## Abstract

In this book the subject of rhizosphere modelling is introduced to help the reader develop skills to improve irrigation and fertigation management. The rhizosphere modelling hinges on solving two partial differential equations, namely Richard’s equation and convection–dispersion equation. Richard’s equation represents the processes involved in flow of water in variably saturated root zone. Convection–dispersion equation represents the processes involved in flow of nutrients in the root zone. Both of them are highly nonlinear equations, and computer software are used for solving them.

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## Appendices

### Appendix I

The detailed procedure for optimization of unsaturated hydraulic parameters through inverse modelling for a double-ring infiltrometer test data using Hydrus-1D is described here.

The soil is sandy loam with sand content of 72.5%, silt content of 16.5% and clay content of 11%. The initial suction pressure head at the top soil and at 75 cm below the surface is − 1250 cm and − 300 cm, respectively. Data from a double-ring infiltration experiment is provided in Tables 10 and 11.

To begin a new project, the project data manager icon in Hydrus main window is opened and the new project is given with a name and a brief description. The project manager window (Fig. 31) consists of some inbuilt projects already existing in Hydrus. The window allows the user to open, rename, copy or delete an existing project and also to create a new project.

Here, the name and brief description given to the new project are ‘inverse modelling’ and ‘inverse modelling using infiltration data’ respectively.

**Preprocessing**

Preprocessing tab has option for entering necessary data (Fig. 32). Then after selecting ‘main processes’, opt for ‘Water flow’ and ‘Inverse solution’ (Fig. 33).

The preprocessing data for inverse solution is shown in Fig. 34. The maximum number of iterations for the inverse solution is to be specified here. If one selects zero, then only the direct simulation is carried out. The maximum number of iterations selected for this problem is ‘20’. The number of data points in objective function is also specified here. In this study, cumulative infiltration data for 13 time intervals is taken as the input data. So, the number of data points is ‘13’.

The geometry information window (Fig. 35) selects the length unit, specifies the depth and inclination of the soil profile to be analysed and determines the number of materials to be used. The ‘decline from vertical axis’ allows the user to choose between a vertical, horizontal or inclined soil profile. The inclination is specified in terms of the cosine of the angle between the vertical axis and the axis of the soil profile. Its value is equal to one for vertical soil columns and zero for horizontal soil columns. In this problem, length units are chosen as centimetres and a soil profile with a single material having 75 cm depth is considered as the domain of study. Since a vertical soil profile is considered, the value of ‘decline from vertical axis’ is specified as 1.

The time information window data used is in Fig. 36. The number of time variable boundary conditions corresponds to number of data in Table 11.

The print information window (Figs. 37 and 38) allows the user to enter the variables governing the output from Hydrus. The T-level information checkbox is used to decide whether certain information concerning the mean pressure head, mean water, cumulative water fluxes and time and iteration information are to be printed after each time step or only at preselected times. One can also specify the number of time steps after which the output is to be printed on the screen.

The iteration criteria window (Fig. 39) specifies the iteration criteria for the solution precision and parameters for the time step control. An iterative process must be used to obtain solutions at each new time step because of the nonlinear nature of the Richards’ equation. The iterative process continues until a satisfactory degree of convergence is obtained, i.e. until at all nodes in the saturated (unsaturated) region the absolute change in pressure head (water content) between two successive iterations becomes less than some small value determined by the imposed absolute pressure head (or water content) tolerance. The water content tolerance represents the maximum desired absolute change in the value of the water content between two successive iterations during a particular time step. Its recommended value is 0.001. The pressure head tolerance represents the maximum desired absolute change in the value of the pressure head between two successive iterations during a particular time step. Its recommended value is 1 cm.

The lower optimal iteration range represents the minimum number of iterations necessary to reach convergence for water flow, below which the time step have to be increased by multiplying it with a lower time step multiplication factor. The recommended and default value for the lower optimal iteration range is 3, and the lower time step multiplication factor is 1.3. The upper optimal iteration range represents the maximum number of iterations necessary to reach convergence for water flow, above which the time step has to be decreased by multiplying it with an upper time step multiplication factor. The recommended and default value for the upper optimal iteration range is 7, and upper time step multiplication factor is 0.7.

