Water and nutrient uptake by plant root systems Pierre-Henri - - PowerPoint PPT Presentation

Water and nutrient uptake by plant root systems

Pierre-Henri Tournier

Laboratoire Jacques-Louis Lions INRIA ´ equipe ALPINES

June 11, 2015

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Goals

Simulate water movement in soil and water uptake by plant roots, together with the transport and uptake of nutrients. Explicitly take into account the geometry of a root system. Study how water and nutrient uptake is affected by the type and shape

• f root systems.

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Sketch of the presentation

1

Mathematical model of soil water and nutrient transport with root uptake

2

A numerical model coupling soil and root water flow

3

Modeling root uptake and root growth using the diffuse domain approach

4

Conclusion

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Mathematical model

Soil water movement - Richards equation Root water uptake and transport

Radial flow Axial flow and transpiration

Soil solute transport and root nutrient uptake

The convection-diffusion equation Michaelis-Menten uptake kinetics

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Soil water movement - Richards equation

The Richards equation represents the movement of water in unsaturated

• soils. It is obtained by combining Darcy’s law with the continuity equation:

   ∂θ(h) ∂t = −∇. q + S

• q = −K(h)∇(h + z).

h is the matric head.

• q is the Darcy flux.

θ(h) is the volumetric water content. K(h) is the hydraulic conductivity. z is the elevation. S represents sources/sinks.

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Soil water movement - Richards equation

The nonlinear relationships θ(h) and K(h) are given by empirical models whose parameters depend on the soil physical properties. Several models can be used, such as the Brooks-Corey model or the van Genuchten model.

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

• 0.1
• 1
• 10
• 100
• 1000

teneur en eau θ (m3.m-3) potentiel h (m) θ(h) 0.02 0.04 0.06 0.08 0.1 0.12 0.14

• 0.1
• 1
• 10
• 100
• 1000

conductivite K (m.j-1) potentiel h (m) K(h)

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Soil water movement - Richards equation

Here we use the Brooks-Corey model: Θ(h) := θ(h) − θm θM − θm = h hb −λ :=

h hb

−λ for h ≤ hb 1 for h ≥ hb, K(h) = Ks h hb −λe(λ) with e(λ) := 3 + 2 λ. θM is the saturated water content. θm is the residual water content. Ks is the saturated hydraulic conductivity. hb is the bubbling pressure head. λ is the pore size distribution index.

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Root water uptake and transport

Radial flow

According to the Ohm’s law analogy and neglecting osmotic pressure, the radial flux per unit area into the root from the soil can be written as: jr = Lr(hs − hr). Lr is the radial conductivity for flow from the root surface to the xylem. hs is the soil matric potential at the root surface. hr is the matric potential in the xylem.

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Root water uptake and transport

Axial flow and transpiration

The driving force for water movement through plants originates in leaves. As water evaporates, negative pressure develops in the leaf and creates a large tension that pulls water through the xylem: this is the transpiration-cohesion-tension mechanism. The longitudinal water flow up the root in the xylem is defined as: jx = −Kx d(hr + z) dl , where Kx is the xylem axial conductance.

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Soil solute transport and root nutrient uptake

The evolution of the concentration c of a nutrient N in the soil solution is governed by the following mechanisms: Diffusion of nutrient ions in the soil solution. Dominant for phosphate. Transport of nutrients by mass flow. Dominant for nitrate. Adsorption of nutrient ions in the soil solid phase. Strong for phosphate, negligible for nitrate. Uptake of nutrients by plant roots.

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Soil solute transport and root nutrient uptake

The convection-diffusion equation

c is solution of the following convection-diffusion equation: ∂ ∂t (θc + ϕ(c)) − ∇.(D∇c − qc) = Sc. ϕ is an adsorption isotherm relating the amount of N in the solid phase to the equilibrium concentration in the soil solution. For example, the Freundlich adsorption isotherm is ϕ(c) = κcb, κ > 0 , b ∈ (0, 1). θ is the volumetric water content.

• q is the Darcy flux.

D is the diffusion coefficient of the nutrient in the soil solution. Sc represents sources/sinks.

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Soil solute transport and root nutrient uptake

Michaelis-Menten uptake kinetics

Active uptake of nutrients by roots can be described by Michaelis-Menten kinetics: the uptake rate h at the root surface is related to the concentration in the soil solution and is given by h(c) = Imc Km + c , Im > 0 , Km > 0.

