1

Dynamic plant uptake modeling

Stefan Trapp

Steady-state considerations: simple & small data need

However: often emission pattern is non-steady, e.g.:

non-steady plant growth (logistic)

pesticide spraying

application of manure and sewage sludge on

agricultural fields

In these scenarios: steady-state solutions are not valid, and

dynamic simulation is required.

Steady-state vs. dynamic models

• Three different types of input, namely

• pulse input (pesticide spraying, sludge application)

• constant input (deposition from air, irrigation)

• irregular input (this we cannot solve --> use numerical integration)

Dynamic input patterns Dynamic models

Designed for

- repeated input

- dynamic growth

- pesticides

- manure or sewage appl.

- 1 year or 10 year

- easy to handle (excel)

1.0E-06

1.0E+00

2.0E+00

3.0E+00

4.0E+00

5.0E+00

0 10 20 30 40 50 60

Real time (d)

C(t

)

C3 Stem C2 root C4 Fruit

C1 soil bioavailable C4a Leaves

● Repeated pulse input from soil or air or constant emission

● Logistic growth of plants (here: summer wheat)

● n variable periods (30 in practice)

Dynamic models Dynamic Model

Differential equation system

In words

soil: change of mass = + Input - degradation - uptake into plants

roots: change of mass = + uptake from soil - loss to stem - degradation

stem: change of mass = + uptake from roots - loss to leaves (fruits) - deg

leaves: change of mass = +uptake from stem ± exchange air - degradation

fruits: change of mass = +uptake from stem ± exchage air - degradation

2

Differential equation system

1 soil

2 roots

3 (stem)

4a

leaves

4b fruits

inputQKM

CCk

dt

dC

SWSoil

SoilSoil

Soil deg

RRRRWRR

SW

SoilR CkMQKCMQK

C

dt

dC ///

StStStStWSt

RW

RSt

St CkMQKCK

CMQ

dt

dC ///

LLL

LLA

LA

L

depL

St

StWL

LL CkCMK

mLgAC

M

vAC

KM

Q

dt

dC

31000

FFF

FFA

FA

F

FSt

StWF

FF CkCMK

mLgAC

M

gAC

KM

Q

dt

dC

31000 Soil

matrix

Soil

water

thick

Roots

xylem

water

Stem

Leaves Fruits

Air

air

diffusive

equilibrium

flux with water

fine

Roots

Kd KRW

KRW

KLA

Standard Model This is again our standard model. Mass balance // differential equations

remain, but mathematical solution is for the dynamic case.

Cascade of compartments

111 mk

dt

dm 2211

2 mkmkdt

dm

Mass balance:

"The change of mass in

tank 2 is what flows out of

tank 1 minus what flows

out of tank 2"

Differential equation system

1 soil

2 roots

3 stem

4a leaves

4b fruits

1111 bCk

dt

dC

2221122 bCkCk

dt

dC

333223

3 bCkCkdt

dC

4443344 bCkCk

dt

dC

4443344 bCkCk

dt

dC

The system written in a schematic

way:

Each DE always relates to the DE

before, but not to any other DE

transfer rate constants kij (d-1)

loss rate constant ki (d-1)

constant external input b (mg kg-1 d-1).

Structure of the multi-cascade crop model Dynamic Model

bC

kk

kk

kk

k

dt

Cd

434

323

212

1

00

00

00

000

Same processes, same differential equations,

but formulated as matrix

1 is soil k1 loss rate k12 transfer rate

2 is roots k2 loss rate k23 transfer rate

3 is stem k3 loss rate k34 transfer rate

4 is leaves or fruits k4 loss rate

b is the input vector

3

tkeCtC 1)0()( 11

tktktk

eCkk

e

kk

eCktC 2

21

)0()()(

)0()( 2

2112

1122

tk

tktk

tktktk

eC

kk

e

kk

eCk

kkkk

e

kkkk

e

kkkk

eCkktC

3

32

321

)0(

)()()0(

))(())(())(()0()(

3

3223

223

231332123121

123123

etc. …

Analytical solution for pulse input, i.e. C(0) ≠ 0

tktkeCe

k

btC 11 01 1

1

11

tktktktkeCeBeeAtC 2221 01 22

tktktktktktkeCeFeeEeeDtC 333231 01 33

112

1121121 0

kkk

bkkkCA

21

21112

kk

bkbkB

13

23

kk

kAD

23

223 0

kk

BACkE

3

323

k

bBkF

Cascade with constant input

Analytical solution for all t

Principle of superposition

Concentrations are additive

We can thus calculate several subsequent periods with different

values, and the output from one period is the input to the next.

