Well Testing Analysis for Variable Permeability Reservoirs
Authors: Zeng F. , Zhao G. 
Published: March 2010

Abstract
The effects of spatial permeability variations of a reservoir on the pressure derivative curve are studied. The presence of a permeability anomaly causes the pressure derivative to locally deviate from that of the corresponding homogeneous reservoir. For a simple variable permeability reservoir that has only one permeability anomaly, the start time, the value and the time of maximum local derivative deviation suggest the originating location, the permeability and the end location of the permeability anomaly, respectively.
To analyze the transient pressure behaviour of a well located in a reservoir with multiple permeability anomalies, an approximate equation is presented. The study shows that the pressure derivative difference, with respect to that of the base homogeneous reservoir, is the summation of the pressure derivative differences of all simple variable permeability reservoirs with respect to the same base. This method has been proved and validated in reservoirs with radial and areal permeability distribution using both analytical and numerical methods.
Applications show that this method provides a useful clue for well testing analysis of heterogeneous reservoirs and the maximum error caused by the proposed approximate equation is less than 3%. The practice guidelines for field test data interpretation are proposed.

Introduction
Traditional well testing analysis tends to determine an overall average permeability. Generally, it cannot reflect the spatial variation in the permeability of a reservoir. Typically, in practical well testing interpretation, the shape and the trend of the pressure derivative curve can be matched very well, but the local waves in the derivative curve cannot be matched at all. Generally, there are two sources producing local waves: the noise of tested data and the heterogeneity of the reservoir. The waves generated by the noise of tested data are random and discontinuous. This kind of wave may lead to interpretation errors. Consequently, many methods, such as wavelet analysis, Schroeter's deconvolution method and the optimal method, have been developed to de-noise the tested data. Different from the noise of tested data, the heterogeneity of the reservoir produces local continuous and smooth waves in the pressure derivative curve. Analyzing this kind of wave may provide more detailed reservoir information than traditional well testing.
Many researchers have discussed some aspects of well testing analysis for heterogeneous reservoirs. Niko(1) presented analytical solutions for stratified and heterogeneous systems with interlayer crossflow. Yaxley(2) studied transient pressure behaviour when a partially communicating fault exists and observed that it can be diagnosed by drawing a semi-log derivative plot. Britto and Grader(3) studied the effects of size, shape and orientation of an impermeable region on transient pressure testing and found that the presence of an impermeable region caused the pressure response to deviate from the homogeneous line source response. They also pointed out that the four major parameters (the shortest distance between the well and the impermeable region, the size, the shape and the orientation of the region) affect the pressure response of the active source well located in such reservoirs. After investigating the transient pressure behaviours of wells located in composite reservoirs, Oliver(4, 5) derived a solution to the problem of a well located in an infinite reservoir with a small arbitrary spatial permeability variation, as demonstrated in the following equation: