U.S. Geological SurveyU.S. Department of the InteriorOpe

Klute,  A.

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F ina l  Report P r o j e c t  Number 65-03G The Analys is  o f  t h e  Uptake o f  Nater by P lan t  Root Systems A p r o j e c t  supported by t h e  Water Resources Cenfer o f  t h e  Un i vers i -by o f  1 l l i no1 s w i t h  funds from t h e  O f f i c e  o f  Water Resources Research, DeparTment o f  I n te r io r ,  Washington, D~CI P r i n c i p a l  Inves t iga tor :  Professor A. K l u t e  Department o f  Agronomy, t h e  U n i v e r s i t y  o f  I l l i n o i s  Out l ine I. Introduction, Backgraund and Object ives I I .  Theory and Analysis A. Flow theory 8. The source funct ion C. Numerical procedures D. Select ion o f  conduct iv i ty  funct ions and parameters E. Results 1. Typical pressure head, source and f l u x  p r o f i l e s  2.  E f f e c t  a f  magnitude o f  t he  source term on pressure head p r o f i l e  3. " S  hor t  i ngM e f f e c t  4. Rooting depth Ill. Experimental System Development 1V. L i t e ra tu re  C i ted  V. Summary, Conclusions I I .  In t roduct ion and Object ives The flow equation f o r  water i n  unsaturated sol l may be w r i t t en  t o  In-  clude a source term which can be used t o  represent t he  uptake o f  water by roots  o f  plants, The roo ts  are  considered t o  be d i s t r i bu ted  i n  the  s o i l  i n  a contin- uous manner and the use o f  water by roo ts  i s  rapresenTed as a negative source. The ob jec t i ve  o f  the  research work described herein was t o  ob ta in  solu- t i o n s  o f  t he  flow equation f o r  s o i l  water f o r  various boundary condi t ions and t o  invesTiga+e the  e f f e c t  o f  the  source Term on the  f low patterns. The choice of  the  source funct ion was studlad. Experimen-l's were planned and s ta r ted  t o  ob ta in  data f o r  v e r i f y i n g  the  pred ic t ions made from the  f low theory and t o  a i d  i n  the  developmenl- o f  a model f o r  t he  source function. Drs. Richard J. M i  l l i ngton end Frank 0. WhiS l e r  have contr ibuted much t o  t h e  work accomplished on t h i s  project.  Dr. M i l l l ng ton  gave counsel on the  nature o f  the source funct ion and contr ibuted t o  the development o f  t he  experi- mental system, Dr. Whisler ca r r ied  ou t  the  numerical camputations w i t h  the  1BM 7094 computer. I n  add l t l on  t o  the  cont r ibut ions of  these two, Drc 0. Be Peters has contr ibuted t o  t he  development o f  the experimental system, 1 I .  Theory and Ana l ys i  s A, Flow theory I n  the  analysis t o  be described below it w i l l  be assumed t h a t  the  modi f ied Darcy equation f o r  f low i n  an unsaturated s o i l  i s  valid, viz.: v = -K(B)VH (I 1 where v Is the  volume flux, 1 .e. the volume of  water passing through u n i t  cross sect ional  area o f  s o i l  i n  u n i t  Pime, VH I s  t he  gradient o f  t h e  hydraul ic  head H, and Kt81 i$ the  conduct iv i ty  o f  the s o i l  t o  water as a funct ion o f  water con- t e n t  (Gardner, 1960a). The hydraul ic  head i s  considered So be t h e  sum o f  a gravita)ional head z and a pressure head h, both expressed i n  length units, The conservation o f  matter p r i n c i p l e  i s  a lso imposed on t he  f low by use of  the  equation o f  cont inu i ty :  a o I n  equation (2) i s  t he  t ime r a t e  o f  increase o f  t he  volumetric water content 8 which i s  considered t o  occur because o f  t h e  excess o f  in f low over out f low and because o f  a source