Water and nutrient uptake by plant root systems

Transcription

1 Water and nutrient uptake by plant root systems Pierre-Henri Tournier Laboratoire Jacques-Louis Lions INRIA équipe ALPINES June 11, 2015 Pierre-Henri Tournier Water and nutrient uptake by plant root systems 1/ 39

2 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 of root systems. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 2/ 39

3 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 Pierre-Henri Tournier Water and nutrient uptake by plant root systems 3/ 39

4 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 Pierre-Henri Tournier Water and nutrient uptake by plant root systems 4/ 39

5 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) =. q + S t 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. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 5/ 39

6 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 θ(h) K(h) teneur en eau θ (m 3.m -3 ) conductivite K (m.j -1 ) potentiel h (m) potentiel h (m) Pierre-Henri Tournier Water and nutrient uptake by plant root systems 6/ 39

7 Soil water movement - Richards equation Here we use the Brooks-Corey model: Θ(h) := θ(h) θ [ ] { m h λ ( ) λ h = := h b for h h b θ M θ m h b 1 for h h b, K(h) = K s [ h h b ] λe(λ) with e(λ) := λ. θ M is the saturated water content. θ m is the residual water content. K s is the saturated hydraulic conductivity. h b is the bubbling pressure head. λ is the pore size distribution index. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 7/ 39

8 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: j r = L r (h s h r ). L r is the radial conductivity for flow from the root surface to the xylem. h s is the soil matric potential at the root surface. h r is the matric potential in the xylem. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 8/ 39

9 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: d(h r + z) j x = K x, dl where K x is the xylem axial conductance. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 9/ 39

10 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. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 10/ 39

11 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) = S c. 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) = κc b, 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. S c represents sources/sinks. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 11/ 39

12 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) = I mc K m + c, I m > 0, K m > h(c) h(c) (mol.m -2.d -1 ) c (mol.m -3 ) Pierre-Henri Tournier Water and nutrient uptake by plant root systems 12/ 39

13 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 Pierre-Henri Tournier Water and nutrient uptake by plant root systems 13/ 39

14 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 interconnected nodes forming a network of root segments Σ, each segment with its own parameters (radius, conductivity,...). Such a representation can be generated by RootBox (Leitner et al., 2010) which implements a root growth model using L-Systems. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 14/ 39

15 Water flow within the root system Water mass balance The radial and longitudinal flow equations J r = L r s r (h s h r ), J x = K x d(h r + 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: K x,i:p (h r,p + z p ) (h r,i + z i ) l i:p = j childs(i) K x,i:j (h r,i + z i ) (h r,j + z j ) l i:j (h s,i h r,i ) + (h s,p h r,p ) + L r,i:p 2πr i:p l i:p. 2 The xylem water potential vector (h r,i ) i is then solution of a linear system, with the right-hand side containing the soil factors h s,i. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 15/ 39 J x, j1 j1 J x,i p i J r,i J x, j2 j2

16 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 f c representative of its geometry and use it to define the sink term as well as to guide the mesh adaptation procedure. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 16/ 39

17 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 d s(x) s Σ where d s (x) is the distance from x to the segment s. Define f c (x) = 1 tanh ( 3d(x) ε ). Thus f c (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): (h s,i h r,i ) + (h s,j h r,j ) J r = L r s r. 2 Build the corresponding sink term S = λf c h l, where h l linearly interpolates h s h r along the segment and with λ > 0 such that S = J r. Ω Pierre-Henri Tournier Water and nutrient uptake by plant root systems 17/ 39

18 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 h t i s be the soil water potential distribution at time t i, hs k and hr k the soil and xylem water potentials at inner iteration k and time t i+1. 1 h 0 s = h t i s. 2 Solve the linear system arising from the problem defined on the tree-like root network with soil factors h k s, obtain h k r. 3 Compute the sink term S using h k s and h k r. 4 Solve the linearized problem corresponding to one Newton step of Richards equation, obtain h s. = hs k + α k (h s hs k ), where 0 < α k 1 is a damping parameter that ensures convergence of the system. 5 h k+1 s 6 If h s h k s > ε, go to 2 with k := k + 1. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 18/ 39

19 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 f c 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 f c (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). Pierre-Henri Tournier Water and nutrient uptake by plant root systems 19/ 39

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

21 Numerical experiment Water uptake of a 20-days-old maize root system Some numbers: Root system composed of 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: 120 Linear speedup # of proc. # of iter. Wall time s s s Wall time (seconds) Number of processors Pierre-Henri Tournier Water and nutrient uptake by plant root systems 21/ 39

22 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 Pierre-Henri Tournier Water and nutrient uptake by plant root systems 22/ 39