At the beginning of a numerical simulation, Hydrus generates for each soil type in the flow domain a table of water contents, hydraulic conductivities and specific water capacities from the specified set of hydraulic parameters. Values of the hydraulic properties are then computed during the iterative solution process using linear interpolation between entries in the table. So, for facilitating the interpolation, a prescribed interval for the pressure head value is to be given. The lower and upper limits of the tension interval represent the absolute value of the lower and upper limits of the pressure head interval for which a table of hydraulic properties will be generated internally for each material. The lower limit of the tension interval is usually selected to be a very small number. The upper limit can be modified so that it encompasses most pressure heads encountered during the simulation.

In the soil hydraulic model window (Fig. 40), van Genuchten–Mualem model and ‘No hysteresis’ are selected.

The user has to provide initial estimates of the soil hydraulic parameters. The Rosetta Lite software can also be accessed from within Hydrus and the soil constituent data to get initial estimate of hydraulic parameters of the soil. The initial parameters used here are as below (Fig. 41).

*θ*_{r} = 0.045*, θ*_{s}=0.3684*, α*=0.0356 cm^{-1}*, l* = 0.5*, n* = 1.4884 and *K*_{s} = 1.4884 cm min^{-1}

In water flow boundary conditions window (Fig. 42), since the ponding depth varies with time, a variable pressure head upper boundary condition is given and the lower boundary condition is free drainage. The initial conditions are specified in terms of pressure head.

The time variable boundary condition window (Fig. 43) allows the user to enter the time dependent values of boundary conditions, and the data of Table 11 is entered.

In the ‘Data for Inverse Solution Window’, the data of cumulative depth of water infiltrated in Table 11 is given as shown in Fig. 44.

The profile information (Fig. 45) module discretizes soil profile into a series of finite elements. If no previous nodal distribution exists, the programme generates automatically an equidistant nodal distribution with a default number of nodes. The number and location of the nodes can then be edited by the user to optimize finite element lengths.

The initial pressure head at the top and bottom are given as per the data as − 1250 cm and − 300 cm, respectively.

After running the Hydrus-1D, the optimized parameters obtained is as follows:

*θ*_{r} = 0.0445*, θ*_{s} = 0.3719*, α* = 0.0251 cm^{−1}*, l* = 0.0003*, n* = 1.5181 and *K*_{s} = 1.5181 cm min^{−1}.

Figure 46 shows the comparison between the simulated cumulative infiltration depth using optimized hydraulic parameters and observed cumulative infiltration depth.

### Appendix II

### Introduction to Finite Difference Methods

The basis of finite difference method (FDM) is expression of partial differential equations like Richard’s equation and convection–dispersion equation into finite difference equation by using truncated form of Taylor series expansion.

Taylor series expansion for pressure head (*h*) can be expressed as follows:

If we neglect second-order terms in the preceding equation and rearrange, we get as follows:

If we express the \(\left(\frac{\partial h}{\partial x}\right)\) of Richard’s equation or any other first-order spatial derivative like this, it is said to have been expressed in forward difference form.

If we substitute \((-\Delta x)\) in the place of \((\Delta x)\) in Eq. 1, we get as follows:

If we neglect second-order terms in the preceding equation and rearrange, we get as follows:

Instead of using Eq. 81, if we use Eq. 83, it is called as backward difference form.

If we add Eqs. 81 and 83, we get as follows:

The preceding form of equation is called as central difference.

By following the same principles followed so far, we can write the following equation for second-order derivatives:

As far as second-order derivative are concerned, Eq. 85 is the only one form, whereas for the first-order derivative, there are three forms.

Let us express the Richard’s equation in one-dimensional form as follows:

Let us assume K (h) and \({C}_{w}\) to be constants for simplicity. Then the preceding equation can be expressed as follows:

When we want to find out the pressure head distribution in a one-dimensional flow field, the flow domain is discretized a shown in Fig. 47. There are two methods by which equations with time derivative are represented in FDM. One is called as explicit method, and another is called as implicit method. Equation 87 is written using explicit method as follows:

In Eq. 88, it must be noted that the time period for which the spatial derivatives are written is for the time period ‘*t*’. Therefore, when Eq. 88 is solved with necessary initial conditions, the pressure head distribution in flow domain for all the nodes at time period ‘*t*’ is known and only unknown is pressure head for time period ‘*t* + 1’. Hence Eq. 88 can be written explicitly as follows:

Equation 89 is solved progressively for all the nodes in the flow domain, and pressure head distributions are found out.

Equation 87 is written using implicit method as follows:

In Eq. 90, it must be noted that the time period for which the spatial derivatives are written is for the time period ‘*t* + 1’. So in Eq. 90, for every time period only \({h}^{t}(x)\) is known and \({h}^{t+1}(x-\Delta x)\), \({h}^{t+1}(x)\) and \({h}^{t+1}(x+\Delta x)\) are unknowns.

