retention c h a r a c t e r i s t i c : Dawson Creek: fallow vs repacked..... 143 - x -Figure Page G.3 Retention c h a r a c t e r i s t i c : Bose s o i l at 2 bulk de n s i t i e s 144 G.4 Retention c h a r a c t e r i s t i c : Beaverlodge uncultivated 145 G.5 Retention c h a r a c t e r i s t i c : Fort St. John annual 146 G.6 Retention c h a r a c t e r i s t i c : Beaverlodge perennial 147 G.7 Retention c h a r a c t e r i s t i c : Dawson Creek perennial 148 G.8 Retention c h a r a c t e r i s t i c : Fort St. John perennial 149 G.9 Retention c h a r a c t e r i s t i c : Beaverlodge repacked s o i l . . . . 150 G.10 Retention c h a r a c t e r i s t i c : Fort St. John repacked s o i l . . 151 - x i -LIST OF SYMBOLS Chapter 1 Ap projected area of S i i n d i r e c t i o n ( 8 , <j>) B radiance [_ ^ ] Sr A P d distance between search ray t i p and plane of f a r b i t r a r y function F12 r a d i a t i o n view factor of 2 as seen from 1 I i n t e n s i t y [^L] l,m g r i d parameters of radiating surface P i , P2 points on r a d i a t i n g , receiving surfaces, r e s p e c t i v e l y r length of search ray r„ T„ shortest distance between S i and So MIN 1 * r . farthest distance between S i and So MAX R emittance [——] 2 m S i , S2 r a d i a t i n g , receiving surfaces, r e s p e c t i v e l y Sr steradian W watt x,y,z rectangular coordinates XT-j-p point at search ray t i p X T I P , Y T I P , ZTIP rectangular coordinates of X i , X 2 , X2 a r b i t r a r y non-coliner points of S2 - x i i -A f i n i t e i n c r e m e n t e e m i s s i v i t y 8 p o l a r a n g l e <(> a z i m u t h a l ang le ft s o l i d a n g l e Chapter 2 C C o n s t a n t i n the e x p o n e n t i a l d i f f u s i v i t y f u n c t i o n D h y d r a u l i c d i f f u s i v i t y E c u m u l a t i v e e v a p o r a t i o n F f l u x f a r b i t r a r y f u n c t i o n L l y s i m e t e r l e n g t h PE p o t e n t i a l e v a p o r a t i o n t t ime t t ime o f t r a n s i t i o n : s t a g e I / s t a g e II a * tn t ime at wh ich f l u x g e n e r a t e d under i n f i n i t e PE w i l l f a l l to PEMEASURED a t t ime at wh ich f i n i t e s o i l column c e a s e s to e v a p o r a t e as a s e m i - i n f i n i t e column z v e r t i c a l c o o r d i n a t e B s l o p e o f e x p o n e n t i a l d i f f u s i v i t y f u n c t i o n 9 v o l u m e t r i c water c o n t e n t 6 a i r - d r y water c o n t e n t a J 6. i n i t i a l water c o n t e n t l - x i i i -^ n r - v r o A o ( t ) measured mean water content at the time of PE MEAS a . . t r a n s i t i o n * 9„„ (t ) predicted mean water content at the time of PE 0 0 a t r a n s i t i o n S i t e and Cropping Practice B/ Beaverlodge D/ Dawson Creek F/ Ft. St. John /A annual crop /F fallow /N uncultivated /P perennial crop - xiv -ACKNOWLEDGEMENTS I wish to thank Mr. Laurenz van V l i e t of Agriculture Canada f o r i n i t i a t i n g the Peace erosion project which evolved into my evaporation study. I thank the s t a f f of Agriculture Canada, Beaverlodge, for providing f a c i l i t i e s for my Peace f i e l d work. I e s p e c i a l l y thank Mssrs. Peter M i l l s and Mervin Hegland for th e i r expertise and help with the project. I thank my colleague and fellow graduate student, Mr. Mohammed Hares, and Mr. Al Neighbors, UBC Dept. of Plant Science, for t h e i r help with the f i e l d experiments performed at UBC. My thanks to Mrs. G a i l Harrop for performing the s o i l p a r t i c l e size analysis that appears i n Appendix F. Many thanks to Mr. Doug Beames for his generous help with el e c t r o n i c s and computer programming problems. Thanks to the s t a f f at Inter-Library Loan (UBC) for t h e i r u n f a i l i n g e f f o r t s i n obtaining obscure publications on view f a c t o r s . I thank my committee members, Dr. Chieng, for reading and c r i t i q u i n g this t h e s i s , and Dr. Black for enthusiasm, i n t e r e s t , and wisdom. And I thank, of course, my advisor, Dr. Novak, for many useful discussions and contributions to this t h e s i s . To Pam - 1 -INTRODUCTION In a micrometeorological study of an ec o l o g i c a l system, i t s physical state i s described i n terms of i t s energy and water regime. Its i n t e r a c t i o n with adjacent systems i s described i n terms of energy, water, and momentum exchange. Energy and water are conserved i n such exchanges, therefore a quantitative analysis of the i n t e r a c t i o n can be undertaken by energy and water balances. The energy balance focuses on heat and l i g h t . Heat exchange can occur i n several modes: conduction, convection, r a d i a t i o n , and mass flow. Radiative heat exchange between bodies depends on t h e i r temperature, emissivity, and t h e i r "view" of each other. Chapter 1 presents a new method to compute ra d i a t i o n view f a c t o r s . The water balance focuses on the processes of water movement i n the soil-plant-atmosphere continuum. Water movement occurs as several processes, among them: p r e c i p i t a t i o n , evaporation, advection, i n f i l t r a t i o n , drainage, r e d i s t r i b u t i o n , overland flow, and plant t r a n s p i r a t i o n . Chapter 2 presents a study of bare s o i l evaporation i n which evaporation rates from microlysimeters are measured, relevant s o i l p h ysical properties are measured and applied i n the testing of a model of evaporation. - 2 -CHAPTER 1 A NEW METHOD TO COMPUTE RADIATION VIEW FACTORS - 3 -A. INTRODUCTION Radiation heat transfer i s an important mode of energy exchange between objects of d i f f e r e n t temperature having an unobstructed view of one another. Calculations of r a d i a t i o n heat transfer are used i n engineering design considerations, and i n micrometeorological studies of urban, a g r i c u l t u r a l , and f o r e s t r y systems. The single greatest obstacle to c a l c u l a t i n g r a d i a t i v e transfer between any two objects i s the computation of the radiation view factor. This purely geometrical e n t i t y governs the f r a c t i o n of r a d i a t i o n emitted from one surface that i s intercepted by another. The computation of r a d i a t i o n view factors for various configurations of elementary surfaces ( i . e . , c i r c l e s , squares, etc.) has been an ongoing f i e l d of research since at least the 1920s. In the more than 60 years since, several d i s t i n c t methods (Siegel and Howell, 1972) have been invented to calculate view factors. Each has i t s strengths i n the p a r t i c u l a r kind of configurations i t can solve. For the most part, a great deal of symmetry i n the configuration i s required. And any symmetry destroying rotation or t r a n s l a t i o n of one surface r e l a t i v e to the other, no matter how small, usually renders the technique unworkable. Presented here i s a new method to compute rad i a t i o n view f a c t o r s . It i s a numerical computer method that has duplicated the results of accepted published work of many other authors, and has solved c e r t a i n configurations not previously solved. This method i s extremely powerful in that i t does not require any symmetry i n the configuration. As such, i t i s expected to have wide a p p l i c a t i o n i n engineering, and micrometeorology. - 4 -The objective of this chapter i s to develop, demonstrate, and va l i d a t e t h i s method to compute rad i a t i o n view factors for configurations of regular geometric objects. B. METHODOLOGY The r a d i a t i o n view f a c t o r i s defined as the f r a c t i o n of energy r a d i a t e d by one surface that i s in t e r c e p t e d by a r e c e i v i n g surface (Appendix A). I f the r a d i a t i n g surface i s pla n a r , i t can be sa i d to have a viewing hemisphere of 2ir s t e r a d i a n s . I f w i t h i n that hemisphere some o b s t r u c t i o n e x i s t s , that o b s t r u c t i o n w i l l subtend a s o l i d angle d>2 82 a = f JQ sine de d<t> ( 1 . 1 ) 91 e l where 9 i s the polar angle and 9 the azimuthal angle, and Q\, 82, <J>i, 92 define the o u t l i n e of the o b s t r u c t i n g object ( F i g . 1.1). There are s e v e r a l techniques ( N u s s e l t , 1928; Johnson and Watson, 1985) that seek to determine the l i m i t s of i n t e g r a t i o n f o r 8 and <j> by p r o j e c t i n g the ob s t r u c t i n g geometry, surface S2, onto a hemispherical s h e l l centered on some point of the r a d i a t i n g surface, S i ( F i g . 1.2). These techniques are d i f f i c u l t , and i f the r a d i a t i n g surface i s f i n i t e , not d i f f e r e n t i a l , the r e c e i v i n g surface needs to be projected onto a s h e l l f o r each point of that sees a s i g n i f i c a n t l y d i f f e r e n t view of S2. However, i f a f i n i t e S i subtends a s u f f i c i e n t l y small s o l i d angle, as seen from points of S2, then Si may be treated as a d i f f e r e n t i a l , or point r a d i a t o r , to good approximation. The technique developed here reasons as f o l l o w s : i f a given s o l i d angle dS2 = sin8 d8 dcj> i s obscured by S2 as seen from some point P i on S i , then i t i s true that a ray o r i g i n a t i n g at P i on Si and along the - 6 -F i g . 1.1 Spherical coordinate system. - 7 -Fig 1.2 S2 projected onto hemispherical s h e l l of S\. - 8 -d i r e c t i o n (8, <(>) w i l l s t r i k e ( i n t e r s e c t ) S2 at some P 2 on S2 ( F i g . 1.3). P 2 w i l l be both a point of S 2 and a point of the ray, consequently i t w i l l s a t i s f y both the equation of the surface S2 and the equation of the ray (or l i n e ) . Hence the question of o b s t r u c t i o n of a given dfi r e s t s w i t h the existence (or not) of P 2 . In 3-dimensional space the equation of a l i n e can be determined from a point on the l i n e and a d i r e c t i o n of the l i n e [Pi (x, y, z) and d i r e c t i o n (9, <(>)]. The equation of a plane can be found i f 3 noncolinear points of i t s surface are known ( X i , X 2 , X3 a l l on S 2 ) . The point of i n t e r s e c t i o n of the ray with S2 can be determined by s o l v i n g the two equations simultaneously. Thus, i f the system of equations has a unique s o l u t i o n , P 2 e x i s t s , and dfi(9, c|>) i s obstructed. This dft i s counted as part of the view of S 2, the d i f f e r e n t i a l view f a c t o r d F 1 2 = cos8 sin9 d9 d<j> (1.2) i s computed, and a running t o t a l / Ja dF 1 ? i s summed fo r a l l 9 and <f> <p o i n the hemisphere of P^. The sum must be normalized by d i v i s i o n by tr to get the view f a c t o r of S2 as seen from P i on S j . I f S 2 i s planar (true of a l l surfaces considered i n t h i s chapter), a simpler c r i t e r i a f o r i n t e r s e c t i o n of the ray (from P i along (9, <)>)) with the plane of S2 can be used. For a given d i r e c t i o n (9, <(>), the ray can begin a "search" f o r the plane by s t a r t i n g at some radius l e n g t h , known from the geometry of the problem to be w e l l short of the minimum p o s s i b l e distance between P i and S2» The t i p of the ray has s p h e r i c a l coordinates ( r , 0 , d)) which can - 9 -F i g . 1.3 Point P^ on Si^ radiates toward point P2 on S2» - 1 0 -be converted to rectangular coordinates (x, y, z) by the coordinate transformation equations: x = r sin9 cos<t> y = r sin9 sin<f> (1.3) z = r cos9 The distance between the ray t i p and the plane of S 2 can be obtained from the standard algebraic geometry formula giving the distance between a point and a plane: | (x_ I p - X X) • [(X 2 - X X) x (X 3 - Xi)] d = |[(X 2 - X x) x (X 3 - X x ) ] | (1.4) where Xj i p = (x, y, z) of the ray t i p . X i , X 2, X3 are three non-colinear points of the plane of S 2, expressed i n t h e i r coordinate t r i p l e t s ( x i , y i , z i ) , ( x 2 , Y2, z 2 ) , and ( X 3 , y 3 , z 3 ) . Scalar products between vectors are designated with (•), and vector products between vectors are designated with ( x ) . The ray can now grow i t e r a t i v e l y , from rMIN t 0 ^ I N + dr. And the distance between the ray t i p and the plane i s recomputed. Geometrically there are only three possible outcomes for a ray o r i g i n a t i n g at a point not on a plane. The ray may grow away from the plane, i t may grow p a r a l l e l to i t , or i t may grow toward i t , eventually i n t e r s e c t i n g i t . Only the l a s t p o s s i b i l i t y i s of i n t e r e s t . In this case, the distance d w i l l decrease upon each i t e r a t i o n of dr. And i f enough i t e r a t i o n s are carried through, d w i l l go through a minimum and begin to increase again — thereby i n d i c a t i n g i n t e r s e c t i o n . Upon esta b l i s h i n g that a minimum i n d has occurred, the i t e r a t i o n i n r i s stopped and the current coordinates of the ray t i p are retained for - 11 -f u r t h e r t e s t i n g . This point i s a candidate f o r P 2 . So f a r , i t has been e s t a b l i s h e d that the d i r e c t i o n ( 9 , 9) i s obscured by the plane of S£» I f S2 i s not i n f i n i t e i n both of i t s dimensions, i . e . , i f S2 i s not i d e n t i c a l l y the plane i t s e l f , i t may be that the candidate p o i n t , while i n the plane of S2, i s outside of the boundary of S2« Now the p r e v i o u s l y stated requirement of regular geometry of S2 comes i n t o play. Tests of (XTIP, YTIP, ZTIP) must be made to determine i f i t i s i n s i d e the boundary of S2» Various simple geometries of S2 w i l l suggest appropriate t e s t s . For example, i f S 2 i s a c i r c l e with center at (XCIR, YCIR, ZCIR) and radius of RCIR, then the t e s t would be: i s /(XTIP - XCIR) 2 + (YTIP - YCIR) 2 + (ZTIP - ZCIR) 2 _< RCIR I f t h i s c o n d i t i o n i s met, then the candidate point i s indeed a point of S 2 , and the d i r e c t i o n ( 9 , 9) i s i n f a c t obscured by S 2 , and so the d i f f e r e n t i a l view f a c t o r d F 1 2 = 7 c o s 6 s i n 9 d6 do> ( 1 . 