Computational Science and Scientific Computing
Computational Models for Root Water Uptake
Supervisor: Prof. Paolo Zunino
Master Thesis of:
Giorgio RAIMONDI, 853037
Abstract - Sommario . . . V
Introduction IX
1 Coupled 3D/1D Soil-Root Problem 1
1.1 Model set up . . . 1
1.2 Coupling root microcirculation with interstitial flow . . . 3
1.3 Derivation of the 1D approximation in case of rectilinear axis . . . 3
1.3.1 Model Assumptions . . . 4
1.3.2 Rectilinear Root Model Derivation . . . 5
1.4 Derivation of the 1D approximation for curved roots . . . 8
1.4.1 Preliminary Model Assumptions . . . 8
1.4.2 Derivation of the average velocity profile . . . 9
1.4.3 Model derivation for curved segments . . . 11
1.5 Richards Equation . . . 13
1.6 Full Model and Dimensional Analysis . . . 15
1.6.1 Dimensional coupled equations . . . 15
1.6.2 Boundary conditions . . . 17
1.6.3 Dimensional Analysis . . . 17
2 Numerical Methods 21 2.1 Mixed Formulation . . . 21
CONTENTS
2.1.2 Weak formulation for the root . . . 22
2.1.3 Coupled weak formulation . . . 26
2.2 Discrete Model . . . 26
2.2.1 Time discretization . . . 26
2.2.2 Finite element approximation . . . 27
2.2.3 Fixed Point Method . . . 28
2.2.4 Algebraic System . . . 29
2.3 Model Validation . . . 32
2.3.1 Steady Root Model . . . 32
2.3.2 Steady Soil Model . . . 33
2.3.3 Unsteady Model of the Soil . . . 36
2.3.4 Unsteady Root-Soil Model . . . 43
2.3.5 Equal order approximation P1/P1 FEM . . . 51
3 Coupled Plant-Root Model 53 3.1 Root Architecture . . . 53
3.1.1 Definition . . . 53
3.1.2 Importance in Plant Productivity . . . 55
3.1.3 Main Root Morphology . . . 58
3.2 Plant-Atmosphere interaction . . . 59
3.2.1 Plant-Atmophere Model . . . 59
4 Simulation and results 63 4.1 Intensive exploitation of the soil . . . 63
4.2 Simulations . . . 63
4.2.1 Plant-Atmosphere Interaction . . . 64
4.2.2 Soil-Root Interaction . . . 64
4.3 Discussion of the results . . . 69
Conclusion and future perspectives 71 Appendices 77 A C++ code Documentation . . . 79
Bibliography 97
water flow field and solute transport processes in the soil. With this work, we want to study the impacts of root architecture, plant water uptake mechanisms, and plant transpiration rate on the water flow field in the soil. Therefore, a fully mechanistic model was used to simulate water flow due to pressure gradients in the root–soil continuum by coupling a three-dimensional Richards equation in the soil with 1D formulation of the root’s flow equation. The calculation of plant transpiration is based on hypotesis that the stomatal conductance of leaves is de-termined by the biochemical demand of photosynthesis.
In order to simulate these effects, we have built and tested a computational solver based on the C++ finite element library GetFEM++. The solver is able to sim-ulate fluid transport in a permeable ground perfused by a root network with arbitrary topology. This software may be a valid support for the analysis of prob-lem of soil-root interactions. Possible applications are: (i) the analysis of the intensive ground exploitation in agriculture and the effects on ground waters, (ii) plants competitions in a semi-arid zone or (iii) modeling plant root growth.
Key words: Richards equation; Root Water Uptake simulation; Curved 1D Flow model; Embedded multiscale approch; Soil-Root interaction; Plants drying effect; Photosynthesis; Root Architecture
tori che influenzano il campo di flusso d’acqua ed i processi di trasporto di sostanze nel terreno. Con questo lavoro, vogliamo andare a studiare l’impatto che l’architettura del sistema radicale, i meccanismi di assorbimento dell’acqua e come la traspirazione delle piante vanno ad influenzare il flusso d’acqua sotter-raneo. Pertanto, abbiamo utilizzato un modello totalmente meccanicistico atto a simulare il flusso lungo gradienti di pressione nel continuum radici-terreno, accop-piano l’equazione tridimensionale di Richards per terreni insaturi con l’equazione monodimensionale per il flusso nelle radici. Il modello sulla traspirazione di pianta invece `e basato sull’ipotesi che la conduttanza stomatica sia determinata tramite una formulazione ottimizzata della domanda biochimica per la fotosintesi.
Per la simulazione di questi modelli, abbiamo programmato un software risolutore basato sulla libreria ad elementi finiti GetFEM++, nel linguaggio di program-mazione C++. Il solutore cos`ı ottenuto risulta in grado di simulare il trasporto di fluidi in un terreno permeabile che presenta al suo interno un rete radicale con topologia arbitraria. Ci`o rende il software un supporto valido per l’analisi di problemi di interazione tra radici e terreno. Possibili applicazioni sono: (i) lo studio dello sfruttamento intensivo del suolo in agricoltura e gli effetti sulle falde sotterranee, (ii) la competizione tra piante in regioni semi aride o (iii) la modell-izzazione della crescita delle radici della pianta.
Parole Chiave: Equazione di Richards; Simulazione di assorbimento radicale; Modello di FLusso1D a rami curvi; Approccio Multiscala incorporato; Interazione Radici-Terreno; Prosciugamento del terreno; Processo di Fotosintesi; Architettura radicale
Forest ecosystems provide many economic, ecological and social benefits and play a key role in regulating the energy, carbon, and water fluxes between the bio-sphere and the atmobio-sphere. Soil water is extracted by plant roots, flows through the plant vascular system and evaporates from the plant leaves thus providing a bridge in which soil water reservoir and atmospheric water vapor concentration in-teract. Root water uptake (RWU) controls the water dynamics in the subsurface, thereby affecting plant water availability, soil water content, and the partitioning of net radiation into latent and sensible heat fluxes thereby impacting atmospheric boundary layer dynamics. Yet, despite its documented importance, a number of issues remain when representing RWU in hydrological and atmospheric models, and addressing a subset of these issues frames this work. Among these issues is the representation of RWU when competition among trees for available root-water occurs. Such competition is rarely accounted for in conventional ecological and hydrological models. One of the barriers to progressing on the root-water competi-tion issue is the inherent three-dimensional nature of the problem. Modeling RWU requires coupling plant transpiration and photosynthesis together with a three-dimensional evolving soil moisture field. The importance of a three-three-dimensional perspective has been recently underlined spawning a number of simulations of water flow through soil and roots using a root hydraulic network. This approach includes detailed plant-scale models based on explicitly resolved root architecture coupled with the three-dimensional Richards equation for water flow in the soil-root system of an isolated single small plant or seedling. Because the precise soil-root
CONTENTS
architecture for multiple interacting trees is rarely known a priori, and given the computational burden involved, a root architecture approach is not yet feasible for large scale hydrological simulations. An intermediate approach that retains the 3D properties of the problem and yet provides a numerically-viable simplified RWU approach is needed when exploring the interplay of hydrological, physiolo-gical, and ecological mechanisms at the watershed scale.
The main goal of this work is to develop a general-purpose finite element solver for large scale simulation of a mechanistic 3D model of RWU. Thanks to dimensional model reduction techniques, the root is described as a one-dimensional (1D) man-ifold immersed in a three-dimesional (3D) model of the soil. Roots can be seen as concentrated sources, to reduce the computational cost of simulations. However, concentrated sources lead to singular solutions that still require computationally expensive meshes to guarantee accurate approximation. The main computational barrier consists in the ill-posedness of restriction operators applied on manifolds with co-dimension larger than one. We overcome the computational challenges of approximating PDEs on manifolds with high dimensionality gap by means of nonlocal restriction operators that combine standard trace with mean values of the solution on low dimensional manifolds.
The outline is as follows. In the first two Chapters we provide the model for the root-soil interaction and the numerical methods at the basis of our software. In particular, Chapter 1 contains the derivation of a generic multiscale PDE model of fluid exchange between roots with general geometry and soil with special at-tention to the 3D/1D reduction of the roots problem and the non linearities of Richards equation. In Chapter 2 we provides the numerical methods with dual mixed formulation for both the roots and the soil flow problems and the the finite elements methods with the algebraic formulation. At the end of the Chapter we discuss some simple results of prototype application of these methods. Chapter 3 is dedicated to description of the importance of root architecture and photos-intensys process on the RWU. At the end of Chapter is shown a model for the computation of the transpiration flow based on the idea that stomatal conduct-ance of leaves is determined by an optimize biochemical demand formulation for photosynthesis. At the end, in Chapter 4 both models are coupled and used to
show the potential of the computational model for an application like the intensive exploitation in agriculture. Futher details about the code are given in Appendix A.
