Introduction
Environmental physics emcompasses a wide field of research and has many links with different subjects, because it studies very complex phenomena. In this field of research one can easily understand the usefulness and the impor-tance to do numerical simulations: by comparing them with the experiments one can realize which are the main physical processes that must be taken into account and which are the neglegible ones, that many times can be very difficult precisely because of complexity.
In this thesis, in particular, I’ll simulate the transport of a conservative solute in the vadose zone of the soil. Studying the solute transport in the soils is interesting for many pratical applications, and mainly because it enables to do predictions of the spreading of contaminants or of nutrients for the plants.
In general, the high complexity of this phenomen is due to the non-trivial structure of the soil, the non-linear dynamics of the water and the solute transport, the chemical and biological interaction of the solute with the soil material. Furthermore, in the experiments, it is especially diffult to measure the quantities of interest without bias.
In particular, this work focuses on the study of the motion of a solute plume with an initial concentration throw zones of different water content in a soil, modeled as porous medium with some material parametrizations.
The code that I used is provided by Olaf Ippisch (Univer. of Heidelberg) and it is called muphi. This code is able to compute the dynamics of the water and the evolution of a solute pulse in a porous medium, given the material parameters that define the kind of porous medium, and the initial condition of both the water and the solute. It also can simulate heterogeneous structures, with different kind of porous media and geometries.
I organized this thesis in four parts: in the first one of this thesis I sum-marize the theory of the water flow and solute transport in porous media.
In the second part I give a very brief description of the code and I intro-duce the quantities that we consider in the result part, like the moments of the solute probability distribution, the dilution index and the reactor ratio.
In the third part I show the results of different numerical simulation in unsaturated media, in order both to check the code and to give an idea of the behaviour of the quantities I am interested in.
In the last part I finally compare my results with the ones of the ex-periments performed by Steffen Heberle and Christina Esterle (Univ. of Heidelberg, Department of environmental physics). To do this, I choose ap-propriate material properties, and I learn that that the code can very well predict the mean motion of the solute pulse, but has some problems with its spreading. In fact the dispersion is, probably, mainly due to the small scale variability of the water velocity field (that is not yet considered in the model), and so, in order to fit the spreading evolution of the pulse it should be inserted in the dispersion coefficient a dependence on the water velocity.
Chapter 1
Theory
A short summary of the theory of soil water dynamics and solute transport will be given here, in order to understand how the simulation is implemented, and what physical laws it uses. It follows mainly [Roth (2006)], that can be refered to for a detailed description.
At its microscopic scale, a soil can be thought as a porous medium with a certain pore-size distribution. Although, all the descriptions that are given here refer to a macroscopic scale. It means that, in the code, only mean properties are taken into account, and not the real, very complex structure of the medium, at its microscopic scale. The unitary volume over which the averaging procedure is done is called R.E.V. (Representative Elementary Volume). This procedure leads to the definition of macroscopic quantities that describe the soil structure at this new scale, as it will be seen.
The first property that should be introduced is the porosity φ, defined as the pore volume over the total one:
φ = Vpor Vtot
(1.1) It can be a function of the space but, in this work, it will be taken as constant over the entire box, with the value φ = 0.34.
1.1
Dynamics of the soil water
Considering water as incompressible, its mass conservation can be stated in the form:
∂θ
where θ is the water content (volume fraction of water in the soil) and ~jw the water flux. Because this water moves through a complicated structure, the flux is not trivially obtained by an empirical law known as Darcy law, that states:
~jw = −K · ∇ψw (1.3)
where K is the conductivity of the soil and ψw is the water potential, that is made explicit afterwards.
Notice that Darcy law states that a flux (i.e. a velocity) is propotional to the gradient of a potential (i.e. a force). It can make one understand that, in the dynamics of the soil water, the viscous friction is the term that dominates in the Navier-Stokes equation.
The Darcy law, stated as in eq. 1.3, is valid only in a completely saturated medium. If the water content is variable, this law continues to be valid, provided that the conductivity is now function of θ. This conjecture is known as Buckingham conjecture, and the new flux law as Buckingham-Darcy law:
~jw = −K(θ) · ∇ψw (1.4)
The water potential, calculated with respect to a reference state with height z0 and pressure p0, can be split in two parts, as you can see in the following equation:
ψw(~x) = (p(~x) − p0) + (z − z0)ρg = ψm(~x) + (z − z0)ρg (1.5) The second term in the equation 1.5 is the gravitational part of the po-tential, while the first term is a pressure difference due to several components (for example to interfacial forces) and it’s called matric potential (ψm). So Buckingham-Darcy law can be rewritten as:
~jw = −K(θ) · [∇ψm− ρ~g] (1.6)
It should be pointed out that K(θ), in general, is a tensor even if it’s now considered only as a scalar (K). Inserting the flux law (1.3) in 1.2 we obtain the equation that describes the soil water dynamics, known as Richards equation:
∂θ
∂t − ∇ · [K(θ)[∇ψm− ρ~g]] = 0 (1.7) For a complete formulation, this equation must be supplemented with the description of the material proprieties: the mentioned K(θ), and the soil-water characteristic, θ(ψm) that gives the water content as a function of the potential. They are described in the next section.
1.2
The material properties
Soil water characteristic describes the water content above the water table, defined as the surface where ψm(x) = 0, if the reference state p0 is chosen equal to the atmospheric pressure. Its profile mainly depends on the pore size distribution of the soil. Because of capillarity, the water rises up from the water table: the capillary fringe, i.e. the zone completely wet (saturated of water) depends on the largest pore radius of the medium, while its profile on the width of the pore size distribution. If one imagines a soil as a bundle of capillaries 1 as represented in the figure 1.1, one can understand why the soil water characteristic has its shape.
Figure 1.1: Soil water characteristic of a porous medium illustrated for a bundle of capillaries. Figure from [Roth (2006)]
In order to describe this function mathematically, different parametriza-tions are used. The most common ones are known as Brooks-Corey and Mualem-Van Genuchten. For convenience, the matric head hm (energy per unit weight ρwg), instead of ψm, is used in following.
Moreover instead of the water content θ, the saturation (Θ) is used. It is defined as:
Θ = θ − θr θs− θr
(1.8) where θsis the water content at saturation, i.e. when the porous medium is completely full of water, and θr is the residual water content, i.e. the water
1
of course, it is a very rough model, because of the high variability of the pore size in a real soil. It causes hysteretical processes that are neglected in this description.
that remains after a first drenaige. They represent the upper and the lower end of the range of water contents; so saturation is a quantity that takes values between 0 and 1 and gives a percentual measure of how much the soil is full of water. When θr = 0, θs is equal to the porosity φ of the medium.
