Thiem equation

Equations [3-8a] to [3-8c] are different applications of the Thiem equation, which estimates drawdown in an aquifer or well under steady-state conditions. As previously mentioned, it is assumed that the changes in saturated aquifer thickness are small compared with the total saturated depth. This is necessarily true in a confined aquifer, but not always in an unconfined (phreatic) aquifer. If drawdown becomes a significant fraction of the saturated aquifer thickness, more complicated expressions for drawdown are obtained see Bear (1979). For an unconfined aquifer in which drawdown is a significant fraction of the saturated thickness, Eq. [3-8a] must be expressed in terms of head instead of drawdown ... 

A successful and effective groundwater control mainly depends on rehable hydrogeological modeling. In the past few decades, under the guidance of groundwater dynamics especially Thiem equation and Theis equation, some analytical models for estimating water inflow rate have been established, which could be utilized to build the... 

In the various applications of the Thiem equation shown in Eqs. (3.8a)-(3.8c), the product Kb) appears. This quantity measures the ability of an aquifer to deliver water to a well and is called transmissivity, T, with units [I /T]. 

In many of the arguments involving chemostats it was shown that the omega limit set had to lie in a restricted set, and the equations were analyzed on that set one simply could choose initial conditions in the restricted set at time zero. The equation defining the restricted set - in effect, a conservation principle - allowed one variable to be eliminated from the system. We want to abstract this idea and make it rigorous. The omega limit set lies in a lower-dimensional set, and the trajectories in that set satisfy a smaller system of differential equations. However, it is not clear (and, indeed, not true [T3]) that the asymptotic behavior of the two systems is necessarily the same. (A very nice paper of Thieme [Tl] gives examples and helpful theorems for asymptotically autonomous systems. A classical result in this direction is a paper of Markus [M].) In this appendix, a theorem is presented which justifies the procedure on the basis of stability. 

Tl] H. R. Thieme (1992), Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, Journal of Mathematical Biology 30 755-63. 

T3] H. R. Thieme (1994), Asymptotically autonomous differential equations in the plane, Rocky Mountain Journal of Mathematics 24 351-80. 

Values taken ftom J. Ginehling, and B. Koibe, Thennodynamik, Thieme verlag, Stuttgart 1988 Antoine equation ... 

At first, by arranging Boolean equations in the operation part of the rules using the techniques described later, the Boolean operations (though they appear to be different at first) are classified into the same functions. Next, expanding, rule by rule, the Boolean equation in the condition part of a rule, we obtain the format for the sum of product terms and replace the original rule with new rules corresponding to each product term. Each new rule has the inherited product term in its condition part and has the same operation part as that of the rule it replaced. Next, we find all the variables (calling thiem... 

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