A Fast And Accurate Calculation of One Dimensional Fronts
Authors: Acs G. , Doleschall S. , Farkas E. 
Published: December 2007

This paper was prepared for the SPE-European Spring Meeting 1974 of the Society of Petroleum Engineers of AIME, held in Amsterdam, the Netherlands, May 29–30, 1974. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made.
Discussion of this paper is invited. Three copies of any discussion should be sent to the Netherland Section of the Society of Petroleum Engineers, P. O. Box 228, The Hague, the Netherlands. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines.

Abstract
A new numerical technique is proposed for the time discretisation of parabolic equations and systems of equations. This technique is shown to be suitable for diffusion-convection equations and is particularly useful for solving problems at high flow rates when a hyperbolic character is dominant and the solution exhibits sharp fronts.
The method is a modification of standard time discretisations. Truncation errors of any time discretisation for any usual space discretisation can be reduced. The method is more successful in computing the translation of sharp fronts, as fewer time steps are required. Consequently, significantly less computer time is needed to reach an approximation of given accuracy than conventional numerical techniques and other high-order-accuracy techniques. None the less, it is particularly suitable for describing the transition from not well-defined fronts to sharp fronts.
The method is applicable to a wide variety of nonlinear equations, reducing the numerical problems resulting from nonlinearity. problems resulting from nonlinearity

Introduction
One of the basic problems in reservoir engineering is the exact description of displacement processes in different layers. Although the equations describing the processes are well known, their analytical solution can be determined for the simplest cases only. In the case of a numerical solution the difficulty lies in the calculation of those fronts that are characteristic of the processes.
At the same time, however, the accuracy of the solution of these displacement problems essentially depends on the exact description of the fronts, since the translational velocity of the front is a function of its shape, both in the case of miscible and nonmiscible displacements; and in the case of miscible displacements and especially of CO2 displacement, the most intensive molecular mass-transport processes are observed in the vicinity of the front. Thus, the results are decisively influenced by the description accuracy of the shape of the front.
Importance and timeliness of the problem is demonstrated by the large number of works dealing with it.