The Brooks-Corey parametrization for the soil water characteristic is: Θ(hm) =
(
[hm/h0]λ; hm < h0
1; hm ≥ h0
(1.9) The parameter h0 takes into account the capillary rise of the water, while the parameter λ the width of the distribution. For various types of soil you can notice, in figure 1.2, that for the coarser material (the sand) the capillary rise is lower and the distribution sharper.
The Van Genuchten parametrization is similar to the previous, but it has not a discontinuity in h0:
Θ(hm) = [1 + (αhm)n]−1+1/n (1.10) where α = 1/h0 and n = λ + 1.
Conductivity describes the resistence experienced by the water flow in a medium and is directly proportional to the Θ, the mean cross-sectional area ℓ2 and inversely to the dynamic viscosity µ. While you can imagine its dependence on ℓ2 and µ, it’s a bit more difficult understand that on Θ: you should think to the fact that, with decreasing saturation, and so a lower quantity of fluid in the medium, this fluid is restricted to a smaller class of pores and also the cross sectional area (ℓ2) decreases.
This is the main dependence explicity inserted in the two parametrizations we are talking about. The K(Θ) for Brooks-Corey has the form:
K(Θ) = K0Θ2+a+2/λ (1.11)
where K0 is saturated conductivity and a a further parameter that takes into account the tortuosity that will be set zero in the simulations. Van Genuchten too provided a parametrization for the conductivity:
K(Θ) = K0Θa1 − (1 − Θn/(n−1))−1+1/n 2
(1.12) Here too, in figure 1.2, you can notice the dependence on various ma-terials, and, for example: the higher conductivity of the coarse material at the same water content (that it is because K is propotional to the pore size squared) and the sharper rise at low water content of the sand, due to its sharper pore size distribution.
Conductivity can also be parametrized as function of the matric head sim-ply inserting into the previous equations the soil water characteristic Θ(hm).
Figure 1.2: Material properties for various kind of soils in Van Genuchten (thick lines) and in Brooks-Corey (dash-dotted lines) parametrization. (figure from [Roth (2006)])
1.3
Solute transport
In the soils, if the solute doesn’t interact with the surrounding material (or better: if these interactions can be neglected and the solute treated as chem-ically and biologchem-ically conservative), its transport is due to two types of physical processes: convection (displacement with the mean water flow) and
dispersion (spreading relative to the center of mass). To formulate an
equa-tion that explains the solute transport, one can begin, as before, from the mass conservation law:
∂Ctot ∂t =
∂θCw
∂t = −∇ · ~js (1.13)
where Ctot is the total concentration of the solute per total volume (while Cw = Ct/θ is the water concentration), and ~js the total flux of the solute. The last is locally given by:
~
js = Cwj~w− θD∇Cw (1.14)
The validity of this flux law is the assumption that is made in the convective-dispersive regime, i.e. that the solute transport is due only to these two processes. It is locally valid in the far field (although not macroscopically, for example in heterogeneous fields).
By inserting this flux law 1.14 into equation 1.13, the equation that de-scribes the solute transport in the soils is obtained:
∂θCw
∂t + ∇ · (θ~vCw) − ∇ · (θD∇Cw) = 0 (1.15) Equation 1.15 is called convection-dispersion equation.
In order to describe how these two processes are inserted in the last equation, one can take a look to eq.1.14: here the first term takes into account
convection, that is the displacement of the solute due to the velocity field of
the liquid in which it’s moving, while the second term the dispersion.
Dispersion can be due to different kinds of physical processes; the infor-mation about them is stored in D, that is, in general, a tensor. The simplest kind of dispersion is molecular diffusion. If this is the only process, and the medium is saturated, D is a scalar, and can be written as:
D = Dm (1.16)
where Dm = 2·10−9m2/s. In case of unsaturated medium D also depends on the water content. This dependence, for the pure molecular diffusion coefficient, is parametrized in the Millington-Quirk relation (eq. 1.17), where Dm is its value in pure water. So D takes the form:
DmMillington-Quirk(θ) = θ 4/3
φ2 Dm (1.17)
In addition, D can depend on the flow velocity too. In this case, it can be written, in cartesian coordinates, as the sum of two terms:
Di,j(~x) = [λl− λt]
vi(~x)vj(~x)
| ~v(~x) | +λt| ~v(~x) | +D
Millington-Quirk
m (θ) δij (1.18)
where the last term (with DMillington-Quirkm (θ)) corresponds to molecular diffusion and the first two to small scale variability of the velocity field (vi) with a certain dispersivity tensor (λi,j), where λtis the transversal component and λl is the longitudinal one.
Chapter 2
Methods and Analysis
2.1
Methods: numerical simulation
In this section a short description of the used code is given, in order to have an idea how it works, and which kind of results is able to give. This program is developed by Olaf Ippisch (Interdisziplin¨ares Zentrum f¨ur Wissenschaftliches Rechnen, Universit¨at Heidelberg).
The code that is used here simulates a two-dimensional sand tank, 0.45m high (y axis) and 0.25m wide (x axis), on a structured rectangular grid of 200 × 360. The y axis points upwards with y = 0 at the bottom. All the lenght measures are given in meters.
This code is composed by two parts: in the first it implements the water transport, given a sand structure (i.e. material properties as function of the space), boundary and initial conditions; the second implements the solute transport in this region, given an initial solute concentration distribution. The first part simulates the hydraulics with a constant flux till the equi-librium is reached. Here a brief description of its various inputs follows: boundary cond. The used boundary conditions are: constant flux coming down from the
upper part, no flux on lateral boundaries (Neumann condition type) and constant pressure on the lower part (Dirichlet condition type). In the code one can also choose to divide the upper part in various segments, and to put in each segment different conditions. The chosen flux rates oscillate between the 10−7 and 10−5 m/s.
p.m. structure The code can implement different types of soil structure, and not only the homogeneous situation. In order to do this, one can use an image with different grey levels: a different parametrization, i.e. a different
kind of porous medium, corresponds to each grey level. These images are shown in the results part.
p.m. parametrization The two parametrizations used to characterize the porous medium, are the ones explained in the previous chapter, the Mualem-Van Genuchten and the Brooks-Corey ones. The inputs to describe them are six pa-rameters, namely: the porosity (φ), the residual water content (θr), the tortuosity (a), the saturated water conductivity (K0) for both, then the numbers α and n for the Van Genuchten one or the equivalent ones h0 and λ for the Brooks-Corey.
The output of this part of the code is the flow velocity and the saturation for each point of the grid. From this output 2-D images with a certain color map can be contructed, to graphically show the results.
The second part simulates the solute transport in the box, obvously using as input the output of the previous part, i.e. the velocity field and the saturation.
