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Hydrologic behavior of a forested mountain slope in coastal British Columbia Tischer, Evelyn 1986

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HYDROLOGIC BEHAVIOR OF A FORESTED MOUNTAIN SLOPE IN COASTAL BRITISH COLOMBIA by EVELYN TISCHER B.Sc. (Meteorology), McGlll U n i v e r s i t y , 1974 M.Sc. (Meterology), Reading Uni v e r s i t y (U.K.), 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (INTERDISCIPLINARY STUDIES) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA © Evelyn Tischer, 1986 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e h e a d o f my d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f 5oc-€ Sc/efta*. The U n i v e r s i t y o f B r i t i s h C o l u m b i a 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3 Da t e S&pkriok&Xr DE-6 n / a n ( i i ) ABSTRACT This i s a study of the hydrologic behavior of a forested west coast mountain slope s o i l . Flow mechanisms were investigated using experimental r e s u l t s , mainly outflow hydrographs, and a simple model of saturated flow over a steep bed, described i n the second chapter and c a l l e d the kinematic wave model. The main experiments consisted of subjecting the forest plot to both concentrated and uniform i r r i g a t i o n . The ration a l e for using concentrated i r r i g a t i o n was that i t was expected i t would enhance flow i n low resistance paths or as fingers i n the unsaturated zone, also c a l l e d s h o r t - c i r c u i t i n g . It was concluded that the soil-water system behaves as i f s h o r t - c i r c u i t i n g were not enhanced by concentration of i r r i g a t i o n . The fact that both the observed hydrograph and the kinematic wave model hydrograph display a straight l i n e r i s e was the rationale for using the kinematic wave model. It indicates that the system behaves as i f the kinematic wave model were v a l i d . Readily v e r i f i e d assumptions are a steep bed slope and a high hydraulic conductivity due to concentration of low resistance paths on top of the bed. As for other assumptions, i t can only be said that the system behaves as i f they were s a t i s f i e d . In p a r t i c u l a r i t behaves as i f no s h o r t - c i r c u i t i n g occurred. (±ii) Using the kinematic wave model, an e f f e c t i v e hydraulic conductivity of 1.6 x 10 to 3.2 x 10" ms was obtained for the saturated zone. F i n a l l y , i t was shown that nonlinear flow, i f i t occurs at a l l , i s probably uncommon i n the unsaturated zone of the Forest plot and i t s v i c i n i t y . It i s not c e r t a i n whether i t can occur i n the saturated zone of the Forest p l o t . - i v -TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES v i i LIST OF FIGURES v i i i LIST OF SYMBOLS x ACKNOWLEDGEMENTS xv INTRODUCTION 1 CHAPTER I - THE INFERENCE OF FLOW MECHANISMS FROM THE OUTFLOW HYDROGRAPH 3 1.1 Introduction 3 1.2 Li t e r a t u r e Review 5 1.3 Materials and Methods 13 1.3.1 Site Description 13 1.3.2 Sprinklers 26 1.3.3 Raingages, Pooled I r r i g a t i o n and Time Averaged Pooled I r r i g a t i o n 26 1.3.4 Neutron Probe Access Tubes 28 1.3.5 Standpipes 29 1.3.6 C o l l e c t i n g Channel and Outflow Tiping Bucket.. 32 1.3.7 Experimental Procedure 34 1.4 Results 39 1.4.1 S o i l Water Content, 9 39 1.4.2 Outflow 44 1.4.3 Water Table 51 1.4.4 Comparison of the Recession with Hewlett and Hibbert's (1963) Recession 69 1.4.5 Vi s u a l Observation of Low Resistance paths i n the Forest Floor and the B Horizon 74 1.4.6 Water Balance 78 1.5 Discussion 81 1.5.1 Water Flow i n the Unsaturated Zone 83 1.5.2 L a t e r a l Saturated Flow i n Low Resistance Paths 100 - v -Page 1.5.3 Discussion of Beven's 1982 Model i n Relation to the Present Work 104 1.5.4 Comparison with other Fast Flow S i t u a t i o n s . . . . 105 1.5.5 The Flow Model 106 1.5.6 V a l i d i t y of the Kinematic Wave Model 107 1.6 Conclusion 109 1.7 References I l l CHAPTER II - DETERMINATION OF AN EFFECTIVE HYDRAULIC CONDUCTIVITY USING THE KINEMATIC WAVE MODEL 114 2.1 Introduction 114 2.2 L i t e r a t u r e Review 115 2.3 The Kinematic Wave Equation for Flow i n a Porous Medium 115 2.4 Use of the Kinematic Wave Model to Calculate an E f f e c t i v e K 123 2.5 Conclusions 131 2.6 References 133 CHAPTER III - NONLINEAR FLOW 134 3.1 Introduction 134 3.2 L i t e r a t u r e Review 138 3.3 Discussion on the Use of F r i c t i o n Factor Versus Reynolds Number Relationships 143 3.4 Occurrence of Nonlinear Flow i n the Forest 153 3.5 Conclusions 161 3.6 References 164 CHAPTER IV - DISCUSSION AND CONCLUSIONS 166 Appendix A Determination of the Stone Ratio 170 - v i -Page Appendix B Calculation of the Minimum Concentration of I r r i g a t i o n Obtained by the P l a s t i c Sheets 172 Appendix C Comparison of the Recession with Hewlett and Hibbert's Recession: Normalization and L i m i t a t i o n s . . . 173 Appendix D Calc u l a t i o n of the Points Used to plot F i g . 1.4.2.6, the Detailed Hydrograph Rises 175 Appendix E E f f e c t of Errors and Limitations i n Outflow Rate 176 Appendix F Calc u l a t i o n of Average Steady State Outflow Rates.... 181 Appendix G D e f i n i t i o n of P o r o s i t i e s 182 Appendix H Calculations for Use i n Chapter I I . . . 183 Note 1 Solution of the kinematic wave equation by the method of c h a r a c t e r i s t i c s 183 Note 2 Problems linked to the evaluation of n 189 Note 3 Taking the water balance into account when ca l c u l a t i n g K gf£ 191 Note 4 Recession with an e f f e c t i v e porosity n r g 191 Appendix I Calculations for use i n Chapter III 193 Note 1 Calc u l a t i o n of j ^ at q by Eq. 3.3.15 d log Re cr J using data obtained by de Vries (1975, 1979). 193 Note 2 Correction of q c to take the temperature difference into account 195 Appendix K References for the Appendices 196 - v i i -LIST OF TABLES Page Table 1.3.1.1 Estimate of satiated hydraulic conductivity K at two outflow points 23 Table 1.3.7.1 F i e l d experiments (summer 1979) 35 Table 1.4.3.1 Standpipes depth 52 Table 1.4.3.2 Steady state values for standpipes (m); r e l a t i v e differences between Ex2 and Ex3 (absolute values) 64 Table 1.4.4.1 Comparison of Hewlett and Hibbert's work with the present work 73 Table 1.4.6.1 I r r i g a t i o n rates, outflow rates, average water table height and r a t i o of outflow to i r r i g a t i o n f o r the 3 experiments 79 Table 2.4.1 Calculations of e f f e c t i v e hydraulic c o n d u c t i v i t i e s using the kinematic wave approximation.. 126 Table 2.4.2 Comparison of the Kg^^ obtained by the kinematic wave model with other values 128 Table 3.3.1 I at q calculated from Eq. 3.3.15 dlog Rg M c r H using data from de Vries (1975, 1979) 152 Table 3.4.1 Comparison of measured flow v e l o c i t i e s with de Vries' c r i t i c a l flow v e l o c i t y 155 dh Table 3.4.2 q and Pg^ from experiments i n 0.4 to 0.6 mm sand by de Vries (1975). K g e n calculated from these values 159 Table C-l Normalization summary for the res u l t s of the present work given i n Table 1.4.4.1 174 Table I - l Calculation of I from Eq. 1-3 194 - v i i i -LIST OF FIGURES Page F i g . 1.3.1.1 Site layout 14 F i g . 1.3.1.2a P a r t i a l hydraulic conductivity curve for the 0.11, 0.15, 0.19 and 0.23 m depths on Nagpal and de V r i e s ' s i t e 18 F i g . 1.3.1.2b K(i|>) for Nagpal and de Vr i e s ' s i t e at o.23 m depth obtained from Nagpal and de Vries (1976) K(6) and 9(i|>) curves 19 F i g . 1.3.1.2c P a r t i a l retention curves for the 0.025, 0.07, 0.15 and 0.23 m depths for Nagpal and de V r i e s ' s i t e 20 F i g . 1.3.1.3 Contours of the bed (bedrock or compacted t i l l ) obtained from pipes depth 25 F i g . 1.3.7.1 P l a s t i c sheets lay-out 37 Fi g . 1.3.7.2 Natural r a i n from c l i m a t o l o g i c a l s t a t i o n 4.8 km away 38 F i g . 1.4.1.1 Water content from the neutron probe readings at the upper neutron probe access tube 40 Fi g . 1.4.1.2 Water content from the neutron probe readings at the lower neutron probe access tube 41 F i g . 1.4.2.1 The three parts of the hydrograph r i s e 44 F i g . 1.4.2.2 Superposition of the 3 hyetographs (pooled i r r i g a t i o n rates calculated without RG11) and of the 3 outflow hydrographs 45 F i g . 1.4.2.3 Pooled i r r i g a t i o n and outflow rate for EX1 46 F i g . 1.4.2.4 Pooled i r r i g a t i o n and outflow rate for EX2 47 F i g . 1.4.2.5 Pooled i r r i g a t i o n and outflow rate f o r EX3 48 F i g . 1.4.2.6 D e t a i l of the three hydrograph r i s e s 50 F i g . 1.4.3.1 Water table behavior at pipe 1 53 F i g . 1.4.3.2 Water table behavior at pipe 2 54 F i g . 1.4.3.3 Water table behavior at pipe 4 55 - i x -Page F i g . 1.4.3.4 Water table behavior at pipe 5 56 F i g . 1.4.3.5 Water table behavior at pipe 6 57 F i g . 1.4.3.6 Water table behavior at pipe 8 58 F i g . 1.4.3.7 Water table behavior at pipe 12 59 F i g . 1.4.3.8 Pipe 2: d e t a i l of r i s e 60 F i g . 1.4.3.9 Pipe 12: d e t a i l of r i s e 62 F i g . 1.4.3.10 Water table shape for EX1 65 F i g . 1.4.3.11 Water table shape for EX2 66 F i g . 1.4.3.12 Water table shape for EX3 67 F i g . 1.4.4.1 Outflow recession for EX1 71 F i g . 1.4.4.2 Recession obtained by Hewlett and Hibbert (1963).. 72 F i g . 1.5.1.1 Schematic hydrograph r i s e s showing the e f f e c t of s h o r t - c i r c u i t i n g i n the unsaturated zone 84 F i g . 2.3.1 Saturated flow i n a porous medium on a sloping bed 117 F i g . 2.3.2 Adapted from Eagleson, 1970, F i g . 15-5. Shape of the water table during r i s e and steady state... 120 F i g . 2.3.3 Shape of the water table during the recession 121 F i g . 2.3.4 Hydrograph given by the kinematic wave model 122 F i g . 3.2.1 Schematic representation of the plot of r versus q obtained by de Vries (1979) 143 F i g . 3.3.1 f versus Rg r e l a t i o n s h i p according to Eqs. 3.3.10 and 3.3.11 149 F i g . H - l . l (a) C h a r a c t e r i s t i c s , (b) Water table p r o f i l e at time t 2 184 - x -LIST OF SYMBOLS Note: Units given indicate the dimensions. They are not always the units used f o r the given symbol. For instance, mm are sometimes used i n place of m. a Constant determined by the properties of the f l u i d and of the porous medium (or possibly of the porous medium only), 9 3 appearing i n Forchheimer's equation (Eq. 3.2.1). (s m ) a' a' = ga. (m" ) a^ Constant determined by the properties of the porous medium and of the f l u i d (or possibly of the porous medium only), appearing i n Eq. I - l . (m_ ) b Constant determined by the properties of the f l u i d and of the porous medium (or possibly of the porous medium only), 9 9 appearing i n Forchheimer's equation (Eq. 3.2.1.). ( s m ) b' b» - & b. (sm- 3) U C A constant i n a f r i c t i o n factor d e f i n i t i o n ( m s - 2 ) . See Eq. 3.3.6. c Wave v e l o c i t y i n the kinematic wave model, (ms - 1) C dt * c Wave v e l o c i t y during the recession, (ms - 1) c Wave v e l o c i t y during the r i s e . (ms - 1) - x i -D Width of the h i l l s l o p e . For the Forest plot i t i s taken as the length of the channel measured perpendicular to the slope. (m) d Some measurement related to the size of the pores, (m) e Constant determined by the properties of the porous medium and of the f l u i d (or possibly of the porous medium only), appearing i n Eq. 1-1. (Dimensionless) f F r i c t i o n f a c t o r . See Eqs. 3.3.6, 3.3.8 and 3.3.9 f o r d e f i n i t i o n s . (Dimensionless) g G r a v i t a t i o n a l a c c e l e r a t i o n . (ms ) h Hydraulic head, (m) or ^— Rate of change of h with s or x. Since i n the present work, q i s i n the main flow d i r e c t i o n and the porous media are considered to be i s o t r o p i c and homogenous, 4-^  or 4— i s the ds dx (macroscopic) hydraulic gradient. It i s not clear whether dh dh homogeneity i s a c t u a l l y required for -^ 7 or — to be the macroscopic hydraulic gradient. (Dimensionless). i Recharge rate per unit area p a r a l l e l to the bed. Can vary with time and be zero. (ms" 1) i Recharge rate per unit area p a r a l l e l to the bed during r i s e and steady state. i Q i s a constant, (ms - 1) K Satiated hydraulic conductivity. W i l l be referred to as hydraulic conductivity for s i m p l i c i t y . Satiated hydraulic conductivity i s defined i n Section 1.2. (ms - 1) E f f e c t i v e hydraulic conductivity obtained by the kinematic wave model. See eq. 2.4.1. (ms" 1) - x i i -General conductivity (ms ). The term i s used for both the l i n e a r and nonlinear range and K n depends on q. K = q A"1 gen M ds 2 Permeability (m ). k i s defined by P Length of the saturated zone, (m) A c h a r a c t e r i s t i c (or representative) length, (m). When a number i s given for the Reynolds number characterizing flow i n a porous medium, the diameter of the p a r t i c l e s forming the porous medium w i l l be used for L. E f f e c t i v e porosity (m m ). See Appendix G for d e f i n i t i o n . 3 3 Porosity (m m~ ). See Appendix G for d e t a i l s . E f f e c t i v e porosity for the recession, ( m m ) E f f e c t i v e porosity during the r i s e . ( m m ) 3 — 1 Outflow rate or discharge (volume per unit time), (m s ) Discharge per unit area or macroscopic flow v e l o c i t y ( m s - 1 ) . In the present work, q i s i n the main flow d i r e c t i o n . C r i t i c a l macroscopic flow v e l o c i t y , that i s , the v e l o c i t y at which flow becomes nonlinear, (ms - 1) Impermeability. (m ) The r used by de Vries (1979) i s defined by dh r = pg- is. u q Time average pooled i r r i g a t i o n rate, (ms - ) Pooled i r r i g a t i o n rate (ms - 1) - x i i i -Re Reynolds number (Dimensionless). In the present work, the following d e f i n i t i o n i s used unless otherwise s p e c i f i e d : v u L u ch ch Re = v R e c r C r i t i c a l Reynolds number (Dimensionless). It i s the Reynolds number at which flow becomes nonlinar. For the discussion of the behavior of the f versus Re r e l a t i o n s h i p s , q d cr Re has been defined as Re = cr cr v s Distance (m) T Thickness of the saturated zone (measured perpendicular to the bed), (m) T Saturated zone thickness seen by the observer at x and t o J o o (m). Used i n the so l u t i o n of the kinematic wave equation by the method of c h a r a c t e r i s t i c s , Note 1, Appendix H. t Time. (s) t' Time from s t a r t of recession, (s) t* Time from s t a r t of the recharge, (s) t Time at which the observer l e f t from x (measured from the o o s t a r t of recharge) ( s ) . Used i n the s o l u t i o n of the kinematic wave equation by the method of c h a r a c t e r i s t i c s , Note 1, Appendix H. t Time at which recharge stopped, (s) v ^ A c h a r a c t e r i s t i c flow v e l o c i t y , generally taken to be q. x Downslope distance from the highest point on the bed where recharge occurs, (m) - xiv -x An observer leaves from x = x . x i s used i n the sol u t i o n o o o of the kinematic wave equation. z Axis perpendicular.to the bed and upward (along T; see F i g . 2 . 3 . 1 . ) . 1 , 2 - 3 s " l i n a l i n = VK- ( s m } 2 3 a ,. a = a . ( s m - ) nonlin nonlin 3 Q B D = b. ( s 2 m - 2) 2 £l, ^2 Variable c o e f f i c i e n t s of q and q respe c t i v e l y i n the extension of Forchheimer's equation, eq. 3 . 2 . 7 . e 1 has 1 9 9 units sm , z<i has units s m . 6 S o i l water content, (m m ) 8 S o i l water content measured by the neutron probe before the 3 3 s t a r t of i r r i g a t i o n . (m m- ) Water content of the s o i l j u s t before the water table r i s e s 3 3 i n i t . (m m- ). Used to calculate Kgff by the kinematic wave model. 6 «. „«. S o i l water content measured by the neutron probe during S L . S u . 3 — 3 steady state, ( m m ). Used to calculate Kgff by the kinematic wave model, p Dynamic v i s c o s i t y of the f l u i d . (kg s - 1 m - 1) v Kinematic v i s c o c i t y of the f l u i d , (m s - ) P P F l u i d density, (kg m - 3) ^ Pressure head, (m) OJ Angle that the bed makes with the ho r i z o n t a l . (Degrees) i n s.z, ( x v ) ACKNOWLEDGEMENTS I w i s h t o t h a n k my s u p e r v i s o r , D r . J a n de V r i e s , f o r g u i d a n c e , a s s i s t a n c e a n d f r i e n d s h i p . D r . d e V r i e s p r o v i d e d m a n y o f t h e i d e a s f o u n d i n t h i s t h e s i s a n d h e l p e d g r e a t l y w i t h f i e l d w o r k . T h a n k s a r e a l s o d u e t o t h e m e m b e r s o f my c o m m i t t e e , D r s . 0. S l a y m a k e r , A . F r e e z e a n d M. Q u i c k f o r a d v i c e a n d s u g g e s t i o n s . D r . Q u i c k ' s a s s i s t a n c e w i t h a n u m b e r o f t e c h n i c a l p r o b l e m s was e s p e c i a l l y a p p r e c i a t e d . I am v e r y g r a t e f u l t o D r . T . A . B l a c k w h o s e d o o r w a s a l w a y s o p e n w h e n I was i n n e e d o f h e l p o r e n c o u r a g e m e n t . E n c o u r a g e m e n t f r o m my c o l l e a g u e s , M e s s r s C o m p t o n P a u l , J o h n H e i n o n e n , T r e v o r M u r r i e a n d M e n s a h B o n s u , a n d f r o m T . D . N g u y e n h a s b e e n a g r e a t h e l p d u r i n g t h e c o m p l e t i o n o f t h i s t h e s i s . I a l s o w i s h t o t h a n k J e e v a J o n a h s f o r p a t i e n t a n d e x p e r t t y p i n g . F i n a l l y , t h a n k s a r e e x p r e s s e d t o t h e N a t u r a l S c i e n c e a n d E n g i n e e r i n g R e s e a r c h C o u n c i l f o r a r e s e a r c h a s s i s t a n t s h i p a n d t o t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a f o r a K i l l a m p r e - d o c t o r a l f e l l o w s h i p . - 1 -INTRODUCTION The purpose of this work i s to shed some l i g h t on the hydrologic behavior of an open and permeable forested mountain slope s o i l such as i s found i n the University of B r i t i s h Columbia Research Forest. A major feature of t h i s s o i l i s the presence of low resistance paths due to root material. Special emphasis w i l l be placed on these features i n r e l a t i o n to the shape of the outflow hydrograph. One hypothesis of the present work i s that fast flow occurring i n the unsaturated zone i s due to the combination of the existence of low resistance paths and of concentrating elements found at or above the s o i l surface, as suggested by Whipkey (1968) and de Vries and Chow (1978). In order to test t h i s hypothesis, experimental data obtained from uniform and concentrated i r r i g a t i o n were compared. In the f i r s t chapter, the r e s u l t s of three experiments conducted on a small h i l l s l o p e plot i n the University of B r i t i s h Columbia Research Forest are discussed. The experiments consisted of i r r i g a t i n g the plot and monitoring the outflow rate, the height of the water table and the s o i l water content. For the second experiment, p l a s t i c sheets were l a i d on the ground i n order to concentrate the i r r i g a t i o n . Experimental r e s u l t s , mainly the outflow hydrograph, w i l l be used to i n f e r flow mechanisms both i n the unsaturated and the saturated zones. One tool used for t h i s purpose w i l l be a model for saturated flow over a steep bed which w i l l be c a l l e d kinematic wave model and presented i n Chapter II. The conclusions drawn w i l l lead to a d e s c r i p t i v e flow model. - 2 -In the second chapter, the kinematic wave model w i l l be presented. While recognizing that the term "kinematic wave model" may be used for a multitude of models using a kinematic wave equation, i n the present work, the term w i l l denote ex c l u s i v e l y the model presented i n Chapter I I . The kinematic wave model w i l l be used to calculate an e f f e c t i v e hydraulic conductivity for the saturated zone of the p l o t . A search of the l i t e r a t u r e showed that this i s the f i r s t time the kinematic wave model has been used as a tool to i n f e r flow mechanisms and to obtain an e f f e c t i v e hydraulic conductivity from the outflow hydrograph. A major assumption of the kinematic wave model i s the presence of a thin saturated zone which represents a water table approximately p a r a l l e l to the bed. One of the requirements for a thin saturated zone i s fast flow. If however, flow i s so fast that i t i s nonlinear, the kinematic wave model f a i l s due to f a i l u r e of Darcy's law. In t h i s respect Chapter III starts where Chapter II ends since nonlinear flow i s the topic of Chapter I I I . In Chapter I I I , a b r i e f t h e o r e t i c a l discussion based on the l i t e r a t u r e i s of f e r r e d . Then, using experimental r e s u l t s obtained by de Vries (1979), the p o s s i b i l i t i e s of nonlinear flow i n both the unsaturated and the saturated zones are investigated. - 3 -CHAPTER I THE INFERENCE OF FLOW MECHANISMS FROM THE OUTFLOW HYDROGRAPH 1.1 Introduction The shape of the outflow hydrograph for a h i l l s l o p e i n part depends on the integrated e f f e c t of the travel of i n f i l t r a t e d rainwater from the ground surface to the stream bank. While i t i s not possible to i n f e r every d e t a i l about flow mechanisms and pathways from the outflow hydrograph, some s a l i e n t features of underground flow can a f f e c t the outflow hydrograph i n a noticeable manner. Indeed, Nagpal and de Vrie s (1976) state that "one of the objectives of research i n s o i l hydrology i s to obtain a better understanding of the r e l a t i o n s h i p between rainstorm c h a r a c t e r i s t i c s , s o i l hydrologic behavior, and streamflow c h a r a c t e r i s t i c s . " Also, Chamberlin (1972), i n a q u a l i t a t i v e fashion, r e l a t e s stream behavior to the nature of the s o i l when he states "an open s o i l explains the very flashy response of coastal streams to p r e c i p i t a t i o n events." Thus, the focus of t h i s chapter i s the inference of water flow mechanisms from the outflow hydrographs measured for an experimental plot located on a mountain slope. This plot i s situated i n the Uni v e r s i t y of B r i t i s h Columbia Research Forest (hereafter c a l l e d the Forest for s i m p l i c i t y ) . At most locations i n the Forest, a permeable and open B horizon rests on a bed of compacted and therefore slowly permeable t i l l . One meter i s a representative depth for the B horizon. - 4 -A major s o i l feature contributing to the open and permeable nature of the B horizon of the Forest s o i l i s the presence of low resistance paths. Some low resistance paths found i n the B horizon are long and continuous. They w i l l be c a l l e d p i p e l i k e low resistance paths, even though they do not n e c e s s a r i l y have a c y l i n d r i c a l shape. P i p e l i k e low resistance paths are due mainly to root material. Other low resistance paths are short and are associated with the presence of stones. In low resistance paths, water moves at zero or near zero pressure head. A number cannot be assigned to the pressure, e s p e c i a l l y because some of the p i p e l i k e low resistance paths are f i l l e d with a high condu c t i v i t y organic material. The rapid flow of water i n these paths, by-passing the s o i l matrix i n the unsaturated zone, i s c a l l e d s h o r t - c i r c u i t i n g . Finger flow w i l l be included i n the term. One of the hypotheses that w i l l be tested i n t h i s chapter i s that s h o r t - c i r c u i t i n g i s an important mechanism of water flow i n the Forest s o i l . Finger flow i s discussed i n Section 1.2, point ( i i i ) . Low resistance paths at the i n t e r f a c e between the B horizon and the bed contribute to quick saturated flow p a r a l l e l to the bed. A p a r t i c u l a r subject of i n v e s t i g a t i o n was flow i n low resistance paths within the unsaturated zone. Water cannot enter a low resistance path unless i t i s at a near zero pressure head. It i s thus expected that an increased supply of free water would f a c i l i t a t e flow of water i n low resistance paths. This hypothesis was tested by concentrating i r r i g a t i o n water and determining whether concentration had any e f f e c t on the behavior of the water table and on the shape of the outflow - 5 -hydrograph. Concentration was achieved by laying p l a s t i c sheets on the ground. The rati o n a l e for supplying free water i n t h i s fashion was the fac t that Whipkey (1968) and de Vries and Chow (1978) observed that free water i s supplied by natural concentrating elements such as the tree canopy and logs. The experimental plot was located in the UBC Research Forest. It was equipped with standpipes and i r r i g a t e d with s p r i n k l e r s . A channel was dug at the bottom of the plot and outflow was measured using a tipp i n g bucket. The s i t e was i r r i g a t e d three times during the summer of 1979. Each i r r i g a t i o n lasted three days. These three experiments w i l l be denoted by EX1, EX2 and EX3. For Ex2, p l a s t i c sheets were l a i d on the ground. In the present chapter, the outflow hydrograph i s used to i n f e r flow mechanisms i n both the unsaturated and the saturated zones. Direct observation of the hydrograph i s one approach that w i l l be used in order to achieve t h i s goal. Another approach that w i l l be used i s the comparison of the observed hydrograph with the hydrograph obtained by the kinematic wave model presented i n Chapter I I . Visual observation of flow and of the s o i l w i l l be used to complement conclusions. 1.2 Literature Review In t h i s section, previous work on the concept of r e l a t i n g the shape of the hydrograph to flow mechanisms w i l l be presented. Beven's (1982) model w i l l be b r i e f l y discussed because i t integrates a model of flow i n the saturated zone, similar to the one used for the present work, with a - 6 -model of flow i n the unsaturated zone. F i n a l l y , l i t e r a t u r e on nonuniform flow, and in p a r t i c u l a r on flow i n low resistance paths w i l l be reviewed. ( i ) Linking the Hydrograph Shape to Basin Properties and Flow  Mechanisms An early example of attempts to l i n k physical c h a r a c t e r i s t i c s of a watershed to geometric aspects of the unit hydrograph i s the synthetic unit hydrograph (see for example Gray, 1970). In unit hydrograph synthesis, the shape of the unit hydrograph i s obtained from physical c h a r a c t e r i s t i c s of the watershed l i k e the length of the main stream. In c e r t a i n cases, authors have associated the shape of the hydrograph with flow processes. A number of authors have interpreted the shape of the recession curve i n terms of physical processes involved. According to Anderson and Burt (1980), Barnes (1939) was able to d i s t i n g u i s h between the parts of the recession due to overland flow, subsurface stormflow and baseflow by p l o t t i n g the outflow hydrograph on semilogarithmic graph paper. The same authors mention that using the recession curve, Ineson and Downing (1964) subdivided baseflow recession into discharge from areas close to the stream and discharge from areas further away from the stream. F i n a l l y , Hewlett and Hibbert (1963) associated the f i r s t major limb observed for the recession on a double logarithmic graph with saturated flow and the second with unsaturated flow. This i n t e r p r e t a t i o n was substantiated by piezometers and tensiometers readings. - 7 -However, Anderson and Burt caution that changes i n the shape of the recession curves can be due to the method of p l o t t i n g and thus be spurious. They also caution that changes i n physical processes may not n e c e s s a r i l y show i n recession curves. They conclude that "runoff processes generating recession flow cannot be inferred from graphical p l o t t i n g techniques." Dunne and Black (1970) inferred the importance of channel p r e c i p i t a t i o n by noticing how se n s i t i v e the outflow hydrograph was to f l u c t u a t i o n s i n r a i n f a l l i n t e n s i t y and by comparing the observed outflow volume with the amount of p r e c i p i t a t i o n f a l l i n g onto the stream's channel. In a s i t e close to the one chosen for the present study, and situated on the same h i l l s l o p e , Nagpal and de Vries (1976) connected the sharp r i s e of the hydrograph with star t of outflow from root channel o u t l e t s along the streambanks. For further reference, the s i t e used by these authors w i l l be c a l l e d "Nagpal and de V r i e s ' s i t e . " ( i i ) Beven's (1982) Model: A Simple Physically Based Model for  Unsaturated and Saturated Flow in Soil Over a Steeply  Sloping Bed Beven (1982) obtained a simple flow model by coupling flow i n the unsaturated zone with flow i n the saturated zone. The model of flow for the saturated zone i s s i m i l a r to the kinematic wave model presented i n Chapter II and which i s used in the present chapter, except that i n Beven's (1982) work, the satiated hydraulic conductivity and the - 8 -porosity decrease with depth. The term "satiated hydraulic conductivity, used by M i l l e r and Bresler (1977) designates the p r o p o r t i o n a l i t y factor between the flow v e l o c i t y and the hydraulic gradient in Darcy's law (Eq. 1.3.1.1), under conditions such that the porous medium has the highest possible water content given entrapped a i r . It w i l l often be referred to as hydraulic conductivity for s i m p l i c i t y . Beven (1982) also compares the outflow hydrograph thus obtained with the outflow hydrograph obtained experimentally by Weytnan (1970, 1973) for a slope segment. He states that the model reproduces the observed hydrograph very well. In Section 1.5.3, Beven's work w i l l be discussed i n r e l a t i o n to the present work. ( i i i ) Nonuniform Flow Two kinds of nonuniform flow have been observed by various authors: finger flow and flow i n low resistance paths. Whereas finger flow i s confined to the unsaturated zone, flow i n low resistance paths can occur in both the saturated and unsaturated zones. The term " s h o r t - c i r c u i t i n g " was introduced by Bouma and Dekker (1978) to describe flow occurring through v e r t i c a l interpedal voids (> 2 mm) i n a clay s o i l , thus bypassing the s o i l matrix. In the present work, the term w i l l be extended to include finger flow i n addition to flow i n low resistance paths within the unsaturated zone. The reason for doing so i s that finger flow, although i t does occur within the s o i l matrix, i s concentrated in such a way that i t bypasses a s i g n i f i c a n t part of the matrix. - 9 -The term "finger" was used by H i l l and Parlange (1972) to describe concentrated flow in sands as a r e s u l t of wetting front i n s t a b i l i t y . In the present work, fingers w i l l be due to concentrated inflow, not to wetting front i n s t a b i l i t y . Fingers are approximately c y l i n d r i c a l regions of concentrated i n f i l t r a t i o n o r i g i n a t i n g from a point source of water. The water content within fingers i s much higher than at l o c a t i o n s where uniform i n f i l t r a t i o n occurs. In f a c t , s o i l within a finger may be saturated. When i r r i g a t i o n occurs as point i r r i g a t i o n , fingers may occur. Smith (1967) observed i n f i l t r a t i o n i n sand due to a point source. He obtained "filaments" or fingers of water formed i n an i n i t i a l l y dry sand by drops of water applied to the s o i l surface with a burette. A finger can i n some way be considered as a low resistance path that water created for i t s e l f since a higher water content means a higher hydraulic conductivity. For c l a r i t y , the term "low resistance path" w i l l , however, be used only in the sense of a s t r u c t u r a l feature. An extensive l i t e r a t u r e review of flow in low resistance paths i s found i n Beven and Germann (1980). It i s i n t e r e s t i n g to note that the e a r l i e s t reference on low resistance paths they quote i s more than one century old; Schumacher (1864) states that "the permeability of a s o i l during i n f i l t r a t i o n i s mainly controlled by big pores. . . ." Both whipkey (1968) and Aubertin (1971) conducted experiments i n p l o t s of coarse textured and of medium to f i n e textured s o i l i n the Allegheny Plateau (USA). Low resistance paths were "roots, root channels, small animal and earthworm burrows, and s t r u c t u r a l cracks" (whipkey, 1968). Their r e s u l t s agree on several points, namely: - 10 -i . flow i n low resistance paths i s more important i n the medium to fine textured s o i l that i n the coarse