Simplification Of The Superposition Principle For Pressure Analysis At Variable Rates
Authors: Brauer E.B. 
Published: December 2007

Publication Rights Reserved
This paper was prepared for the 40th Annual Fall Meeting of the of the Society of Petroleum Engineers of AIME, to be held in Denver, Colo., Oct. 3–6, 1965. 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 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 Society of Petroleum Engineers office. 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
The application of the superposition theorem to define the transient rate behavior in the constant terminal pressure case has been presented by van Everdingen and Hurst. Hurst later stated that the cumulative pressure drop in a reservoir, producing at variable rates, is dependent only on the final rate of production and the total time, if the variable rates have been accounted for as cumulative production. These conditions are applicable to either well or reservoir behavior.
To make the variable rate case general, it will be the purpose of this work to show by the superposition principle that when q = 0 for a period prior to shut-in, the total time is still applicable in determining the pressure build-up. This condition is valid provided total time is sufficiently long when compared to individual producing times at constant rates prior to final shut-in. Previous work in the literature has treated the case when time is very short for the variable rate case.

Introduction
Application of the superposition theorem has been utilized for various engineering problems where transient behavior was to be considered as in electric circuits and heat flow. In the case of petroleum reservoirs, van Everdingen and Hurst applied the superposition principle to solve the variable terminal pressure case and the variable terminal rate case, after first determining the constant terminal rate and pressure cases with the use of the Laplace transformation. The reader is no doubt familiar with the solving of water influx by the inversion of the values of the Q[t] function and multiplying by the pressure differential for the dimensionless time at each influx interval. This practical method of calculating water influx is a direct application of the superposition principle. Conversely, the cumulative pressure drop at the reservoir boundary can be calculated using the P[t] functions if the producing history is known.
A further modification of the variable rate case would be the development of the superposition theorem to apply to the determination of cumulative pressure drop when there has been a brief period of shut-in prior to the final shut-in. This alteration would allow the variable rate case to be made completely general in that it would be extended to accept q = 0 as a rate. The practical implication of this treatment would be that all the data needed in the analysis of the final period of build-up or draw-down would be the final producing rate, the total time of the well [includes prior shut-in and producing time], and the final period of shut-in.