The other input is the initial condition for the solute. This is represented as circolar pulse of solute with three available parameters: the solute con-centration Ctot, chosen equal to 1, the radium r0, and the position (x0,y0) of the center of the circle.
The output of this part are, for each time step calculated, the value of the zeroth moment (total solute mass), of first moment and of the second moment matrix of the solute concentration distribution (see the next section).
For 100 time steps it also gives the exact concentration at each point, that is possible to plot in a 2-D image as done before with the saturation. This is useful, because one can build from those a short video that gives an idea about what is happening in the whole simulation, and what we should roughly expect in a more detailed analysis; and also because from that one can observe better the shape of the pulse, and calculate some other indicators, as the dilution index and the reactor ratio (see the next section).
2.2
Analysis: Description of the transport
In order to analyse the solute transport, different mathematical indicators can be used. In this chapter a short description is given about how they are computed and which meaning they have.
The plume, as said before, is described by a concentration distribution Ctot(~x, t), that gives the total concentration of the solute at each point in the time.
From this one can calculate the normalized one, that represents a proba-bility density function:
p(~x, t) = R Ctot(~x, t) volCtot(~x, t)d~x
(2.1) It gives the probability density to find a particle of the solute at the location (x, t).
From this distribution statistical indicators are constructed, that give a measure, a numerical value, of what we are interested in, and in particular the mean motion of the pulse, its spreading, its dilution and its shape.
2.2.1
Moments analysis
First of all, one can calculate the spatial moments of the distribution. The first moment, i.e. the position of the center of mass, and its time derivative, the mean velocity of the pulse, give a measure of the mean motion of the solute. They are calculated as in the following equations:
Xi(t) = Z vol xip(~x, t)d~x (2.2) Vi(t) = dXi(t) dt (2.3)
where i, j ∈ {x, y}. In the following they will be indicated as X and Y , while the velocities will be indicated Vx and Vy.
The second moments are calculated with respect to the center of mass: σij(t) =
Z
vol
(xi− Xi)(xj − Xj)p(~x, t)d~x (2.4) In the following they will be indicated as σxx, σyy and σxy. The second moment σxx and σyy give measure of the spreading of the pulse in the two different directions, i.e. how much the solute distribution reaches positions far from the center of mass; the last one, σxy, how much x and y are correlated and so how much the shape moves away from the simmetry.
One can also calculate the time derivative of these quantities, i.e. the spreading velocity:
2Dappij = dσij(t)
dt (2.5)
This is called 2Dijapp because in soil physics is double the apparent diffu-sion tensor (Dappij ). In convection-dispersion regime, it is double the effective diffusion tensor. In the following it will always be called Dij.
2.2.2
Dilution index and reactor ratios
The other indicator that it is used here is the dilution index [Kitanidis (1994)]. It represents the dilution of the solute. It should be pointed out that it is a different concept from the spreading: while spreading gives a measure of the stretching and deformation of a plume, dilution gives a measure of the volume occupated by solute in the liquid. A QUESTO PROPOSITO, see the figure 2.1, in which are plotted two distributions that have the same spread-ing, i.e. σ, but different dilution index, and in particular the lower function has a greater one.
Figure 2.1: Mass in two small plug versus mass distributed normally: same σ but different dilution index. Figure from [Kitanidis (1994)]
The dilution index is calculated as in [Kitanidis (1994)]: V (t) = exp Z vol p(~x, t) ln(p(~x, t))dV (2.6) One can notice that the logarithm of this expression is the well-known expression for the entropy of a distribution.
The shape with maximum entropy, i.e. minimum information in an un-bounded domain, is the gaussian one.
This suggests to define some ratios. These are constructed in order to
quantify different features of the shape of the distribution (see [Monahan, Del Sole (2009)]), like “how much” the distribution is compact or gaussian.
In this frame, a non-gaussianity measure can be the ratio of the dilution index of the actual plume and the one of a theoretical perfectly gaussian plume with the same variance matrix (σij). The last one, in this case (two dimentional), is calculated as:
VG = 2πe q
σxxσyy− σxy2 (2.7)
So the reactor ratio is:
R(t) = V (t)/VG (2.8)
This quantity always assume values between 0 and 1, and it is 1 for perfectly gaussian distribution.
It is interesting to calculate this ratio because it enables us to analyse how the plume is distorted by non uniform diffusion or convection, and how it returns to take a gaussian shape when its spreading is due only to uniform processes. In the experiment, it can indicate what kind of processes are mainly driving the transport, and can help to do the comparision with the simulation.
Chapter 3
Results
3.1
Summary of the results
In the first part the results of some simple kinds of structures are shown, in order to give an idea of the behaviour of the quantities that we are going to analyse and of the way the simulation works.
The pulse is injected at the height of y0 = 0.42m on the central vertical line, x0 = 0.125m. For the boundary conditions see the methods part.
The quantities of interest are plotted either as functions of the time or the mean position of the pulse (Y ), i.e. the first moment of the concentration distribution in vertical direction.
The simulations are implemented for:
1. different flux rates, in particular: 3.176 × 10−5, 3.176 × 10−6, 3.176 × 10−7m/s;
2. different initial position of the pulse, in particular: y1(0) = 0.42m and y2(0) = 0.32m;
3. different medium structure, in particular: • a homogeneous sand tank;
• a sand tank with a silt inclusion, and
• a silt tank with a sand inclusion with complementary geometry. All the graphs should be cut, or not considered at the lower end (y ≃ 0.03m), when the pulse goes out the system. In fact, at that point, the moments of the distribution show a behaviour that is not completely due to physical processes but also to the fact that the pulse is not completely inside the system. This happens at different times for the different flux rates.
3.2
Homogeneous structure
3.2.1
Input: material parameters
In this first part, a homogeneous sand tank is considered, in order to show what happens in the simplest situation. A Van Genuchten parametrization is used, with the coefficient n = 8.0 and α = 5.5m−1. You can see the soil water characteristic function with this parameters in fig. 3.2.1.
0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Θ matric head, hm
Figure 3.1: Soil water characteristic function
One can see in fig. 3.2.1 the chosen conductivity parametrization too. The results are shown for three different flux rates, with three different or-ders of magnitude (10−5, 10−6 and 10−7ms−1). Notice that the saturated conductivity K0 (the value of K at Θ = 1) is in the same order of the biggest flux rate, to see which is the output of the simulation with a giant flux.
1e-11 1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 0.0001 0 0.2 0.4 0.6 0.8 1 K Θ 1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 0.0001 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 matric head, hm
Figure 3.2: Conductivity parametrization. On the right as function of the satu-ration, on the left as function of the matric head
3.2.2
Saturation
The plot in fig. 3.3 shows the saturation (Θ) profile only on a vertical line, because Θ is the same at the same heigh.