5 ) should be computed and added to the running t o t a l of / dFi2« Otherwise nothing i s added. Then the search i s begun i n a new d i r e c t i o n Tf ( 9 + d 9 , 9) f o r a l l 9: 0 £ 9 _< —. A s i m i l a r search i s c a r r i e d out i n the range of 9 : 0 < 9 < 2 IT. In t h i s way the e n t i r e viewing hemisphere of ?l i s s y s t e m a t i c a l l y searched. This i s not a Monte Carlo technique. And the p r e c i s i o n of computing view f a c t o r s can be increased by i n c r e a s i n g the number of i t e r a t i o n s i n 9 and 9 . Generally a p a r t i t i o n of the range - 12 -of 8 i n t o 90 d i v i s i o n s and a p a r t i t i o n of the range of <f i n t o 180 d i v i s i o n s gave view f a c t o r s i n agreement with published values to w i t h i n ± 0.02 absolute e r r o r . As a numerical i n t e g r a t i o n with f i n i t e i t e r a t i o n s , one d e t a i l warrants a c l o s e r look. Since planar i n t e r s e c t i o n has required c r o s s i n g the plane i n t h i s technique, the f i n a l coordinates of the ray t i p a f t e r a s u c c e s s f u l i n t e r s e c t i o n are i n f a c t not e x a c t l y i n the plane of S 2. On average the ray w i l l have overshot the plane by 1.0 Ar (where Ar i s the increment length of r) i n the d i r e c t i o n r. Testing the f i n a l value of (XTIP, YTIP, ZTIP) against the boundary l i m i t s of S 2 can give biased or even d r a s t i c a l l y i n c o r r e c t r e s u l t s , depending on the problem, i f the overshoot i s not accounted f o r . I t can be taken i n t o c o n s i d e r a t i o n by modifying the boundaries of S2 such that i f the current ray t i p i s w i t h i n the modified boundaries, then the a c t u a l point of planar i n t e r s e c t i o n was w i t h i n the true boundaries. Additions or s u b t r a c t i o n s are made to the S 2 boundaries i n proportion to the d i r e c t i o n a l bias of the overshooting ray, as i n F i g . 1.4. Here, a t e s t of the y-component of the ray t i p , YTIP, against the true y-boundary of S 2 would y i e l d an a f f i r m a t i v e r e s u l t . The c o r r e c t negative t e s t r e s u l t i s obtained when YTIP i s compared with the modified y-boundary of S 2. In F i g . 1.4, the modified y-boundary has been s h i f t e d r e l a t i v e to the true y-boundary by 1.0 x Ar x sin8 x sin<)>, which i s the average y-overshoot by rays c r o s s i n g the plane along d i r e c t i o n (8, <))). The a c t u a l y-overshoot ranges from 0.5 to 1.5 times t h i s amount. Had the plane of S 2 not been normal to the y-boundary, the modified y-boundary would be s h i f t e d r e l a t i v e to the true y-boundary by - 13 -1,0 * Ar *SIM 6 * SIN 0 ( X T I P , Y T I P ; 2i\?) P U h e o f S S.EZ7 F i g . 1.4 YTIP i s within the true y-boundary, but outside the modified y-boundary. The ray shown crossed the plane of S2 at point Q, outside the true y-boundary of S2. When YTIP i s tested for boundary l i m i t s , the correct negative r e s u l t i s obtained only when the modified y-boundary i s used. - 14 -y - s h i f t = sinS x 1.0 x Ar x sln9 x sine)) (1.6) where 8 i s the acute angle between the y-boundary and the plane of S2« Such a boundary modification needs to be considered for a l l boundary tests — i n general x-, y-, and z-boundary t e s t s , or some other l i n e a r combination thereof. Summarizing then, t h i s technique uses two types of coordinate systems: a global system of rectangular coordinates i n which the surfaces Si and S2 are defined by i n e q u a l i t i e s ( t h e i r boundaries are equations) i n 3-dimensional space; and a l o c a l system of spherical coordinates centered on some Pi of S i , for each such point of Si with a s i g n i f i c a n t l y d i f f e r e n t view of S2. For a point of S i , the l o c a l spherical coordinate system serves to define a l o c a l viewing hemisphere, complete with the di r e c t i o n s (9, <)>) to be probed, i t e r a t i v e l y i n r, by a ray. The coordinates of the t i p of the ray being ( r , 9, <j>), are converted at each i t e r a t i o n into rectangular coordinates (XTIP, YTIP, ZTIP). The global rectangular coordinate system then serves as the framework i n which to conduct the test for planar i n t e r s e c t i o n , and the subsequent test to check i f the i n t e r s e c t i o n point i s within the boundaries of the obstruction S2» The strength of this technique i s the absence of any symmetry requirement i n the configuration. A problem exhibiting l e f t - r i g h t symmetry can be solved as e a s i l y as a similar configuration with one surface translated or rotated r e l a t i v e to the other. Another strength i s that the projected area of S2 need not be known or computed since the p i x e l represented by the search ray can only "see" - 15 -projected areas. Indeed, i n a sense the technique i s an elaborate scheme to project S2 onto the unit hemispherical s h e l l of and then sum up the obscured p i x e l s , weighted with Lambert's cos 8, and normalize the f i n a l sum by d i v i s i o n by IT. In the mechanics of the systematic search of the hemisphere for the obstruction S 2, the range of 9 i s fixed at 2IT, unless symmetry can be exploited to reduce the search to a ha l f or quarter of the hemisphere, and multiply the f i n a l r esult by 2 or 4 respectively. For example, i f an obstruction i s l e f t - r i g h t symmetric with respect to the radiating point, the view factor of the l e f t half w i l l equal that of the right h a l f . Either half may be computed, multiplying the re s u l t by 2. If an obstruction i s c y l i n d r i c a l l y or s p h e r i c a l l y symmetric with respect to the r a d i a t i n g point, only a narrow A<t> s l i c e of the hemisphere need be sampled, multiplying that r e s u l t by 2Tr/A<j>. Care must be taken, however, as symmetry considerations cannot reduce the range of 9 due to Lambert's Law weighting small 9 more than larger 8, i . e . I ( 9 ) " cos9 (1.7) 1(9 = 0) The range of r must be determined from the basic geometry. An estimate or c a l c u l a t i o n of the minimum (r ^ x N ) a n d maximum (r ^ A x ) distance from Pi to S2 must be made pr i o r to s t a r t i n g , and i t i s this range over which i t e r a t i o n s w i l l take place. If inspection of the geometry of the configuration indicate that the obstruction S2 i s limited to some range of 9, 9MAX 2. 9 2. 9MIN' t n e n 3 search of 9 outside of this range i s unnecessary. S i m i l a r l y for 9. - 16 -R e s t r i c t i n g the search ray to ranges of 9 and 9 known to contain the obstruction S2, or more p r a c t i c a l l y eliminating c e r t a i n ranges of 9 and 9 known to be devoid of any obstruction, i s not a necessary step, but i t should be undertaken i n the i n t e r e s t of saving computer time. If the radiating surface i s f i n i t e , then the above procedure must be implemented for each d i f f e r e n t i a l length or area within having a s i g n i f i c a n t l y d i f f e r e n t view of the obstruction, recognizing that each such element has i t s own spherical coordinate system and TMAX a n d rMIN s n o u l d be computed between S 2 and S± o v e r a l l . As well, the f i n a l sum I E E E A F12 (1.8) Si 9 9 must be divided by the length or area of S i . If S2 i s i n f i n i t e i n one of i t s dimensions, a fixed maximum distance between and S2 cannot be given. One a l t e r n a t i v e i s to set l i m i t s on the range of r that depend on the d i r e c t i o n (9, 9) being probed. In i t s implementation this technique i s a numerical integration or summation of f i n i t e integrands, where those integrands are Lambertian-weighted p i x e l s , or s o l i d angles, i . e . , A F 1 2 = cos9 Afi! = cos9 sin9 A9 A9. (1.9) TT/2 2TT The number of di r e c t i o n s probed i s f i n i t e and equals -^g— x . Conceivably d i f f e r e n t l i m i t s on the range of r could be set for each d i r e c t i o n , but the geometry would doubtless suggest s i m p l i f i c a t i o n s . A l l but a few d i r e c t i o n s (those associated with i m p r a c t i c a l l y large r^AX - 17 -could then be set with probe l i m i t s and a d e f i n i t i v e determination of ray in t e r s e c t i o n with S 2 , or not, and hence obstruction of view for that p i x e l , or not, could be made. With most pixels i n the hemisphere reporting t h e i r findings, a good approximation of the view factor could be made. Another al t e r n a t i v e exists for some i n f i n i t e or s e m i - i n f i n i t e S 2. A c e r t a i n class of problems are treatable p r e c i s e l y because the obstruction's perimeter projects onto the unit hemispherical s h e l l such that the projected perimeter can be defined i n a functional r e l a t i o n s h i p between 8 and <f> in the spherical coordinate system of the radi a t i n g surface. For example, consider the view factor of a wall of height H extending to i n f i n i t y i n both d i r e c t i o n s , as seen from a radiating point on the ground some distance d from the wall ( F i g . 1.5). Construct a hemispherical s h e l l , of radius less than d, centered on the radi a t i n g point. Project the l i n e representing the top of the wall onto the hemispherical s h e l l . That projected l i n e , i n the spherical coordinate system of the radiating point, can be expressed i n the functional r e l a t i o n s h i p between 0 and <j>: The base of the wall projects onto the s h e l l as well, the equation of that projected l i n e being t r i v i a l : 9 = (IT/2) - arctan ((H cosc|>)/d), - j < <|> < y. ( 1 . 1 0 ) 8 = TT/2, - < <J> < J . ( 1 . 