The matric flux potential: history and root water uptake

Transcription

1 The matric flux potential: history and root water uptake Marius Heinen, Peter de Willigen (Alterra, Wageningen-UR) Jos van Dam, Klaas Metselaar (Soil Physics, Ecohydrology and Groundwater Management Group, Wageningen-UR)

2 Contents History on the birth and name-giving matric flux potential use, properties etc. of matric flux potential Root water uptake steady-rate solution of RWU with matric flux potential RWU and hysteresis Resume

3 log(k) log(h) Soil water movement Richards equation (93) q q t Darcy-Buckingham (856, 907) q Hydraulic properties h, q, K K h K z matric gravity How to solve the non-linear PDE? q (cm 3 cm -3 ) q

4 History Earlier solutions making use of a transformation e.g., Klute (95), Gardner (958), Philip (966) June 966: Symposium Water in the Unsaturated Zone, Wageningen Discussion session led by G.H. Bolt How to proceed with solving Richards equation

5 The transformation The Kirchhoff transformation: heat transport Gustav Robert Kirchhoff (84-887)

6 Raats (970, SSSAP 34: ) h h ref K h dh q Kh Kz matric gravity q K z

7 Steady-state Richards equation with 0 K non linear z Exponential K(h) relationship (Gardner, 958) Kh K exp h s M h - K h dh K z linear; Laplace (un)saturated

8 and the stream function Stream function : corresponding to any plane flow problem there exists a conjugate problem which is obtained by interchanging and (same Laplace equation) (Raats, 970) z z

9 Raats publications on Total: 67 MFLP: 34

10 Quasi-linear analysis of steady flows () Flows from (sub)surface drip irrigation sources Raats (970, 97, 976) Recently elaborated upon by Communar & Friedman (SSSAJ, 00): coupled source-sink steady water flow model Flows to sinks: porous cup samplers, suction lysimeter Raats (97, 976) Merrill, Raats & Dirksen (978) Raats (00; Developments in soil-water physics since the mid 960s; Geoderma 00: ) Raats (976; Trans ASAE 9: )

11 Quasi-linear analysis of steady flows () Flows from surface disc Wooding (968) Flows from bore-hole permeameters Reynolds et al. (985); Philip (985); see also Heinen & Raats (990) * B Q AK 0 B K0 A Stability of steady flows van Duijn, Pieters, & Raats, 004 And several others... Raats (00; Developments in soil-water physics since the mid 960s; Geoderma 00: ) Q R K 0 RK0

12 Disadvantages Does not apply in layered or heterogeneous soils, unless solved by discretization (Ross,WRR, 990) by averaging ten Berge et al., SAWAH, 99) Cannot be used in combination with hysteresis (e.g. Raats, 990)

13 How to obtain Analytical solution of the integral Numerical integration Analytical approximation e.g. for Mualem-van Genuchten (de Jong van Lier et al., 009; WRR 45 W04) Measure Steady-state evaporation (ten Berge et al., NJAS 987) Non steady-state absorption (Clothier et al., SSSAJ, 983)

14 Example Root water uptake (RWU) single root uptake steady-rate solution in Partial root-soil contact Up-scaling into macro-model Properties of RWU model Model by de Jong van Lier et al.: also

15 Root water uptake Richards equation extended with RWU q t q S w Raats and RWU, e.g.: 974: SSSAP 38:77-7 steady state with plant roots, 990: scaling in soil physics depletion ahead of moving root front review 004: Feddes & Raats 007: TIPM 68:5-8

16 Root water uptake: single root model PhD Thesis 987 Single root with cylindrical shell of soil bulk substrate R L rv T R 0 R q q q q T R 0 h r h rs h soil

17 Steady-rate RWU in q q zl rvk hrs hr zl rv f rs Iteratively find h r, h rs and T act for known h de Willigen and van Noordwijk, 987 T act T pot h h r r,/ a f R R ln Numerical solution Steady-rate approximation

18 Root-soil contact: model soil A: (r,,0 gap root R 0 =R /R 0 van Noordwijk et al., 993; Geoderma, 56:77-86 R

19 Steady-rate solution for partial contact The steady-rate distribution of in the soil cylinder with partial radial contact is given by At A: h is at wilting point (-60 m), so that (,0) = 0 Iso- lines can be constructed sin cos sin ln,0, k k k k k k k k k k k k r r r r r full contact adjustment when 0 < <

20 matric flux potential pressure head Partial contact: and h distributions sand R/R 0 clay loam

21 From a single root to a root system: up-scaling Numerical simulation models: finite elements, control volumes, finite difference grids Assume: roots are regularly distributed within each element, and they are parallel: a given L rv For each time step use single root model for each element: distributed S w Sum of all element uptakes must equal total T act N+ equations with N+ unknowns N values of h rs (+ rs ) + h r (NB: N values of h are known)

