Derivation of the Theis (1935) Equation by Substitution

Derivation of the Theis (1935) Equation by Substitution

“…Equation (21) is called the Boltzmann transformation equation (Bird et al 1960). If n = 2 and C = 1/ √ T then equation 18 of Perina (2010) is obtained for the similarity parameter. Following the same procedure one can obtain equations 5 and 17 of Perina (2010).…”

“…Perina (2010) used the substitution method to obtain the Theis (1935) equation as a supplement to the methods given in Loáiciga (2010). Perina (2010) stated that the substitution method is also called the similarity transformation method.…”

“…Perina (2010) used the substitution method to obtain the Theis (1935) equation as a supplement to the methods given in Loáiciga (2010). Perina (2010) stated that the substitution method is also called the similarity transformation method. However, the substitution method was developed by Bird et al (1960), while the similarity transformation method was first introduced by Boltzman in 1894 (Ames 1965; Debnath 2004).…”

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“…The goal of Perina (2010) was to show a simple method for deriving the solution to the Theis (1935) problem. The substitution (Bird et al 1960) technique when applied to the Theis (1935) problem takes advantage of the fact that the independent variables can be combined into a dimensionless group in only a limited number of ways.…”

“…As is the case with the substitution method, the similarity transformation method when applied to the Theis (1935) problem requires an additional change of variable to arrive at a solution in the form of a tabulated function while simultaneously determining u ; that is, to get from equation 17 of Masoodi and Ghanbari (2011) to the exponential integral in their equation 18. This is analogous to the change of variable required for obtaining the exponential integral from equation 19 or 20 of Perina (2010) derived without the requirement of prior knowledge of u .…”