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.05 0.1 0.15 0.2 0.25 0.3 h(c) (mol.m-2.d-1) c (mol.m-3) h(c)

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A numerical model coupling soil and root water flow

Representation of the root system Water flow within the root system Coupling soil and root water flow Unstructured mesh adaptation Numerical experiment

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Representation of the root system

as a tree-like network of root segments

We consider that the root system is composed of cylindrical root segments. The geometry of the root system can then be represented as a series of in- terconnected nodes forming a network

• f root segments Σ, each segment with

its own parameters (radius, conductiv- ity, ...). Such a representation can be generated by RootBox (Leitner et al., 2010) which implements a root growth model using L-Systems.

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Water flow within the root system

Water mass balance

The radial and longitudinal flow equations Jr = Lrsr(hs − hr), Jx = −Kx d(hr + z) dl can be used to define the following water mass balance for a given root node i of parent node p in the tree-like structure: − Kx,i:p (hr,p + zp) − (hr,i + zi) li:p = −

• j∈childs(i)

Kx,i:j (hr,i + zi) − (hr,j + zj) li:j + Lr,i:p2πri:pli:p (hs,i − hr,i) + (hs,p − hr,p) 2 .

p i j1 j2

Jx, j1 Jx, j2 Jx,i Jr,i

The xylem water potential vector (hr,i)i is then solution of a linear system, with the right-hand side containing the soil factors hs,i.

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Coupling soil and root water flow

through a characteristic function

Root water uptake is taken into account in the soil model by defining a sink term S in the Richards equation. The usual approach (Doussan et al. ,2006; Javaux et al., 2008) is to compute the sink term by summing contributions of root segments to water uptake in each soil voxel. Our approach aims at defining an accurate sink term whose shape matches the geometry of the root system resolving small-scale phenomena at the individual root level. = ⇒ Build a characteristic function of the root system fc representative of its geometry and use it to define the sink term as well as to guide the mesh adaptation procedure.

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Coupling soil and root water flow

Computing the sink term

For a point x in the domain the distance d from x to the root is computed: d(x) = min

s∈Σ ds(x)

where ds(x) is the distance from x to the segment s. Define fc(x) = 1 − tanh

• 3d(x)

ε

• . Thus fc(x) ≈
• 1,

d(x) = 0, 0, d(x) > ε. We can take ε equal to the radius of the root. Consider the case of a single cylindrical root segment (i,j): Jr = Lrsr (hs,i − hr,i) + (hs,j − hr,j) 2 . Build the corresponding sink term S = −λfchl, where hl linearly interpolates hs − hr along the segment and with λ > 0 such that

• Ω

S = −Jr.

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Coupling soil and root water flow

An iterative algorithm

The coupling between the root and soil models consists in iteratively solving the two problems until convergence. Let hti

s be the soil water

potential distribution at time ti, hk

s and hk r the soil and xylem water

potentials at inner iteration k and time ti+1.

1 h0

s = hti s .

2 Solve the linear system arising from the problem defined on the tree-like

root network with soil factors hk

s , obtain hk r .

3 Compute the sink term S using hk

s and hk r .

4 Solve the linearized problem corresponding to one Newton step of

Richards equation, obtain hs.

5 hk+1

s

= hk

s + αk(hs − hk s ), where 0 < αk ≤ 1 is a damping parameter

that ensures convergence of the system.

6 If ||hs − hk

s || > ε, go to

2 with k := k + 1. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 18/ 39

Unstructured mesh adaptation

Unstructured volume mesh adaptation is a flexible and powerful tool in the case of complex geometries. The tetrahedral mesh is adapted to the variations of the characteristic function fc so as to resolve the geometry accurately and capture high gradients and small scale phenomena expected near the roots (local conductivity drop). The mesh adaptation procedure is an iterative algorithm which consists in

◮ computing the characteristic function fc on the current mesh. ◮ defining a nodal-based anisotropic metric tensor field based on the

interpolation error using the reconstructed Hessian of fc (mshmet, P. Frey).

◮ building a unit mesh for which all edges are of unit length in the

prescribed metric, using local mesh modifications and anisotropic Delaunay kernel (mmg3d, C. Dobrzynski and P. Frey).