textured s o i l , i i . outflow from a uniformly wetted zone i s more important i n the coarse textured s o i l , i i i . outflow from the medium to fine textured s o i l occurs e a r l i e r , i v . the s o i l matrix i n the case of the medium to fine textured s o i l wets l i t t l e and/or l a t e . Also, from Whipkey's r e s u l t s , outflow from medium to fine textured s o i l ends much e a r l i e r . As Whipkey (1968) pointed out, a l l these r e s u l t s are consistent with flow i n low resistance paths occurring i n the medium to fine textured s o i l but not i n the coarse textured s o i l . One reason may be that i n the f i n e r material i t i s easier to obtain free water due to the low conductivity of the s o i l matrix. (Water must be at zero or low pressure head i n order to enter low resistance paths since these have a zero or low water entry pressure head.) Both Aubertin and Whipkey suggest that water concentrates on top of the mineral surface and enters low resistance paths there. Beasley (1976) also observed flow i n low resistance paths (decayed roots) i n a forested region i n northern M i s s i s s i p i . In addition to v i s u a l observations, evidence was offered by i . A subsurface flow v e l o c i t y estimated to be 9.3 x 10 ms . i i . "Timing of subsurface flow peaks (...) unrelated to antecedent s o i l moisture". From Beasley's fi n d i n g s , i t appears that flow i n low resistance paths probably occurred within the unsaturated zone. - 11 -Chamberlin (1972) studied water flow i n a forested area i n the Seymour watershed near Vancouver. There was extensive root development and the average s o i l depth was 1 m. He also obtained some evidence of flow i n root channels i n both the unsaturated and the saturated zones. Evidence was i . For the saturated zone: - A hydraulic conductivity of 9.7 x 1 0 - 5 ms - 1. i i . For the unsaturated zone: - Tensiometers located at a depth of 1 m sometimes responded before tensiometers at a depth of 0.70 m. - Tensiometers at 0.70 m and at 1 m responded either very slowly or very r a p i d l y . Note however that flow may have occurred along the tensiometer access tubes, i n v a l i d a t i n g the data, de V r i e s and Chow (1978), i n a plot located i n the Seymour watershed (Greater Vancouver Water D i s t r i c t ) noticed i r r e g u l a r patterns of pressure head during i r r i g a t i o n of a forest s o i l equipped with tensiometers. These authors suggest that these patterns are due to flow through root channels. The patterns could also be due to l o c a l i z e d hydrophobicity. In t h e i r s i t e Nagpal and de Vries (1976) noticed flow out of root channels at the streambank. In p a r t i c u l a r , they calculated a Reynold's number larger than 2000 for flow out of one root channel. They point out that "assuming the nature of flow i n the rootchannel to be s i m i l a r to that i n a pipe", t h i s number indicates "the existence of turbulent flow conditions within the rootchannel." - 12 -Mosley (1979), i n a beech-podocarp-hardwood forest i n New Zealand observed various low resistance paths. Among the works discussed above, the one that o f f e r s the strongest evidence that flow i n low resistance paths can occur within the unsaturated zone i s the work of Whipkey. The difference between time to outflow in the medium to fine textured s o i l and i n the coarse textured s o i l i s so large that i n the medium to fin e textured s o i l the flow must have occurred i n low resistance paths throughout the major part of i t s t r a v e l i n the s o i l . At least part of the flow i n low resistance paths must have occurred i n the unsaturated zone. A high hydraulic conductivity layer within the s o i l p r o f i l e can be another kind of low resistance path. Mosley observed flow of dyes i n the forest mentioned e a r l i e r . At some places, he observed that "a large proportion [of the dye] runs downslope above the surface of the A-horizon [which i s the top of the mineral s o i l ] and within i t s top 1-2 cm, which i s more loosely aggregated and more porous than the rest of the mineral s o i l . " Utting (1978), i n Nagpal and de V r i e s ' s i t e noted the presence of a rootmat on top of the t i l l and root channels i n the B horizon. He stated these organic zones to have a very high satiated hydraulic c o n d u c t i v i t y . Natural pipes are a form of low resistance paths observed by Pond (1971). According to Atkinson (1978), Pond (1971) reported that i n the grass-covered Nant Gerig watershed (U.K.), natural pipes were located less than 0.30 m below the surface. Water flowing in pipes could be - 13 -heard gurgling beneath the surface a f t e r a storm. A natural pipe in the technical sense i s formed by "piping", a "process of continued backward erosion" (Sowers and Sowers, 1970). It i s not c e r t a i n i f the pipes referred to by Atkinson have been formed by this process. Atkinson also stated that "the hydrographs from upland pipes are often flashy." In the Seymour watershed (near Vancouver, B.C.), Chamberlin (1972) noticed "small (...) ephemeral channels (. . . ) , controlled by bedrock morphology, large trees, and rock outcrops." He noted that these channels were "at times exposed and at times underground." 1.3 Material and Methods 1.3.1 S i t e Description Figure 1.3.1.1 shows the experimental plot with the l o c a t i o n of the instruments set-up. The plot (49° 18' N, 122° 35'W) was located i n the University of B r i t i s h Columbia Research Forest which i s 45 km east of Vancouver, B.C., Canada. It i s about 350 m above sea l e v e l . The climate i s P a c i f i c Marine Humid with 2 to 3 m of p r e c i p i t a t i o n annually. More than 85% of the p r e c i p i t a t i o n occurs as r a i n . Vegetation consists of mature western hemlock, balsam f i r , Douglas-fir, and red cedar. The average slope of the plot i s 30°. The plot area i s 50 m . The s o i l i s a humo-ferric podzol. The B horizon i s underlain p a r t l y by compacted t i l l and p a r t l y by bedrock. This low conductivity layer forms the bed. The forest f l o o r depth varies between 0.05 and 0.20 m. F i g . 1.3.1.1 Site layout. See p. 15 for explanation. - 15 -Sprinkler Automatic Raingage Funnel raingage Standpipe Neutron probe access tube Walking board P l a s t i c frame - 16 -There i s no ground cover vegetation. Six logs or groups of logs i n various states of decay are present on the s o i l surface. They cover an 2 area of about 3 m which represents 6% of the plot area. Some of these logs may have acted as natural concentrating elements. There are also seven l i v e trees. The top of the forest f l o o r (L horizon) consists mainly of needles. The F horizon consists of s t r a t i f i e d organic material (from recognizable twigs to amorphous decayed m a t e r i a l ) . The bottom layer of the forest f l o o r (H or humified layer) and the top layer of the mineral s o i l (Ae or eluviated or leached layer) together form a layer of lower conductivity (de Vries and Chow 1978). Pieces of wood create b a r r i e r s to flow i n the forest f l o o r . S o i l depth (including the forest f l o o r ) v aries from 0 to 1.65 m. A minimum depth of 0 was found at the south t i p of the channel at the l o c a t i o n of a rock outcrop, and a maximum depth of 1.65 m i n the middle of the upper part of the p l o t . The outcrop occurs at the channel's bank which was not i r r i g a t e d . In the i r r i g a t e d region, minimum s o i l depth may have been greater. The texture of the B horizon at Nagpal and de Vries' s i t e has been determined to be sandy loam (Nagpal and de V r i e s , 1976) and i s considered to be the same for the plot used for the present study. The B horizon i s very strongly structured with many l i v e and decayed roots, as well as with stones and cemented aggregates. Its structure i s very stable. A larger amount of root material i s found at the i n t e r f a c e of the B horizon with the bed than in the rest of the B horizon. - 17 -At the top of the B horizon, the r a t i o of stones to bulk s o i l by volume was found to be 0.14. Also, the porosity of the top part of the B horizon i s 0.63. (Calculations are shown i n Appendix A. For the d e f i n i t i o n s and symbols used for porosity and e f f e c t i v e porosity, see Appendix G). Studies on two i n s i t u cores gave a satiated hydraulic conductivity of 8 x 10 ms (Heinonen, pers. comm.) for the upper part of the s o i l p r o f i l e , including the forest f l o o r . The cores had a diameter of about 0.25 m and a length of about 0.30 m. The method i s described i n Baker and Bouma (1976). Later mention of " i n s i t u cores" w i l l r e f er to these measurements. The lower part of the forest f l o o r may have a satiated hydraulic conductivity smaller than the one of the upper part of the B horizon, as was noticed by de Vries and Chow (1978) i n the Seymour Watershed of the Greater Vancouver Water D i s t r i c t . Therefore the upper part of the B horizon may have a satiated hydraulic conductivity larger than 8 x 10 - l t ms - 1. 3 1 A s i m i l a r value of 10" ms was found by Harr (1977) at 0.10 m depth i n the s o i l of a Douglas-fir and western hemlock forest i n western Oregon. The s i m i l a r i t y of these values i s i n t e r e s t i n g i n view of the s i m i l a r i t y i n vegetation and climate of the two places. F i g s . 1.3.1.2 a to c show the c h a r a c t e r i s t i c flow curves obtained by Nagpal and de Vries (1976) on t h e i r s i t e . The forest f l o o r was 0.15 m deep. In the present work, K designates the satiated hydraulic conductivity whereas K( ip) and K(9) designate the unsaturated conductivity as a function of the pressure head ^ and of the s o i l water content 6 r e s p e c t i v e l y . For s i m p l i c i t y , the satiated hydraulic - 18 -T 1 1 1 r 11 2 3 i 9 15 mDEPTH F i g . 1.3.1.2a P a r t i a l hydraulic conductivity curve for the 0.11, 0.15, 0.19 and 0.23 m depths on Nagpal and de Vries* s i t e . Adapted from Nagpal and de Vries (1976). (Forest f l o o r was 0.15 m deep.) - 20 -1 • i • • -1*0 -0«8 -0«6 -0»4 -0»S PRESSURE H E A D ( M ) F i g . 1.3.1.2c P a r t i a l retention curves for the 0.025, 0.07, 0.15 and 0.23 m depths for Nagpal and de Vr i e s ' s i t e . Reproduced from Nagpal and de Vries (1976). (Forest f l o o r was 0.15 m deep.) - 21 -conductivity w i l l be referred to as hydraulic conductivity. Porous media w i l l be considered to be i s o t r o p i c and homogeneous whenever Darcy's law i s used. When the uniformity of K i s i n question, the value of K can be considered to be an o v e r a l l value. The assumption of isotropy i s not good for the forest plot but has been considered to be a necessary s i m p l i f i c a t i o n . Also, flow i s assumed to be l i n e a r whenever the phrase "satiated hydraulic conductivity" or "hydraulic conductivity" or the symbol "K" i s used. Otherwise the phrase "general hydraulic conductivity" or the symbol "Kg e n" i s used (see Chapter I I I ) . The K(^) curve ( F i g . 1.3.1.2b) has been obtained from Nagpal and de Vries' K(6) and curves. The satiated conductivity point on the K(^) curve comes from the value obtained in the present work. This i s why a dotted l i n e has been used to j o i n the l a s t 2 points. The lower part of the K(9) curves for 0.19 m and 0.23 m are so steep that small differences i n 6 y i e l d large differences i n K. This suggests that extrapolation of the curves to another s i t e and even from one place to another within a s i t e i s questionable. However, the curves p a r a l l e l each other, suggesting that their general shape i s s i m i l a r . Because the s o i l at Nagpal and de Vries' s i t e i s s i m i l a r to the one at the present s i t e , i t i s suggested that the general shape of the c h a r a c t e r i s t i c curves for the present s i t e i s similar to the general shape of the c h a r a c t e r i s t i c curves obtained by Nagpal and de V r i e s . It i s i n t e r e s t i n g to compare the value of the satiated hydraulic condu c t i v i t y obtained for the Forest plot with values obtained for an a g r i c u l t u r a l watershed. - 22 -Betson et a l . (1968) obtained satiated hydraulic conductivities of 7 x 1 0 _ D to 2 x 10"- ms - for a clay loam A horizon i n a North Carolina a g r i c u l t u r a l watershed. Table 1.3.1.1 shows res u l t s of measurements of satiated hydraulic c o n d u c t i v i t i e s at two outflow points along the channel. Outflow was from a natural storm event. These outflow points were i n the zone r i c h i n root material found at the interface of the B horizon with the bed. It was observed that the outflow points themselves coincided with an accumulation of root material. Hydraulic conductivity values were obtained by measuring the outflow rate and the area of outflow at these points. The water table was assumed to be p a r a l l e l to the bed (see Chapter II) and the slope of the bed was assumed to be the same as the surface slope of the p l o t . It w i l l be shown i n Chapter II that for t h i s case, Darcy's law f o r homogeneous i s o t r o p i c porous media, q = K § ± 1.3.1.1 ds becomes approximately q = K sinw 1.3.1.2 where q = Discharge per unit area or macroscopic flow v e l o c i t y , often c a l l e d Darcy's v e l o c i t y . In the present work, q i s i n the main flow d i r e c t i o n . h = hydraulic head s = distance dh -^g = rate of change of h with s. Since i n the present work, q i s i n the main flow d i r e c t i o n , and the porous media are considered to be i s o t r o p i c and homogeneous, i s the macroscopic hydraulic gradient - 23 -K = s a t i a t e d h y d r a u l i c c o n d u c t i v i t y a) = angle t h a t the bed makes w i t h the h o r i z o n t a l I t i s not c e r t a i n homogeneity i s a c t u a l l y r e q u i r e d f o r Eq. 1.3.1.1 t o h o l d and f o r — to be the m a c r o s c o p i c h y d r a u l i c g r a d i e n t . T a b l e 1.3.1.1 shows the r e s u l t s . The a r e a of o u t f l o w was d i f f i c u l t t o measure and f o r t h i s r e a s o n , two v a l u e s per o u t f l o w p o i n t a r e g i v e n c o r r e s p o n d i n g to a maximum and a minimum a r e a . K i s found t o v a r y from about 5 x l O - 4 to about 5 x 1 0 - 3 m s - 1 . Because p o s s i b l y too s m a l l an o u t f l o w a r e a was used f o r the h i g h e r number, t h i s number may be an upper l i m i t . On the o t h e r hand, the s m a l l e r K, b e i n g s m a l l e r than the K found w i t h the i n s i t u c o r e method may be too s m a l l . Indeed, i f i t i s assumed t h a t at l e a s t a major p r o p o r t i o n of the water f l o w i n g out of an o u t f l o w p o i n t has been t r a v e l l i n g t hrough a low r e s i s t a n c e p a t h , i t i s s u r p r i s i n g t h a t the K measured at the o u t f l o w p o i n t i s s m a l l e r than the K found by i n s i t u c o r e s . I t i s thus p o s s i b l e t h a t the a c t u a l two K's t h a t s h o u l d be o b t a i n e d at the two o u t f l o w p o i n t s are b r a c k e t e d by the two v a l u e s g i v e n h e r e . T a b l e 1.3.1.1 E s t i m a t e of s a t i a t e d h y d r a u l i c c o n d u c t i v i t y K at two o u t f l o w p o i n t s . E s t i m a t e d area of q = d i s c h a r g e D i s c h a r g e ouflow per u n i t a r e a K = —3— = -3— 6 v sinw .5 m s m ms ms O u t f l o w 4.7 x 10~ 6 0.18 x 0.03 8.7 x 1 0 - 4 2 x 1 0 - 3 P o i n t 1 0.18 x 0.01 2.6 x 10~ 3 5 x 10~ 3 r ( 1 ) , , O u t f l o w 2.7 x 10 0.01 2.7 x 1 0 _ q 5 x 1 0 - 4 P o i n t 2 0.26 x 0.01 1.1 x 10~ 3 2.1 x 10~ 3 ^0.26 m x a v a r y i n g h e i g h t . - 24 -Note that i t i s shown i n Chapter III that there i s a p o s s i b i l i t y that flow at these outflow points may have been nonlinear, i n which case Darcy's law should not be used. For the above calc u l a t i o n s of K, i t has been necessary to assume that the n o n l i n e a r i t y , i f existant, i s n e g l i g i b l e . Figure 1.3.1.3 shows the topography of bedrock or compacted t i l l obtained from standpipe depths. Values are v e r t i c a l distances from a datum to the t i p of the pipes. S o i l sampling with a tube indicated the t i p of standpipes 2, 4, 5, 6 and 10 to be restin g on the low conductivity l a y e r . It i s not c e r t a i n what the other pipes are r e s t i n g on (see Section 1.3.5). At Nagpal and de Vries' s i t e , Utting (1978) found the satiated hydraulic conductivity of the compacted t i l l to be 10" ms to 10" -1 ms . The plot used for the present work (see F i g . 1.3.1.1.) was about 2 8 x 7 m . In order to i r r i g a t e a d e f i n i t e area, the plot was surrounded with p l a s t i c . Water f a l l i n g onto the p l a s t i c was routed away from the p l o t . Some walking took place on the plot but very early during the phase of instruments i n s t a l l a t i o n boards were placed a few cm above the ground surface i n order to avoid compaction. I r r i g a t i o n was provided by 2 impact s p r i n k l e r s . Sprinkling rates were measured by 2 tipping bucket raingages and 11 funnel raingages. Water table elevation was measured with 10 standpipes, 2 of which were automatically recording. The outflow c o l l e c t i o n and measurement system consisted of a channel dug at the foot of the plot which was connected to a tipping bucket flow meter with metal troughs. The c o l l e c t i o n channel was covered with p l a s t i c . - 25 -O 20O0 m m F i g . 1.3.1.3 Contours of the bed (bedrock or compacted t i l l ) obtained from pipes depth. Values i n mm. Values are v e r t i c a l distances from a datum situated below the t i p of the pipes to the t i p of the pipes. - 26 -1.3.2 Sprinklers The 2 Impact sp r i n k l e r s were i n s t a l l e d at a spacing of 6.25 m. They were adjusted to sweep approximately half a c i r c l e and the i r ranges overlapped. Range was adjusted for uniformity according to v i s u a l observation at the beginning of Exl and no further adjustments were made. 1.3.3 Raingages, Pooled Irrigation and Time Averaged Pooled  Irrigation Two tipping bucket raingages (denoted by Al and A2) were placed at about 0.40 m (A2) and 0.50 m (Al) above ground l e v e l . The diameter of th e i r c o l l e c t i o n area was 0.20 m. The volume of one tipping bucket was 6 3 approximately 15.5 x 10" m . These raingages were connected to an event recorder. The variance of the time between tip s was zero for constant inflow. This shows that unsteadiness during the experiments was not due to raingage malfunction but to actual unsteadiness i n i r r i g a t i o n rate. The 11 funnel raingages ( i n d i v i d u a l l y denoted by RG1, RG2 etc.) were constructed from 1 l i t e r p l a s t i c b o t t l e s . A hole i n the l i d accommodated a p l a s t i c funnel. The diameter of the c o l l e c t i o n area was about 0.10 m. Funnel raingages were usually read when they were more than ha l f f u l l . Readings were made every 3 to 6 hours or more. Pooled i r r i g a t i o n rates for every .1 hour were calculated using Thiessen polygons and then plotted. For the thir d experiment RG11 had been removed from the plot and this resulted i n a pooled i r r i g a t i o n rate apparently higher for Ex3 than for Exl and Ex2. - 27 -The pooled i r r i g a t i o n rate for Ex3 has been computed by using the remaining raingages on the smaller area covered by these raingages, the area covered by RG11 being l e f t out. Each remaining raingage was covering the same area as during Exl and Ex2. A s i m i l a r method was used to obtain i r r i g a t i o n rates without RG11 for Exl and Ex2. Some of the funnel raingages data were a c c i d e n t a l l y l o s t . When t h i s happened, the average rate measured at a given raingage during a given experiment was used to replace the missing values of t h i s raingage for t h i s experiment. For the water balance, a time average of the pooled i r r i g a t i o n rate was needed. I r r i g a t i o n rate i s not constant with time so that, when an average i s taken for a c e r t a i n period, i t depends upon the period's duration. An e f f o r t was made to choose the period i n a r a t i o n a l way (see Section 1.4.6). Time averages of the pooled i r r i g a t i o n rates were obtained by positioning a thread on the i r r i g a t i o n graphs (Figs. 1.4.2.3 to 1.4.2.5) i n such a manner that the area between Rp and the thread above the thread looked approximately equal to the area between Rp and the thread below the thread, Rp being the pooled i r r i g a t i o n rate. Time average values of pooled i r r i g a t i o n rates showed the r a t i o i r r i g a t i o n rate with RG11 to i r r i g a t i o n rate without RG11 to be approximately 0.8 for Exl and Ex2 (see Table 1.4.6.1). Pooled i r r i g a t i o n rates for Ex3 were corrected by multiplying them by 0.8 for the water balance, since i t was desired to estimate the i r r i g a t i o n rate that would have been obtained had RG11 been a v a i l a b l e . Errors i n the i r r i g a t i o n rate are due to nonuniformity of i r r i g a t i o n and to an i n s u f f i c i e n t number of raingages, e s p e c i a l l y i n the - 28 -plot's f r i n g e s . A lower i r r i g a t i o n rate on the fringes of the plot was e s p e c i a l l y responsible for nonuniformity. This can be appreciated by the fact that when i r r i g a t i o n i s obtained without the area covered by RG11, a time average pooled i r r i g a t i o n rate 1/0.8 larger i s obtained, as shown i n Table 1.4.6.1. The Thiessen polygon covered by RG11 has a size equal to 1/4 the size of the plot and i s the bottom fringe of the p l o t . The d i f f e r e n c e i n pooled i r r i g a t i o n rate due to the absence of t h i s polygon thus gives an i n d i c a t i o n of the importance of recording c o r r e c t l y the i r r i g a t i o n rate i n the fringes of the p l o t . Part of the reason why the Thiessen polygon covered by RG11 i s so important for the o v e r a l l i r r i g a t i o n rate i s that i t represents such a large proportion of the t o t a l area. 1.3.4 Neutron Probe Access Tubes Two neutron probe access tubes were i n s t a l l e d i n the same way as the standpipes (see Section 1.3.5). Readings were made at depths of 0.15, 0.30, 0.50, 0.70, 0.90 and 1.10 m for the upper neutron probe access tube, and 0.15, 0.30, 0.50 and 0.63 m for the lower neutron probe access tube. There may be a systematic error due to the fact that the manufacturer's c a l i b r a t i o n and not a c a l i b r a t i o n s p e c i f i c for the places where measurements were taken was used. This error does not a f f e c t s o i l water content differences at a given point. 1.3.5 Standpipes Eight small (I.D. 16 mm) and 3 large (I.D. 40 mm) standpipes were i n s t a l l e d . (They w i l l be c a l l e d pipes for short). They were - 29 -constructed from galvanized steel conduit pipes. The end of the small pipes was pinched shut to prevent clogging and s l i t s 2 mm wide were cut at t h e i r base. The end of the large pipes was l e f t open. The small pipes were numbered 1 to 9 (7 missing). The large ones were numbered 10 to 12. Pipes 10 and 11 were equipped with Stevens water l e v e l automatic recorders. There may be large time errors associated with these two pipes, therefore only t h e i r steady state readings were used. The standpipes were i n s t a l l e d i n the following manner: a hole was made by hammering down a metal rod (25 mm diameter for the small ones, 50 mm for the large ones) using a sledge hammer. Because of stones i t was not always possible to reach the compacted t i l l or bedrock and i t i s possible that some standpipes rest on stones instead. The metal rod was then removed with a j a c k - a l l . This was sometimes a d i f f i c u l t operation. A gravel f i l t e r was used to prevent clogging of the pipe i n l e t . Small pipes were used because they were easier to i n s t a l l . They also responded faster but were subject to larger measurement errors as w i l l be discussed l a t e r on. The t i p of the pipes reached depths varying from 0.50 m (pipe 8) to 1.66 m (pipe 10). Pipes 1, 11, and 12 rested on bedrock or on a stone. Pipes 2, 4, 5, 6 and 10 rested on t i l l . What the remaining pipes rested on i s unknown. The pipes were not sealed because the s o i l was vented down to the bottom of the B horizon. During E x l , the elevations of the water l e v e l s i n the pipes were measured with a p l a s t i c tube (9.5 mm O.D., 6.3 mm I.D.) holding a f l o a t . In Ex2 and Ex3 an e l e c t r i c water l e v e l detector was used. This detector consisted of a 1.5 m long, 9.5 mm O.D. p l a s t i c - 30 -tube with two bare wire ends mounted at the t i p . Contact of the wire ends with the water i n the standpipe resulted i n closure of an e l e c t r i c a l c i r c u i t and the l i g h t i n g up of an indicat o r l i g h t . The main problem that arose with the data c o l l e c t i o n was lack of r e p r o d u c i b i l i t y . For instance, the water l e v e l i n pipe 1 climbed about 50 mm during a continual series of readings while the general trend given by the graph was a r i s e of at most 12 mm i n 2 hours. Lack of r e p r o d u c i b i l i t y could be traced to 3 causes: extraneous inflow of water into the pipes, clogging of the pipes, and displacement of water inside the pipes when the f l o a t tube was used. At l e a s t part of the v a r i a t i o n can be explained by water flowing along the measuring tube into the pipe. This problem arose from the fact that the spr i n k l e r j e t had a s i g n i f i c a n t horizontal component. It presumably was aggravated by clogging of some pipes. It was found that 5 5 i n a closed small pipe water rose at a rate of 3.3 x 10" to 27 x 10" ms - 1 while the measuring tube was held i n the same way as for a measurement. This source of error i s more important for small pipes than for large ones. In May 1981 "slug tests" were performed to check the pipes for clogging. These tests consisted of pouring water into the pipes and then measuring the water l e v e l as a function of time. They indicated that pipes 4, 12 and 6 responded well and pipes 10 and 11 responded slowly. The slow response of pipes 10 and 11 does not matter since these pipes are used only at steady state. The response times of pipes 2, 8, 5 and 1 were intermediate. - 31 -Clogging i s probably responsible for the fact that pipe recessions for Ex2 are slower than for E x l . Although r a i n f a l l occurred during the recession of Ex2, several pipes show a slower recession rate before r a i n s t a r t s . It i s u n l i k e l y that r a i n f a l l i s responsible for a slower recession of the pipes for Ex2 and Ex3 since the outflow recessions of the 3 experiments are remarkably p a r a l l e l . Gradual clogging i s also supported by the behavior of pipes 3 and 9. Accordingly, these pipes are not used. It i s true that clogging seems u n l i k e l y for such a stable s o i l , and also given the fact that a gravel f i l t e r was used but that i s the only possible explanation for the observed pipe behavior. When the f l o a t tube was used, an addit i o n a l cause of error was introduced. Due to displacement, the water l e v e l inside the pipe was higher than outside the pipe. Water thus would flow out of the pipe and the water l e v e l i n the pipe would be lower for the next reading. This er r o r i s also more important for small pipes than for large ones. One s e r i e s of measurements during EX2 and one during EX3 were taken with both the e l e c t r i c detector and the f l o a t i n order to determine whether r e s u l t s agreed. Differences are less than 0.04 m except for pipes 5 and 8 where they reach 0.10 m. For these large d i f f e r e n c e s , the height given by the e l e c t r i c detector i s higher than the one given by the f l o a t , as i s the case for most of the other d i f f e r e n c e s . For these t e s t s , readings with the f l o a t detector were made aft e r readings made with the e l e c t r i c detector, so that the displacement due to the f l o a t would not influence the e l e c t r i c detector reading. - 32 -For the data a n a l y s i s , the f i r s t reading in a series of readings of water l e v e l was used because i t was considered that i t was the l e a s t affected by the extraneous inflow of water mentioned above and by displacement. Water l e v e l was often obtained in a continual series of readings. In most cases time was recorded at the end of the s e r i e s . Since the f i r s t reading of the series was used there i s a time error estimated to be < 10 minutes. 1.3.6 C o l l e c t i n g Channel and Outflow Tipping Bucket The c o l l e c t i n g channel intercepted subsurface flow over a length of 7 meters measured perpendicular to the slope. Its depth varied from 0 to 1.10 m. The channel bottom consisted of compacted t i l l i n the north reach and bedrock i n the south reach. A cement trough was constructed on top of the bedrock. The compacted t i l l reach was not cemented because i t s conductivity was judged low enough. A p l a s t i c cover prevented i r r i g a t i o n water from entering the channel. Outflow rates were measured with a tipping bucket unit (Chow, 1976) 3 3 having a maximum capacity of 2.2 x 10" m . The following problems should be kept i n mind when considering r e s u l t s obtained from the outflow tipping bucket. 1. There was some antecedent outflow for Ex2 and Ex3. 2. Spurious inflow (overland flow over the hydrophobic forest f l o o r ) may have reached the channel; t h i s happened mainly during E x l . Flow from water f a l l i n g onto the p l a s t i c plot boundaries may also have - 33 -reached the channel. This l a t t e r Inflow was assumed n e g l i g i b l e except for an extra 1.4 x 1 0 - 5 m3 s _ 1 during the r e r i s e of Ex3 t i l l 7:35 a.m., August 23. This part has been suppressed on the outflow graphs. Also, some water col l e c t e d on the p l a s t i c plot boundaries and routed out of the plot may have subsequently reached the channel v i a an underground route. Such a mechanism was observed while using a hose. The fact that the channel outflow point where outflow started during the experiments was not the same one as the one noted to y i e l d outflow while the mechanism j u s t described was operating indicates that the amount of water ( i f any) using t h i s route during the experiment did not influence the time to s t a r t of outflow. 3. Water escaping c o l l e c t i o n because the channel had not been dug far enough. 4. There i s some uncertainty as to whether the correct tipping bucket c a l i b r a t i o n s were used for Exl and Ex2. D e t a i l s are given under point 4, Appendix E. 5. The tipping bucket, i n i t i a l l y adjusted to f i l l to a c e r t a i n mark and c a l i b r a t e d for t h i s mark would gradually f i l l more and even overflow at times. The r a t i o of the volume contained by the tipping bucket when i t i s f u l l to when i t i s f i l l e d to the mark i s 4/3. Thus errors of more than 30% may have occurred since t h e o r e t i c a l l y , errors of at l e a s t 30% could have occurred when the tipping bucket overflowed. This may have happened for Exl at least once, around t = 61 hours. For Ex2, overflowing possibly occurred around t = 19.25 hours and occurred around t = 38.1 hours. This problem was corrected when noticed. No overflow occurred during Ex3. - 34 -6. There i s an inconsistency between i r r i g a t i o n , water table height, and outflow hydrograph r e s u l t s . This w i l l be discussed i n Section 1.4.6. 