Pressure Response at Observation Wells in Fractured Reservoirs

Abstract

Abstract This work examines interference test data in a naturally fractured reservoir. The reservoir model examined here assumes that the reservoir can be represented by a system of horizontal fractures that are separated by the matrix. This model is identical to the deSwaan-Kazemi model. The main contribution of our work is that we combine the parameters of interest in a simple way and present solutions that can be used directly for field application. These solutions can be used to design or analyze interference tests. We also compare the solution for unsteady-state flow in the matrix with the Warren-Root model, which assumes pseudosteady-state fluid flow in the matrix. Introduction This work examines the pressure response at an observation well in a fractured reservoir. Previous works by Kazemi et al. and Streltsova-Adams have examined the pressure response based on the Warren and Root model. In this work, however, we assume unsteady-state fluid transfer from the matrix to the fracture system. We consider the model proposed by deSwaan and Kazemi. This model assumes that the fractured reservoir can be replaced by an equivalent set of horizontal fractures separated by matrix elements (Fig. 1). The results of this study, however, can be applied to other unsteady-state models proposed in the literature. The main contribution of this work is that type curves convenient for analyzing data are presented. We have combined the parameters of interest and correlated results in terms of dimensionless groups that are commonly used in well test analysis. A comprehensive discussion of the pressure behavior at an observation well in a fractured reservoir is presented. Procedures to analyze data by conventional semilog methods also are discussed. New observations on the pressure response at an observation well are presented. For purposes of comparison, we also examined the pressure response in a reservoir that obeys the Warren and Root idealization. Assumptions and Mathematical Model The mathematical model considered here assumes the flow of a slightly compressible (of constant viscosity) fluid in a naturally fractured reservoir. We assume that the matrix-fracture geometry is as shown in Fig. 1. Individually, the fractures and the matrix are assumed to be homogeneous, uniform, and isotropic porous media with distinct properties. Gravitational forces are assumed negligible. The reservoir is assumed to be infinitely large-i.e., the outer boundaries have no effect on the pressure response. The initial condition assumes that the pressure is constant at all points in the reservoir. We assume that the flowing well produces at a constant rate and that the two wells both penetrate the fracture system. In addition, we impose four fundamental assumptions:all production is from the fracture system,flow in the matrix system is one-dimensional-i.e., in the z direction (see Fig. 1),flow in the fracture system is radial, andboth the producing and observation wells are line source wells and wellbore storage and skin effects are neglected. Assumptions 1, 2, and 3 have been used extensively in previous studies of naturally fractured reservoirs. Recent results of Reynolds et al. indicate that these assumptions are valid for the model considered in Fig. 1 if the flow capacity of the matrix system is small relative to the flow capacity of the fracture system (see Ref. 9 for specific details). Assumption 4 is used in virtually all studies of interference testing. (A comprehensive discussion of the influence of wellbore storage and skin effects on interference test data is given in Refs. 10 and 11.) It is important to realize that flee preceding four assumptions (see 2 in particular) imply that the pressure response at the observation well will be equal to the pressure response in the fracture system at the point where the observation well intersects the fractured system. All results given in this study are based on dimensionless variables for purposes of convenience. The dimensionless variables are defined as follows. The dimensionless pressure drop in the reservoir is ...............(1) Here, is the permeability of the fracture system and is the total thickness of the fracture system. If the model of Fig. 1 contains horizontal fractures, then = where is the thickness of each horizontal fracture. The term represents the surface flow rate, is the formation volume factor, is the viscosity of the fluid, and ( ) is the pressure at point at time . The subscript f refers to the fracture. All quantities are expressed in SPE-preferred SI metric units. SPEJ P. 628^