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Overview

root water potential hr defined on the tree-like root network slice through the mesh, showing adaptive refinement relative to the characteristic function fc sink term S in the domain supported by the characteristic function isosurfaces of the characteristic function slice of the solution hs to Richards equation in the soil domain Pierre-Henri Tournier Water and nutrient uptake by plant root systems 20/ 39

Numerical experiment

Water uptake of a 20-days-old maize root system

Some numbers: Root system composed of 10611 segments. Adapted finite element mesh composed of 2.6M vertices and 15.2M tetrahedra. Additive Schwarz overlapping domain decomposition method with a two-level coarse grid preconditioner:

# of proc. # of iter. Wall time

16 17 129.15 s 64 15 13.17 s 140 16 5.54 s

5 10 50 120 16 64 140 Wall time (seconds) Number of processors

Linear speedup

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Modeling root uptake and root growth using the diffuse domain approach

Mathematical model

Configuration of the domain The coupled water problem The nutrient problem

The diffuse domain approach Computing the signed distance function Adaptive meshing Parallel implementation Numerical experiments

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Mathematical model

Configuration of the domain Ωs Γr Γe Γp Ωr

• n

Consider a plant root system Ωr(t) surrounded by the soil domain Ωs(t), t ∈ I := [0, T]. The root surface is represented by the interface between the two domains Γr(t). The root collar is denoted by Γp. Γe is the exterior boundary of the soil domain. The evolution of the domain Ωr(t) over time corresponds to the development of the root system. Let V be the normal velocity of Γr(t). We only consider root growth: V ≤ 0

• n

I × Γr(t).

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Mathematical model

The coupled water problem Ωs Γr Γe Γp Ωr

• n

Soil and root water flow can be coupled in a monolithic way by introducing the following coupled problem: find (h, u) such that                      ∂t(θ(h)) − ∇.(K(h)∇(h + z)) = 0 in I × Ωs, − K(h)∇(h + z). n = 0

• n

I × Γe, − K(h)∇(h + z). n = Lr(h − u) + θ(h)V

• n

I × Γr, h(0, x) = h0(x) in Ωs, − ∇.(Kr∇(u + z)) = 0 in I × Ωr, Kr∇(u + z). n = −Lr(h − u)

• n

I × Γr, u = uc

• n

I × Γp. (1)

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Mathematical model

The nutrient problem Ωs Γr Γe Γp Ωr

• n

In a similar manner, we can define the fol- lowing problem for nutrient transport with root nutrient uptake: find c such that        (θ + ϕ′(c))∂tc − ∇.(D∇c) + q.∇c = 0 in I × Ωs, − D∇c. n = 0

• n

I × Γe, − D∇c. n = h(c) − Lr(h − u)c + ϕ(c)V

• n

I × Γr, c(0, x) = c0(x) in Ωs. (2)

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The diffuse domain approach

Overview

Avoid the generation of a body-fitted mesh by using an implicit representation of the complex geometry of Γr(t) through the use of a phase field function. Replace the sharp boundary Γr(t) with a diffuse layer and reformulate the problem on the regular domain Ω = Ωs(t) ∪ Ωr(t). The phase field function φ approximates the characteristic function of the domain Ωs(t): φ(t, x) := 1 2

• 1 − tanh

3r(t, x) ε

• ,

φ χΩs 1 r ε where x ∈ Ω and r(x) denotes the signed distance from x to the boundary Γr(t), negative in Ωs(t) and positive in Ωr(t). The parameter ε << 1 determines the width of the diffuse interface.

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The diffuse domain approach

The approximate problems

We can then approach the original water problem (1) by the following approximate problem: find (h, u) such that                            ∂t(φθ(h)) − ∇.(φK(h)∇(h + z)) + ε−1B(φ)Lr(h − u) = 0 in I × Ω, − φK(h)∇(h + z). n0 = 0

• n

I × Γ, h(0, x) = h0(x) in Ω, − ∇.(ψKr∇(u + z)) − ε−1B(φ)Lr(h − u) = 0 in I × Ω, − ψKr∇(u + z). n0 = 0

• n

I × Γe, u = uc

• n

I × Γp, (3) with ψ = 1 − φ and where ε−1B(φ) = ε−136φ2(1 − φ)2 is an approximation of the surface delta function δΓr .