7. For Ex3, the tipping bucket had been p r e f i l l e d with one l i t e r . For Exl and Ex2, there i s doubt as to whether i t had been p r e f i l l e d or not. From the above, one sees that the error i n accuracy can, t h e o r e t i c a l l y , be larger than 30%. There i s , however, no way of knowing what the accuracy i s . Therefore the problem of error w i l l be approached from the point of view of whether the conclusions drawn from the data are r e l i a b l e or not. For t h i s purpose, a d e t a i l e d error analysis i s given in Appendix E. 1.3.7 Experimental Procedure A l l 3 experiments were conducted i n the same manner: the plot was i r r i g a t e d for two days i n order to ensure that steady state was reached. I r r i g a t i o n was then stopped for about 2 hours and resumed for about 21 hours. The purpose of stopping the i r r i g a t i o n in the middle of an experiment was to determine the response of the water table and of the outflow to a "dry pulse". I r r i g a t i o n times for the 3 experiments are shown i n Table 1.3.7.1 a). Being performed after two weeks of dry weather, Exl has dry antecedent conditions. Being performed a f t e r E x l , Ex2 has wet antecedent conditions. Ex3 follows Ex2 and has antecedent conditions s l i g h t l y wetter than Ex2. For Ex2, concentrated inflow of free water was supplied by placing p l a s t i c sheets on the ground. Their l o c a t i o n i s shown i n - 35 -Table 1.3.7.1 F i e l d experiments (summer 1979) a) I r r i g a t i o n times I r r i g a t i o n I r r i g a t i o n I r r i g a t i o n I r r i g a t i o n s t a r t s stops r e s t a r t s stops Exl 13:41 July 31 14:00 August 2 15:21 August 2 13:40 August 3 Ex2 11:45 August 11 15:31 August 13 17:23 August 13 14:09 August 14 Ex3 11:55 August 20 13:54 August 22 15:56 August 22 12:40 August 23 b) Main features I r r i g a t i o n Antecedent conditions Exl Uniform dry Ex2 Concentrated wet Ex3 Uniform S l i g h t l y wetter than Ex2 - 36 -Figure 1.3.7.1. The sheets were approximately rectangular and ranged i n 2 2 size from about 0.55 x 0.75 m to 0.80 x 1.40 m . The number of outflow points per sheet ranged from 1 to 4. There were 16 sheets covering 20% of the plot area. Concentration of i r r i g a t i o n for the area covered by p l a s t i c sheets, calculated as area of p l a s t i c  area of inflow from p l a s t i c i s at least 100, as shown in Appendix B. I r r i g a t i o n type and antecedent conditions are summarized in Table 1.3.7.1 b. Figure 1.3.7.2 shows the occurrence of natural r a i n f a l l during the experiments' period. R a i n f a l l records were obtained from the c l i m a t o l o g i c a l s t a t i o n 4.8 km from and about 240 m below the s i t e . Values are from 08:00 of one day to 08:00 of the next day. From what could be observed, r a i n f a l l at the c l i m a t o l o g i c a l s t a t i o n could be quite d i f f e r e n t from r a i n f a l l at the s i t e . It i s expected that t h i s i s p a r t i c u l a r l y so for r a i n f a l l before and during EX3, which was of the convective kind. I t must be noted that, for the same rate, r a i n f a l l has more e f f e c t than i r r i g a t i o n applied to the plot only, because the plot receives subsurface flow o r i g i n a t i n g from r a i n that f e l l u p h i l l . It i s impossible to know the magnitude of t h i s subsurface flow. It w i l l therefore be neglected, keeping i n mind that i n the summer, r a i n f a l l events of the magnitude commonly observed, and occurring after a long dry s p e l l are le s s l i k e l y to generate subsurface flow. xooo miWiwmamtMim mm [ 7 P L A S T I C S H 6 E - T F i g . 1.3.7.1 P l a s t i c sheets lay-out, - 38 -t o -RAIN 5- -0 J J U L Y 12- /6 1* • * ** MO RAIN > 18 20 XL 31 F i g . 1.3.7.2 Natural r a i n from c l i m a t o l o g i c a l s t a t i o n 4.8 km away. Rain i s i n mm from 0800 to 0800 the next day. - 39 -1.4 Results In t h i s section, r e s u l t s are presented. Special emphasis i s placed on the e f f e c t of i n i t i a l s o i l water content and i r r i g a t i o n concentration on the outflow hydrograph and on the behavior of the water table. In addition, recession rates obtained i n the present work are compared with r e s u l t s reported i n the l i t e r a t u r e . V isual observations carried out on low resistance paths are also presented. 1.4.1 S o i l Water Content, 6 Figures 1.4.1.1 and 1.4.1.2 show the p r o f i l e s of 9 obtained with the neutron probe. The neutron probe i t s e l f never reached free water, either because the access tubes did not reach the bed, or because of the bed topography. However, some of the readings at the lowest l e v e l s may have been influenced by a saturated zone ex i s t i n g below the probe (See Note 2, Appendix H). Because another i n i t i a l s o i l water content w i l l be introduced i n Section 2.4, the symbol Q i n i t i a l * o r &in a n ^ t n e phrase " i n i t i a l s o i l water content" w i l l mean s p e c i f i c a l l y " s o i l water content measured by the neutron probe before the start of i r r i g a t i o n . " The symbol ^steady state o r ^ s t . s t . w i l l mean " s o i l water content measured by the neutron probe during steady state." I n i t i a l water content for Exl i s not known exactly because i t was measured before 1.5 hours of preliminary i r r i g a t i o n that took place on Jul y 29, pr i o r to E x l . 40 -B (n3M~3) D E P T H ©•20 0*30 0»40 0-0 J 1 l 0-40 . 0-40-040-o-8o-EL X 1 steady sha^e dr a c r t o «»e Fig. 1.4.1.1 Water content from the neutron probe readings at the upper neutron probe access tube. See p. 42 for d e t a i l s . - 41 -DE P TH CM) o-o • l i " -ao .30 __£ . I 1 o - i o A O-O o - o o-Ao A * # ^ s l - e a c i y s tate B. E X 2 . E X 3 F i g . 1.4.1.2 Water content from the neutron probe readings at the lower neutron probe access tube. See p. 42 for d e t a i l s . - 42 -Det a i l s for Figs. 1.4.1.1 and 1.4.1.2 °initial EX1: July 28, before a preliminary i r r i g a t i o n . EX2: August 10, at 1630 EX3: August 19, at 1100 ^steady state EX1: August 2, at 1100 EX2: August 13, at 1130 EX3: August 22, at 1145 ^drainage EX1: August 5, at 1000, 44 hours af t e r i r r i g a t i o n stopped. - 43 -The 0 p r o f i l e s of Figures 1.4.1.1 and 1.4.1.2 show that 8 i n was lower for Exl than for Ex2 and Ex3, and about the same for Ex2 and Ex3. They a lso ind ica te that 9 s t . s t . w a s near ly the same for a l l three experiments. More s p e c i f i c a l l y , Q±N d i f f erences between Exl and Ex2 range from 2 to 5.5% at the 0.90 and 0.30 m depths r e s p e c t i v e l y . Both these values were obtained at the upper neutron probe access tube. Di f ferences i n 8j_n between Ex2 and Ex3 range from 0% at the 0.50 to 1.10 m depths at the upper neutron probe access tube to 2% at the 0.30 m depth at the lower neutron probe access tube. F i n a l l y , d i f f erences i n 9 ^ n between Exl and Ex3 range from 2 to 7% at the 0.90 and 0.15 m depths r e s p e c t i v e l y . Both these values were obtained at the lower neutron probe access tube. The reasons for these d i f ferences are that Exl was c a r r i e d out a f ter a dry p e r i o d , and that Ex3 followed Ex2 more c l o s e l y than Ex2 followed E x l . For Ex2 and Ex3, data obtained at the upper neutron probe access tube during steady state ind ica te the presence of a 6 s t r a i g h t of 0.35 m 3m~ 3 over the 0.30 to 0.70 m depth i n t e r v a l . For unknown reasons, the corresponding 8 values for Exl are more v a r i a b l e . These water content data ind ica te that over the 0.30 to 0.70 m depth i n t e r v a l , water that was i n t r a n s i t during the simulated r a i n f a l l event of average i n t e n s i t y 3.1 x 1 0 - 6 m s - 1 caused 8 to increase from 0.28 to 0.35 i n Ex2 and from 0.29 to 0.35 i n Ex3. Figure 1.4.1.2 i n d i c a t e s that 8 data for the lower neutron probe access tube fo l low the same t rend . F i n a l l y , i t i s of i n t e r e s t to note that these 8 s t r a i g h t values of 3 —3 3 —3 0.35 m m are wel l below 0.63 m m , which i s the poros i ty of the upper - 44 -part of the B horizon. 1 . 4 . 2 Outflow Before proceeding with the presentation of the outflow results, definitions of the terms used to describe the hydrograph are given. Fig. 1.4.2.1 shows that a hydrograph rise can be schematically broken down into three parts: the early part of the rise, the main limb of rise and the late part of the rise. Figure 1.4.2.2 shows the superposition of the three hyetographs obtained without raingage 11 for Exl, Ex2 and Ex3. It also shows the three hydrographs superposed. Figures 1.4.2.3 to 1.4.2.5 show the rainfall hyetographs obtained with RG11 for Exl and Ex2 but without RG11 for Ex3 together with the outflow hydrographs for the three experiments. RG11 had been removed from the site for Ex3. Fig. 1.4.2.1 The three parts of the hydrograph rise: (a) early part of the rise; (b) main limb of rise; (c) late part of the rise. F i g . 1.4.2.2 Superposition of the 3 hyetographs (pooled i r r i g a t i o n rates calculated without RG11) and of the 3 outflow hydrographs. I ON T I M E F R O M S T A R T O F I R R I G A T I O N ( H O U R S ) F i g . 1.4.2.3 Pooled i r r i g a t i o n and outflow rate for EX1. - Ly -T 1 1 r-1-1 1 1 1 r 2-4 48 72 9 6 T I M E F R O M S T A R T O F I R R I G A T I O N ( H O U R S ) Fig. 1.4.2.5 Pooled irrigation and outflow rate for EX3. - 49 -On Figs. 1.4.2.2 and 1.4.2.5, the r e r l s e of the outflow for Ex3 t i l l 7:35 a.m. August 23 has been omitted because of error, as explained under point 2, Section 1.3.6. Figure 1.4.2.6 shows the detailed r i s e s . The method used to a r r i v e at the data of F i g . 1.4.2.6 i s described i n Appendix D. The e f f e c t of errors on the outflow hydrographs i s discussed i n Appendix E. The e f f e c t of natural r a i n on the hydrograph r i s e s i s discussed in the same Appendix. Examination of Figs. 1.4.2.2 to 1.4.2.5 shows the outflow hydrographs present the following main features: ( i ) The main limbs of r i s e s t a r t a f t e r l i t t l e preliminary outflow. ( i i ) The main limbs of r i s e are straight l i n e s . ( i i i ) The main limbs of r i s e are steep and approximately p a r a l l e l , (iv) The recession limbs are steep and p a r a l l e l . Examination of the det a i l e d r i s e s of F i g . 1.4.2.6 shows that there i s a small i n i t i a l step for Exl and Ex3 but none for Ex2. In order to determine the e f f e c t of i n i t i a l conditions, Exl must be compared with Ex3. As mentioned in Section 1.4.1, the i n i t i a l water content i s lower for Exl than for Ex2 and Ex3. I n i t i a l conditions are also In part expressed by antecedent outflow rates. The l a t t e r was zero for E x l , smaller than 2.78 x 1 0 - 8 m 3s- 1 for Ex2, and 1.5 x 1 0 - 7 m 3 s - 1 nine hours before the s t a r t of i r r i g a t i o n for Ex3. From Figure 1.4.2.2, one can observe that the main limb of r i s e for Exl lags behind the one for Ex3. - OS -- 51 -Hydrophobicity on the surface of the forest f l o o r i s an extreme example of the e f f e c t of i n i t i a l conditions. Overland flow due to hydrophobicity was observed during the preliminary i r r i g a t i o n p r i o r to Exl and during the beginning of E x l . Overland flow was observed i n an area where the s o i l was kept dry under the p l a s t i c covering the channel. It i s not c e r t a i n whether hydrophobicity-related overland flow can take place on s o i l not protected from the r a i n and how s i g n i f i c a n t i t can be under natural conditions. In order to determine the e f f e c t of concentration on i r r i g a t i o n , Ex2 must be compared with Ex3. F i g . 1.4.2.2 shows that the main limb of r i s e for Ex2 i s approximately p a r a l l e l to the one of Ex3 and coincides with i t or lags behind i t by at most one hour. To summarize, steep and l i n e a r main limbs of r i s e s t a r t a f t e r l i t t l e preliminary outflow and are approximately p a r a l l e l . Drier i n i t i a l conditions resulted i n a l a t e r main limb of r i s e . Concentration of i r r i g a t i o n did not change the general shape of the hydrograph r i s e and did not decrease the time lag to the main limb of r i s e . Neither did i t enhance the i n i t i a l step observed for Exl and Ex3. 1.4.3 Water Table The behavior of the water table can be studied from the water l e v e l s measured at the standpipes. Table 1.4.3.1. shows the standpipes depths and indicates whether the pipes are resting on t i l l , or on bedrock or a stone. Figures 1.4.3.1 to 1.4.3.7 show the water table behavior at the 7 pipes superimposed for the three experiments. Figures 1.4.3.8 and - 52 -Table 1.4.3.1 Standpipes depth Pipe depth Resting on (m) 1 0.88 stone or bedrock 2 0.98 t i l l 4 0.97 t i l l 5 1.41 t i l l 6 1.08 t i l l 8 0.50 ? 12 0.58 stone or bedrock 10 1.66 t i l l 11 1.19 stone or bedrock - £5 -W A T E R T A B L E H E I G H T ( M _ vg -* E X JL F i g . 1.4.3.3 Water table behavior at pipe 4. W A T E R T A B L E H E I G H T ( M ) - 95 -W A T E R T A B L E H E I G H T ( M o o o • • • O N - Li -W A T E R T A B L E H E I G H T ( M ) - 85 -* E X J. x o aa U 0.1 -0-0 A LX3 0 12 24 36 48 60 72 T I M E F R O M I R R I G A T I O N S T A R T ( H O U R S ) F i g . 1.4.3.7 Water table behavior at pipe 12. - 09 -- 61 -1.4.3.9 show the r i s e of pipes 2 and 12 drawn at a larger scale for c l a r i t y . Due to the errors linked to the pipes' data, no quantitative comparison can be made of the behavior of the water table from experiment to experiment. Instead, a q u a l i t a t i v e comparison based on the graphs w i l l be made. Because the lags between the r i s e s i n general e i t h e r are large, or are n e g l i g i b l e , the q u a l i t a t i v e comparison between lags i s generally not affected by error. It w i l l be assumed that s i m i l a r i t y i n rates of r i s e i s r e a l and not a "chance outcome" of error. For the purpose of this q u a l i t a t i v e comparison, the r i s e of a pipe w i l l be considered to be the continuous r i s e to steady state. That i s , an i n i t i a l r i s e followed by a f a l l back to zero w i l l be ignored. Figures 1.4.3.1 to 1.4.3.7 show that for the three experiments, there was no water i n the pipes prior to i r r i g a t i o n . Comparison of the r i s e s f o r Exl and Ex3 using Figs. 1.4.3.1 to 1.4.3.7 shows the following: Pipes 1 and 8 r i s e as soon and as fast for Exl as for Ex3. Pipes 4 and 12 r i s e l a t e r for E x l . The rate of r i s e of pipe 4 i s slower for E x l . For pipe 12, the comparison i s d i f f i c u l t to make. Pipes 2 and 6 r i s e l a t e r for Exl but at the same rate. Pipe 5 r i s e s s l i g h t l y e a r l i e r for Exl but at the same rate. Thus the water table has some tendency to r i s e l a t e r for Exl but i n general r i s e s at the same rate. The delay i n r i s e must be due to d r i e r i n i t i a l c onditions. Comparison of the r i s e s for Ex2 and Ex3 shows that r i s e s occur at the same time for Ex2 as for Ex3, except that pipe 5 st a r t s to r i s e WATER TABLE H E I G H T ( M * I O -2 _ i _ to _1_ to H n iTim x r. x " - t o r oo - 39 -- 63 -l a t e r for Ex3. Rates of r i s e are s i m i l a r for the two experiments. Thus concentration of i r r i g a t i o n does not, i n general, y i e l d an e a r l i e r r i s e . Neither does i t y i e l d a faster r i s e . Table 1.4.3.2 gives steady state values obtained from the standpipes for the 3 experiments. These values are averages estimated by eye. Table 1.4.3.2 shows that differences between steady state values for Ex2 and Ex3 are l e s s than 2 cm. Relative differences were obtained as the r a t i o of the absolute value of the difference between the value for Ex2 and the value for Ex3 to the average of these values. The l a r g e s t r e l a t i v e d i f f e r e n c e found was 0.2 and was obtained for the shallowest pipe, pipe 12. Thus, concentrating the i r r i g a t i o n has no s i g n i f i c a n t e f f e c t on the shape of the water table at steady state. Figures 1.4.3.10 to 1.4.3.12 show the water table behavior during r i s e and recession. Pipes are shown on a cross section downslope. That i s , horizontal distances between pipes i n Figs. 1.4.3.10 to 1.4.3.12 are distances between t h e i r projections onto the ordinate of F i g . 1.3.1.1. This ordinate i s the horizontal projection of a l i n e taken in the general d i r e c t i o n of the slope. The times chosen for steady state are not n e c e s s a r i l y the times at which steady state was f i r s t reached. Problems with Figs. 1.4.3.10 to 1.4.3.12 are that the h i l l s l o p e i s not long enough and there are not enough measurement points to make a d e f i n i t e statement about the shape of the water table. Moreover, except at pipes 4 and 6, i t i s not c e r t a i n that the shape of the bed i s correct because i t i s not c e r t a i n the pipes reached the bed. - 64 -Table 1.4.3.2 Steady state values for standpipes (m); relative differences between Ex2 and Ex3 (absolute values) Pipe Exl Ex2 Ex3 Relative Differences Between Ex2 and Ex3* 1 0.114 0.120 0.120 0.0 2 0.150 0.168 0.183 0.08 4 0.066 0.081 0.090 0.10 5 0.354 0.393 0.396 0.008 6 0.174 0.201 0.210 0.04 8 0.300 0.483 0.480 0.006 10 0.318 0.336 0.336 0.0 11 0.126 0.147 0.156 0.06 12 0.030 0.039 0.048 0.20 Average 0.181 0.219 0.224 * |Ex2 - Ex3| (Ex2 + Ex3)/2 - 65 -E X 1 R I S E I X * « f C E S S I O N O IM 2M 3M ® P I P E N F i g . 1.4.3.10 Water table shape for EX1. See p. 68 for explanation. - 66 -t X 2 R I S E E X 2 R E C E S S I O N O IM 2M 3 M <g> PI P E hi Fig. 1.4.3.11 Water table shape for EX2. See p. 68 for explanation. - 67 -/ E X 3 R I S E | X 3 R E C E S S I O N O 1 M 2 M 3 M <g> PIPE N F i g . 1.4.3.12 Water table shape for EX3. See p. 68 for explanation. - 68 -Exl Rise: A: Before s t a r t of i r r i g a t i o n B: 14:18 July 31, i . e . 37 min aft e r s t a r t of i r r i g a t i o n C: 15:12 July 31 i . e . 1 h 31 min aft e r s t a r t of i r r i g a t i o n D: 03:30 August 2nd, steady state (about 38 hours a f t e r s t a r t of i r r i g a t i o n ) Recession E: Before turning i r r i g a t i o n o f f ; (12:00 August 03) F: 14:36 August 03 i . e . 56 min aft e r i r r i g a t i o n stop G: 15:12 August 03 i . e . 92 min a f t e r i r r i g a t i o n stop H: 17:00 August 03 i . e . 200 min a f t e r i r r i g a t i o n stop Ex2 Rise: A: B: C: D: Recession E: F: G: H: Before i r r i g a t i o n s t a r t 12:24 August 11 i . e . 39 min aft e r i r r i g a t i o n s t a r t 16:00 August 11 i . e . 4 h 15 min aft e r i r r i g a t i o n s t a r t Steady state, 09:00 August 12 (about 21 hours a f t e r s t a r t of i r r i g a t i o n ) Before turning i r r i g a t i o n o f f ; (13:30 August 14) 16:00 August 14 i . e . I l l min aft e r i r r i g a t i o n stop 19:00 August 14 i . e . 4 h 51 min a f t e r i r r i g a t i o n stop 21:00 August 14 i . e . 6 h 51 min aft e r i r r i g a t i o n stop Ex3 Before s t a r t of i r r i g a t i o n 12:36 August 20, i . e . 41 min aft e r s t a r t of i r r i g a t i o n 16:12 August 20 i . e . 4 h 17 min a f t e r s t a r t of i r r i g a t i o n 14:00 August 21, steady state (about 26 hours a f t e r s t a r t of i r r i g a t i o n ) Recession E: 1 h before i r r i g a t i o n stop F: 1 h 20 min after i r r i g a t i o n stop G: 3 h 20 min a f t e r i r r i g a t i o n stop H: 4 h 20 min aft e r i r r i g a t i o n stop Rise: A: B: C: D: - 69 -A s m e n t i o n e d i n S e c t i o n 1 . 3 . 5 , d a t a f o r p i p e s 9 a n d 11 a r e n o t r e l i a b l e . H e n c e a d o t t e d l i n e h a s b e e n d r a w n a t t h e s e p i p e s , j o i n i n g t h e w a t e r l e v e l s o f p i p e s 4 t o 6 , t h e n 6 t o 1 2 . I f t h e r e w a s n o w a t e r i n b o t h p i p e s 4 a n d 6 , t h e d o t t e d l i n e was d r a w n t o f o l l o w t h e b e d , e x c e p t b e f o r e s t a r t o f i r r i g a t i o n . The same a p p r o a c h w a s f o l l o w e d b e t w e e n p i p e s 6 a n d 1 2 . To s u m m a r i z e , u n d e r c o n d i t i o n s o f l o w e r i n i t i a l w a t e r c o n t e n t , t h e w a t e r t a b l e h a s some t e n d e n c y t o r i s e l a t e r b u t i n g e n e r a l r i s e s a t t h e s a m e r a t e . C o n c e n t r a t i o n o f i r r i g a t i o n d i d n o t , i n g e n e r a l , c a u s e t h e w a t e r t a b l e t o r i s e e a r l i e r o r f a s t e r . N e i t h e r d i d i t a f f e c t t h e s h a p e o f t h e w a t e r t a b l e a t s t e a d y s t a t e . 1.4.4 Comparison of the Recession with Hewlett and Hibbert's (1963)  Recession I n o r d e r t o d e t e r m i n e h o w s t e e p t h e r e c e s s i o n s a r e r e l a t i v e t o a r e c e s s i o n o b t a i n e d f o r a d i s t u r b e d s a n d y l o a m , t h e r e c e s s i o n o b t a i n e d i n t h e p r e s e n t w o r k w a s c o m p a r e d t o t h e o n e r e p o r t e d i n t h e l i t e r a t u r e b y H e w l e t t a n d H i b b e r t ( 1 9 6 3 ) . T h e s e a u t h o r s b u i l t a s l o p i n g t r o u g h , f i l l e d i t w i t h s a n d y l o a m , s a t u r a t e d i t a n d o b s e r v e d t h e r e c e s s i o n . T h e t r o u g h w a s 0 . 9 1 m d e e p , 0 . 9 1 m w i d e a n d 1 3 . 7 m l o n g . A t t h e s t a r t o f t h e r e c e s s i o n , t h e s a t u r a t e d z o n e w a s 0 . 9 m d e e p , a s c o m p a r e d t o a m a x i m u m d e p t h o f 0 . 5 m i n t h e p r e s e n t w o r k . A l s o , t h e s l o p e w a s 14 m l o n g , a s c o m p a r e d t o a l e n g t h o f 1 0 . 5 m i n t h e p r e s e n t w o r k . The s l o p e a n g l e was s m a l l e r , 22° v e r s u s 3 2 ° . The s o i l ' s t e x t u r e was s a n d y l o a m , t h e same a s t h e o n e o f t h i s s t u d y . S o i l , h o w e v e r , h a d b e e n e x c a v a t e d - 70 -and packed into the trough hence i t had l o s t i t s o r i g i n a l structure. Appendix C discusses the normalization of the r e s u l t s for comparison purposes, and the l i m i t a t i o n s of the comparison. Figure 1.4.4.1 shows the recession for Exl and Figure 1.4.4.2 the recession obtained by Hewlett and Hibbert. The outflow rate for Exl has 3 1 been expressed i n m s per 0.91 m of h i l l s l o p e width i n order to be comparable with Hewlett and Hibbert's work. Outflow rate has been plotted at the middle of the time i n t e r v a l used to calculate t h i s outflow rate. In Table 1.4.4.1, comparisons of some of Hewlett and Hibbert's data with some of those obtained i n the present work for Exl are presented. Data have been extracted from Figs 1.4.4.1 and 1.4.4.2. t' i s the time from s t a r t of recession. Q i s the outflow rate or discharge (volume per unit time). Table C-l i n Appendix C gives the normalization summary for the r e s u l t s of the present work given i n Table 1.4.4.1. The r a t i o of the outflow rate at one day to the one at 0.1 day i s 0.64 for Hewlett and Hibbert and 0.06 for the present work. These numbers have not been corrected for the e f f e c t of a steeper slope for the present work. The recession rate calculated between an outflow rate of 6.6 x 10"" 3 1 6 3 1 m s"" and 4.2 x 10"" m s - i s 16 times larger for the present work. For the c a l c u l a t i o n of t h i s l a t t e r number, the recession rate for the present work has been corrected as indicated i n Table 1.4.4.1 so that i t gives the rate that would be obtained for a h i l l s l o p e of slope s i m i l a r to the one occurring i n Hewlett and Hibbert's work. These r e s u l t s show that within the l i m i t a t i o n s stated i n Appendix - 71 -D A Y S F R O M B E G I N N I N G O F D R A I N A G E F i g . 1.4.4.1 Outflow recession for EX1. - 72 -0 0 X I i i 1 i i L j J 0.1 0.2 0.4 1 2 4 10 20 4 0 100 D A Y S F R O M B E G I N N I N G O F D R A I N A G E F i g . 1.4.4.2 Recession obtained by Hewlett and Hibbert (1963). Adapted from Hewlett and Hibbert's F i g . 2. - 73 -Table 1.4.4.1 Comparison of Hewlett and Hibbert's work with the the present work. Exl was chosen for the present work. Values for the present work are for 0.91 m of h i l l s l o p e width. Drainage from .1 to 1 day Q at .1 day = Qj ( m V 1 ) Q at 1 day = Q 2 (m 3s- 1) Q 2/Ql Hewlett and Hibbert (H&H) 6.6 x 10~ 6 4.2 x IO" 6 0.64 (1963) Present work (P.W) 6.0 x 10~ 6 3.6 x I O - 7 0.06 6 3 1 Drainage from 6.6 x 10" m s to 4.2 x 10~6 m3s" 1 9Q Time at Time at At' 1 At' 9t' H&H P.W. 6.6 x 10~ 6 m 3s _ 1 4.2 x I O - 6 m3s" 1 At' P.W. " H&H Hewlett & 0.1 day 1 day 0.9 day Hibbert 2 22.5 Present 0.09 day 0.13 day 0.04 day work Present work 0.06 day 15.8 3 corrected for slope* *See Appendix C. XAt' = Time at 6.6 x I O - 6 m 3s- 1 - Time at 4. 2 x IO" 6 m 3s~ 1. 20.9/0.04 30.9/0.06 - 74 -C, the recession for the present work is much faster than for Hewlett and Hibbert's. 1.4.5 Visual Observation of Low Resistance Paths i n the Forest Floor  and the B Horizon This section contains descriptive information about the structure of the forest floor and the B horizon. The information i s based on visual examinations carried out in the f i e l d . Emphasis i s placed on the presence of low resistance paths. The purpose of the section is to understand better the soil hydrologic behavior as determined by so i l structural features and as reflected by the shape of the hydrograph. The f i r s t 5 mm or so of forest floor are made up mainly of needles and twigs and are probably sufficiently open for water from individual rain drops to drip down needles. The F layer s t i l l contains large voids but a concentrated inflow is required for water to flow through these voids because water from individual rain drops would be sucked in by the humified material. The H layer contains fewer large voids. In the Seymour Watershed of the Greater Vancouver Water District, de Vries and Chow (1978) indeed noticed that the H layer together with the Ae layer formed a lower conductivity layer. Under a large irrigation rate, larger than the one used for Exl, Ex2 and Ex3, lateral flow was observed to occur within the forest floor. Part of the forest floor can thus act as a low resistance path. Also, the forest floor was irrigated from a distance within one meter of a cut, using a watering can. This irrigation took place during - 75 -a r a i n f a l l occurring after several days of r a i n . During i r r i g a t i o n water was observed to come out flowing on top of a f l a t piece of bark of s i z e approximately 10 by 5 cm i n the lower part of the forest f l o o r . As mentioned i n Section 1.2, Mosley also observed underground l a t e r a l flow. A d i f f e r e n c e with the observation made in the present work i s that he observed flow on top of the mineral layer and in the humified layer whereas i n the UBC Forest, flow probably occurred on top of the composite H-Ae l a y e r . In the B horizon, and at the i n t e r f a c e between the bed and the B horizon, the main source of p i p e l i k e low resistance paths are l i v e or decaying roots. Four kinds of p i p e l i k e low resistance paths due to root material were distinguished: ( i ) The decaying bark of a root. While decaying, the bark has become spongy and hence a high conductivity material, ( i i ) Roots i n which the i n t e r i o r part i s decaying or missing. At places, the i n t e r i o r of roots was noticed to have become a fibrous material with i n s t e r s t i c e s or to have been washed away. Sometimes the bark was hard and b r i t t l e and i f the i n t e r i o r was washed away, a c y l i n d r i c a l conduit resulted, ( i i i ) A rootmat on top of the low condutivity l a y e r . As noted by Utting (1978), roots accumulate over the low conductivity layer because i t i s harder to penetrate. Utting (1978) noted that root accumulation needs a sharp B h o r i z o n - t i l l i n t erface to occur, ( i v ) "Dark material" consisting mainly of decayed organic matter (Martin, 1983, pers. comm.) was observed in the saturated - 76 -zone. I t can c o n t a i n s t a i n e d m i n e r a l matter and r e c o g n i z a b l e r o o t d e b r i s . The decayed o r g a n i c m a t e r i a l i s thought to c o n s i s t of decayed r o o t s ( a s a l s o observed elsewhere by M a r t i n , 1983, p e r s . comm.). P o i n t s ( i i ) and ( i v ) a r e p r o b a b l y what i s meant by " r o o t c h a n n e l s " i n the l i t e r a t u r e . The d i f f e r e n c e between the rootmat and the dark m a t e r i a l i s not a c l e a r c u t one as the l a t t e r can be i n t e r l a c e d w i t h r o o t s and can be found i n a rootmat. In the channel dug at the bottom of the p l o t used f o r the p r e s e n t work, r o o t s w i t h d e c a y i n g bark were found at p l a c e s to occur i n c l u s t e r s and are more common than low r e s i s t a n c e paths d e s c r i b e d under ( i i ) . A t the North t i p of the c h a n n e l , a w e l l d e f i n e d and 0.5 t o 1 cm t h i c k rootmat was found between two l a y e r s of t i l l . I n t e r s t i c e s between the r o o t s were c l e a r l y seen. Some m i n e r a l m a t e r i a l and some decayed o r g a n i c m a t e r i a l was a l s o seen. Because of the l a r g e s i z e of the p o r e s , t h i s rootmat must have had a water e n t r y p r e s s u r e of z e r o . At p l a c e s a l o n g the channel bank, dark m a t e r i a l appeared as l e n s e s o r around r o o t s . A good s i z e d l e n s extended 4 cm above the t i l l . Dark m a t e r i a l and rootmat u s u a l l y c o i n c i d e w i t h c o n c e n t r a t e d o u t f l o w p o i n t s . A f t e r a n a t u r a l event, o u t f l o w was n o t i c e d from l e n s e s of dark m a t e r i a l and from the rootmat o c c u r r i n g at the bottom o f the B h o r i z o n . Because o f the h y d r o l o g i c s i g n i f i c a n c e of the rootmat and of the dark m a t e r i a l i n the s a t u r a t e d zone, i t would be d e s i r a b l e to determine the l a t e r a l e x t e n t and the depth of these f e a t u r e s on the top of the bed which i s -li-the zone where saturation occurs during a storm event. Unfortunately, t h i s i s d i f f i c u l t because the color of dark material grades from dark to almost the same color as the mineral s o i l . A number of pi p e l i k e low resistance paths have been observed i n the 2 v i c i n i t y of the p l o t . On a 60 by 30 cm v e r t i c a l area of a p i t dug u p h i l l from the plot, 6 root channels of type ( i i ) were found, ranging i n size from a few mm to 1 cm. This type of root channels was les s frequent i n the channel dug for the present work than i n this p i t . On the cut at the foot of Nagpal and de Vries' s i t e , water was noticed flowing out through a section of decayed root bark. Pi p e l i k e low resistance paths on the top of the bed are hy d r o l o g i c a l l y more important than i n the unsaturated zone. The f i r s t reason i s that they are probably more widespread. The second i s that since they occur i n the saturated zone, there i s no problem about water being able to enter them l i k e there i s i n the unsaturated zone. Small underground streams are extreme examples of flow i n low resistance paths. At two places near the h i l l s l o p e where the plot for the present work was chosen, small streams were seen to emerge from the ground. Similar features were already observed by Chamberlin (1972) as described i n Section 1.2. It i s speculated that the streams form u p h i l l on low conductivity ground surfaces l i k e rock, then go underground. Some observation of short low resistance paths due to the presence of stones was also made. A few stones were observed to have roots or - 78 -rootmat against them. One had rootmat and fungi hyphae between i t s two s p l i t halves. A rootmat w i l l provide a low resistance path. The few observations made indicated that low resistance paths due to stones are few and far between but observation was d i f f i c u l t due to the disturbance i t necessitated. More work i s required to determine whether free water a c t u a l l y flows around stones and how widespread t h i s mechanism i s . 