As said in the previous paragraphs there is many problems in the oasis of the region that concern the methods adopted to manage the distribution of nutrients and water between plants. The soil of the oasis is isotropic, homogeneous and very porous. We found different kinds of market gardening, different types of fruit trees and many sorts of date palms in the same filed. Those canopies are arranged in different ways and directions. Many questions we must ask which nutrient to use to which type of plant in this kind of soil? How it is distributed and what is the quantity really absorbed by the roots of plants? If the amount of water used to irrigate represents really what the plants need? How the water infiltrates in this type of soil? How to quantify the water lost and how to distribute it between plants with an efficient manner for this special architecture? If the disposition of the existing species is the best for an efficient use of solar radiation in order to minimise evapotranspiration and maximise the photosynthetic process? Which kind and number of plants to add or remove from the existing oasis to ameliorate its architecture for a best use of resources? When renewing the traditional oasis, which species to install for date palms, fruit trees and market gardening, their densities and orientations to reach the best efficiency use of resources (nutrients, water, solar radiation)? How elaborating models to answer those questions only by introducing, physics properties of soil, quality and amount of water that we dispose and climatic parameters of the region (solar radiation, temperature, precipitation, wind velocity.)?
The set of equations that we will present now can be used to reply to those demands, to model and describe those phenomenon and to propose a solution for agricultural water management inside the oasis by only applying the inverse problem and the optimum control theories (Ralf, 1999).
3.1 Equations to Formulate the Management of Salt Leaching and Nutrient Uptake by Plants Inside the Oasis
In this part we will present the equations that describe the solute transport inside the soils for general cases. Those equations are very usefUl because they are concerned by very general conditions. So they can be applied to a vast region (oasis of north Africa) where we can found many type of soils, many boundaries, different climate...After we will give formulas and equations for particular situation. Quantifying and specifying the nutritive elements of plants for every type of soil and for every plant species is an important stage for agricultural management inside the oasis. Also searching indicators for an efficient use of nutrient is very helpful.
3.1.1 Equations Describing Solute Transport Inside the Soil
To simulate the transport of a non-interacting solute in the soil during non-steady infiltration, the effects of convection, ionic diffusion and mechanical dispersion must be investigated (Bresler 1973; Hamdy, A. and Choukr-allah, R. 2002).The one-dimensional solute transport is expressed by the convection-diffusion equation:
This equation can also be represented as follow:
0Soiut is the total flux of solute; C is the solute concentration of the soil solution V is the average interstitial flow velocity x is the flow direction co-ordinate;
o(V,œ) is the combined diffusion-convection coefficient (o(V,œ)=oh(V)+op(œ) ) q is the volumetric flux of solution (q = V œ )
The diffusion coefficient in a clay-water system is considered independent of the salt concentration but dependent on the water content only. Accordingly, it can be written as:
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