Summary
Many reservoirs are faulted, and hydraulic characterization of these faults is essential for the design of field-scale developments. In addition to the effect that a fault has on fluid flow in the reservoir, it may separate two different reservoir regions with distinctly different properties. The detection of the properties on both sides of the fault and the distance to the fault are important in the reservoir-characterization process.
In this study, a linear fault is modeled as an infinitesimal-thickness skin boundary. Analytical solutions for pressure-transient behavior for a line-source, constant-rate well in a composite reservoir are obtained with one Fourier space transformation and time-space Laplace transformations. The solutions are presented for strip and infinite reservoirs. This study examines drawdown-pressure and pressure-derivative behavior and extends and generalizes many studies presented in the past. Correlating parameters for pressure-transient responses are presented. The possibilities of boundary detections are considered, and interference pressure responses in a composite, strip reservoir are briefly discussed.
Introduction
The hydraulic characterization of faults in faulted reservoirs is essential in the design of field-scale developments. Traditionally, the faults have been treated as sealing boundaries. The method of images has been used to study drawdown, buildup, or interference pressure-transient behavior for a well in a homogeneous reservoir, containing single or multiple linear, sealing boundaries. The "doubling of slope" is used to indicate the presence of a linear, sealing boundary. The intersection time of the two semilog straight lines is used to calculate the distance to a barrier. Also, the deviation time at which the pressure response departs from the linesource solution can be used to calculate the distance to a barrier. Log-logs and semilog type-curve-matching methods have been developed to calculate the distance to a barrier. The pressure-transient behavior of a well in a long, narrow reservoir has been studied. Streltsova and McKinley discuss the pressure-transient behavior for a well in a reservoir limited by one or more linear boundaries. They consider both closed and constant-pressure boundaries and discontinuities in reservoir properties. Pressure-transient analysis in bounded reservoirs has also been studied. Prasad and Wong et al. used a Green's function approach to generate multiple sealing boundaries.
Fig. 1 presents a number of homogeneous reservoir configurations with single or multiple linear, sealing boundaries considered in the literature. Recently, the effects of a partially communicating fault in an infinite, homogeneous reservoir have been considered numerically and analytically. Pressure-derivative techniques have been used to analyze pressure-transient data for fault detection. The pressure-transient behavior of an infinitely large composite reservoir with a linear discontinuity in reservoir properties has been considered by Bixel et al. and Streltsova and McKinley. The effects of a partially communicating fault (or a linear leaky discontinuity) in a composite reservoir (Fig. 2a) on transient pressure testing has not been considered. The upper part of Fig. 2a shows a side view of two reservoir regions separated by a fault. Also, the fault may be partially communicating, depending on its characteristics. The reservoir configuration of the upper part of Fig. 2a is simplified by removing the vertical discontinuity caused by the fault, making the two reservoir parts horizontal, and by making the fault discontinuity vertical, as described in the lower part of Fig. 2a. Figs. 2b and 2c are top views of the simplified reservoir configurations that we considered in this paper. Fig. 2b represents a strip reservoir with two parallel no-flow boundaries and Fig. 2c represents an infinite reservoir with a linear discontinuity. The two reservoir regions on both sides of the fault (discontinuity) may have different diffusivities and transmissivities, and the resistance to flow at the fault is modeled as a thin skin, according to the concepts of van Everdingen and Hurst. The active well is located in Region I and is considered as a constant rate line source. The problem is solved along the approach taken by Bixel et al. and Yaxley with one Fourier space transformation and two Laplace time-space transformations. The solutions for the pressure responses for the active and observation wells and drawdown-pressure and pressure-derivative behaviors are presented. Interference-pressure behavior is also briefly discussed.
Theory
The mathematical model and a solution technique for the strip reservoir configuration of Fig. 2b are detailed in Appendix A. In developing the mathematical model, we make the assumptions that we have a single-phase, slightly compressible fluid of constant compressibility, a homogeneous and isotropic reservoir on each side of the partially communicating fault, a constant formation thickness, and a line-source well. The diffusivity equations for both regions along with the appropriate outer and skin boundary conditions are posed. The Laplace transformation with respect to the time variable and the Fourier transformation with respect to the y variable reduce the two partial-differential equations to two ordinary differential equations. The ordinary differential equation for Region II is readily solved. The ordinary differential equation for Region I (the region containing the well), however, is solved by taking the Laplace transformation with respect to the × variable and inverting the resulting solution analytically with respect to the × variable. Thus, the dimensionless pressure drops in the Laplace/Fourier space at any location in Regions I and II at any time are given by Eqs. 1 and 2, respectively: (1) (2)
Eqs. 1 and 2 are of an exponential form in the × variable, and the Laplace time variable and the Fourier y variable are in the a terms as described in Appendix A. The variable S denotes the van Everdingen and Hurst skin assigned to the fault, and M denotes the mobility ratio between Regions II and I. The numerical inversion technique of the Laplace/Fourier solutions is described in Appendix A.