1 1 ) The s o l i d angle defined by the boundaries of the projected perimeter i s the same s o l i d angle subtended by the wall i t s e l f . Calculation of the view factor may now proceed d i r e c t l y : cos6 sin9 d6 d<)>. (1.12) Numerical integration i s used, represented symbolically as cos 6 si n 9 A8 A<|> (1.13) 17 / M / 1 1 9=f(<t>) The necessity of requiring a functional r e l a t i o n s h i p between 9 and <|> can be seen i n the above equations. In a class of problems meeting this requirement, the s o l i d angle subtended by the obstruction i s calculable and the use of the search ray technique (ray i t e r a t i o n , test of i n t e r s e c t i o n with obstruction) i s circumvented. A l l d i f f e r e n t i a l s o l i d angles within the boundaries are integrated in the view factor computation. Or i n the numerical i n t e g r a t i o n language, a l l p i x e l s , i . e . , Aft = sin9 A9 A<j> (1.14) within the projected perimeter boundaries are counted i n the cumulative sum: cos 9 Aft (1.15) - 19 -Figs. 1.5 and 1.6 give the case of the i n f i n i t e l y long wall, with the sky view factor given as a function of radiating source distance from the wall base, that distance expressed i n units of wall-height. The sky view factor i s the complement of the wall view factor, i . e . , VWALL " 1 " V S K Y ( 1 * 1 6 ) The view factor for this configuration was i n c o r r e c t l y computed by Cochran (1969) i n an ap p l i c a t i o n pertaining to longwave r a d i a t i o n exchange i n a forest clearcut. Another structure treated i n this class of problem i s the v e r t i c a l s t r i p of width W, extending v e r t i c a l l y to i n f i n i t y . If the radiating point on the ground i s on the center l i n e of the s t r i p , i . e . , i f the configuration i s l e f t - r i g h t symmetric, then the projection of the perimeter l i n e s onto the hemispherical s h e l l may be expressed, i n the coordinate system of the radiating point, as 9 = -arctan (w/2d) 9 = -farctan (w/2d) 9 = j, - arctan (w/2d) < 9 <_ +arctan (w/2d) (1.17) Figs. 1.7 and 1.8 give the view factor of the i n f i n i t e v e r t i c a l s t r i p of width w as seen from a d i f f e r e n t i a l area along the ground at a distance from the s t r i p given i n units of s t r i p widths. It i s in t e r e s t i n g to note that i n this p a r t i c u l a r geometry, weighting the integrand by the cos8 factor has no e f f e c t , i . e . , the r a d i a t i o n view factor i s i d e n t i c a l l y - 20 -equal to the s o l i d angle f r a c t i o n of the hemisphere subtended by this structure, or i n other words, the "geometric view factor" equals the r a d i a t i o n view factor. Appendix B i s a l i s t i n g of the computer program that generated the view factors for configurations shown i n Figs. 1.9 and 1.11. - 21 -C. DISCUSSION In graphing the view f a c t o r s o l u t i o n s , d i s c r e e t data were connected using a r a t i o n a l s p l i n e i n t e r p o l a t i o n . Data to which the curves were f i t have been i n d i c a t e d . Since values can g e n e r a l l y be read from graphs to no more than 2 s i g n i f i c a n t d i g i t s , with some un c e r t a i n t y i n the second d i g i t , the standard of p r e c i s i o n sought throughout the program was to o b t a i n data not more than 0.02 i n absolute e r r o r , compared to published values. C e r t a i n of the c o n f i g u r a t i o n s given i n the Case Studies have been examined by other researchers, and t h e i r published s o l u t i o n s , often i n the form of a n a l y t i c expressions, have been consulted where p o s s i b l e . The best source of c o l l e c t e d work on view f a c t o r s of various c o n f i g u r a t i o n s i s Howell (1982). F i g s . 1.10, 1.14, 1.18, and 1.20 have been checked against s o l u t i o n s i n Howell and have been found i n agreement. F i g . 1.12 has been checked against Reifsnyder (1967) and found to agree. F i g . 1.30 was checked against Sparrow and Cess (1978) and found i n agreement. And F i g . 1.6 has been checked against an a n a l y t i c s o l u t i o n developed by Novak (personal communication) and found to agree. The remaining c o n f i g u r a t i o n s and t h e i r view f a c t o r s ( F i g s . 1.7, 1.8; 1.15, 1.16; 1.21, 1.22; 1.23, 1.24; 1.25, 1.26; 1.27, 1.28) have not been p r e v i o u s l y published, i n so f a r as a thorough l i t e r a t u r e search has revealed. The view f a c t o r s f o r these c o n f i g u r a t i o n s have been checked with consistency t e s t s , e.g. the view f a c t o r of any f i n i t e o b s t r u c t i o n as seen from a f i n i t e r a d i a t o r i n f i n i t e l y f a r away must be zero. - 22 -While most view factors presented here were well within the pre c i s i o n goal of 0.02 absolute error, F i g . 1.10 (coaxial disks) has somewhat greater error (as much as 0.05 over part of i t s range). This configuration alone features a radiating surface of f i n i t e area. A 2-dimensional gri d must subdivide the f i n i t e area into small enough "elemental" radiators so that each may be reasonably approximated as a d i f f e r e n t i a l radiator. The numerical view factor c a l c u l a t i o n then proceeds for each elementary radiator, the imbedded i t e r a t i v e loops now 5 l e v e l s deep - each of Ar, A9, A<|>; and A£, Am (representing the radiating surface g r i d parameters). Since the required computer time grows exponentially with each a d d i t i o n a l imbedded loop, r e s t r a i n t i n the demanded resolution of each i t e r a t e d parameter i s prudent. Consequently some prec i s i o n may be s a c r i f i c e d i n computing view factors for configurations with f i n i t e dimensioned radiating surfaces. - 23 -D. CASE STUDIES Radiating d i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , i s displaced a distance d from wall base (d expressed in units of wall height), radiates to wall of height H. - 25 -F i g . 1.6 View factor for F i g . 1.5. - 26 -CO \ F i g . 1.7 Radiating d i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , i s displaced along s t r i p c entre-line a distance d from s t r i p base (d expressed i n units of s t r i p width), radiates to s t r i p of width w. - 2 1 -CC 'o-j—I I 11 mil—I 1 11 I I 11 [Jill I I 11 HH ^10" 2 TO"1 10° 101 102 STRIP-WIDTHS DISTANCE [X/W] F i g . 1.8 View f a c t o r f o r F i g . 1.7. - 28 -D - V r F i g . 1.9 Disk of radius r radiates to coaxial disk of Radius R. Separation distance = z. - 29 -r y 4 4 4 4 ) & • 4 4 4 • • • ' * ' 4 9 9 9 9 # i < 9 9 9 9 9 > t »* LEGEND A E = 1 o E = 2 1 • i i i i i i r 10 -1 10° D 10' F i g . 1.10 View factor for F i g . 1.9. - 30 -E -Fig. 1.11 Radiating differential area dA, with outward normal along z direction, is offset from disk axis by x. The planes of the differential area and the disk are separated by a distance z. - 31 -P < UJ > < < cc LEGEND A E = 1 o E = 2 ••• \ % % * * * 1 I " 1 , , , , 1 % % % % « * % * \ \ t \ % % \ » t *.% 'Jl », v. « « -0.33 0.00 0.33 0.66 D 0.99 1.32 1.65 F i g . 1.12 V i e w f a c t o r f o r F i g . 1 . 1 1 . - 32 -F i g . 1.13 Radiating d i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , i s separated from square of side s by distance z, and displaced from square edge by d i s t ance x. Both d i f f e r e n t i a l area and square are centered with respect to y - d i r e c t i o n . - 33 -0.0 0.1 0.2 0.3 0.4 D F i g . 1.14 View factor for F i g . 1.13. - 34 -F i g . 1.15 Radiating d i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , radiates to 45° - i n c l i n e d square of side s, elevated above plane of dA. both d i f f e r e n t i a l area and square are centered with respect to y - d i r e c t i o n . F i g . 1.16 View factor for F i g . 1.15. - 36 -F i g . 1.17 D i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , radiates to square of side s. - 37 -GC 2 UJ > 5 5 Q < cc LEGEND A E = 0.0 o E = 0.5 " X . '"'a. """ ., '•A, • E * = 1.0 * 4_ ''* * % ' ' • J / i - - a — 0.0 0.4 0 .8 1 .2 1 .6 2. D Fig. 1.18 View factor for Fig. 1.17. - 38 -F i g . 1.19 D i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , r adiates to 45° - i n c l i n e d square of side s. - 39 -' ' ' », LEGEND A E = 0.0 X • * M —* •<« r _ _ * o E - 0.5 % * r~r • _E *J.0_ — «->r •>«i •*-V > \ / - 7< / > 10" 10 -1 10u 10' D Fig 1.20 View factor for F i g . 1.19. - 40 -F i g . 1.21 D i f f e r e n t i a l a r e a d A , w i t h outward norma l a l o n g +z d i r e c t i o n , r a d i a t e s to p a r a b o l a : z = - y 2 + 1. - 41 -GC UJ > Z o r—CM Q < rr * "A 0 .0 LEGEND A E - 0 . 0 , ° . E - . 0 . 5 • E = 1.0 * E = 1.5 3 § 1.0 2.0 X 3.0 4 . 0 F i g . 1.22 View factor for F i g . 1.21. - 42 -F i g . 1.23 D i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , radiates to parabola: z = -y +4. - 43 -CC > 9 Q < CC LEGEND A E = 0 .0 o E = 1.0 • E = 1.5 x E = 2.0 \ \ 1 ''*. * V V ' - X i '•. * *, . * '* 1 * *• » • — A 0 .0 1.0 2.0 X 3.0 4.0 F i g . 1.24 View factor for F i g . 1.23, - 44 -F i g . 1.25 D i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , radiates to i s o s c e l e s t r i a n g l e of height H and base B. - 45 -LEGEND * E = 0.5 o E = 1.0 • E = 2.0 * t s , - 4.U 'l , \ N \ ^ — * H; -V -N ^ N r 'i ': % ^ 0.00 0.25 0.50 0.75 1.00 X . 1.26 View factor for F i g . 1.25. - 46 -F i g . 1.27 D i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , radiates to i s o s c e l e s t r i a n g l e of height H and base B. dA i s displaced from centre l i n e by B/4. - 47 cc P C J 56-< Q < CC LEGEND * E = 0.5 o E = 1.0 • E = 2.0 * E - 4 .0 Sv ; \ i \ \ \ * V \ • 0 . 0 0 0.25 0 .50 D 0.75 1.00 F i g . 1.28. View factor for F i g . 1.27. - 48 -F i g . 1.29 D i f f e r e n t i a l area dA, with outward normal along +z d i r e c t i o n , radiates to c i r c l e of radius R i n Y-Z plane. C i r c l e centre i s displaced a distance z from X-Y plane. - 49 -F i g . 1.30 View factor for F i g . 1.29. E. SUMMARY AND CONCLUSION The v e r s a t i l i t y of this new technique for view factor computation i s demonstrated i n Figs. 1.5 - 1.30. Non-symmetric configurations are solved as e a s i l y as symmetric configurations. The accuracy of the generated view factors has been confirmed without exception i n comparison with available published work. A l l view factors presented here have passed " i n t e r n a l " consistency tests of th e i r behavior at various l i m i t i n g geometries of th e i r configurations. Thus the method to compute r a d i a t i o n view factors presented here i s an accurate and powerful r i v a l and complement to established methods. - 51 -CHAPTER 2 A STUDY OF BARE SOIL EVAPORATION USING MICROLYSIMETERS - 52 -A . INTRODUCTION A s t u d y of e v a p o r a t i o n f rom m i c r o l y s i m e t e r s , w i t h i n a t h e o r e t i c a l f ramework o f a 2 - s t a g e e v a p o r a t i o n m o d e l , i s u n d e r t a k e n h e r e . S o i l p h y s i c a l p r o p e r t i e s a re measured and i n c o r p o r a t e d i n a b a r e s o i l e v a p o r a t i o n m o d e l . C o m p a r i s o n i s made w i t h d a t a o b t a i n e d f r o m f i e l d e x p e r i m e n t s , w h i c h s e r v e d b o t h as a check on c e r t a i n f e a t u r e s of the m o d e l , as w e l l as the measured s o i l p h y s i c a l p r o p e r t i e s . The m i c r o l y s i m e t e r s were sampled i n the Peace R i v e r r e g i o n ( F i g . 2 . 1 ) . A q u a n t i t a t i v e model o f e v a p o r a t i o n f rom Peace R i v e r s o i l s i s p a r t i c u l a r l y i m p o r t a n t . W h i l e c l i m a t o l o g i c a l a n a l y s e s i n d i c a t e an e f f e c t i v e water d e f i c i t o v e r the g rowing s e a s o n , a c o m b i n a t i o n of h y d r o l o g i c , p e d o l o g i c , and c l i m a t i c f a c t o r s can produce w a t e r l o g g e d c o n d i t i o n s , e s p e c i a l l y i n s p r i n g and f a l l . S p r i n g me l t r u n o f f o f a c c u m u l a t e d snow, and v e r y low e v a p o r a t i v e p o t e n t i a l s i n s p r i n g and f a l l , pose w a t e r l o g g i n g h a z a r d s and poor t r a f f i c a b i l i t y , thus d e l a y i n g or p r e v e n t i n g s p r i n g p l a n t i n g and f a l l h a r v e s t i n g o p e r a t i o n s . The s o i l s a r e h i g h i n c l a y c o n t e n t and t h u s a re not w o r k a b l e or t r a f f i c a b l e at a p p r e c i a b l e water c o n t e n t s w i t h o u t r i s k i n g damage t o s o i l s t r u c t u r e . T h e i r f i n e t e x t u r e and t y p i c a l l y h i g h b u l k d e n s i t y i m p l y a low h y d r a u l i c c o n d u c t i v i t y w h i c h r e d u c e s i n f i l t r a b i l i t y and impedes i n t e r n a l d r a i n a g e . Add to t h i s the s h a l l o w d e p t h of the Ap h o r i z o n c o u p l e d w i t h a m a s s i v e , low p e r m e a b i l i t y B h o r i z o n , and the p o t e n t i a l f o r w a t e r l o g g i n g becomes a p p a r e n t . In a d d i t i o n to o p e r a t i o n a l p rob lems posed to the f a r m e r , t h e s e s o i l s a r e a t r i s k of e r o d i n g . O v e r l a n d f l o w , f i n e t e x t u r e d and weak ly - 53 -fPrinct Ctorgt tStJetm* I ALBERTA F i g . 