You can see how the equilibrium situation depends on the flux rate: at higher flux higher saturation at the top corresponds. It should be also noticed that for lower flux rates a sharper rise corresponds, even if the material properties are the same.
This plot is important to understand the features of most of the others, because the behaviour of the solute plume, in this simulation, mainly depends on the saturation. 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 T h et a
heigh of the cell [m]
j1≃ 10−5m/s
j2≃ 10−
6
m/s
j3≃ 10−7m/s
Figure 3.3: Saturation profile on the central vertical line
In fig. 3.4 some typical concentration distribution are plotted, at three different times (see the caption); you can look at them to understand the behaviour of the simulation. In the first two ones you can see the pulse really concentrated in a small zone, and a bit stretched along the vertical direction. Then, in the third one, after going throw the saturated zone, it is more spread and does not more show a streched shape. What is done in the following is to analise these features using the moments of the distribution.
Figure 3.4: Three typical concentration distributions in the simulation, from the left: when the pulse is in the unsaturated zone, when it goes throw the edge of the saturated zone and when is in the saturated zone.
3.2.3
Moments analysis
1st moment The first moment in x direction is not plotted because it stays constant: the pulse is always moving on the same vertical line.
The first moment in y is plotted in fig. 3.5 with three different time normalizations (see the caption) to better see the shape of the curves and analyse them together. It shows different pulse velocities depending on the flux rate in the unsaturated part. In all the cases one can notice a velocity change when the pulse reaches the saturated zone. This happens because velocity is proportional to the flux by the factor θ (j = θv). In order to get the flux constant (as it must be in a stationary situation) the velocity has to be in inverse proportion to θ, and reaches a minimum and constant value in the saturated zone.
The kink in the plot is more evident for lower flux rates, because of the more sharp saturation profile. In the saturated zone the pulses moves with the same velocity except for a scaling factor (due to the different flux rates). 2nd moment xx. (fig. 3.6) In the model the spreading of the solute pulse depends only on the water content (because the dispersion coefficient is only function of θ).
In the far field, where the solution of convection-dispersion equation is a gaussian, the evolution of σxx follows the law: σxx(t) = 2Dxxt.
As you can notice in fig. 3.7, Dxx depends mainly on the saturation. It’s more interesting to plot σxx as function of the mean position of the pulse (Y ). In this case the spreading as a function of space is given by:
dσxx dY = dt dY dσxx dt = 2Dxx Vy (3.1) The coefficient D (see fig. 3.7) has always the same order of magnitude for the three fluxes and different water contents, but the differences in V
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.5 1 1.5 2 2.5 3 3.5 M ea n p os it io n of th e p u ls e, Y normalized time j1≃ 10− 5 m/s j2≃ 10−6m/s j3≃ 10− 7 m/s
Figure 3.5: Mean position of the pulse vs normalized time for different flux rates. The three time normalizations are: for j1 1 → 103s, for j2 1 → 104s and j3
1 → 105s
between saturated and unsaturated part are much more pronounced for lower flux rates, and in particular, much higher in the upper part. That explains why in the saturated part the slower pulse is much more spread, while in the unsaturated part the three pulses reach almost the same spreading at the same height, as you can see in fig. 3.6. This underlines that, in a saturated situation, the dispersion implemented in the code is only a time-dependent process.
2nd moment yy. (fig. 3.6) In all the cases a minimum at the upper end of the saturated zone can be seen, but if one takes a look to the dispersion coefficient the slowing down of the spreading is more evident. This means that the pulse is a bit compressed when it reaches that point.
This happens for the same reason explained before about the velocity field. When the pulse reaches a zone where the water content is not constant, its parts are moving with different velocities and cause a change in the shape and the spreading. In this case, it is compressed because the upper part is moving faster than the lower one.
When the pulse is moving in the saturated zone, it spreads with a constant D, following the law: σyy(t) = 2Dyyt; and because in this part the vertical velocity is constant it is also true that: σyy(t) ∝ Y , as you can see in fig. 3.6.
0 5e-05 0.0001 0.00015 0.0002 0.00025 0.0003 0.05 0.1 0.15 0.2 0.25 σx x 0 5e-07 1e-06 1.5e-06 2e-06 2.5e-06 3e-06 3.5e-06 0.25 0.3 0.35 0.4 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.05 0.1 0.15 0.2 0.25 σy y
mean position of the pulse, sat. zone
0 2e-05 4e-05 6e-05 8e-05 0.0001 0.00012 0.00014 0.00016 0.25 0.3 0.35 0.4 mean position of the pulse, unsat. zone
Figure 3.6: Second moment (σxx in the upper two panels and σyy in the lower
ones) vs mean position of the pulse (Y ), for three flux rates (in ms−1): red 3.176 ×
10−5, green 3.176 × 10−6 and blue 3.176 × 10−7. In the right two panels only the
unsaturated part is shown
0 1e-10 2e-10 3e-10 4e-10 5e-10 6e-10 7e-10 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Dx x
mean position of the pulse
-1e-09 0 1e-09 2e-09 3e-09 4e-09 5e-09 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Dy y
mean position of the pulse
Figure 3.7: Dispersion coefficient in xx and yy for three different flux rates (in ms−1): red 3.176 × 10−5, green 3.176 × 10−6 and blue 3.176 × 10−7.
3.2.4
Dilution index and reactor ratio
When the pulse takes almost compact shapes, like in this situation, the dilu-tion index gives no much more informadilu-tions than the second moment matrix. As it can be seen in fig. 3.8, where the three quantities (V , σxx and σyy) are plotted normalized to one at their maximum value, the derivative of the dilution is always like a mean of the two σ.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15 0.2 0.25 V ;σ y y ;σ x x , fo r fl u x 1 0 − 5m / s
mean position of the pulse, sat. zone
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.2 0.25 0.3 0.35 0.4 0.45 mean position of the pulse, unsat. zone 0 0.2 0.4 0.6 0.8 1 0.05 0.1 0.15 0.2 0.25 V ;σ y y ;σ x x , fo r fl u x 1 0 − 7m / s 0 0.01 0.02 0.03 0.04 0.05 0.06 0.2 0.25 0.3 0.35 0.4 0.45
Figure 3.8: Dilution index (blue), σxx (red) and σyy (green) normalized to one
for two different flux rates; 3.176 × 10−5ms−1 for the lower two panels and 3.176 ×
10−7ms−1 for the upper ones shown in saturated (left panels) and unsaturated
(right ones) zone.
The reactor ratio (fig. 3.9) shows for all the flux rates a minimum right above the saturated zone, where the saturation is strongly increasing. It means that there the pulse is not only compressed but also distorted even in this simple situation.