1.4.6 Water Balance Table 1.4.6.1 gives i r r i g a t i o n rates, outflow rates, average water table height and the r a t i o of outflow to i r r i g a t i o n for the three experiments. Steady state values for the outflow have been obtained as follows: for Exl i t i s the eye average of the outflow rate recorded from 22 to 42 hrs a f t e r the s t a r t of i r r i g a t i o n . For Ex2 i t i s the plateau occurring 38 hrs a f t e r the s t a r t of i r r i g a t i o n . For Ex3, the plateau occurring 35 hrs after the sta r t of i r r i g a t i o n has been used. For Ex2 and Ex3, the values used have been dictated by consideration of the tipping bucket behavior. Steady state i r r i g a t i o n rates are time averages of the pooled i r r i g a t i o n rates estimated by eye as explained i n Section 1.3.3 for a period extending from 6 hours before the period of outflow used t i l l the end of this period. The r a t i o n a l e for using 6 hours was that i t took l e s s than 6 hours for the outflow rate to reach steady state again after the dry pulse. It can be assumed that the outflow rate i s influenced by Table 1.4.6.1 I r r i g a t i o n rates, outflow rates, average water table height and r a t i o of outflow to i r r i g a t i o n for the 3 experiments Exl Ex2 Ex3 Average Time average pooled , i r r i g a t i o n rate (ms~ y with RG11 3.17 x I O - 6 3.21 x IO" 6 (2.87 x IO" 6) without RG11 3.99 x 10 _6 4.04 x 10 „ 6 3.63 x 10 _6 Time average steady state out I 3 -IN (m s ) flow rate!, 1.18 x 10"1* 1.29 x 10 - l t 1.30 x I O - 4 2.38 x 10" 6 2.61 x 10~ 6 2.63 x I O - 6 Average steady state water table height (m) 0.18 0.22 0.22 Outflow/Irrigation with RG11 without RG11 0.75 0.81 (0.92) 0.72 0.83 (^Obtained by dividing the above entry by the area of the pl o t , 49.5 m2. - 80 -the i r r i g a t i o n rate occurring not more than 6 hours e a r l i e r , provided the s o i l water content had already adjusted i t s e l f to a s i m i l a r i r r i g a t i o n rate 6 hours e a r l i e r . The period of 6 hours may appear a l i t t l e short f o r Exl for which outflow was just reaching steady state at the beginning of the outflow period used. However, i t turns out that the same averages are obtained i f the time average pooled i r r i g a t i o n rates are taken from the st a r t of i r r i g a t i o n . Values i n brackets are calculated by multiplying the value obtained experimentally without RG11 by 0.8 i n order to take RG11 into account as explained i n Section 1.3.3. The average steady state water table height i s the arithmetic mean of the steady state values given i n Table 1.4.3.2. One puzzling r e s u l t i s the fact that the i r r i g a t i o n rate i s lower f o r Ex3 than f o r Exl and Ex2 but the steady state outflow rate and the average steady state water table height are higher for Ex2 and Ex3 than f o r E x l . The way i n which the steady state value for the outflow rates have been estimated i s perhaps p a r t l y responsible for the i r being inconsistent with the i r r i g a t i o n rate. Errors i n outflow rate mentioned i n Appendix E and i r r i g a t i o n rate are also responsible. Note that multiplying the pooled i r r i g a t i o n rate by 0.8 f o r Ex3 i s possibly ludicrous i n view of the errors attached to the i r r i g a t i o n r a t e . Also, for the time periods chosen f o r Table 1.4.6.1 the value m u l t i p l i e d by 0.8 i s hardly closer to the values obtained for Exl and Ex2 than the value obtained without RG11. - 81 -However, comparing the i r r i g a t i o n rates for the whole lengths of the experiments ( F i g . 1.4.2.2 to 1.4.2.5) shows that the i r r i g a t i o n rate f o r Ex3 i s closer to the one for Exl and Ex2 when a l l three rates are calculated without RG11 than when the rates f o r Exl and Ex2 are calculated with RG11. By the same token, the rate for Ex3 should be closer to the rates f o r Exl and Ex2 when a l l three rates are calculated with RG11 than when only the rates for Exl and Ex2 are calculated with RG11. It i s therefore thought that the correction has some value, e s p e c i a l l y because the r a t i o of time average of the pooled i r r i g a t i o n rate with to without RG11 i s 0.8 for both Exl and Ex2. Table 1.4.6.1 shows that about 10% to 25% of the water i s l o s t . The average r a t i o of outflow to i r r i g a t i o n for the three experiments i s 0.83. A value of 0.80 w i l l be used whenever the water balance w i l l have to be taken into account for c a l c u l a t i o n s . In spite of the uncertainties linked to the outflow and the i r r i g a t i o n rates, the average water loss found for the Forest plot corresponds approximately to the loss of about 20% found by Utting (1978) i n Nagpal and de Vr i e s ' s i t e . Some water was l o s t by tree water uptake, and some was l o s t at the south end of the channel. Utting states that the loss he observed i s due to i n f i l t r a t i o n into the t i l l and possibly into the fractured bedrock. It i s l i k e l y that most of the loss observed for the Forest plot i s due to the same cause. 1.5 Discussion The transformation of a r a i n f a l l hyetograph into an outflow hydrograph r e f l e c t s the integrated e f f e c t of the hydrologic behavior of - 82 -both the saturated and the unsaturated zone (de V r i e s , pers. comm.). This behavior i s related to the s o i l ' s texture and structure, and the mechanism of water flow through i t . In t h i s section, we s h a l l attempt to i n f e r flow mechanisms i n the Forest s o i l from the outflow hydrograph, the behavior of the water table and the v i s u a l observation of low resistance paths. Because the outflow hydrograph r e f l e c t s flow i n both the unsaturated and saturated zones, the transformation of the r a i n f a l l hyetograph into an outflow hydrograph occurs i n two steps. The f i r s t step i s the transformation of the r a i n f a l l hyetograph into a "recharge hyetograph." This transformation r e s u l t s from t r a v e l of water through the unsaturated zone from the ground surface to the bed or to the water table. The second step i s the transformation of the recharge hyetograph into an outflow hydrograph as water flows over the bed to the channel. In the Forest p l o t , the unsaturated zone generally extends through the forest f l o o r and through a l l or part of the B horizon. The saturated zone occurs on top of the bed. The low resistance paths present on top of the bed, although they may be considered to be t e c h n i c a l l y s t r u c t u r a l features of the B horizon, w i l l not be considered to be part of the B horizon. The bed can be considered as a second i n f i l t r a t i n g surface since water reaching i t either flows l a t e r a l l y on top of i t or i n f i l t r a t e s into i t , i n the same way as r a i n f a l l reaching the ground surface. D e f i n i t i o n s of the terms used for various parts of the hydrograph r i s e were given i n Section 1.4.2. Also, the term "steady" i s used with - 83 -the meaning of constant in time while the term "uniform" means constant in space. In the following two subsections, s o i l hydrologic behavior w i l l be inferred from the main hydrograph features presented in Section 1.4.2. Flow of water in the unsaturated zone w i l l be discussed f i r s t , followed by a discussion of the saturated flow taking place laterally on top of the bed. 1.5.1 Water Flow i n the Unsaturated Zone Because an important hypothesis of the present work is that short-circuiting alters the shape of the outflow hydrograph, some time w i l l be devoted to the consideration of short-circuiting and of i t s effect on the outflow hydrograph. The expected effects of short-circuiting within the unsaturated zone on the hydrograph rise are depicted schematically in Figures 1.5.1.1 (a) to (e). For simplicity, a straight line rise has been chosen for the case when flow through the unsaturated zone is uniform. A straight line rise is consistent with the kinematic wave model that w i l l be used below. It is also consistent with the straight line main limbs of rise obtained for the present work. The reasoning presented below can easily be extended to a more general shape of rise. In Fig. 1.5.1.1 i t is also assumed that outflow is recorded as soon as water reaches the bed at the bank of the channel. It ignores the time to break through into the channel and routing in the channel. Soil properties in the saturated zone are assumed to be uniform. Fig. 1.5.1.1 Schematic hydrograph rises showing the ef short-circuiting in the unsaturated zone. - 85 -Figure 1.5.1.1 (a) shows the r i s e expected from uniform flow reaching the bed everywhere at the same time. The main limb of r i s e consists of the whole r i s e , from t^ to t 2« In F i g s . 1.5.1.1 (b) to (e), dashed l i n e s i n d i c a t e the shape the hydrograph r i s e would have i f flow down to the bed occurred uniformly. Figure 1.5.1.1 (b) shows the r i s e expected i f a l l the water s h o r t - c i r c u i t s the s o i l matrix and reaches the bed at the same time. The shape of the r i s e i s the same as for the uniform case because as i n the uniform case, i t Is governed s o l e l y by flow on top of the bed but the r i s e i s shi f t e d to the l e f t . In this case as well, the main limb of r i s e consists of the whole r i s e . This case i s extreme and i n general u n r e a l i s t i c . In Figs. 1.5.1.1 (c) to (e ) , some s h o r t - c i r c u i t i n g occurs but a s i g n i f i c a n t proportion of flow s t i l l occurs through the matrix. Therefore, for these cases, the main limb of r i s e s t i l l s t a r t s when the water having t r a v e l l e d uniformly i n the unsaturated zone reaches the bed. The time at which the main limb of r i s e s t a r t s i s thus independent from s h o r t - c i r c u i t i n g and i s equal to t j l i k e for the uniform case. Also, i t i s l i k e l y that for both the uniform and the nonuniform cases, the water being l a s t to reach the channel w i l l be water having t r a v e l l e d uniformly down the unsaturated zone i n the remotest parts of the watershed or p l o t . Therefore, i t i s l i k e l y that steady state w i l l be reached at the same time t2 whether flow i n low resistance paths and as fingers occurred or not. In Fig s . 1.5.1.1 (a) to (e), because a st r a i g h t l i n e r i s e has been assumed for uniform flow through the - 86 -unsaturated zone, the main limb of r i s e ends when steady state i s reached. In F i g . 1.5.1.1 ( c ) , i t i s assumed that a l l the water s h o r t - c i r c u i t i n g the s o i l matrix i n the unsaturated zone reaches the channel before the water having t r a v e l l e d uniformly. An i n i t i a l step occurs, followed by a main limb of r i s e broken into two by a less steep r i s e . This less steep r i s e occurs when the water that has sh o r t - c i r c u i t e d the s o i l matrix and has already reached the channel would reach the channel had i t t r a v e l l e d uniformly i n the unsaturated zone. It i s possible that i n practice t h i s less steep r i s e w i l l not be discernable. The o v e r a l l slope of the whole r i s e and of the main limb of r i s e w i l l be less steep than for the uniform case. Figure 1.5.1.1 (d) depicts the r i s e expected for the case where water that has s h o r t - c i r c u i t e d the s o i l matrix reaches the channel only a f t e r water that has t r a v e l l e d uniformly. A sharp increase i n flow occurs due to the a r r i v a l at the channel of the water that has sh o r t - c i r c u i t e d the matrix. This increase i s followed by the continuation of the r i s e due to uniform flow, then by a less steep r i s e caused by the same mechanism as i n Figure 1.5.1.1 ( c ) . It i s possible that these i r r e g u l a r i t i e s i n the r i s e would not be s i g n i f i c a n t enough to be noticed i n pr a c t i c e . The o v e r a l l slope of the r i s e and of the main limb of r i s e w i l l be the same as for the uniform case. Figure 1.5.1.1 (e) i s a combination of (c) and (d) and the most l i k e l y to occur i n r e a l i t y for a l l but very short h i l l s l o p e s . The o v e r a l l slope of the r i s e and of the main limb of r i s e w i l l be less steep than for the uniform case. - 87 -In order for flow i n low resistance paths to occur, two things are necessary: the presence of low resistance paths within the unsaturated zone and a supply of free water. If i t occurs, flow i n p i p e l i k e low resistance paths due to root material within the unsaturated zone may have a s i g n i f i c a n t l a t e r a l component as roots grow l a t e r a l l y as well as downward. In order for flow as fingers to occur, a supply of free water i s necessary. In the Forest, free water necessary for flow i n low resistance paths and as fingers i s supplied by concentration of water by concentrating elements. These elements include the tree canopy, logs, l i t t e r on top of the forest f l o o r as suggested for other forest s o i l s by Whipkey (1968), and de Vries and Chow (1978), pieces of wood within the forest f l o o r , stones and water repellent spots. Fungi observed i n the Forest may be responsible for water repellency and corresponding concentration since, according to Bond (1964), Bond and Harris (1964) noted that water repellency may be due to fungi. The presence of both low resistance paths and concentrating elements suggests that some s h o r t - c i r c u i t i n g may take place. Now that a t h e o r e t i c a l basis has been set the observed outflow hydrographs w i l l be used to shed some l i g h t on flow mechanism i n the unsaturated zone. The mechanism of flow i n the unsaturated zone should be inferred from the "recharge hyetographs." However, since these hyetographs are unknown, the outflow hydrographs must be used. The main features of the - 88 -hydrographs presented i n Figs. 1.4.2.2 to 1.4.2.5 w i l l be discussed i n turn. (i) L i t t l e Preliminary Outflow Before the Main Limb of Rise The outflow hydrographs pictured i n Figs. 1.4.2.2 to 1.4.2.5 show l i t t l e preliminary outflow before the main limb of r i s e . This i s confirmed by the detailed r i s e s i n F i g . 1.4.2.6 which show an i n i t i a l step with only a very low outflow rate for Exl and Ex3. For Ex2, the steep small r i s e before the main limb of r i s e has been c a l l e d t e c h n i c a l l y an early r i s e . P r a c t i c a l l y , being steeper than the main limb of r i s e , i t could be considered that i t i s part of i t and that there i s thus no preliminary outflow for Ex2. The only error l i s t e d i n Appendix E that may decrease s i g n i f i c a n t l y the recorded i n i t i a l outflow and shed some doubt as to whether the i n i t i a l outflows before the main limbs of r i s e are indeed small i s the fact that some water escaped c o l l e c t i o n by the channel. However, hydrographs obtained by Utting (1978) and by Nagpal and de Vries (1976) on the l a t t e r authors' s i t e also display very l i t t l e i n i t i a l outflow. Therefore i t i s l i k e l y that there i s l i t t l e i n i t i a l outflow from the Forest p l o t . An error that l i m i t s the comparison of the hydrographs of the 3 experiments i s that they were not normalized with respect to i r r i g a t i o n r a t e . An i n i t i a l step i s due to water reaching the channel before the main limb of r i s e s t a r t s . This water could have reached the bed v i a low - 89 -resistance paths or by finger flow, thus " s h o r t - c i r c u i t i n g " the s o i l matrix. The fact that the outflow rate during the i n i t i a l step of Exl and Ex3 shown i n Fig. 1.4.2.6 i s so small suggests that only very l i t t l e s h o r t - c i r c u i t i n g took place. One r e s u l t that perhaps appears to confirm the occurrence of some s h o r t - c i r c u i t i n g i s the fact that pipes 1, 5 and 8 displayed a r i s e as early and as fast for Exl as for Ex3. Indeed, i f flow were to occur a l l the way through a low resistance path, i n i t i a l conditions would have no e f f e c t . Appendix B shows that for Ex2 the p l a s t i c sheets concentrated water at l east 100 times. P l a s t i c sheets covered 20% of the Forest p l o t . It was expected that such a large concentration of i r r i g a t i o n water and hence increase i n free water supply would trig g e r or enhance s h o r t - c i r c u i t i n g , possibly causing an i n i t i a l step to occur, or causing an i n i t i a l step already occurring under uniform i r r i g a t i o n to occur e a r l i e r or be l a r g e r . Contrary to this expectation, the hydrograph for Ex2 does not display an i n i t i a l step although the ones for Exl and Ex3 do d i s p l a y a small one. Note that i t i s possible that no enhanced early step was observed f o r Ex2 because the p l a s t i c sheets were not close enough to the channel. Consequently, a r i s e of the type depicted in F i g . 1.5.1.1 (d) might be expected for Ex2, rather than of the type of F i g . 1.5.1.1 ( c ) . Due to the fact that i r r e g u l a r i t i e s i n the main limb of r i s e may not be observable i n p r a c t i c e , i t could be argued that a r i s e s i m i l a r to the - 90 -one depicted i n F i g . 1.5.1.1 (d) may have occurred. However, the p o s s i b i l i t y of a r i s e of type 1.5.1.1 (d) w i l l be rejected in favor of what i s a c t u a l l y observed. Also, the p l a s t i c sheets were close to the pipes and no e f f e c t of concentration was noted on the pipes, although some e f f e c t might have been noticed had the r e s o l u t i o n been better. Hence the r e s u l t that the p l a s t i c sheets did not enhance s h o r t - c i r c u i t i n g w i l l be accepted as an apparent r e s u l t . Further experiments, better controlled and monitored should be performed in order to confirm that concentration has no e f f e c t . In p a r t i c u l a r , p l a s t i c sheets should be placed uniformly. The presence of an area of shallow s o i l could be another cause of preliminary outflow. Whatever the reason for the preliminary outflow, the hydrographs of Figs. 1.4.2.2 to 1.4.2.6 show that the amount of water involved i s very small. Thus i t can be concluded that the system behaves as i f the front were uniform. To summarize, i t appears that the s o i l and water system in the integrated unsaturated and saturated zones domain behaves as i f very l i t t l e s h o r t - c i r c u i t i n g , i f any, occurred at the beginning of the outflow, and as i f , within the l i m i t a t i o n s of the experimental design, s h o r t - c i r c u i t i n g at the beginning of the outflow was not enhanced by concentration of i r r i g a t i o n . ( i i ) L i n e a r i t y of the Main Limb of Rise A s t r i k i n g feature displayed by the hydrographs of Figs. 1.4.2.2 to 1.4.2.5 i s the l i n e a r i t y of the main limb of r i s e . Outflow hydrographs - 91 -obtained by Nagpal and de Vries (1976) and by Utting (1978) on Nagpal and de Vries' s i t e also display l i n e a r r i s e s . One way to i n f e r a flow mechanism from an outflow hydrograph i s to consider a p h y s i c a l l y based model of flow y i e l d i n g a sim i l a r outflow hydrograph and to speculate that the r e a l flow occurs according to the same mechanism as the one described by the model. There i s , however, no guarantee that the speculation w i l l be correct. Even so, the s i m i l a r i t y of the th e o r e t i c a l and the experimental r e s u l t s i s i n t e r e s t i n g i n that i t indicates the r e a l system behaves as i f the assumptions of the model were met. A very simple model of saturated flow i n a porous medium over a sloping bed i s offerred by Beven (1981). This model, hereafter referred to as "the kinematic wave model," w i l l be described i n more d e t a i l below and i n Chapter I I . For the moment i t w i l l be mentioned that i t assumes among other things a recharge rate constant i n time and uniform over the bed. It also assumes the v a l i d i t y of Darcy's law which means that flow must be l i n e a r . The kinematic wave model yields a straight l i n e r i s e , that i s , a constant 4T" during the r i s e . The v a l i d i t y of Darcy's law a t for the Forest plot i s discussed i n Chapter I I I . It i s not c e r t a i n whether flow i n the saturated zone of the Forest plot i s l i n e a r . In order to apply the kinematic wave model to the Forest pl o t , i t i s necessary to assume flow Is l i n e a r , or that n o n l i n e a r i t y i s n e g l i g i b l e . Since both the observed hydrograph and the hydrograph obtained by the kinematic wave model y i e l d a straight l i n e main limb of r i s e , i t can - 92 -be speculated (de V r i e s , pers . comm.) that the assumptions of steady and uniform recharge rate are v a l i d for the h i l l s l o p e p l o t . A steady and uniform recharge rate means that the recharge hyetograph i s a step funct ion and i s the same step funct ion everywhere on the bed. This would mean that apart from the small amount of water involved i n the i n i t i a l step for Exl and E x 3 , a l l the water reaches the bed at the same time and that the recharge rate i s uniform over the bed. A number of conclusions about the hydrolog ic behavior of the unsaturated zone w i l l now be drawn from a uniform and steady recharge rate during the main limb of r i s e . F i r s t , uniform and steady recharge rate during the main limb of r i s e i s consistent with no s h o r t - c i r c u i t i n g taking place during t h i s period of time. In p a r t i c u l a r , t h i s appears to ind icate that flow of the type depicted by F i g . 1.5.1.1 (d) does not occur. That i s , no water from s h o r t - c i r c u i t i n g flow reaches the channel af ter the a r r i v a l of water o r i g i n a t i n g from uniform flow. One reason for the absence of s h o r t - c i r c u i t i n g could be that there are not enough p ipe l ike low res i s tance paths for t h e i r entrances to rece ive concentrated inf low from e i t h e r na tura l concentrating elements or from the p l a s t i c sheets. Poss ib ly water spreads into the forest f l oor or the B horizon before reaching a p i p e l i k e low res i s tance path. Another reason may be that the wal ls of p i p e l i k e low res i s tance paths are s u f f i c i e n t l y permeable for water to flow qu ick ly into the matr ix . This reasoning of course appl ies to flow i n low res i s tance paths at a l l t imes, not only during the main l imb of r i s e . Note that i t i s poss ib le that s h o r t - c i r c u i t i n g does occur - 93 -but that a straight main limb of r i s e i s due to an averaging process (de V r i e s , pers. comm.). Second, a uniform and steady recharge rate i s consistent with flow within the unsaturated zone being approximately v e r t i c a l downward. V e r t i c a l flow i n the unsaturated zone i s t h e o r e t i c a l l y sound since Childs (1969) notes that in the unsaturated zone, flow of water due to p r e c i p i t a t i o n i s mainly v e r t i c a l . He explains that i t i s so because unsaturated s o i l s have such low hydraulic c o n d u c t i v i t i e s that pressure p o t e n t i a l gradients larger than those "commonly observed i n the horizontal d i r e c t i o n , are required to produce appreciable movements of water." Note however that Hewlett and Troendle (1975) mention the occurrence of l a t e r a l flow i n the unsaturated zone of a sloping s o i l model. Whether l a t e r a l flow occurred in the unsaturated zone of the Forest plot should be determined experimentally. Third, a uniform and steady recharge rate i s consistent with a uniform s o i l thickness. The s o i l within the pl o t , however, i s not uniformly thick since at the pipes i t s thickness v a r i e s from 0.50 m at pipe 8 to 1.66 m at pipe 10. There can be two ways of explaining the fact that nonuniformities in s o i l thickness do not a f f e c t the main limb of r i s e . One p o s s i b i l i t y i s that nonuniformity and unsteadiness i n the recharge rate due to nonuniformities i n s o i l thickness occur only for a short period of time at the beginning of the recharge and do not influence the main limb of r i s e . Another p o s s i b i l i t y i s that a straight l i n e main limb of r i s e i s due to an averaging process. Continuous monitoring of the water table - 94 -height during the early part of i r r i g a t i o n would help determine whether the recharge a c t u a l l y started everywhere at the same time. To summarize, a l i n e a r main limb of r i s e indicates that the system apparently behaves as i f the recharge hyetograph were a uniform step function. That i s , the system behaves as i f recharge were steady and uniform from the moment i t begins. ( i i i ) Steep and Approximately Parallel Main Limbs of Rise The main limbs of r i s e i n Figs. 1.4.2.2 to 1.4.2.5 have steep slopes and are approximately p a r a l l e l . Moreover, the main limbs of r i s e of Ex2 and Ex3 are more or le s s on top of each other while the main limb of r i s e of Exl occurs l a t e r . A slope can appear to be steep due to the scale chosen. In order to be able to say that the slope of the r i s e i s steep, one should be able to compare i t with the r i s e obtained for flow within l e s s open s o i l s such as a g r i c u l t u r a l s o i l s . It was not possible to find i n the l i t e r a t u r e outflow hydrographs together with r a i n f a l l hyetographs adequate for comparison purposes, or with the i n d i c a t i o n as to whether or not overland flow took place. Later on, i t w i l l be shown that due to low resistance paths on top of the bed and to a r e l a t i v e l y high recharge rate, the slopes of the main limbs of r i s e can be expected to be steep. It i s important to note that for the case where water t r a v e l s v e r t i c a l l y down through the unsaturated zone, a steep main limb of r i s e i s not a proof that s h o r t - c i r c u i t i n g occurred in the unsaturated part of - 95 -the B horizon, as can be seen from Fig. 1.5.1.1. Figure 1.5.1.1 (b) for instance shows that i f a l l the water t r a v e l l e d down to the low condu c t i v i t y layer through low resistance paths, and reached i t at the same time, the slope of the outflow hydrograph r i s e would not be steeper than for the corresponding case where a l l the water would tr a v e l down through the s o i l matrix. Therefore, an open s o i l i n the unsaturated zone does not mean a steeper r i s e . In the remainder of this discussion, i t should be kept i n mind that the hydrographs have not been normalized with respect to i r r i g a t i o n r a te, and errors mentioned in Appendix E may a f f e c t the time of occurrence of the main limbs of r i s e and their slopes. The average steady state outflow rates for Ex2 and Ex3 are the same (1.25 x 10~ 4 and 1.24 x I O - 4 m3 s - 1 r e s p e c t i v e l y . See Appendix F). There i s no physical reason why the r a t i o of outflow to i r r i g a t i o n should not be the same at steady state for the three experiments. Therefore, on the average, the i r r i g a t i o n rate for Ex2 must be the same as for Ex3, although F i g . 1.4.2.2 shows i t i s not the case, probably due to error. Since the i r r i g a t i o n rate i s not constant with time, normalization of the outflow rate using a time dependent i r r i g a t i o n rate should s t i l l have been performed i n order to compare s p e c i f i c parts of the hydrograph such as the r i s e s but i t has not been performed. The lag of the main limb of r i s e of Exl with respect to the main limb of r i s e of Ex3 i s at least one hour. Moreover, the fact that the main limb of r i s e of Exl lags the main limb of r i s e of Ex3 i s v e r i f i e d to some extent by - 96 -the fact that 4 pipes out of 7 r i s e l a t e r for Exl than for Ex3. Therefore the lag of the main limb of r i s e of Exl with respect to the one of Ex3 i s r e a l , although i t s numerical value may have been d i f f e r e n t , had the outflow rate been normalized with respect to r a i n f a l l and were there no errors. The exact p o s i t i o n of the main limb of r i s e of Ex2 with respect to Ex3 i s unknown due to the same l i m i t a t i o n s . The e f f e c t of i n i t i a l conditions on the timing and slopes of the main limbs of r i s e are as follows: The main limb of r i s e of Exl i s delayed with respect to the one of Ex3 due to d r i e r i n i t i a l conditions. This delay must be due to storage i n the unsaturated zone. The main limb of r i s e of Exl i s p a r a l l e l to the one of Ex3, showing that a d i f f e r e n t i n i t i a l water content did not a f f e c t the slope of the main limb of r i s e . In order to determine i f concentration has any e f f e c t on the shape and timing of the main limbs of r i s e , l e t us f i r s t review what e f f e c t s are expected from s h o r t - c i r c u i t i n g . As shown in F i g . 1.5.1.1, the only modifications of the hydrograph r i s e that are expected from s h o r t - c i r c u i t i n g are an i n i t i a l step, i r r e g u l a r i t i e s i n the main limb of r i s e , and le s s steep o v e r a l l r i s e s and main limbs of r i s e . If a more general hydrograph shape, including a l a t e part of r i s e , i s used, i r r e g u l a r i t i e s may appear i n the l a t e part of the r i s e . The extreme case of F i g . 1.5.1.1.b where the o v e r a l l r i s e gets s h i f t e d to the l e f t i s considered to be u n r e a l i s t i c . Within the l i m i t a t i o n s mentioned above and the r e s o l u t i o n of the graph, the hydrograph for Ex2, as can be observed i n F i g . 1.4.2.2, does - 9 7 -not display any of the features expected from s h o r t - c i r c u i t i n g . Indeed, the i n i t i a l step i s suppressed rather than enhanced, as can be observed also from F i g . 1.4.2.6. Moreover, no i r r e g u l a r i t i e s are noticed i n the main limb of r i s e and the ones occurring l a t e r can be due to other causes than s h o r t - c i r c u i t i n g . F i n a l l y , the slope of the main limb of r i s e of Ex2 i s approximately equal to the one of Ex3. It can also be noticed from F i g . 1.4.2.2 that the main limb of r i s e of Ex2 does not occur e a r l i e r than the one of Ex3. It would, however, be desirable to observe what would happen when the i n i t i a l conditions are exactly the same for the concentrated and the uniform i r r i g a t i o n s . I n i t i a l conditions for Ex2 were s l i g h t l y d r i e r than for Ex3, perhaps masking the e f f e c t of concentration. However, except f o r F i g . 1.5.1.1.b which represents an u n r e a l i s t i c case, s h o r t - c i r c u i t i n g i s not expected to s h i f t the main limb of r i s e . From F i g . 1.4.2.2, i t can also be observed that Ex2 does not reach steady state e a r l i e r than Ex3, but s h o r t - c i r c u i t i n g i s not expected to decrease the time to steady state. The fact that concentration has apparently no e f f e c t on flow i n the unsaturated zone i s i n agreement with the standpipes data given i n Section 1.4.3 In summary, within the l i m i t s of experimental design and of e r r o r s , the hydrograph r i s e due to concentrated i r r i g a t i o n obtained for Ex2 does not display the features expected from s h o r t - c i r c u i t i n g . Moreover, i t i s not steeper, nor s h i f t e d e a r l i e r than the hydrograph for Ex3, obtained from uniform i r r i g a t i o n and s l i g h t l y wetter i n i t i a l conditions. - 98 -(Iv) Steep and Parallel Recessions Recessions as shown i n Figure 1.4.2.2 are remarkably p a r a l l e l . Errors l i k e l y to influence the recessions are d r i f t i n the tipp i n g bucket c a l i b r a t i o n , and for Ex2 and Ex3, natural r a i n f a l l . However, given the p a r a l l e l nature of the recessions, i t i s l i k e l y that these errors were n e g l i g i b l e during the recessions for the graphs at the scale used f or F i g . 1.4.2.2. It i s l i k e l y that flow i n low resistance paths and as fingers down to the low conductivity layer w i l l p r e c i p i t a t e and steepen the beginning of the recession. Thus, in contrast with the r i s e s , recessions may be steeper due to the structure of the unsaturated zone. From F i g . 1.4.2.2, one can note that recessions occur soon a f t e r i r r i g a t i o n i s stopped and are f a s t . It might thus be concluded that a s i g n i f i c a n t flow In low resistance paths occurs. However, reaching a conclusion i s d i f f i c u l t i n view of the fac t that there i s no means of determining the shape a recession should have i f s h o r t - c i r c u i t i n g e x i s t s . It should also be noted that, as i n the case of the ris e s , t h e fact the recessions are fast must be proved by the presence of low resistance paths on top of the bed. Because of water drainage i n the unsaturated zone, i t i s not clear i f a r e l a t i v e l y high recharge rate i s also responsible for fast recessions. Figure 1.4.2.2 shows that the recessions f o r Ex2 and Ex3 are s i m i l a r . There i s thus no i n d i c a t i o n that concentration p r e c i p i t a t e s the beginning of the recession. - 99 -Concentration of i r r i g a t i o n had thus no noticeable influence on the recessions, i n d i c a t i n g that concentration did not a f f e c t the proportion of water s h o r t - c i r c u i t i n g the s o i l matrix to a point where the e f f e c t was observable. This i s i n agreement with the observations made for the r i s e s . No e f f e c t from i n i t i a l conditions should be expected since i n i t i a l conditions are not expected to have any e f f e c t on recessions occurring a f t e r steady state has been reached. To summarize, although the recessions occur soon af t e r the end of i r r i g a t i o n , and are f a s t , they cannot be used as a proof of s h o r t - c i r c u i t i n g . Moreover, concentration has no detectable e f f e c t on the shape of the recession of Ex2, as can be observed from the p a r a l l e l nature of the recession limbs for Ex2 and Ex3. It apparently has no ef f e c t either on the timing of the recession. After having inferred the apparent hydrologic behavior of the unsaturated zone from the major features of the outflow hydrograph i n points ( i ) to ( i v ) above, one must note the following. Although i t was concluded that the s o i l behaves almost as i f the wetting front were uniform, the actual mechanism of flow in the unsaturated zone has not been uncovered. One observation concerning t h i s mechanism i s that the e f f e c t of i n i t i a l conditions i s important. This suggests that a s i g n i f i c a n t proportion of the flow must occur within the s o i l matrix. As a summary, from the outflow hydrograph, i t i s i n f e r r e d that the s o i l and water system i n the integrated unsaturated and saturated zones - 100 -domain apparently behaves as i f l i t t l e or no s h o r t - c i r c u i t i n g occurred. More generally, i t behaves as i f the wetting front, except for a l i t t l e i n i t i a l flow, were uniform and sharp. The actual flow mechanism within the unsaturated s o i l however remains unknown. For instance, the lack of e f f e c t of i n i t i a l conditions on pipes 1,5 and 8 i s puzzling. 