2.1 Peace R i v e r R e g i o n - 54 -aggregated s o i l , and the practice of summer fallowing, lead to serious erosion of the very shallow cu l t i v a t e d horizon. The amount of runoff, and hence erosion, from the high i n t e n s i t y convective r a i n f a l l events that t y p i c a l l y occur i n the region depends upon many f a c t o r s , among them the s o i l moisture content at the beginning of the rainstorm. This i n turn depends upon evaporative losses since the l a s t rainstorm. Thus the evaporation process impinges i n d i r e c t l y upon the serious s o i l erosion problem e x i s t i n g i n the Peace River region. Drainage systems are an active but expensive amelioration of the farmers problems. Plant t r a n s p i r a t i o n and bare s o i l evaporation are natural and ongoing processes working to the a l l e v i a t i o n of these problems. Thus the quantitative modelling of evaporation for Peace s o i l s i s a contribution to the solu t i o n of the unique problems a f f e c t i n g Peace region a g r i c u l t u r e (Chanasyk et a l . , 1983). The objectives of this study were: (1) To measure evaporation under f i e l d conditions using undisturbed microlysimeter cores. (2) To measure hydrologic properties relevant to the evaporation process (hydraulic d i f f u s i v i t y , water retention c h a r a c t e r i s t i c ) . (3) To use the measured hydraulic d i f f u s i v i t y to predict Stage I and Stage II evaporation using a 2-stage evaporation model. - 55 -B. THEORETICAL CONSIDERATIONS Water movement through unsaturated s o i l during evaporation i s w e l l described by: where z i s the v e r t i c a l coordinate ( p o s i t i v e downward), t i s time, 8(z,t) i s the volumetric water content, and D(9) i s the h y d r a u l i c d i f f u s i v i t y f u n c t i o n . Equation (2.1) i s a n o n - l i n e a r , second order, time dependent, p a r t i a l d i f f e r e n t i a l equation. The second order and time dependence r e q u i r e the s p e c i f i c a t i o n of 2 boundary c o n d i t i o n s and an i n i t i a l c o n d i t i o n f or complete s o l u t i o n . The dependence of d i f f u s i v i t y on water content makes (2.1) non-linear. The e f f e c t of g r a v i t y i s neglected i n (2.1) (Gardner, 1959), and the s o i l i s assumed to be isothermal. On a d i u r n a l b a s i s the evaporation process i s s t r o n g l y non-isothermal (Idso et a l . , 1974), but P h i l i p (1957) concluded that the isothermal assumption i s an acceptable approximation on t h e o r e t i c a l grounds provided that the surface 30 cm i s not extremely dry. This requirement i s f e l t to have been s u b s t a n t i a l l y met throughout t h i s study. The drying of a s o i l subject to a constant p o t e n t i a l evaporation (PE) proceeds i n 2 stages (Gardner and H i l l e l , 1962). Stage I , synonomously p o t e n t i a l or energy-limited evaporation, i s defined here as the adherence to the boundary c o n d i t i o n : - 56 -- 0 (6 ) | | 8z |z=0 = F(0,t) = - PE (2.2) where F(z,t) i s introduced as a notational shorthand for the fl u x . Stage II , synonomously f a l l i n g - r a t e or s o i l - l i m i t e d evaporation, i s defined here as adherence to the boundary condition: where 6 a i s the air - d r y s o i l water content. During stage I evaporation the surface dries from i t s i n i t i a l value, 9^, to 9 a. The hydraulic d i f f u s i v i t y at the surface decreases by orders of magnitude during t h i s surface drying while the moisture gradient at the surface increases - exactly o f f s e t t i n g the f a l l i n g d i f f u s i v i t y such that t h e i r product i s constant, as i n (2.2). When the surface water content just reaches 9 a, the d i f f u s i v i t y at the surface equals D(9 a). The surface moisture gradient i s then at a maximum, and further drying from subsurface s o i l r e s u l t s i n a smaller gradient at the surface. Thus the product of the surface d i f f u s i v i t y with the surface moisture gradient must decrease i n magnitude, and boundary condition (2.2) i s v i o l a t e d . Therefore drying of the surface to 9 a (boundary condition (2.3)), marks the beginning of stage II evaporation. The time at which this t r a n s i t i o n occurs i s designated t a , and may be referred to as the t r a n s i t i o n time. Stage II evaporation proceeds independent of the actual PE, provided PE remains greater than the s o i l ' s capacity to d e l i v e r water to the 9(0,t) = 9 a (2.3) - 57 -surface. Indeed, the mathematical constraint of stage II evaporation, solving (2.1) subject to (2.3), makes no reference to PE. (The e f f e c t of PE i n d i r e c t l y enters through the " i n i t i a l condition" for stage I I , i . e . the p r o f i l e of 9 at t = t a ) . Consequently, any PE (PE >| F(0,t) | ) , regardless of i n i t i a l conditions and actual PE, may be substituted for the actual PE i n a simulation of stage II evaporation. In p a r t i c u l a r , PE = °° may be chosen as pr e v a i l i n g during stage I I . The l i m i t a t i o n s usually associated with any lysimeter study l i e i n the a r t i f i c i a l lower boundary imposed, i . e . - D ( 9 ^ | Z = L = F ( L ' t ) = 0 ( 2 ' 4 ) where L i s the lysimeter length. The a r t i f i c a l lower boundary w i l l , a f t e r some time, reduce evaporation at the lysimeter surface below that from undisturbed s o i l . S o i l s of the Peace River region consist on average of a 15 cm A-horizon overlying a low conductivity B-horizon, thus boundary condition (2.4), with L = 15 cm, i s a very good approximation to the region's hydrology. Lysimeter core extractions extending to the B-horizon, under si m i l a r PE and i n i t i a l water content, are expected to f a i t h f u l l y reproduces the f i e l d s i t u a t i o n , with no l i m i t on the run time. - 58 -C. METHODOLOGY 1. S o i l s and Sampling S o i l samples were col l e c t e d from 3 locations i n the Peace River region: Beaverlodge, Alberta; Dawson Creek, B r i t i s h Columbia; and Fort St. John, B r i t i s h Columbia ( F i g . 2.1). The s o i l s at a l l 3 locations are c l a s s i f i e d (Van V l i e t , 1984) as Landry series s i l t loams, of the Solodized Solonetz great group, moderately to imperfectly drained and developed on l a c u s t o - t i l l parent material. The surface Ah horizon averages 15 cm and i s f a i r l y high i n organic matter. Sampling consisted of extracting i n t a c t cores for subsequent laboratory analysis and f i e l d experimentation to be performed at the University of B r i t i s h Columbia (UBC). These cores were taken from the surface s o i l of erosion plots at a l l 3 s i t e s , under various cropping system: perennial grasses, annual row crops, and fallow. Uncultivated s o i l s from adjacent areas were also sampled for comparison. S o i l samples were also c o l l e c t e d from the UBC Dept. of Plant Science Research Station. This s o i l i s c l a s s i f i e d (Luttmerding, 1980) as a Bose series sandy loam of the Duric Humo-Ferric Podzol sub-group, with a 20-70 cm Ap below which l i e s a hard pan of compacted g l a c i a l t i l l . Samples from the surface s o i l were extracted for lab analysis and f i e l d experiments concurrent with the Peace Region s o i l s . Cores of dimension 3.0 cm high by 5.4 cm di a . were extracted for use i n pressure plate extractor measurements of moisture retention c h a r a c t e r i s t i c (Appendix G). Cores of dimension 6.0 cm high by 5.4 cm - 59 -d i a . were e x t r a c t e d i n t a c t f o r use i n Tempe c e l l o u t f l o w e x p e r i m e n t s and s u b s e q u e n t c o m p u t a t i o n of h y d r a u l i c d i f f u s i v i t y ( A p p e n d i x C ) . C o r e s o f d i m e n s i o n 15.3 cm h i g h by 7.3 cm d i a . were e x t r a c t e d i n t a c t f o r use as m i c r o l y s i m e t e r s under n a t u r a l v a r i a t i o n s of PE i n a f i e l d e x p e r i m e n t c o n d u c t e d at UBC ( F i g s . 2 .5 - 2 . 9 ) . 2 . Laboratory Methods H y d r a u l i c d i f f u s i v i t y f u n c t i o n s were d e t e r m i n e d f rom the a n a l y s i s o f d a t a f rom o u t f l o w e x p e r i m e n t s u s i n g the method of P a s s i o u r a ( 1 9 7 6 ) . From the raw d a t a of Tempe c e l l o u t f l o w v s . t i m e , the h y d r a u l i c d i f f u s i v i t y was c a l c u l a t e d a t s e l e c t e d water c o n t e n t s . A d i f f u s i v i t y f u n c t i o n o f the f o r m : D(8) = exp[B6 + C] ( 2 . 5 ) where 3 and C a re c o n s t a n t s , was f i t to the measured v a l u e s . The measured v a l u e s ranged f rom a wa te r c o n t e n t n e a r s a t u r a t i o n down to a water c o n t e n t e q u i l i b r a t e d to 1 bar t e n s i o n ( t h e a p p l i e d p r e s s u r e ) . S i n c e d i f f u s i v i t y v a l u e s o b t a i n e d n e a r the wet end were d i s t o r t e d by the f i n i t e c o n d u c t i v i t y of the c e r a m i c p l a t e , the d i f f u s i v i t y f u n c t i o n was f i t to o n l y t h o s e p o i n t s f o r w h i c h the d rop i n p r e s s u r e a c r o s s the p l a t e was l e s s than one t e n t h of a p p l i e d p r e s s u r e . T h i s c r i t e r i a a s s u r e d t h a t the d e s o r p t i o n p r o c e s s had p r o g r e s s e d to the s t a g e where P a s s i o u r a ' s a s s u m p t i o n s were v a l i d and the a n a l y s i s f o r d i f f u s i v i t y was c o r r e c t . Once a d i f f u s i v i t y f u n c t i o n had been d e t e r m i n e d , the range of water c o n t e n t o v e r w h i c h the d i f f u s i v i t y f u n c t i o n was a p p l i e d was e x t r a p o l a t e d to c o v e r the range measured i n l y s i m e t e r s i n f i e l d e x p e r i m e n t s . W h i l e no - 60 -d i f f u s i v i t y measurements were made near a i r - d r y moisture content, Hanks and Gardner (1965) showed that the exact form of the d i f f u s i v i t y function at the dry end has l i t t l e e f f e c t on simulated evaporation f l u x . These measured d i f f u s i v i t i e s , categorized by s i t e and cropping p r a c t i c e , were then used to simulate stage II evaporation from lysimeters of the same category, and comparison with actual measured flux rates were made. As an example, F i g . 2.2 gives the experimental d i f f u s i v i t y data to which was f i t an exponential function of the form (2.5) for the Fort St. John perennial s i t e . Appendix C contains a l l the experimental d i f f u s i v i t y data, and Appendix D l i s t s a l l the d i f f u s i v i t y functions obtained from the data. The hypothesized exponential r e l a t i o n s h i p between d i f f u s i v i t y and water content i s s u b s t a n t i a l l y supported throughout the measurements, as indicated by the good str a i g h t l i n e f i t to d i f f u s i v i t y data plotted on a semi-log graph. As supplementary information, moisture retention c h a r a c t e r i s t i c s were measured for the undisturbed samples of various cropping p r a c t i c e from the 3 Peace River l o c a t i o n s , as well as the UBC Bose s o i l . The measurements were made using a pressure plate extractor, applying pressures ranging from 0.025 bar to 15.0 bar. Measurements were also made for repacked (disturbed). cores. Appendix G discusses the retention curves i n d e t a i l . 3. F i e l d Experiments The f i e l d study was conducted on the UBC Dept. of Plant Science Research Station. The study plot served as a host medium for the Peace —' 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 VOLUMETRIC MOISTURE CONTENT 0.60 F i g . 2.2 D i f f u s i v i t y f u n c t i o n : F o r t S t . J o h n - P e r e n n i a l . - 62 -microlysimeters, as well as an evaporating s o i l i n which flux and moisture p r o f i l e measurements were made. The plot was r o t o - t i l l e d followed by compaction on half of the s i t e with a weighted r o l l i n g compactor. The s i t e (approx. 30 m by 30 m) was then sprinkle i r r i g a t e d for 2 days followed by 2 days of drainage i n an e f f o r t to achieve f i e l d capacity conditions. ( A l l references to day 1 of the study r e f e r to the day a f t e r the 2 drainage days). With the host s o i l at f i e l d capacity, advective enhancement to the lysimeters was reduced. The lysimeters from the Peace region were brought to saturation i n the laboratory, then allowed to drain for 1 day. The lysimeters were then placed i n the center of the f i e l d (concurrent with day 1 of the study) into pre-formed holes of the same dimensions as the lysimeters. Lysimeters cored from the host Bose s o i l were run concurrently with the Peace lysimeters. The bulk densities of the compacted and r o t o - t i l l e d treatments were 1.2 g/cm and 1.0 g/cm , respectively. Lysimeters were cored from both treatments. However, i n this experiment, the lysimeters could not be l e g i t i m a t e l y used for more than 2 days (Boast and Robertson, 1982), due to the problem with boundary condition at the bottom of a lysimeter. Consequently, new lysimeters of the Bose s o i l were re-cored on alternate days. This provided a comparison of evaporation rates from the same s o i l at 2 bulk d e n s i t i e s . Moisture p r o f i l e s of the top 20 cm of both treatments were measured at the conclusion of each day. The pre-formed lysimeter holes were li n e d with a p l a s t i c bag to prevent any d i r e c t contact between lysimeter and host s o i l , and the - 63 -lysimeter i t s e l f was sealed (except at i t s evaporating surface) i n a p l a s t i c bag. This double bag method (Boast and Robertson, 1982) ensured that no s o i l or water from the host s o i l contaminated the lysimeter. The lysimeters were then subject to the ambient evaporative regime for 13 days (10 Aug 85 - 22 Aug 85). Weighings were taken at least twice per day over the period on a Sartorius (model 1264 MP) e l e c t r o n i c balance, g i v i n g readings to the nearest hundreth gram. The measured weight included the inner p l a s t i c bag. Differences i n weight were due to evaporation losses. P o t e n t i a l Evaporation was measured (PE^^g) from the weight changes of a lysimeter of the Bose sandy loam s o i l kept near saturation by d a i l y additions of l o s t water. 