Pressure Transient Analysis of Partially Penetrating Wells in a Naturally Fractured Reservoir
Authors: Slimani K. , Tiab D. 
Published: March 2010

Abstract
Often and for many reasons the wellbore does not completely penetrate the entire formation, yielding a unique early-time pressure behaviour. Some of the main reasons for partial penetration, in both fractured and unfractured formations, are to prevent or delay the intrusion of unwanted fluids into the wellbore, i.e., water coning. The transient flow behaviour in these types of completions is different and more complex compared to that of a fully penetrating well.
This paper proposes a method for identifying, on the pressure and pressure derivative curves, the unique characteristics of the different flow regimes resulting from these types of completions and to obtain various reservoir parameters, such as vertical and horizontal permeability, fracture properties and various skin factors. Both naturally fractured and unfractured (homogeneous) reservoirs have been investigated.
For a naturally fractured formation, the type curves of the pressure and pressure derivative reveal that the combination of partial penetration and dual-porosity effects yields unique finger prints at early and transition periods. These unique characteristics are used to calculate several reservoir parameters, including the storage capacity ratio, interporosity flow coefficient, permeability and pseudoskin. Equations have been developed for calculating the skin for three partial completion cases: top, centre and bottom. The analytical solution was obtained by combining the partially penetrating well model in a homogeneous reservoir with the pseudo-steady model for a naturally fractured reservoir (NFR).
The interpretation of pressure tests is performed using the TDS (Tiab's Direct Synthesis) technique for analyzing log-log pressure and pressure derivative plots. The TDS technique uses analytical equations to determine reservoir and well characteristics without using type-curve matching. These characteristics are obtained from unique fingerprints, such as flow regime lines and points of intersection of these lines, which are found on the log-log plot of pressure and pressure derivative. Two numerical examples are included to illustrate the application of the. proposed technique.

Introduction
Consider a vertical well partially penetrating a naturally fractured reservoir, i.e., only a portion of hydrocarbon-bearing formations is perforated. The naturally fractured reservoir has an infinite radial extent. The Warren and Root(1) model is used in which the matrix blocks are replaced by a system of uniform rectangular parallelepipeds with identical properties. The fractures are assumed to be parallel with the principal axes.
FIGURE 1: Different types of partially penetrating vertical wells based on the position in the perforated interval hw. Available in Full Paper
The pressure solution is derived using the Laplace transformation and the separation of variables technique as proposed by Bui et al.(2). This solution is expressed as an infinite Fourier-Bessel series in Laplace domain. The theory for a partially penetrating well in a homogenous reservoir developed by Yildiz and Bassiouni(3) is used for comparison purposes.
The analytical solution for constant flow rate in Laplace space was inverted into real dimensionless pressure using the Stehfest algorithm(4).

Pressure Derivative Behaviour
Four types of partial penetration or completion schemes are considered (as shown in Figure 1). A plot of the dimensionless pressure derivative tD * P'D versus tD is shown in Figures 2 and 3.