1.5.2 L a t e r a l Saturated Flow i n Low Resistance Paths In t h i s section, the hydrologic behavior of the material on top of the bed and i n which saturated flow takes place w i l l be i n f e r r e d from the shape of the outflow hydrograph. The kinematic wave model w i l l again play a central role i n the discussion. The conditions that must be met for the kinematic wave model to be applicable are a low, steady, and uniform recharge rate, a steep and uniform bed slope, a high, uniform and steady hydraulic conductivity, a steady and uniform e f f e c t i v e porosity, and the v a l i d i t y of Darcy's law. Also, flow must be u n i d i r e c t i o n a l . A steep bed slope and a high hydraulic conductivity y i e l d fast flow over the bed. Together with a low recharge rate they y i e l d a thin saturated zone, hence a water table approximately p a r a l l e l to the bed. The model yi e l d s a hydrograph r i s e which i s a straight l i n e , and a water table p a r a l l e l to the bed while i t i s r i s i n g . From the st r a i g h t l i n e r i s e observed i n Figs. 1.4.2.2 to 1.4.2.5, i t appears that the s o i l of the Forest plot behaved as i f the assumptions necessary for the kinematic wave model were s a t i s f i e d . Section 1.5.1 already treated - 101 -steadiness and uniformity of the recharge rate. The recharge rate was f a i r l y large. However, a low recharge rate i s required for the saturated zone to be t h i n , and hence for the water table to be approximately p a r a l l e l to the bed. Consequently, a high recharge rate can be compensated by a large hydraulic conductivity and a steep bed slope. The bed slope of 30° i s indeed steep. Whether a high hydraulic conductivity exists within the saturated zone w i l l now be established. The kinematic wave model allows the c a l c u l a t i o n of an e f f e c t i v e hydraulic conductivity for the saturated zone, using the slope of the outflow hydrograph r i s e . Values of 1.6 x 10 _ l + to 3.2 x I O - 4 ms - 1 are obtained i n Chapter II for t h i s conductivity. As shown i n Section 2.4, these values correspond to the hydraulic c o n d u c t i v i t i e s of fine or f i n e to medium sand. Proofs that t h i s can be considered to be a high hydraulic conductivity are offered by a recession rate about 16 times the one obtained by Hewlett and Hibbert for disturbed sandy loam, by v i s u a l observations and hydraulic c o n d u c t i v i t i e s measured at outflow points. In Section 1.4.4, a comparison of the outflow hydrograph recession obtained for the present work with the one obtained for a s o i l of s i m i l a r texture by Hewlett and Hibbert was presented. It was concluded that, within the l i m i t a t i o n s linked to the comparison, the recession for the present work i s much faster than the one obtained by Hewlett and Hibbert. Since the s o i l used by Hewlett and Hibbert had been excavated and had l o s t i t s o r i g i n a l structure, i t can be concluded that the higher conductivity of the s o i l i n the saturated zone of the Forest plot i s due to the low resistance paths. - 102 -The rootmat and dark material found on top of the bed are obviously materials responsible for high c o n d u c t i v i t i e s . Within the saturated zone, flow i n low resistance paths i s not r e s t r i c t e d by lack of free water l i k e i n the unsaturated zone, therefore the e f f e c t of low resistance paths occurring i n this zone can be large. Flow on top of the bed takes place according to topography and the various low resistance paths water can f i n d . It does not take place uniformly downhill but as a network of concentrated flow and thus v i o l a t e s the assumptions of uniform hydraulic conductivity and u n i d i r e c t i o n a l flow made i n the kinematic wave model. Yet the system behaves as i f the hydraulic conductivity were uniform and flow u n i d i r e c t i o n a l . Points of concentrated outflow observed at the bottom of the channel bank are proof of a network of p i p e l i k e low resistance paths. Since outflow was not observed to s t a r t from a l l outflow points at the same time, not a l l the low resistance paths on top of the low conductivity layer contribute to flow at the beginning. As explained e a r l i e r , hydraulic co n d u c t i v i t i e s at two outflow points at the bottom of the channel may be bracketed by 5 x 10" 4 and 5 x 1 0 - 3 ms - 1. The lower of these conductivities i s of the same order of magnitude as the one of medium sand (0.25 - 0.5 mm). Quick, saturated flow over the bed must be a general mechanism i n places s i m i l a r to the p l o t . This flow, and on a larger scale, streams flowing a l t e r n a t i v e l y over the surface or underground, must be the main causes of creek f l a s h i n e s s . It may also be a major cause of nonlinear flow, as discussed i n Chapter I I I . - 103 -The slope of the main limb of r i s e and of the recession i s influenced by saturated flow over the bed. Due to fast saturated flow over the bed, i t can be expected that r i s e and recession slopes are steep. If i t can be assumed that the water v e l o c i t y i n the saturated zone i s not influenced by the thickness of the saturated zone, the time from the st a r t of the main limb of r i s e to steady state w i l l be independent of the i r r i g a t i o n rate. Since a higher i r r i g a t i o n rate means a higher outflow rate, i t w i l l also mean a steeper hydrograph r i s e . Thus a r e l a t i v e l y high recharge rate i s also responsible for a steep r i s e . The ef f e c t of the recharge rate on the recession i s more d i f f i c u l t to determine due to the drainage of the unsaturated zone. The presence of low resistance paths on top of a steep bed and hence fast flow should be expected to y i e l d a thin saturated zone and therefore a water table approximately p a r a l l e l to the bed i f the recharge rate i s steady and uniform. Technically, a water table p a r a l l e l to the bed while i t r i s e s i s yielded by the kinematic wave model. It i s not possible to conclude from Figs. 1.4.3.10 to 1.4.3.12 that the water table rose p a r a l l e l to the bed because of the small s i z e of the h i l l s l o p e plot and the small number of measurement points. The system however behaves as i f the water table were r i s i n g p a r a l l e l to the bed. A water table r i s i n g p a r a l l e l to the bed would be the outcome of approximately uniform and v e r t i c a l downward flow i n the unsaturated zone, a large hydraulic conductivity i n the saturated zone, and a steep bed slope. To summarize, two of the assumptions necessary for the kinematic wave model are r e a d i l y v e r i f i e d : a steep bed slope and a high hydraulic - 104 -conductivity. One difference with the kinematic wave model, however, i s the presence of discre t e paths of low resistance rather than of a uniform layer of high conductivity. 1.5.3 Discussion of Beven's 1982 Model in Relation to the Present Work In Section 1.2, i t was mentioned that Beven (1982) compared the hydrograph obtained experimentally by Weyman (1970, 1973) for a slope segment with the hydrograph obtained using a model based on the kinematic wave model described i n Chapter I I . Beven's work w i l l now be discussed i n more d e t a i l . F i r s t , the difference between Beven's work and the use of the kinematic wave model i n the present work should be noted. Beven used Weyman's f i e l d data i n order to obtain the parameters necessary f o r the simulation of a hydrograph by his model. He also compared t h i s hydrograph with a hydrograph obtained experimentally by Weyman. In the present work, the s i m i l a r i t i e s between the general shape of the hydrographs obtained experimentally and the shape of the hydrograph yielded by the kinematic wave model are used to i n f e r flow mechanisms i n the saturated and the unsaturated zones and to cal c u l a t e an e f f e c t i v e hydraulic conductivity. Second, i t i s l i k e l y that the s o i l i n Weyman's plot i s less suited for the kinematic wave model or the extension thereof used by Beven (1982) than the one of the Forest p l o t . Although i t i s possible to obtain a hydraulic conductivity d i s t r i b u t i o n from Weyman's data, i t was not possible to obtain a d i s t r i b u t i o n that agreed with the one Beven - 105 -obtained. Therefore i t was considered preferable not to attempt to use either d i s t r i b u t i o n . This means that the hydraulic conductivity of Weyman's plot cannot be compared q u a n t i t a t i v e l y with the one of the Forest p l o t . From the s o i l d e s c r i p t i o n given by Weyman (1970, 1973), i t can however be inferred that the s o i l i n Weyman's plot has a lower hydraulic conductivity. E s p e c i a l l y , i t does not have the s p e c i a l feature of an accumulation of pi p e l i k e low resistance paths over the bed, l i k e the one found i n the Forest p l o t . One expected e f f e c t of a lower hydraulic conductivity i s that, i n order to have the water table p a r a l l e l to the bed, the recharge rate has to be smaller. 1.5.4 Comparison with Other Fast Flow Situations It i s i n t e r e s t i n g to determine whether the presence of low resistance paths on top of the bed, which contributes to a high hydraulic conductivity for the saturated zone, has been observed by other authors. As mentioned i n Section 1.2, Utting, i n a s i t e close to the Forest p l o t , noted the presence of a rootmat on top of the t i l l . A s i t u a t i o n possibly close to the one observed i n the Forest plot has been observed by Mosley (1979). He noticed "points of concentrated seepage, usually at the base of the B horizon, at which high rates of outflow were observed during storms." Although he does mention the existence of flow i n root channels higher up in the p r o f i l e , he does not say i f the seeps are due to root material. He also notes that " i t seems probable that e l u v i a t i o n has increased hydraulic c o n d u c t i v i t i e s [ i n the - 106 -seepage zone at the Interface between the B horizon and the bed]" and that i n some locations of this i nterface pipes e x i s t . It i s not c l e a r , however, i f the hydraulic conductivity at the base of the B horizon i s larger than higher up i n the p r o f i l e . Seeps were also observed by Harr (1977) and by Weyman (1970). The seeps observed by Harr apparently were on top of the bed. Downslope flow in pipes (e.g., Atkinson, 1978) also resembles the fast flow above the bed occurring i n the Forest, although i n a more remote way. 1.5.5 The Flow Model As a conclusion, based on the outflow hydrograph and v i s u a l observations, the following model of flow i s presented for the Forest p l o t . A l i t t l e water reaches the channel before the main limb of r i s e s t a r t s . This water may have reached the bed early because i t flowed through low resistance paths or as f i n g e r s . Another p o s s i b i l i t y i s that i t came from an area where the s o i l was shallow. Because the amount of water involved i n this i n i t i a l flow i s small and because the main limb of r i s e i s a st r a i g h t l i n e , one can conclude that the system behaves as i f the recharge rate were a step function and the same step function at every l o c a t i o n of the bed. Water having reached the bed flows r a p i d l y towards the channel i n a network of low resistance paths offered by the rootmat and by dark material present on top of the steep bed. In spite of this nonuniform - 107 -flow, the system behaves as i f the hydraulic conductivity of the saturated zone were uniform. The fast flow on top of the low conductivity layer i s a major cause of flashiness and may even be nonlinear. 1.5.6 V a l i d i t y of the Kinematic Wave Model One may wonder whether other models rather than the kinematic wave model could y i e l d the straight l i n e r i s e of the hydrographs obtained for the Forest plot and i f another model would be more s a t i s f a c t o r y than the kinematic wave model. While answering this question f u l l y i s beyond the scope of the present work, some attention w i l l now be devoted to i t . F i r s t , the saturated zone w i l l be examined. The kinematic wave model assumes a water table approximately p a r a l l e l to the bed. Because of a high hydraulic conductivity and a steep bed slope, one can expect a thin saturated zone, and hence a water table approximately p a r a l l e l to the bed for the Forest p l o t . The kinematic wave model thus appears a reasonable model for the Forest plot i n t h i s respect. Not a l l the assumptions necessary for the kinematic wave model are met by the saturated zone, though. In p a r t i c u l a r , the hydraulic conductivity and the e f f e c t i v e p o r o s i t y as defined i n Appendix G are not uniform and the flow i s not u n i d i r e c t i o n a l . As said e a r l i e r , a straight l i n e r i s e could be due to an averaging process. The unsaturated zone w i l l now be examined. The recharge rate i s nonuniform and unsteady, at least at the very - 108 -beginning of the recharge. Here too an averaging process can be invoked unless nonuniformity and unsteadiness occur only for a short period of time at the beginning of the recharge and not during the main limb of r i s e . The p o s s i b i l i t y of a s i g n i f i c a n t proportion of flow taking place i n low resistance paths within the unsaturated zone i s not t o t a l l y r ejected, e s p e c i a l l y due to the lack of response of pipes 1, 5 and 8 to change i n i n i t i a l conditions. A straight l i n e r i s e would be due to averaging. More attempts should be made i n order to check the occurrence of s h o r t - c i r c u i t i n g experimentally. The only value of the kinematic wave model for understanding the flow mechanism i n the unsaturated zone i s that the integrated flow i n the unsaturated and saturated zones occurs as i f i t were v a l i d . As a summary, the kinematic wave model applies to the Forest p l o t only to the extent that the observed hydrograph r i s e i s a st r a i g h t l i n e and that the saturated zone exhibits a high hydraulic conductivity and rests on a steep bed. Other assumptions for the saturated zone are not met and i t can only be said that the system behaves as i f they were. In Chapter I I I , i t i s shown that i t i s not c e r t a i n whether flow can be nonlinear and what the implications of n o n l i n e a r i t y would be. In order to apply the kinematic wave model, i t i s necessary to assume that n o n l i n e a r i t y , i f existant, i s n e g l i g i b l e . The assumptions of a uniform and steady recharge rate are probably not met, and therefore a l l that can be said i s that the unsaturated zone and the saturated zone together behave as i f the kinematic wave model - 109 -were v a l i d . Formulation of a model that represents r e a l i t y more c l o s e l y would require more experimental work. 1.6 Conclusions V i s u a l observations showed that the s o i l has a very stable structure. It i s open due to the presence of low resistance paths. P i p e l i k e low resistance paths are due mainly to root material. Uniform and concentrated i r r i g a t i o n of the small forested h i l l s l o p e p l o t chosen for the present work yielded the following r e s u l t s : ( i ) Drier i n i t i a l conditions increased the time lag to the main limb of r i s e . They also delayed the water table r i s e . They did not a l t e r the slope of the main limb of r i s e . ( i i ) Concentration of i r r i g a t i o n did not change the general shape of the outflow hydrograph nor did i t decrease the time lag to the main limb of r i s e . It did not cause the water table to r i s e e a r l i e r , neither did i t a f f e c t the shape of the water table at steady state. It did not enhance the i n i t i a l step of the outflow hydrograph observed for uniform i r r i g a t i o n . Rather, i t suppressed i t . ( i i i ) Outflow hydrographs are characterized by l i t t l e or no preliminary outflow, by steep and straight main limbs of r i s e and by steep recessions. These observations led to the conclusion that the s o i l system behaves as i f the kinematic wave model were v a l i d . It should be remembered that the term "kinematic wave model" denotes e x c l u s i v e l y the model described i n the second chapter. Transformation of the outflow hydrograph into a recharge hyetograph using the kinematic wave model indicates that the s o i l and water system - 1 1 0 -i n the integrated unsaturated and saturated zones domain apparently behaves as i f the recharge hyetograph were a uniform step function. In other words i t behaves as i f l i t t l e or no s h o r t - c i r c u i t i n g occurred and as i f recharge started everywhere at the same time and reached steady state instantaneously. The lack of e f f e c t of i n i t i a l conditions on pipes 1, 5 and 8 may however suggest s h o r t - c i r c u i t i n g . Comparison between Ex2 and Ex3 of the outflow hydrograph and of the water table indicates that the system behaves as i f s h o r t - c i r c u i t i n g were not enhanced by concentration. The fact that the system behaves as i f the kinematic wave model were v a l i d also suggests a steep bed and a high hydraulic conductivity i n the saturated zone. Both these features are r e a d i l y v e r i f i e d , a high hydraulic conductivity on top of the bed being due to a network of low resistance paths. It i s possible that a large hydraulic conductivity over the bed has also been noticed by Mosley (1979). About the value of the kinematic wave model for the Forest p l o t , i t can be said that i t i s reasonable for the saturated zone to the extent that two major assumptions are s a t i s f i e d . For the unsaturated zone, however, i t i s probable that i t s only value i s the fact that the system behaves as i f i t were v a l i d . More experimental work i s necessary to determine flow conditions there. For instance, the lack of e f f e c t of i n i t i a l conditions on pipes 1, 5 and 8 i s puzzling and suggests s h o r t - c i r c u i t i n g . - 1 1 1 -1.7 References Anderson, M.G. and T.P. Burt. 1980. Interpretation of recession flow. Jour, of hydrology, Vol. 46: 89-101. Atkinson, T.C. 1978. Techniques for measuring subsurface flow on h i l l s o p e s . In Kirkby, M.J. (ed.) H i l l s l o p e hydrology. John Wiley. 389 pp. Aubertin, G.M. 1971. Nature and extent of macropores in forest s o i l s and t h e i r influence on subsurface water movement. U.S. Forest  Exp. Sta. Res. Paper, NE-192. Baker, F.G. and J . Bouma. 1976. V a r i a b i l i t y of hydraulic conductivity i n two subsurface horizons of two s i l t loam s o i l s . S o i l S c i . Soc.  Amer. Jour. V o l . 40: 219-222. Barnes, B.S. 1939. The structure of discharge-recession curves. Amer. Geophys. Union Trans., V o l . 20: 721-725. Cited by Anderson and Burt. Beasley, R.S. 1976. Contribution of subsurface flow from the upper slopes of forested watersheds to channel flow. S o i l S c i . Soc.  Amer. Jour., Vol. 40: 955-957. Betson, R.P., J.B. Marius and R.T. Joyce. 1968. Detection of saturated interlow i n s o i l s with piezometers. S o i l S c i . Soc. Amer. Proc., Vol. 32: 602-604. Beven, K. 1981. Kinematic subsurface stormflow. Water resources  research, V o l . 17, No. 5: 1419-1424. Beven, K. 1982. On subsurface stormflow: predictions with simple kinematic theory for saturated and unsaturated flows. Water  resources research, V o l . 18, No. 6: 1627-1633. Beven, K. and P. Germann. 1980. The role of macropores i n the hydrology of f i e l d s o i l s . I n s t i t u t e of hydrology report 69, Wallingford, England. Bouma, J. and L.W. Dekker. 1978. A case study on i n f i l t r a t i o n into dry clay s o i l . 1. Morphological observations. Geoderma, Vol. 20: 27-40. Bond, R.D. 1964. The influence of the microflora on the physical properties of s o i l s . I I . Australian Jour. S o i l Res., Vol. 2: 123-131. Bond, R.D. and J.R. H a r r i s . 1964. The influence of the microflora on the physical properties of s o i l s . I. E f f e c t s associated with filamentous algae and fungi. A u s t r a l i a n Jour. S o i l Res., V o l . 2: 111-122. Cited by Bond (1964). - 112 -Chamberlin, T.W. 1972. Interflow i n the mountainous forest s o i l s of coastal B r i t i s h Columbia. In Mountain Geomorphology:  Geomorphological processes i n the Canadian C o r d i l l e r a . B.C. geographical s e r i e s , number 14, Tantalus Research Ltd., Vancouver, B.C., Canada. C h i l d s , E.C. 1969. An introduction to the physical basis of s o i l water  phenomena. Wiley. 493 pp. Chow, T.L. 1976. A low-cost tipping bucket flow meter for overland flow and subsurface stormflow studies. Canadian Journal of S o i l  Science, Vol. 56: 197-202. de V r i e s , J . 1979. Pred i c t i o n of non-Darcy flow i n porous media. American Society of C i v i l Engineers, Journal of the i r r i g a t i o n and  drainage d i v i s i o n , Vol. 105, No. IR2, Proc. Paper 14610: 147-162. de V r i e s , J . and T.L. Chow. 1978. Hydrologic behavior of a forested mountain s o i l i n coastal B r i t i s h Columbia. Water Resources  Research, Vol. 14, No. 5: 935-942. Dunne, T. and R.D. Black. 1970. P a r t i a l area contributions to storm runoff i n a small New England watershed. Water resources research, Vol. 6, No. 5: 1296-1311. Gray, D.M. ( e d i t o r ) . 1970. Handbook on the p r i n c i p l e s of hydrology. Harr, R.D. 1977. Water flux i n s o i l and subsoil on a steep forested slope. Jour, of hydrology, Vol. 33: 37-58. Hewlett, J.D. and A.R. Hibbert. 1963. Moisture and energy conditions within a sloping s o i l mass during drainage. Jour, of geophysical  research, Vol. 68, No. 4: 1081-1087. Hewlett, J.D. and C.A. Troendle. 1975. Non-point and diffused water sources: a va r i a b l e source area problem. American Society of C i v i l  Engineers, Committee on watershed management. Watershed Management, Utah State University (Symposium): 21-46. H i l l , D.E. and J.Y. Parlange. 1972. Wetting front i n s t a b i l i t y i n layered s o i l s . S o i l S c i . Soc. Amer., V o l . 36, No. 5: 697-702. H i l l e l , D. 1971. S o i l and water: physical p r i n c i p l e s and processes. Academic Press. 288 pp. Ineson, J . and R.A. Downing. 1964. The groundwater component of r i v e r discharge and i t s r e l a t i o n s h i p to hydrogeology. I n s t i t u t i o n of  water engineers j o u r n a l . 18: 519-541. Cited by Anderson and Burt. - 113 -Mehra, O.P. and M.L. Jackson. 1960. Iron oxide removal from s o i l s and clays by a d i t h i o n i t e - c i t r a t e system buffered with sodium bicarbonate, Clays and clay minerals. V o l . 5: 317-325. Inter-national series of monographs on earth science, Pergamon Press. M i l l e r , R.D. and E. Bresler. 1977. A quick method for estimating s o i l water d i f f u s i v i t y functions. S o i l S c i . Soc. Amer. Jour., V o l . 41: 1020-1022. Mosley, M.P. 1979. Streamflow generation i n a forested watershed, New Zealand. Water resources research, V o l . 15, No. 4: 795-806. Nagpal, N.K. and J . de V r i e s . 1976. On the mechanism of water flow through a forested mountain slope s o i l i n coastal western Canada. Unpublished m a t e r i a l . Pond, S.F. 1971. Qua l i t a t i v e i n v e s t i g a t i o n into the nature and d i s t r i b u t i o n of flow processes i n Nant Gerig. Subsurface  hydrology, Rept. No. 28, I n s t i t u t e of hydrology, Wallingford, England. Cited by Atkinson. Schumacher, W. 1864. Die Physik des Bodens. B e r l i n , ( c i t e d by Beven and Germann). Smith, W.0. 1967. I n f i l t r a t i o n i n sands and Its r e l a t i o n to groundwater recharge. Water resources research, V o l . 3, No. 2: 539-555. Sowers, G.B. and G.F. Sowers. 1970. Introductory s o i l mechanics and  foundations. Macmillan. 556 pp. Utting, M.G. 1978. The generation of stormflow on a glaci a t e d h i l l s o p e  i n coastal B r i t i s h Columbia. M.Sc. th e s i s . U niversity of B r i t i s h Columbia, Vancouver. Weyman, D.R. 1970. Throughflow on h i l l s l o p e s and i t s r e l a t i o n to the stream hydrograph. International association of s c i e n t i f i c  hydrology. B u l l e t i n , XV e Annee, No. 3: 25-33. Weyman, D.R. 1973. Measurements of the downslope flow of water i n a s o i l . Jour, of hydrology, Vol. 20: 267-288. Whipkey, R.Z. 1968. Storm runoff from forested catchments by subsurface routes. Int. Assoc. S c i . Hydrology, publ. 85: 773-779. - 114 -CHAPTER II DETERMINATION OF AN EFFECTIVE HYDRAULIC CONDUCTIVITY USING THE KINEMATIC WAVE MODEL 2.1 Introduction One of the p i l l a r s of the discussion of the experimental r e s u l t s presented i n Chapter I i s the kinematic wave model. The term "kinematic wave model" denotes ex c l u s i v e l y the model described i n the present chapter. This model allowed us to make inferences with respect to s o i l hydrologic behavior. It i s a very simple model of saturated flow over a sloping bed, described by an equation that can be solved a n a l y t i c a l l y . It has been developed by Henderson and Wooding (1964) and Beven (1981). A major underlying assumption of the kinematic .wave model i s that the water table i s approximately p a r a l l e l to the bed. Using a reasoning s i m i l a r to the one made by Henderson and Wooding (1964) for the case of overland flow, the water table i s approximately p a r a l l e l to the bed i f the water depth i s small. Moreover, a thin saturated zone occurs when the recharge rate i s low, the bed slope steep and the hydraulic conductivity high. In t h i s chapter, the kinematic wave equation for saturated flow i n a porous medium over a sloping bed w i l l be derived and solved following mainly Beven (1981) whose work i s based on Henderson and Wooding's. This w i l l include a presentation of the behavior of the water table and of the outflow hydrograph yielded by the solu t i o n . The equation describing the hydrograph r i s e w i l l then be used to obtain an e f f e c t i v e satiated hydraulic conductivity for the Forest p l o t . - 115 -2.2 L i t e r a t u r e Review A number of approximate equations have been proposed to describe flow i n a porous medium over a sloping bed. For flow taking place between an approximately horizontal free water surface and an approximately horizontal impermeable bed, i t can be assumed that flow l i n e s are h o r i z o n t a l . Hence the hydraulic gradient i s equal to the slope of the water table. This i s the Dupuit-Forchheimer theory ( C h i l d s , 1969). On steep slopes, one should rather assume that the flow l i n e s are p a r a l l e l to the bed. This was done by Henderson and Wooding (1964), Childs (1971) and Beven (1981). Henderson and Wooding derived the equation of flow for t h i s case, equation that Beven c a l l s "extended Dupuit-Forchheimer equation." From th i s equation they obtained a n a l y t i c a l l y the p r o f i l e of the water table at steady state. Using a numerical method they obtained the p r o f i l e during r i s e and recession. Beven obtained the r i s i n g limb of the hydrograph for the "extended Dupuit-Forchheimer equation" by the f i n i t e d i f f e r e n c e s method. Henderson and Wooding, then Beven (1981), s i m p l i f i e d the equation of flow further by assuming the water table i s approximately p a r a l l e l to the bed. The equation thus obtained w i l l be c a l l e d the kinematic wave equation and describes kinematic wave motion. 2.3 The Kinematic Wave Equation f o r Flow i n a Porous Medium In t h i s section, the assumptions linked to the kinematic wave model w i l l be presented. The kinematic wave equation describing the model - 116 -w i l l be derived and the solution of t h i s equation presented. For doing so, the work of Henderson and Wooding (1964), Beven (1981) and Eagleson (1970) w i l l be followed. ( i ) Assumptions The kinematic wave model i s based on the following assumptions found e x p l i c i t l y or i m p l i c i t l y i n Beven's work. The fact that a t h i n saturated zone i s a necessary assumption has been stated by Henderson and Wooding. (a) The saturated zone i s t h i n , leading to a water table approximately p a r a l l e l to the bed (Henderson and Wooding). A th i n saturated zone i s yielded by a steep bed slope, a high hydraulic conductivity and a low recharge rate. Flow l i n e s are p a r a l l e l to the bed. (b) Recharge occurs as a uniform step function. It i s equal to i Q during r i s e and steady state, and zero during recession. The recharge rate i s thus both steady and uniform. Note that, as said i n Chapter I, a uniform recharge rate i s consistent with flow occurring v e r t i c a l l y i n the unsaturated zone. (c) The e f f e c t i v e porosity and the hydraulic conductivity are steady and uniform. Flow i s u n i d i r e c t i o n a l and the bed slope uniform. The medium i s i s o t r o p i c . (d) Flow i s l i n e a r hence Darcy's law can be used. ( i i ) D e rivation The kinematic wave equation used by the kinematic wave model can be derived from Darcy's law and a continuity equation. - 117 -For saturated flow over steep beds Henderson and Wooding assumed flow lines to be parallel to the bed, which enabled them to write Darcy's law as 3T qT « -KT (-g^  cos u> - sinu)) 2.3.1 where q • discharge per unit area or macroscopic flow velocity, often called Darcy's velocity. In the present work, q i s in the main flow direction. T • thickness of the saturated zone measured perpendicular to the bed. Thus 3 1 1 qT = discharge per unit width of hillslope (m s~ m~ ) K = satiated hydraulic conductivity. x = downslope distance from the highest point on the bed where recharge occurs. oi m angle of the bed with respect to the horizontal. Flow over a steep slope is depicted in Fig. 2.3.1. Figure 2.3.1 Saturated flow in a porous medium resting on a sloping bed. o> i s the bed slope angle with the horizontal. - 118 -R i s the i r r i g a t i o n rate. Note that by assuming qT i s discharge per unit width of h i l l s o p e , i t i s assumed that K i s uniform with respect to z, where, as shown i n Fig . 2.3.1, z i s upward and perpendicular to the bed. K w i l l be assumed uniform i n the kinematic wave model. Also, the medium i s assumed to be i s o t r o p i c . If the water table i s approximately p a r a l l e l to the bed, 3T T T — cosco « sincu 9x and Eq. 2.3.1 becomes (Beven, 1981) qT = TK sinco 2.3.2 Note that Eq. 2.3.2 i s equivalent to Darcy's law with the hydraulic gradient taken to be sino). The equation of c o n t i n u i t y for flow i n the x d i r e c t i o n i s (Beven, 1981) l | B l ) = . | _ ( T q ) + i 2.3.3 where n = e f f e c t i v e porosity; see Appendix G for d e f i n i t i o n , t = time i = recharge rate per unit area p a r a l l e l to the bed. i can vary with time and be zero. Let y be a d i r e c t i o n perpendicular to both x and z, and along the h i l l s l o p e width. Assuming n steady and uniform with respect to z and - 119 -y, assuming also K and to uniform with respect to x and y, and combining Eqs. 2.3.2 and 2.3.3. yi e l d s the kinematic wave equation (Beven, 1981) 9T 9T = -Ksinu) - r - + i 2.3.4 d t d X Adapting for the flow i n a porous medium treated here the d e r i v a t i o n offered by Henderson and Wooding for overland flow, the wave v e l o c i t y c i s dx K sinco „ „ c c = - T — = 2.3.5 dt n Using Eq. 2.3.5 and assuming n uniform with respect to x, Eq. 2.3.4 can be rewritten as i C g l + c iOjIi = i 2.3.6 d t dX ( i i i ) Solution Eq. 2.3.6 i s solved i n Note 1, Appendix H. F i g . 