1
Coupled 3D/1D Soil-Root Problem
In this work we are going to study the fluid transport problem in a permeable ground perfuse by a draining root network with arbitrary topology. In the first section we state the problem in the bulk and on the immersed manifold as a system of PDE’s. In the Section 1.2 we explain the embedded multiscale approach to couple the two problems leaving on separate scales. Then in the Sections 1.3, 1.4 we simplify the root problem by proposing a reduced 1D model in place of a generic 3D description, able to take in account of the curved geometry of the network and the non linearity due to the Navier-Stokes equation. In Section 1.5 we are exploit the non linearities for the porous media’s equation due to the unsaturated problem. Finally, in Section 1.6 we derive the ultimate dimensionless formulation of the coupled problem, that is the starting point for the upcoming analysis.
1.1
Model set up
The domain in R3where the model is defined is composed by two parts, Ω
tand Ωv,
denoting the ground interstitial volume and the root bed respectively. Assuming thet the roots can be described as cylindrical tubes, we denote with Γ the outer surface of Ωv while Λ indicates the 1D manifold representing the root centerline.
The root radius R is generally subject to change in the network.
CHAPTER 1. COUPLED 3D/1D SOIL-ROOT PROBLEM
medium, descibed by the Richard’s law. It reads: ut= −
1
µK(ψt) 5 (ψt+ z); (1.1)
∂θ
∂t(ψt) + 5 · ut = 0 (1.2)
where ut is average filtration velocity vector in the tissue, K is the non linear
permeability tensor, µ is the viscosity of the fluid, θ is thewater content of porous medium, z is the vertical coordinate, and ψt is the pressure head in the porous
medium which is defined as ψt =
pt
ρg, where pt is the pressure of the fluid, ρ is the density of the fluid and g is the gravitational accelleration. Recall that in the simple isotropic case the permeability tensor is given by K = kI, being k the scalar permeability function and I the identity tensor.
Concerning the roots, we start assuming a steady incompressible Navier-Stokes model for water flow with a reactive term due to friction forces, namely:
ρ(uv · 5)uv− µ 4 uv+ 5pv + Kauv = 0 (1.3)
5 ·uv = 0 (1.4)
where uv is the flow velocity, pv is the pressure, ρ the density of the fluid,
µ is the fluid viscosity and Ka is the resistence coefficient due to friction. We
notice that not all the terms of equation 1.3 have the same order of magnitude. On one hand, if the inertial forces are relevant, then the first term dominates. On the other hand if the friction forces are important, we expect that the first term becomes small and the last one is large. The latter scenario is the most appropriate to modeling the plant roots.
1.2
Coupling root microcirculation with
inter-stitial flow
At this stage the two problems are completely uncorrelated. Indeed to close the problem we need to impose the continuity of the flow at interface Γ = ∂Ωv∩ ∂Ωt,
namely:
uv· n = ut· n = Lp(pv − pt), uv· τ = 0, (1.5)
where n and τ are the outward unit normal vector and the unit tangent vector on surface Γ, respectively, while Lpis the hydraulic conductivity of the root wall. This
equation together to the previous ones identify a fully three-dimensional model able to capture approximation of the phenomena we are interested in. However, many technical difficulties arise in the numerical approximation of the coupling between a complex network with surrounding volume. To this purpose, we adopt the multiscale approach to avoid the complex 3D geometry of the network. We exploit the Immersed Boundary Method (IBM) combined with the assumption of large aspect ratio between root radius and capillary axial length. More precisely, we apply a suitable rescaling of the equation and let the capillary radius go to zero (R → 0).
As a consequence, the three-dimensional description of the roots is reduced to a simplified one-dimensional representation by replacing the immersed interface and the related interface conditions with an equivalent mass source, namely:
ut· n = f (pv, pt) on Γ, (1.6)
being f the flux per unit area released through surface Γ: it is a point-wise constitutive law for the capillary leakage in term of the fluid pressure.
1.3
Derivation of the 1D approximation in case
of rectilinear axis
Here we introduce the simplest non-linear 1D flow in compliant roots.
The basic equations are derived for a track of root free of bifurcations, which is idealised as a cylindrical compliant tube.
CHAPTER 1. COUPLED 3D/1D SOIL-ROOT PROBLEM
geometry it is handy to introduce a cylindrical coordinate system. Therefore, in the following we indicate with r, θ and z the radial circumferential and axial
unit vectors, respectively, (r, θ, z) being the corresponding coordinate.
1.3.1
Model Assumptions
The basic model is deduced by making the following assumptions:
1. Axial symmetry. All quantities are independent from the angular coordin-ate θ. As a consequence, every axial section z = const is assumed circular on all the root. The tube radius R is a function of z.
2. Fixed cylinder axis. This simply means that the axis is assumed as rectilinear for the whole root. This hypothesis is indeed consistent with that of axial symmetry.
3. Constant pressure on each section. We assume that the pressure P is constant on each section. This comes as result of the following count: using the axial symmetry we have uv(r, θ, z) = uz(r, θ, z)z and splitting the
equation [1.2] by components we obtain (u · ∇)ur− ν4ur+ ∂rp/ρ + Kaur/ρ = 0 (u · ∇)uθ− ν4uθ+ ∂θp/ρ + Kauθ/ρ = 0 (u · ∇)uz− ν4uz+ ∂zp/ρ + Kauz/ρ = 0 ⇒ ∂rp = 0 ∂θp = 0
ρ(u · ∇)uz− µ4uz+
+∂zp + Kauz = 0
(1.7) So we see that p depends only on z.
4. No body forces. We neglect body forces.
5. Dominance of axial velocity. The velocity components orthogonal to the z axis are negligible compared to the component along z. The latter is indicate by uz and its expression in cylindrical coordinates is supposed to
be of the form
uz(r, z) = u(z)s(rR−1(z)) (1.8)
where u is the mean velocity on each section, s : R → R is the velocity profile and reads as
s(y) = γ−1(γ + 2)(1 + yγ). (1.9)
The fact that the velocity profile does not vary in space is in contrast with ex-perimental observations and numerical results carried out with full-scale models. However, it is a necessary condition for the derivation of the reduced model. One may think to s as being a profile representative of an average flow configuration.
A generic axial section will be indicated by S = S(z). Its measure A is given by A(z) =
Z
S(z)
dσ = πR2(z). (1.10)
The mean velocity u is given by
u = A−1 Z
S
uzdσ. (1.11)
We will indicate with α the momentum-flux correction coefficient, (sometimes also called Coriolis coefficent) defined as
α = R Su 2 zdσ Au2 = R Ss 2dσ A (1.12)
where the dependence of various quantities on the spatial coordinates is easy to understand. In general this coefficient will vary in time and space, yet in our model it is taken constant as a consequence of [1.7].
1.3.2
Rectilinear Root Model Derivation
To derive our model, we use an approach consisting of integrating the Navier-Stokes equations on a generic portion P of the root. Moreover we can provide the root wall in parametric form:
CHAPTER 1. COUPLED 3D/1D SOIL-ROOT PROBLEM
where L is the length of the root in the axial direction and n is the out oriented normal to ∂Ωv.