22 Properties of the RWU model Under wet conditions: RWU distribution is proportional to L rv distribution In case of local dry conditions: compensation through uptake by roots under more favorable conditions Hydraulic lift Root resistance is dominating The model is extended with osmotic hindrance (cf. Dalton, Raats & Gardner, 975; Agr. J. 67: ) q zl rvk hrs hr ho,rs ho,r

23 RWU model and hysteresis () Raats: what is the effect of hysteresis on your RWU model? As said before: an analysis with can only be done when hysteresis is disregarded Numerical analysis around a single root alternating d drying and d wetting: effect on uptake extremely large difference in drying wetting retention curves: sand

24 RWU model and hysteresis ()

25 de Jong van Lier, Metselaar, van Dam Assume h h wilting at root soil interface Steady-rate solution for Confirmed steady rate assumption by numerical modeling Derived: T T a p crit Relative transpiration scales as relative crit : at the on-set of transpiration reduction dependent on root density, and atmospheric demand (no root!)

26 Resume Peter Raats introduced and promoted the term matric flux potential, and used it for numerous analytical analyses of flow problems. Others have used it as well. We have successfully applied the matric flux potential concept in steady-rate root water uptake models up-scaling to root systems compensation; hydraulic lift root resistance mostly dominating Hysteresis is not of great importance with respect to RWU

27 I wonder if the matric flux potential of this soil can explain the presence of these sand mushrooms... Thank you Wageningen UR

28 The transformation The Kirchhoff transformation: heat transport Gustav Robert Kirchhoff (84-887) Soil physics M h K h dh θ D θ dq h ref θ ref

29 Experimental evidence Raats (980; Report visit New Zealand)

30 Some aspects concerning Analytical expressions for : Raats & Gardner (97): 6 K(h) relationships constant, exponential, power Critical pressure head (Bouwer, 964): K-weighted mean pressure head between h = 0 and h = - h Stream function : corresponding to any plane flow problem there exists a conjugate problem which is obtained by interchanging and (same Laplace equation) (Raats, 970) h crit 0 - K K s dh

31 Raats (00; Geoderma 00: ) Quasi-linear analysis of steady flows () Flows involving specified extraction patterns by roots Raats (98) Flows around obstructions: solid object or air-filled cavities Philip

32 Advantages Disadvantages Do revisit analytical solutions (Laplace equation) Robust and stable numerical solutions - known as integrated average (e.g. Shaykewich and Stroosnijder; NJAS, 977) Does not apply in layered or heterogeneous soils, unless solved by discretization Ross (WRR, 990); by averaging ten Berge et al. (SAWAH, 99) Do exclude hysteresis, e.g. Raats (990)

33 Measurement of ten Berge et al., NJAS, 987 Steady-state evaporation (ten Berge et al., NJAS 987) measure flux and weight ( q) A( q*) B q * Non steady-state (Clothier et al., SSSAJ, 983) measure water content and Boltzmann transform l d dq D( q) dl dq ldq

34 Root water uptake Richards equation extended with RWU q t q S w Raats and RWU, e.g.: 974: SSSAP 38:77-7 steady state, 990: scaling in soil physics depletion ahead of moving root front review 004: Feddes & Raats 007: TIPM 68:5-8 i v 0 W

35 Root-soil contact van Noordwijk et al., 993; Geoderma, 56:77-86

36 Fractional depletion z = 0 cm T = 5 mm d - de Willigen and van Noordwijk, 987

37 Some observations Steady-rate solution in confirmed by numerical validation Highest resistance to flow mostly across the root wall the soil is (mostly) not limiting Except for coarse sands, only at small root-soil contact problems with uptake occur

38 for Mualem van Genuchten () ; ) ( / s r s r n n m S S K S K h h S m m m n l q q q q S D S S S C S K D S h S S C d d d a ref S S r s r s q q q q

39 for Mualem van Genuchten (),,, F,,, F /m ref /m ref ref /m a /m a a s a l m x m x f x m x f S f S f S S f S f S m K S m m After: de Jong van Lier et al. (009; WRR 45 W04) Note: their Eq. [A-0] contains an error F (Michel & Stoitsov, 008; Computer Physics Communications 78: ) power series expansion around x = 0 series converges always for x < singularities for x = 0, x = : approximations available Fortran and C++ routines available

40 Matric flux potential (cm d - ) for Mualem van Genuchten (3) Analytic Numeric Degree of saturation S (-)

41

42 O ut[ 0 ]=

43 RWU model and hysteresis (3)

44 RWU model and hysteresis (4)

45 RWU model and hysteresis (5)

46 RWU model and hysteresis (6)