2.3.2, adapted from Eagleson (1970) depicts the behavior of the water table during r i s e and steady state. I n i t i a l conditions are T = 0 for a l l x's before the s t a r t of recharge. Boundary conditions are T = 0 at x = 0 for a l l times. Recharge occurs from x = 0 to x = L. In other words, L i s the length of the saturated zone. Recharge rate i s a constant for the kinematic wave model. At time t ^ , the water table shape i s given by the curve OAE; at time t2 i t i s given by the curve OBF. F i n a l l y , when steady state i s - 120 -reached by the e n t i r e saturated zone, the water table i s the s t r a i g h t l i n e OD. Figure 2.3.2 Adapted from Eagleson, 1970, F i g . 15-5. Shape of the water table during r i s e and steady state. It i s shown i n Appendix H that at a time t* after the s t a r t of the recharge, the thickness of the saturated zone for the part of the saturated zone having reached steady state i s given by (Beven, 1981) where I D i s the recharge rate per unit area p a r a l l e l to the bed during r i s e and steady s t a t e . For the part s t i l l r i s i n g (Beven, 1981) D O X I n, O X T = n c 2.3.7 2.3.8 - 121 -where t* = time from the s t a r t of recharge. Figure 2.3.3 depicts the shape of the water table during recession. It Is i n t e r e s t i n g to note that the assumption that the water table i s approximately p a r a l l e l to the bed y i e l d s a water table p a r a l l e l to the bed during the r i s e but not at steady state and neither during the recession. T O L * Figure 2.3.3 Shape of the water table during the recession * Figure 2.3.4 pictures the outflow hydrograph. It i s very simple, with s t r a i g h t l i n e r i s e and recession. As shown in Appendix H, Note 1, the discharge at the bottom of the h i l l s l o p e i s i Q = Ksinu) D-°- t* during the r i s e 2.3.9 Q = D i D L at steady state 2.3.10 - 1 2 2 -Figure 2.3.4 Hydrograph given by the kinematic wave model i D Q - 1 L D - Ksinui D— t' during the recession 2.3.11 x o n where _ . . . ,volumev Q • discharge (——; ) tinie - width of the hillslope t* - t - t» r t r - time at which recharge stopped The effective porosity n has been assumed to be the same for the rise and the recession. In Appendix H, Note 4, i t is shown that i f the effective porosity during the recession is n r e , then n in Eq. 2.3.11 should be replaced by n r e> It is of interest to note that Eqs. 2.3.9 and 2.3.11 indicate that the rise and recession rates are larger for a larger recharge rate. - 1 2 3 -F i n a l l y , note that although Beven (1981) does not give Eqs. 2.3.6, 2.3.9, 2.3.10 and 2.3.11, he gives equations used in the d e r i v a t i o n of Eqs. 2.3.2, and 2.3.4 and plots the hydrograph r i s e and steady state. 2 . 4 Use of the Kinematic Wave Model to Calculate an Effective K It was shown in Chapter I that the s o i l and water system i n the Forest plot behaved as i f the kinematic wave model assumptions were s a t i s f i e d . In p a r t i c u l a r , two of these assumptions are r e a d i l y v e r i f i e d : a steep bed slope and a high hydraulic conductivity. In Section 3.4 i t i s shown that i t i s not c e r t a i n i f the kinematic wave model can be applied to the Forest plot as far as nonlinear flow i concerned. It must be assumed that n o n l i n e a r i t y , i f present, i s n e g l i g i b l e . Note that use of the kinematic wave model w i l l always present the problem of the need for flow to be s u f f i c i e n t l y f a s t , yet not too f a s t . A number of the assumptions necessary for the kinematic wave model have not been considered in Chapter I. They are a uniform bed slope, steady hydraulic conductivity and e f f e c t i v e porosity and a uniform e f f e c t i v e porosity. These w i l l now be b r i e f l y examined. The bed slope i s not uniform. However, i t w i l l be assumed that, as a r e s u l t of an averaging process, nonuniformity e f f e c t s can be neglected. The hydraulic conductivity and the e f f e c t i v e porosity are probably steady since the s o i l structure i s very stable. The e f f e c t i v e porosity i s not uniform. However, i t can be argued, l i k e i n the case o - 124 -nonuniformities i n the hydraulic conductivity, that the system behaves as i f i t were uniform. The water table behavior i s an important c r i t e r i o n for the use of the kinematic wave model. The shape of the water table was given i n F i g s . 1.4.3.10 to 1.4.3.12. As mentioned i n Section 1.4.3, the shape of the bed i s uncertain and the data i n s u f f i c i e n t to make a d e f i n i t e statement as regard to the shape of the water table. However, since the system behaves as i f the kinematic wave model were v a l i d , i t behaves as i f the water table were approximately p a r a l l e l to the bed. D i f f e r e n t i a t i n g Eq. 2.3.9 with respect to t* and solving the equation thus obtained for K y i e l d s an e f f e c t i v e K K = n , L 2.4.1 eff Di sino) o ^ e f f can thus be calculated from measured data. Since Eq. 2.4.1 i s derived from the hydrograph r i s e equation, n i n Eq. 2.4.1 should be the e f f e c t i v e porosity of the s o i l the water table i s r i s i n g i n . The water content of the s o i l j u s t before the water table r i s e s i n i t w i l l be c a l l e d 8^n s > z.> s.z. standing for "saturated zone." This w i l l allow us to d i s t i n g u i s h i t from the i n i t i a l water content found before i r r i g a t i o n s t a r t and denoted by 6^n. A maximum value of n has been obtained by using f o r 6^n g < z the minimum 9 i n . A minimum value has been s i m i l a r l y obtained by using for ^ i n S m Z the maximum 6 s t . s t . Th e minimum and the maximum values of n - 125 -lead to a minimum and a maximum value for an e f f e c t i v e K. Since the recharge rate i s not known, i t has to be obtained from the i r r i g a t i o n rate, taking the water balance into account. This i s done i n Note 3, Appendix H. D i s the length of the channel measured perpendicular to the slope. The angle of the bed with the horizontal has been taken to be equal to the angle of the ground surface with the h o r i z o n t a l . Table 2.4.1 shows the c a l c u l a t i o n of Keff. For E x l , Kgff i s found to be between 1.6 x 10~^ and 3.1 x 10 - 1 + ms - 1. For Ex2 i t i s between 1.8 x 10 - l + and 3 x 10~h ms - 1 and for Ex3 i t i s between 2.1 x l O - 4 and 3.2 x ICT* ms - 1. A number of errors are attached to the estimate of Kgff. They are errors due to possible n o n l i n e a r i t y and to inaccuracies i n the values used for c a l c u l a t i o n . The magnitude of these errors i s unknown. Also, i t should be remembered that Keff i s calculated using s i m p l i f y i n g assumptions. An e f f e c t i v e satiated hydraulic conductivity should be found to be the same for any r a i n f a l l event. It i s not clear why the minimum K e f f i s larger for Ex2 than for Exl and i s larger for Ex3 than for Ex2. In Table 2.4.2, the e f f e c t i v e K obtained by the kinematic wave model i s compared with the K's obtained at two outflow points, with an e f f e c t i v e K obtained by Utting (1978) on Nagpal and de V r i e s ' s i t e , with the K's obtained by the i n s i t u cores, and with the K's of medium and f i n e sand. The e f f e c t i v e K obtained by Utting (1978) was mentioned by t h i s author to be probably a lower estimate. In order to obtain an e f f e c t i v e Table 2.4.1 Calculations of effective hydraulic conductivities using the kinematic wave approximation Exl Ex2 Ex3 Max 6 steady state 0.40 (at 0.63 m depth, lower neutron probe access tube) 0.38 (at 0.63 m depth, lower neutron probe access tube) 0.38 (at 0.63 m depth, lower neutron probe access tube) Smallest n (a) 0.23 0.25 0.25 Minimum 9 . in 0.18 (at 0.15 m depth, lower neutron probe access tube) 0.23 (at 0.15 m depth, lower neutron probe access tube) 0.24 (at 0.15 m depth, lower neutron probe access tube) Largest n (b) 0.45 0.40 0.39 R (ms - 1) from (c) ap / start of irrigation to end of main limb of rise 3.19 x 10 - 6 2.8 x 10 - 6 2.9 x 10 - 6 (f) Slope of the main (d) n o limb of rise (m s - ) 5.5 x 10 - 9 5.1 x 10 - 9 5.8 x 10 - 9 Minimum K (e) etr using the main limb of rise (ms - 1) 1.6 x 10~k 1.8 x IO - 4 2.1 x 10 - l t Maximum K .... ef f using the main limb of rise (ms - 1) 3.1 x I0~k 3.0 x 10~k 3.2 x 10 - l t a. Smallest n = n a - Max 9 steady state; n a = 0.63 = porosity b. Largest n = n a - Minimum 9-^ n c R ap = Time averaged pooled i r r i g a t i o n rate d. Slope of the main limb of r i s e = read off the outflow hydrograph n 8t* e. K e f f sinoo D i o D = Length of channel measured perpendicular to the slope = 7 m i = recharge rate per unit area p a r a l l e l to the bed. i i = 0.80 R coso) _ o ap ro R a p C o s o ) i s multiplied by 0.80 i n order to take into account the loss found by the water 1 balance to = Angle of the ground surface with the horizontal, assumed to be equal to the angle of the bed with the horizontal, to = 30° Thus: i Q = (0.85)(0.80)R a p R f l p = time averaged pooled i r r i g a t i o n rate f . Corrected to the value that would have been observed, had RG11 been present. - 128 -Table 2.4.2 Comparison of the Kgff obtained by the kinematic wave model with other values. K ,,. by kin wave ef f K at 2 outflow points E f f e c t i v e K obtained by Utting on Nagpal and de Vries' s i t e 1.6 x I O - 4 ms - 1 to 3.2 x IO-1* ms"1 Between 5 x IO-1* ms"1 and 5 x I O - 3 ms - 1 8 x I O - 5 ms - 1 K by i n s i t u cores K medium sand K f i n e sand 8 x IO-1* ms"1 5 x I O - 4 ms - 1 I O - k ms - 1 - 129 -K, Utting used Eq. 2.3.2 and values of Q and T at steady state. His solution i s thus only p a r t l y independent of the solution obtained by the kinematic wave model. The solutions d i f f e r in that both Q and T used for the estimate of K were observed values for Utting, whereas when the kinematic wave model i s used, the observed -ir^r i s used, but T i s 3t* obtained t h e o r e t i c a l l y . They are a l i k e i n that for both estimates, Eq. 2.3.2 i s used. Table 2.4.2 shows that Kgff i s 2.5 to 5 times smaller than the K's obtained by i n s i t u cores. This i s perhaps a l i t t l e too small, considering the concentration of low resistance paths at the bottom of the saturated zone. It would indicate that, on the average, the conductivity of the top of the s o i l p r o f i l e , where the i n s i t u core measurements have been made, i s s t i l l larger than the conductivity of the saturated zone. Table 2.4.2 also shows that the value of Keff corresponds to the one obtained for fine or fine to medium sand. F i n a l l y , Table 2.4.2 shows that the values obtained by the kinematic wave model are up to four times the e f f e c t i v e K found by Utting and are one order of magnitude smaller than the larger of the two K values obtained from outflow points. It i s thus l i k e l y that i t s order of magnitude of 10 - 1 + ms - 1 i s reasonable. Also, the v a r i a t i o n of the calculated value of Kgff from 1.6 x 10" to 3.2 x 10- m s - i s acceptable. Most of the v a r i a t i o n i s due to the fact that, as explained e a r l i e r , for one experiment, a minimum or a maximum value of Keff i s obtained depending on what value - 130 -i s used f or 9 l n s < z > . To conclude, use of the kinematic wave equation i n order to calculate an e f f e c t i v e hydraulic conductivity for the Forest p l o t indicates that K e f f may be bracketed by 1.6 x 10 and 3.2 x 10" ms - 1. Comparison of the value obtained by the kinematic wave model with other values indicates that the order of magnitude obtained can be considered to be a reasonable estimate. One important note must be made concerning no n l i n e a r i t y and the kinematic wave model. If the water table i s p a r a l l e l to the bed, the hydraulic gradient i s constant since i t i s equal to sinw. Hence there i s no physical reason why q should vary at one given place since the dr i v i n g force i s constant. It i s possible, however, that flow i s fa s t enough for Darcy's law to f a i l . Instead of Darcy's law, q = K g e n sintu would apply. K g e n i s defined i n Section 3.1. As shown for instance by de Vr i e s ' (1979) experimental r e s u l t s , K g e n i n the nonlinear range i s smaller than K. It would however be constant since q and sinu) are constant. Theoretical research i s needed i n order to investigate further the ef f e c t s of nonl i n e a r i t y on the kinematic wave model. In p a r t i c u l a r , i t i s not clear whether the equations used for the kinematic wave model, and hence the kinematic wave model, would s t i l l be v a l i d . It i s not cle a r e i t h e r , whether t h e o r e t i c a l l y K e f f would vary due to n o n l i n e a r i t y . One reason why -jjp q, and hence K e f f might vary i s that the kinematic wave model does not assume the water table i s exactly - 131 -p a r a l l e l to the bed, as seen i n Eq. 2.3.4. 2.5 Conclusions In t h i s chapter, the kinematic wave equation for flow i n a porous medium over a steep bed and subject to a number of simplifying assumptions was derived and solved using the l i t e r a t u r e . The term kinematic wave model denotes ex c l u s i v e l y the model described i n t h i s chapter. The main features of the solution are straight l i n e r i s e s and recessions for the outflow hydrograph and a water table r i s i n g p a r a l l e l to the bed. S i m i l a r i t y of the hydrograph r i s e given by the kinematic wave model with the hydrograph r i s e s obtained experimentally was the basis for determining an e f f e c t i v e hydraulic conductivity for the saturated zone of the Forest p l o t . As mentioned i n Chapter I, there are several reasons for t h i s s i m i l a r i t y . Two important ones are the steepness of the h i l l s l o p e ' s bed and the high hydraulic conductivity of the saturated zone, due to the presence of pip e l i k e low resistance paths on top of the bed. It i s not certain flow i n the Forest plot i s l i n e a r , and the kinematic wave model requires l i n e a r flow. In order to apply the kinematic wave model to the Forest pl o t , i t was necessary to assume that n o n l i n e a r i t y , i f present, i s n e g l i g i b l e . Further research i s needed to determine the e f f e c t s of no n l i n e a r i t y on the kinematic wave model. Calculations show that the e f f e c t i v e K obtained using the kinematic wave model i s bracketed by 1.6 x 10 - l + and 3.2 x 1 0 - 4 ms - 1. - 132 -The order of magnitude of the Kgff thus obtained i s within the range of other values measured in the Forest plot or nearby and hence appears reasonable. - 133 -2.6 References Beven, K. 1981. Kinematic subsurface stormflow. Water resources  research, Vol. 17, No. 5: 1419-1424. Chi l d s , E.C. 1969. An introduction to the physical basis of s o i l water  phenomena. Wiley. 493 pp. Childs, E.C. 1971. Drainage of groundwater resting on a sloping bed. Water resources research, Vol. 7, No. 5: 1256-1263. Eagleson, P.S. 1970. Dynamic hydrology. McGraw-Hill, New York. Henderson, F.M. and R.A. Wooding. 1964. Overland flow and groundwater flow from a steady r a i n f a l l of f i n i t e duration. Journal of  geophysical research, Vol. 69, No. 8: 1531-1540. Utting, M.G. 1978. The generation of stormflow on a gla c i a t e d h i l l s l o p e i n coastal B r i t i s h Columbia. M.Sc. Thesis, U n i v e r s i t y of B r i t i s h Columbia, Vancouver. - 134 -CHAPTER III NONLINEAR FLOW 3.1 Introduction F l o w i n p o r o u s m e d i a i s g e n e r a l l y l i n e a r . T h i s m e a n s i t c a n b e d e s c r i b e d b y a l i n e a r r e l a t i o n s h i p b e t w e e n t h e d r i v i n g f o r c e w h i c h i s t h e m a c r o s c o p i c h y d r a u l i c g r a d i e n t ^ a n d t h e m a c r o s c o p i c f l o w v e l o c i t y q . T h i s l i n e a r r e l a t i o n s h i p i s r e p r e s e n t e d b y D a r c y ' s l a w . M a n y a u t h o r s , h o w e v e r , h a v e n o t i c e d t h a t D a r c y ' s l a w c e a s e s t o b e v a l i d f o r e i t h e r v e r y s m a l l f l o w v e l o c i t i e s ( e . g . , D u d g e o n , 1 9 6 6 ) o r l a r g e f l o w v e l o c i t i e s ( e . g . d e V r i e s , 1 9 7 9 ) . O n l y t h e l o s s o f l i n e a r i t y a t l a r g e f l o w v e l o c i t i e s w i l l b e d i s c u s s e d h e r e . F l o w m u s t b e s a t i a t e d i n o r d e r f o r v e l o c i t i e s t o b e l a r g e e n o u g h f o r t h i s k i n d o f n o n l i n e a r i t y t o b e n o t i c e a b l e . One p r a c t i c a l i m p l i c a t i o n o f n o n l i n e a r f l o w i s t h a t t h e h y d r a u l i c c o n d u c t i v i t y c a n n o t b e o b t a i n e d f r o m D a r c y ' s l a w . U s i n g D a r c y ' s l a w d h _ 1 w o u l d y i e l d a " g e n e r a l " c o n d u c t i v i t y ^-gen = qOjg*) d e p e n d i n g o n q . M o r e o v e r , t h e v a l i d i t y o f m a t h e m a t i c a l m o d e l s u s i n g D a r c y ' s l a w w o u l d b e q u e s t i o n a b l e s i n c e t h e y u s e c o n s t a n t h y d r a u l i c c o n d u c t i v i t i e s . I n t h i s c h a p t e r , i t w i l l b e a s s u m e d t h a t K a n d K g e n a r e u n i f o r m a n d t h a t t h e m e d i u m i s i s o t r o p i c . When t h e u n i f o r m i t y o f K a n d K g e n i s i n q u e s t i o n t h e y c a n b e t h o u g h t o f a s o v e r a l l v a l u e s . T h e a s s u m p t i o n o f i s o t r o p y i s n o t a g o o d o n e f o r t h e f o r e s t p l o t b u t h a s b e e n c o n s i d e r e d t o b e a n e c e s s a r y s i m p l i f i c a t i o n . - 135 -High hydraulic c o n d u c t i v i t i e s are found i n the forest s o i l used for the present work, (see Chapters I and I I ) . Also, i n t h e i r s i t e (see Chapter I ) , Nagpal and de Vries (1976) calculated the Reynolds number for flow occurring i n a root channel, assuming flow was s i m i l a r to flow i n a pipe. They obtained a Reynolds number of 2000, large enough for turbulent flow to occur. A study of nonlinear flow therefore appears relevant to the present work. In t h i s chapter, a l i t e r a t u r e review w i l l f i r s t be presented. One technical problem, namely the e f f e c t of p l o t t i n g the data as a f r i c t i o n factor versus Reynolds number r e l a t i o n s h i p w i l l then be addressed. F i n a l l y , the p o s s i b i l i t y of occurrence of nonlinear flow i n the Forest plot w i l l be examined. A number of terms used i n the following sections need to be defined. These d e f i n i t i o n s w i l l now be given. The f r i c t i o n factor i s treated i n d e t a i l i n Section 3.3, therefore i t s d e f i n i t i o n w i l l not be given below. ( i ) Reynolds Number The Reynolds number can be defined i n sev e r a l ways. One group of d e f i n i t i o n s uses v L ch ch Re = where L ,: a c h a r a c t e r i s t i c (or representative) length ch - 136 -v c h : a c h a r a c t e r i s t i c flow v e l o c i t y , generally taken to be q v : kinematic v i s c o s i t y of the f l u i d = — P p : dynamic v i s c o s i t y of the f l u i d p : f l u i d density This d e f i n i t i o n of Re w i l l be used i n the present work. When a number i s given for the Re characterizing flow i n a porous medium, the diameter of the p a r t i c l e s forming the porous medium w i l l be used for L c h and the macroscopic flow v e l o c i t y w i l l be used for v c ^ . ( i i ) Flow Velocity The macroscopic flow v e l o c i t y , or discharge per unit area q (volume per unit area per unit time) i s often c a l l e d Darcy's v e l o c i t y . Because t h i s term i s confusing when Darcy's law i s not v a l i d , i t w i l l not be used i n t h i s chapter. Also, as mentioned e a r l i e r , i n the present work, q i s taken i n the main d i r e c t i o n of flow. ( i i i ) Hydraulic Conductivity and General Hydraulic Conductivity The general hydraulic conductivity or for short general conductivity K g e n i s calculated from K = 4 - 3.1.1 gen dh ds where ^— i s the hydraulic gradient. - 137 -It i s c a l l e d "general" because although i t i s equal to the true hydraulic conductivity i n the l i n e a r range, i t i s not equal to i t anymore in the nonlinear range. As shown for instance by de V r i e s ' (1979) experimental r e s u l t s , K g e n i n the nonlinear range i s smaller than the hydraulic conductivity. (iv) Permeability Whereas the hydraulic conductivity K i s a function of both the properties of the porous medium and the f l u i d , the permeability k i s only a function of the properties of the porous medium. The two are related by K - - ^ - * 3.1.2 u where g = g r a v i t a t i o n a l acceleration (v) Linear and Nonlinear Range Some authors consider that there i s no s t r i c t l y l i n e a r range but admit the existence of an approximately l i n e a r range at small q. Whether authors consider this range to be s t r i c t l y or only approximately l i n e a r , i t w i l l be c a l l e d l i n e a r . When judged necessary, l i n e a r w i l l be put i n quotations marks to mean "so-called l i n e a r . " The term nonlinear range w i l l be used for the decidedly nonlinear range following the " l i n e a r " range. Because i t w i l l be suggested turbulence i s possibly not necessary for n o n l i n e a r i t y , nonlinear means "laminar nonlinear." - 138 -Note also that since the n o n l i n e a r i t y s p e c i f i c to very small q i s not considered, the type of n o n l i n e a r i t y mentioned i n the " l i n e a r " range w i l l be the same as the one at large q. The flow v e l o c i t y and the Reynolds number at which flow becomes nonlinear are c a l l e d " c r i t i c a l . " Careful note should be taken of t h i s since authors often use the term " c r i t i c a l " for the apparition of turbulence (e.g., Chauveteau and T h i r r i o t , 1967). 3.2 L i t e r a t u r e Review Loss of l i n e a r i t y was probably f i r s t observed by Forchheimer (1901a,b) (Bear, 1972) using data obtained by Masoni (1896) (de V r i e s , 1979). According to Engelund (1953), n o n l i n e a r i t y for porous media has been noticed to star t at a Reynolds number Re varying from 1 to 10. Lindquist (1933) using leadshot 1 to 5 mm in diameter found that n o n l i n e a r i t y started at a Re of 4. de Vries (1979) using sand and glass beads of diameter 0.4 - 0.6 mm and 0.8 - 1.4 mm observed n o n l i n e a r i t y s t a r t i n g at a Re varying from 1.6 to 5.4. Schneebeli (1955) noticed that n o n l i n e a r i t y occurred at a Reynolds number of about 5 in 27 mm diameter glass beads. Since Forchheimer's work, a f a i r amount of research has been done i n order to determine the causes of n o n l i n e a r i t y , and the equations governing flow both i n the l i n e a r and the nonlinear range. The rest of t h i s Section w i l l be devoted to a b r i e f review of t h i s research. - 139 -( i ) Causes of Nonlinearity Although Ward (1964) at t r i b u t e s n o n l i n e a r i t y to turbulence, there may be a p o s s i b i l i t y that turbulence i s not necessary for n o n l i n e a r i t y . This p o s s i b i l i t y i s offered by both experimental (Schneebeli, 1955; Chauveteau and T h i r r i o t , 1967; Dudgeon, 1966; Wright, 1968) and t h e o r e t i c a l work (Hubbert, 1956; Ahmed and Sunada, 1969; Stark, 1969; Lindquist, 1933; Irmay, 1964). Experimental evidence that turbulence i s not necessary for n o n l i n e a r i t y i s based on the fact that n o n l i n e a r i t y has been observed at flow v e l o c i t i e s smaller than the ones at which turbulence has been observed. It i s not e n t i r e l y convincing, however, since i t i s r e s t r i c t e d , or even contradicted. F i r s t , experimental evidence i s r e s t r i c t e d by the fact that turbulence has been detected either on a large scale (Schneebeli, 1955) or at one or several points (Dudgeon, 1966; Wright, 1968). In both these cases, the f i r s t occurrence of turbulence in the porous medium may have been at a flow v e l o c i t y smaller than the one at which turbulence was f i r s t detected since the flow domain was not monitored at every point. The experiments of Wright o f f e r more evidence that turbulence i s not necessary for n o n l i n e a r i t y than those of the other two authors, since he monitored flow at 9 points. Yet even from his work, the observation that n o n l i n e a r i t y started at a flow v e l o c i t y smaller than the one at which turbulence started may be debated. One reason i s that the e n t i r e flow domain was not monitored. Another reason i s that he did not monitor the flow for turbulence and for no n l i n e a r i t y appearance - 140 -using the same materials but assumed he could combine the r e s u l t s invoking s i m i l a r i t y . Apparently Chauveteau and T h i r r i o t were able to observe the onset of turbulence i n i n d i v i d u a l pores. However, they observed flow i n a two-dimensional model b u i l t to reproduce a porous medium. Second, and more importantly, for some of Dudgeon's runs, turbulence was detected at the point where flow was monitored when the o v e r a l l flow was s t i l l l i n e a r . (Contrary to the other three authors, Dudgeon does not claim n o n l i n e a r i t y s t a r t s at flow v e l o c i t i e s smaller than the ones at which turbulence s t a r t s but some of his r e s u l t s show i t i s the case.) It i s i n t e r e s t i n g that Dudgeon found turbulence s t a r t i n g at a Reynolds number as low as 1 whereas Schneebeli observed turbulence s t a r t at a Re of 60 and Chauveteau and T h i r r i o t at a Re of 80. It appears that more experimental work i s needed. A t h e o r e t i c a l approach used by several authors (Irmay, 1964; Ahmed and Sunada, 1969; Stark, 1969) to prove that laminar flow can be nonlinear i s to derive Forchheimer's equation from Navier-Stokes equation. The most common cause of n o n l i n e a r i t y proposed by authors for nonturbulent flow i s micro-acceleration, the acceleration of water p a r t i c l e s within the pores. It was not possible to determine whether the t h e o r e t i c a l evidence that n o n l i n e a r i t y i s not neces s a r i l y due to turbulence i s e n t i r e l y s a t i s f a c t o r y . If i t i s accepted that turbulence i s not necessary for n o n l i n e a r i t y , the turbulent range occurs at flow v e l o c i t i e s larger than - 141 -the ones at which n o n l i n e a r i t y s t a r t s (e.g. Schneebeli, 1955) and follows the nonlinear range as q increases. ( i i ) Mathematical Description of Linear and Nonlinear Flow The following equation, suggested by Forchheimer i s probably the most commonly used to describe nonlinear flow: Q = vaq + b q 2 (3.2.1) where a,b are constants determined by the properties of the f l u i d and the porous medium (or possibly of the porous medium only. This was not f u l l y investigated in the present work.) This equation i s generally c a l l e d Forchheimer's equation. Two other types of equation, a power law also c a l l e d Missbach equation (Dudgeon 1966; Trollope et a l . 1971) and an i n f i n i t e series equation (Rummer and Drinker 1966) have been proposed. Authors using Forchheimer's equation have done so i n various ways. Ward, Schneebeli, and Ahmed and Sunada consider that the same quadratic equation i s v a l i d for both the l i n e a r and the nonlinear ranges, the Darcy's law followed by the data in the " l i n e a r " range being an approximation of the quadratic equation at small q. Other authors (Engelund 1953; de Vries 1979) consider that data follow two d i s t i n c t equations. These l a t t e r two authors consider that the data points follow Darcy's law i n the l i n e a r range and Forchheimer's equation i n the nonlinear range. What makes the two equations d i s t i n c t - 142 -i s that the equation describing the l i n e a r range i s not an approximation of the equation describing the nonlinear range. This point w i l l now be examined i n more d e t a i l using de Vries' work. de Vries (1979), following Lindquist (1933) and Engelund, plots his data as impermeability r versus q where r i s defined by (The r's used by Lindquist and Engelund d i f f e r s l i g h t l y from de V r i e s ' ) . de Vries' data, schematically shown i n Figure 3.2.1, f i r s t follow approximately Darcy's law dh r = Pg ds P q (3.2.2) dh ds (3.2.3) recast as 1 k " r (3.2.4) then Forchheimer's equation recast as r = a* + b'q (3.2.5) where a' = ga b' = PS b y - 143 -Fig. 3.2.1 Schematic representation of the plot of r versus q obtained by de Vries (1979). Data follow approximately the solid line. A similar behavior was observed earlier by Lindquist plotting data from Zunker (1920). de Vries stresses that, as can be seen from Fig. 3.2.1, for small q, r is equal to l /k and not to a', which would be the case were Eq. 3.2.4 an approximation of Eq. 3.2.5 for small q. It is interesting that both Engelund and de Vries find from experimental data that -£ - 1.07 ga (3.2.6) Incidently, Fig. 3.2.1 shows particularly well what is meant by "linear" range, the range during which r Is approximately constant, and by nonlinear range, the range where r obeys approximately Forchheimer's equation. The nonlinear range starts at the crit ical flow velocity <Icr» - 144 -The t h i r d way in which Forchheimer's equation has been used consists of extending i t to an equation similar i n form, but with v a r i a b l e c o e f f i c i e n t s . This equation can be written as | | = e i q + e 2 q 2 (3.2.7) where and e 2 vary with q. Stark (1969) and Trollope et a l . (1971) consider that, s t r i c t l y speaking, such an equation applies as soon as q i s non-zero. As they point out, a consequence of th i s i s that the hydraulic conductivity i n Darcy's law i s only approximately constant. Eq. 3.2.7 i s i n t e r e s t i n g from a t h e o r e t i c a l point of view. However, i t does not say much about the actual behavior of the data points since, due to the fact that and e 2 are allowed to vary with q, i t can f i t any data points. Using Darcy's law i n the " l i n e a r " range and Forchheimer's equation i n the non-linear range appears i n reasonable agreement with the data. In addition, i t shows that at a ce r t a i n q, some mechanism, possibly microacceleration, modifies the versus q r e l a t i o n s h i p . Whether Forchheimer's equation can be used i n both the l i n e a r and the nonlinear ranges depends on how accurate a representation of the data i s needed. - 145 -3.3 Discussion on the Use of Friction Factor Versus Reynolds Number  Relationships As mentioned in the preceding section, some authors (Lindquist, p l o t t i n g data from Zunker, and de V r i e s ) , obtained two d i s t i n c t -=— as versus q r e l a t i o n s h i p s , one for the " l i n e a r " range, and one for the nonlinear range. The two r e l a t i o n s h i p s were d i s t i n c t i n that the Darcy's law obtained for the l i n e a r range was not an approximation of Forchheimer's equation at small q. On the other hand, Schneebeli apparently obtained a unique f r i c t i o n factor versus Reynolds number r e l a t i o n s h i p for both ranges. This fin d i n g i s equivalent to finding that the Darcy's law followed by the data i n the l i n e a r range i s an approximation of the Forchheimer's equation they follow i n the nonlinear range. Since Lindquist and de Vries plot data as a r versus q r e l a t i o n s h i p whereas Schneebeli plots them as a f r i c t i o n factor versus Reynolds number r e l a t i o n s h i p , i t i s i n t e r e s t i n g to determine whether the d i f f e r e n c e in the r e s u l t s i s due to di f f e r e n c e s i n the data, or may be due to the d i f f e r e n t methods used to plot the data. In order to do so, we s h a l l express Darcy's law and Forchheimer's equation as f r i c t i o n factor r e l a t i o n s h i p s . Then, using data from de V r i e s , we s h a l l determine whether i t i s possible for authors using a f r i c t i o n factor versus Reynolds number representation to notice that the equation describing the data's behavior in the l i n e a r range i s not an approximation of the equation describing behavior i n the nonlinear range. - 146 -(i) Expressing Darcy's Law and Forchheimer's Equation as Friction  Factor Versus Reynolds Number Relationships In order to f a c i l i t a t e the following discussion, Darcy's law Eq. 3.2.3 and Forchheimer's Equation Eq. 3.2.1 w i l l be rewritten as -T- = VOL , q (3.3.1) ds l i n n and dh 2 .. „ „. — = va q + 3 q (3.3.2) ds nonlin o Comparison of Eqs. 3.3.1 and 3.3.2 with Eqs. 3.2.3 and 3.2.1 shows that v a 1 ± n = 1 (3.3.3) a = a (3.3.4) nonlin 3 = b (3.3.5) o Authors using a f r i c t i o n factor f and a Reynolds number Re define them as f = C (3.3.6) q R e ^ (3.3.7) where d i s some measurement related to the size of the pores and C i s a constant. - 147 -In the present work, a nondimensional f r i c t i o n f a c t o r w i l l be defined as e i t h e r dh ds a l i n « 2 d (3.3.8) or dh — — (3.3.9) a . q d n o n l i n When used i n conjunction with the f r i c t i o n f a c t o r , the Reynolds number w i l l be defined by Eq. 3.3.7. Eqs. 3.3.8 and 3.3.9 are seemingly d i f f e r e n t from Eq. 3.3.6 i n that d occurs e x p l i c i t l y i n the denominator rather than i n the numerator. However, by Eqs. 3.1.2 and 3.3.3 1 ot a — l i n k The p e r m e a b i l i t y k c a r r i e s the u n i t s of l e n g t h 2 and /k has indeed been used by Ward as d. Hence, i t i s reasonable to assume that ot « —— l i n d2 and thus, although occur r i n g e x p l i c i t l y i n the denominator of f as given by Eq. 3.3.8, d i s r e a l l y i m p l i c i t l y i n the numerator. - 148 -Because va . . i s not equal to i t i s not cer t a i n a s i m i l a r nonlin n K' reasoning can be made for Eq. 3.3.9. Eqs. 3.3.1 and 3.3.2 can now be rewritten as f versus Re rel a t i o n s h i p s using Eq. 3.3.7 and ei t h e r Eq. 3.3.8 or Eq. 3.3.9. Eq. 3.3.9 w i l l not be used here. Dividing Eq. 3.3.1 by a±±n q2 d, using Eqs. 3.3.7 and 3.3.8 and taking logarithms y i e l d s the r e l a t i o n s h i p v a l i d i n the l i n e a r range as log f = log jj£ (3.3.10) (Natural logarithms w i l l be used. However, the discussion remains the same i f common logarithms are used). S i m i l a r l y , Eq. 3.3.2 can be written to y i e l d the r e l a t i o n s h i p i n the nonlinear range as 0 1 14 ! B i c i r nonlin 1 o v ,„ . , , . log f - l o g ( - — + j ) (3.3.11) a, . Re O L . a l i n l i n ( i i ) Discussion of the Behavior of the f Versus Re Relationships Eqs. 3.3.10 and 3.3.11 are schematically represented i n F i g . 