Under the previous assumption, the momentum along z and continuity equations, in the hypothesis of constant viscosity, are
(
ρ(u · ∇)uz− µ4uz+ ∂zp + Kauz = 0
divu = 0 (1.14)
The convective term in the momentum equation has been taken in divergence form because it simplifies the further derivation. Now we are ready to derive our reduced model and we start first from the continuity equation
0 = Z P [divu]dΩ = Z ∂P [u · n]dσ = = Z S(z1) [u · n]dσ + Z S(z2) [u · n]dσ + Z Γ [u · n]dσ = = − Z S(z1) [uz]dσ + Z S(z2) [uz]dσ + Z Γ [f (pt, pv)]dσ (1.15)
using the fact that n = z on S(z1) and S(z2) and the equation [1.4]. Now using
equation [1.5] for the first two integrals and a change of variable on the third one we obtain:
[1.15] = −u(z1)A(z1) + u(z2)A(z2) +
Z z2 z1 [f (pt, pv)]dz = = Z z2 z1 [f (pt, pv) + ∂z(Au)]dz (1.16) since pt = 1 2πR R
∂Sptds is the mean interstitial pressure on the boundary of a
section S and the fundamental theorem of calculus hold. At the end we use the arbitrary of z1 and z2 to obtain
∂z(Au) + f (pt, pv) = 0 (1.17)
Now it’s the turn of the momentum equation. Lets start with the Stokes term: Z P [∂zp]dΩ = Z z2 z1 dz Z S(z) [∂zp]dσ = Z z2 z1 [A(z)∂zp]dz (1.18)
An other simple manipulation, thank to the simple assumption of constant reac-tion coeffincent on each secreac-tion, can be done for the reactive term, which becomes:
Z P [Kauz]dΩ = Z z2 z1 dz Z S(z) [Kauz]dσ = Z z2 z1 [A(z) Kau]dz (1.19) 6
Now we manipulate the advective term in divergence form: Z P [div(uzu)]dΩ = Z ∂P [uzu · n]dσ = − Z S(z1) [u2z]dσ + Z S(z2) [u2z]dσ + Z Γ [uzu · n]dσ (1.20) In order to eliminate the boundary integral we exploit the fact that uz = 0 on
Γ. Subsequently using equation [1.8] and the fundamental theorem of calculus we obtain: − Z S(z1) [u2z]dσ+ Z S(z2)
[u2z]dσ = α[−u(z1)2A(z1)+u(z1)2A(z1)] =
Z z2
z1
α∂z[A(z)u(z)2]dz
(1.21) We finally consider the viscous term:
Z P [4uz]dΩ = Z ∂P [∇uz· n]dσ = − Z S(z1) [∂zuz]dσ + Z S(z2) [∂zuz]dσ + Z Γ [∇uz· n]dσ. (1.22) Here, we neglect the ∂zuz term by the fact that 0 = div(u) = ∂zuz almost
every-where in Ωv. Moreover we split the n vector into two components, the radial
component nr = (n · r)r = nrr and axial component nz = n − nr. Remember
that owing to the cylindrical geometry, n has no component along the circumfer-ential coordinate and, consequently, nz is indeed oriented along z. Thus, we can
write: Z P [4uz]dΩ = Z Γ [∇uz· nr+ ∇uz· nz]dσ. (1.23)
Again, we neglect the ∇uz· nz, which is proportional to ∂zuz. We recall now the
relation [1.8] to write: Z Γ [∇uz· nr]dσ = Z Γ [nr∇uz· r]dσ = Z Γ [uR−1s0(1)nr]dσ = Z z2 z1 [2πs0(1)u]dz (1.24) where we used nrdσ = 2πRdz and indicating s0 as the first derivative of s. Then
using the arbitrary of the extreme on [1.18], [1.22], [1.24] and calling Kr = 2πνs0(1)
we obtain
(Kr+ A Ka)u +
A
ρ∂zp + α∂z(Au
CHAPTER 1. COUPLED 3D/1D SOIL-ROOT PROBLEM
1.4
Derivation of the 1D approximation for curved
roots
1.4.1
Preliminary Model Assumptions
Now we want to derive similar results in the case of curvilinear axis.
To do this we define the parametric arc length Ψ : R → R3 as the axis trajectory, taken such that
Ψ ∈ C3(R)
|Ψ0(z)| = 1 ∀z ∈ [0, L], (1.26)
where L is the length of the arc and the norm is compute as the Euclidean norm. In the following we will use parametric cylindrical coordinate, and we indicate r(z), θ(z) and ψ(z) respectively, the radial, circumferential and axial unit
vec-tors, (r, θ, ψ)(z) being the corresponding coordinates at each position z and, due to the choose of Ψ, we obtain that ψ(z) = Ψ0(z). From now on we will omit the parametric dependence. To derive the model we need these preliminary assump-tions:
1. Circular Section. For each value of the parameter z the intersection between the orthogonal plane to ψ and the root wall is assumed as circular. 2. Constant pressure on each section. The pressure is assumed constant
on each section due to the same reason of the rectilinear case. 3. No body forces. We neglect all forces effect.
4. Dominance of axial velocity. The velocity components orthogonal to the ψ axis are negligible compared to the component along ψ. The latter is indicate by uψ and its expression is supposed to be of the form:
uψ(r, θ, ψ) = u(ψ)Φ(r, θ, ψ), (1.27)
where u is the mean velocity on each axial section and Φ : R3 → R is the
velocity profile on the curvilinear root. As for the linear case, this profile can also be seen as the average flow configuration.
1.4.2
Derivation of the average velocity profile
To derive the velocity profile we are going to study it in the simpler case of a curve root with fixed curvature and we will generalize it in a more complete problem adding more hypothesis.
Surely there are many possible ways to choose the configuration of Φ, but we are going to use the following second order approximation:
Φ(r, θ, ψ) = s(rR−1)(1 + ar cos θ + br sin θ + cr2cos θ sin θ + dr2cos2θ + er2sin2θ), (1.28) where a,b,c,d and e are constants depending on the curvature and s is the velo-city profile of the linear case. To improve the significance of this configuration is possible to add higher order term and compute the respectively coefficients. Now to better understand the procedure, let’s take for example a uniform cur-vilinear motion. On it we have that the tangential velocity proportional to the distance from the center, so in our problem we are going to expect similar result. Given κ, the curvature of the arc, defined as κ = |Ψ00|, and the centripetal unit vector N := Ψ00/κ, we have that the center C0 of the osculating circle is the point
in direction N(z) whose distance from Ψ(z) is R0, where R0 is the curvature
CHAPTER 1. COUPLED 3D/1D SOIL-ROOT PROBLEM
Now we have to impose suitable condition to our velocity profile in order to com-pute the value of all the constants:
1. Choice of θ. To simplify our calculation we assume that on each section all the vectors with θ = 0 are in N direction.
2. Symmetry of the profile. In each section the profile must be symmetric with respect to the axes connecting C0 and Ψ(z). So:
Φ(r, θ, ψ) = Φ(r, −θ, ψ) ∀r, θ, ψ and we obtain that b = c = 0.
3. Zero velocity in the Center. For the linear dependence of the velocity profile with the distance from the center of the osculating circle, our profile must be zero in C0 = (r = 1/κ, θ = 0, ψ).
Φ(C0) = 0 → (1 + a/κ + d/κ2) = 0 → d = −aκ − κ2
4. Linear dependence. Due to the assumptions that the velocity profile is linear dependent to the distance from the center of the osculating circle we have that all the points with distance 1/κ from it must have the same dependence.
All these are the points having the following property: W = {(r, θ) : r = 2 cos θ κ , θ ∈ [− π 2; + π 2]}. Moreover we have that
Φ(r = 0, θ, ψ) = s(0) 10
so ∀(r, θ) ∈ W then Φ(r, θ, ψ) = s(r/R). It follows that ∀(r, θ) ∈ W : 0 = ar cos θ+dr2cos2θ+er2sin2θ = 2a
κcos 2 θ+4 d κ2 cos 4 θ+4 e κ2 cos 2 θ sin2θ. Now for θ = ±π
2 the equation is verified. In the other cases we can divide all by 2 cos2θ/κ2, to obtain:
0 = aκ + 2d cos2θ + 2e sin2θ ∀θ ∈ (−π 2; +
π 2).
To find the value of the constant, we test it on two particular cases: θ = π/4, θ = π/3.
For θ = π/4, using assumption 3: 0 = aκ + 2d(1
2) + 2e( 1
2) = aκ + d + e = aκ − κ
2− aκ + e = e − κ2.
For that e = κ2. Instead for θ = π/3, using assumption 3 and the previous
result: 0 = aκ + 2d(1 4) + 2e( 3 4) = aκ + d 2 + 3e 2 = aκ − κ2 2 − aκ 2 + 3κ2 2 = aκ 2 + κ 2 So a = −2κ and d = κ2.
Due to this assumptions, our velocity profile is of the form:
Φ(r, θ, ψ) = s(rR−1)(1 + r2κ2− 2κr cos θ). (1.29) In a general geometry we have that the curvature is dependent on the trajectory position: κ = κ(ψ). But thanks to the assumption of regularity on the trajectory, Ψ ∈ C3(R), then κ(ψ) ∈ C1(R) and our velocity profile still holds, by continuity of κ:
Φ(r, θ, ψ) = s(rR−1)(1 + r2κ2(ψ) − 2κ(ψ)r cos θ). (1.30)
1.4.3
Model derivation for curved segments
Now that we have the velocity profile, the derivation of the equation is the same of the rectilinear case, the only difference is for the coefficients which depend on
CHAPTER 1. COUPLED 3D/1D SOIL-ROOT PROBLEM
Figure 1.1: κR = 0 Figure 1.2: κR = 0.1
Figure 1.3: κR = 0.3 Figure 1.4: κR = 0.5
Figure 1.5: κR = 1.0
Figure 1.6: Velocity profile with imposition of Poiseuille flow (γ = 2) for different curvature value.
the profile.