4 . M o d e l l i n g E v a p o r a t i o n Modelling of the evaporation process e n t a i l s predicting the duration of stage I, and simulating the f a l l i n g flux rate of stage II. Covey (1963) and l a t e r Zachman et a l . (1980) predict the t r a n s i t i o n time f o r f i n i t e depth s o i l columns given the (exponential) d i f f u s i v i t y function, PE, the i n i t i a l and a i r - d r y s o i l water content, and the s o i l depth. They numerically solved (2.1) subject to (2.2) and (2.4). The approach taken here i n modelling stage II evaporation from a f i n i t e column lysimeter i s a v a r i a t i o n on a Gardner and H i l l e l (1962) technique. They had shown that a reasonably good f i t to stage II cumulative evaporation (E) can be obtained by t r a n s l a t i n g (along the time axis) the t h e o r e t i c a l drying curve associated with an i n f i n i t e PE acting - 64 -on the same column ( F i g . 2.3). This t r a n s l a t i o n w i l l give the correct t o t a l cumulative evaporation but only approximate evaporation rates, given by the time d e r i v a t i v e , or slope, of the cumulative evaporation curve. Fi g . 2.4 shows a hypothetical data set of a lysimeter evaporative flux rate vs. time. The data (X's) are divided into the 2 stages of evaporation. Superimposed on the plot i s shown the simulated flux curve that would r e s u l t had that same lysimeter been subjected to an i n f i n i t e PE. Analogous to the Gardner and H i l l e l procedure, t h i s simulated flux curve ( l a b e l l e d PE = °°) i s translated along the time axis (shown) u n t i l i t i n t e r s e c t s the point: ( t a , PEMEASURED)* The flux function generated under i n f i n i t e PE i s an approximate a n a l y t i c s o l u t i o n of (2.1) subject to (2.3) and (2.4), with 9 = 9-^  at t = 0. The so l u t i o n , developed by Novak (personal communication), requires a known exponential d i f f u s i v i t y function, the i n i t i a l and ai r - d r y s o i l water content, and the lysimeter length. According to Novak, for t < t m F(0,t) = -D(9 ) (0 - 6 ) (e G - 1) a l a (2.6a) and for t > t ^ , F(0,t) = f ( t ) L (2.6b) where f ( t ) i s i m p l i c i t l y given by T I me. F i g . 2.4 Modelling stage II f l u x . - 67 -0 . 9 9 9 8 r L 2 ( 1 - e ) ° 2 D ( 9 a ) ( 6 . - 9 a ) < - - - s n r y ^ T ^ T ; ^ G 3„ L 2 f ( t ) 1 - G 2D(9 ) (9 - 8 ) + * * [ 0 . 9 9 9 3 , ] > ( 2 - 7 > e G The t i m e , t j , , i s a m a t c h i n g t ime t h a t d e t e r m i n e s when the column can no l o n g e r be c o n s i d e r e d s e m i - i n f i n i t e , i . e . when the " d r y i n g f r o n t " r e a c h e s the bot tom of the column ( d e f i n e d by [ 9 ( L , t ) - 9 ] = 0 .999 (9., - 9 ) ) . m a i a F o r the s e m i - i n f i n i t e p a r t o f the s o l u t i o n ( t < t^j), G a r d n e r ' s ( 1 9 5 9 , e q . 14) s o l u t i o n i s assumed, w h i c h y i e l d s an i m p l i c i t e q u a t i o n f o r t m , i . e . , 0 . 9 9 9 3 r T e - 1 e r f [ ^ £ ±- ( 2 . 8 ) e G f~V e " - 1 / 2 ( e G + I] a m The p a r a m e t e r 3Q IS d e f i n e d by the g i v e n e x p o n e n t i a l d i f f u s i v i t y f u n c t i o n , i . e . 8 ( 9 - 9 ) D(9) = D ( 9 a ) exp[ ^ e _ Q a ^ ( 2 . 9 ) i a so t h a t 8 G = ( 9 ± - 9 a ) 3 . - 68 -D. RESULTS 1. General Comments The 24-hour average evaporation rates measured from the lysimeters are presented in Figs. 2.5-2.9. The 2 stages of evaporation emerge c l e a r l y for most samples. During stage I, the samples evaporated at rates within 20% of PE and followed the day to day v a r i a t i o n s i n PE. Fluxes were computed assuming that the f u l l surface area of the lysimeter core (42 cm ) acted as an evaporating surface. Since these samples were extracted as intact cores, stones or other obstructions to the s o i l conductivity may have e f f e c t i v e l y reduced the usable evaporation surface. This may account for stage I fluxes being consistently below measured PE. Gardner and H i l l e l (1962) observed in t h e i r studies that a s a l t crust often forms, even i n r e l a t i v e l y s a l t - f r e e s o i l s , causing a reduction i n evaporation rate. The Peace s o i l s are t y p i c a l l y s a l t - a f f e c t e d , although no obvious crust was observed on lysimeter surfaces. 2 . P r e d i c t i n g Stage I Duration The duration of stage I evaporation could be resolved by graphic i n t e r p o l a t i o n to within 8 hours at best, and usually no worse than to within 24 hours. The measured t r a n s i t i o n time was then compared to a predicted t r a n s i t i o n time generated from the numerical r e s u l t s of Zachman et a l . (1980). Fig. 2.10 compares the agreement of measured values with o o 00 o X LU > QC o Q. o to o o CO o or o d o • x o • X o X X • o 8 o I LEGEND ^ EVAP. POTENTIAL o B ANNUAL 1 • BANNUAL 2 x B FALLOW 1 o B FALLOW 2 • 8 i i I l l I I I I I I I I 0.0 24.0 48.0 72.0 96.0 120.0 144.0 168.0 192.0 216.0 240.0 264.0 288.0 312.0 HOURS F i g . 2 .5 B e a v e r l o d g e annua l and f a l l o w s a m p l e s . 9 • • o • o • o o • I § I a LEGEND A EVAP. POTENTIAL o B UNCULTIVATED 1 • B UNCULTIVATED 2| • B PERENNIAL 1 • B PERENNIAL 2 o I e 1 1 1 1 1 1 1 1 1 1 1 1 0.0 24.0 48.0 72.0 96.0 120.0 144.0 168.0 192.0 216.0 240.0 264.0 288.0 312.0 HOURS Fig. 2.6 Beaverlodge uncultivated and perennial samples. o CO O CO o <5 l h X 3 q ET! 1 0 UJ > GC O CL m o c\i O O co O ci o X + -tr-8 X o • X o • + o o • + o o • + o X LEGEND A EVAP. POTENTIAL o D UNCULTIVATED o D PERENNIAL 1 x D PERENNIAL 2 o D FALLOW 1 + D FALLOW 2 o o o X o + 1 1 1 — 0.0 24.0 48.0 72.0 8 + —T 1 1 1 1 1 1 1 1 96.0 120.0 144.0 168.0 192.0 216.0 240.0 264.0 288.0 312.0 HOURS Fig. 2.7 Dawson Creek uncultivated, perennial, and fallow samples. X • © o • o X o • X o X o o X 8 o LEGEND A EVAP. POTENTIAL o F UNCULTIVATED o F PERENNIAL 1 o F PERENNIAL 2 x F ANNUAL 1 * F FALLOW 2 • 8 + T T 0.0 24.0 48.0 72.0 I 1 I 1 1 1 1 1 1 96.0 120.0 144.0 168.0 192.0 216.0 240.0 264.0 288.0 312.0 HOURS Fig. 2.8 Fort St. John uncultivated, perennial, annual, and fallow samples. -A <r LEGEND A EVAP. POTENTIAL o UBC COMPACT 1 o UBC COMPACT 2 • UBC TILLED 1 • UBC TILLED 2 —I 1 1 1 1 1 1 1 1 96.0 120.0 144.0 168.0 192.0 216.0 240.0 264.0 288.0 312.0 0.0 24.0 48.0 72.0 HOURS Fig. 2.9 UBC Bose series s o i l at 2 bulk d e n s i t i e s . - 74 -p r e d i c t e d v a l u e s u s i n g the s e p a r a t e l y measured d i f f u s i v i t y f u n c t i o n s . The p r e d i c t e d t imes a re of the c o r r e c t o r d e r of m a g n i t u d e . The v a r i a n c e 2 ( a ) o f ( p r e d i c t e d t ime minus measured t ime) was c a l c u l a t e d to be 1.74 2 days , about a mean of 1.13 d a y s , w i t h the d a t a t a k e n to be a p o p u l a t i o n ( N - w e i g h t i n g was used as the p o p u l a t i o n p a r a m e t e r ) . D e f i n i t e b i a s can be seen w i t h the p r e d i c t e d v a l u e s b e i n g o v e r - e s t i m a t e d . The a c t u a l v a l u e s o f t r a n s i t i o n t ime f o r Peace s o i l s i n s i t u may d i f f e r f rom t h o s e measured h e r e due to PE b e i n g d i f f e r e n t i n the 2 r e g i o n s . Chanasyk and Woytowich (1984) r e p o r t mean PE f o r the Peace R i v e r r e g i o n to be i n the 3 .5 - 4 .0 mm/d range d u r i n g the k i l l i n g f r o s t f r e e p e r i o d . 3. Modelling Stage I I Flux M o d e l l i n g s t a g e I I e v a p o r a t i v e f l u x i s shown i n F i g s . 2 . 1 1 - 2 . 1 8 . The s i m u l a t e d f l u x c u r v e s were t r a n s l a t e d a l o n g the t ime a x i s and f i x e d at the t ime t h a t gave the b e s t f i t t o the f i r s t few d a t a of s t a g e I I . As the d a t a were 2 4 - h o u r a v e r a g e f l u x e s , the t r a n s i t i o n t ime c o u l d g e n e r a l l y be s p e c i f i e d to w i t h i n a one day i n t e r v a l . The a c t u a l t r a n s i t i o n t ime c o u l d have o c c u r r e d anywhere w i t h i n t h a t r a n g e . C o n s e q u e n t l y , the t ime to w h i c h the c u r v e was f i x e d was a v i s u a l judgement of the t r e n d f rom s e v e r a l d a t a near the s t a g e I / s t a g e II i n t e r f a c e . The model o f s t a g e II e v a p o r a t i o n s h o u l d be j u d g e d then more on i t s a b i l i t y to p r e d i c t the o v e r a l l c u r v a t u r e o f the f a l l i n g f l u x , and on i t s agreement w i t h measured f l u x at l o n g t i m e s , when f l u x a p p e a r s n e a r l y c o n s t a n t . - 75 -0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 OBSERVED TRANSITION TIME [days] F i g . 2 .10 D u r a t i o n o f s t a g e I e v a p o r a t i o n . - 76 -The curves i n Figs. 2.11-2.18 are modelled from a s t a r t i n g water content equal to that i n the corresponding lysimeter. The agreement i s remarkably good. Both the shape of the simulated f a l l i n g rate curve, and i t s magnitude near the end of the 13 day study period are strongly supported by the data. 4 . Modelling Moisture P r o f i l e s The measured time course of volumetric moisture content for 7 depths of the Bose f i e l d s o i l during the study i s shown i n Figs. 2.19 and 2.20 for the 2 bulk d e n s i t i e s . The f i e l d was assumed to have a uniform i n i t i a l moisture content over i t s extent and to the depth of the c u l t i v a t e d horizon, which varied from approximately 40 to 70 cm. This assumption allowed random p r o f i l e sampling d a i l y during the study. Further assumptions were made to allow the use of an approximate so l u t i o n to predict moisture p r o f i l e developed by Gardner (1959, eq. 14), i . e . The s o i l depth was taken to be s e m i - i n f i n i t e , and PE was taken to be i n f i n i t e . Figs. 2.21 and 2.22 show the predicted values of 8 at the 1 and 17 cm depths. Agreement i s poor, judging from a v i s u a l inspection, with the predicted p r o f i l e much wetter than the measured at long times. It i s not thought that the above assumptions caused such gross discrepancies. Indeed, the assumption of i n f i n i t e PE i s the more at 1 c 8 = 6 + -i- + (e - 1) erf [ z 1} (2.10) o CO o CO o . c . « o X Z> o ul LU > O CC ° S-m o o d o •. o o" o LEGEND EVAP. POTENTIAL B ANNUAL 1 B-A-1 THEORETICAL • B ANNUAL 2 B-A-2 THEORETICAL a... •'b~ Q • 1 1 1 1 1 1 1 1 1 1 1 1 1 0.0 24.0 48.0 72.0 96.0 120.0 144.0 168.0 192.0 216.0 240.0 264.0 288.0 312.0 HOURS F i g . 2.11 B e a v e r l o d g e a n n u a l . © X x>-. LEGEND EVAP. POTENTIAL x B FALLOW 1 . J?:^ :.1 .THE_9.R.ITi9A|r. o B FALLOW 2 B-F-2 THEORETICAL 8 © x 1 1 1 1 1 1 1 1 1 1 1 1 0.0 24.0 48.0 72.0 96.0 120.0 144.0 168.0 192.0 216.0 240.0 264.0 288.0 312.0 HOURS F i g . 2.12 B e a v e r l o d g e f a l l o w . o. » o '"ti-LEGEND A EVAP. POTENTIAL o B UNCULTIVATED 1 B-N-1 THEORETICAL • B UNCULTIVATED 2 B-N-2 THEORETICAL -o • e 1 1 1 1 1 1 1 1 1 1 1 1 0.0 24.0 48.0 72.0 96.0 120.0 144.0 168.0 192.0 216.0 240.0 264.0 288.0 312.0 HOURS F i g . 2.13 Beaverlodge uncultivated. o o 00 o I — 1 l < T3 i" .C. CD X erf 1 0 LU > O CC 0 S-CL 00 Qj O oi O b LEGEND EVAP. POTENTIAL B PERENNIAL 1 B-P-1 THEORETICAL B PERENNIAL 2 B-P-2 THEORETICAL • 1 1 1 1 1 1 1 1 1 1 1 1 0.0 24.0 48.0 72.0 96.0 120.0 144.0 168.0 192.0 216.0 240.0 264.0 288.0 312.0 HOURS 00 o F i g . 2.14 Beaverlodge perennial. o o 0 0 O i — i l< .C. CO X UJ > o GC o S Cu n  m O CN o 0 o «o .+.. LEGEND EVAP. POTENTIAL o D FALLOW 1 D-F-1 THEORETICAL + D FALLOW 2 D-F-2 THEORETICAL mr>-+ © + "1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.0 24.0 48.0 72.0 96.0 120.0 144.0 168.0 192.0 216.0 240.0 264.0 288.0 312.0 HOURS I CO F i g . 2 .15 Dawson Creek f a l l o w . q co O 00 o X UJ > o GC ° S CL 00 DJ q CN O b A -I • • • X • • X LEGEND EVAP. POTENTIAL • D PERENNIAL 1 . .P:f;lTli§9.^IlP.AL. x D PERENNIAL 2 D-P-2 THEORETICAL • X . 1 1 1 I 1 1 1 1 1 1 1 1 0.0 24.0 48.0 72.0 96.0 120.0 144.0 168.0 192.0 216.0 240.0 264.0 288.0 312.0 HOURS 00 S3 Fig. 2.16 Dawson Creek perennial. x» + •a + LEGEND A EVAP. POTENTIAL x F ANNUAL .F.-A ™ORETICAL + F FALLOW F-F THEORETICAL X X + X + 1 1 1 1 1 1 1 1 1 1 1 1 1 0.0 24.0 48.0 72.0 96.0 120.0 144.0 168.0 192.0 216.0 240.0 264.0 288.0 312.0 HOURS F i g . 