Pressure Buildup From a Vertically Fractured Low-Permeability Formation(includes associated paper 12938 )
Authors: Chen Techien , Kry Richard 
Published: December 2007

Summary
Buildup theory for a vertically fractured gas well was reexamined assuming planar flow against constants and face pressure. The resultant pressure buildup is an arcsine function of a time group. If this relation is used to analyze buildup, it is possible to calculate the product of permeability and fracture half-length squared. This analysis gives better agreement with a set of field data and numerical simulation as compared with that from constant-rate theories using a Homer plot and atande msquare-root time plot.

Introduction
Pressure buildup data for a vertically fractured well can be analyzed by the usual Homer method, provided that the drawdown has reached pseudo steady state. For low-permeability reservoirs, the entire well test may lie within the initial flow period. The flow behavior of aportion of this period can be predominantly planar. Millheim and Cichowicz showed that the pressure buildup varies as a tandem square root of time. One ofthe assumptions they used was the constant-ratedrawdown, which usually is not valid for low- permeability reservoirs. In fact, it generally is accepted that as soon as a well is opened to flow in these reservoirs, the sand face pressure drops rapidly to a constant. This boundary condition has been recognized as valid fora large number of well tests. All these papers considered cylindrical radial flow. Agarwal et al. evaluated numerically the drawdown behavior of avertically fractured well under both constant-rate and constant-pressure boundary conditions. However, the rate/time type curves they generated for the drawdown against the constant-pressure case cannot be used to analyze buildup.
The appropriate formulation considers the pressure buildup analysis of a vertically fractured well when the drawdown is against a constant sand face pressure. The buildup is derived by use of Duhamel's integrals withthe variable rate being the drawdown rate and the subsequent shut-in rate of zero. The integrand reduces to a simple form at the fracture face. The resultant closed form expression enables the pressure/time data to be plotted into a straight line, the slope of which is characterized by the initial reservoir pressure and the sand face drawdown pressure. Alternately, this slope can be expressed in terms of the last flow rate and the other reservoir/fracture parameters. One of them is the product of the formation permeability and the fracture half-lengthsquared. Without other information, it is possible to calculate only this product and not its individual components.
Application of the derived equation to one example gave a result in good agreement with drawdown analysis and numerical simulation. The effects of wellbore storage and fracture damage also were shown to be negligible.

Planar flow Period
The geometry of the fracture plays an important role in the early-time flow data. Cinco-L. and Samaniego-V. divided the entire flow into four periods. Initially there isa fracture linear flow period in which the flow inside the fracture is the dominant factor. As the fluid from the formation starts its contribution, the flow enters the bilinear flow period. If the fracture is highly conductive, then there will be a period when planar flow from the formation into the fracture is a good approximation. Eventually, pseudo radial flow takes over in any fracture/formation system of sufficient extent.
In analyzing a set of well test data using the plan arapproximation, care must be exercised to ensure that these data points fall within the planar flow period. It turns out that the determining factor in each of the early periods is the fracture flow capacity.
JPT
P. 917^