In fact the integral of an integrable function f : R3 → R on trajectory Ψ is
Z Ψ [f (ψ)]dψ = Z L 0 [f (Ψ(z))|Ψ0|]dz = Z L 0 [f (Ψ(z))]dz. (1.31)
To denote our portion P we can use two arbitrary position z1 and z2 ∈ [0, L],
z1 < z2 and we obtain the same results.
The first difference is on equation [1.10]: α∗∗= R S[u 2 ψ]dσ Au2 = R S[Φ 2]dσ A = R S[s 2(1 − 4κr cos θ + 4κ2r2cos2θ + 2κ2r2+ κ4r4)]dσ A .
Now for the periodicity of cos θ: α∗∗ = R S[s 2(1 + 4κ2r2cos2θ + 2κ2r2+ κ4r4)]dσ A = α + κ 2β + κ4γ, (1.32) 12
where α is the momentum-flux correction coefficient for the rectilinear case, β = R S[2s 2r2(1 + cos2θ)]dσ A γ = R S[s 2r4]dσ A (1.33)
On the other hand, for the permeability term we have: Z P [4uψ]dΩ = Z Γ [u∂rpnr]dσ = Z Γ {u∂r[s(rR−1)(1 − 2κr cos θ + κ2r2]}nrdσ = = Z Γ
{u[R−1s0(rR−1)(1 − 2κr cos θ + κ2r2) + (κ2r − 2κ cos θ)s(rR−1)]} =
= Z
Ψ
Z 2π
0
{uR[R−1s0(1)(1 − κ cos θ + κ2R2) + (κ2R − 2κ cos θ)s(1)]}dθdψ. Now using the fact that s(1) = 0, the periodicity of cos θ and doing a change of variable we obtain Z P [4uψ]dΩ = Z z2 z1 [2πs0(1)(1 + κ2R2)u(t)]dt, (1.34)
so the permeability term becomes Kr∗∗= Kr(1 + κ2R2), where Kr is the
permeab-ility coefficient of the rectilinear case. Assembling all these terms we obtain:
(Kr∗∗+ A Ka)u + A ρ∂zpv+ ∂z(α ∗∗ Au2) = 0. (1.35)
1.5
Richards Equation
In this section we manipulate the Richard equation in order to have only two variable for the porous media, the filtration velocity ut and the pressure pt.
Sub-sequiently we will address the non linear terms of these equations.
Flow in the vadose zone has many complications as the parameters that control the flow are dependent on the saturation of the media, leading to a non-linear problem. As we said in Section 1.1 in this zone the flow is referred to as un-saturated flow and it is described by Rihcards equation. The groundwater flow equation has a diffusion term , as well as an advection term that is related to gravity and only acts in the z-direction. There are several forms of this equations, but in this paper we considered only two: the Richards mixed-formulation, given
CHAPTER 1. COUPLED 3D/1D SOIL-ROOT PROBLEM
with equations [1.1],[1.2], and the most commonly used Head-Based form, which reads as: ut= − k(ψt) µ ∇(ψt+ z); (1.36) C(ψt) ∂ψt ∂t + ∇ · ut = 0. (1.37) where C(ψt) = ∂θ ∂ψt
and is called the specific moisture capacity function. The popularity of this second formulation is due to the fact that the time-stepping is explictly written in terms of ψt.
What we need to do now is to adapt this formulation to our coupled problem. Passing fastly from the integral form to the differential form, from the divergence term we can exploit the continuity condition at the root’s wall. So the Richards Head-Based equations, after the substitution ψ = pt
ρg are rewritten as: ut= − k(pt/ρg) µρg ∇(pt+ ρgz); (1.38) C(pt/ρg) ρg ∂pt ∂t + ∇ · ut− 2πRLp(pv− pt) = 0. (1.39) The last thing to close the interstial problem is to introduce suitable empirical funtions for the Hydraulic Conductivity and the Water Content. In fact an import-ant aspect of unsaturated flow is noticing that both water content and hydraulic conductivity are functions of pressure head (ψ). There are many empirical ex-pressions used to relate these parameters, including the Brooks-Corey model and the Van Genuchten model. The Van Genuchten model is slightly more popular because there are no discontinuities in the functions, unlike the Brooks-Corey model. A version of this model, proposed by Celia [Celiaetal., 1990], is written as: θ(ψ) = α(θs− θr) α + |ψ|β + θr (1.40) k(ψ) = Ks A A + |ψ|γ (1.41)
where the parameters α, β, γ and A are fitting parameters that are often assumed to be constant in the media; θr and θsare the residual and the saturated moisture
contents; and Ks is the saturated hydraulic conductivity. An example of the
curves using Van Genucten parameters from Celia is shown in figure 1.7. Small 14
changes in pressure head can change the hydraulic conductivity several orders of magnitude; as such, k(ψ) is a highly non linear function. The water content curve is also highly nonlinear as saturation can change dramatically over a small range of pressure head values. It shoul be noted that these functions are only valid when the pressure-head ψ is negative. This condition is directly impose by the moisture capacity function, which now reads as:
C(ψ) = − α(θs− θr) (α + |ψ|β)2β|ψ|
β−1
sign(ψ) (1.42)
Remember that when the media is fully saturated k = Ks and θ is equal to the
porosity.
Figure 1.7: Water content and Conductivity depending on the pressure head.
1.6
Full Model and Dimensional Analysis
In this last section we resume the results of the derivation of the model and for the generalization of the problem we propose some suitable boundary condition and we adimensionalize the equations.
1.6.1
Dimensional coupled equations
Now that we have derived the 1D model equations for a network with only a single branch, we need to generalize them to a more complex topology.To this purpose,
CHAPTER 1. COUPLED 3D/1D SOIL-ROOT PROBLEM
we decompose the network in Λi branches, i = 1, ..., N . The branch is
paramet-rized by the arc length Ψi; and the tangent unit vector defined as λi := Ψ0i over
each branch, accounting for an arbitrary branch orientation. Differentiation over the branch is defined using the tangent unit vector, namely ∂si := ∇ · λi on Λi,
i.e. ∂si is the projection of ∇ along λi.
The last thing we need to take in account, it is that the root is a filled media, which doesn’t allow the water to flow freely, where the radical filling can be mod-elled as porous medium, with almost constant velocity profile and high friction term. As said in the first section, that make the advective term of root equations negligible, in order to mantain the laminar condition of Darcy. Moreover, through our model, we can catch the almost flat profile by imposing an high exponent γ (γ ≥ 10) for the velocity profile (as the one shown in Figure 1.8).
So the coupled soil-root 3D-1D problem’s equations read as: ut+ 1 µρgK(pt)∇(pt+ ρgz) = 0 on Ωt 1 ρgC(pt) ∂pt ∂t + div(ut) − 2πRLp(pv− pt) = 0 on Ωt (Kr∗∗+ A Ka)uv + A ρ∂sipv = 0 on Λi ∀ i = 1, ..., N ∂si(Auv) + 2πRLp(pv− pt) = 0 on Λi ∀ i = 1, ..., N (1.43)
Figure 1.8: Velocity profile for filled root (γ = 20).
1.6.2
Boundary conditions
For the well-poseness of problem [1.23], we have to specify suitable boundary con-ditions (BCs) on both the tissue and root boundary, i.e. ∂Ω and ∂Λ respectively.
Since we aim to present the most generic setting, we assume the tissue interstitium boundary to be partitioned as follows:
∂Ω = Γp∪ Γu, ˚Γp∩ ˚Γu = ∅. (1.44)
As suggested by the apices p and u, we enforce a given pressure distribution on Γp and/or a fixed value for the normal flux over Γu, namely:
pt= gt on Γp, (1.45)
ut· n = β(pt− p0) on Γu, (1.46)
Here p0 represent far field pressure value, while β can be interpreted as an
effect-ive conductivity accounting for layers of tissue surrounding the considered sample. Assuming that the interstitial pressure decay from pt to p0 over a distance
com-parable to the sample characteristic size, D, dimensional analysis shows that a rough estimate of conductivity is β = kt/D. For the pressure datum we require
gt∈ L2(Γp).
Concerning the network, we split the collection of extrema into two subset: on the boundary extrema, εp, we enforce a pressure distribution, on immersed extrema ,
εu, we enforce the flux (hence the roots velocity); namely:
pv = gv on εp (1.47)
πR02uv = β(pv − p0) on εu (1.48)
where gv is a boundary datum for which is required measurability and
square-summability, namely gv ∈ L2(εp), while p0 and β are as above. In particular, in
future applications we will always enforce constant pressure drop Pout
v − Pvin, that
means we will adopt piecewise-constant boundary data.