23 Mathematical model Configuration of the domain Γ e Ω r n Γ p Γ r Ω s Consider a plant root system Ω r (t) surrounded by the soil domain Ω s (t), t I := [0, T ]. The root surface is represented by the interface between the two domains Γ r (t). The root collar is denoted by Γ p. Γ e is the exterior boundary of the soil domain. The evolution of the domain Ω r (t) over time corresponds to the development of the root system. Let V be the normal velocity of Γ r (t). We only consider root growth: V 0 on I Γ r (t). Pierre-Henri Tournier Water and nutrient uptake by plant root systems 23/ 39

24 Mathematical model The coupled water problem Γ p Γ e n Γ r Soil and root water flow can be coupled in a monolithic way by introducing the following coupled problem: Ω r Ω s find (h, u) such that t (θ(h)).(k(h) (h + z)) = 0 in I Ω s, K(h) (h + z). n = 0 on I Γ e, K(h) (h + z). n = L r (h u) + θ(h)v on I Γ r, h(0, x) = h 0 (x) in Ω s,.(k r (u + z)) = 0 in I Ω r, K r (u + z). n = L r (h u) on I Γ r, u = u c on I Γ p. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 24/ 39 (1)

25 Mathematical model The nutrient problem Γ p Γ e n Γ r In a similar manner, we can define the following problem for nutrient transport with root nutrient uptake: Ω r Ω s find c such that (θ + ϕ (c)) t c.(d c) + q. c = 0 in I Ω s, D c. n = 0 on I Γ e, D c. n = h(c) L r (h u)c + ϕ(c)v on I Γ r, c(0, x) = c 0 (x) in Ω s. (2) Pierre-Henri Tournier Water and nutrient uptake by plant root systems 25/ 39

26 The diffuse domain approach Overview Avoid the generation of a body-fitted mesh by using an implicit representation of the complex geometry of Γ r (t) through the use of a phase field function. Replace the sharp boundary Γ r (t) with a diffuse layer and reformulate the problem on the regular domain Ω = Ω s (t) Ω r (t). The phase field function φ approximates the characteristic function of the domain Ω s (t): 1 χ Ωs φ(t, x) := 1 ( ( )) 3r(t, x) 1 tanh, 2 ε φ r 0 0 ε where x Ω and r(x) denotes the signed distance from x to the boundary Γ r (t), negative in Ω s (t) and positive in Ω r (t). The parameter ε << 1 determines the width of the diffuse interface. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 26/ 39

27 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 1 B(φ)L r (h u) = 0 in I Ω, φk(h) (h + z). n 0 = 0 on I Γ, h(0, x) = h 0 (x) in Ω,.(ψK r (u + z)) ε 1 B(φ)L r (h u) = 0 in I Ω, ψk r (u + z). n 0 = 0 on I Γ e, u = u c on I Γ p, (3) with ψ = 1 φ and where ε 1 B(φ) = ε 1 36φ 2 (1 φ) 2 is an approximation of the surface delta function δ Γr. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 27/ 39

28 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)) t c.(φd c) + φ q. c + ε 1 B(φ) (h(c) L r (h u)c) + ϕ(c) t φ = 0 in I Ω, φd c. n 0 = 0 on I Γ, c(0, x) = c 0 (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. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 28/ 39

29 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 Γ r (t (n) ) is then given by r(t (n), x) = min s Σ (n) (d s (x) r s ), where d s (x) is the distance of x to the segment s and r s is the radius of segment s. b Γ r a r s Root surface Γ r of a root tip represented by segment (a, b) Pierre-Henri Tournier Water and nutrient uptake by plant root systems 29/ 39

30 Adaptive meshing The tetrahedral mesh T h 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). Pierre-Henri Tournier Water and nutrient uptake by plant root systems 30/ 39

31 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 Pierre-Henri Tournier Water and nutrient uptake by plant root systems 31/ 39

32 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 T h 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. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 32/ 39

33 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 of 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 objective function f o as f o = 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 f o 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. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 33/ 39

34 Numerical experiments Nitrate uptake by a growing maize root system with chemotropism Pierre-Henri Tournier Water and nutrient uptake by plant root systems 34/ 39

35 Numerical experiments Nitrate uptake by a growing maize root system with chemotropism Pierre-Henri Tournier Water and nutrient uptake by plant root systems 35/ 39

36 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 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

37 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 Pierre-Henri Tournier Water and nutrient uptake by plant root systems 37/ 39

38 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. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 38/ 39

39 Thank you for your attention! Pierre-Henri Tournier Water and nutrient uptake by plant root systems 39/ 39