Use Of Normal Analysis In The Description Of Heterogeneous Reservoirs
Authors: Perez-Rosales Candelario 
Published: December 2007

Abstract
A method for obtaining the continuous areal distribution of the flow capacity of heterogeneous reservoirs is presented. For its application, it requires the knowledge of a set of discrete kh values. The fundamental assumption of the proposed method establishes that all natural random distributions can be considered as the superposition of elementary distributions, whose mathematical model is the normal or Gaussian distribution curve. The mathematical too used by the method is normal analysis. This is a technique that allows the representation of random distributions as a linear combination of normal density functions. For illustrating the operational aspect of normal analysis, two examples are presented.

Introduction
A problem that frequently arises in the description of heterogeneous reservoirs can be stated as follows: given a set of discrete kh values corresponding to a reservoir, find an analytical function that represents efficiently the continuous areal distribution of the flow capacity of the reservoir.
Similar problems are found in other areas, for example, when it is desired to obtain a contour map of the top of a geological formation from well data, or when it is intended to construct topographic surfaces from measurements made on some control points. The same problem is met when trying to points. The same problem is met when trying to represent the porosity distribution within a reservoir.
To deal with this type of problems, different methods have been developed. Among the wellknown techniques are those that make use of trend-surface analysis based on polynomial approximations and those based on the use of Fourier series. Another technique is kriging, and recently Hardy developed multiquadric analysis for the construction of topographic surfaces.
A method based on a new mathematical technique is presented in this paper. The proposed method allows the representation of the areal distribution of a random variable as a linear combination of normal density functions.
PREVIOUS WORK PREVIOUS WORK In a previous paper, the author proposes a new technique for determining flow capacity. This technique makes use of pressure buildup tests and is based on the assumption that oil reservoirs are essentially heterogeneous systems, where the heterogeneities have a random distribution. In that paper it is shown that the characteristic flow capacity of the influence zone of a well can be obtained by means of the relationship.
kh = ,..........(1)
where mw is a weighted average of the slopes of the buildup curve obtained upon plotting the pressure in psi vs the natural logarithm of the closed-in time. psi vs the natural logarithm of the closed-in time. The value of mw is given by
m w = ,..........(2)
where mi is the slopes along the buildup curve and (delta t)i is the corresponding shut-in times.
The kh values thus obtained are of little interest when it is desired to have an overall view of the areal distribution of the reservoir flow capacity. To obtain this areal distribution, interpolation procedures are necessary. In the referenced paper, use was made of a simple interpolation technique for finding the kh distribution of an actual reservoir.