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The diffuse domain approach

The approximate problems

Similarly, the original nutrient problem (2) is approximated by the following diffuse domain problem: find c such that        φ(θ + ϕ′(c))∂tc − ∇.(φD∇c) + φ q.∇c + ε−1B(φ) (h(c) − Lr(h − u)c) + ϕ(c)∂tφ = 0 in I × Ω, − φD∇c. n0 = 0

• n

I × Γ, c(0, x) = c0(x) in Ω. (4) We can show using the method of matched asymptotic expansions that solutions of the reformulated problems (3) and (4) converge to those of the original problems (1) and (2) when ε → 0.

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Computing the signed distance function

Consider a representation of the root system by a set of segments Σ(t). At the discrete level, new root segments are added by RootBox at each time step as the root system develops. The signed distance r(t(n), x) of a point x to the root surface Γr(t(n)) is then given by r(t(n), x) = − min

s∈Σ(n) (ds(x) − rs) ,

where ds(x) is the distance of x to the segment s and rs is the radius of segment s.

a b rs Γr

Root surface Γr of a root tip represented by segment (a, b)

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Adaptive meshing

The tetrahedral mesh Th of the domain Ω is adapted to the variations of the phase field function φ, so as to resolve the diffuse interface as well as the potentially high gradient of the solution in the vicinity of the interface. Define a metric tensor field based on the interpolation error using the reconstructed Hessian of φ (mshmet, P. Frey). Build a unit mesh for which all edges are of unit length in the prescribed metric (mmg3d, C. Dobrzynski and P. Frey).

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Overview

tree-like network Σ isosurface r = 0 ⇔ φ = 0.5 u at the soil-root interface vertical slice of the mesh horizontal slice of h

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Parallel implementation

Since the mesh is modified at each time step as the root system expands due to root growth, using a domain decomposition method presents additional difficulties and requires efficient load balancing and repartitioning algorithms. Instead, the linear systems are solved by the multifrontal parallel sparse direct solver Mumps. The phase field function evolves at each time step due to new segments being added as the root system develops, and thus at time t(n+1) the mesh Th has to be adapted only in a neighborhood of each new segment in S := Σ(n+1) \ Σ(n). We can then devise an iterative algorithm which consists in computing a subset M ⊂ S of segments that are sufficiently distant from each other, extracting submeshes corresponding to the neighborhoods of the segments in M and performing mesh adaptation on each submesh in

• parallel. This procedure is repeated a few times until all segments in S

have been processed.

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Numerical experiments

Nitrate uptake by a growing maize root system with chemotropism

This example shows the effects of chemotropism on root growth. Chemotropism is included by coupling the model with the implementation

• f growth and tropisms in RootBox.

RootBox simulates root tip response to various types of tropisms through random minimization of an objective function. In this example, we consider a combination of gravitropism and chemotropism by defining the

• bjective function fo as

fo = −λc + z, (5) where λ > 0 represents the relative strength of chemotropism. At time t(n+1), the coupling algorithm simply consists in computing for each active root tip the value of fo at each new potential tip position by linear interpolation of c(n) on the mesh. Then, RootBox generates new root segments based on the best growth direction for each root tip.

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Numerical experiments

Nitrate uptake by a growing maize root system with chemotropism

Vertical slice showing nitrate concentration at different time steps of the simulation

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Numerical experiments

Nitrate uptake by a growing maize root system with chemotropism

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Numerical experiments

Nitrate uptake by a growing maize root system with chemotropism

Some numbers for the last time step: The root system is composed of 10024 root segments. The mesh is composed of 3.9M vertices and 22.7M tetrahedra. Parallel mesh adaptation algorithm: 16 processors, execution time of 1066 s. Assembling and solving the linear systems on 64 processors:

◮ Computing the signed distance r for each quadrature point: 907 s. ◮ 7 and 10 nonlinear iterations for the water and nutrient problems

respectively.

◮ Assembling the linear systems: 6 s on average. ◮ Solving the linear system for the water problem: 96 s on average. ◮ Solving the linear system for the nutrient problem: 33 s on average. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 36/ 39

Numerical experiments

Water and phosphate uptake by a growing maize root system with hydrotropism

Horizontal slice (left) and isosurfaces (right) of the concentration of phosphate in the soil solution

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Conclusion

Such numerical models at the whole root system scale which also resolve local phenomena at the single root scale can help us improve our understanding of plant-soil relationships and can be used for benchmarking simpler, less costly models. Perspectives in regard to parallel computing: extend the domain decomposition method to the diffuse domain problems involving root growth and transient mesh adaptation and design efficient load balancing, repartitioning and parallel remeshing algorithms.

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Thank you for your attention !

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