term St which represents t he  volume o f  water produced a t  a po in t  i n  the  f low system per u n i t  volume o f  s o i l  per u n i t  timer I n  the present analysis t he  source funct ion S w i l l  be used t o  represent the uptake o f  water by p l an t  roo ts  i n  the  so i l ,  The roo ts  are  considered t o  be d i s t r i bu ted  i n  +he s o i l  i n  a continuous (but no t  necessari ly uniform) manner, The volume o f  water extracted from the  s o i l  per u n i t  volume o f  s o i l  per u n i t  t ime can be represented as a negative source. The present macroscopic approach t o  t h e  uptake o f  water by p lan t  m o t s  stands i n  qontrast  t o  the  microscopic approach i n  which f low t o  a s ing le  r o o t  i s  analyzed (Gardner, 1960b). The analysis i s  thus made on t he  macroscopic level;  t he  level a t  which observations and measure- ments a re  made, The f l u x  may be eliminated as a var iab le  by combining equstions (I) and (2): If it i s  fu r the r  assumed t h a t  a defined r e l a t i o n  between t he  water content and t h e  pressure head o f  the  s o i l  water exlsts, i.e, t h a t  0 = 0(h) ,  then one may w r i t e  equatton (3) as: where t he  water capaci ty C(0) i s  deflned as d~/dh, .the r a t e  o f  change o f  water content w i t h  pressure head. 4 t  Is the  ob jec t i ve  o f  +he present analysis t o  apply equation (4) t o  t h e  problem o f  ex t rac t ion  of water by roo ts  from a uniform v e r t i c a l  s o i l  column wi th  a water t ab le  a t  i t s  lower end and w i t h  evaporatlon from the  s o i l  surface occurr ing af the upper end. For one-dimensional v e r t i c a l  f low equation !4) be- comes: where we have regarded the  water capaci ty and conduct iv i ty  as funct ions o f  t he  pressure head. A coordinafe system w i l l  be chosen so t h a t  z equals zero a t  t he  upper end o f  the  s o i l  column and -L a t  the lower end. There are two types o f  s i tua t ions  t h a t  may be treated, viz., the  steady s ta te  and unsteady state. We sha l l  t r e a t  on ly  t he  steady s ta te  f low here and leave the  unsteady stafe case f o r  fu tu re  analysis. I n  the  steady s ta te  we have: so t h a t  equatlon ( 5 )  becomes: The boundary condiPlons chosen f o r  t h i s  analysis are: h(-L) = 0 The f i r s t  o f  these represents the  presence o f  a water t a b l e  a t  t he  lower end o f  t he  column and the  second represents t h e  f l u x  o f  water vo a t  the  upper end o f  the  s o i l  column. !4e sha l l  regard v, as a p o s i t i v e  constant, ioe,$ evaporation w i l l  be assumed t o  occur a t  a constan+ r a t e  from the  s o i l  surface, I n  pr lncip le,  a pa r t i cu l a r  so lu t ion  o f  (6) i s  determined by (7) and (8) when t he  conduct iv j ty  K(h) and source funct ion S are specif ied, The con- d u c t i v i t y  funct ion can be obtained by separate measurements on representat ive samples of  t he  so i l .  These measurements, whi le d i f f i c u l t ,  a re  i n  p r i n c i p l e  q u i t e  c l e a r l y  defined (Klute$ 1965). The source funct ion i s  not  so wel l  known and understood. One o f  t he  ob ject ives of  t h i s  analysts i s  t o  assess t he  e f f e c t  of  4 varlous assumptions f o r  the  source funct ion on t h e  flow i n  t he  soi  I. For purposes o f  analysis it w i l l  be convenient t o  use dimensionless varfables. We def ine a dimensionless pressure head: a distance var iable:  a conduct i v i  ty :  and a source term: I I r = s  L K, (121 j