1.6.3
Dimensional Analysis
Now, let’s rewrite now the problem in dimensionless form in order to highlight the most significant mechanisms governing the flow between microcirculation and
CHAPTER 1. COUPLED 3D/1D SOIL-ROOT PROBLEM
biological tissue, under the assumption of constant radius over each branch of the root. First, we indentify the characteristic dimension of our problem: length, velocity and pressure are chosen as primary variables for the analysis. The corres-ponding characteristic values are: (I) the average spacing between capillary root d, (II) the average velocity in the capillary bed U and (III) the average pressure in the interstitial space P .
Correspondingly, the dimensionless groups affecting our equations are:
R0 = R d dimensionless radius, κ0 = κ d dimensionless curvature, kt= kP
µρgU d dimensionless interstitial permeability,
Q = 2πR0L p
P
U dimensionless wall permeability,
kv = πR04 Kr P d U = πR04 2µ(γ + 2) P d
U dimensionless root permeability,
ka=
1 Ka
P d
U dimensionlees reactive root permeability,
c = C
ρg dimensionless moisture capacity function.
Therefore, the coupled dimensionless problem of microcirculation and tissue interstitium reads as follows:
a) 1 kt(pt) ut+ ∇(pt+ ρgz) = 0 on Ωt b) c(pt) ∂pt ∂t + div(ut) − Q(pv − pt) = 0 on Ωt c) πR 02 kv (1 + κ02R02) + πR 02 ka uv+ ∂sipv = 0 on Λi ∀ i = 1, ..., N d) ∂si(uv) + Q 2πR02(pv− pt) = 0 on Λi ∀ i = 1, ..., N (1.49)
2
Numerical Methods
For complex geometrical configurations explicit solutions of the problem [1.49] are not available. The only way to apply such a model to real cases is to resort to a more general variational framework and consequiently to numerical simulations. That is done in the first two section, in which we exploit the Finite Element methods for the space discretization and the Euler Backward method for the time discretization. Over this discrete model we built a solver, written in C++ language, and thanks to it, in the last section, we present the numerical results for the validation of the model.
2.1
Mixed Formulation
The aim of this section is to propose a mixed finite element approximation of the coupled differential problem discribed in Chapter 1. Specifically, we derive a dual weak formulation of both the tissue and root problems within the context of the Saddle-Point Theory.
2.1.1
Weak Formulation for ground interstitium
First of all we need to take suitable spaces for the velocity and the pressure in the tissue.
Vt := Hα,βdiv(Ωt) and Qt = L2α(Ωt) (2.1)
with α, β ∈ (−1, 1).
suffi-CHAPTER 2. NUMERICAL METHODS
ciently smooth functions and integrate over the volume Ωt, namely:
Z Ωt h1 kt ut· vt i dΩ + Z Ωt h ∇pt· vt i dΩ = − Z Ωt h ρg∇z · vt i dΩ (2.2) Z Ωt h c∂pt ∂tqt i dΩ + Z Ωt h div(ut)qt i dΩ − Q Z Ωt h (pv − pt)δΛqt i dΩ = 0 (2.3)
Note that, to simplify the notation, from now on we will omit the space de-pendece of the Conductivity and the Moisture Capacity Function. Now applying the Green’s theorem to [2.2] we obtain an anti-symmetric formulation of Darcy’s problem in the tissue:
Z Ωt h1 kt ut·vt i dΩ− Z Ωt h div(vt)pt i dΩ+ Z ∂Ωt h ptvt·n i dσ = − Z Ωt h ρg∇z·vt i dΩ (2.4) Finally, let’s discuss the treatment of the boundary term. Therefore, thanks to the linearity of the integral we can rewrite the integral on Γp and Γu, which are
the boundary where are imposed respectively the pressure and velocity boundary conditions. Z ∂Ωt h ptvt· n i dσ = Z Γp h gtvt· n i dσ + Z Γu h p0vt· n i dσ+ +1 β Z Γu h (ut· n)(vt· n) i dσ − Z Ωt h ρg∇z · vt i dΩ (2.5)
So the weak formulation for the tissue interstitium reads as: Z Ωt h1 kt ut· vt i dΩ + 1 β Z Γu h (ut· n)(vt· n) i dσ − Z Ωt h div(vt)pt i dΩ = = − Z Γp h gtvt· n i dσ − Z Γu h p0vt· n i dσ (2.6) Z Ωt h c∂pt ∂tqt i dΩ + Z Ωt h div(ut)qt i dΩ − Q Z Ωt h (pv − pt)δΛqt i dΩ = 0 (2.7)
2.1.2
Weak formulation for the root
As for the root problem we start giving a general functional framework. At this point we only require regularity for the root velocity and pressure over each branch in a separate way. Vv = N [ i=1 H1(Λi) and Qv = N [ i=1 L2(Λi). (2.8) 22
Now at first thing we rescale equation [1.49 c] and [1.49 d] by the function πR02(z), therefore we multiply the resulting equations by sufficiently smooth function, vv
and qv, and integrate over the root domain Λ:
Z Λ h 1 kp uvvv i ds + Z Λ h πR02∂spvvv i ds = 0 (2.9) Z Λ h ∂s(uv)qv i ds + Z Λ h Q 2πR02(pv− pt)qv i ds = 0 (2.10)
where, in order to have a simplify notation, we use 1 kp = π2R04 kv (1 + κ02R02) + π2R04 ka
with kp is a resulting permeability. The integration by parts is not trivial
in such a case because the root variables uv and pv may be discontinuous at
multiple junctions. Specifically, we assume that roots pressure is continuous while the difficulty associate with the root velocity remains. Let us consider the second integral of [2.9]. We will derive a proper Green’s formula for the network problem. First, we rewrite the integral over the whole network as a summation of the integrals over single branches, namely:
Z Λ h πR02∂spvvv i ds = N X i=1 Z Λi h πR02∂spvvv i ds.
Let’s assume the root radius to be a step function of the arc length Ψ with constant values over the one-dimensional varieties of Λi:
R0(s) =
N
X
i=1
R0iδΛi,
being δΛi the Dirac delta function on the i − th branch. Thus the integral becomes
Z Λ h πR02∂spvvv i ds = N X i=1 πR02i Z Λi h ∂spvvv i ds
and we can finally apply the standard Green’s formula over the branch Λi:
Z Λ h πR02∂spvvv i ds = N X i=1 πR02i n− Z Λi h pv∂svv i ds +hvvpv iΛ+i Λ−i o = = − Z Λ h πR02i ∂spv i ds + N X i=1 h vvpv iΛ+i Λ−i , (2.11)
CHAPTER 2. NUMERICAL METHODS
were Λ+i and Λ−i represent the inflow and the outflow extrema of Λi according
to the orientation of λi. At this point, we re-organize the local boundary term
in order to collect contributions of different branches affecting the same junction point. Let’s define the set of indexes of junction points, namely:
J := {j ∈ N : sj ∈ Λ, #(Psj) ≥ 2}
where Psj is the patch of the j −th junction node, i.e. the collection of all branches
joining at the node, and # indicates the counting measure. We also need the following disjoint partition of the indexes in Psj. According to the orientation
of λi, for any branching points sj we distinguish branches that are entering the
node, whose contribution to mass conservation is positive, from branches leaving the node, whose contribution is negative. The former are branches whose outflow region coincides with the point sj, while for the latter it is the inflow region:
Pjout := {i ∈ {1, ..., N } : Λ+i ≡ {sj}}, (2.12)
Pjin := {i ∈ {1, ..., N } : Λ−i ≡ {sj}}, (2.13)
for all j ∈ J . At this point we can rewrite the boundary term by isolating the terms relative to inner junction nodes to those relative to outer inflow and outflow nodes, namely: N X i=1 πRi02hvvpv iΛ+i Λ−i = h πR02i vvpv iΛout Λin + X j∈J h X i∈Pout j πR02i (pvvv)|Λ+i − X i∈Pin j πR02i (pvvv)|Λ−i i = =hπR02i vvpv iΛout Λin + X j∈J pv(sj) h X i∈Pout j πR02i vv|Λ+i − X i∈Pin j πR02i vv|Λ−i i (2.14) where Λout and Λinidentify the collection of the inflow and outflow tips of the root network, i.e. non junction points where the tangent unit vector is inward-pointing and outward-pointing, respectively. This collection contains the boundary points, i.e. extrema belonging to ∂Ω, but the inclusion may be strict. Indeed we have to address the issue of immersed tips, i.e. of network extrema belonging to ˙Ω. As for the tissue boundary, we used the apex p and u to identify the subset on which we will enforce pressure and velocity boundary conditions, respectively. For the
sake of simplicity, we also implicitly assume that boundary points and bifurcation or branching points can not coincide. Obviously this is not a strong limitation, it only allows us to easily write the equations by avoiding further indexes.
