Pressure Transient Testing of the Heterogeneous Reservoirs
Authors: Pecher R. , Stanislav J.F. , Easwaran C.V. 
Published: March 2010

Abstract
Over the years, attempts have been made to include system heterogeneities into mathematical descriptions and to determine their effect on pressure response. In this study, a general case of a system with layers of non-uniform thicknesses is considered. The mathematical description is based on assuming radial flow within individual layers and pseudosteady-state cross flow between the adjacent layers. Since the angle of the interphasing plane may assume any angle between 0 and 90 degrees, layered (uniform thicknesses) and composite systems constitute the lim iting cases.
The analytical solution is found in Laplace space and the numerical inversion to real space is obtained by using the Stehfest algorithm. The model is verified for several simpler subsystems and its sensitivity to reservoir parameters is investigated. Solutions are presented in a graphical form for a selected set of parameters.
Based on this model a method of well test data analysis is proposed.

Introduction
Well testing is a valuable source of information on reservoir properties and well conditions. Test data are interpreted on the basis of pressure transient theory, which is deve1oped for various physical concepts of the reservoir flow conditions. In this study, modeling of complex reservoir systems is attempted in order to enhance the quality of the well test data analysis.
It is well recognized that all petroleum reservoirs are heterogeneous to a certain degree. The heterogeneity manifests itself-in terms of variable rock and/or fluid properties, which occur due to either natural processes or human activities. Heterogeneities may be observed both in the radial and the vertical directions.
Modelling of heterogeneous systems is usually approached by considering several subsystems each with properties lumped at some characteristic value. Such an approach has been used over the years and has given rise to the concepts of composite and the layered reservoir models.

Literature Review
In this section, only the key papers are mentioned.
A complete review of the work done on the subject of multilayer pressure transient theory was published by Ehlig-Economides and Joseph(l) in 1985; the contributions made by individual authors are categorized according to the solution method, number of layers, existence of the formation crosstlow and types of the boundary conditions.
The models describing the pressure response of layered reservoirs may be classified into two groups depending whether or not the interlayer flow is considered.
One of the first descriptions of crossflow between two layers was introduced by Barrenblatt et al(2), who proposed the double porosity model for naturally fractured reservoirs. Such systems are featured by pressure imbalance between high diffusivity fractures and low diffusivity matrix blocks: the resultant fluid transfertends to equalize pressures in the two subsystems. Assuming the lump parameter concept, the flow between a matrix and a fracture is proportional to their pressure difference. This approach has been successfully used in simulating crossflow between layers of different properties.