For lower flux rates, the pulse can take more gaussian shape, mainly be-cause it has more time to spread and the diffusion part of transport equation is more important.
0.98 0.982 0.984 0.986 0.988 0.99 0.992 0.994 0.996 0.998 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 re ac to r ra ti o
mean position of the pulse
j1≃ 10− 5 m/s j2≃ 10−6m/s j3≃ 10− 7 m/s
Figure 3.9: Reactor ratio for three different flux rates
3.2.5
The transport is convection-dominated
Here two simulations with different initial position of the pulse, (y1(0) = 0.42m and y2(0) = 0.32m) with a flux rate of 3.176 × 10−6m/s are shown.
The behaviour of the two first moments is easily explainable: they coincide with a time translation: Y1(t) = Y2(t + T ) where T is the time at which the first pulse reaches the initial position of the second pulse.
For the second moment (fig. 3.10) the same considerations on time dependence can be done. It’s better visible the compression for the pulse with the higher initial position. If the second moments are plotted vs the mean position of the pulse, one can see that, in the saturated zone, the two spreading are almost the same for the same mean position. This is maybe due to the fact that the spreading is almost convective, so, in a stationary unsaturated situation, it doesn’t depend on the time but on the position, because in this case convection is not a dissipative (and so time dependent) process. In such a situation the spreading of solute injections is independent from their initial position.
Peclet number The indicator that is typically used to mathematically quantify the strenght of the convective transport with respect to the diffusion transport is the Peclet number. This number is here computed as:
0 5e-06 1e-05 1.5e-05 2e-05 2.5e-05 3e-05 3.5e-05 0 500010000150002000025000 σx x 0 5e-06 1e-05 1.5e-05 2e-05 2.5e-05 3e-05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 5e-06 1e-05 1.5e-05 2e-05 2.5e-05 3e-05 3.5e-05 0 500010000150002000025000 σyy time 0 5e-06 1e-05 1.5e-05 2e-05 2.5e-05 3e-05 3.5e-05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 mean position of the pulse
Figure 3.10: Second moment (σxx in the upper two panels and σyy in the lower
ones) vs time (in the two right panels) and mean position of the pulse (Y , in the two left panels) for two different initial position: red y1(0) = 0.42m, green
y2(0) = 0.32m
P e = r0V
D (3.2)
where r0 = 0.45m is a typical scale of our problem, V is the local mean velocity, D the local dispersion coefficient.
In the saturated zone, where both the velocity and the dispersion coef-ficient are almost constant, an almost constant Peclet number (fig. 3.11) is obtained, for the all the three flux rates much bigger than one. That means that the trasport is convection dominated. For lower fluxes lower Peclet number is obtained. It means that for lower fluxes the trasport undergoes more the influence of the diffusive regime, because the velocity is smaller.
In the unsaturated zone notice that P e is much bigger. That is always because of the velocity; one can also understand it by looking how much the dispersion is influenced by the velocity field in the pictures of the
concentra-100 1000 10000 100000 1e+06 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 P e
mean position of the pulse
Figure 3.11: Peclet number computed as r0V /D as function of the mean position
of the pulse
tion distribution.
3.3
Heterogeneous structure
In this section the results obtained with some simple kinds of heterogeneous structures are shown. What is done in the following is the comparision between the transport in the homogeneous structure and the heterogeneous one and between different kinds of heterogeneous structures.
The considered structures are: • a homogeneous sand tank;
• a sand tank with a silt inclusion, and
• a silt tank with a sand inclusion with complementary geometry. This small inclusion is a little squarely area with the following borderlines on x and y: x1 = 0.111, x2 = 0.126, y1 = 0.276 and y2 = 0.296 (all in m). In all the graphics shown the border of this area are indicated by black lines.
For the sand the same parameters of the previous section are used, for the silt a Brooks-Corey parametrization with λ = 0.3 and h0 = 0.714m is used.
3.3.1
Comparision: with and without inclusion
Here the behaviour of the solute pulse moving in a homogeneous sand box is compared with the behaviour of the one moving in a sand box with a silt inclusion.
The first plot shown (fig. 3.12) is the saturation profile. You can notice how the profile is distorted by the influence of the inclusion. In particular the silt retains more water and cause a little rise in the water content. This also effects the lower part: the saturation rise is traslated a bit lower. This should be regarded as the simplest input in a homogeneous sand box.
Figure 3.12: Saturation in the cell for the sand with silt inclusion. blue Θ = 0, red Θ = 1
In figure 3.13 you can see the pulse moving throw the inclusion (from left to right). In the first time step considered it crashes on the different material, that retains more water. In the second you can see that is quite sharply divided in two parts, and that in the lower part it is already much more spread. In the third image it is again moving in a homogeneous environment and takes the previos compact shape.
1st moment In figure 3.14 both mean position and mean velocity are plotted. The effect of the inclusion is to increase the water content and so to decrease velocity. The pulse in the heterogeneous environment is there slowed down, then it takes a velocity a bit higher than the one moving in the homogenous background. When it reaches the completely saturated zone the two velocity are equal. Although, the complessive effect of such a small
Figure 3.13: Concentration distributions in the simulation in three time steps, in which the pulse is moving throw the inclusion.
inclusion is to speed up the vertical transport.
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 5000 10000 15000 20000 25000 Y time 5e-06 1e-05 1.5e-05 2e-05 2.5e-05 3e-05 3.5e-05 4e-05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Vy
mean position of the pulse
Figure 3.14: On the left panel: mean position of the pulses, on the right panel their velocities. Green: without inclusion, red: with inclusion
2nd moment in xx (fig. 3.15, left panels). When the pulse goes throw the inclusion it undergoes a strong spreading in the horizontal direction. Looking at the dispersion coefficient graphic, one can notice how the two ones coincide again in the completely saturated zone. The complessive effect of this small inclusion is here to make the solute more spread.
2nd moment in yy When the pulse reaches the inclusion, it’s strongly compressed. In the saturated part it spreads with almost the same time derivative in both the cases, and it seems that, independently on what hap-pened in the part right above, it reaches almost the same second moment in y, while, as you have seen before the x moment is evidently increased in the heterogeneous soil. Indeed, at the bottom of the simulation box, it’s found that σyy is increased only by less of 8% with respect to the one of the homogeneous structure, while σxx by more than 25%. This is due to the
fact that a more spread pulse in the vertical direction undergoes a stronger vertical compression when it reaches the edge of saturated zone, that restores a situation similar to the previous one.