Finally we have the following integration formula: Z Λ h πR02∂spvvv i ds = − Z Λ h πR02i pv∂svv i ds + h πRi02vvpv iΛout Λin + +X j∈J pv(sj) h X i∈Pout j πR02i vv|Λ+i − X i∈Pin j πR02i vv|Λ−i i (2.15)
After this manipulation, we gain a term that looks like the weak counterpart of the mass conservation constrain (written for a generic test function vv). It is now
explained why we pre-multiplied the roots problem with the function πR02. In fact, thanks to that trick, the desired constraint come to light in a natural way. The conservation of the local flow reate at root junctions can be indeed expressed in term of the above notation as follows:
X i∈Pout j πRi02vv|Λ+i = X i∈Pin j πR02i vv|Λ−i ∀j ∈ J (2.16)
Therefore, it is reasonable to weakly enforce it in the variational formulation by multiplying it with the pressure test functions qv, which act as a Lagrangian
multiplier for this constraint, namely: X j∈J qv(sj) h X i∈Pout j πRi02vv|Λ+i − X i∈Pin j πR02i vv|Λ−i i = 0 (2.17)
So the full weak formulation of the root problem reads as: Z Λ h 1 kp uvvv i ds + 1 β h π2R04uvvv i εu − Z Λ h πR02i pv∂svv i ds+ +X j∈J pv(sj) h X i∈Pout j πR02i vv|Λ+i − X i∈Pin j πRi02vv|Λ−i i = = −hπR02p0vv i εu −hπR02gvvv i εp , (2.18) Z Λ h ∂s(uv)qv i ds + Z Λ h Q 2πR02(pv− pt)qv i ds+ +X j∈J qv(sj) h X i∈Pout j πR02i vv|Λ+i − X i∈Pin j πR02i vv|Λ−i i = 0 (2.19)
CHAPTER 2. NUMERICAL METHODS
2.1.3
Coupled weak formulation
At this point combining all the equation we obtain the whole weak formulation of our 3D/1D coupled model of fluid exchange between the roots network and the ground intestitium. The variational formulation of the problem read as:
Find ut ∈ Vt, pt∈ Qt, uv ∈ Vv, pv ∈ Qv s.t.: a) (1 kt ut, vt)Ωt + 1 β(ut· n, vt· n)Γu− (pt, div(vt))Ωt = = −(gt, vt· n) − (p0, vt· n)Γu− ρg(∇z, vt)Ωt ∀vt∈ Vt b) (c∂pt ∂t, qt)Ωt + (div(ut), qt)Ωt − (Q(pv− pt), qt)Λ= 0 ∀qt ∈ Qt c) (1 kp uv, vv)Λ+ 1 β[π 2R04 uvvv]εu− (πR 02 pv, ∂svv)Λ+ hvv, pviJ = = −[πR02p0vv]εu− [πR 02 gvvv]εp ∀vv ∈ Vv d) (πR02∂suv, qv)Λ+ (Q(pv − pt, qv)Λ− huv, qviJ = 0 ∀qv ∈ Qv (2.20) Where hvv, pviJ := X j∈J pv(sj) h X i∈Pjout πR02i vv|Λ+i − X i∈Pin j πR02i vv|Λ−i i
2.2
Discrete Model
In this section we purpose a time discretization for the partial time derivative present in the formulation [2.20 b]. After that we set the finite elements spaces for the discretization of spatial coordinates, in order to write the full discrete model.
2.2.1
Time discretization
How to discretize in time the Richards equation was argument of several studies vadose ground. In this work we are going to use the Backward Euler Method, which reads as:
cn+1p n+1 t − pnt ∆t + div(u n+1 t ) − Q(p n+1 v − p n+1 t ) = 0 (2.21)
where the apex indicate the time step node and cn+1 = c(pn+1t /ρg). The advant-age of this choose are that it has an easy implementation and it doesn’t require
any stabilazation conditions. The drawback is that its convergence order is not optimal, so to optain an accurate solution we will need a small discretization in time, which increase the computational cost. So our variational form reads as:
a) ( 1 ktn+1u n+1 t , vt)Ωt + 1 β(u n+1 t · n, vt· n)Γu− (p n+1 t , div(vt))Ωt = = −(gt, vt· n) − (p0, vt· n)Γu− ρg(∇z, vt)Ωt ∀vt∈ Vt b) 1 ∆t(c n+1 pn+1t , qt)Ωt + (div(u n+1 t ), qt)Ωt − (Q(p n+1 v − p n+1 t ), qt)Λ= = 1 ∆t(c n+1pn t, qt)Ωt ∀qt ∈ Qt c) (1 kp un+1v , vv)Λ+ 1 β[π 2 R04un+1v vv]εu − (πR 02 pn+1v , ∂svv)Λ+ +hvv, pn+1v iJ = −[πR02p0vv]εu− [πR 02 gvvv]εp ∀vv ∈ Vv d) (πR02∂sun+1v , qv)Λ+ (Q(pn+1v − ptn+1, qv)Λ− hun+1v , qviJ = 0 ∀qv ∈ Qv (2.22)
2.2.2
Finite element approximation
The Finite Element Method is adopted to approximately solve [2.22]. We denote by Tth an admissible family of partitions of Ω into tetrahedrons K that satis-fies the usual conditions of a conforming triangulation of Ω. We use discontinu-ous piecewise-polynomial finite elements for pressure, Hdiv-conforming
Raviart-Thomas for velocity, namely
Ykh := {wh ∈ L2(Ω) : wh|K ∈ Pk−1 ∀K ∈ Tth},
RTkh := {wh ∈ H(div, Ω) : wh|K ∈ Pk−1(K, Rd) ⊕ Pk−1(K) ∀K ∈ Th, t},
for every integer k ≥ 0, where Pk identifies the standard space of polynomials of
degree ≤ k in the variables x = (x1, ..., xd). For the simulations presented later on,
the lowest order Raviart-Thomas approximation has been adopted, corresponding to k = 0 above.
Concerning the capillary network, we split the discrete domain into the segments, Λh =SNi=1Λhi, where Λhi is a finite element mesh on the one-dimensional manifold
Λi. The solution of [2.22c] and [2.22d] over the branch Λi is approximated using
continuous piecewise-polynomial finite element spaces for both pressure and velo-city. Since we want velocity to be discontinuous at multiple junctions, we define
CHAPTER 2. NUMERICAL METHODS
the related finite element space over the whole network as the collection of the local spaces of the single branches. We have the following trial spaces for root pressure and velocity, respectively:
Xk+1h (Λ) := {wh ∈ C 0(Λ) : w h|S ∈ Pk+1(S)∀S ∈ Λh}, Wk+2h (Λ) := N [ i=1 Xk+2h (Λi),
for every integer k ≥ 0. As a result, we use generalized Taylor-Hood elements on each network branch, satisfying in this way the local stability of the mixed finite elements pair for the network Λ.
In order to reduce the computational cost of the root system, we decide to test also the following non conforming mixed finite element,
Y1h(Λ) := {wh ∈ C0(Λ) : wh|S ∈ P1(S)∀S ∈ Λh}, Z1h(Λ) := N [ i=1 Y1h(Λi).
Usually this choose is avoid, because it doesn’t respect the IN-SUP condition for flow problems and the stability can’t be prove. Instead, what we notice from the results is that for this choose we don’t have any kind of instabilty, also in case of complex root system. Surely more sofisticate analysis are required, but from these results we can expect a stable solution also for this choose of finite element. The results of the simulation of this problem are in the last section of this chapter.
2.2.3
Fixed Point Method
Before the derivation of the discrete model we have to take into account the non linear terms of Richards equations. To manage this problem we decided to use a fixed point method, which, for each fixed time step, iterate the solution untill convergence is reached. So we need to rewrite the equations [2.22a], [2.22b] as :
( 1 ktm,n+1u m+1,n+1 t , vt)Ωt + 1 β(u m+1,n+1 t · n, vt· n)Γu− (p m+1,n+1 t , div(vt))Ωt = = −(gt, vt· n) − (p0, vt· n)Γu− ρg(∇z, vt)Ωt∀vt ∈ Vt 28
1 ∆t(c m,n+1pm+1,n+1 t , qt)Ωt + (div(u m+1,n+1 t ), qt)Ωt − (Q(p m+1,n+1 v − p m+1,n+1 t ), qt)Λ = = 1 ∆t(c m,n+1pn t, qt)Ωt
where pm+1,n+1t and um+1,n+1t are the pressure and the filtration field at the time step n+1 for the iteration m+1, cm,n+1= c(pm,n+1
t /ρg) and k m,n+1
t = kt(pm,n+1t /ρg).