2.17 Fort St. John annual fallow. o oi O oo O 1SH LTJ 1X5 LU > O t r O P Q_ CO m o CN o o a o o 'A \ 0 • LEGEND EVAP. POTENTIAL • F PERENNIAL 1 F-P-1 THEORETICAL F PERENNIAL 2 F-P-2 THEORETICAL • © 8 9 e —I 1 1 1 1 1 1 1 1 1 1 1 0.0 24.0 48.0 72.0 96.0 120.0 144.0 168.0 192.0 216.0 240.0 264.0 288.0 312.0 HOURS Fig. 2.18 Fort St. John perennial. - 85 -v a r i a n c e w i t h r e a l i t y , y e t t h i s e r r o r would tend t o p r o d u c e a p r o f i l e t h a t i s too d r y , e s p e c i a l l y n e a r the s u r f a c e . The measured d i f f u s i v i t y f u n c t i o n , used i n the G a r d n e r a n a l y s i s , i s s u s p e c t . The d i s c r e p a n c y between measured and p r e d i c t e d p r o f i l e s c o u l d be a c c o u n t e d f o r i f the p a r a m e t e r 3 i n the d i f f u s i v i t y f u n c t i o n were too l a r g e . Such a l a r g e B would p r e d i c t the d e l i v e r y o f water to the e v a p o r a t i n g s u r f a c e at e x c e s s i v e r a t e s , g i v i n g a m o i s t u r e p r o f i l e t h a t was too wet n e a r the s u r f a c e . LEGEND A 0-2 CM 1 l 1 1 1 1 1 1 1 1 1 1 1 1 1 -24.0 0.0 24.0 48.0 72.0 96.0 120.0 144.0 168.0 192.0 216.0 240.0264.0288.0 312.0 HOURS Time c o u r s e of v o l u m e t r i c water c o n t e n t f o r Bose s o i l at b u l k d e n s i t y 1.2 g / c m ^ . LO d ti CO CJ CO CJ I-!-r CO cj LU GC (/) d O CJ cc in o O > o d LEGEND A 0-2 CM o 2-4 CM • 4-6 CM • 6-8 CM a 8-10 CM B 10-14 CM • 14-20 CM — i r~ 1 1 1 1 1 1 1 1 1 1 1 -24.0 0.0 24.0 48.0 72.0 96.0 120.0 144.0 168.0 192.0 216.0 240.0264.0288.0 312.0 HOURS F i g . 2 .20 Time c o u r s e of v o l u m e t r i c water c o n t e n t f o r Bose s o i l a t b u l k d e n s i t y 1.0 g / c m ^ . ID d d CO d CM d -| C C h- £ U J o -O > o o J LEGEND 0-2 CM o 14-20 CM 00 00 -24.0 0.0 24.0 48.0 72.0 — I 1 1 1 1 1 1 1 I 1 96.0 120.0 144.0 168.0 192.0 216.0 240.0264.0288.0 312.0 HOURS F i g . 2.21 Predicted time course of volumetric water content for Bose s o i l at bulk density 1.2 g/cm^. i 1 1 1 1 1 1 ~\ 1 I r -24.0 0.0 24.0 48.0 72.0 96.0 120.0 144.0 168.0 192.0 216.0 240.0264.0288.0 312.0 00 V O HOURS F i g . 2.22 Predicted time course of volumetric water content for Bose s o i l at bulk density 1.0 g/cm^* - 90 -E. DISCUSSION 1. D i f f u s i v i t y Measurements P a s s i o u r a (1976) found 8 - 5 0 f o r a c l a y s o i l . G a r d n e r and H i l l e l (1962) found 6 = 18.5 f o r t h e i r sandy l o a m . The mean 8 f o r the 13 d i f f u s i v i t i e s measured f o r Peace s i l t loam s o i l was 3 2 . 7 , w i t h a sample s t a n d a r d d e v i a t i o n of 8 . 3 . T h i s v a l u e i s w e l l p l a c e d on the b a s i s o f i t s i n t e r m e d i a t e t e x t u r e . The 3 measurements f o r Bose sandy loam s o i l had a mean 3 = 44 .9 w i t h a sample s t a n d a r d d e v i a t i o n of 8 . 0 . T h i s i s i n c o m p l e t e d i s a g r e e m e n t w i t h the G a r d n e r and H i l l e l 3 = 18.5 f o r the same t e x t u r e . I n s p e c t i o n of the sandy loam d i f f u s i v i t y c u r v e s t o g e t h e r w i t h the d a t a to wh ich t h e y were f i t , F i g s . 2 . 2 3 - 2 . 2 5 , shows t h a t a l l 3 c u r v e s were f i t to o n l y 4 p o i n t s , and i n a l l c a s e s the d r i e s t end d a t a p o i n t s t r o n g l y b i a s e s the c u r v e toward g r e a t e r s l o p e , w h i c h i s the p a r a m e t e r 3. The a c c u r a c y o f d e t e r m i n i n g the d i f f u s i v i t y f u n c t i o n f rom a n a l y s i s o f o u t f l o w e x p e r i m e n t s w a r r a n t s f u r t h e r s c r u t i n y h e r e . The P a s s i o u r a (1976) a n a l y s i s p r o v i d e s a method t o f i t a d i f f u s i v i t y f u n c t i o n to t h a t s u b s e t o f the o u t f l o w d a t a f o r w h i c h the c o n d u c t i v i t y o f the s o i l was more l i m i t i n g than the c e r a m i c p l a t e , and f o r w h i c h the d r y i n g p u l s e had t r a v e l l e d to the f a r b o u n d a r y , i . e . , the sample no l o n g e r behaved as s e m i - i n f i n i t e . I t was c o n s i d e r e d t h a t i g n o r i n g a l l d a t a f o r w h i c h the p r e s s u r e d rop a c r o s s the c e r a m i c p l a t e was g r e a t e r than one t e n t h o f t h e a p p l i e d p r e s s u r e would be s u f f i c i e n t to i n s u r e t h a t r e m a i n i n g d a t a were v a l i d under P a s s i o u r a ' s method . T h i s c u t - o f f c r i t e r i a p roved so 0.20 0.25 0.30 0.35 0.40 0.45 0.50 VOLUMETRIC MOISTURE CONTENT 0.55 0.60 F i g . 2.23 D i f f u s i v i t y function: Bose s o i l @ 1.2 g/cm3. H* o H-i-h H i C CO < C o O 3 W O CD fD CO O o B 1 0 4 HYDRAULIC DIFFUSIVITY [CM2/S] 10"3 10"2 10'' 10" o ro o o Ko CJl < o m —I O O o co o o CO OI o o DO m o O £ o o o cn O O - 36 -CO CN b > coN L J _ Q CJ —I <'o. CC «-Q >-0.20 0.25 0.30 0.35 0.40 0.45 0.50 VOLUMETRIC MOISTURE CONTENT 0.55 0.60 Fig. 2.25 Diffusivity function: Bose so i l @ 1.0 g/cm-5. - 94 -conservative in some samples that diffusivity functions were f i t to as few as four points. Coarse textured soils, having a higher conductivity than finer textured soils near the saturated end of water content, would be expected to yield more of their outflow, hence more data, while supporting a smaller pressure gradient across the s o i l column than finer textured s o i l s . Thus more data would be excluded from a coarse textured s o i l outflow experiment. This explains why measurement of sandy loams was apparently poor, while that of the s i l t loams apparently good. This problem with coarser textured soils can be remedied both by using higher conductivity plates (which would support a smaller pressure gradient for any given flux rate), and by operating at pressure greater than 1 bar (Passioura operated at 6 bar). The dry end points were generated as the sample water content approached 1 bar tension, which was the applied guage pressure. Passioura's assumption of approximate spacially uniform desorption, |f- * f(z) (2.11) (i.e., the rate of desorption is not a function of position within the sample) begins to break down at this time, resulting in an over-estimation of diffusivity. Passioura suggests that a method of Gupta et a l . (1974), applicable during this f i n a l stage of outflow, be supplemented. This suggestion has been implemented. However, in 14 out of 17 measured d i f f u s i v i t i e s , the datum at the lowest water content f e l l below the curve, thus tending to increase i t s slope, hence 8. While the - 95 -i m p a c t on samples w i t h an abundance of d a t a i s no t s i g n i f i c a n t , d i f f u s i v i t y f u n c t i o n s f i t to o n l y 4 p o i n t s have been a f f e c t e d . The c a u s e of t h i s d r y end b i a s i s not known. I t may be t h a t the Gupta e t a l . supplement o v e r - c o r r e c t e d P a s s i o u r a ' s p o s i t i v e b i a s . I t may be t h a t s m a l l a i r l e a k s i n the Tempe c e l l a p p a r a t u s o c c u r r i n g between the b r a s s c o r e and the r u b b e r o - r i n g c o n v e c t i v e l y d r i e d the s a m p l e , thus downward b i a s i n g t h o s e d i f f u s i v i t y p o i n t s r e q u i r i n g the l o n g e s t measurement t i m e s , 1. e . , the d r y end p o i n t s . 2. Improvements to Stage I I Flux Modelling G a r d n e r and H i l l e l (1962) s u c c e e d e d i n m o d e l l i n g t o t a l c u m u l a t i v e e v a p o r a t i o n ( E ) . However t h e i r p r o c e d u r e of t r a n s l a t i n g the PE = 0 0 c u m u l a t i v e e v a p o r a t i o n c u r v e to t = t a ( F i g . 2 .3 ) does not m a i n t a i n c o n t i n u i t y o f s l o p e a t t = t a . A n a l y s i s of t h e i r t e c h n i q u e , i f one assumes a c o n s t a n t average d i f f u s i v i t y and a s e m i - i n f i n i t e s o i l , has shown t h a t the r a t i o of the s l o p e (hence f l u x ) at t = t a + to the s l o p e a t t = t a ~ i s 0 . 8 1 1 , i . e . , dE _1_ d t T t = t a dE d t t = t ~ a where t = t + d t and t = t - d t . a a a a C o n s e q u e n t l y the G a r d n e r and H i l l e l method o f m o d e l l i n g t o t a l c u m u l a t i v e e v a p o r a t i o n i s not c o m p l e t e l y s u i t a b l e as a model of f l u x . - 96 -The converse problem arises i n the d i r e c t modelling of fl u x . Theory suggests that a hypothetical lysimeter, with i n i t i a l water content Q±, i f subjected to PE = 0 0 (boundary condition (2.3) for a l l t _> 0), would lose less water (and hence be wetter) by the time ( t a * ) i t s flux rate had dropped to some nominal l e v e l (PEMEASURED)» than i f that same lysimeter had instead been subjected to PEMEASURED ^ o r duration t a . Thus i n modelling f l u x , the simulated flux can be chosen to match PEMEASURED at t = t a , but the cumulative evaporation (which i s the i n t e g r a l of the flux) during stage II w i l l be over-estimated, i . e . , 9PE - <0 > 9PE MEAS ( ta> < 2' 1 3> * — * where F(0,t ) = PE.„.„, 0_.„ i s the mean water content (at t ) of a ' a MEAS' PE 0 0 a lysimeter subjected to PE = 0 0, and 8_„ „_.„ i s the mean water content J PE MEAS (at t a ) of a lysimeter subjected to PE^E^S* Table 2.1 gives the measured (~8~pE ( t g ) and "8"pE ^  ( t * ) computed from the simulation model. An improvement in flux modelling methodology i s suggested here for those lysimeters for which i n e q u a l i t y (2.13) holds. By treating the i n i t i a l water content, 9^, of the simulated lysimeter as an unknown to be determined such that (2.13) i s an equality, the t o t a l cumulative evaporation during stage II can be c o r r e c t l y matched with the measured value. Thus simulated stage II flux can be forced to match the measured values of the surface flux at the time of t r a n s i t i o n , the surface moisture content, the surface moisture gradient at the time of t r a n s i t i o n , the vanishing flux at the impermeable boundary at L, as well as the t o t a l cumulative evaporation. Stated - 97 -Table 2.1 Microlysimeter mean water content at the time of transition: Measured vs Modelled Sample 6 p E ^  ( t f t ) 9 p E M (t f l) BPI 0.294 0.263 BP 2 0.306 0.265 BF1 0.253 0.325 BF2 0.274 0.328 BAl 0.327 0.250 BA2 0.285 0.255 BN1 0.308 0.333 BN2 0.347 0.328 DF1 0.351 0.374 DF2 0.342 0.363 DPI 0.396 0.357 DP2 0.332 0.342 FF 0.352 0.258 FP1 0.311 0.319 FP2 0.329 0.321 FA 0.338 0.315 - 98 -mathematically: l-D(6) | i ] d z z=0 t=t = PE. ME AS (2.14) Ko,t) = e a , t > t a (2.15) _99 3z PE MEAS z=0 t=t -D(8 a) (2.16) -D(9) _96 9z z=L = 0 (2.17) T ^ 8 > l | l z - O d t - - [ 6 P E M E A S ( t a ) - V ' L a (2.18) In Figs. 2.11-2.18, only constraints (2.14-2.17) were implemented i n the simulation. As stated previously, constraint (2.18) can only be implemented i f i n e q u a l i t y (2.13) holds. A possible reason that (2.13) did not hold i n 9 of 16 lysimeters (Table 2.1) i s s o i l v a r i a b i l i t y . The d i f f u s i v i t y functions were not measured d i r e c t l y from the lysimeter samples, but from samples i n physical proximity and for the same s i t e and cropping p r a c t i c e . The 7 of 16 samples (Table 2.1) that dried under simulated °° PE according to (2.13), also behaved much as predicted i n the stage I analysis of Zachman et a l . (1980). The i d e n t i f i c a t i o n of the stage I predicted vs modelled data i n F i g . 2.10 i s found i n Appendix E ( t a : measured vs predicted). Thus i f a d i f f u s i v i t y function successfully - 99 -modelled a lysimeter i n stage II (both q u a l i t a t i v e l y (Figs. 2.11 to 2.18), and q u a n t i t a t i v e l y (Table 2.1)), then i t modelled stage I t a ( F i g . 2.10) reasonably well also. This indicates that the theory i s consistent. And those d i f f u s i v i t i e s that poorly modelled " t h e i r " lysimeters i n stage I, were also i n error i n stage II (Table 2.1). V a r i a b i l i t y could account for some success mixed with some f a i l u r e . If the exponential d i f f u s i v i t y parameter 3 i s treated as an unknown, to be determined by applying yet another constraint i n the f l u x simulation, then an independent ( f i e l d method) of determining 3 i s introduced, as well as improving the responsiveness of the simulated f l u x to measured flux parameters. Stating this l a s t constraint mathematically: [-D(6) |i] o z z=o - ' M E A S ^ ^ ( 2 ' 1 9 ) t=t' where F J ^ E A S (0,t') i s the flux rate measured at some time t' > t a . - 100 -F . SUMMARY AND CONCLUSION The essential features of bare s o i l evaporation from Peace River region shallow soils is well modelled using a modification of a Gardner and H i l l e l method. Hydraulic diffusivity measurements required by the model may be measured by application of the Passioura analysis to outflow experiments performed on intact f i e l d samples. The effects of v a r i a b i l i t y of structure in the f i e l d do not preclude making meaningful measurements of hydraulic di f f u s i v i t y , which may be applied to the modeling of evaporation from adjacent field-extracted lysimeters. The greatest limitation in extrapolating the evaporation modelling procedure outlined here to a regional scale [i.e., the Peace region] rests with further improvements in the measurement of the diffusivity function, and an assessment of the scope and importance of s o i l variability in i t s affect on s o i l physical properties. i - 101 -BIBLIOGRAPHY Boast, C.W. and T.M. Robertson. 