This allows to simulate non-constant conditions.

Our "cascade model" has by default 24 periods to 5 days (= 120

days, i.e. one vegetation period), but this is variable.

Principle of superposition

Figure: Concentrations are additive

That's it with the math!

Questions?

Default data used in the standard model

Plant mass per m2

roots 1 kg (wet weight)

leaves 1 kg

fruits ½ kg

Transpiration

1 L d-1 m-2 (365 L/m2/year)

Growth rate

0.1 d-1 (doubling in 1 week) for field crops

0.035 d-1 (doubling in 3 weeks) for meadows

I admit hereby that this is simplified :-)

4

Real Growth Dynamic model

max

1M

MMk

dt

dM

tk

eM

M

MtM

110

max

max

logistic growth function

k = growth rate, Mmax is final growth

Plant mass at time t

Most annual crops show a logistic growth curve

initial growth is exponential

towards ripening, growth slows down

and finally stops

Change of plant mass M [kg]:

Plant growth

max

1M

MMk

dt

dM k First-order rate constant

(for exponential growth)

[1/d]

Mmax Maximum plant mass [kg]

tk

eM

M

MtM

110

max

max

Plant mass as a function of time

M0 Initial plant mass [kg]

Real Data for Growth

Measured data (EC stages) Total biomass, fitted

Real Data for Growth

Mass of roots, stem, leaf, corn Total biomass, fitted

Growth and transpiration of plants are related by

the water use efficiency (kg plant / L water) or the

transpiration coefficient TC (L water / kg plant).

Typical values range between 200 and 1000 L/kg dry weight

Default value for TC is 100 L/kg fresh weight.

Plant growth and transpiration

max

1M

MMkT

dt

dMTQ CC

Q Transpiration [L/d]

TC Transpiration coefficient [L/kg dw]

In our model, transpiration takes place only when plants are growing

Data obtained from agricultural handbooks

(summer wheat)

Annual seed plant

- Initial mass 10-4 kg (0.1 g for seeds)

- Growth rate constant k = 0.1 d-1

(doubling time ≈ 1 week)

- Final mass 1 kg

data related to 1 m2

Transpiration coefficient TC = 50 L/kg fw (water content green plans ≈ 90%)

Plant growth and transpiration

5

Standard scenario: summer wheat

tk

eM

M

MtM

110

max

max

max

1M

MMkT

dt

dMTQ CC

Maximum transpiration Qmax is at ½ Mmax (inflection point)

with maxmax 4

1 MkTQ C

at time

1

1ln

1

0max MMkt

Plant growth and transpiration

Annual seed plant

Plant growth and transpiration

Growth is exponential

for t < 70 d

Absolute growth & transpiration

peak at t = 92 d

Growth almost stops for t > 135 d

= phase in which fruit or corn ripe

leaves decay and plants dry out

Biomass M and transpiration Q

of summer wheat

Dynamic model: Default scenario

Reading:

0.0

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30 35

Time (d)

C(t

)

C3 Stem C2 root C4 Fruit C1 soil

Example simulation for a repeated pesticide

application on pepper

Repeated application of insecticide

by drip irrigation to soil

Comparison to measured data

Model result before calibration

0

0.05

0.1

0.15

0.2

0.25

0 5 10 15 20 25 30 35

Time (d)

C F

ruit

Measured Model

Comparison to measured data

After fit of two parameters (temperature, soil depth)

00.05

0.10.15

0.20.25

0.3

0 5 10 15 20 25 30 35

Time (d)

C F

ruit

Measured Model

6

More reading:

you survived this part

Questions?

Coupled Model for Water and Solutes in

Soil and Plant Uptake

Stefan Trapp

Differential equation system

soil

roots

(stem)

leaves

fruits

RRRRWRR

SW

SoilR CkMQKCMQK

C

dt

dC ///

StStStStWSt

RW

RSt

St CkMQKCK

CMQ

dt

dC ///

LLL

LLA

LA

L

LSt

StWL

LL CkCMK

mLgAC

M

gAC

KM

Q

dt

dC

31000

FFF

FFA

FA

F

FSt

StWF

FF CkCMK

mLgAC

M

gAC

KM

Q

dt

dC

31000

this is what comes now in this lecture

Decades we waited that some encouraged soil

transport modelers would integrate the four equations

for plant uptake.

