Theoretical Analysis of Pressure Phenomena Associated with the Wireline Formation Tester
Authors: Moran J.H. , Finklea E.E. 
Published: December 2007

Abstract
The pressure build-up technique is a recognized method of determining permeability from conventional drillstem tests. In this paper an effort is made to extend such techniques to the interpretation of data obtained from the wireline formation tester. Such a study is necessary because of the differences, for this case, in the magnitude of the flow parameters (rate of flow, amount of recovered fluids) and in the flow geometry (flow through a perforation vs flow across the face of the wellbore, etc.) involved in the solution of the equations of flow for compressible fluids. The perforation is replaced by a spherical hole, and the effect of the borehole is neglected, so that the flow can be considered to be radial in a spherical co-ordinate system. Arguments are presented to justify this idealization. Assuming single-phase flow, general relations between pressure and flow rate are developed for a homogeneous medium. The study is then extended to permeable beds of finite thickness. It is shown that the early stages of pressure build-up tend towards spherical flow, while the later stages tend towards cylindrical flow. The thinner the bed, the more quickly flow approaches the cylindrical model. The prevalence of thin beds in practical work makes this analysis quite important. Cases involving permeability anisotropy are treated.

Introduction
From wireline formation tester operation, two types of data are obtained:the nature and amount of recovered fluids, andthe pressure history recorded during the test.
A number of papers have been written dealing with the interpretation of formation production on the basis of the recovered fluids. In general, the methods described have been quite accurate for both high- and low-permeability formations. The present paperwill deal with an analysis of the pressures observed. An analysis of the pressure build-up curves obtained in hard-rock country has already been attempted on the basis of the formula proposed by Horner. Although this approach has met with success in many instances, some questions have been raised as to its validity. It is the aim of the present study to place the analysis of pressure build-up in the formation tester on a firmer basis, from which more detailed methods of interpretation can evolve. Because of the great differences between the operation of the wireline formation tester and the conventional drillstem test, modifications are necessary in the interpretation. The major difference relates to the flow geometry. Once the flow geometry has been established other features such as multiphase flow, skin effect, afterflow, etc., well described in the literature, can he introduced. It will be assumed that the mechanical operation of the formation tester is already known to the reader. It will suffice here merely to state that the tester provides the means for taking a relatively small sample of the fluid immediately adjacent to the borehole, and for recording the subsequent pressure response. In comparison with conventional drillstem tests, the time required for a satisfactory pressure build-up response is much shorter, because of the relatively small quantity of fluid withdrawn by the wireline tester. This feature is highly desirable in the case of low-permeability formations. For an analysis of the pressure response within the formation, three simple flow geometries are considered-linear, cylindrical and spherical. The spherical and cylindrical flow geometries are most pertinent to the formation tester; therefore, they will receive the major emphasis. Since the configuration of the borehole and the perforation made by the tester complicate the flow geometry, it is necessary to allow for them in the drawdown response. However, because of the volume of formations contributing to the pressure- response, the details of the perforation shape are unimportant in the build-up period. Since relatively small amounts of fluid are withdrawn from the formation, in contrast to a conventional drillstem test, a study of the "depth of investigation" and the significance of drawdown as well as build- up data will be included. Because the "depth of investigation" will be shown to be rather large, the effect on the build-up curves of the finite thickness of the permeable bed is considered. It is this consideration that leads to the importance of cylindrical flow geometry. Also included is a discussion of permeability anisotropy and its effect on the interpretation of the tester results. The pressure curves recorded by the formation tester will follow two general patterns, depending upon whether the formation is of high or low permeability. Fig. 1 (a and b) schematically illustrates these two responses. In Fig. 1(a), the high pressure recorded during fill-up of the tool is essentially the pressure differential across the choke in the system.
JPT
P. 899^