The Error Function and its Complement: A Comparison of Some Approximate Models

Abstract Applying the conservation laws to the transient transport problems of most processes, including diffusion, heat and momentum transfer, normally yield parabolic partial differential equations which solutions, for semi-infinite systems, usually include either the error function or the complementary error function. While tabulated values are available for real arguments of these functions, the need for look-up tables and interpolations makes them computationally intensive for implementation in computer programs and other applications. Although ‘simple’ models have been fitted to these values, it is imperative that such models preserve the key properties of these functions. Combining the criteria of accuracy, differentiability, and integrability with mathematical consistency, this paper examines the approximate models of Kalkaja (2009), Winitzki (2008), van Halen (1989) and Greene (1989), premised on different assumptions and of varied forms and mathematical elegance. In terms of accuracy, the van Halen (1989) model is the most reliable, successively trailed by Winitzki (2008), Kalkaja (2009), and Greene (1989) models. Despite being the most accurate, the van Halen (1989) model is the most complex and computationally expensive. Although they are all continuous and differentiable, obtaining their derivatives analytically, involves varying complexities, with van Halen (1989) most demanding and Greene (1989), the least intensive. With current knowledge, none of the models is amenable to analytic (direct) integration. Considering all performance indices, while the Winitzki model is the most consistent, van Halen approximation is the least consistent. In conclusion, whereas the accuracies of these approximate models are satisfactory for most engineering applications, their robustness to mathematical (analytic) operations is not convincing. Although there is scope for improvement, it is recommended that greater efforts be directed at the development of theoretically rigorous models, which ironically, may not necessarily be more complex. Typical applications of this work include analysis of reservoirs under thermal or miscible floods.