0 5e-06 1e-05 1.5e-05 2e-05 2.5e-05 3e-05 3.5e-05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 σxx 0 5e-06 1e-05 1.5e-05 2e-05 2.5e-05 3e-05 3.5e-05 4e-05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 σyy 0 5e-10 1e-09 1.5e-09 2e-09 2.5e-09 3e-09 3.5e-09 4e-09 4.5e-09 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Dx x
mean position of the pulse
-2e-08 -1.5e-08 -1e-08 -5e-09 0 5e-09 1e-08 1.5e-08 2e-08 2.5e-08 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Dy y
mean position of the pulse
Figure 3.15: Upper panels: second moments (σxx on the left and σyy on the
right), lower panels: dispersion coefficients (Dxx on the left and Dyy on the right)
vs mean position of the pulse, Y . Green: without inclusion, red: with inclusion.
Dilution index and reactor ratio are plotted in fig. 3.16. Reactor ratio indicates distorsion when the pulse moves in non homogeneous environments. When it moves in homogenous background it tends to take again a gaussian shape.
3.3.2
Comparision: complemetary structures
I analyse here two complementary structures: the first is a sand tank with a little silt inclusion, the second is a silt tank with a little sand inclusion, with the same geometry of the previous. For the silt tank, with this huge flux,
0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 D il u ti o n in d ex
mean position of the pulse
0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 R ea ct o r R a ti o
mean position of the pulse
Figure 3.16: Dilution index (on the right panel) and Reactor ratio (on the left panel). Green: without inclusion, red: with inclusion.
the medium is completely saturated with the exception of the sand inclusion, that has a saturation of Θ ≃ 0.11.
As done with the other simulation, some concentration distributions are plotted here too (fig.3.17. Notice that the pulse (i) does almost not penetrate the sand inclusion, where the saturation is much lower; (ii) shows a tail in the last image, caused by the release of little amount of solute that remained for a longer time entrapped in the sand zone.
Figure 3.17: Concentration distributions in the simulation in three time steps, in which the pulse is moving throw the inclusion.
1st moment and its derivative. In fig. 3.18 mean position and velocity of the pulse for the two structures are plotted. For the main trend, one should notice that the velocity in the unsaturated part is different in the two materials, in particular is bigger in the sand, as expected, while it’s the same in the saturated part. The last rising of the moment in the pulse travelling in silt is due to the slow release of solute from the sand inclusion.
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 5000 10000 15000 20000 25000 Y time 1e-05 1.5e-05 2e-05 2.5e-05 3e-05 3.5e-05 4e-05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Vy
mean position of the pulse
Figure 3.18: On the left panel: mean position of the pulses, on the right panel their velocities. Green: silt box with sand inclusion, red: sand box with silt inclusion
Inclusion: If you take a look on the second graph of fig. 3.18, you can easily see how the pulse is slowed down in the silt inclusion for the sand tank. The complementary structure simulation shows, as expected, a complemen-tary behaviour, even if less pronounced in the second.
2nd moment xx The silt, that retains more water, has the effect to in-crease the diffusion at the same flux rate. In fact, in this case, the pulse is moving in an almost saturated environment and spreads always with the same derivative except when it’s passing the inclusion.
At that position, σxx signals an expansion, due to the fact that the pulse is moving away from the sand inclusion, where its diffusion is really slowed down. Then, right after it leaves the sand, it undergoes a compression and takes again the previous compactness.
On the other hand, the pulse of the previous simulation spreads slower in the upper part, because of lower saturation. When it reaches the silt, now it spreads more not because it is moving away from inclusion but because it is passing throw it.
Anyway, for two different reasons a pulse reaching an inclusion with coarser or finer material increases its second moment.
2nd moment yy For σyy a bit different comment should be done; in fact while the pulse experiencing a silt inclusion is compressing, the pulse expe-riencing the sand inclusion continue to spread.
is supported, the velocity is lower; while the second one, as said before, moves away from the sand.
The pronounced expansion that, in the saturated zone, characterizes the first pulse, is due to presence of a big tail (see the precedent paragraph).
0 5e-06 1e-05 1.5e-05 2e-05 2.5e-05 3e-05 3.5e-05 4e-05 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 σx x 0 2e-05 4e-05 6e-05 8e-05 0.0001 0.00012 0.00014 0.00016 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 σyy
mean position of the pulse
-2e-09 -1e-09 0 1e-09 2e-09 3e-09 4e-09 5e-09 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Dx x -2e-08 -1.5e-08 -1e-08 -5e-09 0 5e-09 1e-08 1.5e-08 2e-08 2.5e-08 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Dy y
mean position of the pulse
Figure 3.19: Upper panels: second moments (σxx on the left and σyy on the
right), lower panels: dispersion coefficients (Dxx on the left and Dyy on the right)
vs mean position of the pulse, Y . Green: silt box with sand inclusion, red: sand box with silt inclusion
Dilution index and reactor ratio In this case is possible to find a dif-ference in the behaviour of σij and the dilution index (see figure 3.20) The presence of a tail, in fact, with respect to the opposite case (in the comple-mentar structure), increases more the vertical component of the σ matrix (see the precedent paragraph) than the dilution index.
The reactor ratio too signals the tail with a pronounced decrease, due to less gaussian shape: this indicator sees a pulse very stretched on the y axis.
0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 D il u ti o n in d ex
mean position of the pulse
0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 R ea ct o r R a ti o
mean position of the pulse
Figure 3.20: Dilution index (left) and reactor ratio (right). Green: silt box with sand inclusion, red: sand box with silt inclusion
Chapter 4
Comparision with the
experiment
An other issue of this work is to have a better understanding on what is happening in the experiments perfomed by Steffen Heberle and Christina Esterle , and to learn what is reproducible and what is not. This comparison is done for the lowest flux of the experiment of Christina Esterle, that is j = 4.85 × 10−5m/s.
4.1
The chosen parameters and the
satura-tion profile
In the experiment, the water phase concentration Cw could be measured and it is used to compute the stauration profile, dividing by Ctot. In fact, recalling the definition of Ctot (that is given in par.1.3):
θ(x) = Ctot Cw(x)
(4.1) For Ctot is used the value of Cw at the bottom of the cell, where it is sure that the medium is completely saturated, i.e. θ = 1, and so Ctot = Cw.
Trusting the measured value for the flux, one cannot reproduce well both the saturation profile and the first moment. The two measures in the ex-periment are indipendent, and, in particular, because of the difficulty of the measure of the concentration in the water phase, the first one is less trustable than the second one. Because of this, and because the material parameters for the experiment are not known, the parameters of the simulation (i.e. the Van Genuchten n, the conductivity, the porosity) are chosen in order to fit the evolution of the first moment, as it can be found in the next paragraph.
Although an analysis of the saturation profile can be done. In the sat-urated part, one finds that Θ increases and eventually become greater than one; this is obviously an artefact and is due to the non-zero detection limit: because of it the measured Cw is lower, and so Θ higher. In the simulation, of course, the profile in that part is constant, and takes the value 1.