When we write Eq. 90 for all the nodes considering initial and boundary conditions for every time step, the number of linearly independent simultaneous equations obtained will be equal to the number of unknowns. These simultaneous equations can be solved using matrix methods. These matrices are diagonally dominant matrices, and hence they can also be solved by iterative methods.

### Truncation Errors and Numerical Dispersion

Since in FDM, truncation of Taylor series is done, these methods are bound to have errors. These errors are called truncation errors, and they can be minimized by selecting finer grids to the extent possible. When we solve the convection–dispersion equation, due to truncation of Taylor series, the error caused is termed as numerical dispersion. From the following derivation, we would be able to understand why this error is called as numerical dispersion.

The following is the two-dimensional form of convection–dispersion equation:

Let us assume dispersive flux is zero and velocity of flow is also a constant. Then the convection–dispersion equation becomes a pure convection equation which is as follows:

When we solve Eq. 92 using FDM by using the implicit method for time derivative and backward difference for the spatial derivative, we get as follows:

Equation 93 is obtained by truncating Taylor series expansion. When we solve Eq. 93, what partial differential equation is exactly solved is little different. It may sound very odd at this place. But continuing the derivation would dispel all our doubts.

Taylor series expansion for time derivative and neglecting third-order terms is as follows:

Similarly, we can write using Taylor series expansion for writing the spatial derivative as follows:

Put Eqs. 95 and 96 in Eq. 93 and we get,

We can write following equations by partially differentiating Eq. 92 with respect to time once and with respect to *x* once:

Putting Eq. 100 in Eq. 97, we get as follows:

Rearranging the preceding equation we get,

Rearranging the preceding equation, we get,

We intended to use FDM for solving the partial differential Eq. 92 (shown in bold in Eq. 103). But while doing so, we are solving the partial differential Eq. 103 which has an additional dispersive flux terms with artificially induced dispersion coefficient of \(\left({v}_{x}^{2}\frac{\Delta t}{2}+\frac{\Delta x {v}_{x}}{2}\right)\).

Hence numerical dispersion is defined as the dispersion caused due to application of FDM to the convective terms. Obviously, in order to keep numerical dispersion at a satisfactory level, the spatial discretization and temporal discretization level must be as low as possible. This fact can also be verified by doing a numerical experiment by solving a pure convective equation for modelling a sharp concentration front. We will be able to see from the results that after some time period from the initial time, the sharp concentration front would have got diffused.

Oscillation of results is another term used in modelling. When the concentration results go above the maximum expected level and go below the minimum expected level, the solution is said to be oscillating. Oscillation can be totally eliminated by adopting upstream weighting methods. Upstream weighting methods represent the convective term for each node based on the direction of flow at each node from the surrounding nodes. For instance in Fig. 48, four possible flow patterns that may occur at a node are shown. While writing the FD terms for \((v\frac{\partial c}{\partial x})\), direction of velocity of flow is considered and the convective flow is computed based on the node from which flow occurs. Hence, this method is called as upstream weighting method. Upstream weighting methods severely suffer from numerical dispersion.

A non-dimensional called as Peclet number (Pe) is used to characterize spatial discretization. It is defined as follows:

If Peclet number is maintained below 5, the oscillations can be kept at a satisfactory level for the other FDM methods.

Stability of numerical solution is another term commonly used. Stability is an issue for explicit methods. When the magnitudes of spatial and temporal discretization are above certain critical level, the solution from explicit method becomes unstable. Instability may be identified by observing the solution, and the solution will have very big positive and negative concentration values. The reason for instability in explicit method is due to the error caused due to truncation, and the error gets propagated unbounded. Implicit methods do not suffer from instability.

Courant number (Cr) is used to characterize the spatial and temporal discretization and is as follows:

A Courant value of less than one is one necessary condition for avoiding instability in explicit methods. Another condition called as Neumann condition must also be satisfied for getting a stable solution which is as follows:

Studying the subject of numerical modelling rigorously is always useful in modelling. However, the material presented here will be very useful for the readers who do not intend to write any computer code for modelling and use the standard software.

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Ravikumar, V. (2023). Rhizosphere Modelling. In: Sprinkler and Drip Irrigation. Springer, Singapore. https://doi.org/10.1007/978-981-19-2775-1_16

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