3.3.1, as well as the asymptote of Eq. 3.3.11 for small Re. This asymptote i s log f = log (^2BiiH 1 ) (3.3.12) l i n and i s below the l i n e 3.3.1 f versus Re relationship according to Eqs. 3.3.10 and 3.3.11. o o o hypothetical data points assuming these 2 equations to hold. - 150 -log f = log -L. because log n o n l l , n ^ s negative as w i l l now be shown. a l i n Using Eqs. 3.3.3, 3.3.4, 3.1.2 and 3.2.6 " l i n = 1.07 °honlin l T nonlin . , Hence < 1. a i • l i n This r e s u l t can also be observed more generally from F i g . 3.2.1, using Eqs. 3.3.3, 3.3.4, 3.1.2 and the fact that a' = ga. The hypothetical data points obtained assuming Eq. 3.3.10 to be v a l i d i n the l i n e a r range and Eq. 3.3.11 to be v a l i d i n the nonlinear range are also shown. These data points stop following Eq. 3.3.10 to follow Eq. 3.3.11 at the c r i t i c a l Reynolds number R e c r defined by where q c r i s the c r i t i c a l flow v e l o c i t y , that i s , the flow v e l o c i t y at which flow becomes nonlinear. A graph s i m i l a r to the one depicted i n F i g . 3.3.1 would be obtained were f defined by Eq. 3.3.9. The behavior of the f versus Re curve followed by the hypothetical data w i l l now be examined. - 151 -dh ds Since r , as depicted by F i g . 3.2.1 i s continuous, the — at q c r dh ds obtained from Eq. 3.3.1 can be equated to the — obtained at q from q cr Eq. 3.3.2. This y i e l d s OL . v = a v + B q (3.3.13) l i n nonlin o cr Dividing by q c r c i l i n d: 1 _ nonlin 1 o ,~ - , , >. — — = — — + r (3.3.14) Re a. . Re OL . d cr l i n cr l i n It follows that the straight l i n e given by Eq. 3.3.10 and the curve given by Eq. 3.3.11 cross each other at Re c r. Thus, i f r i s continuous, the f versus Re r e l a t i o n s h i p followed by the data i s continuous. The slope of the log f versus log Re curve followed by the data, however, i s discontinuous at q c r , as w i l l now be shown. Using the chain r u l e of d i f f e r e n t i a t i o n , ^1°^ ^ can be found to & ' dlog Re be -1 for Eq. 3.3.10. For Eq. 3.3.11: d l ° g f = Zl = zl (3 3 15) dlog Re 3 OL, Re g ^J.J.I-V ! + _° _ _ i i H i + _ 2 .a oi d ot a v l i n nonlin nonlin Experimental values obtained by de Vries (1975, 1979) for 0.4 - 0.6 mm sand and 0.8 - 1.4 mm sand can be used to obtain ^ ^ at q from dlog Re cr Eq. 3.3.15. The d e t a i l of the ca l c u l a t i o n s i s given i n Appendix I, Note 1. The r e s u l t s are shown in Table 3.3.1. - 152 -Table 3.3.1. jj"°g ^ at q calculated from Eq. 3.3.15 using data dlog Re n c r M * from de Vries (1975, 1979). Material dlog f j dlog Re q cr 0.4 - 0.6 mm sand - 0.97 0.8 - 1.4 mm sand - 0.93 It can be seen from Table 3.3.1 that there i s indeed a break at q c r , since there the slope changes from -1 to -0.97 for the 0.4 - 0.6 mm sand and from -1 to -0.93 for the 0.8 - 1.4 mm sand. The change in slope for the materials used for Table 3.3.1 i s so small that i t i s l i k e l y that authors w i l l not notice i t , e s p e c i a l l y when there i s some scatter due to experimental error. Also, authors w i l l not be able to see that the data follow two d i s t i n c t equations and that therefore the asymptote of the curve the data follow i n the nonlinear range i s not the str a i g h t l i n e they follow i n the l i n e a r range. They w i l l deduce that the data follow the same quadratic equation i n both the l i n e a r and the nonlinear ranges. Note that one may wish to check whether the f r i c t i o n factor as i t i s defined i n the present work i s close enough to the f r i c t i o n factors used by other authors for the reasoning presented above to be v a l i d . Such a check might be done by considering the works of other authors i n d i v i d u a l l y . - 153 -As a summary, i f data following two d i s t i n c t equations, a l i n e a r one i n the l i n e a r range and a quadratic one i n the nonlinear range, are plotted as a f r i c t i o n factor versus Reynolds number r e l a t i o n s h i p , the following problem i s l i k e l y to r e s u l t : It w i l l not be possible to observe on the f versus Re plot that the straight l i n e followed by the data i n the l i n e a r range i s not the asymptote of the curve followed i n the nonlinear range. Rather, data w i l l apparently follow a unique quadratic r e l a t i o n s h i p . As a r e s u l t , authors w i l l erroneously conclude that Darcy's law i s an approximation of Forchheimer's equation at small q. 3.4 Occurrence of Nonlinear Flow i n the Forest In t h i s section, the p o s s i b i l i t y of occurrence of nonlinear flow i n the Forest plot and i t s v i c i n i t y w i l l be examined. Implications of nonlinear flow w i l l be b r i e f l y examined and suggestions for further research presented. ( i ) Investigation of the Possibility of Occurrence of Nonlinear  Flow In and Near the Site For the purpose of inves t i g a t i n g the occurrence of nonlinear flow i n and near the s i t e , observed flow v e l o c i t i e s w i l l be compared with the c r i t i c a l flow v e l o c i t y obtained by de Vries for 0.4 to 0.6 mm sand. The reason for using t h i s sand f r a c t i o n i s that, among the materials used by de V r i e s , i t i s the one considered to be the closest to the Forest plot s o i l . In order to compare the various flow v e l o c i t i e s observed i n the - 154 -f i e l d with the c r i t i c a l flow v e l o c i t y obtained for 0.4 to 0.6 mm sand, i t i s necessary to assume the Forest s o i l behaves l i k e the sand. Comparisons are shown i n Table 3.4.1. Because the temperature and hence the v i s c o s i t y of the water used by de Vries was d i f f e r e n t from the temperature of the water i n the Forest, i t was determined whether i t was necessary to correct the c r i t i c a l v e l o c i t y for temperature. In order to do so, i t has been assumed that nonlinear flow w i l l s t a r t at the same Re i n the sand and i n the Forest material and that t h i s i s true i r r e s p e c t i v e of differences i n temperatures. Also, i t was assumed that the representative length L c h appearing i n the Reynolds number i s the same for the sand and for the Forest material. This assumption i s not l i k e l y to be v e r i f i e d , but i s necessary for c a l c u l a t i o n purposes. Using the above assumptions, a q c r of 5 x 1 0 - 3 ms - 1 for flow i n the Forest has been calculated f o r a temperature of 5°C, taken to be a lower estimate for the temperature of the water during the measurements made i n the Forest. This c a l c u l a t i o n , shown i n Note 2, Appendix I, uses the q c r of 3 x 10 ms and the y given by de Vries (1979 and 1975). Because the q c r at 5°C i s close to the one at 22°C, i t w i l l be considered that the difference between the two values i s within _ 3 _ 1 experimental error and a q c r of 3 x 10 ms w i l l be used as a conservative estimate. In order to determine i f the conditions for nonlinear flow are f u l f i l l e d , i t i s necessary to separate the flow path i n two parts: f i r s t , flow down to the bed and second, l a t e r a l saturated flow over the Table 3.4.1 Comparison of measured flow v e l o c i t i e s with de Vries' c r i t i c a l flow v e l o c i t y V e r t i c a l Ir i f i l t r a t i o n q at 2 outflow points along the channel's bank q c r i t i c a l at 22°C q c r i t i c a l at 5°C concentrated inflow rate from two logs K for the upper part of the s o i l p r o f i l e by i n s i t u cores 8 x 1 0 - 5 3 1 to 4 x 10 ms 8 x 10_1* ms - 1 Between 2.7 x I0~k and 1.1 x 10~ 3 ms - 1 Between 8.7 x 10 - l t and 2.6 x 10" 3 ms - 1 3 x 10~ 3 ms"1 3 1 5 x 10 ms - 156 -bed. Although the f i r s t part of the flow path i s i n the generally unsaturated zone, l o c a l saturation i s necessary for nonlinear flow to occur. For nonlinear flow during the f i r s t part, s u f f i c i e n t l y high hydraulic c o n d u c t i v i t i e s and s u f f i c i e n t l y large inflow rates from concentrating elements are required. For nonlinear flow during the second part, a s u f f i c i e n t l y large d r i v i n g force and a s u f f i c i e n t l y large hydraulic conductivity are required. Measurements of inflow rates from natural concentrating elements were made in the v i c i n i t y of the Forest plot during and a f t e r r a i n f a l l . These measurements are l i m i t e d by d i f f i c u l t i e s i n measuring the i n f i l t r a t i n g area. Also, i t i s not known how representative they are for an average r a i n f a l l event. As mentioned in Chapter I, logs i n various states of decay are present on the ground. They are considered to be the major sources of concentrated inflow as they act as more or le s s impermeable elements. A large inflow rate was found from a rock but rocks are sparse. From two logs, inflow rates at various points were 8.1 x 10~ 5, 1.6 x 10~h, and 4.5 x 10" 3 ms - 1. The l a s t estimate i s possibly too large. It i s puzzling that i t i s one order of magnitude larger than the minimum concentrated inflow rate due to the p l a s t i c sheets (see Appendix B). If i t i s c o r r e c t , however, i t shows the inflow rate may at places be large enough for nonlinear flow to occur. For flow i n the unsaturated zone, the K of 8 x 10 _ l + ms - 1 obtained by the i n s i t u cores method for the top part of the p r o f i l e has been used. Apparently, i f g r a v i t y driven v e r t i c a l flow takes place, t h i s K - 157 -i s not large enough to allow nonlinear flow at steady state, since at steady state K = q for g r a v i t y driven v e r t i c a l flow and thus a K of 3 x 3 1 IO - ms - or larger i s required. Since the K obtained by i n s i t u cores i s an average value, i t i s s t i l l possible that water can find pathways with c o n d u c t i v i t i e s large enough for flow to be nonlinear. Hence nonlinear flow may occur i n the unsaturated zone. Because spots where large inflow rates occur are not widespread, i t can however be concluded that under natural conditions nonlinear flow down the p r o f i l e i s not common. Also, i t probably occurs, i f at a l l , only for short distances since i t i s l i k e l y that i n many cases water t r a v e l l i n g i n a low resistance path would soon get sucked into the s o i l matrix. As for l a t e r a l flow over the bed, flow v e l o c i t y was d i r e c t l y measured at 2 outflow points along the channel's bank in the Forest plot (see Section 1.3.1). Outflow was from a natural storm event. As mentioned i n Section 1.3.1, there are inaccuracies i n the measurements, the true flow v e l o c i t i e s being possibly bracketed by 2.7 x 10 _ 1 + and 2.6 3 1 x 10" ms . The largest value i s equal to q c r , i n d i c a t i n g that flow i s close to being non-linear. The flow v e l o c i t i e s calculated at the two outflow points chosen may not be the highest v e l o c i t i e s occurring in the Forest p l o t . In p a r t i c u l a r , they may not be the highest v e l o c i t i e s having occurred during the a r t i f i c i a l events E x l , Ex2 and Ex3. Indeed, during i r r i g a t i o n , the saturated zone thickness i s larger and more low resistance paths are used. This increases the chance of finding low - 158 -resistance paths i n which nonlinear flow occurs. Thus flow i n the Forest plot may be nonlinear. As for flow i n the v i c i n i t y of the Forest p l o t , one should remember that i n a rootchannel i n Nagpal and de Vries' s i t e , "assuming the nature of flow...to be sim i l a r to that i n a pipe, a Reynolds number larger than 2,000 was calculated [presumably as with d the diameter of the root channel], i n d i c a t i n g the existence of turbulent flow conditions within the rootchannel" (Nagpal and de V r i e s , 1976). The Reynolds number at which turbulence i s expected to st a r t i s not d e f i n i t e but for a pipe, the lower l i m i t i s 2000 (Ginzburg, 1963). Since i t i s expected that any small disturbance of flow renders flow turbulent at a Reynolds number of 2000 and that the root channels are not smooth, i t i s indeed expected that flow w i l l become turbulent at a Reynolds number of 2000. Note that i n this case, pipe flow theory rather than experimental r e s u l t s on sand was used to determine the nature of the flow. To conclude, i t i s suggested that both i n the Forest plot and i n i t s v i c i n i t y , only a small proportion of flow, i f any, can be nonlinear i n the unsaturated zone under natural conditions. It i s not cer t a i n whether nonlinear flow can occur i n the saturated zone of the Forest p l o t . ( i i ) implications of nonlinear flow in the Saturated Zone One implication of nonlinear flow over the low conductivity layer i s that hydraulic conductivities determined using outflow to a trench may be "general" hydraulic c o n d u c t i v i t i e s (see Sections 2.4, and 3.1), - 159 -i n fact smaller i n the nonlinear range than the true hydraulic c o n d u c t i v i t i e s . When used i n a model using Darcy's law, general hydraulic c o n d u c t i v i t i e s obtained from nonlinear flow would r e a l l y be v a l i d for only one value of q except i f the hydraulic gradient can be taken to be equal to the bed slope, that i s , i f q = K g e n sinu) (see section 2.4). In an e f f o r t to determine how s i g n i f i c a n t the e f f e c t of n o n l i n e a r i t y i s on K g e n , experiments by de Vries (1975) have been used. These experiments showed the v a r i a t i o n of K g e n with q, for 0.4 to 0.6 ram sand to be as given by Table 3.4.2. Table 3.4.2. q and Pg^r from experiments i n 0.4 to 0.6 mm sand by de Vries (1975). K _ „ calculated from these values. q ms - 1 dh P gds" 2 2 kg s~ m~ Kgen - q p g dh PSds" (p = 10 3 kg m - 3; g = 9.8 ms - 2) ms - 1 3.0 x 10" 3 9.2 x 10~ 3 2.0 x 1 0 - 2 2.7 x 10 4 8.9 x IO 4 2.2 x 10 5 0.11 x I O - 2 0.10 x 10~ 2 0.089 x I O - 2 Table 3.4.2 shows that as q _3 _ 1 9.2 x 10 ms , K g e n changes by changes from 9.2 x 10~ 3 to 2.0 x changes from 3 x 10" (q c r i t i c a l ) to 10%. It changes by another 10% as q I O - 2 ms - 1. - 160 -The value of K g e n at q c r , 1.1 x 10" ms - can be taken as K since K g e n i s constant i n the l i n e a r range. For the purposes of discussion, i t w i l l be assumed that K, a and b are the same for the material i n the pi p e l i k e low resistance paths over the bed as i n the 0.4 to 0.6 mm sand. Although such an assumption cannot be e n t i r e l y correct, i t i s not unreasonable since the K for the sand i s i n between the K's measured at the outflow points. Table 3.4.2 shows that i f q in the p i p e l i k e low resistance paths over the bed i s about 10 times the highest flow rate observed for the 2 — 3 1 outflow points used (2.6 x 10 ms~ , Table 1.3.1.1), K g e n i s about 20% less than K. Table 3.4.2 w i l l not be used to i n f e r e f f e c t s of n o n l i n e a r i t y on the Kgff's determined for the forest plot because there are too many unc e r t a i n t i e s . Note that use of the kinematic wave model w i l l always present the problem of the need for flow to be s u f f i c i e n t l y f a s t , yet not too f a s t . It should be remembered that the term "kinematic wave model" denotes e x c l u s i v e l y the model described i n Chapter I I . As far as the Forest plot i s concerned, i t i s concluded that i t i s not c e r t a i n what the implications of nonlinear flow i n the Forest plot saturated zone are on K g e n , and hence on Darcy's law and on the kinematic wave model. The reasons are the following: ( i ) It i s not ce r t a i n what proportion of the flow, i f any, can be nonlinear. ( i i ) It i s not ce r t a i n how severely nonlinear the flow can be. ( i i i ) It i s not c e r t a i n what the e f f e c t of the n o n l i n e a r i t y w i l l - 161 -be on K g e n , on Darcy's law and on the use of the kinematic wave model. One reason i s that the cal c u l a t i o n s of Table 3.4.2 were based on 0.4 to 0.6 mm sand and not on the Forest plot s o i l . Another reason i s the note made on the e f f e c t of n o n l i n e a r i t y on the kinematic wave model made at the end of Section 2.4. ( i i i ) Suggestions for Further Research There i s a need for further research on no n l i n e a r i t y i n the f i e l d . F i r s t , a way of determining whether flow i s nonlinear i n the f i e l d should be found. Second, i t i s suspected that flow i n underground "streams" may be nonlinear. This should be checked, and more generally, the proportion of nonlinear flow i n the saturated zone and the s e v e r i t y of t h i s n o n l i n e a r i t y should be determined. F i n a l l y , the implications of nonlinear flow on satiated general c o n d u c t i v i t i e s determined i n the f i e l d and on p h y s i c a l l y based mathematical models should be determined. As a summary, i t i s not cer t a i n whether nonlinear flow occurs i n the Forest plot saturated zone. The Implications of nonlinear flow on use of the kinematic wave model for the Forest plot are also uncertain. It i s also suggested that only a small proportion of flow, i f any, can be nonlinear i n the unsaturated zone under natural conditions. 3.5 Conclusions A l i t e r a t u r e review showed that there i s a p o s s i b i l i t y that turbulence i s not necessary for n o n l i n e a r i t y . However, more - 162 -i n v e s t i g a t i o n s are needed since some r e s u l t s are contradictory. Microacceleration i s often recognized as a cause of n o n l i n e a r i t y for laminar flow. Forchheimer's equation has been widely used to describe nonlinear flow. Its use, however, va r i e s from author to author. Three d i f f e r e n t theories have been proposed i n the l i t e r a t u r e : 1. Data obey Darcy's law i n the " l i n e a r " range and Forchheimer's equation i n the nonlinear range. 2. Data obey Forchheimer's equation in both the " l i n e a r " and the nonlinear ranges. 3. Data obey an equation similar to Forcheimer's, but i n which the 2 c o e f f i c i e n t s of q and q vary with q as soon as q i s nonzero. Theory (1) appears to be the one to be preferred from a p r a c t i c a l point of view as being simple and yet i n reasonable agreement with experimental data. In t h i s chapter, i t has been shown that i t i s l i k e l y that authors using a f r i c t i o n factor versus Reynolds number representation of th e i r data w i l l assume theory (2) to be v a l i d even i f data follow the graph given by theory ( 1 ) . The p o s s i b i l i t y of occurrence of nonlinear flow in the Forest plot and i t s v i c i n i t y was investigated. It was suggested that, either in the Forest plot or i t s v i c i n i t y , only a small proportion of flow, i f any, can be nonlinear i n the unsaturated zone under natural conditions. It was shown that i t i s not c e r t a i n whether nonlinear flow can occur in the saturated zone of the Forest p l o t . - 163 -It was also concluded that i t i s not ce r t a i n what the implications of nonlinear flow i n the saturated zone of the Forest plot would be on Darcy's law and hence on the use of the kinematic wave model. - 164 -3.6 References A h m e d , N . a n d D . K . S u n a d a . 1 9 6 9 . N o n l i n e a r f l o w i n p o r o u s m e d i a . A m e r i c a n S o c i e t y o f C i v i l E n g i n e e r s , J o u r n a l o f t h e h y d r a u l i c s  d i v i s i o n , V o l . 9 5 , N o . H Y 6 : 1 8 4 7 - 1 8 5 7 . B e a r , J . 1 9 7 2 . D y n a m i c s o f f l u i d s i n p o r o u s m e d i a . A m e r i c a n e l s e v i e r . 764 p p . C h a u v e t e a u , G . a n d C l . T h i r r i o t . 1 9 6 7 . R e g i m e s d ' e c o u l e m e n t e n m i l i e u p o r e u x e t l i m i t e d e l a l o i d e D a r c y . L a h o u i l l e b l a n c h e , N o . 2 : 1 4 1 - 1 4 8 . d e V r i e s , J . 1 9 7 5 . U n p u b l i s h e d d a t a . d e V r i e s , J . 1 9 7 9 . P r e d i c t i o n o f n o n - D a r c y f l o w i n p o r o u s m e d i a . A m e r i c a n S o c i e t y o f C i v i l E n g i n e e r s , J o u r n a l o f t h e i r r i g a t i o n a n d  d r a i n a g e d i v i s i o n , V o l . 1 0 5 , N o . I R 2 , P r o c . P a p e r 1 4 6 1 0 : 1 4 7 - 1 6 2 . D u d g e o n , C R . 1 9 6 6 . A n e x p e r i m e n t a l s t u d y o f t h e f l o w o f w a t e r t h r o u g h c o a r s e g r a n u l a r m e d i a . L a H o u i l l e b l a n c h e , N o . 7 : 7 8 5 - 8 0 0 . E n g e l u n d , F . 1 9 5 3 . On t h e l a m i n a r a n d t u r b u l e n t f l o w s o f g r o u n d w a t e r t h r o u g h h o m o g e n o u s s a n d . T r a n s a c t i o n s o f t h e D a n i s h A c a d e m y o f  T e c h n i c a l S c i e n c e s , N o . 3 , C o n t r i b u t i o n f r o m t h e H y d r a u l i c L a b o r a t o r i e s T e c h n i c a l U n i v e r s i t y o f D e n m a r k , C o p e n h a g e n , D e n m a r k , B u l l e t i n , N o . 4 . F o r c h h e i m e r , P . 1 9 0 1 a . W a s s e r b e w e g u n g d u r c h B o d e n . Z . v e r . D e u t s c h . I n g . 4 5 : 1 7 8 2 - 1 7 8 8 . C i t e d b y B e a r . F o r c h h e i m e r , P . 1 9 0 1 b . W a s s e r b e w e g u n g d u r c h B o d e n . Z e i t s c h r i f t d e s  v e r e i n e s D e u t s c h e r I n g e n i e u r e , N o . 4 9 : 1 7 3 6 - 1 7 4 9 ; N o . 5 0 : 1 7 8 1 - 1 7 8 8 . C i t e d b y d e V r i e s ( 1 9 7 9 ) . G i n z b u r g , I . P . 1 9 6 3 . A p p l i e d f l u i d d y n a m i c s . T r a n s l a t e d f r o m R u s s i a n . I s r a e l P r o g r a m f o r S c i e n t i f i c T r a n s l a t i o n s , J e r u s a l e m . H u b b e r t , M . K . 1 9 5 6 . D a r c y ' s l a w a n d t h e f i e l d e q u a t i o n s o f t h e f l o w o f u n d e r g r o u n d f l u i d s . T r a n s a c t i o n s o f t h e A m e r i c a n i n s t i t u t e o f  m i n i n g , m e t a l l u r g i c a l , a n d p e t r o l e u m e n g i n e e r s , V o l . 2 0 7 : 2 2 2 - 2 3 9 . I r m a y , S . 1 9 6 4 . M o d e l e s T h e o r i q u e s d ' e c o u l e m e n t d a n s l e s c o r p s p o r e u x . R i l e m s y m p o s i u m , P a r i s . e r L i n d q u i s t , E. 1 9 3 3 . On t h e f l o w o f w a t e r t h r o u g h p o r o u s m e d i a , J. c o n g r e s d e s g r a n d s b a r r a g e s , S t o c k h o l m , S w e d e n , V o l . 5 : 8 1 - 1 0 1 . M a s o n i . 1 8 9 6 . D i a l c u n e d e t e r m i n a z i o n i s p e r i m e n t a l i s u i c o e f f i c i e n t i d i f i l t r a z i o n e . N a p l e s , I t a l y . C i t e d b y d e V r i e s ( 1 9 7 9 ) . - 165 -Nagpal, N.K., and J . de V r i e s . 1976. On the mechanism of water flow through a forested mountain slope s o i l i n coastal western Canada. Unpublished m a t e r i a l . Rumer, R.R., J r . , and P.A. Drinker. 1966. Resistance to laminar flow through porous media. American Society of C i v i l Engineers, Journal  of the hydraulics d i v i s i o n , V o l. 92, No. HY5: 155-163. Schneebeli, G. 1955. Experiences sur l a l i m i t e de v a l i d i t e de l a l o i de Darcy et 1*apparition de l a turbulence dans un ecoulement de f i l t r a t i o n . La H o u i l l e blanche, No. 2: 141-149. Stark, K.P. 1969. A numerical study of the nonlinear laminar regime of flow i n an i d e a l i z e d porous medium. International association f o r  hydraulic research. International symposium on the fundamentals of transport phenomena i n porous media, Haifa: 86-102. Trollope, D.H., K.P. Stark and R.E. Volker. 1971. Complex flow through porous media. The A u s t r a l i a n Geomechanics j o u r n a l : 1-10. Ward, J.C. 1964. Turbulent flow i n porous media. American Society of  C i v i l Engineers, Journal of the hydraulics d i v i s i o n , V o l. 90, No. HY5, Proc. Paper 4019: 1-12. Wright, D.E. 1968. Nonlinear flow through granular media. American  Society of C i v i l Engineers, Journal of the hydraulics d i v i s i o n , V o l . 94, No. HY4, Proc. Paper 6018. Zunker, F. 1920. Das allgemeine Grundwasserfliessgesetz, Gasbeleuchtung und Wasserversorgung. Cited by de Vries (1979). - 166 -CHAPTER IV DISCUSSION AND CONCLUSIONS The observation of low resistance paths due mainly to root material i n the Forest plot prompted the in v e s t i g a t i o n of flow mechanisms i n both the unsaturated and the saturated zones. One tool used for i n v e s t i g a t i o n of flow mechanism i n the unsaturated zone was the kinematic wave model for saturated flow over a steep bed. The term "kinematic wave model" denotes e x c l u s i v e l y the model described i n the second chapter. The kinematic wave model requires a steep bed slope, a high hydraulic conductivity and a low recharge rate. It also assumes steady and uniform recharge rate, e f f e c t i v e porosity and hydraulic conductivity, u n i d i r e c t i o n a l flow and a uniform bed slope. Flow must be l i n e a r . The porous medium must be i s o t r o p i c . The kinematic wave model yi e l d s s t r a i g h t l i n e r i s i n g limb and recession. The f i e l d observation that the main limb of hydrograph r i s e for the Forest plot i s a straight l i n e indicates that there i s some j u s t i f i c a t i o n for applying the kinematic wave model to the Forest p l o t . S o i l thickness i s not uniform. Hence i t might be expected that the recharge rate could be somewhat nonuniform and unsteady. It i s not clear for how long nonuniformity and unsteadiness would l a s t , but i f they l a s t for longer than for a short period of time at the beginning of the recharge, the main limb of r i s e would not be a stra i g h t l i n e . Yet the main limb of r i s e i s a straight l i n e , i n d i c a t i n g that the s o i l and water system i n the unsaturated and the saturated zones behaves as i f - 167 -the kinematic wave model were v a l i d , and hence as i f recharge were uniform and steady. In p a r t i c u l a r , a s t r a i g h t l i n e main limb of r i s e together with the fact that there was l i t t l e preliminary outflow before the main limb of r i s e shows that the system behaves as i f l i t t l e or no s h o r t - c i r c u i t i n g occurred. A straight l i n e main limb of r i s e i s perhaps due to an averaging process. Another means used to investigate flow in the unsaturated zone was to compare r e s u l t s obtained from concentrated i r r i g a t i o n with r e s u l t s obtained from uniform i r r i g a t i o n . It was found that concentration of i r r i g a t i o n did not change the general shape of the outflow hydrograph and did not decrease the time lag to the main limb of r i s e . Also, rather than enhancing the i n i t i a l step of the outflow hydrograph obtained for uniform i r r i g a t i o n , i t suppressed i t . It was also observed that concentration did not cause the water table to r i s e e a r l i e r , and did not a f f e c t the shape of the water table at steady state. From these observations, i t was concluded that the s o i l and water system i n the unsaturated and saturated zones of the Forest plot behaves as i f s h o r t - c i r c u i t i n g were not enhanced by concentration. Thus, the outflow hydrograph indicates that within the l i m i t s of experimental design the system behaves as i f l i t t l e or no s h o r t - c i r c u i t i n g occurred and as i f s h o r t - c i r c u i t i n g were not enhanced by concentration. Some aspects of the water table behavior may suggest that s h o r t - c i r c u i t i n g occurred. Indeed, pipes 1, 5 and 8 were not affected - 168 -by i n i t i a l conditions. Further experimental work i s needed to learn more about flow mechanisms in the unsaturated zone. The saturated zone in the Forest plot i s characterized by low resistance paths which together with a steep bed slope and a high recharge rate are responsible for steep r i s e s and recessions of the outflow hydrograph. Thus two of the assumptions necessary for the kinematic wave model, a high hydraulic conductivity and a steep bed slope are r e a d i l y v e r i f i e d . A number of the assumptions necessary for the saturated zone are not s a t i s f i e d , however. In p a r t i c u l a r , the hydraulic conductivity and the e f f e c t i v e porosity are not uniform and the flow i s not u n i d i r e c t i o n a l . Yet the system behaves as i f these assumptions were s a t i s f i e d . Here also an averaging mechanism could be invoked. On the basis that the system behaves as i f the kinematic wave model were v a l i d , e f f e c t i v e hydraulic c o n d u c t i v i t i e s were calculated for the saturated zone, using the kinematic wave model and the r i s i n g limb of the outflow hydrographs. Calculations showed that the e f f e c t i v e — 4 —4 — 1 hydraulic conductivity i s bracketed by 1.6 x 10 and 3.2 x 10 ms In order for the kinematic wave model to be v a l i d , flow i n the saturated zone must be reasonably f a s t . If i t i s too f a s t , i t becomes nonlinear and Darcy's law and hence the kinematic wave model f a i l s . Because i t i s suggested that possibly turbulence i s not necessary for flow to be nonlinear, nonlinear means "laminar nonlinear". The p o s s i b i l i t y of occurrence of nonlinear flow in the Forest plot and i t s v i c i n i t y was investigated. It was suggested that, either i n the - 169 -Forest or i t s v i c i n i t y , only a small proportion of flow, i f any, can be nonlinear i n the unsaturated zone under natural conditions. It was shown that i t i s not ce r t a i n whether nonlinear flow can occur i n the saturated zone of the Forest p l o t . It was also concluded that i t i s not ce r t a i n what the implications of nonlinear flow i n the saturated zone of the Forest plot would be on Darcy's law and on the use of the kinematic wave model. As a summary, the main conclusions about flow mechanisms are as follows: the f i e l d observation that the main limb of r i s e i s a straight l i n e indicates that the s o i l and water system i n the unsaturated and the saturated zones behaves as i f the kinematic wave model were v a l i d . In p a r t i c u l a r , i t indicates that the system behaves as i f l i t t l e or no s h o r t - c i r c u i t i n g occurred. Two of the assumptions linked to the kinematic wave model are r e a d i l y v e r i f i e d : a high hydraulic conductivity and a steep bed slope. If the outflow hydrograph obtained with a r t i f i c i a l concentration i s compared to the one obtained without a r t i f i c i a l concentration, i t i s seen that s h o r t - c i r c u i t i n g i s apparently not enhanced by concentration. A s i m i l a r conclusion i s obtained from the behavior of the water table. It was suggested that only a small proportion of flow, i f any, can be nonlinear i n the unsaturated zone under natural conditions. This i s true for the Forest plot and i t s v i c i n i t y . It i s not c e r t a i n whether nonlinear flow can occur i n the saturated zone of the Forest p l o t . - 170 -APPENDIX A DETERMINATION OF THE STONE RATIO 1. Method to Separate Stones from Concretions by Breaking Down the  Concretions This i s a method to dissol v e Fe oxides but i t can also be used for Al oxides. Once the cementing agents ( i r o n and aluminum oxides) are removed, the concretions break down. The method has been adapted from Mehra and Jackson, 1960. - Place the s o i l into a beaker - Add enough c i t r a t e bicarbonate solution (8:1 r a t i o by volume of sodium c i t r a t e and sodium bicarbonate resp.) to cover the s o i l . - Heat up the beaker to 75°C in a water bath. - Add 1 teaspoon per 100 g of s o i l of sodium hydrosulfite (sodium d i t h i o n i t e ) low i n iron and p u r i f i e d . - S t i r slowly at f i r s t then vigorously, for about 1 min. Heat for an a d d i t i o n a l 15 min, but do not allow the temperature to r i s e above 80°C. - Cool. - Pass through a 2 mm sieve. This method i s adequate for a rough estimate. 