Even if, as it is already said, it was not possible to reproduce the exact value, both the simulation and experimental saturation profile have the same behaviour in the unsaturated part, with a different initial value.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.05 0.1 0.15 0.2 0.25 0.3 0.35 S a tu ra ti o n T he ta 1st moment (y) [m] Saturation profile: comparison
exp. sim.
Figure 4.1: Saturation profile in simulation and experiment
4.2
The first moment
The quite good agreement of the simulated first moment with the experi-mental one shows that the simple processes taken into account are enough to obtain a good description of the mean motion of a solute drop in the soil. The main feature, the change in the velocity at the upper edge of the capillary fringe is well visible and explainable (see par. 3.2.3). If one takes a look to the velocity plot, two main characteristic can be noticed: the high variability in the unsaturated part, due to remaining heterogenity of the almost uniform medium, and the fact that the experimental velocity doesn’t tend to an asymptotic value, as the simulated one does, but seems to decrease quite linearly.
An explanation for the first problem could be that here the saturation profile is not so constant as in the simulation, because of heterogenity and to physical effects like entrapped air, that cannot be reproduced, play some role and obviously effect the motion of the solute drop.
For the second one can think to a slow loss of mass, caused by a leaking boundary. In fact, in this case, the condition ∇ · v = 0 is probably not fulfilled, and so the vertical velocity need not be constant, as in a closed system, but can decrease.
5e-05 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 0.00045 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 v el o ci ty [m / s]
Mean position of the pulse
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 200 400 600 800 1000 1200 1400 1600 1 st m o m en t [m ] Time exp. sim. exp. sim.
Figure 4.2: 1st moment vs time (s) and velocity of the pulse vs its mean position (m) in simulation and experiment.
4.3
Second moments
More problems rise when one considers the evolution of the second moments. In the simulation with only molecular diffusion one can see that, even if the initial values are equal, the simulated second moment (both σxx and σyy) take values much smaller than the experimental one.
The two main component of the second moment matrix, σxx and σyy, are plotted in fig. 4.3, while their derivative, the dispersion coefficient Dxx and Dyy are plotted in fig. 4.4.
Although it is reproduced, as expected, the compression at the upper edge of the saturated zone, for the longitudinal component. The follow-ing expansion, that one sees in the simulation in the saturated zone, is not visible in the experiment because of presence of a detection limit. When the concentration at the edge of the plume become lower than this value, any further spreading cannot be seen. Infact, at a certain point in the experi-mental values it seems even to compress, that, in a homogenous medium, is quite surely an artefact.
The transversal component, the experimental one shows a compression on the upper edge of the saturated zone (as the longitudinal does) while the simulated one doesn’t. Even if in the first part the behaviour is not reproduced, in the second the simulation reaches almost the experimental value. 0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0 200 400 600 800 1000 1200 1400 1600 1800 2 n d m o m en t in y y [m ] Time 2.5e-05 3e-05 3.5e-05 4e-05 4.5e-05 5e-05 5.5e-05 6e-05 6.5e-05 7e-05 0 200 400 600 800 1000 1200 1400 1600 1800 2 n d m o m en t in x x [m ] Time sim. exp. sim. exp.
-1.5e-06 -1e-06 -5e-07 0 5e-07 1e-06 1.5e-06 0 100 200 300 400 500 600 700 800 Dy y [m / s] -1.5e-07 -1e-07 -5e-08 0 5e-08 1e-07 1.5e-07 0 100 200 300 400 500 600 700 800 Dx x [m / s] Time -6e-08 -4e-08 -2e-08 0 2e-08 4e-08 6e-08 8e-08 800 90010001100120013001400150016001700 -3e-08 -2e-08 -1e-08 0 1e-08 2e-08 3e-08 4e-08 5e-08 800 90010001100120013001400150016001700 Time
Figure 4.4: Diffusion coefficient Dxx and Dyy in simulation and experiment.
4.4
Reactor ratio
The reactor ratio (fig. 4.5) too shows some problems linked with the second moment’s ones, as it will be explained in the next section.
In fact, in the unsaturated part, you can notice a quite linearly descrease of this quantity in the experiment, while in the simulation it remains almost constant.
In the experiment, the reactor ratio increases only in corrispondence to the compression of the pulse, when it comes in the saturated part. For the same reason, but more slowly, the simulated one has a positive derivative at that point.
Then, in the saturated part, the simulated one tends to the value 1 while the experiment does not show this.
This could be taken as an evidence that the molecular diffusion is not the main process driving the dispersion: if it was true, the pulse would tend to be gaussian shaped, i.e. take to the value 1. Maybe one can more easily
understand it in the unsaturated medium, where simulation too tells us that convection dominates, but, as you can see, this happens also in the saturated part. A possible explanation of these phenomena is given in the following section. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.15 0.2 0.25 0.3 0.35 0.4 sim exp.
Figure 4.5: Reactor ratio in simulation and experiment vs mean position of the pulse.
4.5
Dispersion model. Conclusion.
The differences that rise between the simulation and the experiment are probably mainly due to the fact that in the simulation is used a dispersion model too rough, that doesn’t take into account dispersion caused by small scale variability of the velocity field of the porous medium. That means that in the model of D a dependence on the flow velocity should be added: D(θ, v) (look at eq. 1.18).
To understand where is the problem, in figure 4.6, a graph is shows us the dependence of Dapp on the flow velocity, and indicates the presence of different regimes. In particular, the inverse of the dynamical Peclet number (P edyn = dgV /Dapp) is plotted versus the molecular one (P emol = dgV /Dm), where dg is the grain size.
Among the regimes found, that differ by the dependence of Dapp on V , take a look in particular to the two pure ones, i.e. (I) and (IV) in the figure.
Figure 4.6: Different regimes in solute dispersion. On the x axis the molecular Peclet number (Pemol ∝ 1/Dm), on the y axs the inverse of the dynamical Peclet
number (1/Pedyn∝ Dapp). Figure from [Pfannkuch (1963)]
(I) represents the pure molecular diffusion, while (IV) the pure dynamical dispersion, with the following equation for Dapp:
(I): Dapp = DmMillington-Quirk(θ) (4.2)
(IV): Dapp = P dgV (4.3)
where V is the flow velocity, dg the grain size and P is an empirical factor. This study tells us that for bigger flow velocity, the importance of the dynamical dispersion rises, and eventually the molecular diffusion plays no role.
In our case, in the unsaturated part of the cell V is bigger than in the second, as we expect, and the discrepance between the simulated dispersion with the experimental one is bigger.
The issue is so to add this new process to the model, that can give already very good result on the mean motion.