1982. A "Microlyslmeter" Method for Determining Evaporation for Bose S o i l : Description and Laboratory Evaluation. S o i l S c i . Soc. Am. J. 46: 689-696. Chanasyk, D. and Woytowich. 1984. A Hydraulic Model for the Soil/Land Management i n the Peace River Region, F i n a l Report for Farming for the Future Project #79-0034. Dept. S o i l S c i . , U. of Alberta, Edmonton. Chanasyk, Woytowich, Crown, Verschuren, and Rapp. 1983. Factors Causing Delays i n Farming Operations i n the Peace River Region of Alberta. Canadian A g r i c u l t u r a l Engineering, 25: 5-9. C l o t h i e r , B.E. and I. White. 1981. Measurement of S o r p t i v i t y and S o i l Water D i f f u s i v i t y i n the F i e l d . S o i l S c i . Soc. Am. J. 45: 241-245. C l o t h i e r , B.E. and I. White. 1982. Water D i f f u s i v i t y of a F i e l d S o i l . S o i l S c i . Soc. Am. J. 46: 155-158. Cochran, P.H. 1969. Lodgepole Pine Clearcut Size Affects Minimum Temperatures Near the S o i l Survae. U.S.D.A. Forest Service Research Paper PNW-86. Covey, W. 1963. Mathematical Study of the F i r s t Stage of Drying of a Moist S o i l . S o i l Science Society Am. Proceedings, 27: 130-134. Gardner, W.R. 1957. Some Steady-State Solution of the Unsaturated Moisture Flow Equation with Application to Evaporation from a Water Table. S o i l S c i . 83: 295. Gardner, W.R. 1959. Solution of the Flow Equation for the Drying of S o i l and Other Porous Media. S o i l S c i . Soc. Am. J. Proc. 23: 183-187. Gardner, W.R. 1962. Note on the Separation and Solution of D i f f u s i o n Type Equations. S o i l S c i . Soc. Am. Proc. 26: 404. Gardner, W.R. and D.I. H i l l e l . 1962. The Relation of External Evaporation Conditions to the Drying of S o i l s . J. Geophysical Res. 67(11): 4319-4325. Hanks, R.J. and H.R. Gardner. 1965. Influence of Di f f e r e n t Diffusivity-Water Content Relations on Evaporation of Water Free S o i l s . S o i l S c i . Soc. Am. Proc. 29: 495-498. Howell, J.R. 1982. A Catalogue of Radiation Configuration Factors. McGraw-Hill. Idso, S.B., R.J. Reginato, R.D. Jackson, B.A. Kimball and F.S. Nakayama. 1974. The Three Stages of Drying of a F i e l d S o i l . S o i l S c i . Soc. Am. Proc. 38: 831-837. - 102 -Johnson, G.T., and I.D. Watson. 1985. Modelling Longwave Radiation Exchange Between Complex Shapes. Boundary Layer Met. 33: 363-378. Luttmerding, H.A. 1980-84. S o i l s of the Langley-Vancouver Map Area. Vols. 1,3,6. Assessment Planning Div., Ministry of Environment, Province of B.C. Nusselt, W. 1928. Graphishe Bestimmung des Winkelverhaltnisses bie der Warmestrahlung, VDI Z., 72: 673. Passioura, J.B. 1976. Determining S o i l Water D i f f u s i v i t i e s for One-Step Outflow Experiments. Aust. J. S o i l Res. 15: 1-8. P h i l i p , J.R. 1957. Evaporation, and Moisture and Heat F i e l d s i n the S o i l . J. Meterol. 14: 354-366. Reifsnyder, W.E. 1967. Radiation Geometry i n the Measurement and Interpretation of Radiation Balance. Agr. Meteorol. 4: 255-265. Siegel and Howell. 1972. Thermal Radiation Heat Transfer. McGraw-Hill. Sparrow and Cess. 1978. Radiation Heat Transfer. Hemisphere Publishing Co. Van V l i e t , L.J.P., A.M.F. Hennig, P. M i l l s , and M.D. Novak. 1984. S o i l erosion loss monitoring and prediction under semi-arid a g r i c u l t u r e i n the Peace region of northwestern Canada. Inter. Workshop on Land Evaluation for Land Use Planning and Conservation in Sloping Areas, at I.T.C., Enschede, Netherlands, Dec. 17-21, 1984. W a l l i s , C.H., T.A. Black, 0. Hertzman and V.J. Walton. 1983. Application of a Water Balance Model to Estimating Hay Growth i n the Peace River Region. Atmosphere-Ocean. 21(3): 326-343. Zachmann, D.W., H.R. Gardener, and P.C. DuChateau. 1980. A Mathematical Treatment of the I n i t i a l Stages of Drying of a S o i l Column. S o i l S c i . Soc. Am. J. 44: 235-237. - 103 -APPENDIX A LONGWAVE RADIATION AND THE CONCEPT OF VIEW FACTOR - 104 -Planck's law of blackbody r a d i a t i o n states that an i d e a l blackbody w i l l radiate electromagnetic energy i n a wavelength band dX per unit time and per unit surface area of the blackbody given by d R b X C l (A.I) dX x5{eC2/\T_l) d RbX W where . ^ = blackbody emittance i n the band dX per band width dX [—-] m dR^ = blackbody emittance i n the band dX [—] m X = wavelength [m] T = temperature [°K] Ci = 3.740 x IO" 1 6 [W m2] C 2 = 1.4387 x I O - 2 [m °K] The t o t a l blackbody emittance, which i s the energy radiated per unit time and per unit surface area over a l l wavelengths i s obtained by integra t i n g Eq. (A.I), y i e l d i n g the Stefan-Boltzmann Equation for blackbody r a d i a t i o n : R b = a T* (A.2) where o = 5.67 x 10~ 8 [ — - — ] . m K - 105 -Real materials have emissivities e less than unity and the term grey body applies. Equation (A.2) is modified for real materials as follows: R = e o T * (A.3) Non-glossy grey surfaces radiate energy with spacial-directional characteristics described by Lambert's cosine law which states that the W ratio of the radiant intensity I [ — ] , in the direction 9 to the radiant D L intensity of the source along the direction of the surface normal is given by 1(9) 1(9 =0) cos 0 (A.4) (see Fig. 1). Because the projected area of a surface along a direction 9 goes as cos9 as well, the intensity per projected area of emitting surface is independent of 9. The radiance, B [ g r ^ ] > is defined as B - g - (A.5) P where Ap is the projected area. Thus B is isotropic for Lambertian surfaces. In considering radiative energy exchange between two surfaces, the geometric configuration of those surfaces relative to one another, as well as the emittance and surface area of the surfaces, will determine - 106 -qu a n t i t a t i v e l y the energy exchanged. Since the energy per unit time, or power, output of a surface of area dAi i s given by where Rj = o T i i t follows that some f r a c t i o n of th i s output w i l l be intercepted by a surface of area dA 2 i f that surface subtends some s o l i d angle i n the hemisphere defined by the outward normal of dAi ( F i g . 1.2). This f r a c t i o n , c a l l e d v a r i o u s l y the ra d i a t i o n view fa c t o r , configuration f a c t o r , or angle factor, i s a function both of the magnitude of the s o l i d angle subtended, as well as i t s 6-value. This l a t t e r dependence i s a d i r e c t consequence of Lambert's Law. The power output of dAi which i s intercepted by dA 2 can now be written where d F j _ 2 i s the ra d i a t i o n view factor. Analogously, the power output of dA 2 which i s intercepted by dA^ i s written dQ : = dA : (A.6) d 2 Q l - 2 = d F i _ 2 R i d A i (A.7) d 2 Q 2 - l = d F 2 - l R 2 d A 2 (A.8) The net exchange from dAi to dA 2 can be written :NET 1-2 = d F i _ 2 R i d A i - d F 2 _ i R 2 d A 2 (A.9) Making use of the r e c i p r o c i t y theorem, A i F i _ 2 = A 2 F 2 _ i (A.10) equation (A.9) can be rewritten - 107 -d 2 Q N E T = d F l _ 2 R l d A l - d A l d ^ l ^ R 2 dA 2 1-2 = d F i _ 2 Ri dAi - d F i _ 2 R2 dAi d 2 Q N E T = dFi _ 2 dAi (Ri - R 2) (A.11) 1-2 The net r a d i a t i v e exchange between two f i n i t e surfaces i s obtained by an area int e g r a t i o n over surface 1 and a s o l i d angle integration over that f r a c t i o n of the hemisphere of dAi obscured by d A 2 QNET = / . L (Ri - R 2) dF i _ 2 dAl (A.12) 1-2 A l " - 108 -APPENDIX B LISTING OF PLANE-PARALLEL DISKS VIEW FACTOR PROGRAM - 109 -1 C PROGRAM COMPUTES RADIATION VIEW FACTORS FOR P L A N E - P A R A L L E L 2 C DISKS (AS IN CONFIGURATIONS: F I G S . 1.9 AND 1 . 1 1 ) . 2 . 5 C #**RUN USING FORTRAN-G.H C O M P I L E R * * * 3 C RADIATING DISK HAS RADIUS=RRAD, RECEIVING DISK HAS RADIUS=RDISK. 4 C THE PLANES OF THE DISKS ARE SEPARATED BY DISTANCE=ZNOT. 5 C DISK AXES ARE OFFSET BY DISTANCE=ROFSET. 5 . 5 C 6 C THUS 4 PARAMETERS MUST BE S P E C I F I E D BY THE USER: 7 C 8 C PARAMETER: RRAD RESTRICTIONS: RRAD >= 0 . 0 9 C RDISK RDISK > 0 . 0 10 C ZNOT ZNOT > 0 . 0 10 .5 C ROFSET ROFSET >= 0 . 0 10.7. C 10 .8 C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 11 REAL INTLM2 12 REAL INTLAM 13 REAL INT 14 DIMENSION D I S T ( 1 0 0 0 0 ) 15 P I=3 .14159 16 C 19 .5 C * * * * * * * * * * * * * * USER S P E C I F I E D PARAMETERS: * * * * * * * * * * * * * * * * * * * 19 .7 C 20 Z N O T M . O 22 RDISK=2.0 26 RRAD=0.0 28 R 0 F S E T = 1 . 0 2 8 . 5 C 2 8 . 7 C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 29 X0FSET=R0FSET 30 C 3 0 . 5 C CENTER COORDS OF RECEIVING DISK (RDISK) ARE ( 0 . 0 . 0 . 0 . ZNOT) 3 0 . 7 C CENTER COORDS OF RADIATING DISK (RRAD) ARE ( X O F S E T , 0 . 0 , 0 . 0 ) 3 0 . 8 C 31 NR1=>(2*IFIX(RRA0/ZN0T))+5 32 NPHI=180 33 NTHETA=90 34 NALPHA=6 35 C 36 I F ( R R A D . E O . O . O ) GOTO 10 37 I F ( R O F S E T . N E . O . O ) G O T O 5 38 JJALPH=NALPHA 39 GOTO 7 40 5 J J A L P H = 1 41 7 IIR1 = 1 42 GOTO 15 43 10 JJALPH=NALPHA 44 IIR1=NR1 45 C 46 15 RMAX=SQRT(ZN0T**2+(RDISK+RRAD+R0FSET)**2) 47 DR1°RRAD/NR1 48 NR=1O*IF IX(RMAX/ZN0T) 49 DR=(RMAX-ZNOT)/NR 50 C 51 I F ( ( R O F S E T - R D I S K ) . L T . R R A D ) GOTO 100 52 P H I H A F = P I / 2 . 0 53 GO TO 120 54 100 PHIHAF=PI - 110 -55 120 D P H I = 2 . » P H I H A F / N P H I 56 C 57 THETMX=(P I /2 . ) -ATAN(ZNOT/ (ROFSET+RDISK+RRAD) ) 58 DTHETA=THETMX/NTHETA 59 C 60 INTLM2=0.0 61 INT=0.0 62 INTLAM=0.0 63 ALPHA=0.0 64 C 65 OALPHA=PI/NALPHA 66 DO 275 Jd=JJALPH,NALPHA 67 ALPHA=ALPHA+DALPHA 68 R1=0.0 69 00 270 I I= I IR1.NR1 70 R1=R1 + 0R1 71 A N N H A F = ( ( R 1 - ( D R 1 / 2 . ) ) * 2 . » P I * D R 1 ) / 2 . 72 A N N S E G = ( R 1 - ( D R l / 2 . ) ) * D A L P H A * D R 1 73 T H E T A = ( 0 . 0 ) - ( 0 . 5 ) * D T H E T A 74 DO 260 1=1,NTHETA 75 THETA=THETA+DTHETA 76 C 77 C ANY OFFSET OF RRAO IS IN +X D IRECTION. SO RDISK WILL BE 78 C "BEHIND" I T , AT PHI VALUES SYMMETRIC ABOUT P I , NOT 0 . 0 79 C 80 P H I = P I - P H I H A F - ( 0 . 4 ) * D P H I 81 DO 240 J=1,NPHI 82 PHI=PHI+DPHI 83 R=ZNOT-DR 84 ZT IP1=R*C0S(THETA) 85 D I S T ( 1 ) = A B S ( Z N 0 T - Z T I P 1 ) 86 MR=NR+3 87 DO 180 K=2.MR 88 R=R+DR 89 X T I P = R * S I N ( T H E T A ) * C O S ( P H I ) + XOFSET + R1*COS(ALPHA) 90 Y T I P = R * S I N ( T H E T A ) * S I N ( P H I ) + R1*S IN (ALPHA) 91 ZT IP=R*COS(THETA) 92 D I S T ( K ) = » A B S ( Z N O T - Z T I P ) 93 I F ( D I S T ( K ) . L T . D I S T ( K - 1 ) ) G 0 T 0 180 94 C CANDIDATE FOR PLANE CROSSING 96 C I F ( D I S T ( K - 1 ) . G T . ( D R / 2 . 0 ) ) G O T O 190 97 C CROSSING CONFIRMED 98 I F ( X T I P . L T . O . O ) G 0 T 0 140 99 I F ( ( X T I P . E Q . 0 . 0 ) . A N D . ( Y T I P . G E . 0 . 0 ) ) G 0 T 0 135 100 I F ( ( X T I P . E Q . 0 . 0 ) . A N D . ( Y T I P . L T . O . O ) ) G 0 T 0 137 101 GOTO 138 102 135 0 M E G A = P l / 2 . 103 GOTO 145 104 137 0 M E G A = - P I / 2 . 105 GOTO 145 106 138 OMEGA=ATAN(YTIP/XTIP) 107 GOTO 145 108 140 OMEGA=PI + A T A N ( Y T I P / X T I P ) 109 145 F U Z Z X = 1 . 0 * D R * S I N ( T H E T A ) * C O S ( P H I ) 110 SCAFZX=FUZZX*COS(OMEGA) 111 F U Z Z Y = 1 . 0 * D R » S I N ( T H E T A ) * S I N ( P H I ) 112 SCAFZY=FUZZY*SIN(OMEGA) 113 160 I F ( ( R D I S K + S C A F Z X + S C A F Z Y ) . G E . S Q R T ( X T I P * * 2 + Y T I P * * 2 ) ) G 0 T 0 200 - I l l -114 GOTO 190 115 180 CONTINUE 116 190 HVYSID=0.0 117 GO TO 210 118 200 HVYSID=1.0 119 210 DINT=HVYSID*SIN(THETA)*DTHETA*DPHI 120 I F ( R R A D . N E . 0 . 0 ) G 0 T 0 223 121 INT=INT + DINT 122 INTLAM = INTLAM + D I N T * ( C O S ( T H E T A ) ) 123 GOTO 240 124 223 I F ( R O F S E T . N E . O . O ) GOTO 230 125 INT=INT + DINT*ANNHAF 126 INTLAM=INTLAM + D I N T * ( C O S ( T H E T A ) ) * A N N H A F 127 GOTO 240 128 230 INT=INT + DINT*ANNSEG 129 INTLAM=INTLAM + D I N T * ( C O S ( T H E T A ) ) * A N N S E G 130 240 CONTINUE 131 260 CONTINUE 132 270 CONTINUE 133 275 CONTINUE 134 I F ( R R A D . E Q . O . O ) G O T O 290 135 V F G E 0 = 2 . * I N T / ( 2 . * P I * P I * ( R R A D * * 2 ) ) 136 V F R A D = 2 . * I N T L A M / ( P I * P I * ( R R A D * * 2 ) ) 137 GO TO 294 138 290 V F G E 0 = I N T / ( 2 . * P I ) 139 VFRAD=INTLAM/PI 140 294 WRITE(1 ,295 ) 141 295 F O R M A T ( 5 X , ' R D I S K ' , 5X, ' Z N O T ' , 6 X , ' R R A D ' , 6 X , ' R O F S E T ' ) 142 W R I T E ( 1 , 3 0 0 ) R D I S K , Z N O T , R R A D , R O F S E T 143 300 FORMAT(7F10 .3 ) 143 .5 WRITE(1 ,303 ) 143 .7 303 FORMAT( ' ' ) 144 WRITE(1 ,305 ) 145 305 FORMAT( 'RADIATION V . F . ' ) 146 WRITE(1 ,310)VFRAD 147 310 FORMAT(1F10 .3 ) 148 WRITE(1 ,315 ) 149 315 FORMAT( ' ' ) 153 STOP 154 END - 112 -- 113 -APPENDIX C HYDRAULIC DIFFUSIVITY DATA 10 -4 HYDRAULIC DIFFUSIVITY [CM2/S] 10 .