Water and nutrient uptake by plant root systems Pierre-Henri Tournier Laboratoire Jacques-Louis Lions INRIA ´ equipe ALPINES June 11, 2015 Pierre-Henri Tournier Water and nutrient uptake by plant root systems 1/ 39
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
Sketch of the presentation Mathematical model of soil water and nutrient transport with root 1 uptake A numerical model coupling soil and root water flow 2 Modeling root uptake and root growth using the diffuse domain 3 approach Conclusion 4 Pierre-Henri Tournier Water and nutrient uptake by plant root systems 3/ 39
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
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
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. 0.45 0.14 θ (h) K(h) 0.4 0.12 teneur en eau θ (m 3 .m -3 ) conductivite K (m.j -1 ) 0.35 0.1 0.3 0.08 0.25 0.06 0.2 0.04 0.15 0.02 0.1 0 -1000 -100 -10 -1 -0.1 -1000 -100 -10 -1 -0.1 potentiel h (m) potentiel h (m) Pierre-Henri Tournier Water and nutrient uptake by plant root systems 6/ 39
Soil water movement - Richards equation Here we use the Brooks-Corey model: � h � − λ � � � − λ Θ( h ) := θ ( h ) − θ m h for h ≤ h b = := h b θ M − θ m h b 1 for h ≥ h b , � h � − λ e ( λ ) with e ( λ ) := 3 + 2 K ( h ) = K s λ. h b 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
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
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
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
Soil solute transport and root nutrient uptake The convection-diffusion equation c is solution of the following convection-diffusion equation: ∂ ∂ t ( θ c + 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 ) = κ c 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
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 I m c h ( c ) = K m + c , I m > 0 , K m > 0 . 0.008 h(c) 0.007 0.006 h(c) (mol.m -2 .d -1 ) 0.005 0.004 0.003 0.002 0.001 0 0 0.05 0.1 0.15 0.2 0.25 0.3 c (mol.m -3 ) Pierre-Henri Tournier Water and nutrient uptake by plant root systems 12/ 39
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
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 of 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. Pierre-Henri Tournier Water and nutrient uptake by plant root systems 14/ 39
Water flow within the root system Water mass balance The radial and longitudinal flow equations d ( h r + z ) J r = L r s r ( h s − h r ) , J x = − K x 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: p ( h r , p + z p ) − ( h r , i + z i ) − K x , i : p J r , i l i : p J x , i ( h r , i + z i ) − ( h r , j + z j ) � = − K x , i : j l i : j i j ∈ childs ( i ) J x , j 1 J x , j 2 ( h s , i − h r , i ) + ( h s , p − h r , p ) + L r , i : p 2 π r i : p l i : p . j 1 j 2 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
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
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 d s ( x ) where d s ( x ) is the distance from x to the segment s . � 1 , d ( x ) = 0 , � � 3 d ( x ) Define f c ( x ) = 1 − tanh . Thus f c ( x ) ≈ ε 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 = − λ f 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
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