At each time step the starting condition for our iterative solver is always set as p0,n+1t = pnt. Moreover as stopping condition, it is used the incremental relative error of the pressure:
||pm+1,n+1t − pm,n+1t ||L2(Λ)
||pm+1,n+1t ||L2(Λ)
≤ M ax Error (2.23)
2.2.4
Algebraic System
We can now derive a discrete formulation of our problem by projecting the con-tinuous infinite-dimensional PDE’s over the discrete spaces defined above.
Given the bases of the finite elements base functions:
RTkh = n ϕito i=1:Nt , Ykh =nψtio i=1:Mt , Xk+1h = n ϕivo i=1:Nv , Wk+2h =nψivo i=1:Mv ,
where Nt, Mt, Nv and Mv are the dimension of the finite element spaces.
Then, by writing the discrete unknowns as linear combinations of the finite element base functions, we can deduce that at each iteration of any time step we have to solve the following linear system:
NLMmtt −DTtt O O Dtt Btt+ NLMmpp O −Btv O O Mvv −DTvv− JTvv O −Bvt Dvv+ Jvv Bvv Um+1,n+1t Pm+1,n+1t Um+1,n+1v Pm+1,n+1 v = Ft NLFmt Fv 0 (2.24)
CHAPTER 2. NUMERICAL METHODS
Submatrices and subvectors are defined as follows:
[NLMmtt]ij := ( 1 km t ϕjt, ϕit)Ω+ 1 β(ϕ j t · n, ϕit· n)Γu NLMtt ∈ R Nt×Nt, [Dtt]ij := (∇ · ϕ j t, ψ i t)Ω Dtt∈ RNt×Mt, [NLMmpp]ij := (cmψtj, ψ i t)Ω NLMpp∈ RNt×Mt, [Btt]ij := (Qψ j tδΛh, ψ i t)Ω Btt ∈ RMt×Mt, [Btv]ij := (QψvjδΛh, ψ i t)Ω Btv ∈ RMt×Mv, [Bvt]ij := (Qψ j t, ψ i v)Λ Bvt ∈ RMv×Mt, [Bvv]ij := (Qψvj, ψ i v)Λ Bvv ∈ RMv×Mv, [Mvv]ij := ( 1 kp ϕjv, ϕiv) + 1 β[π 2R04 ϕjvϕiv]εu Mvv ∈ R Nv×Nv, [Dvv]ij := (πR02∂ϕjv, ψ i v)Λ Dvv ∈ RNv×Mv, [Jvv]ij := h[ϕjv], ψ i viJ Jvv ∈ RNv×Mv, [Ft]i := −(gth, ϕ i t· n)Γp− (p h 0, ϕ i t· n)Γu− (ρg∇z, ϕ i t)Ωt Ft ∈ R Nt, [NLFmt ]i := X j (Pnt)j(cmψtj, ψit)Ωt NLFt∈ R Mv, [Fv]i := −[πR02gvϕiv]εp− [πR 02p 0ϕiv]εu Fv ∈ R Nv,
where ψjt is the average of ψjt, cm = c(pm,n+1/ρg) and kmt = kt(pm,n+1/ρg) are the
water content and the hydraulic conductivity computed at the previous iteration. Now for the implementation of exchange matrices, namely Btt, Bvt, Btv and Bvv,
we define two discrete operators: the first one is the mean value of a generic basis of Qh
t, while the second is an interpolation between Qhv and Qht. For each
node sk ∈ Λh we define Tγ(sk) as the discretization of the perimeter of the root
γ(sk). For simplicity, we assume that γ(sk) is a circle of radius R defined on
the orthogonal plane to Λh at this point sk. The set of points of Tγ(sk) is used
to interpolate the basis functions ψi
t. Let us introduce the interpolation matrix
πγ(sk) which returns the values of each test functions ψti on the set of points
belonging to Tγ(sk). Then, we consider the average operator πvt : Qht → Qhv such
that each row is defined as,
πvt|k = wT(sk)πγ(sk) k = 1, ..., Mvh, (2.25)
Figure 2.1: Illustration of the vessel with its center line Λh, a cross section, its perimeter γ(s k) and
its discretization Tγ(sk) used for the definition of the interface operators πvt: Qht → Q h v
and πtv : Qhv → Qht
where w are the weights of the quadrature formula used to approximate the integral qt(s) = 1 2πR Z 2π 0 qt(s, θ)Rdθ,
on the nodes belonging to Tγ(sk). The discrete interpolation operatorπtv : Qhv →
Qh
t returns the value of each basis function belonging to Qht in the corresponding
nodes of Qh
v. In algebraic form it is expressed as an interpolation matrix πtv ∈
RMvh×Mth. Using these tools we obtain:
Btt = πvtTM P vvπvt, Btv = πvtTMPvv, Bvt = MPvvπvt, Bvv = MPvv being MP
vv the pressure mass matrix for the root problem defined by
[MPvv]ij := (Qψvj, ψ i v)Λ.
Concerning the implementation of junction compatibility conditions, we intro-duce a linear operator giving restriction with sign of a basis function of Vvh over a given junction node. For a given k ∈ J , we define Rk: Vvh → R such that
Rk(ϕjv) := ( +πRl02ϕj v(sk) j ∈ Λhl ∧ l ∈ Pkout −πR02 l ϕjv(sk) j ∈ Λhl ∧ l ∈ Pkout , (2.26)
CHAPTER 2. NUMERICAL METHODS
for all j = 1, ...Nvh, where the expression ”j ∈ Λhl” means the the j − th dof is linked to some vertices of the l − th branch. Note that we are implicitly using the usual property of Lagrangian finite element basis functions, i.e. that they vanish on all nodes except the related one. As a consequence, our definition is consistent for all junction vertices. Indeed, Rk may only assumes values {−πR02l , 0, +πR
02 l }
for some l and in particular Rk(ϕjv) = 0 for all couples of indexes (k, j) that are
uncorrelated. Furthermore, the definition of Rk can be trivially extended to all
network vertices. Using this operator, the generic (i, j) element of Jvv may be
computed as follows: [Jvv]ij = − X k∈J Rk(ϕjv)ψ i v(sk).
2.3
Model Validation
When developing models, the most critical metric regards how well the model does in some critical cases. For the validation we propose four main test cases, in order to point out different aspects of the model: a Validation of the Root model, one for the steady Ground model, one for the unsteady Ground model, the validation for the coupled Root-Ground model and finally a test case on the P1/P1 non comforming finite elements.
2.3.1
Steady Root Model
For the first test case we present the simulation of a water flow in single root branch, not interacting (Q = 0) with the surrounding ground, modeled as an ho-mogeneous sature porous medium. As single root we chose to take a branch cross-ing the ground from side to side. Different roots shapes are considered, namely we simulate a straight segment and two circular arcs with different intermediate curvature, respectively κR = 0.06 and κR = 0.11 (remember that κR is the dimen-sionless curvature). Concerning the boundary conditions, we apply the pressure at the endpoints of the network that in this case we set as Pinlet = 32 mmHg and
Poutlet = 28.5 mmHg and for the velocity profile we imposed a Pouiseuille Profile
(γ = 2), without loss of generality.
Thank to the assumption of not interaction between the ground and the root, our 32
two models look completely uncoupled and we can compute easily the analytical solution inside the root using the mixed form of equations [1.49 c] and [1.49 d] which read as:
div(kp∂spv) = 0 on Γ (2.27)
Under the assumption of linear and circular branch we have that the effective conductivity has no space dependece so our solutions read as:
pv(s) = ∆P · s + Pinlet (2.28)
uv(s) = kp∆P (2.29)
where ∆P = Poutlet− Pinlet is the drop of pressure imposed by the boundaty
con-dition Poutlet and Pinlet.
In Figure 2.2 (right-hand panel) we compare the computed and analytically
de-Figure 2.2: The left-hand panel shows the geometrical configuration of the test case (κR = 0.11 highest curvature) with variantion of the pressure inside the root. The right-hand panel compares the velocity profile (constant) and magnitute along the axis of the root, in case of impermeable wall.
terminated values of the velocity magnitude. Obviously, since the radix is imper-meable, the velocity profile is constant along the axis, but the velocity magnitude decreases with the curvature, because the culved roots oppose higher resistence to flow. We observe that this behavior is correctly captured by the model.
