Determination of Aquifer Influence Functions From Field Data
Authors: Coats K.H. , Rapoport L.A. , McCord J.R. , Drews W.P. 
Published: December 2007

Abstract
Water movement about an oil or gas reservoir can be predicted provided an aquifer influence function is known. This function is generally determined from an electric analyzer study or by fitting an idealized mathematical model to field data. A problem arises in determining that influence function which best reproduces field data and satisfies certain smoothness requirements, without making any idealizations concerning aquifer geometry and heterogeneity. This study shows the problem to be solvable through the linear programming technique and presents applications to one oil reservoir and two gas storage reservoirs. In addition, a theorem is stated and proven which confirms influence function smoothness requirements heretofore accepted on physical intuition and establishes additional smoothness constrains formerly not recognized.

Introduction
The pressure decline accompanying production results in water encroachment into oil and gas reservoirs situated adjacent to aquifers. The importance of this water movement derives from the significant dependence of production rate upon reservoir pressure and of pressure, in turn, upon water encroachment.
Pressure and rate of water movement are uniquely related through an "influence" function which may be in the form of reservoir pressure response to a unit water encroachment rate or of water movement caused by a unit reservoir pressure drop. This influence function reflects the heterogeneity and geometry of the aquifer and is therefore particular to each reservoir. If the influence function is known, then reservoir pressure may be predicted from a production rate schedule, or vice-versa, with the aid of the material balance equation which relates water influx to reservoir production and pressure.
The problem of determining this influence function for a given reservoir has been approached in three different ways. Van Everdingen and Hurst calculated and tabulated certain influence functions pertinent to idealized mathematical models which satisfy simplifying assumptions such as aquifer homogeneity and elementary reservoir-aquifer geometries. Other studies developed additional tables for a variety of elementary reservoir-aquifer geometrical configurations and demonstrated their validity and utility. Hicks, et al. described the use of an electrical network to determine an influence function giving "best" match of field history and allowing prediction of future performance. Finally, Hutchinson and Sikora and Katz, Tek and Jones attempted to develop a calculation method for deriving the influence function directly from field data with no idealizing assumptions concerning aquifer geometry and homogeneity. They encountered difficulties with inaccuracies in field data and with satisfaction of well- known smoothness requirements of the influence function.
This study is concerned with the determination of influence functions from field data and, more specifically, treats the following problem: given field pressure- production data of arbitrary accuracy, how does one determine the influence function which, subject to satisfaction of known smoothness requirements, gives best agreement with field history? A rigorous solution to this problem is obtained through linear programming as described below. Example applications to one oil and two gas storage reservoirs are presented. In addition, a theorem is proved which confirms influence function smoothness requirements heretofore accepted on physical intuition and establishes additional smoothness requirements formerly not recognized.

DEFINITION AND PROPERTIES OF THE INFLUENCE FUNCTION
If the influence function F(t) denotes the reservoir pressure response to a unit rate of water influx, then the reservoir pressure response to a varying rate of water movement, e (t), is given by
(1)
The integral may be approximated by a summation to give
(2)
or, equivalently,
(3)
where. In these equations, e and F are both zero. We will now consider the situation where F(t) is unknown but p(t) and e(t) are known from field data.
JPT
P. 1417ˆ