-3 10" 10 -1 10" - <7lT " 0.20 0.25 0.30 0.35 0.40 0.45 0.50 VOLUMETRIC MOISTURE CONTENT 0.55 0.60 F i g . C.2 B e a v e r l o d g e f a l l o w . 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 VOLUMETRIC MOISTURE CONTENT 0.60 Fig. C.3 Beaverlodge perennial. HYDRAULIC DIFFUSIVITY [CM2/S] 10T 10 .-3 10 -2 10 -1 10° a o o - LU -CO > jJL Q y _i Q >-0.20 0.25 0.30 0.35 0.40 0.45 0.50 VOLUMETRIC MOISTURE CONTENT 0.55 0.60 F i g . C .5 B e a v e r l o d g e u n c u l t i v a t e d . 10"4 HYDRAULIC DIFFUSIVITY [CM2/S] 10"3 10"2 10'' 10° - 6TT -HYDRAULIC DIFFUSIVITY [CM2/S] 1CT 10 -3 10 .-2 10 10° - ozi -0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 VOLUMETRIC MOISTURE CONTENT 0.60 F i g . C.8 Dawson Creek p e r e n n i a l . 0.20 0.25 0.30 0.35 0.40 0. VOLUMETRIC MOISTURE 45 0.50 0. CONTENT 55 0.60 F i g . C.9 Dawson Creek perennial. •• #' • V * 4 , 0.20 0.25 0.30 0.35 0.40 0.45 0.50 VOLUMETRIC MOISTURE CONTENT 0.55 0.60 F i g . C . 1 0 Fort St. John annual. 10"4 HYDRAULIC DIFFUSIVITY [CM2/S] 10"3 10'2 10"' 10° - nzi -10 -4 HYDRAULIC DIFFUSIVITY [CM2/S] 10 -3 10" 10 10° - SZT -1 O o l-h M i C CD < Tl O o 3* 3 i-d fD i-( fO 3 3 H* o m O O Z H m HYDRAULIC DIFFUSIVITY [CM2/S] 1CT 10 -3 10" 10 .-1 10° o fO O O ro cn < O P C co C o ^ m ^ H p 3 2 co ft O w O 3 o o o CTI o CTI o cn cn O CD O - 9ZT -CO >-fc > LL, Q O _ J Q 0 . 2 0 0 . 2 5 0 . 3 0 0 . 3 5 0 . 4 0 0 . 4 5 0 . 5 0 VOLUMETRIC MOISTURE CONTENT 0 . 5 5 0 . 6 0 F i g . C.14 Beaverlodge B-horizon. HYDRAULIC DIFFUSIVITY [CM2/S] 10"3 10"2 10"1 10° - 8ZT -CO CN tz > co„ LL. Q u _ l Q > -X 0.20 0.25 0.30 0.35 0.40 0.45 0.50 VOLUMETRIC MOISTURE CONTENT 0.55 0.60 F i g . C.16 D i f f u s i v i t y function: Bose s o i l @ 1.2 g/cmJ. CO >-fc > LL. Q u /CO <'o. CC "~ Q >-O-0.20 0.25 0.30 0.35 0.40 0.45 0.50 VOLUMETRIC MOISTURE CONTENT 0.55 0.60 F i g . C.17 D i f f u s i v i t y f u n c t i o n : Bose s o i l @ 1.0 g / c m 3 . - 131 -APPENDIX D HYDRAULIC DIFFUSIVITY FUNCTIONS - 132 -Sample Fig. # Beaverlodge - Perennial (C.3) (C.4) D = exp [37.24 0 - 15.94] D = exp [43.01 0 - 19.17] Beaverlodge - Fallow Beaverlodge - Annual Beaverlodge - Uncultivated Beaverlodge - B Horizon (C.2) ( C l ) (C.5) (C.14) D = exp [43.99 0 - 21.20] D = exp [31.11 0 - 15.03] D = exp [25.26 0 - 15.76] D = exp [30.99 6 - 15.97] Dawson Creek - Perennial (C.8) (C.9) D = exp [27.79 8 - 16.83] D = exp [15.37 0 - 13.41] Dawson Creek - Fallow (C.6) (C.7) D = exp [35.70 0 - 21.01] D = exp [35.01 0 - 19.57] Fort St. John - Perennial Fort St. John - Fallow (C.13) (C.12) ( C l l ) D = exp [29.85 6 - 17.44] D = exp [40.68 0 - 19.35] D = exp [36.99 0 - 16.93] Fort St. John - Annual (CIO) D = exp [23.43 8 - 14.91] UBC - Compacted UBC - Rototilled (C.15) (C.16) (C.17) D = exp [38.82 8 - 16.85] D = exp [41.97 8 - 16.79] D = exp [53.92 8 - 20.65] - 1 3 3 -APPENDIX E MICROLYSIMETER SOIL PARAMETERS - 134 -Sample B u l k D e n s i t y M o i s t u r e P o r o s i t y S a t u r a t i o n [—Sj] I n i t i a l F i n a l I n i t i a l F i n a l cm BP1 1.28 .41 .23 .51 .82 .45 BP2 1.23 .44 .24 .53 .83 .46 BF1 1.26 .40 .20 .52 .78 .39 BF2 1.17 .44 .21 .55 .80 .26 BA1 1.32 .38 .25 .49 .77 .51 BA2 1.31 .42 .22 .50 .85 .44 BN1 1.31 .36 .24 .50 .72 .48 BN2 1.21 .39 .27 .54 .72 .51 DF1 1.08 .50 .26 .59 .86 .44 DF2 1.12 .38 .28 .57 .66 .49 DPI 1.18 .50 .31 .55 .91 .57 DP2 1.14 .44 .25 .56 .79 .45 *DN 1.11 .45 .31 .57 .79 .54 FF 1.32 .38 .28 .49 .77 .57 FP1 1.23 .43 .23 .53 .82 .44 FP2 1.27 .45 .25 .51 .87 .49 FA 1.14 .44 .25 .56 .78 .45 * F N 1.12 .48 .30 .57 .84 .53 * L C 1.23 .37 .21 .53 .70 .39 * L R 1.02 .33 .21 .61 .55 .35 - 135 -Sample Da 2 f c m }  1 s J 9 I 9a PE[f] todays] ( m e a s ) ( p r e d ) BP1 .388 1.18 X 1 0 ~ 7 .414 .040 20 .83 3 .7 5.7 40.1 BP2 .939 1.18 X I O " 7 .436 .040 22 .53 3.7 5 .5 40.1 BF1 .0272 3.61 X I O - 9 .400 .040 20 .99 4 . 5 2 .7 4 4 . 0 BF2 .173 3.61 X 10~ 9 .442 .040 23.97 4 . 5 3 .9 44 .0 BA1 .0404 1.03 X I O " 6 .380 .040 20 .12 1.7 4 .6 31.1 BA2 .140 1.03 X I O " 6 .420 .040 23.39 3 .7 5.1 31.1 BN1 .00124 3.93 X I O " 7 .359 .040 21 .99 1.5 1.2 25 .3 BN2 .00245 3.93 X i o - 7 .386 .040 19.42 1.3 2 .5 25 .3 DF1 .0818 9 .02 X io- 9 9 .503 .050 21 .72 4 . 5 4 .6 35 .4 DF2 .000984 9.02 X i o - 9 .378 .050 16.31 1.4 0 .9 35 .4 DPI .0118 7.98 X I O " 7 .495 .050 24 .49 2 .6 4 .1 21 .6 DP 2 .00369 7.98 X I O " 7 .441 .050 17.50 4 .0 4 . 3 21 .6 FF .0506 2.82 X I O " 7 .377 .050 14.73 1.1 4 . 3 3 7 . 0 FP1 .0438 5.01 X I O " 8 .433 .040 21.21 3.7 4 .0 35 .3 FP2 .0669 5.01 X I O - 8 .445 .050 20.20 3.7 4 .8 35 .3 FA .00958 8 .54 X I O " 7 .438 .040 20 .02 3.2 4 .3 23 .4 * L C .153 1.66 X i o " 7 .373 .030 23 .74 7 .0 9 .5 40 .4 * L R .0575 3.16 X I O " 9 .332 .020 19.56 5 .0 6 .6 53 .9 * N o t m o d e l l e d . - 136 -APPENDIX F PARTICLE SIZE ANALYSIS - 137 -% Sand % S i l t % Clay Beaverlodge Dawson Creek Fort St. John 18.1 13.6 10 .9 4 5 . 8 64.1 52 .9 36 .1 22 .2 36 .2 - 138 -APPENDIX G RETENTION CHARACTERISTIC AND STRDCTDRE - 139 -Clo t h i e r and White (1982) found that repacking s o i l samples for subsequent hydrologic measurements a l t e r s the pore size d i s t r i b u t i o n at le a s t on a scale greater than 0.2 mm, and disrupts continuous networks of pores created by b i o l o g i c a l processes. Repacking would also destroy any anisotropy introduced by the e f f e c t of gravit y . Thus repacking the s o i l a l t e r s i t s structure even beyond the range of s t r u c t u r a l v a r i a b i l i t y found i n the f i e l d . Therefore measured hydrologic properties from repacked samples may not represent f i e l d "matric" properties. The e f f e c t of repacking on water retention i s seen i n F i g . G.1. The B-horizon sampled from the Beaverlodge area i s high i n clay content and r e l a t i v e l y impermeable to flow. The bulk density of the samples was the same. Genetic structure, expected to become generally more important and more complex with f i n e r texture, i s c l e a r l y affected by repacking. E f f e c t s of repacking are also seen i n Dawson Creek fallow vs repacked ( F i g . G.2). This comparison i s i n s t r u c t i v e i n showing that even a fallowed s o i l , presumed to have the least structure of any f i e l d s o i l , behaves d i f f e r e n t l y from one repacked to the same bulk density. The Bose sandy loam s o i l at 2 bulk densities i s seen i n F i g . G.3. The compact s o i l held more water at a l l tensions, and had an avai l a b l e water storage capacity (AWSC = 9Q.I bar - Q\^^Q bar) of 0.25 cm R^O/cm s o i l , compared with the r o t o - t i l l e d s o i l AWSC of 0.18 cm H2O/cm s o i l . R o t o - t i l l i n g seems to increase the number of large pores which empty at small tensions (le s s than f i e l d capacity). Although lower bulk density r e s u l t s i n higher t o t a l porosity, the usable porosity i s reduced. This suggests the existence of some optimum bulk density to maximize AWSC. - 140 -Clothie r and White have suggested that f i e l d heterogeneity l i e s p r i m a r i l y i n the inhomogeneous d i s t r i b u t i o n of voids greater than 0.75 mm d i a . Excluding the p a r t i c i p a t i o n of these large voids during measurement of hydrologic properties would then characterize the i n t r i n s i c s o i l matrix. Voids of th i s diameter remain empty at tensions greater than 4 cm H 2 O . Since hydrologic properties such as retention c h a r a c t e r i s t i c s are measured over a range of water content, most of which i s at tensions greater than 4 cm H 2 O , i t might be expected that such measured properties are, except at the extreme wet end, i n t r i n s i c properties of the s o i l matrix. Proceeding with this l i n e of reasoning, intact s o i l samples -i d e n t i c a l i n a l l respects except possibly i n structure - would be expected to behave i n d i v i d u a l l y at the wet end, but thereafter to behave i d e n t i c a l l y . In terms of water retention c h a r a c t e r i s t i c , a l l differences established between saturation and some c r i t i c a l tension (corresponding to the largest pore diameter unaffected by structure) would be expected to vanish at greater tensions. This hypothesis i s not supported i n Figs . G. 1-G.10. The ordering of the curves established at the wet test measurement (^ = -0.025 bar) endures over the range of matric suction (-i|0 measured. The e f f e c t of cropping practice may impart to water retention c h a r a c t e r i s t i c i s surveyed i n Figs. G .4 - G . 8 . A trend emerges with uncultivated s o i l retaining the most water over the measured range of tensions ( F i g . G . 4 ) , followed by annual row cropped s o i l s and fallowed s o i l s as intermediates, and perennially cropped s o i l (grasses) r e t a i n i n g the least water over the measured range. The water retention - 141 -c h a r a c t e r i s t i c i s a measure of pore s i z e d i s t r i b u t i o n . The f i g u r e s show t h a t the u n c u l t i v a t e d s o i l had the g r e a t e s t p o r o s i t y i n the range of p o r e r a d i u s <^  0.1 pm ( c o r r e s p o n d i n g t o d r a i n a g e t e n s i o n s >^  15 b a r ) , f o l l o w e d by the a n n u a l l y c r o p p e d and f a l l o w e d s o i l s , w i t h p e r e n n i a l l y c r o p p e d s o i l s h a v i n g the s m a l l e s t p o r o s i t y i n the r a n g e . Throughout many of the r e t e n t i o n c u r v e s , p l o t t i n g v o l u m e t r i c wa te r c o n t e n t a g a i n s t the l o g of m a t r i c s u c t i o n y i e l d s a n e a r l y s t r a i g h t l i n e . T h i s s u g g e s t s a r e l a t i o n s h i p f o r water r e t e n t i o n c h a r a c t e r i s t i c of the f o r m : -i> = a exp[b ( f - 6) ( G . l ) where i s m a t r i c s u c t i o n , a and b a re c o n s t a n t s , f i s t o t a l p o r o s i t y , and 9 i s v o l u m e t r i c water c o n t e n t . - 142 -LEGEND ..•..J-HORjZONJ^ . . _B-HOf^ ZON R ' V . —i—i i i i n i | i i i 11 n i | 1—i i i 11 ii| 1—i i i 11 u| 10"2 10"1 10° 101 102 MATRIC SUCTION [BARS] Fig. G.1 B-horizon: intact vs repacked. - 143 -CD LEGEND A DC FALLOW A A .?..J??.JiAkLPw. • DC FALLOW Y,XJ\P * Y • DC REPACKED ik *, ^ \ DC REPACKED i^ X *°*-e —I—I I I 11 i i | 1—I I I 11111 1—I l l 11 i i | 1—I I I 11 II 10"2 10"1 10° 101 102 MATRIC SUCTION [BARS] F i g . G. 2 Dawson C r e e k : f a l l o w vs r e p a c k e d . - 144 -LEGEND •....y.B.^ :c9!y!.PACI o UBC-T1LLED a UBC-T1LLED 1—i i i 1111| 1—i i i 1111| 1—i i 11111| 1—i i i 1111| 10"2 10"1 10° 101 102 MATRIC SUCTION [BARS] F i g . G.3 Bose s o i l at 2 bulk d e n s i t i e s . - 145 -CD • K . LEGEND v v» " • " B UNCULTIVATED •v .• , — i — i i i 11 ii| 1—i i i 11111 1—i i i 11 ii| 1—i i i i m i 10"2 10"1 10° 101 102 MATRIC SUCTION [BARS] F i g . G.4 Beaverlodge uncultivated. - 1 4 6 -CO CO ° CJ CO «>. 2 ° CJ 111 o I-Z o cj m cc ° UJ 5 2 CJ cc O o > o LEGEND * FSJ Af^UAL o FSJ ANNUAL "a ' FSJ ANNUAL • | 1 I I I I l l l | 1 I 1 I I I M III 1 I I I Mil l 10*2 10'1 10° 101 102 MATRIC SUCTION [BARS] F i g . G.5 Fort St. John annual. - 147 -CO U CO w cj U J o O CJ « o c c H _ 3 O o > o LEGEND ..^ ...^ .PCTENNIAL A o _ B PERENNIAL "a... • B PERENNIAL X ^ . * . . ' X . • B PERENNIAL • * » f - . » » % 0 •••..A X ^ i - l x • B PERENNIAL A , — i — i i II i i i | 1—i i i 11 i i | 1 i i i 11 i i | 1 i i i 11111 10"2 10"1 10° 101 102 MATRIC SUCTION [BARS] F i g . G.6 B e a v e r l o d g e p e r e n n i a l . - 148 -CO LEGEND • DC PERENNIAL — i i i i 11 i i | 1—i i i 11111 i—i i i 11 i i | 1—i i i 11 n 10"2 10"1 10° 101 102 MATRIC SUCTION [BARS] F i g . G.7 Dawson Creek p e r e n n i a l . - 149 -co cj CO w U J o O CJ co LU CJ cc ISS-O o > o LEGEND A FSJ PERENNIAL o FSJ PERENNIAL D FSJ PERENNIAL %N o» ''v. ^H. V I I I I I I III I I I I I I III I I I I I I III I I I I I I I ll 10"2 10"1 10° 101 102 MATRIC SUCTION [BARS] F i g . G.8 F o r t S t . John p e r e n n i a l . - 150 -CO CO d CJ CO ^ UJ O Z o CJ co <r° LU Is CJ cc 5 ° 3 O o > d LEGEND " B RP.P.A.C.^.D. •, • B REPACKED s — i — i i 111 ii| 1—i M i n n 1—i i i 11 ii| i—i i i 111 10"2 10"1 10° 101 itf MATRIC SUCTION [BARS] F i g . G.9 Beaverlodge repacked s o i l . - 151 -LEGEND • FSJ REPACKED • FSJ REPACKED "'••..\B '••••.•» \ *. \ *• % IN • % ; \ IN *. % i o 2 i o 1 io° io' 10 MATRIC SUCTION [BARS] .2 F i g . G.10 Fort St. John repacked s o i l . 

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