2.3.2
Steady Soil Model
In the second test case we are going to test our solver only for ground model, taking the root uncoupled by the earth. Therefore we take as domain a cube of 0.4m side of a sandy ground. All the parameters are given in the Table 2.1.
CHAPTER 2. NUMERICAL METHODS
As boundary conditions to our ground domain, we impose No-flux condition on the laterals boundaries and Dirichlet boundary condition for the top and bottom surfaces of our cube, namely Pt(h = 0) = −6027 P a and P (h = 0.4) = −5948 P a
SYMBOL PARAMETER UNIT VALUE
d characteristic length m 0.4
K ground hydraulic conductivity m/s 9, 44 10−5
A constant for non linear conductivity 1.61 106
γ exponent for non linear conductivity 3.96
ρ density of the fluid kg/m3 1000
Table 2.1: Table of value, given by Celia et al, for the steady simulation of a sandy ground.
Figure 2.3: Rappresentation of the pressure solution in the 3D ground.
Figure 2.4: Here is shown the value of the pressure take on a vertical line.
Figure 2.5: Plot of the velocity field magnitude in the 3D domain, with the flow vectors.
Thanks to these boundary conditions and the absence of any forcing term due to the root draining, we can calculate the analytical solution, for the saturated problem. Again, using the mixed formulation of the equations [1.49 a],[1.49 b], and assuming constant conductivity in all the domain, we have to solve the following equations:
∆pt= 0
ut= −kt∇(pt+ ρgz)
Now using the No-Flux conditions on the later boundary and the absence of horizontal forcing term, we can assume ∂xpt = 0 and ∂ypt = 0. So our solution
depend only to the vertical coordinate and, solving the equation, read as: pt(x, y, z) = (ptopt − pbott ) · z + p bot t (2.30) ut(x, y, z) = 0 0 kt(pbott − p top t − ρg) (2.31)
where ptopt and ptopt are the pressure boundary condition on the top and the bottom of the cube.
As we can see from Figures 2.3 and 2.4, the numerical solution for the pressure fit well our exact solution even if the conductivity is no more assumed constant. That is due to the fact that the pressure drop is not so high to point out the non
CHAPTER 2. NUMERICAL METHODS
linearity of the equation. In Figure 2.5 we observe that for the numerical solution, the velocity magnitude is constant in all the domain and the vector field is going down, exactly as expected from the analytical solution.
2.3.3
Unsteady Model of the Soil
With this third test case we begin to see how our model can catch the time depending solution. As for the previous case we test our solver on a domain cube of 0.4m side of sandy ground. The parameters for this case are written in the Table 2.2. For this test we impose as starting condition constant pressure in all
SYMBOL PARAMETER UNIT VALUE
d characteristic length m 0.4
K ground hydraulic conductivity m/s 9, 44 10−5
A constant for non linear conductivity 10
γ exponent for non linear conductivity 3.96
ρ density of the fluid kg/m3 1000
α constant for non linear water content 10
β exponent for non linear water content 3.96
θs water content for saturated ground m/s 0.287
θr water retention m/s 0.075
Table 2.2: Table of value, given by Celia et al, for the unsteady simulation of a sandy ground.
the domain, pt = −6027 P a , and for all the time t > 0 we impose the same
boundary condition of the previous test case, and we simulate it for 5 seconds. What we expect from this simulation, is that shock, imposed by the non matching of the starting condition and the boundary conditions for t > 0, will generate an unstable solution which will become stable after a certain time.
Figure 2.6: Plot of the pressure over a section for different time. T = 0s (upper left-hand), T = 0.3s (upper hand), T = 1s (middle left-hand), T = 1, 5s (middle right-hand), T = 3s (down left-right-hand), T = 4s (down right-hand).
CHAPTER 2. NUMERICAL METHODS
Figure 2.7: Plot of the pressure over a vertical line for different time. T = 0s (upper left-hand), T = 0.3s (upper right-hand), T = 1s (middle left-hand), T = 1, 5s (middle right-hand), T = 3s (down left-hand), T = 4s (down right-hand).
Figure 2.8: Plot of the velocity field magnitude over a section for different time. T = 0s (upper left-hand), T = 0.3s (upper right-hand), T = 1s (middle left-hand), T = 1, 5s (middle right-hand), T = 3s (down left-hand), T = 4s (down right-hand).
CHAPTER 2. NUMERICAL METHODS
Figure 2.9: Plot of the velocity field magnitude over a vertical line for different time. T = 0s (upper left-hand), T = 0.3s (upper right-hand), T = 1s (middle left-hand), T = 1, 5s (middle right-hand), T = 3s (down left-hand), T = 4s (down right-hand).
Figure 2.10: Plot of the water content over a section for different time. T = 0s (upper left-hand), T = 0.3s (upper right-left-hand), T = 1s (middle left-left-hand), T = 1, 5s (middle right-hand), T = 3s (down left-hand), T = 4s (down right-hand).
CHAPTER 2. NUMERICAL METHODS
Figure 2.11: Plot of the water content over a vertical line for different time. T = 0s (upper left-hand), T = 0.3s (upper right-left-hand), T = 1s (middle left-left-hand), T = 1, 5s (middle right-hand), T = 3s (down left-hand), T = 4s (down right-hand).
From the Figures 2.6, 2.7, 2.8 and 2.9 it is clear that for time t > 0 our solu-tion perturbate by the starting solusolu-tion and after few seconds it reaches a stable solution, that, in this case, coincides with the steady solution of the problem. Thank to the lower coefficients of the non linear function, we can show in figures 2.10 and 2.11 the variation of the water content in the volume. Slightly interessant is the fact that from the upper surfaces of the domain we have a recharge of water content, which will distribute in all the domain in few seconds.
2.3.4
Unsteady Root-Soil Model
In the fourth test case, we take the same ground problem of the previous example in which we put a single root draining water. This radix is penetrating vertically from the top of the ground to the depth of three quater of the cube length, and it traspirate a constant rate for all the times t > 0. As boundary condition on the root we impose a constant traspiration rate at the top and no-flux condition at the bottom, using Neuman Boundary condition on pressure. This choose is due to fact that the radix can drain water only through the wall. The remaining parameters, choosen for this test, are shown in Table 2.3 and the results are collected in the following figures.
SYMBOL PARAMETER UNIT VALUE
d characteristic length m 0.4
K ground hydraulic conductivity m/s 9, 44 10−5
kv root hydraulic conductivity m/s 4, 0 10−4
R root Radius m 0.02
Lp wall hydraulic conductivity m2 s kg−1 1, 75 10−7
T root traspiration rate m3/s 5, 02 10−6
A constant for non linear conductivity 10
γ exponent for non linear conductivity 3.96
ρ density of the fluid kg/m3 1000
α constant for non linear water content 10
β exponent for non linear water content 3.96
θs water content for saturated ground m/s 0.287
θr water retention m/s 0.075
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Figure 2.12: Plot of the solution for the one-dimensional root respectively the mean velocity field (upper) and the mean pressure (down).
Figure 2.13: Plot of the Velocity Field in the ground over an horizontal section for time t = 3s.
Figure 2.14: Plot of the velocity field magnitude over a section for different time. T = 0s (upper left-hand), T = 0.2s (upper right-hand), T = 1s (middle left-hand), T = 2, 5s (middle right-hand), T = 4s (down left-hand), T = 6s (down right-hand).
CHAPTER 2. NUMERICAL METHODS
Figure 2.15: Plot of the pressure over a section for different time. T = 0s (upper left-hand), T = 0.2s (upper hand), T = 1s (middle left-hand), T = 2, 5s (middle right-hand), T = 4s (down left-right-hand), T = 6s (down right-hand).
Figure 2.16: Plot of the water content over a section for different time. T = 0s (upper left-hand), T = 0.2s (upper right-left-hand), T = 1s (middle left-left-hand), T = 2, 5s (middle right-hand), T = 4s (down left-hand), T = 6s (down right-hand).
CHAPTER 2. NUMERICAL METHODS
Figure 2.17: Plot of the water content over an orizzontal line for different time. T = 0s (upper ), T = 0.2s (middle), T = 1s (down).
Figure 2.18: Plot of the water content over an orizzontal line for different time. T = 2, 5s (upper), T = 4s (middle), T = 6s (down).
CHAPTER 2. NUMERICAL METHODS
Figure 2.19: Plot of the conductivity over a section for different time. T = 0s (upper left-hand), T = 0.2s (upper hand), T = 1s (middle left-hand), T = 2, 5s (middle right-hand), T = 4s (down left-right-hand), T = 6s (down right-hand).