With this improvement, a more accurate simulation can maybe fit quite well experimental values. In any case, further experimental research, with different kind of porous medium and improved measurement technique are required.
List of Figures
1.1 Soil water characteristic of a porous medium illustrated for a bun-dle of capillaries. Figure from [Roth (2006)] . . . 5 1.2 Material properties for various kind of soils in Van Genuchten
(thick lines) and in Brooks-Corey (dash-dotted lines) parametriza-tion. (figure from [Roth (2006)]) . . . 7 2.1 Mass in two small plug versus mass distributed normally: same σ
but different dilution index. Figure from [Kitanidis (1994)] . . . . 12 3.1 Soil water characteristic function . . . 15 3.2 Conductivity parametrization. On the right as function of the
saturation, on the left as function of the matric head . . . 16 3.3 Saturation profile on the central vertical line. . . 17 3.4 Three typical concentration distributions in the simulation, from
the left: when the pulse is in the unsaturated zone, when it goes throw the edge of the saturated zone and when is in the saturated zone. . . 18 3.5 Mean position of the pulse vs normalized time for different flux
rates. The three time normalizations are: for j1 1 → 103s, for j2
1 → 104s and j
3 1 → 105s . . . 19 3.6 Second moment (σxx in the upper two panels and σyy in the lower
ones) vs mean position of the pulse (Y ), for three flux rates (in ms−1): red 3.176 × 10−5, green 3.176 × 10−6and blue 3.176 × 10−7.
In the right two panels only the unsaturated part is shown . . . . 20 3.7 Dispersion coefficient in xx and yy for three different flux rates (in
ms−1): red 3.176 × 10−5, green 3.176 × 10−6 and blue 3.176 × 10−7. 20 3.8 Dilution index (blue), σxx (red) and σyy (green) normalized to one
for two different flux rates; 3.176 × 10−5ms−1 for the lower two
panels and 3.176×10−7ms−1for the upper ones shown in saturated
(left panels) and unsaturated (right ones) zone. . . 21 3.9 Reactor ratio for three different flux rates . . . 22
3.10 Second moment (σxx in the upper two panels and σyy in the lower
ones) vs time (in the two right panels) and mean position of the pulse (Y , in the two left panels) for two different initial position: red y1(0) = 0.42m, green y2(0) = 0.32m . . . 23 3.11 Peclet number computed as r0V /D as function of the mean position
of the pulse . . . 24 3.12 Saturation in the cell for the sand with silt inclusion. blue Θ = 0,
red Θ = 1. . . 25 3.13 Concentration distributions in the simulation in three time steps,
in which the pulse is moving throw the inclusion. . . 26 3.14 On the left panel: mean position of the pulses, on the right panel
their velocities. Green: without inclusion, red: with inclusion . . . 26 3.15 Upper panels: second moments (σxx on the left and σyy on the
right), lower panels: dispersion coefficients (Dxx on the left and
Dyyon the right) vs mean position of the pulse, Y . Green: without
inclusion, red: with inclusion. . . 27 3.16 Dilution index (on the right panel) and Reactor ratio (on the left
panel). Green: without inclusion, red: with inclusion. . . 28 3.17 Concentration distributions in the simulation in three time steps,
in which the pulse is moving throw the inclusion. . . 28 3.18 On the left panel: mean position of the pulses, on the right panel
their velocities. Green: silt box with sand inclusion, red: sand box with silt inclusion . . . 29 3.19 Upper panels: second moments (σxx on the left and σyy on the
right), lower panels: dispersion coefficients (Dxx on the left and
Dyy on the right) vs mean position of the pulse, Y . Green: silt
box with sand inclusion, red: sand box with silt inclusion . . . 30 3.20 Dilution index (left) and reactor ratio (right). Green: silt box with
sand inclusion, red: sand box with silt inclusion . . . 31 4.1 Saturation profile in simulation and experiment . . . 33 4.2 1st moment vs time (s) and velocity of the pulse vs its mean
posi-tion (m) in simulaposi-tion and experiment. . . 34 4.3 2nd moments in simulation and experiment. . . 35 4.4 Diffusion coefficient Dxx and Dyy in simulation and experiment. . 36 4.5 Reactor ratio in simulation and experiment vs mean position of
the pulse. . . 37 4.6 Different regimes in solute dispersion. On the x axis the
molec-ular Peclet number (Pemol ∝ 1/Dm), on the y axs the inverse
of the dynamical Peclet number (1/Pedyn ∝ Dapp). Figure from
Contents
Introduction 1
1 Theory 3
1.1 Dynamics of the soil water . . . 3
1.2 The material properties . . . 5
1.3 Solute transport . . . 7
2 Methods and Analysis 9 2.1 Methods: numerical simulation . . . 9
2.2 Analysis: Description of the transport . . . 10
2.2.1 Moments analysis . . . 11
2.2.2 Dilution index and reactor ratios . . . 12
3 Results 14 3.1 Summary of the results . . . 14
3.2 Homogeneous structure . . . 15
3.2.1 Input: material parameters . . . 15
3.2.2 Saturation . . . 17
3.2.3 Moments analysis . . . 18
3.2.4 Dilution index and reactor ratio . . . 21
3.2.5 The transport is convection-dominated . . . 22
3.3 Heterogeneous structure . . . 24
3.3.1 Comparision: with and without inclusion . . . 25
3.3.2 Comparision: complemetary structures . . . 27
4 Comparision with the experiment 32 4.1 The chosen parameters and the saturation profile . . . 32
4.2 The first moment . . . 33
4.3 Second moments . . . 34
4.4 Reactor ratio . . . 36
Bibliography
[Roth (2006)] K. Roth Soil Physics. Lectures notes Institute of envi-ronmental physics, Heidelberg University, 2006. web: www.iup.uni-heidelberg.de/institut/forschung/groups/ts/student
[Kitanidis (1994)] P. Kitanidis The concept of the dilution index. Water Re-sources Research, vol. 30, no. 7 (july 1994) 2011-2026
[Monahan, Del Sole (2009)] A. H. Monahan, T. Del Sole Information
theo-retic measures of the dependence, compactness and non-gaussianity for multivariate probability distributions Nonlinear processes in Geophysics
16 (2009), 57-64
[Rossi, Ippisch, Fl¨uhler (2008)] M. Rossi, O. Ippisch, H. Fl¨uhler Solute
di-lution under imbibition and drainage conditions in a heterogenous struc-ture: modeling of a sand tank experiment. Advanced in Water Resources
31 (2008) 1242-1252
[Pfannkuch (1963)] H. O. Pfannkuch Contribution `a l’´etude des d´eplacements de fluide miscible dans un milieu poreux Rev. Inst.