where 9, i s  the  volumefrio water content a t  saturation, and K, i s  t he  conduc- I t i v i t y  a t  saturat lon, Using these var iables equation (6) becomes: and t he  boundary conditions become: ca( -1)  = 0 B, The source funct ion The source funct ion i s  a t  t h l s  po ln t  unspecjfied. This representat ion o f  r o o t  uptake o f  water has no t  been widely used and knowledge of  i t s  behavior j I s  l imited. The theory ou t l i ned  above forms t h e  basis f o r  a considerable amount 1 o f  experimentation t o  determine t he  fac tors  t h a t  may a f f e c t  the  source funct ion , and es tab l i sh  t h e i r  importance. A t  t h l s  t ime we can on l y  invest igate  t he  e f fec f  : i o f  t h e  source funcf ion on the  f low by assuming various forms f o r  t he  funct ion and examining t h e i r  e f f ec t  on fhe flow, Several such models o f  t he  gource funct ion \ I have been examined. I n  terms o f  t h e  dimensionless source funct ion r these were: r = ~ ( c )  ( ( P ~ - I ~ )  ( 19) I n  e q u a t l ~ n  (191 I s  the  pressure head i n  t h e  p lan t  r o o f  t i s sue  and a(6) i s  C P ~  a r o o t  densi ty funct ion* I n  equations (16) and (17) A, and Al are a r b i t r a r y  constants. The development of  t he  source f unc+ion given i n  equation ( 1  9) proceeds as follows: It w i l l  be assumed t h a t  +he source i s  a funct ion o f  depth because o f  i t s  dependence on ( I )  a roo* densi ty funct ion ACz), (2) t h e  conduct iv i ty  o f  t he  s o i l  t o  water and 13) +he d i f ference between the  pressure head o f  t he  water i n  the  p l an t  r o o t  t l s sue  h and the  pressure head o f  the  water i n  the  s o i l  h. P Thus we def ine the source as fol lows: S ( Z )  = A(z) K(h) (hp - h l  (20) The dimensionless f o r m o f  t h l s  source func+ion is given i n  equation (19). An a l t e rna te  formulat ion o f  t h e  source term i s  as fol lows: where L, i s  some e f f ec t f vo  length over which t h e  pressure head d i f ference h;h - (h -h) r i s  presumed t o  act, Thus the  product K(h) --.I?--- i s  t h e  f lux o f  water t o  t h e  Le root, vlz,, volume o f  water passing through u n i t  area o f  absorbing surface o f  t he  roo ts  per u n i t  time* Vlewed i n  t h i s  context the  funct ion a(z)  has t he  dimensions o f  r o o t  absorbing surface per u n i t  volume o f  so i l .  Since Le i s  essen t ia l l y  unknown we combine It w i t h  a tz)  t o  g ive a r oo t  densi ty funct ion A(z) and hence ob ta in  equation (201. The Aiz) funetlon i s  used t o  represent t he  depth d i s t r i b u t i o n  o f  a+ouaa *sa3e~~ns +ueld pue 110s wo~j uoj.+e~odena 40 ~+QJ paulqwo9 +ue+suoa e s! aJaq+ *a4! 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Je+eM aq+ 40 peaq a~nssa~d eq+ cdq gu~ouyun MBU Q samp -OJ+U! (61) uo!+enbe Aq ue~!6 lapad wJa+ aDJnos aq+ 40 uo!&dunsse eq~. *uo!+Dun+ (z)~ ua~!6 12 40 seauanbasuo3 ey~ asnpap o+ AJ+ pue u~ouy s! (z)~ +eq+ apn+!++B aq+ +dope l IQYS 3M *aJay paJap!suo:, aq +ou 11 !M asaq+ CJ~A~M~ *uo!+s+aJdJa+ul pue +uawaJnseom s+l o+ p~~6a-1 q+!~ Ala+e!pawu! as!Je suo!+sano *I !os 40 awnion +lun dad s4oo-1 ~U~~JOS~Q Ja+eM 40 q+6ual eq+ se +! p~e6a~ o+ s! +! 30 uo!+e+aJdJa+u! le~!sAqd elqlssod suo *s+oo~ 

Analysis of the uptake of water by plant root systems

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https://www.ideals.illinois.edu/bitstream/2142/89956/2/Klute_1966.pdf

Analysis of the uptake of water by plant root systems

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