2. Calculation of the Stone Ratio In the B horizon, the r a t i o "mass of concretions, cemented aggregates, cemented sand + stones > 2 mm" to "mass of s o i l " was found - 171 -to be 0.49. By treating the f r a c t i o n > 2 mm with a c i t r a t e bicarbonate sol u t i o n and with sodium h y d r o s u l f i t e (see above), the r a t i o "mass of stones" to "mass of f r a c t i o n > 2 mm" was found to be 0.84 hence the r a t i o "mass of stones" to "mass of s o i l " i s 0.41. Note that because the c i t r a t e bicarbonate treatment removed cementing agents, some of the p a r t i c l e s i n the stone f r a c t i o n may have been smaller than 2 mm. The porosity for the top of the B horizon calculated by _ bulk density p a r t i c l e density i s 0.63 m3 m - 3 The p a r t i c l e density i s 2512 ± 254 kgm" (based on 9 samples; the number given after the ± i s 2 standard de v i a t i o n s ) . This means that approximately 95% of the p a r t i c l e d e n s i t i e s are between 2258 and 2766 kgm~ 3 . The bulk density for the top of the B horizon determined by the excavation method i s 931 ± 326 kgm (based on 7 samples). This means that approximately 95% of the bulk densities are between 605 and 1257 kgm"3. Using the bulk density given above, and using for stone density the density of granite (2650 kgm- ), the r a t i o of stones to bulk s o i l by volume can be found to be , , r , ^ • bulk density . „ , , ( r a t i o of stones by mass) \—. — - r — ) = 0.14 J density of granite where .- _ , mass of stones , , r a t i o of stones by mass = — = 0.41 J mass of s o i l - 172 -APPENDIX B CALCULATION OF THE MINIMUM CONCENTRATION OF IRRIGATION OBTAINED BY THE PLASTIC SHEETS Calculation of the minimum concentration of i r r i g a t i o n by the p l a s t i c sheets i s obtained by minimum concentration = "minimum estimate of area of p l a s t i c sheets" divided by "maximum estimate of area of i n f i l t r a t i o n for the water flowing from the p l a s t i c sheets." The minimum area of a p l a s t i c sheet i s 55 x 75 cm ; there are 16 p l a s t i c sheets. A maximum estimate for the area of i n f i l t r a t i o n of 2 water flowing from one p l a s t i c sheet outflow point i s 10 cm . There i s a maximum of 4 outflow points per p l a s t i c sheet. Therefore: 2 • • ^ 16 x 55 x 75 cm , minimum concentration = = 100 2 16 x 4 x 10 cm C 1 For the applied i r r i g a t i o n rate of about 2.8 x 10" ms" , a concentration of 100 times y i e l d s an inflow rate of 2.8 x 10 _ l + ms - 1. - 173 -APPENDIX C COMPARISON OF THE RECESSION WITH HEWLETT AND HIBBERT'S RECESSION: NORMALIZATION AND LIMITATIONS In order to compare the recession obtained for the present work with the one obtained by Hewlett and Hibbert, i t would be desirable to normalize the r e s u l t s with respect to a number of parameters. The kinematic wave model (see Note 4, Appendix H), gives the outflow rate during the recession as K sinw D i t' Q = Di L - - (C-l) o n re where D = width of the h i l l s l o p e Q = discharge (volume/time) i = a constant, nonzero recharge rate per unit area p a r a l l e l to ° „, volume . the bed ( — — ) area x time K = satiated hydraulic conductivity n = e f f e c t i v e porosity for the recession re t- J t' = time from stop of recharge = t* - t r t = time at which recharge stopped L = length of the saturated zone K and n r e are properties of the s o i l . Since the e f f e c t s of the s o i l on Q are sought, i t i s not desirable to normalize Q with respect to K and n r e . Moreover n r e would be d i f f i c u l t to obtain in p r a c t i c e . - 174 -Normalization with respect to i Q cannot be achieved since i 0 for Hewlett and Hibbert i s not known. Q for the present work has been calculated for a D of 0.91 m, which i s equal to the D i n Hewlett and Hibbert's work. Also, obtaining = from Eq. C-l and solving for At' indicates At' i s proportional to l/sinto. Hence At' for the present work must be divided by 0.70 i n order to be the At' that would be obtained for a slope of 22°. Normalization with respect to L cannot be performed for Q since L appears only i n one term. Normalization with respect to sinai cannot be performed for Q because sinu appears only i n one term. Table C-l indicates which values of the present work given i n Table 1.4.4.1 have been normalized. Table C-l Normalization summary for the r e s u l t s of the present work given i n Table 1.4.4.1. Normalized with respect to Q Time At' D Yes Yes Yes i o No No No L No No No sinio No No Yes, where indicated The comparison of the present work with Hewlett and Hibbert's should be considered with some caution since only p a r t i a l normalization has been performed. Moreover, this normalization assumes the kinematic wave theory can be applied to both Hewlett and Hibbert's work and the present work. - 175 -APPENDIX D CALCULATION OF THE POINTS USED TO PLOT FIG. 1.4.2.6, THE DETAILED HYDROGRAPH RISES Most of the early points i n F i g . 1.4.2.6 are based on two t i p s of the outflow tipping bucket f i r s t , then on 4 t i p s . The f i r s t point i s given by t i p s one to three, t i p one being the f i r s t t i p a f t e r the s t a r t of i r r i g a t i o n , the second point by tip s three to f i v e , and so on. Rates of outflow are plotted i n the middle of the i n t e r v a l between the tips which i s not accurate but i s s a t i s f a c t o r y for comparison purposes. Some of the points, e s p e c i a l l y l a t e r points, have been taken from the small scale hydrographs ( F i g s . 1.4.2.3 to 1.4.2.5). - 176 -APPENDIX E EFFECT OF ERRORS AND LIMITATIONS IN OUTFLOW RATE The outflow rate i s subject to a number of errors and l i m i t a t i o n s . Their e f f e c t on the main r e s u l t s of outflow are discussed below. 1. There i s an antecedent outflow smaller than 2.8 x 10~ 8 m3 s - 1 for Ex2 and smaller or equal to 1.5 x 10" m s for Ex3. For Ex3, the f i r s t t i p i s due to antecedent conditions only. The antecedent outflow does not a f f e c t the comparison of the timing of the main limb of r i s e (e.g. whether the main limb of r i s e Exl lags behind the main limb of r i s e Ex3) because a s h i f t of 1.5 x 10 m s for the main limb of r i s e i s n e g l i g i b l e . It i s expected to have a n e g l i g i b l e e f f e c t on the shape of the main limb of r i s e and no e f f e c t on the time to steady state. A 7 3 _ 1 rate of 1.5 x 10" m s i s n e g l i g i b l e with respect to some of the rates occurring during the i n i t i a l step of Ex3. Hence the i n i t i a l step of Ex3 does occur before the early r i s e part of the r i s e of Ex2 and i s not due uniquely to antecedent outflow. 2. Some spurious inflow to the channel occurred. ( i ) Some inflow due to overland flow when the forest f l o o r was hydrophobic and some inflow from water f a l l i n g on the p l a s t i c plot boundaries occurred. Attempts were made to prevent overland flow due to hydrophobicity from reaching the channel and confusing the interflow data. This spurious inflow i s n e g l i g i b l e once the main limb of r i s e has s t a r t e d . - 177 -Flow over the p l a s t i c frame may have been responsible for part of the e a r l y parts of the r i s e s but t h i s does not change the statement that the i n i t i a l steps are small for Exl and Ex3 and non-existant for Ex2. It i s probably n e g l i g i b l e a f t e r the main limbs of r i s e have started, except during the r e r i s e of Ex3. This spurious inflow during the r e r i s e of Ex3 does not a f f e c t the r e s u l t s considered. ( i i ) Extraneous inflow from the water c o l l e c t e d on the p l a s t i c plot boundaries reaching the channel by an underground route. This inflow, i f i t a c t u a l l y occurs, i s presumably the same for the three experiments and i s constant. Hence i t should not a f f e c t the comparison between the 3 experiments. Neither should i t a f f e c t the slope of the main limb of r i s e . It has no e f f e c t on the time to steady state. I t would a f f e c t the water balance. If i t p a r t i c i p a t e s i n the outflow during the i n i t i a l step, i t would however not change the statement that t h i s outflow i s small. 3. Water escaping c o l l e c t i o n because the channel had not been dug f a r enough. This a f f e c t s the water balance. It might a f f e c t the comparison between Ex2 and the other two experiments i f the outflow missed i s affected by concentration. Since the outflow missed i s probably only a small proportion of the t o t a l outflow, chances that i t i s affected by concentration are smaller. It w i l l be assumed that t h i s error does not a f f e c t the comparison between the three experiments. It would not a f f e c t the time to steady state. It i s doubtful whether i t can a f f e c t s i g n i f i c a n t l y the shape of the main limb of r i s e . S i m i l a r l y , i t i s doubtful whether i t can a f f e c t the water balance s i g n i f i c a n t l y . - 178 -4. Uncertainty as to whether the right tipping bucket c a l i b r a t i o n has been used for Exl u n t i l 3.5 hours before f i n a l i r r i g a t i o n stop and for Ex2 u n t i l 39 hours a f t e r the st a r t of i r r i g a t i o n . Uncertainty i n c a l i b r a t i o n may have some e f f e c t on the hydrographs. The error i n c a l i b r a t i o n however would not be large enough to be responsible for the lack of i n i t i a l step and the delay of the early part of the r i s e for Ex2. Errors i n c a l i b r a t i o n for Exl and Ex2 may a f f e c t the shape of the main limb of r i s e . Since i t i s a straight l i n e for Ex3 as well as for Exl and Ex2, i t i s l i k e l y that the straight l i n e r i s e for Exl and Ex2 i s r e a l . Also, i t i s rather u n l i k e l y that errors i n c a l i b r a t i o n would by chance y i e l d a straight l i n e r i s e for both Exl and Ex2. Moreover, stra i g h t l i n e main limbs of r i s e were also observed by Utting (1978) and by Nagpal and de Vries (1976) on Nagpal and de Vries' s i t e . An error i n c a l i b r a t i o n may a f f e c t the slopes of the main limbs of r i s e , the lags of the main limbs of r i s e and the steady state values used for the water balance. It would have no e f f e c t on the time to steady state. 5. D r i f t i n tipping bucket c a l i b r a t i o n occurred during the experiments. Although some d r i f t occurred during the main limbs of r i s e , the tipping bucket did not f i l l to the rim during t h i s period hence the error was less than 30%. Following a reasoning si m i l a r to the one made under point 4, one can reach the conclusion that the straight l i n e r i s e for Exl and Ex2 i s r e a l . Error due to d r i f t i n g may a f f e c t the lags and slopes of the main limbs of r i s e . It may also have reduced the - 179 -outflow rate of the i n i t i a l steps although d r i f t at t h i s stage i s rather u n l i k e l y . 6. Inconsistency between i r r i g a t i o n rate, steady state water table height, and steady state outflow rate was observed: the i r r i g a t i o n rate fo r Exl and Ex2 i s higher than for Ex3, yet steady state outflow rates and steady state heights of the water table are lower for Exl than f o r Ex2 and Ex3. However, here also, i t can be concluded that the straight l i n e main limbs of r i s e must be r e a l because i t i s u n l i k e l y that errors y i e l d three straight l i n e limbs of r i s e by chance. 7. The tipping bucket had been p r e f i l l e d with one l i t e r for Ex3. There i s doubt whether i t was also p r e f i l l e d for Exl and Ex2. This error does not a f f e c t the detailed hydrographs because i t does not a f f e c t the outflow rate after the f i r s t t i p . It does not a f f e c t the other outflow hydrographs e i t h e r . 8. For the detailed p l o t , outflow rates have been plotted i n the middle of the i n t e r v a l of the t i p s they correspond to, which i s not quite accurate. This however cannot be the cause for the lack of i n i t i a l step observed for Ex2. For both Exl and Ex3 there are several points before the f i r s t point of Ex2, thus the delay of the early part of the r i s e of Ex2 cannot come from the way the points are plotted. 9. Natural r a i n f a l l occurred now and then. No antecedent outflow was observed before Exl and no r a i n was observed on the s i t e during t h i s experiment. Therefore the r e s u l t s of t h i s experiment are free from the influence of natural r a i n f a l l . No r a i n f a l l was recorded at the meterological s t a t i o n between Exl - 180 -and Ex2. No r a i n f a l l was recorded on the site for at least 16 hours before the start of irrigation for Ex2. No r a i n f a l l was observed during the main limb of rise of this experiment. Therefore, i f rain occurred between Exl and Ex2, i t s effect would be included in the negligible antecedent outflow. Rainfall was observed on the site between Ex2 and Ex3. However, no r a i n f a l l was observed on the Forest plot for at least 21 hours before the start of irrigation until most of the main limb of rise has occurred for Ex3. It is therefore likely that the only effect of the r a i n f a l l that occurred prior to Ex3 is included in the very small outflow of 1.5 7 3 — 1 x 10" m s noted 9 hours before the start of irrigation. It can therefore be concluded that natural r a i n f a l l on the plot has no influence on the shape and slope of the main limbs of rise for the three experiments. It is assumed that i t did not have any influence on the early parts of the rises. Even i f i t did, i t would decrease the proportion of outflow due to irrigation during the i n i t i a l steps and hence make the statement that the system behaves as i f recharge were a uniform step function even stronger. As mentioned in Section 1.3.7, i t is assumed that no subsurface flow reached the plot from uphill. Steady state outflow values may have been influenced by natural r a i n f a l l f a l l i n g on the plot. This natural r a i n f a l l does not affect the water balance, though, since i t is included in the irrigation rate. Similarity between the small scale recessions (Figs. 1.4.2.3 to 1.4.2.5) for the three experiments indicates that they were not influenced by natural r a i n f a l l . - 181 -APPENDIX F CALCULATION OF AVERAGE STEADY STATE OUTFLOW RATES Average steady state outflow rates have been calculated as I x. 1 N where x^ = a data point within the steady state period on the outflow hydrograph N = number of data points used. - 182 -APPENDIX G DEFINITION OF POROSITIES Because the e f f e c t i v e or a i r f i l l e d porosity i s used extensively Chapter I I , i t i s designated by n i n order to s i m p l i f y the notation. The porosity w i l l be designated by n a . Thus ( H i l l e l , 1971) V + V _ a w a " V t V a where V = volume of a i r a V = volume of water w V = bulk volume of s o i l - 183 -APPENDIX H CALCULATIONS FOR USE IN CHAPTER II Note 1 Solution of the Kinematic Wave Equation by the Method of C h a r a c t e r i s t i c s  Eq. 2.3.6 can be rewritten as I t 1 H U The d e f i n i t i o n of a t o t a l d i f f e r e n t i a l l i k e the one given i n Eq. H - l . l means that this equation can be considered to describe the rate of change of nT with respect to t as seen by a wave or observer moving at a v e l o c i t y Wave or "observer" motion w i l l now be used to solve Eq. (2.3.6) by the method of c h a r a c t e r i s t i c s (e.g. Eagleson, 1970). Figure H - l . l (a) shows the plots of x versus t for observers. The plot of x versus t for one observer i s c a l l e d a c h a r a c t e r i s t i c . Figure H-l.l(a) and H-l.l(b) are based on figures given by Eagleson for a more general case. The following discussion and de r i v a t i o n of the thickness T of the saturated zone i s adapted from the discussion for overland flow found i n Eagleson (1970). Some of the r e s u l t s are found in Beven (1981). Eagleson himself followed Henderson and Wooding's (1964) and Wooding's - 184 -F i g . H - l . l (a) C h a r a c t e r i s t i c s , (b) Water table p r o f i l e at time t 2 . - 1 8 5 -(1965a, b, 1966) analysis which according to Eagleson i s a s i m p l i f i c a t i o n of the methods of L i g h t h i l l and Whitham (1955) and Iwagaki (1955). The recharge rate i i s equal to i Q during r i s e and steady state; i t i s zero for the recession. (i) Rise and Steady State If i Q i s steady and uniform, Eq. H - l . l can be integrated to y i e l d i T(t*) - T = — ( t * - t ) H-1.2 o n o where t* = time from the st a r t of recharge t = time at which the observer l e f t from x (measured from the o o st a r t of recharge) T = saturated zone thickness seen by the observer at x and t . o o o Observers t r a v e l on the c h a r a c t e r i s t i c s shown i n Fi g . H - l . l (a) at a v e l o c i t y x - x c = — ^- H-1.3 t* - t o Having K and n steady and uniform with respect to x and z renders c steady and uniform with respect to x and z. The boundary condition T Q = 0 at x = 0 i s assumed. Therefore observers leaving from x Q = 0 observe a saturated zone thickness T Q = 0 at t* = t Q and for t h i s case, Eq. H-1.2 y i e l d s - 186 -I T(t*) = — ( t * - t ) H-1.4 n o For the observers leaving from x Q = 0 at t Q = 0, that i s , f o r observers t r a v e l l i n g on the l i m i t i n g c h a r a c t e r i s t i c , l T(t*) = — t* H-1.5 n For observers leaving from x Q = 0 at t 0 > 0 (Beven, 1981) i i T ( t * ) = -° ( t* - t ) = — - H-1.6 n o n e Thus, once the observer t r a v e l l i n g on the l i m i t i n g c h a r a c t e r i s t i c has reached a point x, T at this point i s constant since n and c are steady, and i t i s given by Eq. H-1.6. For observers leaving at t Q = 0 from x Q > 0, Eq. H-1.2 becomes (Beven, 1981) i T(t*) = — t* H-1.7 n This means that, at time t * , a l l the observers having l e f t at time t D = 0 from a point further downhill than x = 0 and hence having reached a point further downhill than the observer on the l i m i t i n g c h a r a c t e r i s t i c , see the same height of water. In other words, the water L table r i s e s p a r a l l e l to the bed. Eq. H-1.7 i s v a l i d for t* _< — . - 187 -These re s u l t s are depicted i n F i g . H - l . l (b). In th i s F i g . , which shows the shape of the water table at time t 2 , the observer t r a v e l l i n g on the l i m i t i n g c h a r a c t e r i s t i c i s at x 2 . Steady state has thus been reached by a l l points for which x < x 2 . In p a r t i c u l a r , point x i reached steady state at t j , as can be seen from F i g . H - l . l ( a ) . The water table height at points for which x < x 2 i s given by Eq. H-1.6. F i g . H - l . l (b) shows that the water table height i s the same for a l l the points for which x > x 2, i n p a r t i c u l a r for X 3 . For these points, the water table height at time t 2 i s given by Eq. H-1.7 with t* = t 2 . The discharge Q (volume/time) f or a h i l l s l o p e of saturated zone thickness T and width D i s at the foot of the h i l l s l o p e during the r i s e . The discharge i s thus l i n e a r with respect to time during the r i s e . When the whole h i l l s l o p e has reached steady state, l e t t i n g x = L i n Eq. H-1.6, and using Eqs. 2.3.2 and H-1.8 y i e l d s Q = qTD H-1.8 Combining Eqs. 2.3.2, H-1.5 and H-1.8 y i e l d s i H-1.9a Q - o L Ksinin D = D 1 L n c o H - l . 9 b where L i s the length of the h i l l s o p e . - 188 -H e n d e r s o n a n d W o o d i n g i n d e e d o b t a i n Q = D i Q L f o r o v e r l a n d f l o w . E q s . H - 1 . 9 a a n d H - 1 . 9 b c a n b e d e r i v e d f r o m e q u a t i o n s f o u n d i n B e v e n ( 1 9 8 1 ) . The h y d r o g r a p h r i s e a n d s t e a d y s t a t e a r e a l s o d e p i c t e d i n B e v e n ' s p a p e r . ( i i ) Recession F o r t h e r e c e s s i o n , i = 0 h e n c e f r o m E q . H - l . l , d T / d t = 0 ( H e n d e r s o n a n d W o o d i n g , 1 9 6 4 ) . I t w i l l be a s s u m e d t h a t t h e r e c h a r g e s t o p s a t t = t r . A l s o , t h e t i m e d u r a t i o n t ' a f t e r r e c h a r g e s t o p p e d i s d e f i n e d b y t * - t r = t ' I t i s a l s o a s s u m e d t h a t n d u r i n g t h e r e c e s s i o n i s t h e same a s d u r i n g t h e r i s e . I n N o t e 4 , t h e e q u a t i o n w i t h a n e f f e c t i v e p o r o s i t y n r e d u r i n g t h e r e c e s s i o n i s o b t a i n e d . d T B e c a u s e = 0 , a n o b s e r v e r l e a v i n g x Q w h e n i r r i g a t i o n j u s t s t o p s a n d t r a v e l l i n g a t c = d x / d t w i l l o b s e r v e a c o n s t a n t w a t e r d e p t h i 0 x 0 / ( n c ) f r o m x Q t o L . T h i s m e a n s t h a t a t a p o i n t x a n d a t a t i m e t ' < — a f t e r t h e — c r e c h a r g e s t o p p e d , i x T ( x , t + t ' ) = -2-^ H - 1 . 1 0 r n c w h e r e h e r e x Q i s f o u n d f r o m x - x = c t ' o - 189 -(The reason why t 1 must be < — i s that when t' > —, ct' > x — c c and l e t t i n g x Q = x - ct' i n Eq. H-1.10 would y i e l d a negative T. P h y s i c a l l y , the observer having st arted from X Q = 0 at t' = 0 would have reached x and thus T ( x , t r + t') would be zero). Thus, at time t r + t' , i (x - ct' ) i x i t' T(x, t + t ' ) = — = ^ 5 — H - l . l l r nc Ksinto n i L i t ' At x = L, T(t + t') = ^  - — H-1.12 r Ksinco n At a time t' after recession started, a l l observers w i l l be a distance ct' further downslope and as explained above, they w i l l s t i l l see the same T as when they l e f t . Therefore during the recession, the water table s l i d e s p a r a l l e l to i t s e l f down the bed. Using Eqs. 2.3.2, H-1.8 and H-1.12, the discharge during the recession i s i D Q(t + t') = i LD - Ksinco—2- t' x r o n Note 2 Problems Linked to the Evaluation of n Estimating the e f f e c t i v e porosity n for Eq. 2.4.1 i s subject to the following problems: (a) The water content 9j_ n s . z . observed i n the s o i l j u s t before the water table r i s e s i n i t i s larger than 6j_n but may be s l i g h t l y smaller than 6 s t > s t . To overcome this d i f f i c u l t y , two estimates of n have been calc u l a t e d : a maximum, obtained by using the minimum 9^n and a minimum, s i m i l a r l y obtained by using the maximum 9 s t e S t . - 1 9 0 -(b) The kinematic wave model assumes that at the water table the water content increases abruptly by n whereas in r e a l i t y there must be some t r a n s i t i o n zone i n which the water content reaches s a t i a t i o n gradually. It i s not clear what the e f f e c t of th i s t r a n s i t i o n zone i s . (c) Neutron probe data can be used to obtain the maximum unsaturated 6 since a l l the water contents detected by the neutron probe were lower than saturated. However, the water content obtained by the neutron probe at the deepest l e v e l s (more s p e c i f i c a l l y at the 110 cm depth for the upper neutron probe access tube, and at the 63 cm and 50 cm depth l e v e l s for the lower neutron probe access tube) i s subject to the following errors: F i r s t , the neutron probe does not detect s o i l water content at a point but an average content over a region with a 15 cm radius. Thus the water content read at the deepest l e v e l s may be too high i f the access tube rests on t i l l and i f the probe detects the water retained i n the t i l l . In the case the probe did not reach the bed, the water content obtained at the deepest l e v e l s may s t i l l be too high i f the probe detected a saturated zone or a t r a n s i t i o n zone above the bed. Second, because the access tube may not have reached the bed, the reading on the neutron probe when i t was i n i t s lowest p o s i t i o n may not correspond to the moisture content at the bottom of the B horizon. In p a r t i c u l a r the reading may be too low. These errors may in p a r t i c u l a r a f f e c t the maximum water content used to calculate the smallest Kgff (at the 63 cm depth l e v e l of the lower neutron probe access tube). - 191 -Note 3 Taking the Water Balance Into account When Calculating K g j j In general, the recharge rate w i l l not be known and w i l l have to be obtained from the r a i n f a l l rate. Since i t i s l i k e l y that some water i s l o s t , the water balance must be used to obtain the recharge rate that would have yielded the observed outflow rate, had no loss taken place. From the water balance (Chapter I, Section 1.4.6), the measured outflow rate i s about 80% of the time average pooled i r r i g a t i o n rate. Thus the recharge rate per unit area p a r a l l e l to the bed ( i Q ) i s equal to 0.80 times the i r r i g a t i o n rate per unit area p a r a l l e l to the bed. Note 4 Recession with an Effective Porosity n re In Eq. 2.3.11, i t was assumed that n during the recession was the same as during the r i s e . Q during the recession w i l l now be derived assuming n during the recession i s n r e , where n r e i s equal to the porosity minus the re s i d u a l water content and i s assumed constant. For the recession, Eq. 2.3.3 yields n | I + 4 a I I = o re 3t 3x The wave v e l o c i t y during the recession, c r e i s defined by K sino) As shown i n Note 1, during the recession, an observer having l e f t x Q at t 1 =0 and t r a v e l l i n g downhill observes a constant height of saturated zone - 192 -i x T = ° ° n . c . r i r i where n ^ = e f f e c t i v e porosity during the r i s e . c r ^ = wave v e l o c i t y during the r i s e K sinio c . = . r i n . r i x - x Q I x Q Therefore, at time t' = the height of water at x i s c n . c . re r i r i Since c x - x o re t' i (x - t' c ) ™ / i \ o re T(x, t') = n c r i r i i x i t ' o o K since n re i Q t' Hence, during the recession, Q = i LD - K sinio D o n re - 1 9 3 -APPENDIX I CALCULATIONS FOR USE IN CHAPTER III Note 1: Calculation of ^ J"° S 5 at q by Eq. 3.3.15 using data dlog Re cr J  obtained by de Vri e s (1975, 1979)  Because de Vries (1979) uses a notation s l i g h t l y d i f f e r e n t from one used here, i t i s necessary to express the variables used here i n terms of the ones he used. The equation used by de Vries for the nonlinear range i s dh ^ 1 / 2 2 pg — = a d p q + ea d pq (1-1) where g = g r a v i t a t i o n a l a c celeration y = dynamic v i s c o s i t y of the f l u i d p = f l u i d density a^ and e: constants determined by the properties of the porous medium and of the f l u i d (or possibly of the porous medium onl y ) . Rewriting the corresponding equation for the present work, dh Eq. 3.3.2, i n terms of pg - r - y i e l d s Q S dh 2 pg T - = li ct , . g q + pg 3 q 1-2 & ds nonlin b o Comparison of Eq. 1-1 with Eq. 1-2 shows that - 194 -nonlin g e a,2 9 = —A o g e a .1/2 Hence a n o n l i n 3 .1/2 — d Table I - l shows the c a l c u l a t i o n of dlog f dlog Re -1 i q c r + e_ cr / a d (1-3) using the values obtained by de Vries (1975, 1979) for 0.4 - 0.6 mm sand and 0.8 - 1.4 mm sand. dlog f Table I - l . C a l c u l a t i o n of , v „ from E q . 1-3 for de Vr i e s ' dlog R e (1975, 1979) r e s u l t s . Material a d ( n r 2 ) e q c r (Ws) e q M c r /a, V a dlog f dlog Re q c r 0.4-0.6 mm 9.2 x 10 9 0.9 3.0 x I O - 3 0.96 x I O - 5 0.029 -0.97 sand 0.8-1.4 mm 1.7 x 10 9 0.9 3.4 x IO" 3 0.98 x IO" 6 0.076 -0.93 sand - 195 -Note 2: Correction of q c to take the temperature difference into account. If 2 Reynolds numbers Re\ and Re2 are equal, then q l L c h l P l _ q2 Lch2 p2  U l " y2 where q has been used as c h a r a c t e r i s t i c v e l o c i t y . Subscript 1 w i l l r e f e r to de V r i e s ' runs i n 0.4 to 0.6 mm sand at 22°C. Subscript 2 w i l l r e f e r to the experimental flow conditions i n the Forest plot presented i n Table 3.4.1 ( v e r t i c a l i n f i l t r a t i o n and q at 2 outflow points along the channel's bank). As mentioned i n Section 3.4, a temperature of 5°C w i l l be used for the Forest and i t w i l l be assumed that L c h l = L c h 2 . Also, the change i n p i s n e g l i g i b l e f o r the temperatures considered. For 5°C, p = 1.5 x 1 0 - 3 k g s - ^ - 1 . The y given by de Vries (1975) i s about 0.96 x 1 0 - 3 k g s - 1 m - 1 . Thus — = — = 9 ' ^ . q, i s q , found by q y 1.5 M l M c h 3 — 3 — 1 de Vr i e s and i s equal to 3 x 10 ms . Hence „ .._3 1.5 _ l q2 = 3 x 10 x -Q-gft m s o 1 q2 = 4.7 x 10 ms - 196 -APPENDIX J REFERENCES FOR THE APPENDICES Beven, K. 1981. Kinematic subsurface stormflow. Water resources  research, Vol. 17, No. 5: 1419-1424. de V r i e s , J . 1975. Unpublished data. de V r i e s , J . 1979. P r e d i c t i o n of non-Darcy flow i n porous media. American Society of C i v i l Engineers, Journal of the i r r i g a t i o n and  drainage d i v i s i o n , V o l . 105, No. IR2, Proc. Paper 14610: 147-162. Eagleson, P.S. 1970. Dynamic Hydrology. McGraw-Hill. Henderson, F.M. and R.A. Wooding. 1964. Overland flow and groundwater flow from a steady r a i n f a l l of f i n i t e duration. Journal of  geophysical research, Vol. 69, No. 8: 1531-1540. H i l l e l , D. 1971. S o i l and water: physical p r i n c i p l e s and processes. Academic Press. New York, 288 pp. Hewlett, J.D. and A.R. Hibbert. 1963. Moisture and energy conditions within a sloping s o i l mass during drainage. Jour, of geophysical  research, Vol. 68, No. 4: 1081-1087. Iwagaki, Y. 1955. Fundamental studies on the runoff analysis by c h a r a c t e r i s t i c s , Disaster Prevent. Res. Inst., B u l l . 10, Kyoto Un i v e r s i t y . Cited by Eagleson. L i g h t h i l l , M.H. and G.B. Whitham. 1955. On kinematic waves. I. Flood movement i n long r i v e r s . Proc. Roy. S o c , Ser. A, Vol. 229: 281-316. Cited by Eagleson. Mehra, O.P. and M.L. Jackson. 1960. Iron oxide removal from s o i l s and clays by a d i t h i o n i t e - c i t r a t e system buffered with sodium bicarbonate. Clays and clay minerals. V o l . 5: 317-325. International series of monographs on earth science, Pergamon Press. Nagpal, N.K. and J . de V r i e s . 1976. On the mechanism of water flow through a forested mountain slope s o i l i n coastal western Canada. Unpublished material. - 197 -Utt i n g , M.G. 1978. The generation of stormflow on a glaci a t e d h i l l s l o p e i n coastal B r i t i s h Columbia. M.Sc. Thesis, University B r i t i s h Columbia, Vancouver. Wooding, R.A. problem. 254-267. 1965a. A hydraulic model for the catchment stream I. Kinematic-wave theory. Jour, of hydrology, V ol. 3: Cited by Eagleson. Wooding, R.A. problem. 268-282. 1965b. A hydraulic model for the catchment-stream I I . Numerical so l u t i o n s . Jour, of hydrology, V o l . 3: Cited by Eagleson. Wooding, R.A. problem. hydrology, Vol. 4: 21-37. 1966. A hydraulic model for the catchment-stream I I I . Comparison with runoff observations. Jour, of Cited by Eagleson. 

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