lL is the longitudinal dispersivity of the medium; is the transversal dispersivity of the medium;

Sij is Kronecker delta;

Vi is the component of the average interstitial solution velocity V in the direction i

Op(ro) is the soil diffusion coefficient

Many authors (Brandt et al. (1971); Bresler (1975); Lomen and Warrick (1976)) consider that in presence of a trickle source the water flow and solute transport can be treated as a two-dimensional problem using axi-symmetric cylindrical model where the cylindrical coordinates are considered.

If we assume, under irrigation from a trickle source, that the soil is stable, isotropic, and homogeneous porous medium(for the same oasis the soil has generally those characteristics), the differential equation that describe the flow of water in the system can be expressed in terms of diffusivity form as follows (Brandt et al (1971):

da=d dt dx

I must signal that the same equation can be used in the case of a single trickle nozzle or a number of nozzles spaced far enough to prevent overlapping and when a set of trickle sources are placed at equal and close distance so that their wetting fronts overlap after a short time of irrigation.

The one-dimensional transient soil moisture and solute in the vertical direction is described by the Richards equation as follows:

ro is the volumetric water content;

t is the time;

z is the soil depth;

K(ro) is the hydraulic conductivity;

h is the soil hydraulic head pressure;

A(z,t) is a root-extraction term

The root extraction term, that intervene as a sink, is given by the following equation:

Proot is the root water potential at the soil surface;

Rroot is the root resistance term

Psoil(z,t) is the soil matrix potential;

Psolute(z,t) is the solute potential;

r(z,t) is the proportion of the total active roots in the depth increment Az;

Ax is the distance between the plant roots and the point in the soil where Psoii and Psoiute are measured.

3.2.3 Distribution of Water in Drip Irrigated Row Crop

We will give here simple expressions (Coelho and Or (1996) describing and predicting uptake patterns within the wetted soil volumes of drip-irrigated row crop. The simple expression for describing and predicting root uptake patterns within the wetted soil volume could improve drip irrigation design and management inside the new oasis. The background of those expressions is based on field observations, the introducing of a sink term into the Richards' equation to model the influence of water uptake on unsaturated flow regime and the use of Bivariate Gaussian density function as a parametric models.

The water distribution patterns around a dripper is influenced by: 1) the total value of applied water; 2) dripper flow rate, source configuration and initial and boundary conditions; 3) the soil physical properties and their distribution; 4) the root activity; 5) irrigation management.

\Proot +zRroot) Psoil(z,t) ^solute(z-, t)r (z,t)K(a)

AxAz

By considering the co-ordinates system originate at the base of plant, we can made three different solutions for three different way of placing the drippers. We can then write the fraction of total uptake intensity occurring at any point inside the field by the following expressions.

If the surface drip line is on the crop row the maximum uptake intensity is given by the estimated mean deviation in the radial direction and by the estimated mode in the vertical direction we write:

2nzAst,rad Ast, z

st,rad

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