Nobody did it. We had to do ourselves.

In soil

- water moves

- compounds move

Both is connected. More and less complex models exist to predict

movement of solution and solutes (pesticides):

PRZM Pesticide root zone model

PELMO Pesticide leaching model

MACRO focus on macropore transport

etc.

Soil Transport Models

7

Water Transport Substance Transport

Tipping Buckets Transport Model

Each soil layer is a "bucket" that is between empty

(PWP) and full (FC).

The "Tipping Buckets" model needs only two soil

parameters to describe water transport:

FC Field capacity

above this water content (L/L), water flows deeper

PWP Permanent Wilting Point

below PWP (L/L), plants stop to take up water.

Simple and discrete - perfect to connect to our cascade

model approach.

Trapp & Matthies 1998 Chemodynamics

Legind et al. 2012 PLoS one

Trapp & Eggen 2013 EnvironSciPollutRes

Coupling of tipping buckets and cascade model

In each discrete time step, plants extract the water required for transpiration.

Plants will always take water where they find it, upper layer first - no root

growth is calculated, we assume that within the period (2 weeks) roots grow

towards the water.

Equations of the Buckets Model (1) Equations of the Buckets Model (2)

8

Equations of the Buckets Model (3)

new C = old C + from air - to air + from rain + emission

- leaching and transpiration

into plant = sum of transpiration

from all soil layers and groundwater

Differential equation system

1 roots

2 (stem)

3a

leaves

3b fruits

airCkMQKCIdt

dCRRRRWR

R //

airCkMQKCK

CMQ

dt

dCStStStStWSt

RW

RSt

St ///

LLL

LLA

LA

L

depL

St

StWL

LL CkCMK

mLgAC

M

vAC

KM

Q

dt

dC

31000

FFF

FFA

FA

F

FSt

StWF

FF CkCMK

mLgAC

M

gAC

KM

Q

dt

dC

31000

input from soil

Data for the Buckets Model

Input data were taken from a ten-years field study done by INRA in Feucherolles,

France, 30 km west of Paris (see Legind et al. 2012).

Water balance

+ Precipitation (sum of rain, snow, fog and irrigation = input data)

- Transpiration (calculated from plant growth)

- Evaporation from soil surface (Penman-Monteith equation)

- run-off (input data)

= leaching to next layer or GW = calculated

Looks easy - but it is best done day by day.

Water balance - simulation results field study

Legind et al. 2012

Figure 4. Simulated water balance and content of soil. (a) Simulated

annual water balance, control scenario, August 1998 to October 1999; (b)

simulated water content of the five soil layers, same simulation event.

Figure 5. Leaching of water from soil layer 2. Model compared to

measurement for three treatments. Model is average of all predictions, min and

max is minimum and maximum lysimeter measurements.

Validation

(from Legind et al. 2012)

Simulation of leaching (depth 40

cm) versus lysimeter results.

Experimental data were only

available for year 2005.

Both water and substance

leaching (not shown) were pretty

well predicted.

Nice surprise :-)

measured vs. simulated

9

Buckets model implementation

file

"Field TCPP with air mit Graphs ORIGINAL.xlsx"

Field TCPP with air mit Graphs.xls

Exercise day 2: Cascade and buckets model

You need the file Exercise day 2 and the paper Trapp and Eggen ESPR 2013.pdf

as well as the SI

1) The file named Field TCPP with air mit Graphs ORIGINAL.xlsx is the original

file used to make the figures in the paper. You do not need it, except for

comparison.

The file Field TCPP with air mit Graphs DAMAGED.xlsx is a file where some

student (or your teacher) messed around, changed numbers, changed data and

then saved it. Now we need to come back to the original file.

Read the paper. The chemical simulated is TCPP, and it is the field case with

application of 40 tons sewage sludge.

The input data you need are listed in Table 3, and Tables SI2,SI3,SI4.

Damaged are chemical input with sludge (cell H21, Table 3), concentration in rain

(cell F8 to AD8, Table 3), amount of precipitation (m3/d), cells F35 to AD 35

contain false values, see Table SI2, several values of the soil layers A296 to

B375, see Table 3, TC Transpiration coefficient B15 or Table SI 4, final plant

mass cells B23, 31, 41, 48, Table SI4